MACROECONOMIC CONSEQUENCES OF UNCERTAIN SOCIAL SECURITY
REFORM
by
ERIN COTTLE HUNT
A DISSERTATION
Presented to the Department of Economics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2018
DISSERTATION APPROVAL PAGE
Student: Erin Cottle Hunt
Title: Macroeconomic Consequences of Uncertain Social Security Reform
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Economics
by:
George Evans Chair
Bruce McGough Core Member
Shankha Chakraborty Core Member
Brandon Julio Institutional Representative
and
Sara Hodges Interim Vice Provost and Dean of the
Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded June 2018
ii
c© 2018 Erin Cottle Hunt
All rights reserved.
iii
DISSERTATION ABSTRACT
Erin Cottle Hunt
Doctor of Philosophy
Department of Economics
June 2018
Title: Macroeconomic Consequences of Uncertain Social Security Reform
The U.S. social security system faces funding pressure due to the aging
of the population. This dissertation examines the welfare cost of social security
reform and social security policy uncertainty under rational expectations and under
learning. I provide an overview of the U.S. social security system in Chapter I.
In Chapter II, I construct an analytically tractable two-period OLG model
with capital, social security, and endogenous government debt. I demonstrate the
existence of steady states depends on social security parameters. I demonstrate
a saddle-node bifurcation of steady states numerically, and demonstrate a
transcritical bifurcation analytically. I show that if a proposed social security
reform is large enough, or if the probability of reform is high enough, the economy
will converge to a steady state.
In Chapter III, I develop a three-period lifecycle model. The model is
inherently forward looking, which allows for more interesting policy analysis. With
three periods, the young worker’s saving-consumption decision depends on her
expectation of future capital. This forward looking allows analysis of multi-period
uncertainty. Analysis in the three-period model suggests that policy uncertainty
may have lasting consequences, even after reform is enacted.
iv
In Chapter IV, I develop two theories of bounded rationality called life-cycle
horizon learning and finite horizon life-cycle learning. In both models, agents use
adaptive expectations to forecast future aggregates, such as wages and interest
rates. This adaptive learning feature introduces cyclical dynamics along a transition
path, which magnify the welfare cost of changes in policy and policy uncertainty.
I model policy uncertainty as a stochastic process in which reform takes place in
one of two periods as either a benefit cut or a tax increase. I find the welfare cost
of this policy uncertainty is less than 0.25% of period consumption in a standard,
rational expectations framework. The welfare cost of policy uncertainty is larger
in the learning models; the worst-off cohort in the life-cycle horizon learning
model would be willing to give up 1.98% of period consumption to avoid policy
uncertainty.
v
CURRICULUM VITAE
NAME OF AUTHOR: Erin Cottle Hunt
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
Utah State University, Logan, UT
DEGREES AWARDED:
Doctor of Philosophy, Economics, 2018, University of Oregon
Master of Arts, Political Science, 2011, Utah State University
Bachelor of Arts, Economics, 2009, Utah State University
Bachelor of Arts, Political Science, 2009, Utah State University
AREAS OF SPECIAL INTEREST:
Macroeconomics
Public Economics
Behavioral Economics
GRANTS, AWARDS AND HONORS:
Kleinsorge Summer Research Fellowship, University of Oregon, 2016
Kimble First Year Teaching Award, University of Oregon, 2016
Economics Distinctive Scholar Award, University of Oregon, 2013-2015
Graduate Teaching Fellowship, University of Oregon, 2013-2018
Valedictorian, Utah State University, 2009
vi
ACKNOWLEDGEMENTS
I would like to express appreciation to my committee chair, George Evans,
who provided guidance and encouragement during each step of this research. I
would also like to thank Bruce McGough, Shankha Chakraborty, Brandon Julio,
and University of Oregon Macroeconomics Group for providing helpful feedback
during the many iterations of this project. I would also like to thank Scott Findley
and Frank Caliendo for introducing me to economics and mentoring me throughout
my education. Finally, I would like to thank my family for providing love and
support during this entire process. I am especially grateful for Pat and Aggie.
vii
To my mother
viii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 1
II.POLICY UNCERTAINTY IN AN ANALYTICAL TWO-
PERIOD OLG MODEL . . . . . . . . . . . . . . . . . . . . . . . 10
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Household Problem . . . . . . . . . . . . . . . . . . . . . 11
Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Government . . . . . . . . . . . . . . . . . . . . . . . . 13
Markets . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 14
Saddle-node Bifurcation and Steady States . . . . . . . . . . . 16
Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . . 20
Dynamic Efficiency . . . . . . . . . . . . . . . . . . . . . 24
Explosive Paths and the Basin of Attraction . . . . . . . . . . 26
Explosive Paths . . . . . . . . . . . . . . . . . . . . . 27
Parameterization . . . . . . . . . . . . . . . . . . . . . . 28
Policy Experiments . . . . . . . . . . . . . . . . . . . . . . . 29
Announced Policy Changes . . . . . . . . . . . . . . . . . . 29
Tax Rate Uncertainty . . . . . . . . . . . . . . . . . . . . 32
Replacement Rate Uncertainty . . . . . . . . . . . . . . . . 33
Replacement Rate Uncertainty: Fixed Probability of Reform . . . 34
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Chapter Page
Policy Timing Uncertainty: Uniform Reform . . . . . . . . 37
Policy Uncertainty: History Dependent Reform . . . . . . . 39
Welfare Analysis . . . . . . . . . . . . . . . . . . . . . . . . 49
Welfare Analysis with Policy Uncertainty . . . . . . . . . . . 54
Possible Extensions . . . . . . . . . . . . . . . . . . . . . . . 58
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 59
III.POLICY UNCERTAINTY IN A MULTI-PERIOD OLG
MODEL WITH RATIONAL EXPECTATIONS . . . . . . . . . . . . . 62
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Household problem . . . . . . . . . . . . . . . . . . . . . 62
Production . . . . . . . . . . . . . . . . . . . . . . . . . 64
Government . . . . . . . . . . . . . . . . . . . . . . . . 65
Market Clearing . . . . . . . . . . . . . . . . . . . . . . 68
Market Clearing Equations in Per-Labor-Hour Terms . . . . . 68
Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 69
Steady State Analysis . . . . . . . . . . . . . . . . . . . . . . 71
Linearization and Eigenvalue Analysis . . . . . . . . . . . . . 72
Examples of Steady States . . . . . . . . . . . . . . . . . . 74
Parameterization . . . . . . . . . . . . . . . . . . . . . . . . 78
Policy Experiments: Pension Reforms . . . . . . . . . . . . . . . 80
Perfect Foresight Expected Policy Changes . . . . . . . . . . . 80
Stochastic Policy Changes . . . . . . . . . . . . . . . . . . 85
Simple Two-Period Timing Uncertainty . . . . . . . . . . . 86
Timing and Policy Uncertainty . . . . . . . . . . . . . . 97
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Chapter Page
Long-run Consequences of Policy Uncertainty . . . . . . . . . . . 104
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 105
IV.LIFE-CYCLE HORIZON LEARNING AND POLICY UNCERTIANTY . . 111
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Households . . . . . . . . . . . . . . . . . . . . . . . . . 113
Demographics . . . . . . . . . . . . . . . . . . . . . . . 114
Production . . . . . . . . . . . . . . . . . . . . . . . . . 115
Government . . . . . . . . . . . . . . . . . . . . . . . . 116
Markets . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Rational Expectations Equilibrium . . . . . . . . . . . . . . 118
Life-cycle Horizon Learning . . . . . . . . . . . . . . . . . . . 121
Finite Horizon Life-cycle Learning . . . . . . . . . . . . . . . . 125
Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Announced Social Security Reform . . . . . . . . . . . . . . 130
Announced Change, Finite Horizon Life-cycle Learning . . . . . 136
Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Welfare Cost of Social Security Reform . . . . . . . . . . . . 137
Welfare Cost of Learning . . . . . . . . . . . . . . . . . . . 140
Welfare Costs of Policy Uncertainty . . . . . . . . . . . . . . . 145
Welfare Cost . . . . . . . . . . . . . . . . . . . . . . . . 151
Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Forecast tax burden . . . . . . . . . . . . . . . . . . . . . 156
xi
Chapter Page
Forecast Policy Adaptively . . . . . . . . . . . . . . . . . . 157
Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . 160
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 160
V.CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A. TWO-PERIOD MODEL APPENDIX . . . . . . . . . . . . . . . . 166
Derivation of Golden Rule Level of Capital . . . . . . . . . . . . 166
Balanced Budget . . . . . . . . . . . . . . . . . . . . . . . . 166
Transcritical Bifurcation of the Balanced Budget Model . . . . . 170
Analytical Transcritical Bifurcation . . . . . . . . . . . . 171
Special Case: Univariate Model . . . . . . . . . . . . . . 175
Endogenous Government Surpluses with Dynamic Inefficiency . . . . 177
Special Case with Dynamic Efficiency . . . . . . . . . . . . . . . 178
Leeper Tax . . . . . . . . . . . . . . . . . . . . . . . . . 178
Replacement Rate Uncertainty: Time Varying Probability
of Reform . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
Derivation of Social Welfare Function . . . . . . . . . . . . . . . 182
B. THREE PERIOD MODEL APPENDIX . . . . . . . . . . . . . . . 184
Solution Method . . . . . . . . . . . . . . . . . . . . . . . . 184
Time Path Iteration . . . . . . . . . . . . . . . . . . . . . 184
Examples of Steady States in the Three-Period Model . . . . . . . 185
Additional Types of Uncertainty . . . . . . . . . . . . . . . . . 187
Simple three-period timing uncertainty . . . . . . . . . . . 187
Two Special Cases with Balanced Social Security Budget . . . . . . 192
Balanced Budget, No Government Debt . . . . . . . . . . . . 192
xii
Chapter Page
Balanced Budget, Constant Debt . . . . . . . . . . . . . . . 195
C. LIFE-CYCLE HORIZON LEARNING APPENDIX . . . . . . . . . . 197
Stability of REE under Learning . . . . . . . . . . . . . . . . . 197
Recessions and Life-cycle Learning . . . . . . . . . . . . . . . . 198
Note on Golden Rule . . . . . . . . . . . . . . . . . . . . . . 203
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . 205
xiii
LIST OF FIGURES
Figure Page
1. Saddle-node bifurcation diagram. . . . . . . . . . . . . . . . . . . 17
2. Stability of roots for a planar system . . . . . . . . . . . . . . . . 21
3. Phase diagram, two-period model. . . . . . . . . . . . . . . . . . 23
4. Phase diagrams for saddle node bifurcation. . . . . . . . . . . . . . 25
5. Phase diagram initial conditions. . . . . . . . . . . . . . . . . . . 27
6. Time paths for fixed probability of benefit reform. . . . . . . . . . . 40
7. Time paths for fixed probability of tax reform . . . . . . . . . . . . 41
8. Time paths for history dependent reform. . . . . . . . . . . . . . . 44
9. Time paths for history dependent tax reform. . . . . . . . . . . . . 48
10. Time paths for capital and bond under four different scenarios . . . . . 57
11. Steady state contour graphs. . . . . . . . . . . . . . . . . . . . . 76
12. Steady state contour graphs of model with Leeper tax. . . . . . . . . 77
13. Time paths for introduction of social security. . . . . . . . . . . . . 81
14. Steady state contour graphs of model with Leeper tax. . . . . . . . . 82
15. Time paths for three period model with benefit or tax reform. . . . . . 84
16. Time paths for tax increase in period 3 or 4. . . . . . . . . . . . . . 93
17. Time paths for benefit reduction in period 3 or 4. . . . . . . . . . . . 94
18. Time paths for unsustainable policy, corrected via tax increase. . . . . 96
19. Time paths for unsustainable policy, corrected via benefit reduction. . . 98
20. Time paths for an economy with 4 option uncertainty. . . . . . . . . 102
21. Time paths for an economy that is on an explosive path
until social security benefits are cut. . . . . . . . . . . . . . . . . 106
xiv
Figure Page
22. Equilibrium paths for the realization of policy A and the
realization of policy B in a model with uncertainty . . . . . . . . . . 107
23. Ratio of beneficiaries to retirees . . . . . . . . . . . . . . . . . . 129
24. Time paths for capital and bonds under RE and LCH. . . . . . . . . 132
25. Time paths for expected interest rates, wages, and
government debt levels . . . . . . . . . . . . . . . . . . . . . . 133
26. Time paths for the savings accounts under RE and LCH . . . . . . . 134
27. Planned future savings and actual future savings. . . . . . . . . . . . 135
28. Time paths for capital and bonds for RE, LCH, and FHL . . . . . . . 137
29. Consumption equivalent variation measure of the welfare
cost of an announced social security benefit cut . . . . . . . . . . . 139
30. Consumption equivalent variation measure of the welfare
cost of an announced social security tax increase . . . . . . . . . . . 140
31. Consumption equivalent variation measure of the welfare
cost of an announced social security tax increase: learning
models compared to RE . . . . . . . . . . . . . . . . . . . . . . 142
32. Consumption equivalent variation measure of the welfare
cost of an announced social security benefit cut: learning
models compared to RE . . . . . . . . . . . . . . . . . . . . . . 143
33. Consumption equivalent variation measure of the welfare
cost of using adaptive learning along the transition path of
an announced social security benefit cut . . . . . . . . . . . . . . . 145
34. Transition paths for asset holdings in the RE model with
policy uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 148
35. Transition paths for capital and bonds in the LCH model
and RE model with policy uncertainty . . . . . . . . . . . . . . . 149
36. Transition paths for asset holdings in the LCH model and
RE model with policy uncertainty. . . . . . . . . . . . . . . . . . 150
37. Consumption equivalent variation measure of the welfare
cost of policy uncertainty in the RE model . . . . . . . . . . . . . 152
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Figure Page
38. Consumption equivalent variation measure of the welfare
cost of policy uncertainty in the LCH model . . . . . . . . . . . . . 154
39. Equilibrium paths for the rational expectations economy
and learning economies that forecast the tax burden . . . . . . . . . 158
40. Consumption equivalent variation paths learning economies
that forecast the tax burden . . . . . . . . . . . . . . . . . . . . 158
A.41.Phase diagram for balanced budget φˆ = (1 + n)τˆ . . . . . . . . . . . 168
A.42.Phase diagrams demonstrating the transcritical bifurcation
with a balanced budget . . . . . . . . . . . . . . . . . . . . . . 171
A.43.Phase diagrams demonstrating the transcritical bifurcation
with a balanced budget . . . . . . . . . . . . . . . . . . . . . . 172
A.44.Endogenous government surpluses with dynamic inefficiency . . . . . . 178
A.45.Leeper tax in two-period model. . . . . . . . . . . . . . . . . . . 180
A.46.Graph of p(bt) equation (A.15). . . . . . . . . . . . . . . . . . . 181
B.1. No social security . . . . . . . . . . . . . . . . . . . . . . . . 187
B.2. Social security, φ = 0.05 . . . . . . . . . . . . . . . . . . . . . 187
B.3. Time paths for an economy is in a steady state and
experiences a large tax increase in period 3, 4, or 5 . . . . . . . . . . 191
B.4. Time paths for an economy is in a steady state and
experiences a large benefit reduction in period 3, 4, or
5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
C.5. Convergence to a steady state for the Finite Horizon
Lifecycle Learning model . . . . . . . . . . . . . . . . . . . . . 199
C.6. Equilibrium paths for capital and bonds in an economy with
a recession . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
C.7. Consumption equivalent variation measure of the welfare
cost of a recession . . . . . . . . . . . . . . . . . . . . . . . . 201
C.8. Path of consumption for the rational and learning models
for a recession . . . . . . . . . . . . . . . . . . . . . . . . . . 202
xvi
LIST OF TABLES
Table Page
1. Baseline parameterization for two-period model . . . . . . . . . . . . 29
2. Last possible reform dates. . . . . . . . . . . . . . . . . . . . . . 31
3. Smallest possible benefit cut . . . . . . . . . . . . . . . . . . . . 31
4. Possible time paths of the replacement rate. . . . . . . . . . . . . . 35
5. Possible benefit reductions . . . . . . . . . . . . . . . . . . . . . 45
6. Possible tax reforms . . . . . . . . . . . . . . . . . . . . . . . . 49
7. Welfare of various reforms. . . . . . . . . . . . . . . . . . . . . . 51
8. Welfare of various reforms on an explosive path . . . . . . . . . . . . 53
9. Minimum reform needed. . . . . . . . . . . . . . . . . . . . . . 55
10. Baseline parameters in 3-period model. . . . . . . . . . . . . . . . 79
11. Summary of perfect foresight policy experiments. . . . . . . . . . . . 83
12. Policy experiments with simple two-period timing uncertainty. . . . . . 97
13. Policy experiment with timing and policy uncertainty. . . . . . . . . . 103
14. Baseline parameters for 6 period model . . . . . . . . . . . . . . . 128
B.1. Steady States for figures in section III and B . . . . . . . . . . . . . 186
B.2. Policy Experiments with three-period timing uncertainty . . . . . . . . 192
xvii
CHAPTER I
INTRODUCTION
The Old Age Survivor and Disability Insurance program (OASDI),
commonly referred to as social security, is one of the largest transfer programs
in the world, accounting for roughly twenty percent of U.S. federal government
spending. OASDI currently provides retirement benefits to 44 million people and
collects payroll taxes from 171 million workers. Social security’s cost has exceeded
its tax income in each year since 2010. For 2017, the program is expected to
have a primary deficit of $59 billion. The Social Security Administration (SSA)
projects this relationship to extend through the short and long-term. Similarly, the
Congressional Budget Office (CBO) projects federal government deficits to grow
from 2.9% of GDP in 2017 to 9.8% in 2047, driven in large part by entitlement
spending. Concurrent with an increase in government deficits, the stock of federal
debt is expected to reach 150% of GDP by 2047 (CBO, 2017).
The SSA has built up a trust fund to finance the difference between tax
receipts and social security benefit payments; however, the fund is projected to
be depleted by the year 2034. Following the depletion of the trust fund, the SSA
will not be able to fully pay scheduled social security benefits. The SSA Board of
Trustees project that an immediate and permanent payroll tax rate increase of 2.76
percentage points (12.4% to 15.16%) or an immediate and permanent benefit cut
of 17% applied to all current and future beneficiaries would be needed to insure
sufficient tax revenue to cover promised benefits over the long-term (SSA, 2017).
Given the projected shortfall between social security tax revenues and
promised benefits, reform seems likely. However, it is not clear if or when the
1
government will act. The uncertainty regarding when and how social security will
be reformed is interesting and economically important. A survey conducted by the
American Life Panel in 2011 found that 56% of respondents are “not too confident”
or “not confident at all” that the social security system will be able to provide
them the level of benefits currently promised. Wariness in the government’s ability
to pay benefits is decreasing in age, with 75% of respondents under the age of 40
reporting they are not confident they will receive benefits (47% “not too confident”
and 28% “not confident at all”).1 Social security benefits are a major source of
retirement income for many Americans, and uncertainty regarding future benefits
could impact savings decisions. The economy wide uncertainty regarding social
security deficits and debt accumulation is also interesting, as rapid debt growth
could strain the economy (see, for example, Davig, Leeper, and Walker (2010)).
Social security reform uncertainty has been examined from the perspective
of the government.2 Additionally, a growing literature examines the welfare
implications of uncertain social security reform from the perspective of the
agent. However, little research has been done on the long-run macroeconomic
consequences of social security policy uncertainty.
Bu¨tler (1999) calibrates a partial equilibrium life-cycle model to the Swiss
economy and examines the impact of expectations on social security policy.
The underlying model does not have any policy uncertainty, but agents form
(potentially incorrect) beliefs about future government policy. Bu¨tler finds
that the beliefs of agents have large impacts on the aggregate economy through
1The RAND American Life Panel is a nationally representative, probability-based panel of
over 6000 members ages 18 and older who are regularly interviewed over the internet for research
purposes. This question comes from the 2011 survey Well-being ms179. More information about
the ALP is available at alpdata.rand.org.
2See Auerbach (2014) and Auerbach and Hassett (2007).
2
precautionary savings. The welfare cost of social security reform uncertainty
is largest for the older cohorts in her model. Gomes, Kotlikoff, and Viceira
(2012) builds a more complex life-cycle model with social security reform timing
uncertainty. They find that agents would be willing to sacrifice up to 0.6% of life-
time resources to have policy uncertainty resolved at age 35 rather than discovering
that social security benefits have been cut at age 65.3
Recent work by Caliendo, Gorry, and Slavov (2015), Kitao (2016),
and Nelson (2016), examine the welfare cost of timing and social security
policy type uncertainty. Caliendo et al. (2015) build a continuous-time, partial
equilibrium model that can accommodate rich timing and policy uncertainty. The
(heterogenous) agents form expectations over a two-dimensional distribution of
reform type and reform dates.4 The welfare cost of policy uncertainty is small
(about 0.01% of lifetime consumption), but varies across the income distribution
with largest costs for low-income agents. Kitao (2016) considers a general
equilibrium discrete-time model with benefit cut timing uncertainty, calibrated
to the Japanese economy.5 Kitao’s model incorporates the changing population
growth rate of Japan, and finds welfare costs of over two percent of life-time
consumption for older generations. Nelson (2016) uses a smaller-scale general
equilibrium discrete-time model to examine joint timing and policy uncertainty.
3Note, the agent’s in the Gomes et al. model do not face actual policy timing uncertainty–
benefit are cut with certainty at age 65–but, agents do not know their benefits level with certainty
until a given age.
4Caliendo et al. (2015) consider a continuous distribution of reforms, governed by a parameter
α that indicates the share of the reform that is a tax increase, with (1−α) of the reform is a
benefit cut. Timing uncertainty is modeled using a continuous distribution of potential reform
dates.
5The main analysis of Kitao’s paper focuses on policy timing uncertainty. An extension of the
model considers timing and policy type uncertainty with a given benefit cut occurring in either
2020, 2030, or 2040; or a smaller benefit cut occurring in 2040.
3
Pension reform either occurs early or late, and either takes the form of a benefit cut
or a tax increase. Nelson focuses his analysis on varying the probability that reform
takes a certain type. He find welfare costs of reform of similar magnitude to other
authors.
Fiscal policy uncertainty more broadly, often motivated by a story about
entitlement spending growth has also been studied in a variety of cases. Davig,
Leeper, and Walker (2010, 2011) examine the long-run interactions of fiscal and
monetary policy in a New Keynesian economy with growing government debt and
uncertain fiscal policy. They find that hyperinflation is a possible, but unlikely
outcome of growing entitlements, and that the government is likely to renege
on promised future transfers. Davig and Leeper (2011b, 2011a) build similar
models with fiscal policy uncertainty to examine the role of expectations and
debt accumulation on the long-run path of the economy. Davig and Foerster
(2014) examine the interaction of fiscal policy uncertainty and a fiscal cliff. The
main weakness of these New Keynesian papers is the reliance on a representative,
infinitely lived consumer to model household behavior. The authors have to
abstract away from key details of a social security system (namely the transfer of
wage income from the workers to retirees) because the models do not incorporate
aging or retirement.
One natural alternative to the infinite horizon, representative agent models,
is the overlapping generations (OLG) model, which explicitly models the aging and
retirement of agents. OLG models provide a natural framework to study the impact
of social security in an economy. Several authors have examined the desirability
and feasibility of various social security reforms (for example, see Bouzahzah,
De la Croix, and Docquier (2002) or Bohn (2009)). The implications of postponing
4
reform have also been considered. R. Evans, Kotlikoff, and Phillips (2013) show
that the economy breaks down if government debt becomes explosive. Similarly,
Chalk (2000) demonstrates if government deficits are too large, government debt
spirals out of control and convergence to a steady state is not possible.
The long-run implications of social security policy uncertainty have not been
examined in an overlapping generations framework with rational agents (or with
boundedly rational agents with model-consistent expectations). I begin this work in
Chapter II of this disseration. I construct an analytically tractable two-period OLG
model with capital and government debt. The government runs a social security
transfer system that taxes the working generation to pay benefits to retirees. If
retiree benefits exceed tax revenue, the government finances the short-fall with
one period bonds. Thus, the government deficits are endogenous. I find that the
existence of and convergence to steady states depends on the size of government
deficits and the stock of government debt. I demonstrate two additional policy
results with the two-period model. two-period First, if the economy is on an
explosive trajectory, either because the existing stock of government debt is large
relative to the capital stock or because the endogenous government deficits are too
large, the government can change the social security system to put the economy
on a stable path. Specifically, the government can increase social security taxes or
decrease benefits to reduce the deficit and pay down the government debt. This
intuitive result is important, because it highlights the government’s ability to
avoid catastrophic debt accumulation by making straight-forward (albeit likely
unpopular) policy changes.
Second, if the economy is on an explosive path and agents believe that
policy will change in the next period, these beliefs can be sufficient to put the
5
economy on a stable path. Specifically, if the young generation believes that their
social security benefits will be small, they increase their savings, which increases
the aggregate capital stock, increases future wages, and increases future tax
revenue. Using my simple two-period model, I can show that if the social security
reform is large enough or if the probability of reform is high high enough, the
economy can converge to a steady state, even if the government runs deficits (or
accumulates debt) that would otherwise be explosive. This result highlights the role
of expectations in the economy.
The main mechanism in the model is the precautionary savings of agents.
Young agents who believe they may receive smaller pension benefits in the future,
respond by increasing their savings today. The precautionary savings increases
the capital stock. The general equilibrium feedback effects of higher capital
(higher wages and lower interest rates) influence the social security deficit and
thus government debt levels. Changes in capital and government debt alter the
equilibrium paths of the economy.
In Chapter III, I extend my two-period model to a three-period model in
which agents work for two-periods and then are retired for the third period. The
appeal of a two-period model is that it can be solved analytically. The downside
of a two-period model, is that agents only care about policy changes that happen
in the next period (absent a bequest motive). This limits the range of policy
experiments that can be considered. It is necessary to extend the model to three-
periods in order to entertain reforms, or expected reforms, that are more than
one period in the future or reforms that involve changes in taxes. The three-
period model cannot be solved analytically, but it can be solved using standard
computational methods (described in Appendix B).
6
The three-period model is inherently forward looking, which allows for
more interesting policy analysis than the simple two-period model. With three-
periods, the young worker’s saving-consumption decision depends both on the
predetermined capital and bonds for the next period, and also on her expectations
over capital and bonds in the following period. This forward looking element of the
model allows multi-period uncertainty analysis in the fashion of Davig et al. (2011)
or Davig and Leeper (2011a).
The main policy implications of the two-period model can be replicated in
the three-period model. In addition, the three-period model illustrates that certain
types of policy uncertainty can dampen the precautionary savings motive of agents
who expect reform in (or near) their lifetime. Additionally, analysis in the three-
period model suggests that policy uncertainty may have lasting consequences, even
after reform is enacted.
In Chapter IV, I explore the macroeconomic consequences of social security
policy uncertainty in a multi-period lifecycle model. Agents’ expectations about
future social security policy drive agent-level decision making. These decisions,
combined with realized government policy, determine macroeconomic output
and prices. The way in which agents form expectations has a large impact on
both the short-run responses to changes in policy (or possible changes in policy)
and the long-run level of capital and output in the economy. I consider rational
expectations and adaptive expectations. This chapter contributes to a growing
literature that examines the response to anticipated fiscal policy in models with
adaptive expectations (see G. W. Evans, Honkapohja, and Mitra (2009), Mitra,
Evans, and Honkapohja (2013), Gasteiger and Zhang (2014), and Caprioli (2015)).
7
The rational expectations hypothesis is standard in macroeconomics and I
develop a rational expectations model as the baseline for my analysis. I also relax
the assumption of rational expectations and explore the consequences of agents
using adaptive expectations in a model with an aging society and social security
reform. One criticism of the rational expectations hypothesis is that it requires
agents to possess more knowledge about the structure of the economy than any
econometrician possesses about the actual economy. Models of adaptive learning
relax that requirement, and allow for boundedly rational agents (Sargent (1993),
G. W. Evans and Honkapohja (2001)).
In this chapter, I introduce two new frameworks for modeling bounded
rationality that I call life-cycle horizon learning, and finite horizon life-cycle
learning. In both models, agents use adaptive expectations to forecast future
aggregates, such as wages and interest rates. The expectations are updated each
period as agents acquire additional information. This adaptive learning feature
introduces cyclical dynamics along a transition path, which magnifies the welfare
cost of changes in policy and policy uncertainty. I model policy uncertainty as a
stochastic process in which reform takes place in one of two-periods as either a
benefit cut or a tax increase. I find the welfare cost of this policy uncertainty is
equivalent to less than 0.3% of period consumption for the cohort of agents most
harmed in a standard, rational expectations framework. The welfare cost of policy
uncertainty is larger in the learning models; the worst-off cohort in the life-cycle
horizon learning model would be willing to give up 1.98% of period consumption
to avoid policy uncertainty, and would be willing to give up 0.8-1.2% of period
consumption to avoid the cyclical dynamics introduced via learning.
8
In all chapters of this dissertation, I model policy uncertainty as policy risk.
That is, I consider various stochastic processes for government policy parameters.
The possible realizations of policy parameters and their relative probabilities are
known to all agents in the model. I refer to this type of risk as “uncertainty.” A
growing body of economic and finance research has distinguished between risk
(stochastic process with known probabilities) and uncertainty or ambiguity (process
in which either probabilities or possible realizations are unknown). I will abstract
from ambiguity in this dissertation; modeling the response to agents to social
security policy risk provides an interesting baseline for thinking about the impact of
government policy uncertainty on the macroeconomy. I may consider richer models
of uncertainty or ambiguity in my future research.
9
CHAPTER II
POLICY UNCERTAINTY IN AN ANALYTICAL TWO-PERIOD OLG MODEL
Motivation
The need for pension reform in aging societies, like the United States, is
well documented. Despite the looming need, uncertainty remains regarding the
timing and structure of eventual reform. In this chapter, I construct an analytically
tractable two-period OLG model with capital, government debt, and uncertain
social security reform. I find that existence of and convergence to steady states
depends on the size of endogenous government deficits and the stock of government
debt. I demonstrate three key policy results with my model.
First, if the economy is on an unstable trajectory, either because the existing
stock of government debt is large relative to the capital stock or because the
endogenous government deficits are too large, the government can change the
social security system to put the economy on a stable path. This intuitive result
highlights the government’s ability to avoid catastrophic debt accumulation by
making straight-forward (albeit likely unpopular) policy changes. The model also
highlights the tradeoff between the size and timing of reform. If the government
acts sooner, it can make relatively small changes. If the government delays reform,
a larger benefit cut (or tax increase) is needed to achieve stability.
Second, if the economy is on an unstable path and agents believe that policy
will change in the next period, these beliefs can be sufficient to put the economy
on a stable path. If the probability of reform is large enough, the economy can
converge to a steady state, even if the government runs deficits (or accumulates
10
debt) that might otherwise be explosive. If agents believe that benefits might be
cut in the future, they engage in precautionary savings. This savings can increase
the capital stock enough to delay the need for reform. This result parallels the
results of Davig et al. and highlights the role of expectations in the economy.
Third, the optimal policy reform from the perspective of a benevolent
planner depends on the length of the planner’s time horizon, and the social
discount factor used. If the planner looks many generations into the future,
reforms that take place sooner will be socially preferable to later reforms. It also
appears that agents are worse off in an economy that faces uncertain policy reform
compared to an economy with known policy changes.
The remainder of the chapter is organized as follows. I begin by presenting
the model in section II. The model has several interesting features, including
a saddle-node bifurcation, which is discussed in section II. Phase diagrams are
presented in section II. Several special cases are presented at the end of section
II. Policy analysis is presented in section II. The paper ends with welfare analysis in
section II.
Model
Household Problem
Households live for two periods and choose savings and consumption
to maximize their lifetime utility. An agent who is young in period t, chooses
savings (or assets), at, and consumption, c1,t, for the first period, and retirement
period consumption, c2,t+1, to maximize utility, taking prices, and government
social security policy (tax rate τt, and benefit zt+1) as given. Agents supply labor
inelastically in the first period and are retired during the second period. Agents
11
receive the wage Atwt where wt is the wage in effective units for labor provided in
period t. The gross real return on savings is given by Rt+1.
The labor force (and thus, the population) grows at rate n such that Lt =
(1 + n)Lt−1. Labor-augmenting technology grows at rate g such that At = (1 +
g)At−1.
The household’s problem can be written as follows:
max
c1,t, at, c2,t+1
u(c1,t) + βEtu(c2,t+1)
s.t. c1,t + at = (1− τt)wtAt
c2,t+1 = Rt+1at + zt+1
Here the notation Et is used to denote the time t conditional expectation.
Optimization leads to the household first order condition:
u′(c1,t) = βEtRt+1u′(c2,t+1).
Assuming log-preferences, the first order condition implies the optimal
savings choice:
a∗t =
β
1 + β
(1− τt)wtAt − Et
(
zt+1
(1 + β)Rt+1
)
(2.1)
where a∗t indicates the optimal choice.
Firms
Firms produce output using a Cobb-Douglass production with labor-
augmenting technology (At). Aggregate output, Yt is given by:
Yt = F (Kt, LtAt) = K
α
t (LtAt)
1−α.
This production function can be written in per-effective worker or efficient
terms as yt = f(kt) = k
α
t where kt = Kt/(AtLt) and yt = Yt/(AtLt).
12
Firms are competitive which leads to interest rate and wage:
Rt = f
′(kt) + 1− δ (2.2)
wt = f(kt)− ktf ′(kt) (2.3)
For convenience, I assume full depreciation (δ = 1). This assumption allows
me to solve the non-stochastic model analytically.
Government
The government runs a modified pay-as-you-go social security system funded
via pay-roll taxes (rate τt) and the issuance of bonds Bt. Social security benefits
are designed to replace a fraction, φt ∈ [0, 1], of an agent’s wage. Social security
benefits are wage-indexed, such that benefits in time period t are calculated as the
replacement rate φt multiplied by the wage at time t:
zt = φtwtAt (2.4)
The government issues one-period bonds, Bt+1 (issued in period t, repaid in period
t+ 1) to satisfy the following flow budget constraint:
Bt+1 = RtBt + φtAtwtLt−1 − τtAtwtLt (2.5)
where φtAtwtLt−1 − τtAtwtLt is the period t deficit (transfers minus revenues).
Markets
This economy has four markets: the labor market, the capital market, the
bond market, and the goods market. In equilibrium, all markets clear. Agents
supply labor inelastically. The wage rate adjusts to clear the labor market. Capital
market clearing requires that the savings of all agents in period t is equal to
13
aggregate capital and government debt in period t + 1. Agents are indifferent
between capital and debt, since both assets offer the same, safe return. The capital
market clearing equation can be written as follows:
Kt+1 +Bt+1 = a
∗
tLt
Bond market clearing is given by the government’s flow budget constraint 2.5, and
the goods market clears by Walras law.
Along the balanced growth path, effective labor hours (AtLt), output (Yt),
capital (Kt), and bonds (Bt) all grow at rate (1 + n)(1 + g). Therefore, it will be
convenient to rewrite the market clearing equations in per-effective-hours terms
by defining bt = Bt/(AtLt), and kt = Kt/(AtLt). I will refer to variables in per-
effective-hours terms as “efficient.”
The market clearing equations can be written in efficient terms as:
(1 + n)(1 + g)(kt+1 + bt+1) =
a∗t
At
(2.6)
(1 + n)(1 + g)bt+1 = Rtbt +
(
φ
1 + n
− τ
)
wt (2.7)
Equilibrium
Given initial conditions k0, b0, a−1, an initial cohort of young and old,
population growth rate n, and rate of technological progress g, a competitive
equilibrium is a sequences of functions for the households {at}∞t=0, production
plans for the firm, {kt}∞t=1, government bonds {bt}∞t=1 factor prices {Rt, wt}∞t=0,
government policy variables {τt, φt}∞t=0, that satisfy the following conditions:
1. Given factor prices and government policy variables, individuals’ decisions
solve the household optimization problem (2.1)
14
2. Factor prices are derived competitively according to (2.2) and (2.3)
3. All markets clear according to (2.6), and (2.7)
The equilibrium path of the state variables, capital and bonds, can be characterized
as a non-linear system of first order difference equations. These are called the
transition equations. The equilibrium paths of the remaining variables are simply
functions of the state variables and exogenous parameters.
The transition equation for bonds is found by plugging the competitive
market prices into the government’s flow budget constraint (2.7) and rearranging:
bt+1 =
1
(1 + n)(1 + g)
[
αkα−1t bt +
(
φ
1 + n
− τ
)
(1− α)kαt
]
(2.8)
The equilibrium equations for capital is found by plugging the optimal
savings decision into the capital market clearing equation, subbing in competitive
prices and the social security benefit, and finally subbing in transition equation for
bonds (2.8). This leads to the transition equation listed below.
Etkt+1 =
[
(1 + n)(1 + g) +
φ(1− α)(1 + g)
(1 + β)α
]−1
[
β
1 + β
(1− τ)(1− α)kαt − αkα−1t bt −
(
φ
1 + n
− τ
)
(1− α)kαt
] (2.9)
The two transition equations (2.8) and (2.9) govern the dynamics of the
system. The transition equations can be written compactly as follows:
Etxt+1 = G(xt) (2.10)
where xt = (kt, bt)
′.
15
Saddle-node Bifurcation and Steady States
The steady state is the set {k, b} for which the system (2.10) is constant.
The non-stochastic steady state is a pair {k, b} that solves:
(1 + n)(1 + g)b = αkα−1b+
(
φ
1 + n
− τ
)
(1− α)kα (2.11)
(1 + n)(1 + g)(k + b) =
β
1 + β
(1− τ)(1− α)kα − φ(1 + g)(1− α)
(1 + β)α
k (2.12)
The steady state equations are both non-linear in k and can only be solved
numerically. Both the existence and number of nontrivial steady states depend on
the parameter values. The social security system deficit or surplus is determined
endogenously. If the deficit becomes too large, the model no longer has a non-zero
steady state. This relationship between steady states and the size of the deficit is
a common feature in planar OLG models.1 The key innovation in this model is
that the deficit is endogenous, as described in equation (2.5). The deficit depends
directly on the parameters τ , φ and n, as well as indirectly on the other parameters
(α ,β, δ) through the steady state level of capital.
As an example, consider a parameterization of the model with two steady
states. At each of these steady states capital is positive and the government holds
a positive amount of bonds. As τ decreases, the government’s deficit increases. At
a threshold critical value τc, the government’s deficit is as large as the economy can
bear, and only one steady state of the model exists. If τ falls below the threshold
τc, the deficit is unsustainably large and government debt becomes explosive; there
are no steady states. Thus, as τ increases from zero to one, the number of steady
states increases from zero (on the interval τ ∈ [0, τc]) to one (when τ = τc) to two
1See Azariadis (1993) Chapters 19 and 20, and De La Croix and Michael (2002) Chapter 4
section 3 for a discussion.
16
0.10 0.12 0.14 0.16 0.18 0.20
tax rateτ0.00
0.05
0.10
0.15
0.20
steady state
captial
τc
FIGURE 1. Saddle-node bifurcation diagram. For illustrative purposes, this graph
plots steady state intensive capital for each tax rate τ . For τ < τc no steady states
exist; for τ > τc two steady states exist. When τ = τc one steady state exists.
(for τ > τc). This relationship is called a saddle-node bifurcation and is illustrated
in Figure 1.
Similarly, the other parameters that directly affect the deficit (n and φ) can
be treated as the bifurcation variable, holding all other variables constant. The
diagram for n would look qualitatively the same as the diagram for τ presented
above. The bifurcations for φ moves in the opposite direction as the τ bifurcation;
that is, it is an inverse saddle-node bifurcation. At low values of the φ, below the
bifurcation threshold, two steady states exist; and above the threshold, no steady
states exist.
Formally, the saddle-node bifurcation can be described by the equation
below:
xt+1 = G(xt; τ) (2.13)
17
This is equation is exactly the same as the transition equation (2.10); the
dependence of the steady state on the parameter τ is emphasized in equation (2.13)
for notational ease. As τ varies, so will the fixed points of the system (2.13).
Let DxG(xc; τc) denote the Jacobian matrix of G evaluated at the steady
state (xc, τc). At the critical value τc, the Jacobian has a single eigenvalue equal
to +1, with the other eigenvalue strictly less than one in modulus. Then, for a
sufficiently small neighborhood of (xc; τc), the system G has two hyperbolic steady
states for τ > τc, exactly one steady state at τc, and no (non-trivial) steady states
for τ < τc. In general, the steady states for this model do not have closed-form,
analytical solutions, thus the eigenvalues of the Jacobian cannot be evaluated
analytically. I use numerical methods to verify the saddle-node bifurcations for
each of the relevant bifurcation parameters.
In a special case of the model when the government runs a balanced budget,
the steady state equations can be solved analytically. In this special case, the model
under goes a transcritical bifurcation rather than a saddle-node bifurcation. A
transcritical bifurcation occurs when the stability properties of the fixed points of
a system change as an underlying parameter changes. At a critical value of the
parameter, a single steady state exists and a single eigenvalue of the Jacobian
evaluated at that steady state is equal to +1. On either side of the critical value
of the parameter, two steady states exist. The stability properties of the steady
states switch as the parameter moves through the critical value. The transcritical
bifurcation is demonstrated analytically in appendix A.
18
Stability
The stability of a steady state can be verified by examining the eigenvalues
of the Jacobian matrix evaluated at the steady state. Capital and bonds are both
predetermined in this model; steady states with two eigenvalues with modulus
less than one are stable (the steady state is a sink). Given appropriate initial
conditions, the model converges to the stable steady state.
A steady state with one or more eigenvalue greater than one in modulus
is unstable, or explosive. Any deviation from the unstable steady state will either
cause bonds to grow infinitely, or will converge to the stable steady state (if there
are two steady states). At the saddle-node bifurcation (where there is only one
steady state), one eigenvalue of the Jacobian will be equal to one and the other
eigenvalue will be less than one in modulus.2
The eigenvalues and stability of steady states can be depicted graphically
for a planar system, such as (2.10). The eigenvalues of a planar system are the
solutions (λ) to the characteristic equation which can be written as
p(λ) = (λ− λ1)(λ− λ2) = 0
where λ1 and λ2 are the eigenvalues. Note if one of the eigenvalues of the Jacobian
is equal to 1, that p(1) = (1− λ1)(1− λ2) will be equal to zero. Thus, if we plot the
eigenvalues in Trace-Determinant space (plotting the Trace on the horizontal axis,
and the Determinant on the vertical axis), the saddle-node bifurcations appear on
the segment of p(1) = 0 between (0,-1) and (2,1), which indicated by a black line in
Figure 2.
2See Azariadis (1993) chapter 6 for a discussion of the stability of steady states in two-period
OLG models. See Laitner (1990) for a discussion of stability in multi-period OLG models.
19
In the baseline parameterizations of the model with two steady states, the
low capital steady state is explosive and the high capital steady state is stable.3
The eigenvalues for an example case are plotted in Trace-Determinant space in
Figure 2 (recall the product of the eigenvalues is equal to the determinant, and
the sum of the eigenvalues is equal to the trace in a two-by-two system). The
low-capital steady state is an (unstable) saddle and appears in the right-most
region of the Trace-Determinant graph. The high-capital steady state is a sink
and appears in the center.4 I have numerically verified that the eigenvalues of the
system approach p(1) = 0 along the relevant segment as the bifurcation variable
approaches its threshold value.
Phase Diagrams
The dynamics of the system (2.10) can be displayed in (kt, bt) space using a
phase diagram. The vector fields are determined by the following expressions:
bt+1 − bt (2.14)
kt+1 − kt (2.15)
The constant bond locus is the combination of capital and bonds for which bonds
are unchanging and occurs when bt+1 − bt = 0. Similarly, the constant capital locus
plots the points for which kt+1 − kt = 0. Using the transition equation for bonds
(2.8), expression (2.14) implies changes in bonds are given by:
R(kt)bt +
(
φ
1 + n
− τ
)
w(kt)− (1 + n)(1 + g)bt (2.16)
3I have verified the stability of steady states numerically.
4Chalk (2000) refers to the low capital steady state in a planar model as the saddle steady
state, and refers to the high capital steady state as a sink. This dichotomy is also reflected in
Azariadis (1993). I follow their word-choice in this paper and will refer to the high capital (stable)
steady state as a sink. I’ll also refer to the low capital, explosive steady state as a saddle.
20
p(-1)=0p(1)=0
Det=1
Det2 +4Tr=0
saddle node bifurcation
complex roots (source)
complex roots (sink)
real roots (sink)
real roots (saddle)
real roots (saddle)
real roots (source)
-3 -2 -1 1 2 3 Tr
-3
-2
-1
1
2
Det
FIGURE 2. Stability of roots for a planar system. The eigenvalues for the
standard case of the model are plotted (as Tr-Det pairs). As the policy parameter
approaches threshold value for the bifurcation, the eigenvalues approach the
line p(1)=0 which indicates the saddle-node bifurcation. When a saddle-node
bifurcation occurs, the roots will fall along the segment p(1)=0 between (0,-1)
and (2,1), which is indicated by a thicker line. This graph also includes the line
p(−1)=0 which indicates a Flip bifurcation (along the segment (-2,1), (-1,0)), and
the line Det(DxG)= 0 along which the Hopf bifurcation occurs (along the segment
(-2,1) and (1,2)). The graph also includes the parabola Tr(DxG)
2−4Det(DxG)=0.
Eigenvalue pairs that fall above (inside) the parabola have complex roots; those
that fall below (outside) the parabola have real roots. See Azariadis Figure 6.6 and
8.4 for further discussion.
21
Where R(kt) = αk
α−1
t and w(kt) = (1 − α)kαt for notational ease. If we set (2.16)
equal to zero, this gives the equation for the constant bond locus. The sign of this
expression depends on the value of kt. If the level of capital, kt is less than the
golden rule level of capital, kgr, then the gross interest rate R(kt) is greater than
the golden rule gross interest rate R(kgr) = (1 + n)(1 + g) and the sign of the
denominator in (2.17) below is positive. When kt > kgr is denominator of (2.17)
is negative. The inequality in (2.17) below indicates when the change in bonds is
positive (when bt+1 − bt > 0). This inequality depends on the value of capital.
bt

>
−( φ
1+n
− τ)w(kt)
R(kt)− (1 + n)(1 + g) if kt < kgr
<
−( φ
1+n
− τ)w(kt)
R(kt)− (1 + n)(1 + g) if kt > kgr
(2.17)
When kt is equal to the golden rule level of capital, kgr, the interest rate R(kgr) =
(1 + n)(1 + g) and the denominator in (2.17) is zero.5
Using the transition equation for capital (2.9), expression (2.15) implies:
ψ−1
[
β
1 + β
(1− τ)(1− α)kαt − αkα−1t bt −
(
φ
1 + n
− τ
)
(1− α)kαt
]
− kt (2.18)
Where ψ = (1+n)(1+g)+φ(1−α)(1+g)/(1 + β)α. When (2.18) is equal to
zero, this expression give the constant capital locus. Using the factor price notation
discussed above, the change in capital is positive (i.e., kt+1 − kt > 0) if
bt <
β
1+β
(1− τ)w(kt)− ( φ1+n − τ)w(kt)− ψkt
R(kt)
(2.19)
Note that the direction of the inequality of (2.19) holds as long as the interest rate
R(kt) > 0, which will always be the case if kt ≥ 0.
Figure 3 displays the vector fields implied by (2.14) and (2.15), along with
the corresponding constant capital and constant bond loci. The constant bond
5The derivation of the golden rule level of capital is given in Appendix A.
22
0.00 0.05 0.10 0.15
0.00
0.02
0.04
0.06
k
b
stable ss
saddle ss
Δk=0
Δb=0
saddle
FIGURE 3. Phase diagram, with parameters chosen to show two non-trivial steady
states. The constant capital loci is the solid gray line. The constant bond loci is
depicted as a black line. The saddle path is indicated by a dashed line. The arrows
represent the vector field for the dynamics of the system. The steady states occur
at the intersections of the constant capital and constant bond loci.
locus has an asymptote at the golden rule level of capital. The graph depicts the
dynamics for a set of parameters that permit two steady states. For this baseline
case, I have chosen parameter values that generate steady states with positive
values for both bonds and capital.6
As Figure 3 illustrates, two dynamically inefficient steady states can arise in
this model because of the downward sloping constant capital locus (∆k = 0) and
the downward sloping constant bond locus (∆b = 0). The constant capital locus
is downward sloping (in the relevant region when k > kgr) because of crowding
out. Increased bond holdings by the government crowds out dynamically inefficient
capital and reduces the capital stock. The constant bond locus is downward sloping
(in the relevant region when k > kgr) because of an interest rate effect. As the
6It is also possible to choose parameters such that one of the steady states has negative bonds
(see section II).
23
capital stock decreases, the interest rate increases. This increases the government’s
debt burden and they must issue new bonds.
If the government changes policy in such a way that deficits increase (either
by raising social security benefits, increasing φ, or by decreasing the social security
tax rate τ), this shifts the constant bond locus out. As the bond locus shifts out,
the two steady states move close together. At a certain threshold, the bond locus
and capital locus are tangent, and only one steady state exists.7 The threshold
deficit level for a steady state to exist is endogenous. For deficit levels above this
threshold, no steady state exists. This saddle-node bifurcation is depicted in Figure
4.
Dynamic Efficiency
I have assumed dynamic inefficiency and endogenous government deficits
as the baseline case for this model. This results in steady state(s) with positive
bonds and positive capital. In OLG models, debt is only valued if the interest
rate is less than the population growth rate; that is, if the economy is dynamically
inefficient. Some economists, notably Chalk (2000) and Azariadis, have argued
that dynamic inefficiency is representative of the postwar U.S. experience.8 I am
agnostic about this assumption, and allow for dynamic efficiency in Appendix A
(with a balanced social security budget) and in Appendix A (with endogenous
government surpluses).
7Note, that there are several combinations of parameters that can generate this threshold
deficit level.
8 Martin and Ventura (2012) and Gali (2014) also consider dynamically inefficient OLG models
with asset bubbles. In these models, agents over-accumulate capital as they save. Asset bubbles
(or public pension with deficits), can be used to crowd out dynamically inefficient capital.
24
0.05 0.10 0.15 0.20
k
-0.02
0.02
0.04
0.06
b
kgr
stable ss
saddle ss
Δk=0
Δb=0
0.05 0.10 0.15 0.20
k
-0.02
0.02
0.04
0.06
b
kgr
saddle ss
Δk=0
Δb=0
0.05 0.10 0.15 0.20
k
-0.02
0.02
0.04
0.06
b
kgr
Δk=0
Δb=0
FIGURE 4. Phase diagrams for a dynamically inefficient economy running a deficit
in the social security program. As the size of the deficit endogenously grows as an
underlying parameter is changed (such as the tax rate being lowered), the constant
capital locus shifts out (up and to the left). At a critical value of the parameter,
only one steady state exists. When deficits are large enough, the capital locus shifts
so that it no longer crosses the bond locus and no steady states exist. This is an
illustration of the saddle-node bifurcation.
25
Explosive Paths and the Basin of Attraction
The parameters of the model govern both the number of steady states in the
model, and the basin of attraction for the stable steady state. For the baseline case
of the model (with dynamic inefficiency and government deficits), two non-trivial
steady states exist as depicted in Figures 3. The basin of attraction for the stable,
high capital steady state depends on the saddle path for the saddle steady state.
Initial conditions that lie below (to the right of) the saddle path will converge to
the stable steady state, and initial conditions above (to the left) of the saddle will
not converge. For example, in Figure 5, initial conditions A lead to an explosive
path where capital goes to zero in finite time and bonds approach infinity. The
initial conditions B converge.
The size of the basin of attraction also depends on the parameters of the
model. If the government runs a large (enough) deficit, no stable steady state exists
and the basin of attraction disappears. Chalk (2000) emphasizes the existence and
size of the basin of attraction is endogenous in a planar OLG model with deficits.
Similarly, in my set-up, the possibility of converging to a steady state depends on
government policy. Changes in policy alter the basin of attraction and eliminate
(or add) initial conditions that otherwise might converge to a steady state. This
has clear policy implications; there is not a single level of government debt that is
too high and therefore explosive. Rather, the government’s ability to carry a (large)
debt burden depends on it current and past deficit policies. Running large deficits
reduces the basin of attraction and reduces the feasible capital-bond pairings that
are convergent in the long-run.
26
0.05 0.10 0.15 0.20
k
-0.02
0.02
0.04
0.06
b
kgr
saddleΔk=0
Δb=0
A
A'
B
B'
FIGURE 5. Parameters chosen such that two non-trivial steady states exist. Initial
conditions above the saddle path, such as points A and A′ will not converge to a
steady state. Initial conditions below the saddle path, such as points B and B′,
will converge to the stable, high capital, steady state. The size of the government
deficit (or surplus) determines the size of the basin of attraction. Smaller deficits
(or larger surpluses below a certain threshold) create a larger basin of attraction.
Explosive Paths
Initial conditions that lie outside the basin of attraction cannot be on an
equilibrium path. Similarly, for a given set of initial conditions, government policy
may be such that the economy will not converge to a steady state (either because
taxes are too low or benefit are too high). I will call paths with rapidly growing
government debt “explosive.” This naming convention reflects that fact that bonds
will explode to infinity, if government policy is not changed.
If the economy is on an explosive path and policy is not corrected, capital
will equal zero (or cross the zero threshold) in finite time. Generally, these
explosive paths are ruled out of scalar OLG models by arguing that capital cannot
go to zero. As capital approaches zero, the interest rate will approach infinity,
and increase the incentive to save. This will increase capital and prevent capital
from hitting zero. The argument against explosive paths that lead to zero capital
27
in planar models is more nuanced. Although the interest rate increases as capital
decreases, savings does not directly translate into capital. Capital can be crowded
out by increased government borrowing along the explosive path. Thus, capital can
approach zero in the planar model.
Suppose the economy is on an explosive path so that the path dictated by
equation (2.10) leads to KT ≤ 0 for some T . Clearly, KT < 0 is infeasible, so
suppose KT = 0. Then, YT = 0 and the government must default on its debt, i.e.,
the old at T will not receive either their promised returns on savings or promised
benefits. Hence, the young at T − 1 will instead consume all of their wage income
at T − 1. There will be no demand by the young at T − 1 for the government debt
held by the old at T − 1 and bought by the young at T − 2, etc. Hence, this path is
infeasible. Ultimately, an explosive path with zero capital is not an equilibrium.
In the set-up of this model, policy ensures a rational expectations
equilibrium for a given set of initial conditions (k0, b0) even though the paths
initially follow an unstable trajectory in the sense that capital is falling towards
zero and bonds are increasing. Later in section II, I will consider the case where
agents anticipate the government will change policy before capital goes to zero. In
this way, I’ll be able to examine the dynamics of a path that starts out explosive,
but ultimately converges to a steady state.
Parameterization
For the policy analysis that follows, the model will be parameterized to be
dynamically inefficient and have two positive bond steady states. The baseline
parameters are summarized in Table 1. Agents are assumed to enter the model at
28
Parameter Baseline Note
α Capital share of output 0.22 Ensures two steady states
β HH discount rate 0.86 Annual discount factor of 0.995
δ depreciation rate 1 Allows analytical results
n population growth rate 0.37 Annual population growth rate
of 1.06% (average in US since
1960)
g growth rate of technology 0.1 Annual wage growth of 0.3%
τ social security payroll tax 0.106 Corresponds to the OASI
portion of social security
φ benefit replacement rate 0.147 Ensures two steady states
TABLE 1. Baseline parameterization for two-period model
age 25; each period lasts 30 years such that retirement occurs at age 55, and agents
live to be 85.
Policy Experiments
Before considering the impacts of social security uncertainty on the long-run
stability of the economy, it is useful to summarize the effects of announced, perfect-
foresight changes to the pension program. Following the presentation of announced
policy changes, I will consider uncertain tax changes and uncertain benefit changes.
Announced Policy Changes
If economy is on an unstable path with explosive government debt, either
because the initial stock of debt was large relative to the initial capital stock or
because the government is running large deficits, the government might be able to
change policy in a way that will lead to steady state convergence. The reverse is
29
also true; the government can derail the evolution of the economy by changing the
social security system to include large deficits.
There is a natural trade-off between the size of government deficits and how
long reform can be delayed. If the government runs a large deficit (or begins with
a large initial stock of debt), they will need to make policy changes sooner than a
government that runs a small deficit (or begins with a small stock of debt).
If the economy is on an explosive path and policy is not corrected, capital
will equal zero (or cross the zero threshold) in finite time. When capital crosses
zero (i.e., becomes negative) between periods T and T + 1, then period T is the
last possible date of reform, since it is the last date with positive capital. Finding
the last possible date for reform is a simple numerical exercise. As an illustration, I
provide six examples in table 2.9
An alternative way to view the trade-off between deficits and reforms needed
to restore solvency, is to consider reform at a fixed date T , and then numerically
compute the smallest benefit cut that could be used to ensure convergence to a
steady state. This simply requires searching for the largest possible benefit rate
(that is smaller than the initial benefit rate) such that the model converges. If the
initial deficit (or stock of debt) is large, than a larger benefit cut is needed for a
given T to ensure solvency. Table 3 illustrates the minimum benefit cut needed for
four explosive paths.
9This exercise is similar to R. Evans et al. (2012). They construct a stochastic OLG model
with a fixed transfer from the young to the old and run simulations to see how many periods
it takes for the economy to shut down (i.e., zero capital). As expected, they find that larger
government transfers result in more rapid economic shutdown.
30
baseline Explosive paths Explosive paths
(high benefits) (high initial debt)
b0 0.01 0.01 0.01 0.01 0.03 0.04 0.05
τ 0.106 0.106 0.106 0.106 0.106 0.106 0.106
φ 0.147 0.15 0.17 0.2 0.147 0.147 0.147
last reform date 1139 7 4 8 5 3
TABLE 2. The latest possible reform date for various explosive paths. The initial
values of capital and bonds, and policy parameters are listed in the table. The
other parameters are held constant at k0 = 0.1 (initial capital value), τ = 0.106,
α = 0.22, β = 0.86, n = 0.37 and g = 0.1. See table 1 for calibration details. The
model has two steady states with initial conditions of (0.1, 0.01) when τ = 0.106
and replacement rate of φ = 0.147. The stable steady state associated with these
baseline parameters is (0.1054, 0.0034).
baseline Explosive paths Explosive paths
(high benefits) (high initial debt)
b0 0.01 0.01 0.01 0.03 0.04
φ 0.147 0.15 0.17 0.147 0.147
given reform date 5 5 5 5
Min new benefit φ2 0.149996 0.11101 0.12508 <0
TABLE 3. The smallest possible benefit increase for various explosive paths.
The initial values of capital and bonds, and policy parameters are listed in the
table. The other parameters are held constant at k0 = 0.1 (initial capital value),
τ = 0.106, α = 0.22, β = 0.86, n = 0.37 and g = 0.1. See table 1 for calibration
details. The model has two steady states with initial conditions of (0.1, 0.01) when
τ = 0.106 and replacement rate of φ = 0.147. The stable steady state associated
with these baseline parameters is (0.1054, 0.0034).
31
Tax Rate Uncertainty
Suppose that taxes are stochastic and time varying, but that the benefit
rate φ is constant. Changing the social security tax rate changes the endogenous
deficits of the model and alters both debt and capital accumulation, as discussed
above. Somewhat surprisingly, allowing taxes to change over time does not alter
household behavior. In other words, the household’s decision does not depend on
the expectation of future tax rates. Stochastic changes in the tax rate affect the
model exactly the same as the announced changes discussed in section II.
Young agents in period t make their saving consumption decision based on
the expected rate of return Rt+1 and their expected social security benefit zt+1
according to equation (2.1). The expectation drops out of the right hand side of
the equation since second period consumption is predetermined (because the rate of
return and the social security benefit are both predetermined). The savings decision
of the young facing tax rate uncertainty, is the same as the savings decision of the
young in the non-stochastic model. Similarly, the retired agents make the same
decision under tax rate uncertainty as they do under certainty. This is because
retired agents do not make any decisions; they simply consume their savings
and social security benefit. Together, the household decisions in period t are not
influenced by the tax rate in period t + 1. Any objective or subjective uncertainty
about future tax rates will have the same (lack of) effect on the households. This
result is sensitive to the model specification and could be changed if agents had a
bequest motive, faced a labor-leisure choice, or if the model was extended to three
periods.
The remainder of the paper will focus on policy changes that alter the
benefit rate. Although tax changes do impact the model and can be used to correct
32
explosive paths, they are somewhat less interesting than benefit changes, since
there is no forward looking component with tax changes (in this simple set-up).
The emphasis on benefit changes could also be viewed as political intolerance for
higher taxes (although benefit cuts also seem political unpalatable).
Replacement Rate Uncertainty
Changes in the social security replacement rate, φt, change the size of
government deficits, and alter young agents savings decisions. Specifically, the
savings decision of the young is implied by the first order condition equation
reprinted here:
u′(c1,t) = βEtRt+1u′(c2,t+1) (2.20)
The agent’s expectation of second period consumption c2,t+1 (and therefore,
the marginal utility of second period consumption), depend on her expectations
about the replacement rate. The right hand side of 2.20 is the expectation of
a non-linear function of φt+1. Suppose that the realization φt+1 depends on φt,
and is contained in the set Φ. Suppose further that the probability of realizing a
particular value of φt+1 given φt is described by pi(φt+1|φt). Then, the expected
value in equation (2.20) can be written as:
u′(c1,t) = β
∑
φt+1∈Φ
pi(φt+1|φt)Rt+1u′(c2,t+1) (2.21)
I will consider a simple stochastic processes for φt+1: the government lowers
the social security benefit rate φ to a known level φnew with a fixed probability p
each period. In Appendix A I outline the process for reform that depends on the
government’s stock of debt.
33
Replacement Rate Uncertainty: Fixed Probability of Reform
Suppose the economy is on an explosive path such that benefits will need
to be cut in the future (for simplicity, we will assume the government is unable
or unwilling to change the tax rate). Suppose the current benefit rate φ is given
by φold. The government can delay reform until period T , at which point it will
cut benefits to rate φl (where the l indicates “low” or “last”). This last minute
intervention is needed to prevent capital from going to zero, the government
defaulting on its debt, and the economy unwinding in the preceding periods (as
outlined earlier in the paper). If the government acts sooner than period T , the
benefit cut need not be as large.
Suppose that there is fixed probability p that the government will cut
benefits to known rate φnew ≥ φl next period (up to period T − 1). If the
government hasn’t cut benefits by period T − 1 (that is, if φT−1 = φold), everyone
in the economy knows that benefits will be cut with certainty in period T . If the
government cuts benefits at or before period T − 1, call that period Tˆ .
For all period t < Tˆ ≤ T − 1, there is a fixed probability p that φt+1 = φnew.
Thus, for periods t < Tˆ , φt follows a two-state Markov switching process with an
absorbing state. The transition matrix Π for the Markov process is given by:
Π =
(1− p) p
0 1

Consider the following example. Suppose that the last possible date for
reform is four periods in the future, T = 4. If reform has not happened by period
t = 3, agents know that φ4 = φl. If reform happens in period t = 3 or earlier,
the new benefit level is φnew. There are four possible paths for the evolution of the
benefit rate given Table 4.
34
Time period
Probability 1 2 3 4
p φnew φnew φnew φnew
(1− p)p φold φnew φnew φnew
(1− p)2p φold φold φnew φnew
(1− p)3p φold φold φold φl
TABLE 4. Illustrative example of possible time paths of the replacement rate. In
this example, the last possible date for reform is T = 4, at which point the benefit
rate is cut to φl if reform hasn’t already happened. In the proceeding periods, there
is a probability p that benefits get cut to φnew, and a (1 − p) chance that benefits
remain unchanged at level φold. The table shows the four possible time paths. The
left column shows the probability of each path.
Given this simple process for the evolution of φt, the household’s first order
condition (2.21) for periods before reform t < Tˆ is given by:
u′(c1,t) = β [pRt+1u′(c2,t+1(φnew)) + (1− p)Rt+1u′(c2,t+1(φold))]
Plugging in the equations for c1,t and c2,t+1 and emphasizing that the social security
benefit received zt+1 is a function of the realized replacement rate:
u′((1− τ)wtAt − at) = β[pRt+1u′(Rt+1at + zt+1(φnew))
+ (1− p)Rt+1u′(Rt+1at + zt+1(φold))]
(2.22)
1
(1− τ)wtAt − at = β[
pRt+1
Rt+1at + zt+1(φnew)
+
(1− p)Rt+1
Rt+1at + zt+1(φold)
]
The optimal savings choice under uncertainty a∗t is defined implicitly by
equation (2.22). Let the first order condition be written as follows:
F = u′(c1,t)− β [pRt+1u′(c2,t+1(φnew)) + (1− p)Rt+1u′(c2,t+1(φold))] = 0
The implicit function theorem can be used to evaluate the the partial derivatives of
the savings function. The savings function is increasing in p as shown in equation
(2.23) below. The numerator and denominator are both positive, since the interest
rate and consumption in both periods are positive as long as capital is greater than
35
zero. If the probability of a benefit cut tomorrow were increased, the agents in the
model would save more to off-set the benefit cut.
∂at
∂p
=
−∂F/∂p
∂F/∂at
=
βRt+1
(
1
c2,t+1(φnew)
+ 1
c2,t+1(φold)
)
1
c21,t
+ βR2t+1
(
p
(c2,t+1(φnew))2
+ 1−p
(c2,t+1(φold))2
) > 0 (2.23)
The savings function is also increasing in the new benefit rate, φnew, as show
in equation (2.24). If the future benefit rate is large, the agents save less, since they
anticipate larger benefits. If the future benefit is low, they save more.
∂at
∂φnew
=
−∂F/∂φnew
∂F/∂at
=
βRt+1p
(c2,t+1(φnew))2
At+1wt+1
1
c21,t
+ βR2t+1
(
p
(c2,t+1(φnew))2
+ 1−p
(c2,t+1(φold))2
) > 0 (2.24)
Note, that the future benefit rate influences savings even if the probability
of reform, p is quite low. Agents change their behavior even if the threat of a
benefit cut is small. The rate of change in savings in response to changes in the
benefit rate decreases as the probability of reform increases. Similarly, the increase
in savings due to increases in the probability of reform decreases as the as size
of future benefits increases. This is evidenced by the cross-partial derivative in
equation (2.25).
∂2at
∂φnew∂p
=
−βRt+1At+1wt+1
(c2,t+1(φnew))2
βR2t+1
(
1
(c2,t+1(φnew))2
− 1
(c2,t+1(φold))2
)
(
1
c21,t
+ βR2t+1
(
p
(c2,t+1(φnew))2
+ 1−p
(c2,t+1(φold))2
))2 (2.25)
The partial derivative is negative if φnew < φold. This is because second period
consumption is an increasing function of the benefit rate. If the new benefit rate
is lower, new second period consumption is also lower. When this is true, the
numerator of equation (2.25) is negative, since it is the product of a negative term
and a positive term. The first term is negative because of the negative sign. The
36
second (last) term of the numerator is positive (if φnew < φold) as show below:
φnew < φold
=⇒ (c2,t+1(φnew))2 < (c2,t+1(φold))2
=⇒ 1
(c2,t+1(φnew))2
>
1
(c2,t+1(φold))2
=⇒ 1
(c2,t+1(φnew))2
− 1
(c2,t+1(φold))2
> 0
The denominator of (2.25) is always positive. Thus the fraction is negative if
φnew < φold. These comparative static exercises confirm the basic intuition that
agents will save more if there is a risk that they will receive smaller social security
benefits when they retire.
The optimal savings choice under uncertainty a∗t cannot be solved
analytically; however, it can be easily approximated using a numeric solver.
The numeric approximation of at can then be used with the market clearing
conditions, and price equations, to numerically approximate the transition path
for the economy. After reform takes place (either in period Tˆ or at the last possible
minute in period T ), the dynamics of the system are governed by the non-stochastic
transition equations, since the uncertainty has been resolved.
Policy Timing Uncertainty: Uniform Reform
As a beginning example, consider an economy that faces uncertainty about
the timing of a reform. Agents know that the government will cut benefits to a
given amount φnew in the future, they are just unsure about when. For simplicity,
assume also that φnew = φl. That means, that regardless when reform happens,
benefits get cut to the level φl which the smallest benefit cut that could take place
in the last period to put the economy on a stable path. Thus, the economy evolves
37
to the same steady state, regardless of when the reform takes place. If reform
takes place sooner, the economy evolves to the new steady state sooner. If reform
is delayed, it takes longer to reach the steady state. Since reform requires social
security benefits to be cut, agents would prefer the reform to be after their lifetime,
all else equal.
Figure 6 shows the evolution of capital and bonds for an economy facing
uncertain reform with constant probability p. The last possible date for reform,
for the given parameterization of the model, is T = 7. The benefit cut required
in period 7 is relatively large (the new benefit rate is lower than the tax rate,
so the government runs a small surplus each period to pay down the debt it
accumulated), so the economy evolves to a steady state with negative bonds. This
is just a result of parameterization; it’s also possible to have reforms that lead to
steady states with positive bonds (especially if the government reforms sooner).
The graphs include the time paths of each possible evolution of the benefit rate φ.
The economy reaches the same steady state regardless of when the reform takes
place. The graph also includes consumption and savings by the young, and the
consumption of the old.
The final graph included in Figure 6, depicts the lifetime utility of each
generation. Recall lifetime utility for an agent born in period t is defined as
Log(c1,t) + βLog(c2,t+1). Agents who retire in the first period after reform (the
first period that benefits are cut), have the lowest lifetime utility, compared to
those born before or after the reform. These agents fare the worst, since they were
hit with a benefit cut, and so their second period consumption is lower than they
would have preferred. Agents who retire after the reform have higher lifetime utility
that those in the moment of reform since the aggregate capital stock is higher, so
38
wages and social security benefits are higher. The agents alive after the reform live
in a world with low benefits, but higher capital; essentially a good economy with
low benefits. In contrast, in the moment of reform, agents experience low benefits
and low capital stock. Finally, if there are multiple periods before reform, each
generation alive prior to the reform has lower utility than the previous generation,
because the capital stock is falling. Agents have the highest utility in the first
generation; this is because the capital stock is (relatively) high, and they escape
the reform.
The government can achieve similar results implementing tax changes. As
discussed in section II, tax changes do not alter behavior in advance; tax changes
only alter contemporaneous choices by agents. Figure 7 depicts the time paths
for the economy that faces a fixed probability p of tax reform in any period. The
tax increase is the same, regardless of when it takes place. The last period for
tax reform (given the initial conditions), is in period 7. Thus, if reform has not
occurred by period 6, it occurs with certainty in period 7. The tax increase is very
large, and leads to a steady state with negative bonds. Note, that the timing of
tax reform is different than benefit reform. A tax reform that is announced in
period t, changes the tax rate in period t and impacts the decisions during period
t. A benefit reform that is announced in period t, takes place in period t + 1, and
impacts decisions during t.
Policy Uncertainty: History Dependent Reform
In the example above, the government made the same, relatively large,
benefit cut regardless of when reform occurred. This example fails to capture the
inherit trade-off between the minimum size of reform needed to ensure stability
39
0 5 10 15 20 25
0.00
0.05
0.10
0.15
time
C
ap
ita
l
0 5 10 15 20 25
-0.04
-0.02
0.00
0.02
0.04
time
B
on
ds
0 5 10 15 20 25
0.170
0.175
0.180
0.185
0.190
0.195
time
sa
vi
ng
s
0 5 10 15 20 25
0.23
0.24
0.25
0.26
0.27
0.28
time
Y
ou
ng
C
on
su
m
pt
io
n
0 5 10 15 20 25
0.00
0.02
0.04
0.06
0.08
0.10
time
O
ld
C
on
su
m
pt
io
n
0 5 10 15 20 25
-4.4
-4.2
-4.0
-3.8
-3.6
generation
Li
fe
tim
e
U
til
ity
T=1 T=2 T=3 T=4 T=5 T=6 T=7
FIGURE 6. Possible time paths for economy given fixed probability p of benefit
reform in each period. The reform is announced in period T . The top left shows
the time paths for capital, top right is bonds, middle left is savings, middle right is
young consumption, and bottom left is old consumption. The final plot (the lower
right), graphs the lifetime utility of the various generations before, after, and during
reform. For this example, the initial values of capital and bonds are (0.1, 0.01),
tax rate τ = 0.106, the initial benefit replacement rate φold = 0.17, and the new
replacement rate is φnew = φl = 0.068. The probability of reform next period is
p = 0.9. The steady state values of capital and bonds are (0.17,−0.04). The other
parameters in the model (β, n, g) have the same values as in all simulations in the
paper.
40
0 2 4 6 8 10 12 14
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
time
C
ap
ita
l
0 2 4 6 8 10 12 14
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
time
B
on
ds
0 2 4 6 8 10 12
-0.10
-0.05
0.00
0.05
0.10
0.15
time
S
av
in
gs
0 2 4 6 8 10 12
0.05
0.10
0.15
0.20
0.25
time
Y
ou
ng
C
on
su
m
pt
io
n
0 2 4 6 8 10 12
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
time
O
ld
C
on
su
m
pt
io
n
0 2 4 6 8 10 12
-5.0
-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
time
Li
fe
tim
e
U
til
ity
T=1 T=2 T=3 T=4 T=5 T=6 T=7
FIGURE 7. Possible time paths for economy given fixed probability p of tax reform
in each period. The reform takes place in period T . The top left shows the time
paths for capital, top right is bonds, middle left is savings, middle right is young
consumption, and bottom left is old consumption. The final plot (the lower right),
graphs the lifetime utility of the various generations before, after, and during
reform. For this example, the initial values of capital and bonds are (0.1, 0.01),
th benefit replacement rate φ = 0.17. The initial tax rate is 0.106, and the new tax
rate is τnew = 0.90. The probability of reform next period is p = 0.9. The steady
state values of capital and bonds are (0.39,−0.46). The other parameters in the
model (β, n, g) have the same values as in all simulations in the paper.
41
and the length of time before reform (discussed in section II). The model above can
be modified slightly to allow for a more sophisticated policy process. Suppose, as
before, that the economy is on an unstable trajectory and the last possible period
for reform is period T . If the government waits to reform pensions until period
T − 1, they will cut the benefit rate to φl, effective in period T . However, if the
government reforms before period T − 1, they will make a smaller benefit cut. If
the government acts, they will make the smallest possible benefit cut that ensures
long-run stability, φnew,t. As in the model above, in period t, there is a probability
p chance that the government changes the replacement rate in t + 1 (for all periods
before T − 1). The forward-looking rational expectation agents can accurately
predict what the benefit cut will be for any given reform date. The young make
their saving-consumption decision based on the first order condition:
1
(1− τ)wtAt − at = β[
pRt+1
Rt+1at + zt+1(φnew,t)
+
(1− p)Rt+1
Rt+1at + zt+1(φold)
] (2.26)
Equation (2.26) is identical to equation (2.22), except that the new benefit
rate φnew,t is time indexed. The evolution of the economy under this type of policy
reform can be solved recursively, moving forward. If reform happens in period t =
1, then benefits are cut to φnew,1 in the next period. The uncertainty is resolved,
and the path of the economy is known. If reform does not happen, the state vector
(k1, b1) can be used as an initial condition to solve for the minimum needed reform
in period t = 2. This process can be repeated until the economy reaches period
T − 1.
Figure 8 and Table 5 show the possible time paths for an economy that
starts with the same initial conditions as the economy in section II, but that faces
time-dependent minimum reform φnew,t with probability p. Table 5 shows the
critical value of φ that the government must enact in the following period to ensure
42
convergence to a steady state. The longer the government waits to reform, the
larger benefit cut they must make (φnew,t is smaller). The table also lists the steady
state values of capital and bonds associated with each possible reform. Reforms
that take place soon result in higher capital and lower accumulation of bonds.
As reform is delayed, the capital stock is depleted and bonds are accumulated,
resulting in lower capital, higher debt steady states. However, if reform is delayed
long enough, the government must make such a large benefit cut that it starts
running endogenous surpluses. This dramatic reduction in social security benefits
drives up the savings rate and leads to a steady state with higher capital. The
endogenous surpluses lead to a steady state with negative bonds. This happens
if reform is delayed until period 6 or 7 in this numeric example.10 Note that the
government must make large benefit cuts (that lead to surpluses) if reform is
delayed until period 6 or 7 partially as a result of the types of policies I consider.
I have only allowed the government to make a one-time, permanent change to a
single parameter. In reality, the government might pursue a more nuanced policy in
which they cut benefits dramatically for a short period of time to pay down some
outstanding government debt, and then gradually increase benefits again when the
debt is at a sustainable level.
The model with fixed probability of a time-dependent benefit cut (section
II) provides a sharp contrast to the model with a fixed probability of the same
benefit cut (section II). The two numeric examples presented have the same initial
conditions, the same probability of reform next period, and the same constant
policy parameters. The only difference is the uniform benefit cut enacts the same
10Reform in period 5 requires a benefit cut that creates small surpluses each period. The
surpluses are small enough that they never fully offset the debt accumulated in the first four
periods. Thus, the steady state still has positive bonds. Reform in period 6 or 7 requires a large
enough benefit cut that the surpluses drive bonds to a negative steady state.
43
0 5 10 15 20
0.00
0.05
0.10
0.15
0.20
0.25
time
C
ap
ita
l
0 5 10 15 20
-0.10
-0.05
0.00
0.05
0.10
time
B
on
ds
0 5 10 15 20
0.16
0.18
0.20
0.22
0.24
time
S
av
in
gs
0 5 10 15 20
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
time
Y
ou
ng
C
on
su
m
pt
io
n
0 5 10 15 20
0.1
0.2
0.3
0.4
0.5
time
O
ld
A
ge
C
on
su
m
pt
io
n
0 5 10 15 20
-3.0
-2.8
-2.6
-2.4
-2.2
cohort
Li
fe
tim
e
U
til
ity
T=1 T=2 T=3 T=4 T=5 T=6 T=7
FIGURE 8. Time paths for the economy facing a fixed probability p of history
dependent benefit cut φnew,t. The benefit cut required in each period is listed in
Table 5. As reform is delayed, larger benefit cuts are required to ensure convergence
to a a steady state. Reform that takes place in the first five periods results in
qualitatively similar outcomes: capital falls and bonds increase until the reform is
enacted. Reform in period 6 or 7 are qualitatively different since the government
is required to cut benefits enough that they run a (large) surplus each period.
This results in steady state negative bonds, and very high savings which drives up
the capital stock. In this parameterization p is 0.9. The initial values for capital
and bonds are (0.1, 0.01) for this example. Initial benefits,φold, are set to to 0.17.
Taxes,τ , are set at 0.106. All other parameters are equal to their baseline value
(α = 0.22, β = 0.86, n = 0.37, g = 0.1).
44
Reform time 1 2 3 4 5 6 7
φnew,t 0.150 0.148 0.144 0.137 0.124 0.086 -0.022
k steady state 0.093 0.089 0.084 0.079 0.073 0.162 0.247
b steady state 0.014 0.019 0.023 0.029 0.035 -0.038 -0.082
TABLE 5. This table lists the rational expectations next period benefit level for
each possible period of reform φnew,t. The table also includes the steady state
associated with each reform. The benefit cut required for stability increases as
the time to reform gets longer. Likewise, the steady state capital after reform gets
smaller if reform is delayed (at first). If reform is delayed long enough, the benefit
cut will be so large that the government will run a surplus each period, which
will increase the steady state capital and result in negative steady state bonds.
This is the case if reform doesn’t happen until period 6 or 7. Note, that reform in
period 6 or 7 is unlikely in this parameterization, since p=0.9. The initial values
for capital and bonds are (0.1, 0.01) for this example. Initial benefits are set to
φold=0.17. Taxes are τ=0.106. All other parameters are equal to their baseline
value (α = 0.22, β = 0.86, n = 0.37, g = 0.1).
policy change regardless of when the change happens, while the time-dependent
model makes the smallest benefit cut possible each period to ensure long-run
stability. An obvious difference between the two models are the steady states
after each reform. The uniform model results in the same steady state, regardless
of when reform happens. In contrast, the time-dependent model converges to a
different steady state for each possible reform date.
An additional differences between the models is the evolution of the optimal
savings choice. The uniform model results in essentially the same savings decision
being made each period, while the history-dependent model has much different
savings choices if reform is delayed. In the uniform model, an agent in period
t = 1 awaiting reform, makes her savings choice on her expected social security
benefit which depends on next period’s wage, w2, as well as the benefit rate,
φ2. The savings rate is either equal to the old rate φold or the new rate φnew. If
reform doesn’t take place, an agent in period t = 2 makes almost exactly the
same choice. The young agent in period 2 also saves based on expected benefits,
45
which depend on the next period’s wage, w3, and the benefit rate, φ3. Again, the
benefit will either remain unchanged at φold or will fall to φnew. The only difference
between the savings choice of a young agent awaiting reform in period 1 and a
young agent awaiting reform in period 2, is that the next-period wage will be
different. The expected benefit rate is the same. This results in a savings choice
that looks qualitatively similar regardless of when reform takes place. If reform
doesn’t happen until period 7, first period consumption and savings falls for each
generation until the reform takes place.
This contrasts the optimal savings choice in the time-dependent model,
which changes each period that reform is delayed. An agent in period 1 awaiting
reform makes a savings decision based on expected benefits which depend on the
wage, w2, and the benefit rate. The benefit rate is either the old rate, φold, or
the new rate associated with this reform, φnew,1. If reform doesn’t happen, the
young agent in period 2, makes a savings choice based on expected prices, w3, and
expected benefit rate which either stays the same at φold, or is cut to the lower rate
φnew,2. Thus, the agent awaiting reform in period 2, saves more than the agent
awaiting reform in period 1, since the potential benefit cut is larger. If reform
doesn’t happen until the 7th period, savings increases each time period (with each
generation of young), while the youth consumption falls.
The final difference I want to highlight, is the lifetime utility of each
generation in both models. Agents in the uniform reform model have qualitatively
similar changes in utility. Agents that live during the reform (who experience the
benefit cut) have a dramatic reduction in lifetime utility compared to their parents.
The generations that come after the reform see their utility increase compared to
the reform generation. In contrast, in the model with time-dependent reform, the
46
lifetime utility of agents can be very different depending on when reform occurs. If
reform occurs within the first five period, agents alive before the reform have higher
utility that those alive after. Agents alive during the change have lower lifetime
utility than their parents, but they have higher utility than their children. However,
if reform is delayed until period 6 or 7, agents alive during the policy change have
much lower utility. Their utility is much lower than their parents, and is also lower
than their children’s. Lifetime utility by generation does not monotonically decline
if reform takes place in period 6 or 7.
It is also possible for the government to enact history dependent tax reform.
Future tax rates do not effect the agents behavior, so history dependent tax reform
is computationally equivalent to perfect foresight comparative dynamics. Suppose
the government increases taxes this period with probability p; if taxes are changed,
they are increased to the minimum amount to ensure convergence to a steady state.
If taxes are not reformed by period t = 7, taxes are changed with certainty. The
time paths for a numeric example are presented in Figure 9. The example is similar
to the history-dependent benefit cut example above. The initial conditions are the
same, and the government takes actions with the same probability p = 0.9. The
only difference is in this example, taxes are increased rather than benefits being
cut. As in the benefit case, the history-dependent tax changes converge to different
steady states for each possible reform. Delaying reform until the last period, results
in very large tax hike that actually leads to young agents choosing to borrow in the
first period. They are able to borrow against their future social security benefits,
which are known with certainty. If reform happens earlier, the new tax rate is low
enough that agents still choose to save a positive amount in steady state.
47
0 2 4 6 8 10 12 14
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
time
C
ap
ita
l
0 2 4 6 8 10 12 14
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
time
B
on
ds
0 2 4 6 8 10 12
-0.10
-0.05
0.00
0.05
0.10
0.15
time
S
av
in
gs
0 2 4 6 8 10 12
0.05
0.10
0.15
0.20
0.25
time
Y
ou
ng
C
on
su
m
pt
io
n
0 2 4 6 8 10 12
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
time
O
ld
C
on
su
m
pt
io
n
0 2 4 6 8 10 12
-5.0
-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
time
Li
fe
tim
e
U
til
ity
T=1 T=2 T=3 T=4 T=5 T=6 T=7
FIGURE 9. Time paths for the economy facing a fixed probability p of history
dependent tax increase τnew,t. The tax increase required in each period is listed
in Table 6. As reform is delayed, larger tax increases are required to ensure
convergence to a a steady state. Reform that takes place in the first six periods
results in qualitatively similar outcomes: capital falls and bonds increase until
the reform is enacted. Reform in period 7 is different since the tax increase is
so large. In this parameterization p is 0.9. The initial values for capital and
bonds are (0.1, 0.01) for this example. benefits, φ, are set to 0.17. Taxes, τ ,
are 0.106 before reform. All other parameters are equal to their baseline value
(α = 0.22, β = 0.86, n = 0.37, g = 0.1).
48
Reform date 1 2 3 4 5 6 7
τnew 0.123 0.125 0.13 0.14 0.164 0.242 0.901
k stead state 0.099 0.105 0.112 0.122 0.139 0.177 0.395
b steady state 0.004 -0.002 -0.01 -0.021 -0.042 -0.095 -0.469
TABLE 6. This table lists the rational expectations tax increases needed if reform
takes place in a given period. Reform takes place during the current period with
probability p=0.9. If reform does not take place, there is a p=0.9 probability it
takes place next period. If reform has not happened by period 7, it occurs with
certainty. If reform is delayed, the benefit cut must be larger. The required tax cuts
are large enough that the government runs an endogenous surplus, which leads to
steady states with negative bonds. The reforms in this table correspond to Figure
9.
Welfare Analysis
The policy analysis above reveals a trade-off between the size of policy
reform (i.e., the magnitude of benefit cuts or tax increases) and the timing of the
reform (i.e., how long reform can be postponed). Current agents prefer larger policy
changes in the future, and future agents would prefer smaller changes that happen
sooner. It is not clear a priori which type of reform would be welfare maximizing
for society.
Consider a benevolent social planner than must choose a new social
security benefit rate φnew and a reform date T to ensure long-run stability for an
economy. For simplicity, assume that the tax rate τ is given and the planner cannot
change it.11 The parameters φnew and T are chosen jointly to ensure the economy
converges to a steady state. The planner’s problem can be maximizing a social
welfare function:
11This assumption could be relaxed to allow the Planner to change either the tax rate, the
benefit rate, or both.
49
W =
T−1∑
t=0
ρt(1 + n)t [(1 + n)c1,t(φ) + c2,t(φ)]
+
T∑
t=T
ρt(1 + n)t [(1 + n)c1,t(φnew) + c2,t(φnew)]
(2.27)
where T > T represents the end of the planning horizon. The discount factor ρ
represents both a social discount factor, to weight future generations less, and
a discount factor to offset wage growth due to technological innovation. Thus,
ρ ≤ 1
1+g
. There is some variation in the OLG literature on what should be used
as the social discount factor. Feldstein (1985) considers a social discount factor of
1 (which he calls additive welfare) and also ρ = 1/(1 + η) for η ≤ n. When the
discount factor is set equal to 1/(1 + n), all generations are counted equally, but
individuals in larger generations are weighted less (the equivalent discount factor in
a model with growth is ρ = 1/((1 + n)(1 + g)) ) . Sanchez-Marcos and Sanchez-
Martin (2006) use a social discount factor of ρ = 1
R
. Olovsson (2010) and Bohn
(2009) both use discount factors equivalent to ρ = u′(c2,t)/u′(c1,t). Bohn (2009)
explains that welfare weights of ρ = 1/E0 [u
′(c1,t)/u′(c2,t)] are the only weights that
maximize welfare.
The social welfare function W captures the trade-off between delaying
reform and the size of reforms. For example, consider an economy on an explosive
path such that last possible date of reform is in t = 6 = T , as discussed in section
II. The social planner could wait until period t = 6 to make policy changes, or
the planner could make smaller changes sooner. The welfare associated with the
minimum reform at time period t = 1, 2, · · · , 6, is presented in Table 7. If the
social planner has a long planning horizon, reforming as soon as possible maximizes
welfare. This is unsurprising, since reforming sooner allows the planner to make
50
Social Welfare of various reforms
reform date 1 2 3 4 5 6
φnew 0.1497 0.1477 0.1430 0.1333 0.1110 0.0432
T 100 2538 2537 2508 24812 24275 20771
90 11372 11389 11261 11128 10880 9309
80 5093 5100 5068 4995 4875 4172
70 2278 2283 2283 2246 2184 1869
60 1016 1019 1020 1016 977.6 837.0
50 451.1 452.2 453.1 453.7 436.8 374.3
40 197.6 198.1 198.5 198.8 195.2 166.8
30 84.02 84.24 84.43 84.58 84.65 74.32
20 33.08 33.18 33.27 33.35 33.42 33.44
10 10.25 10.29 10.34 10.38 10.45 10.55
TABLE 7. Welfare for various policy reforms, given different planning horizons.
In each case, the economy is on an explosive path. The benefit cut considered for
each period is the smallest benefit cut that converges to a steady state. Reforming
earlier allows the government to make a smaller cut. The bolded values indicate
the welfare maximizing reform. For long planning horizons (110 periods) making
the small reform in the first period maximizes the welfare function. For a short
planning horizon (10 periods), waiting until the last possible minute to reform is
the best choice. For planning horizons in between, choosing a policy change date in
the middle is the best. The initial benefit rate φ = 0.17. The tax rate τ = 0.106.
The initial values for capital and bonds are (0.1, .0.01). The discount factor is
ρ=β/(1+g), although the qualitative results are invariant to a variety of discount
factors. The other parameters of the model are set to their baseline values.
a smaller benefit cut, which helps future generations, relative to a delayed reform
that would require a larger benefit cut. Conversely, if the social planner has a
short planning horizon, then delaying reform can maximize welfare, since the
older generation(s) who receive large benefits before the reform account for a large
fraction of the total welfare. The table uses the discount factor ρ = β/(1 + g). The
results are qualitatively the same if the discount factor is changed to ρ = 1/(1 + g)
or ρ = β˜/(1 + g) for β˜ near β (the numeric values change, but welfare is still
maximized by the same reform as in the table). Note, that the welfare analysis is
different if a generational discount factor is used. This will be discussed below.
51
If the social discount factor is small enough (ρ < 1/(1 + n)(1 + g)),
the planner may use an infinite planning horizon, since the welfare function will
converge12. After c1,t and c2,t reach their steady state values, the welfare function
collapses into a simple geometric series. Suppose the consumption terms reach
their steady states in t = Tss, and that the steady state value are c1ss and c2ss.
Let ρ = ρˆ < 1/(1 + n)(1 + g). Then, the last terms of the infinite horizon welfare
function can be computed as follows:
∞∑
t=Tss
ρˆ [(1 + n)c1,ss + c2,ss]
=
∞∑
t=0
ρˆ [(1 + n)c1,ss + c2,ss]−
Tss∑
t=0
ρˆ [(1 + n)c1,ss + c2,ss]
=
(1 + n)c1,ss + c2,ss
1− ρˆ − [(1 + n)c1,ss + c2,ss]
1− ρˆTss+1
1− ρˆ
= [(1 + n)c1,ss + c2,ss]
ρˆTss+1
1− ρˆ
The first terms of the infinite horizon welfare function are calculated according
to equation (2.27). As an example, the infinite horizon welfare is presented
for six different possible reforms in Table 8. The social discount factor used is
ρ = 0.999/(1 + n)(1 + g). The results are qualitatively unchanged if a discount
factor of β/(1 +n)(1 + g) is used. Under this specification, using an infinite horizon,
welfare is maximized by reforming in the 5th period. Note, that these are the same
reforms as those considered in Table 7; the only change is the discount factor.
Table 8 also includes welfare analysis for shorter planning horizons, using
the same discount factor of ρ = 0.999/(1 + n)(1 + g). In contrast to the welfare
12Note, that a discount factor of this size is similar to the generational discount factor used by
Feldstein. The discount factor accounts for both population growth and technology growth. If
ρ = 1/(1 + n)(1 + g), each generation is treated equally; a discount factor of ρ < 1/(1 + n)(1 + g),
implies that the social planner discounts for population and technology growth and also discounts
the future.
52
Social Welfare of various reforms
reform date 1 2 3 4 5 6
φnew 0.1497 0.1477 0.1430 0.1333 0.1110 0.0432
T ∞ 6.359 6.387 6.412 6.436 6.451 6.423
100 6.359 6.387 6.412 6.436 6.451 6.422
90 6.358 6.386 6.411 6.435 6.451 6.422
80 6.356 6.384 6.409 6.434 6.449 6.420
70 6.351 6.379 6.405 6.429 6.444 6.416
60 6.339 6.367 6.393 6.417 6.433 6.404
50 6.308 6.335 6.361 6.386 6.402 6.373
40 6.224 6.252 6.277 6.302 6.323 6.291
30 6.006 6.033 6.058 6.083 6.109 6.080
20 5.435 5.461 5.484 5.508 5.534 5.567
10 3.938 3.961 3.981 4.003 4.028 4.067
TABLE 8. Welfare for various policy reforms, given different planning horizons.
In each case, the economy is on an explosive path. The benefit cut considered for
each period is the smallest benefit cut that converges to a steady state. Reforming
earlier allows the government to make a smaller cut. The bolded values indicate
the welfare maximizing reform. The initial benefit rate φ=0.17. The tax rate
τ=0.106. The initial values for capital and bonds are (0.1, .0.01). The discount
factor is ρ=0.999/(1+n)(1+g). This discount factor was chosen to be smaller than
1/(1+n)(1+g), to ensure that the infinite sum converges when considering a social
planner with an infinite planning horizon. The other parameters of the model are
set to their baseline values.
analysis in Table 7 above, delaying reform until period 5 is almost always welfare
maximizing using a discount factor of ρ = 0.999/(1 + n)(1 + g). This is because
future generations do not count more than current generations, even though they
are larger. So, the social planner effectively puts more weight on the current
generations who benefit from delaying reform.
The welfare analysis in the non-stochastic model emphasizes the trade-
off between reform size and reform timing. Depending on the planner’s discount
function and planning horizon, any reform timing can be optimal. This illustrates
the importance of the social discount factor when there are intergenerational
effects. If the planner discounts future generations heavily; then optimal policy
53
will current generation. Similarly, if the planner has a short planning horizon, she
will favor current generations. In the numeric examples considered, the planning
horizon had to be very long (over 100 periods) and the social discount factor could
not be too small in order for reforming in the first period to be welfare maximizing.
In a two-period model, each period corresponds to roughly 30-40 years; so a 100
period planning horizon is roughly 300 years. The S.S.A. makes 75-year long-run
projections, and the C.B.O. publishes a 10-year projection each year. Perhaps a
planning horizon of 20 periods or less would better represent government behavior.
When a short planning horizon is used, delaying reform is the optimal choice for a
policymaker.
Welfare Analysis with Policy Uncertainty
In the previous section, the social welfare function was used to demonstrate
the trade-off between making large reforms several periods in the future from
making smaller reforms sooner. The social welfare function can also be used to
compare the trade-off between facing a given reform at a certain date, with facing
uncertain reform.
Suppose the economy is on an explosive path, and the government will
reform social security benefits next period with probability p. If the benefit cut
takes place, it will be the minimum possible benefit cut. Agents in the model are
able to predict the specific benefit cut that would happen in each period, were
reform to occur (as in section II). Suppose further that the probability of reform,
p = 0.25, so that the expected amount of time before reform is four period
(expected duration before reform = p−1). The minimum benefit cuts needed for
periods 1-4 are presented in Table 9.
54
period 1 2 3 4
φnew,t 0.1497 0.1477 0.1434 0.1345
TABLE 9. Minimum reform needed in each period, given that there is a probability
p=0.25 chance of reform in each period. Table includes the benefit cuts for periods
1-4 only, as the expected duration of the old benefit rate is 4 periods. The path of
the economy that faces uncertainty and enacts the benefit cut φnew,4 is depicted
in Figure 10 as the blue line (circles). The steady state following this reform is
(0.077, 0.030). The initial values for capital and bonds are (0.1, 0.01), the tax rate
is τ=0.106, and the initial benefit rate is φ=0.17. All other parameters are set as in
previous examples.
We would like to compare the welfare of the agents in this economy (with
policy uncertainty) to the welfare of agents who face a similar policy change with
certainty. One natural comparison would be to compare the welfare of agents
along the expected time path with uncertainty (that is, the time path when the
reform takes place in period 4), with agents who face reform with certainty in the
same period (that is, agents who know they will face the policy change in period
4). Since the minimum possible reform in any period is history dependent, the
minimum reform for the economy with uncertainty that happens to reform in
period 4, will not the be same as the minimum possible reform for an economy
that faces reform in period 4 with certainty. This is illustrated in Figure 10. Thus,
the welfare comparison is between agents in an economy facing uncertain reform
that happens to occur in period 4 (benefits are cut to φnew,4), with the welfare
of agents in an economy facing a larger certain reform in period 4 (benefits are
cut to φnew,certain < φnew,4). I compared the the welfare values for these two
economies using different discount factors and different planning horizons, and in
each case, welfare was higher in the economy with certainty. This result is driven
by uncertainty; not by the fact the benefit cut in the certain case is slightly larger
than in the uncertain case. Recall, taxes aren’t changed in either scenario; the
55
only policy change is a reduction in benefits. Reducing benefits lowers old age
consumption and lifetime utility. (Reducing benefits also has general equilibrium
impacts by changing the trajectory of the state variables, but these effects appear
to be smaller near the policy change). I also verified numerically that welfare is
lower in the economy with certainty if benefits are cut further. That is, welfare is
lower if benefits are cut to φlower < φnew,certain in period 4, as compared to when
benefits are cut to φnew,certain in period 4.
Figure 10 depicts the time path for capital and bonds for four different
scenarios. First, the graph depicts the explosive path of the economy with no
reform (capital quickly goes to zero, and bonds increase; this is green line with
diamond markers). Second, the graph shows the time-path for the economy that
faces uncertainty, and happens to have reform in period 4 (capital and bonds
converge to a steady state; this is the blue line depicted with circle points).
Benefits are cut to φnew,4 following the fourth period. Third, the graph shows the
time path for an economy that cuts benefits to φnew,4 with certainty after 4 periods
(yellow line, small square markers). This path traces the explosive path for the
first four periods. After the reform, bonds grow slowly and capital still approaches
zero. The benefit cut φnew,4 was not enough to make the economy converge since
the agents in the first 4 periods did not face any uncertainty and did not engage in
an precautionary saving. Finally, the graph depicts the time path for the minimum
possible reform that can be enacted in period 4 that will converge (orange line,
triangle markers). This benefit cut is larger than the benefit cut under uncertainty
(that is, φnew,certain < φnew,4).
The welfare analysis of this section is only an illustrative example. It is
possible (although unlikely) that welfare could be higher for an economy facing
56
●
● ● ●
● ● ● ● ● ● ● ● ● ● ●
●●
■
■
■
■
■ ■ ■ ■ ■ ■ ■ ■
■
◆
◆ ◆ ◆ ◆
◆
◆
▲
▲ ▲ ▲
▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
0 2 4 6 8 10 12 14
0.05
0.06
0.07
0.08
0.09
0.10
time
C
ap
ita
l
●
●
●
●
● ● ● ● ● ● ● ● ● ● ●
●●
■
■
■
■
■ ■ ■ ■ ■
■ ■
■
◆ ◆
◆
◆
◆
◆
▲ ▲
▲
▲
▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
0 2 4 6 8 10 12 14
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
time
B
on
ds
FIGURE 10. Time paths for capital and bond under four different scenarios.
The green path (diamonds) shows the economy with no policy change. The blue
path (circles), shows the economy with uncertainty that happens to have reform
in period 4 (benefits are cut to φnew,4 = 0.1345). This path converges to the
steady state (0.0772, 0.302). The yellow path (squares), shows an economy with
certainty that enacts the same policy change (φnew,4) in period 4, but does not
have any uncertainty. This path is still explosive, since the benefit cut is not large
enough to ensure convergence. The orange path (triangles) depicts the path for
an economy that enacts the minimum possible reform in period 4 with certainty
(φnew,certain = 0.1332 < φnew,4 = 0.1345). This path converges to the steady state
(0.0765, 0.0307).
57
uncertain reform than an economy facing a comparable certain reform under some
parameterizations. This seems unlikely, however, given the way policy uncertainty
impacts the savings decision of the young agents. Agents who face a potential
benefit cut over-save compared to those who know their benefit in advance. This
extra precautionary saving moves the agents away from their optimal bundle of
young vs. old consumption.
Finally, the welfare analysis of this and the previous section both rely on
aggregate, intergeneration measures of well-being. I have shown that in aggregate,
some reforms are preferable to others (or that reform with certain can be preferable
to uncertain reform). This does not mean that every generation prefers the same
reform or the same level of uncertainty. Individual generations would prefer their
own taxes to be as low as possible and their social security benefits to be as high as
possible. Thus, even if benefit cuts in period x are preferable to benefit in in period
y > x, the agents alive in period x would still have higher utility if the reform
were delayed until period y. The agents in this model are only forward-looking
one period, so they don’t care if benefits are cut in two periods. A richer model is
needed to fully capture the potential trade-offs within and between generations.
Possible Extensions
A planar OLG model with endogenous government deficits and debt
provides an interesting framework to examine the existence and stability of steady
states in relation to government policy. A natural extension of this paper would be
to examine a typology of possible steady states in the model. Specially, it would
be interesting to examine the stability properties of steady states (determinant vs.
explosive) for steady states with positive bonds (b > 0), negative government bonds
58
(b < 0), dynamic inefficiency (with either positive or negative bonds), and dynamic
efficiency (with either positive or negative bonds).
In Appendix A, I introduce a Leeper tax to the model which allows social
security taxes to respond endogenously to government debt. A possible extension
of this model would be to examine the stability properties of steady states under
various Leeper taxes. It may be possible to use a Leeper tax to stabilize a zero-
bond steady state that is explosive in the absence of the Leeper tax. The Leeper
tax might also be used as an automatic adjustment to avoid explosive dynamics
that result from large government debt holdings. A non-linear Leeper tax may also
provide interesting results.
Conclusion
In this chapter, I developed an analytical OLG framework to examine
endogenous government deficits (surpluses) arising out of a social security system.
The model highlights the risk of running large deficits that may result in explosive
paths. The model can be used to evaluate different policy reforms for economies on
unstable trajectories.
The policy analysis of section II highlights the welfare impacts of delaying
reform. If the economy is on an unstable path, it is essentially paying retirees
benefits that it (the government) cannot afford. Long-term, benefits need to be
cut or taxes need to be raised. Before the reform takes place, retirees are receiving
artificially high benefits. Their consumption is higher than the counterfactual
where benefits paid are smaller. Because of this, agents who live during or after
the reform, have lower lifetime utility than those who live before reform. The
artificially high consumption of retirees before reform is financed by borrowing
59
against the consumption of future generations. Moreover, if reform is delayed, the
ultimate benefit cut (or tax increase) enacted will be larger in magnitude, as the
economy will have moved further out of the basin of attraction of the stable steady
state. Thus, the utility loss to agents after a delayed reform is even larger than the
utility loss to agents alive after a more quickly implemented reform. This is seen
most easily by the utility by generation graphs in Figures 6 and 7. Governments
wishing to avoid a large drop in generational utility have an incentive to reform
sooner.
The key takeaway from this policy analysis is the trade-off between delaying
reform (which benefits current generations) and making a smaller reform now
(which benefits future generations). Policy makers that are concerned about the
welfare of future generations have an incentive to act sooner rather than later to
minimize the negative impact on future generations. However, the reverse is also
true. Policy makers who are concerned about current generations should delay
reform since it is painful. If policy makers are responding to current generations,
perhaps by seeking election, they have strong incentives to delay reform. Yet,
reform cannot be delayed forever. Thus, even policy makers who are motivated
to maintain the benefits of current generations, need to credibly promise reform in
the future.
This model also allows me to consider the macroeconomic impacts of social
security policy uncertainty. Young agents who face a potential social security
benefit cut, engage in precautionary saving. This raises the capital stock and
allows the government to perpetuate what would have been explosive policies for a
longer period of time. However, even in the model with uncertainty, eventual policy
change is needed to resolve explosive paths. In section II, I discuss the tradeoffs
60
between living in a world with certain policy reform and uncertain reform. For the
numeric example considered, (most) agents would be better off in a world with
certainty, rather than a world with uncertainty. Agents only prefer uncertainty if
the realization of the uncertain policy process means that they don’t experience
a benefit cut (or tax increase) in their lifetime (when they would have otherwise
in the certain model). All other agents are worse off, since they weren’t able to
perfectly smooth their consumption between periods.
61
CHAPTER III
POLICY UNCERTAINTY IN A MULTI-PERIOD OLG MODEL WITH
RATIONAL EXPECTATIONS
Introduction
In this Chapter, I extend the two-period model to three periods. The three
period model is inherently forward looking, which allows for more interesting policy
analysis. With three periods, the young worker’s saving-consumption decision
depends both on the predetermined capital and bonds for the next period, and also
on her expectations over capital and bonds in the following period. This forward
looking element of the model allows multi-period uncertainty analysis in the fashion
of Davig et al. (2011).1
I describe the model and its equilibrium in Section III. In Section III, I
discuss the number and stability of steady states. The model parameters are
described in section III. In section III, I use the model to examine the effects of
changes to social security policy. I examine both perfect foresight changes and
stochastic policy changes. I consider the long-run implications of social security
policy uncertainty in section III, and Section III concludes.
Model
Household problem
Households live for three periods, choose asset allocation (savings) when
young a1, assets in middle age a2, and consumption in all three stages of life, c1,
1See also Davig and Leeper 2011b, 2011a.
62
c2, c3, to maximize utility, taking price, and government social security policy (tax
rates2 τ , and benefit z) as given. Agents receive the wage wt for labor provided in
period t. The gross real return on savings in period t is given by Rt+1. Note that
superscripts refer to life-cycle stages (i.e., age), and subscripts refer to time periods.
Thus a2t+1 is second period savings in time t+ 1.
The household’s problem can be written as follows:
max
a1t , a
2
t+1
u(c1t ) + βEtu(c
2
t+1) + β
2Etu(c
3
t+2) (3.1)
subject to
c1t + a
1
t ≤ (1− τt)wtl1t (3.2)
c2t+1 + a
2
t+1 ≤ Rt+1a1t + (1− τt+1)wt+1l2t+1 (3.3)
c3t+2 ≤ Rt+2a2t+1 + z3t+2 (3.4)
Where Et(x) indicates the time t expectation of x. I will normalize the (exogenous)
age-labor endowments to unity for all generations such that l1t = l
2
t = 1 for all t. I
will impose this normalization in all of the equations that follow.
Labor is supplied inelastically and preferences are given as the standard
CRRA function:
u(c) =
c1−σ
1− σ if σ 6= 1
u(c) = ln(c) if σ = 1
(3.5)
The inverse IES for this utility function is simply σ.
The household first order conditions are:
u′(c1t ) = βEt[Rt+1u
′(c2t+1)] (3.6)
((1− τt)wt − a1t )−σ = βEt
[
Rt+1(Rt+1a
1
t + (1− τt+1)wt+1 − a2t+1)−σ
]
2The model can accommodate time-varying policy parameters where τt 6= τt−1
63
u′(c2t+1) = βEt+1[Rt+2u
′(c3t+2)] (3.7)
(Rt+1a
1
t + (1− τt+1)wt+1 − a2t+1)−σ = βEt+1
[
Rt+2(Rt+2a
2
t+1 + z
3
t+2)
−σ]
Equations (3.6) and (3.7) show the intertemporal trade off between
consumption today and consumption tomorrow. The agent (trivially) chooses to
consume all of her resources in the final period of life, that is c3t+2 = Rt+2a
2
t+1 + z
3
t+2
and a3t+2 = 0.
Note that equations (3.6) and (3.7) have been printed above for an
individual who is young in time t and middle aged in time t + 1. Equation (3.7)
can be iterated back one period to show the decision of a middle aged agent in time
t.
To start off the economy, I assume that in period zero, there are three
households (young, middle, and old) who enter the economy with given asset
holdings according to their age. I assume that the initial young enter the
economy with zero assets, and the initial middle and old enter with a1−1 and a
2
−1,
respectively. Successive cohorts enter the model with zero assets when they are
young. Nt indicates generation t and is given by number of young at time t. the
population (and hence the labor force) grows at rate n such that
Nt = (1 + n)Nt−1 (3.8)
Production
The consumption good in the economy (Yt) is produced by single firm (or
equivalently many small firms) using a constant elasticity of substitution technology
that takes aggregate capital (Kt) and labor hours (Ht) as inputs and produces the
64
consumption good according to:
Yt = F (Kt, Ht) =
(
αK
1− 1
ε
t + (1− α)H1−
1
ε
t
) 1
1− 1ε
The parameter α measures the intensity of use of capital in production.
The parameter ε measures the elasticity of substitution in production. When this
parameter is set to one, this production function collapses to the familiar Cobb-
Douglass form: Yt = K
α
t H
1−α
t . For the analytical expressions in the remainder
of this paper, I will assume the Cobb-Douglass functional form. The numerical
exercises are qualitatively similar when a different value of ε is selected.3
Factor markets are competitive and capital and labor (hours worked) are
paid their marginal products. The gross real interest rate Rt is given by:
Rt = FK(Kt, Ht) + 1− δ (3.9)
where δ is the rate of depreciation. The wage rate wt is given by
wt = FH(Kt, Ht) (3.10)
Government
The government runs a modified pay-as-you-go social security system. The
government pays retirement benefits to the old generation by taxing the working
generations and by (possibly) issuing debt.
The pay-roll tax rate is τt. The tax has two components, a baseline tax rate
τ 0t , and a Leeper tax τ
1
t that (potentially) responds to the level of government debt.
Although actual social security taxes do not change based on government debt, a
Leeper tax can be thought of as a way to capture legislative unease with increasing
3We abstract from technological growth, though it is straightforward to incorporate this into
the model. If technological process was included in the model, aggregate variables would grow
that the rate (1 + n)(1 + g) along the balanced growth path (if g indicates the growth rate of
technology). In all other regards, the model is identical.
65
debt. A Leeper tax is also a useful modeling tool, since it can be used to calibrate
the model such that a dynamically efficient steady state with positive bonds is
possible. I consider a Leeper tax of the following form:
τt = τ
0
t + τ
1
t (Bt/Ht). (3.11)
Here τ 0 ∈ [0, 1] is the baseline tax rate when government debt is zero, τ 1 ≥ 0 is the
incremental tax, Bt/Ht is government debt per labor hours. When τ
1 = 0 the taxes
are exogenous and do not respond to government borrowings. When τ 1 > 0, taxes
increase when government debt increases. This slows the accumulation of debt and
increases the basin of attraction of the stable steady state.4 Notationally, a tax rate
without a superscript (τt) will refer to the entire pay-roll tax τt = τ
0
t + τ
1
t (Bt/Ht).
5
Social security benefits zt are paid according to a benefit earning rule:
zst = φtwt. (3.12)
where zst represents the individual benefit paid to an agent of age s in time t. The
parameter φt is the replacement rate and shows how much of a workers’ wage-
indexed average period earnings are replaced by social security benefits. The
current wage (wt) represents the average wage-indexed income of a retired person in
4Leeper taxes are generally modeled as responding to the previous period’s debt (see Davig
et al. (2011) as an example). However, in this model, government debt (Bt) and capital (Kt) are
both predetermined. Thus, a tax rate in time t that responds to bond in time t is reasonable,
since agents with perfect foresight know Bt at the beginning of time t.
5The Leeper tax given by equation (3.11) leaves open the possibility of a total tax rate greater
than one hundred percent if bonds levels are high or if τ1 is large (i.e., τ0t + τ
1
t (Bt/Ht) > 1,
for large Bt/Ht and/or large τ
1). I avoid this by imposing the restriction τ1 is either the
exogenous parameter value chosen by the economist, or the (smaller) parameter value such
that τ0t + τ
1
t (Bt/Ht) = 1. In practice, the calibration of the model results in numeric values
for capital and bonds that are both very small. In the calibrated model, bonds are generally in
the range [-0.03, 0.01], and thus it is never necessary to impose the restriction. For example, if
τ0 = 0.1, τ1 = 0.8 and bonds are 0.01, then the total payroll tax burden for the consumer is
0.1 + 0.8 × 0.01 = 0.108. In all of the policy experiments that follow, the majority of the tax
burden will come from the baseline tax rate τ0, and the Leeper tax τ1 will be used only to create
the possibility of a dynamic efficiency with positive bonds.
66
period t. It shows a retired person’s average period income in current wages. The
model can accommodate time-varying replacement rates φt 6= φt+1.
The benefit earning rule (3.12), is similar to the actual benefit earning
rule used by the Social Security Administration. There is no within-generation
heterogeneity in this model, so there is no need for the benefit earning rule to
be a piecewise-linear function of earnings (as is the case in the United States).
All individuals within the same generation work the same number of hours
(inelastically) for the same wage, thus their average earnings and social security
benefits are the same.
The government revenue generated from the social security payroll tax may
be greater or less than the government transfer to retirees in any period. If social
security taxes are less than social security benefits, the social security program runs
a deficit. The time t deficit is defined as the time t social security benefits less the
pay-roll tax revenue:
Deficitt = Nt−2φtwt − (Nt−1 +Nt)(τ 0t + τ 1t (Bt/Ht))wt
The number of tax payers is Nt + Nt−1 and represents the young and middle aged.
The number of old is Nt−2 and they receive benefits according to (3.12).
The government issuance of bonds is equal to the gross interest on
outstanding debt plus the social security deficit from the previous period.
Bt+1 = RtBt +Nt−2φtwt − (Nt−1 +Nt)(τ 0t + τ 1t (Bt/Ht))wt (3.13)
67
Market Clearing
There are four markets: labor, capital, bonds, and goods. Prices adjust in
equilibrium to ensure all markets clear.
Labor market clearing requires the total number of hours worked by the
young and middle aged to equal the labor input of the representative firm:
Ht = Nt +Nt−1 (3.14)
Asset market clearing requires that aggregate capital and bonds next period
are equal to total savings of the young and middle aged today:
At+1 = Kt+1 +Bt+1 = a1,tNt + a2,tNt−1 (3.15)
Goods market clearing requires that aggregate consumption and next period
capital are equal to aggregate production and undepreciated capital:
Ntc
1
t +Nt−1c
2
t +Nt−2c
3
t +Kt+1 = K
α
t H
1−α
t + (1− δ)Kt (3.16)
The goods market clears by Walras law, and so it is not necessary to track
this equation as part of the equilibrium.
Bond market clearing is ensured by the government’s flow budget constraint,
reprinted below:
Bt+1 = RtBt +Nt−2φtwt − (Nt−1 +Nt)(τ 0t + τ 1t (Bt/Ht))wt (3.13)
Market Clearing Equations in Per-Labor-Hour Terms
Along the balanced growth path, cohort size (Nt), labor hours worked (Ht),
output (Yt), capital (Kt), and bonds (Bt) all grow at rate n. Therefore, it will be
convenient to rewrite the market clearing equations (3.13), (3.15), and (3.16) in
68
per-hours terms by defining bt = Bt/Ht, and kt = Kt/Ht. I will refer to variables in
per-hours terms as “efficient.”
The government equation can be written in efficient terms by dividing both
sides by Ht and multiplying the left hand side by the fraction Ht+1/Ht+1.
6
Bt+1
Ht
Ht+1
Ht+1
=
RtBt
Ht
+
Nt−2φtwt
Ht
− (Nt−1 +Nt)(τ
0
t + τ
1
t (Bt/Ht − b∗))wt
Ht
bt+1(1 + n) = Rtbt +
1
(2 + n)(1 + n)
φtwt − (τ 0t + τ 1t (bt − b∗))wt (3.17)
The capital market clearing equation can be written in efficient terms
by dividing both sides by Ht and multiplying the left hand side by the fraction
Ht+1/Ht+1.
Kt+1 +Bt+1
Ht
Ht+1
Ht+1
= a1t
Nt
Ht
+ a2t
Nt−1
Ht
(kt+1 + bt+1)(1 + n) = a
1
t
1 + n
2 + n
+ a2t
1
2 + n
(3.18)
The goods market clearing can be written in efficient terms by dividing both
sides by Ht
Nt
Ht
c1t +
Nt−1
Ht
c2t +
Nt−2
Ht
c3t +
Kt+1
Ht
=
Kαt H
1−α
t + (1− δ)Kt
Ht
1 + n
2 + n
c1t +
1
2 + n
c2t +
1
(1 + n)(2 + n)
c3t + (1 + n)kt+1 = Ak
α
t + (1− δ)kt (3.19)
Equilibrium
Given initial conditions k0, b0, a
1
−1 and a
2
−1, an initial cohort of young N0,
middle N−1 = N01+n , and old N−2 =
N0
(1+n)2
, and a population growth rate of n such
that Nt = (1 + n)Nt−1, a competitive equilibrium is a sequences of functions for the
households {a1t , a2t}∞t=0, production plans for the firm, {kt}∞t=1, government bonds
6Note, Ht can be written as Ht =
2+n
1+nNt, which implies Ht = (2+n)Nt−1 and Ht = (1+n)(2+
n)Nt−2. These identities are used to simplify 3.17, and 3.18.
69
{bt}∞t=1 factor prices {Rt, wt}∞t=0, government policy variables {τ 0t , τ 1t , φt}∞t=0, that
satisfy the following conditions:
1. Given factor prices and government policy variables, individuals’ decisions
solve the household optimization problem (3.2), (3.3), and (3.4)
2. Factor prices are derived competitively according to (3.9) and (3.10)
3. All markets clear according to (3.14), (3.18), and (3.17)
The equilibrium equations hold for t = 0, 1, ...∞ are reprinted below
(substituting in the Leeper tax τt = τ
0
t + τ
1
t bt).(
(1− τ 0t − τ 1t bt)w(kt)− a1t
)−σ
= βEt[R(kt+1)
(
R(kt+1)a
1
t + (1− τ 0t+1 − τ 1t+1(bt+1))w(kt+1)− a2t+1
)−σ
]
(3.20)
(
R(kt)a
1
t−1 + (1− τ 0t − τ 1t bt)w(kt)− a2t
)−σ
= βEt+1[R(kt+1)
(
R(kt+1)a
2
t + φt+1w(kt+1)
)−σ
]
(3.21)
bt+1(1 + n) = R(kt)bt +
1
(2 + n)(1 + n)
φtw(kt)− (τ 0t + τ 1t bt)w(kt) (3.22)
(kt + bt)(1 + n) = a
1
t−1
1 + n
2 + n
+ a2t−1
1
2 + n
(3.23)
Note that equation (3.21) has been iterated back to show the decision of
a middle aged person in period t, that happens simultaneously with the decision
of a young person in period t characterized by (3.20). Equation (3.23) has also
been iterated back one period so that it holds for t = 0, , 1, ...∞. Note finally that
k−1 = 0.
A stationary equilibrium is a competitive equilibrium in which per capita
variables and functions, as well as prices and policies, are constant, and aggregate
variables grow at the constant growth rate of the population n.
70
The steady state is a collection {k, b, a1, a2} that solves:
((1− τ 0 − τ 1b)w(k)− a1)−σ
= βR(k)(R(k)a1 + (1− τ 0 − τ 1b)w(k)− a2)−σ
(3.24)
(
R(k)a1 + (1− τ 0 − τ 1b)w(k)− a2)−σ = βR(k) (R(k)a2 + φw(k))−σ (3.25)
b(1 + n) = R(k)b+
1
(2 + n)(1 + n)
φw(k)− (τ 0 + τ 1b)w(k) (3.26)
(k + b)(1 + n) = a1
1 + n
2 + n
+ a2
1
2 + n
(3.27)
where R(k) and w(k) are given by (3.9) and (3.10).
Steady State Analysis
The number and stability of steady states in this model depends crucially
on the government’s social security policy. Government deficits and debt are
both endogenous and depend on the underlying policy parameters. When the
government runs relatively small deficits or surpluses, the model has two steady
states, one of which is stable, the other of which is explosive. As government
deficits (or surpluses) increase, the steady states move closer together, until at a
critical policy value, only one steady state exists. If deficits (surpluses) increase
beyond this critical size, no steady states exist. This change in the number of
steady states is called a saddle-node bifurcation, and is discussed in great detail in
Chalk (2000) and Azariadis (1993) (Chapter 8 and Chapter 20). Specific examples
of steady states, along with stability analysis, will be presented below in section
III. Before examining examples, however, I will present the general framework for
stability analysis in this model.
71
Linearization and Eigenvalue Analysis
The stability of steady states can be studied using eigenvalue analysis of the
linearized system, following Laitner (1990).
Define Xt = (a
1
t−1, a
2
t , kt, bt)
′. Given initial conditions (a1−1, a
2
−1, k0, b0),the
non-linear system governing the economy can be described as
G(Xt, EtJ(Xt+1)) = 0 (3.28)
where the notation EtJ(Xt+1) indicates the time t expectation of a function J of
Xt+1.
The system G(Xt, EtJ(Xt+1)) = 0 is given by:
((1− τt)w(kt)− a1t )−σ − βEt[R(kt+1)(R(kt+1)a1t + (1− τt+1)w(kt+1)− a2t+1)−σ] = 0
(R(kt)a
1
t−1 + (1− τt)w(kt)− a2t )−σ − βEt[R(kt+1)(R(kt+1)a2t + φt+1w(kt+1))−σ] = 0
bt+1(1 + n)−R(kt)bt − 1
(2 + n)(1 + n)
φtw(kt) + τtw(kt) = 0
(kt+1 + bt+1)(1 + n)− a1t
1 + n
2 + n
− a2t
1
2 + n
= 0
Equation (3.28) implicitly defines EtXt+1 as a function of Xt, using the
chain rule.7 Define F (Xt) as the implicit function such that EtXt+1 = F (Xt). Non-
linear solution techniques, outlined in Appendix B are used to approximate F (Xt).
Alternatively, The system F (Xt) can be approximated linearly. The linearized
system is useful in assessing the stability of steady states.
Define Zt = (τ
0
t , τ
0
t+1, τ
1
t , τ
1
t+1, φt, φt+1)
′. Total differentiation of equation
(3.28) using the chain-rule, yields a system that is linear in dEtXt+1, dXt, and dZt:
H1dXt +H2dEtXt+1 +H3dZt = 0 (3.29)
7Recall that the tax rate without a subscript refers to the entire tax τt = τ
0
t + τ
1
t bt. The tax
rate is written the suppressed form τt for this section to save space.
72
where H1, H2, and H3 are matrices of partial derivatives.
8 Multiplying through by
the negative inverse of H2 yields:
dEtXt+1 = M1dXt +M2dZt (3.30)
where M1 = −H−12 H1 and M2 = −H−12 H3.
The system EtXt+1 = F (Xt) can be approximated at a steady state X
∗
using a first order Taylor approximation. Using the matrix notation from the total
differentiation above,
F (X) ≈ F (X∗) +M1dX∗(X −X∗) (3.31)
Following Laitner (1990), the stability of a given steady state depends on the
eigenvalues of matrix M1 evaluated at the steady state. Three cases are possible.
Case 1 stability: the matrix M1 has three distinct, nonzero eigenvalues λi,
with modulus less than one for i = 1, 2, 3; and one distinct, nonzero eigenvalues λ4,
with modulus greater than one.
Case 2 instability (explosive): the matrix M1 has distinct, nonzero
eigenvalues λi with modulus greater than one for more than one eigenvalues.
Case 3 indeterminacy: the matrix M1 has distinct, nonzero eigenvalues λi
with modulus less than one for all four eigenvalues (i = 1, 2, 3, 4).
Stability (Case 1) requires three eigenvalues lie inside the unit circle because
the model has three predetermined variables. The predetermined variables are
capital, bonds, and asset level of the middle aged (a2t ). Capital and bonds are
backward-looking variables. Period t + 1 capital and bonds (kt+1, bt+1) are defined
as functions of time t variables only (kt, bt, a
1
t , and a
2
t ). Thus, period t + 1 capital
and bonds are known in period t. The asset holdings chosen by the middle aged
8H1 and H2 are size 4× 4, and H3 has size 4× 6.
73
in period t (a2t ) is also a predetermined variable. The middle aged savings choice
depends on last period’s young saving choice (a1t−1), current capital (kt), and future
capital (kt+1). Since period t + 1 capital is known in time t, everything in the a
2
t
equation is known in time t. The remaining variable, asset choice of the young (a1t ),
is a free variable that is not known in period t. The savings decision of the young
depends on their forecast of their savings decision in the next period, which in turn,
depends on their forecast of the capital stock two periods ahead.
Examples of Steady States
In the previous chapter, I illustrate the existence and stability of steady
states in the two-period model using a phase digram. In a three-period model, a
simple phase diagram is not possible, but the steady state equations can still be
displayed in two dimensions in a special case.
In general, the optimal savings choices of the young and middle-aged in the
model cannot be solved analytically. However, in the special case of Log Utility
(σ = 1) and Cobb-Douglass Production (ε = 1), the optimal savings of both
generations have closed form solutions that are functions of capital and bonds
only. Substituting the steady state versions of these expressions into the steady
state bond equation (3.27) and asset market clearing equation (3.26) yields a two-
dimensional system in (k, b):
b(1 + n) = R(k)b+
1
(2 + n)(1 + n)
φw(k)− (τ 0 + τ 1b)w(k) (3.32)
(k + b)(1 + n) = a1∗(k, b)
1 + n
2 + n
+ a2∗(k, b)
1
2 + n
(3.33)
where a1∗(k, b) and a2∗(k, b) are the analytical solution to (3.24) and (3.25) when
σ = 1, and R(k) and w(k) indicate competitive factor prices.
74
Contour plots of the steady state bond equation (3.32) and the steady state
asset market clearing equation (3.33) each plot the pairs (k, b) for which the specific
equation holds. The intersection of the two equations show the steady states of the
system.
Government policy and dynamic efficiency both play a key role in the
determining the number and stability of steady states. The model has at most
two steady states; however, one or zero steady states is also possible. The number
of steady states results from a saddle-node bifurcation. The model has two
steady states when government deficits are not “too large” (that is, if the benefit
replacement rate φ is not too high relative to the tax rate τ and the population
growth rate n). The steady state with higher capital is stable (case 1) and the
lower capital steady state is explosive (case 2). As the size of the government deficit
increases (due to changes in φ, τ , or n), the steady states more closer together.
When the deficit is just large enough, a single steady state exists.9 When deficits
are too large, no steady states exist.
I illustrate the saddle-node bifurcation with two dynamically inefficient
steady states and increasing deficits in Figure 11. This case is comparable to the
set-up of Chalk 2000 and my own two-period model analysis. Note the saddle node
bifurcation can occur at other parameterizations; the graph is presented was chosen
because it is qualitatively similar to Chalk (2000), and Figure 4 of the previous
chapter. The steady state bond equation (3.32) is a hyperbola with an asymptote
9This steady state is not unique–there are many possible parameter combinations that yield a
single steady state.
75
Low benefits
0.09 0.10 0.11 0.12 0.13 0.14 0.15
k
-0.01
0.01
0.02
0.03
0.04
0.05
0.06
b
kgr
stable ss
saddle ss
Δk=0
Δb=0
Higher benefits
0.09 0.10 0.11 0.12 0.13 0.14 0.15
k
-0.01
0.01
0.02
0.03
0.04
0.05
0.06
b
kgr
saddle ss
Δk=0
Δb=0
Even higher benefits
0.09 0.10 0.11 0.12 0.13 0.14 0.15
k
-0.01
0.01
0.02
0.03
0.04
0.05
0.06
b
kgr
Δk=0
Δb=0
FIGURE 11. Steady state contour graphs of equations (3.32) and (3.33) in (k, b)
space (capital is plotted on the horizontal axis, bonds on the vertical). These
contour plots illustrate the steady states for an economy that is dynamically
inefficient and running a social security program with a deficit. As the size of
the deficit is increased (because the tax rate is decreased), the constant bond loci
shifts out (up and to the left). When taxes are small enough, the bond locus shifts
so that it no longer crosses the asset locus and no steady states exist. This is an
illustration of the saddle-node bifurcation. The parameters for this example have
been selected to generate graphs that look qualitatively similar to Chalk 2000.
as the golden rule level of capital.10 The steady state asset equation (3.33) is an
inverted parabola. In the figures the top of the parabola is not visible.
The steady state contour plots for an economy with positive bonds and a
Leeper tax are presented in Figure 12. The graph looks qualitatively similar to
the contour plots for dynamic inefficiency with a deficit in Figure 11. The key
similarities are that the bond locus intersects the asset locus above the zero-bond
axis, and the steady states both occur on the same “arm” of the bond locus.
The bond locus shifts out (up and to the right) as the social security benefits
increase (or taxes decrease). If policy changes are large enough, the saddle node
bifurcation occurs and the steady states disappear. The key feature of a positive
Leeper tax (τ 1 > 0) is that a positive bond steady state is possible at a dynamically
efficient level of capital. I find this to be a compelling parameterization, since the
10The asymptote is at the golden rule level of capital when the Leeper tax τ1 = 0; the
asymptote moves to the left if a positive Leeper tax is introduced.
76
0.06 0.08 0.10 0.12 0.14
k
-0.01
0.01
0.02
0.03
b
kgrstable ss
saddle ss
Δk=0
Δb=0
FIGURE 12. Steady state contour graphs of model with Leeper tax in (k, b) space
(capital is plotted on the horizontal axis, bonds on the vertical). These contour
plots illustrate the steady states for an economy that is dynamically efficient and
running a social security program with a small surplus.
United States holds debt and the economy is (likely) dynamically efficient. If the
Leeper tax is not included (or if τ 1 is too small), then dynamically efficient steady
states only occur with negative bonds (government asset holdings), which is less
compelling as a model of the U.S. I will use the parameterization of Figure 12 as
the backdrop for policy experiments.
The baseline parameterization presented above in Figure 12 has two steady
states. The high capital steady state is determinant (or stable); three of the
eigenvalues of the linearized system evaluated at this steady state fall inside the
unit circle and one falls outside. The low capital steady state is explosive (or
unstable); two of the eigenvalues of the linearized system evaluated at the steady
state fall outside the unit circle. The Appendix section B includes additional
examples of steady states. The steady states and the eigenvalues of the linearized
system evaluated at the steady state are presented in a summary Table B.1.
77
Parameterization
Agents are economically active as soon as they enter the model and live for
a total of three periods. Each period can be thought of as twenty years. I assume
that agents enter the model at age twenty-five. Thus, young agents enters the
model at time t when they are twenty-five years old, become middled aged at time
t + 1 when they are forty-five years old, retire at time t + 2 when they are sixty-five
years old, and die at the end of period t+1 when they are eighty-five years old. The
remaining model parameters correspond the baseline model presented in Auerbach
and Kotlikoff (1987) with two exceptions.
In this paper, the utility preference parameter, σ, is set to one in the
baseline. This corresponds to log utility. This choice is for analytical simplicity,
which allows the four steady state equations to be consolidated into two equations
that can then be graphed in (k, b) space. It is common for this parameter to be
higher in discrete time models. For example, Auerbach and Kotlikoff (1987) set this
parameter to 4, and Krueger and Kubler (2006) target an IES of 0.5 (which would
correspond to σ = 2). However, log utility is not uncommon in the literature and is
within the range of empirical estimates (see Gourinchas and Parker (2002),Bullard
and Feigenbaum (2007), and Feigenbaum (2008)). The policy experiments are
qualitatively similar if a higher value for σ is chosen.
Second, I have assumed full deprecation for the baseline in this paper. This
assumption is not crucial to the results, although it does increase convergence speed
of the solution algorithm.11 Qualitatively similar results occur with lower rates of
depreciation (including the extreme case of no deprecation δ = 0). The baseline
11Full deprecation is a reasonable assumption for an OLG model with 20 year periods.
Assuming an annual deprecation rate of 5% corresponds to δ = 1 − 0.9520 = 0.64 in this model.
Annual deprecation of 10% corresponds to δ = 1− 0.920 = 0.88.
78
Parameter Value Notes
α capital share of output 0.25 AK γ = 0.25
β discount factor
(
1
1+0.015
)20
AK ρ = 0.015
n population growth rate 1− (1− 0.01)20 AK n = 0.01
δ depreciation 1 chosen for
computational speed
σ Inverse Elasticity
of Intertemporal
Substitution
1 Log utility chosen for
simplicity of graphs
TABLE 10. Baseline parameters. “AK” in the right column refers to the baseline
parameters of Auerbach and Kotlikoff (1987). In the three period model, and single
period corresponds to roughly 20 years; thus annual parameters from AK have been
converted to 20-year parameters.
parameters are presented in Table 10. Social security parameters (taxes τ 0, τ 1 and
benefit replacement rate φ) are generally set to correspond to the US system which
has a payroll tax rate of 10.6% to fund the Old Age Survivor portion of social
security, and an average benefit replacement rate of 38% (see Olovsson (2010) for
an example for calibrating social security in an OLG model with few periods).
Note, in the absence of social security (when τ 0 = τ 1 = φ = 0), the steady
states of the model are dynamically inefficient. As social security is introduced,
this crowds out capital, and results in steady state that are dynamically efficient.
This parameterization loosely follows Imrohoroglu, Imrohoroglu, and Joines
(1995), which also assumes dynamic inefficiency in the absence of social security
and dynamic efficiency with the program. The assumption of dynamic efficiency
with social security is also consistent with many computational life-cycle models,
including Krueger and Kubler (2002), (2004), and (2006).
79
Policy Experiments: Pension Reforms
Perfect Foresight Expected Policy Changes
Before examining policy uncertainty, it is useful to describe how the model
responds to expected, perfect foresight changes in policy. As a simple example,
consider an economy in a steady state without public pensions. Social security is
introduced in period t = 6, this decreases the savings of the young and middle aged
and drives down the capital stock. Under the specific parameterization considered,
the government owns debt at the final steady state, and the social security system
runs a small surplus to pay the interest on the debt. The transition paths are
presented in Figure 13.
The policy changes are anticipated with perfect foresight, so we see changes
in the savings decisions prior to the introduction of social security. We observe the
largest changes in the two periods immediately preceding the policy change (since
agents experience the policy change in their lifetimes). Very small changes also
occur in the periods before that as a result of the general equilibrium feedback.
For example, consider the generation who retires at time t = 6. This
generation is young at time t = 4, middle aged at time t = 5, and retired with
social security at time t = 6. This generation experiences a windfall gain relative
to other generations, since they do not pay taxes, yet they receive social security
benefits. Thus, they save less in both working periods of their life than previous
generations (a14 < a
1
3 and a
2
5 < a
2
4 ).
The generation that retires in t = 7 also experiences a windfall gain: they
only pay taxes in middle age, and still receives a social security benefit. This
generation increases savings while they are young, to help smooth the drop in
80
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FIGURE 13. Time paths for an economy that starts in a steady state without
public pensions (τ 0=τ 1=φ=0). Social security is introduced in period t=6; the
social security system has a Leeper tax and runs a small surplus to pay interest
on the outstanding government bonds (τ 0=0.077, τ 1=0.8, φ=0.2). All other
parameters set at baseline values.
consumption they will experience by paying taxes in middle age (a15 > a
1
4). Finally,
generations that retire at or after t = 8 are taxed when young and middle aged
and receive a social security benefit. These generations save less compared to their
predecessors.
Next, consider an economy that is on a (temporarily) explosive trajectory
because the social security system is running an unsustainable deficit, such that
no steady states exist for the economy.12 Suppose also that the government has
12Perhaps the government was running a sustainable social security system (with taxes that
were high enough to cover benefits), but then the growth rate of the population slowed, causing
the total taxes collected each period to be lower than the total benefits paid.
81
0.06 0.08 0.10 0.12 0.14
k
capital
-0.01
0.01
0.02
0.03
b
bonds
kgr
Δk=0
Δb=0
initial value
FIGURE 14. Steady state contour graphs of model with Leeper tax in (k, b) space
(capital is plotted on the horizontal axis, bonds on the vertical). No steady states
exist for this parameterization. The taxes raised are insufficient to cover benefits
paid each period. τ 0 = 0.065, τ 1 = 0.8, and φ = 0.2. The social security system has
the same benefit level as in Figure 12, but the taxes are lower. All other parameters
are at the baseline values. The initial values of capital and bonds for the policy
experiment presented in Figure 15 is also included in the graph. Note, that the
initial value is not a steady state.
existing government debt (b > 0). It is possible for the government to remedy
the social security system, and ensure convergence to a steady state, by simply
raising taxes or lowering benefits as in Figure 15. The size of the benefit cut was
chosen such that the steady state capital is the same after the benefit cut or the
tax increase.13 In both examples, initial government policy is unsustainable with
taxes insufficient to cover benefits. If policy were left unchecked, government bonds
would increase infinitely. The contour plots of the steady state equations for the
unsustainable policy are presented in Figure 14. Note that no steady states exist.
Increasing taxes or decreasing benefits shifts the bond locus down and to the left,
which allows for the possibility of steady states.
13The steady states, initial values, and policy parameters for these experiments are presented in
Table 11.
82
Fig Initial policy New policy Initial values Final steady state(s)
τ0 τ1 φ τ0 τ1 φ k b a1 a2 k b a1 a2
13 0 0 0 0.077 0.8 0.2 0.129 0 0.085 0.233 0.103 0.002 0.078 0.178
15 0.065 0.8 0.2 0.077 0.8 0.2 0.103 0.002 0.078 0.177 0.103 0.002 0.078 0.178
15 0.065 0.8 0.2 0.065 0.8 0.173 0.103 0.002 0.078 0.177 0.103 0.006 0.081 0.185
TABLE 11. Summary of perfect foresight policy experiments. Experiment 13:
Economy in steady state, social security introduced in t=6. Experiment 15
(circles): Economy on explosive path, taxes increased in period t=4. Experiment 15
(triangles): Economy on explosive path, benefits cut in period t=4.
In Figure 15, I start the economy off with positive bonds and unsustainable
social security policy. The government reforms social security in period t =
4 by raising taxes (indicated on the graph with circles) or cutting benefits
(presented in graph as triangles). This tax increase or benefit cut results in a
small social security surplus each period that is sufficient to pay the interest on
the government’s debt. During the first three periods, before the reform, the
capital stock is falling which drives down wages. Agents respond by increasing their
savings relative to their parents to smooth their consumption across their lifecycle.
Following either policy reform, capital, bonds, and the savings of both generations
converge to the new steady state.
Consider first the response to an announced tax increase in period t = 4.
Young agents in period t = 3 increase their savings (a13) to offset the loss in
consumption in their middle age due to higher payroll taxes. Middle-aged savings
a2, also increases leading up to the tax cut, but this seems to be driven mainly by
the falling capital stock. In contrast, if social security benefits are cut in period
t = 4, we see a large response in the middle-aged savings of agents in period t = 3.
These agents see that their social security benefits will be lower than previous
generations, and so they save more. The savings of the young also increases before
a benefit cut. The young in period t = 2 see that their benefits will be cut in t = 4,
and so they save more.
83
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FIGURE 15. Time path of endogenous variables for an economy that starts on
an explosive trajectory (bonds increasing infinitely and capital falling). Taxes are
increased (circles) or benefits are cut (triangles) in period t=4 and the economy
converges to the steady state. Agents respond prior to the change in policy,
since they fully anticipate the reform. The social security parameters before the
reform are τ 0=0.065, τ 1=0.8, and φ=0.2. The tax reform increases τ 0 to 0.077. The
benefit reform cuts φ to 0.173. All other parameters are set to the baseline values.
Initial values and steady states are listed in Table 11.
84
Stochastic Policy Changes
Let ωt = (τ
0
t , τ
1
t , φt)
′ describe the government policy parameters at time t.
Suppose that the social security program will be reformed (permanently) at some
unknown future date, such that ωnew 6= ωold. Suppose that the realization of ωt+1
depends on ωt, and is contained in the finite set Ω. Suppose also that each possible
reform converges to a steady state (the path is non-explosive). Suppose also, that
if reform is not enacted by period Tˆ , that policy ωˆ will be enacted with certainty
in period Tˆ + 1. (This final assumption is to guarantee that debt does not become
explosive in the long-run).
Let the probability of realizing a particular value of ωt+1 given ωt be
described by pi(ωt+1|ωt). Using this notation, the expected value in the household
first order equations (3.6) and (3.7) can be written as:
u′(c1t ) = β
∑
ωt+1∈ Ω
pi(ωt+1|ωt)Rt+1u′(c2t+1) (3.34)
u′(c2t ) = β
∑
ωt+1∈ Ω
pi(ωt+1|ωt)Rt+1u′(c3t+1) (3.35)
These first order conditions, combined with the asset market clearing
equation (3.18) and the bond transition equation (3.17), define the equilibrium
path of the economy with policy uncertainty.
Introducing aggregate uncertainty to the model raises some concerns.
Krueger and Kubler (2004) explain that in a OLG model with aggregate
uncertainty, a steady state wealth distribution will not exist in general.
Additionally, even when “exogenous aggregate shocks can take only a finitely
many values the equilibrium allocations in general do not have finite support
85
(p.2).” I overcome this problem by modeling aggregate policy uncertainty in a
simple, stylized way. Policy uncertainty only exists for a small number of periods,
Tˆ , and policy reform can only be one of a small number of policy options. These
simplifications, combined with the fact agents only live three periods, allow me to
calculate the equilibrium paths for the economy with policy uncertainty. To begin,
I consider policy uncertainty that spans two-periods only (section III). Next, I
consider simultaneous timing and policy uncertainty (section III). In the appendix,
I present examples with timing uncertainty that spans three periods (section B).
Simple Two-Period Timing Uncertainty
As the simplest possible type of uncertainty, suppose government policy
ω = (τ 0t , τ
1
t , φt)
′ will either be changed in period S with probability p, or if it is
not changed in period S, it is changed with certainty in period S + 1. Let ωold
indicate the policy parameters before the reform, and ωnew indicate policy after.
Under this simple type of policy uncertainty, there are only two possible paths for
the economy: either reform occurs in period S, or it occurs in period S + 1. For
notational ease, I will distinguish the two paths using tildes and hats: x˜ for the
path of the economy when reform occurs in period S, and xˆ for the path of the
economy when reform happens in S + 1.
This simple uncertainty only impacts two generations: the generation who
is young at time S − 2 (and thus isn’t sure what the value of φS will be when they
retire), and the generation who is young at time S − 1 (who isn’t sure about the
value of τ 0S and τ
1
S for their middle age). Decisions made by all other generations
are determined by the non-stochastic Euler equations (3.6) and (3.7).
86
The generation who is young at time S − 2 chooses a1S−2 using the non-
stochastic Euler equation (3.36) and a2S−1 using the stochastic Euler equation (3.37)
or (3.38).
((1− τS−2)w(kS−2)− a1S−2)−σ
= β(R(kS−1)(R(kS−1)a1S−2 + (1− τS−1)w(kS−1)− a2S−1)−σ)
(3.36)
For path x˜:
(R(k˜S−1)a˜1S−2 + (1− τ˜S−1)w(k˜S−1)− a˜2S−1)−σ
= β[p(R(k˜S)(R(kS)a˜
2
S−1 + φ˜neww(k˜S))
−σ)
+ (1− p)(R(kˆS)(R(kˆS)aˆ2S−1 + φˆoldw(kˆS))−σ)]
(3.37)
For path xˆ:
(R(kˆS−1)aˆ1S−2 + (1− τˆS−1)w(kˆS−1)− aˆ2S−1)−σ
= β[p(R(k˜S)(R(kS)a˜
2
S−1 + φ˜neww(k˜S))
−σ)
+ (1− p)(R(kˆS)(R(kˆS)aˆ2S−1 + φˆoldw(kˆS))−σ)]
(3.38)
The agents in either state of the world have the same information set and
form the same expectations thus:
a˜1S−2 = aˆ
1
S−2
a˜2S−1 = aˆ
2
S−1
The generation who is young at time S − 1 choices a1S−1 using the stochastic
Euler equation (3.39) or (3.41). Uncertainty is resolved in period S (either reform
occurs, or it occurs in period S + 1 with certainty). The agents choose a2S using the
equation (3.40) or (3.42).
87
Path x˜:
((1− τ˜S−1)w(k˜S−1)− a˜1S−1)−σ
= β[p(R(kS˜)(R(k˜S)a˜
1
S−1 + (1− τ˜S)w(k˜S)− a˜2S)−σ)
+ (1− p)(R(kˆS)(R(kˆS)aˆ1S−1 + (1− τˆS)w(kˆS)− aˆ2S)−σ)]
(3.39)
p((R(k˜S)a˜
1
S−1 + (1− τ˜S)w(k˜S)− a˜2S)−σ)
+ (1− p)((R(kˆS)aˆ1S−1 + (1− τˆS)w(kˆS)− aˆ2S)−σ)
= β(R(k˜S+1)(R(k˜S+1)a˜
2
S + φ˜S+1w(k˜S+1))
−σ)
(3.40)
Path xˆ:
((1− τˆS−1)w(kˆS−1)− aˆ1S−1)−σ
= β[p(R(kS˜)(R(k˜S)a˜
1
S−1 + (1− τ˜S)w(k˜S)− a˜2S)−σ)
+ (1− p)(R(kˆS)(R(kˆS)aˆ1S−1 + (1− τˆS)w(kˆS)− aˆ2S)−σ)]
(3.41)
p((R(k˜S)a˜
1
S−1 + (1− τ˜S)w(k˜S)− a˜2S)−σ)
+ (1− p)((R(kˆS)aˆ1S−1 + (1− τˆS)w(kˆS)− aˆ2S)−σ)
= β(R(kˆS+1)(R(kˆS+1)aˆ
2
S + φˆS+1w(kˆS+1))
−σ)
(3.42)
Prior to reform, agents along either path of the economy have the same
information set and form the same expectations; however, in period S, the agents
will make different choices depending on the realization of the stochastic policy
process. Thus:
a˜1S−1 = aˆ
1
S−1
a˜2S 6= aˆ2S
Following the resolution of uncertainty in period S, all subsequent
generations make decision under perfect foresight using (3.6) and (3.7). The choices
88
of the agents differ between the two paths.
a˜1S+j 6= aˆ1S+j
a˜2S+j 6= aˆ2S+j for j ∈ [1, T ]
Note that decisions made by agents prior to period S− 1 will be the same on
either path of the economy.
a˜1S−i = aˆ
1
S−i
a˜2S−i = aˆ
2
S−i for i ≥ 1
Note finally that the path of the state variables will the same for the two
paths until the reform is enacted.
I solve for the paths {a˜1t−1, a˜2t−1, k˜t, b˜t, aˆ1t−1, aˆ2t−1, kˆt, bˆt}Tt=1 simultaneously
given initial conditions a1−1, a
2
−1, k0 and b0.
Although this simple policy uncertainty is a small deviation from the
perfect foresight model, it is enough to produce discernible changes to equilibrium
dynamics. I will present four examples below to highlight the impact of this
simple policy timing uncertainty. Before presenting examples, I will discuss the
expectation formation process for the individuals in the economy.
The majority of agents in the model do not form expectations, as they do
not face any uncertainty. They simply solve the household maximization problem
under perfect foresight and choose asset holdings a1 and a2 to maximize their
utility. Only two cohorts of agents face uncertainty in this simple set up. These
agents choose asset allocations according to the equations listed above. The agents
choose assets to maximize the mathematical expected value of their lifetime utility.
The agents expectations are consistent with the underlying stochastic process that
89
governs policy. The agents are fully rational and act according to expected utility
theory.
This is an important difference between the policy experiments in Bu¨tler
(1999) and the policy experiments presented in this paper. Bu¨tler examines how
agents’ expectations about social security policy influence macroeconomic variables.
However, as Phelan (1999), points out, the policy expectation of the agents in
Bu¨tler’s paper are not model consistent. Bu¨tler’s agents are not rational and do
not form expectations based on an underlying stochastic process that governs
policy. The agents expectations are effectively different “perceptions” or beliefs
regarding how and when policy will change. The beliefs are not tied to actual
policy process. The analysis presented in this paper can be viewed a response to
Phelan’s comment.
Using the simple two-period timing uncertainty presented in this section,
there are only two possible time paths for the economy. Either policy is changed
in period S, or it is changed in period S + 1. I solve the equilibrium equations
for both possible time paths (including the agent level expectations) and present
them graphically. In more complex models with greater aggregate uncertainty (like
Davig et al. (2011), Davig and Leeper (2011b), or Krueger and Kubler (2004)), it
is common to present a time path for the economy that is constructed by taking
the average of several, possibly thousands, draws from the stochastic process. Since
the number of possible stochastic paths is so small in this model, I simply present
the evolution of the economy for all paths. I also present the time paths for an
alternate economy that does not have any policy uncertainty, as a benchmark to
isolate the impact of the uncertainty.
90
First, I consider an economy that is in a steady state that experiences a
large social security tax increase (τ 0 increases, τ 1 remains the same) in either
period t = 3 or period t = 4 (Figure 16). Agents alive prior to period t = 3
know there is a probability p = 0.5 that taxes will be increased in t = 3, and
that if taxes are unchanged in t = 3, they will be increased the following period.14
Young agents who expect that they may have to pay higher taxes when they are
middled aged respond by increasing their savings. Thus, regardless of when the
policy change is implemented, we see the young increase their savings in period
t = 2 in anticipate of possibly higher taxes in period t = 3. This is particularly
evident when comparing the savings of the young who experience the tax increase
in period t = 4 under uncertainty (that is, they thought the tax increase was
possible in either period 3 or 4, indicated in the graph with diamonds) compared to
the savings of the young who expect the tax increase with perfect foresight in t = 4
(indicated in the graph with squares). The agents in the model with uncertainty
save more in periods 2 and 3 leading up to the tax increase. After the tax increase
is implemented, savings fall for both the young and middle aged, capital increases,
and bonds decrease as the government runs large surpluses.
It is also interesting to compare the middle aged savings of an agent who
knows taxes will be increased in her old age to the middle aged savings of an agent
who faces a probability p chance that taxes are increased in her old age and a
probability 1 − p chance that taxes are increased the following period. Although
agents in the model do not pay taxes in retirement, they anticipate the general
equilibrium feedback effects of younger generations decreasing savings in response
14The tax increase creates a large social security surplus and leads the government to
accumulate assets. This does not represent a policy that is likely to occurs in the U.S., but it
provides a useful experiment to examine the responses to agents to tax rate uncertainty.
91
to a tax increase (reduced savings lowers the capital stock, driving down the wage
and social security benefit). Thus, middle aged agents who see taxes increasing
the following period with certainty save more than agents who have the possibility
that taxes won’t be increased until after their death. This is evident in the graph
by comparing the middle aged savings a22 of agents facing certain reform in period
t = 3, with the middle aged savings of agents who expected reform was possible in
t = 3 or t = 4. The initial values, policy parameters, and final steady state for this
experiment are listed in Table 12.
I consider a similar example in Figure 17. The economy starts in a steady
state with social security and the government greatly reduces social security
benefits in either period t = 3 or period t = 4. Young agents who think that
their benefits might be cut relative to their parents, increase their savings. This
is particularly evident comparing young savings in period t = 1 (a11) for reform
uncertainty with young savings in period t = 1 for an economy that doesn’t cut
benefits until period 4 with certainty. The young agents in the world with certainty
know their benefits will not be cut, so they do not increase a11 compared to the
initial generation of young. However, the young agents who think benefits will be
cut in period t = 3 with probability p = 0.5, save more to offset the possible loss
in retirement income. We also see large effects in the savings choices of the middle
aged. Middle aged agents in period t = 2 who face a probability p = 0.5 that
their social security benefit will be cut (indicated with circles and diamonds) save
more than the prior generation and more than agents who know benefits will not be
reduced until period t = 4 (indicated with squares). However, these agents save less
than agents in the counterfactual world who know their benefits will be cut with
92
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FIGURE 16. Time paths for an economy that is in a steady state and then
experiences a large social security tax increase either in period 3 or 4. Time paths
are plotted for an economy that faces probability p=0.5 of reform in period 3. If
reform is not enacted in period 3, it happens with certainty in period 4. There are
two possible paths for the economy under this type of stochastic policy, either the
reform takes place in period 3 (these paths are labeled uncertain reform in 3), or it
happens in period 4 (these paths are labeled uncertain reform in 4). Time paths are
plotted in addition for reform that happens with certainty in either period. The top
left panel shows capital, top right shows bonds, bottom left shows the savings (or
asset allocations) of the young (a1), and the bottom right shows the savings of the
middle aged (a2).
93
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FIGURE 17. Time paths for an economy that is in a steady state and then
experiences a large social security benefit reduction either in period 3 or 4.
Time paths are plotted for an economy that faces probability p=0.5 of reform in
period 3. If reform is not enacted in period 3, it happens with certainty in period 4.
There are two possible paths for the economy under this type of stochastic policy,
either the reform takes place in period 3 (these paths are labeled uncertain reform
in 3), or it happens in period 4 (these paths are labeled uncertain reform in 4).
Time paths are plotted in addition for reform that happens with certainty in either
period. The top left panel shows capital, top right shows bonds, bottom left shows
the savings (or asset allocations) of the young (a1), and the bottom right shows the
savings of the middle aged (a2).
certainty in period t = 3 (indicated in the graph with triangles). The initial values,
policy parameters, and final steady state for this experiment are listed in Table 12.
94
Next I consider an economy that is on an unsustainable path (deficits are
too large and no steady states exist), that remedies the situation by increasing
the social security tax rate in either period t = 3 or period t = 4 (Figure 18).15
Reform is expected in period t = 3 with probability p = 0.5, and if reform does
not happen in period t = 3, it happens in period t = 4. The tax change in this
example is relatively small compared to the tax change in examined in Figure
16, thus the time paths for uncertain reform are closer to the time paths with no
uncertainty. There is still a discernible increase in young savings (a12) for agents
who think they might be taxed higher next period, compared to those who know
then tax increase won’t happen until after they retire. The most visible difference
between uncertain and certain reform in this example is evident in middle aged
savings in period 3 (a23). If reform takes place in period t = 4 with certainty, we
observe middle aged savings increase dramatically in t = 3. If reform is possible in
period t = 3, but doesn’t actually take place until period t = 4, the middle aged
savings increases more slowly. Specifically a23 is lower in the model with uncertainty.
The key difference is that when they were young (in period t = 1), the agents in
the model with uncertainty saved more because they though the tax increase was
possible in their middle age. They entered period t = 2 with more assets than the
agents in the counterfactual world where the reform occurs in period t = 4 with
certainty. Thus, they did not need to save as much as the agents in the model with
certainty. Although it is not visible in the graph, all four time paths converge to
the same steady state that is determined by the policy parameters after the tax
increase. The initial values, policy parameters, and steady state are listed in Table
12.
15The initial parameter values are identical to the perfect foresight example presented in Figures
14 and 15. The magnitude of the tax increase is also identical to the example in Figure 15.
95
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FIGURE 18. Time paths for an economy that has unsustainable policy that is
reformed via tax increase either in period 3 or 4. Time paths are plotted for an
economy that faces probability p=0.5 of reform in period 3. If reform is not enacted
in period 3, it happens with certainty in period 4. There are two possible paths for
the economy under this type of stochastic policy, either the reform takes place in
period 3 (these paths are labeled uncertain reform in 3), or it happens in period 4
(these paths are labeled uncertain reform in 4). Time paths are plotted in addition
for reform that happens with certainty in either period. The top left panel shows
capital, top right shows bonds, bottom left shows the savings (or asset allocations)
of the young (a1), and the bottom right shows the savings of the middle aged (a2).
96
Fig Initial policy New policy Initial values Final steady state(s)
τ0 τ1 φ τ0 τ1 φ k b a1 a2 k b a1 a2
16 0.076 0.8 0.2 0.1 0.8 0.2 0.098 0.009 0.081 0.18 0.121 -0.031 0.058 0.163
17 0.076 0.8 0.2 0.077 0.8 0.18 0.098 0.009 0.081 0.18 0.113 -0.012 0.071 0.176
18 0.065 0.8 0.2 0.077 0.8 0.2 0.103 0.002 0.078 0.177 0.103 0.002 0.078 0.178
19 0.065 0.8 0.2 0.065 0.8 0.173 0.103 0.002 0.078 0.177 0.103 0.006 0.081 0.185
TABLE 12. Policy Experiments with simple two-period timing uncertainty.
Experiment 16: Economy in steady state with social security, taxes raised in either
period t=3 or t=4. Experiment 17: Economy in steady state with social security,
benefits cut in either period t=3 or t=4. Experiment 18: Economy on explosive
path, taxes increased in in either period t=3 or t=4. Experiment 19: Economy
on explosive path, benefits cut in either period t=3 or t=4. Two-period timing
uncertainty.
The final example I consider with simple two-period timing uncertainty is
an economy that is on an explosive path that reforms social security via benefit
reduction in either period t = 3 or t = 4 (Figure 19). The initial values and the size
of the benefit cut are identical to the perfect foresight example in Figure 15. Agents
who are young in period t = 1 and face the possibility of a benefit cut during their
retirement in t = 3 save more than agents who know benefits will not be cut until
t = 4. Conversely, young agents who aren’t sure if benefits will be cut in their
lifetime save less than agents who know benefits will be cut in t = 3 with certainty.
We see a similar pattern looking at the savings of the middle age in period t = 2.
Agents who might have their benefits cut in t = 3 save more than the agents who
know benefits won’t be cut until t = 4, and less than the agents who know benefits
will be cut in t = 3. The initial values, policy parameters, and steady state are
listed in Table 12.
Timing and Policy Uncertainty
The analysis of the previous section only includes policy timing uncertainty.
In Appendix B, I also consider simple policy type uncertainty (with a known date
of reform). In reality, Americans face uncertainty about the future of social security
97
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FIGURE 19. Time paths for an economy that has unsustainable policy that is
reformed via benefit reduction either in period 3 or 4. Time paths are plotted
for reform that happens with certainty in either period. Time paths are plotted
in addition for an economy that faces probability p=0.5 of reform in period 3. If
reform is not enacted in period 3, it happens with certainty in period 4. There are
two possible paths for the economy under this type of stochastic policy, either the
reform takes place in period 3 (these paths are labeled uncertain reform in 3), or it
happens in period 4 (these paths are labeled uncertain reform in 4).
98
policy that span time and policy options. Social security could be reformed by
cutting taxes, raising benefits, or both. Additionally, that reform could take place
today, or tomorrow, or ten years from now. In this final section, I explore simple
examples of timing and policy uncertainty. For example, consider a government
that will implement either reform A or reform B, and the reform will either take
place in period S, or in period S + 1. Under this regime, four policy paths are
possible. Suppose the probability of each path is as follows: probability p of reform
A in period S, probability q of reform A in S + 1, probability r of reform B in S,
and probability s of reform B in S + 1.16
The generation who is young at time S − 2 chooses a1S−2 using the non-
stochastic Euler equation (as in previous sections, see 3.36) and a2S−1 using a
stochastic Euler equation such as (3.43). There will be four distinct Euler equations
(one for each path of the economy). Only one path is presented below.17
For path x˜:
(R(k˜S−1)a˜1S−2 + (1− τ˜S−1)w(k˜S−1)− a˜2S−1)−σ
= β[p(R(k˜S)(R(kS)a˜
2
S−1 + φ˜Sw(k˜S))
−σ)
+ q(R(kˆS)(R(kˆS)aˆ
2
S−1 + φˆSw(kˆS))
−σ)
+ r(R(k¯S)(R(k¯S)a¯
2
S−1 + φ¯Sw(k¯S))
−σ)
+ s(R(k˘S)(R(k˘S)a˘
2
S−1 + φ˘Sw(k˘S))
−σ)]
(3.43)
The generation who is young at time S − 1 choices a1S−1 using a stochastic
Euler equation such as (3.44). Uncertainty is resolved in period S only if reform
16p+ q + r + s = 1, and p < 1, q < 1, r < 1, and s < 1
17The distinct path are denoted using the following notation: reform A in S as x˜, reform A in
S + 1 as xˆ, reform B in S as x¯, and reform B in S + 1 as x˘, for x = k, b, a1, a2, τ, φ.
99
occurs in that period; if not there is still uncertainty about which reform will take
place in period S + 1. The agents choose a2S using the equation (3.45).
Path x˜:
((1− τ˜S−1)w(k˜S−1)− a˜1S−1)−σ
= β[p(R(k˜S)(R(k˜S)a˜
1
S−1 + (1− τ˜S)w(k˜S)− a˜2S)−σ)
+ q(R(kˆS)(R(kˆS)aˆ
1
S−1 + (1− τˆS)w(kˆS)− aˆ2S)−σ)
+ r(R(k¯S)(R(k¯S)a¯
1
S−1 + (1− τ¯S)w(k¯S)− a¯2S)−σ)
+ s(R(k˘S)(R(k˘S)a˘
1
S−1 + (1− τ˘S)w(k˘S)− a˘2S)−σ)]
(3.44)
p(R(k˜S)a˜
1
S−1 + (1− τ˜S)w(k˜S)− a˜2S)−σ)
+ q(R(kˆS)aˆ
1
S−1 + (1− τˆS)w(kˆS)− aˆ2S)−σ)
+ r(R(k¯S)a¯
1
S−1 + (1− τ¯S)w(k¯S)− a¯2S)−σ)
+ s(R(k˘S)a˘
1
S−1 + (1− τ˘S)w(k˘S)− a˘2S)−σ)
= β(R(k˜S+1)(R(k˜S+1)a˜
2
S + φ˜S+1w(k˜S+1))
−σ)
(3.45)
Generations born in period S face uncertainty if reform did not take place in
period S. They know reform will occur in period S+1, but they don’t know form it
will take. Along path x˜ (reform A in period S) and path x¯ (reform B in period S),
young agents in period S choose savings using non-stochastic Euler Equations.
Young agents in period S using the following first order equations for path xˆ
(reform A in period S + 1, there is a similar set of FOC for path x˘):
((1− τˆS)w(kˆS)− aˆ1S)−σ
= β[q/(q + s)(R(kˆS+1)(R(kˆS+1)aˆ
1
S + (1− τˆS+1)w(kˆS+1)− aˆ2S+1)−σ
+ s/(q + s)(R(k˘S+1)(R(k˘S+1)a˘
1
S + (1− τ˘S+1)w(k˘S+1)− a˘2S+1)−σ]
(3.46)
100
q/(q + s)(R(kˆS+1)aˆ
1
S + (1− τˆS+1)w(kˆS+1)− aˆ2S+1)−σ
+ s/(q + s)(R(k˘S+1)a˘
1
S + (1− τ˘S+1)w(k˘S+1)− a˘2S+1)−σ
= β(R(kˆS+2)(R(kˆS+2)aˆ
2
S+1 + φˆS+2w(kˆS+2))
−σ
(3.47)
As an example, consider an economy that is in a steady state, and then
experiences a change of policy in either period t = 3 or t = 4. Suppose there is
probability p that taxes are raised in period t = 3, probability q that taxes are
raised in period t = 4, probability r that benefits are cut in t = 3, and a probability
s that benefits are cut in period t = 4 (where p = q = r = s = 0.25). The time
paths for the four possible outcomes are plotted in Figure 20. The four time paths
for the same policy changes occurring with certainty have been excluded from the
graph for simplicity. This exercise combines the timing uncertainty of Figures 16
and 17 with the simple policy option uncertainty.
Young agents in period t = 1 who aren’t sure if reform will happen in their
lifecycle and also aren’t sure if benefits will be cut or if taxes will be raised engage
in a small amount of precautionary savings. Only one of the four possible policy
paths increases the young agents need to save (a benefit reduction in period 3).
Thus, the young agent doesn’t respond very much. In period t = 2, the young
agents increases savings to offset the possible tax increase during their middle age
and the possible benefit cut. If reform occurs in period t = 3, agents respond
accordingly. If reform does not take place in period t = 3, the young agent
still faced some uncertainty; benefits might be cut in period 4 or taxes might be
increased. The young agent in period t = 3 facing this uncertainty increases savings
to offset the possibility of higher taxes next period.
This example has important policy implications. Survey data suggests that
many American households expect social security to be reformed and only expect
101
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FIGURE 20. Time paths for an economy that begins in a steady state and then
either experiences i) a tax increase in t=3, ii) a tax increase in t=4, iii) a benefit
cut in t=3, or iv) a benefit cut in t=4. There are four possible time paths possible
given this type of uncertainty.
102
Figure Initial policy New policy Initial values Final steady state(s)
τ0 τ1 φ τ0 τ1 φ k b a1 a2 k b a1 a2
20 0.0756 0.8 0.2 0.1 0.8 0.2 0.098 0.009 0.081 0.18 0.121 -0.031 0.058 0.163
0.0756 0.8 0.18 0.113 -0.012 0.071 0.176
TABLE 13. Policy experiment with timing and policy uncertainty. Experiment 20:
Economy in a steady state. Possible reforms (each with probability 0.25): taxes
raised in t=3, benefits cut in t=3, taxes raised in t=4, or benefits cut in t=4. Policy
option and two-period timing uncertainty.
to receive a fraction of promised benefits.18 Yet, in contrast to standard economic
theory, these households are not increasing their private savings.
Standard lifecycle models suggest that households who expect to receive
smaller social security benefits should save more. In the context of this model,
the relationship between savings and social security benefits is negative, that is
∂a1t/∂φt+2 < 0 and ∂a
2
t/∂φt+1 < 0. Similarly, young agents who expect to pay
higher taxes next period, save more today; ∂a1t/∂φt+1 > 0.
The introduction of policy and timing uncertainty can partially explain the
paradox that households expect social security reform, yet they do not increase
their savings. The time t = 1 young agents in Figure 20 do not increase their
savings very much, even though they believe there is a 50% chance of reform in
their lifetime. The incentive to increase savings to offset a potential benefit cut is
offset by the incentive to consume now and hope for policy that doesn’t directly
impact the lifetime resources of the agent.19
I intend to explore this relationship further in my future work.
18See Dominitz and Manski (2006) for an overview of the social security expectations literature
19 Young savings is only a function of the benefit rate two period ahead; ∂a1t/∂φt+i = 0 for all
i 6= 2 and only responds to current and one period ahead taxes: ∂a1t/∂τt+i = 0 for all i <
0 and i ≥ 2. Middle aged savings responds to benefit rate the next period, but no other periods
∂a2t/∂φt+i = 0 for all i 6= 1. Middle aged savings does not depend on future taxes ∂a2t/∂τt+i = 0
for all i ≥ 2.
103
Long-run Consequences of Policy Uncertainty
The analysis of the previous sections focused on the short-run,
intergenerational consequences of policy uncertainty. The savings decisions
of agents who face uncertainty also have interesting long-run, macroeconomic
consequences. The possibility of unstable government policy (low taxes and/or
high benefits that lead to explosive government debt) can destabilize the economy.
Unsurprisingly, the possibility of low taxes (or high benefits) decreases the agents’
incentives to save and decreases the capital stock, even if the unstable policy is not
realized.
Interestingly, the reverse is also possible. The potential for tax increases (or
benefit cuts) in the future, may be enough to stabilize the path of the economy
even if the tax cut (or benefit increase) is not realized. The precautionary savings
of the agents facing uncertainty can be enough to increase the capital stock to lie
within the basin of attraction of the stable steady state. That is, policy uncertainty
can generate a rational expectations equilibrium that converge to a steady state for
given initial conditions that would not converge to a steady state in the absence of
policy uncertainty.
As an example, consider an economy that is on a (temporarily) explosive
path. The government reforms social security in period t = 4 by implementing a
benefit cut. In the absence of policy uncertainty, benefit cut A is insufficient to
stabilize the economy; that is, benefit cut A is not an equilibrium path. Benefit cut
B, on the other hand, is larger, and is sufficient to cause the economy to converge
to a steady state.
If there is a probability p that the government implements policy A, and
a probability (1 − p) of policy B, the realization of either policy can be a stable
104
equilibrium path, as long as p is small enough. This is depicted graphically in
Figure 21 and Figure 22. In this example, the initial benefit replacement rate is
φ = 0.2. Policy A (which is unstable on it’s own) reduces the benefit replacement
rate to φ = 0.1743. Policy B cuts the benefit replacement rate to φ = 0.17. The
probability of benefit cut A is quite small at p = 0.05.
Figure 21 focuses on the first few periods of the economy. Note that the
implementation of policy B with certainty, and realization of policy B under
uncertainty are nearly identical. This is because the probability of policy B was
quite high in the model with uncertainty. Figure 22 shows the long-run converge of
the model. Benefit cut B converges to a higher level of capital and a lower level of
bonds, since it is a larger benefit cut. The realization of benefit cut A in the model
with uncertainty converges to a steady state with lower capital and higher bonds.
The realization of benefit cut A also takes much longer to converge than policy B.
The capital stock is depressed for generations following the realization of policy A.
Conclusion
Uncertainty about social security policy impacts agents over their lifecycle
and impacts the economy for generations following a reform. I develop a three-
period overlapping generations model with endogenous government deficits and
debt to examine the interaction between social security policy and the long-run
stability of the economy. The general equilibrium model is forward looking and
agents care about future policy. The model is simple enough to accommodate
aggregate uncertainty regarding the government’s social security policy and can
be solved computationally. Agents who face uncertain social security policy in
their lifetime behave different than agents in a perfect foresight model who know
105
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FIGURE 21. Time paths for an economy that is on an explosive path until
social security benefits are cut in period t=4. There is a probability p=0.05 of
policy A where benefits are cut to 0.1743 and a probability (1−p)=0.95 policy
B where benefits are cut to 0.17. Policy A does not converge to a steady state if
it is enacted with certainty. However, where there is a large enough probability
of benefit but B, the precautionary savings of agents leading up to the reform
increases the capital stock enough that the realization of policy A does, in fact,
converge to a steady state. The graph also includes the equilibrium path for
an economy that implements policy B with certainty. The equilibrium paths
for realization of policy B in the uncertain model are nearly identical to the
equilibrium paths for policy B implemented with certainty.
106
0 20 40 60 80 100 120
0.096
0.098
0.100
0.102
0.104
0.106
Time
C
ap
ita
l
0 20 40 60 80 100 120
0.000
0.005
0.010
0.015
Time
B
on
ds
0 20 40 60 80 100 120
0.080
0.082
0.084
0.086
Time
S
av
in
gs
of
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ou
ng
0 20 40 60 80 100 120
0.180
0.182
0.184
0.186
0.188
0.190
Time
S
av
in
gs
of
M
id
dl
e
uncertain policy, reform A realized
uncertain policy, reform B realized
certain reform B
FIGURE 22. This is the same policy experiment as Figure 21. These graphs show
the entire equilibrium paths for the realization of policy A and the realization of
policy B in a model with uncertainty (where there is a 0.05 probability that policy
A is enacted). The time paths for an economy where policy B is enacted with
certainty is also depicted. The time paths for the realization of policy B under
uncertainty are so similar to the time paths for the economy that enacts B with
certainty, that the graphs lie directly on top of each other.
107
the future path of all policy parameters. Agents increase their savings if there is a
possibility that their taxes will be increased or if their social security benefits will
be decreased. Agents save less if there is a possibility that reform doesn’t happen
until after their lifetime, or the possibility that reform is small. The introduction
of uncertainty regarding future policy partially explains the empirical finding that
many American households simultaneously predict social security will be reformed
and yet do not engage in precautionary saving.
The choices of agents who face policy uncertainty have lasting consequences
even after the policy uncertainty has been resolved. The savings choices of agents
determine the capital stock the following period, which in turn influences the
savings and consumption decisions of agents. Even a small amount of uncertainty,
for example a reform that takes place in either period S or in period S + 1, can
change the capital stock and influence the transition path of the economy for
generations.
Policy uncertainty can destabilize an otherwise stable transition to a steady
state, and policy uncertainty can make an otherwise unsustainable path converge
to a steady state. As illustrated in section III, the possibility of a large reform
can make the realization of a small reform stable in the long-run. This result is
similar to the findings of Davig et al. (2011) who show in a New-Keynesian model
with unsustainably large government transfers, that dire policy outcomes like
hyperinflation can be avoided if there is a small probability that actual transfers
will be less than promised transfers. In the New-Keynesian model the possibility
of the government reneging on promised transfers increases the incentive for agents
to save. In the OLG framework of this paper, the potential for a larger (or sooner)
reform also causes agents to save more, which increases the basin of attraction for
108
a given steady state and opens the possibility that more paths will converge in the
long-run.
The short-run impacts of policy uncertainty, specifically the changes in
savings decisions of agents prior to a reform, are similar to the results in Bu¨tler
(1999); expectations leading up to a reform have large impacts on aggregate
variables and on consumption across generations. The analysis of the paper
solidifies the results of Bu¨tler, by providing a baseline model will forward-looking,
fully rational agents. The agents in this paper do not suffer from “misperception”
nor are they surprised by unannounced changes to policy. The expectations of
agents are consistent with the (simple) stochastic process that governs the policy
parameters.
Caliendo et al. (2015) calculate the welfare cost to a worker of facing
uncertainty about the future of social security. They find that the welfare cost of
uncertainty to be up to one percent of total lifetime consumption for agents who do
not save (agents who save optimally experience a smaller welfare loss). The model
includes many realistic details of the lifecycle (like stochastic survival probability)
and can accommodate rich policy uncertainty that spans a distribution of potential
reforms that take place in many possible periods. The partial equilibrium model
focuses on a single agent, and does not consider the transition of the economy
to a steady state following a reform. I have shown that policy uncertainty can
have aggregate impacts for generations following a policy reform. The general
equilibrium feedback of changes in saving can cause the capital stock to be lower
(or higher) for fifty of more periods as the economy transitions to a steady state.
Combining the insights from my three-period model with the results of Caliendo
109
et al. (2015) suggest that welfare of the economy could be impacted for many
generations if there is any uncertainty regarding the future path of social security.
110
CHAPTER IV
LIFE-CYCLE HORIZON LEARNING AND POLICY UNCERTIANTY
Introduction
In this chapter, I continue my analysis of policy uncertainty in a richer,
multi-period lifecycle model. The way in which agents form expectations has a
large impact on both the short-run responses to changes in policy (or possible
changes in policy) and the long-run level of capital and output in the economy. I
consider rational expectations and adaptive expectations. My paper contributes
to a growing literature that examines the response to anticipated fiscal policy in
models with adaptive expectations (see G. W. Evans et al. (2009), Mitra et al.
(2013), Gasteiger and Zhang (2014), and Caprioli (2015)).
The rational expectations hypothesis is standard in macroeconomics and I
develop a rational expectations model as the baseline for my analysis. I also relax
the assumption of rational expectations and explore the consequences of agents
using adaptive expectations in a model with an aging society and social security
reform. One criticism of the rational expectations hypothesis is that it requires
agents to possess more knowledge about the structure of the economy than any
econometrician possesses about the actual economy. Models of adaptive learning
relax that requirement, and allow for boundedly rational agents (Sargent (1993),
G. W. Evans and Honkapohja (2001)).
Adaptive learning has several benefits. Models of adaptive learning produce
more realistic dynamics that better fit observed data. Eusepi and Preston (2011)
demonstrate that infinite-horizon learning amplifies technology shocks in a real
111
business cycle model and creates more realistic dynamics. W. A. Branch and
Mcgough (2011) show that a calibrated model with heterogenous beliefs (some
agents are fully rational, others use N-step ahead adaptive learning) provides a
closer fit to business cycle data than the rational expectations baseline. Models
of adaptive learning are also better able to match survey data on expectations
(W. A. Branch (2004)), asset price volatility (Bullard and Duffy (2001), Adam,
Marcet, and Nicolini (2016)), and experimental data on expectation formation
(Adam (2007) and Pfajfar and Santoro (2010)). Additionally, adaptive learning can
be viewed as a robustness check to examine the stability properties of a rational
expectations equilibrium (see G. W. Evans and Honkapohja (2001) and (2009)).
In this paper, I introduce a two new frameworks for modeling bounded
rationality that I call life-cycle horizon learning, and finite horizon life-cycle
learning. I simulate the underlying demographic changes and the two policy
changes that the SSA suggests would be sufficient to fund future benefits. I find
the welfare cost of implementing a reform is much larger for some cohorts in the
learning models, particularly in the life-cycle horizon learning model, compared
to a with model fully rational agents. Cohorts of agents alive before and after
reforms can be negatively impacted by the cyclical dynamics adaptive expectations
introduce into the model.
I also explore the consequences of policy uncertainty by modeling reform
that can take place in one of two different dates and can take the form of a
tax increase or a benefit cut. I find that the welfare cost of this type of policy
uncertainty is small in the rational expectations model. The consumption
equivalent variation that makes an agent indifferent between an announced policy
reform or policy uncertainty (between four different options, tax increases or benefit
112
cuts possible in two different periods) is at most 0.3% of period consumption. I find
that the welfare cost to be much larger for agents in a life-cycle horizon learning
model. The most harmed agents who experience an announced policy change in
a rational expectations framework would have to give up 1.9% of their period
consumption to be indifferent between announced policy in a life-cycle horizon
model and policy uncertainty in a life-cycle horizon learning model.
I present the model in section IV and introduce life-cycle horizon learning
and finite horizon life-cycle learning in sections IV and IV. The model is calibrated
in section IV. Policy applications, including social security reform uncertainty
are explored in section IV. Welfare analysis is in sections IV and IV. Section IV
includes alternative modeling assumptions, and section IV concludes.
Model
Households
Households live for J periods, choose asset allocation (aj for j = 1, · · · J − 1)
in the first J − 1 periods of life, and consumption (cj for j = 1, · · · , J ) in all
J stages of life, to maximize utility, taking price, and government social security
policy (tax rates τ , and benefit z) as given. Agents receive wage wt for labor
provided in period t, and retire at date T ≤ J . The gross real return on savings
in period t is given by Rt+1. Superscripts on variable indicate life-cycle stage (i.e.,
age), and subscripts indicate time period.1
Household’s maximize discounted expected lifetime utility (4.1) subject to
period budget constraints (4.2). Here E∗t (x) indicates the time t expectation of x.
1As an example, a2t+1 is age 2 savings (saving in the second stage of life), in time period t+ 1.
113
The star indicates that the expectations need not be rational.
max
ajt+j−1
E∗t
J∑
j=1
βj−1u(cjt+j−1) (4.1)
cjt+j−1 + a
j
t+j−1 ≤ Rt+j−1aj−1t+j−2 + yjt+j−1 for j = 1, · · · , J (4.2)
Here yj indicates period labor income during working life and social security
income after retirement:
yjt+j−1 = (1− τt+j−1)wt+j−1 for j < T
yjt+j−1 = z
j
t+j−1 for j ≥ T
The lifespan is certain and agents do not have a bequest motive, so they
exhaust all of their resources in the final period of life.
aJt+J−1 = 0 (4.3)
The household first order conditions are:
u′(cjt+j−1) = βE
∗
t [Rt+ju
′(cj+1t+j )] for j = 1, · · · , J − 1 (4.4)
The agent (trivially) chooses to consume all of her resources in the final period of
life, that is cJt+J−1 = Rt+J−1a
J−1
t+J−2 + z
J
t+J−1.
Labor is supplied inelastically and preferences are given as the standard
constant elasticity of substitution function:
u(c) =
c1−σ
1− σ if σ 6= 1
u(c) = ln(c) if σ = 1
Demographics
To start off the economy, I assume that in period zero, there are J cohorts
who enter the economy with given asset holdings according to their age. I assume
that the initial young enter the economy with zero assets, and all other cohorts
114
enter with aj−1 for j = 1, · · · , J − 1. Successive cohorts enter the model with zero
assets when they are young.
Nt indicates generation t and is given by number of young (i.e., the
generation born) at time t. The population grows at rate nt such that
Nt = (1 + nt)Nt−1 (4.5)
The population at any time t is the sum of all living cohorts. If the growth
rate n is constant, the population can be expressed relative to the initial young or
initial old:
J∑
j=1
Nt+1−j =
J∑
j=1
(1 + n)1−jNt =
J∑
j=1
(1 + n)J−jNt+1−J .
Production
The consumption good in the economy (Yt) is produced by single firm (or
equivalently many small firms) using a constant elasticity of substitution technology
that takes aggregate capital (Kt) and labor (Ht) as inputs and produces the
consumption good according to:
Yt = F (Kt, Ht) = K
α
t H
1−α
t
The parameter α measures the intensity of use of capital in production.2 Factor
markets are competitive and capital and labor (hours worked) are paid their
marginal products. The gross real interest rate Rt is given by:
Rt = FK(Kt, Ht) + 1− δ (4.6)
2I abstract from technological growth, although it is straightforward to incorporate this into
the model. If technological process was included in the model, aggregate variables would grow
that the rate (1 + n)(1 + g) along the balanced growth path (if g indicates the growth rate of
technology). In all other regards, the model is identical.
115
where δ is the rate of depreciation. The wage rate wt is given by
wt = FH(Kt, Ht) (4.7)
Government
The government runs a modified pay-as-you-go social security system.
The government pays retirement benefits to the retired generations by taxing the
working generations and by (possibly) issuing debt.
The pay-roll tax rate is τt. The tax has two components, a baseline tax rate
τ 0t , and a Leeper tax τ
1
t that responds to the level of government debt. Although
actual social security taxes do not change based on government debt, a Leeper tax
can be thought of as a way to capture legislative unease with increasing debt. I
consider a Leeper tax of the following form:
τt = τ
0
t + τ
1
t (Bt/Ht). (4.8)
where τ 0 ∈ [0, 1] is the baseline tax rate when government debt is zero, τ 1 ≥ 0 is
the incremental tax, and Bt/Ht is government debt per labor hours. Notationally,
a tax rate without a superscript (τt) will refer to the entire pay-roll tax τt = τ
0
t +
τ 1t (Bt/Ht).
3
Social security benefits zjt are paid according to a benefit earning rule:
zjt = φtwt+T−j. (4.9)
3The Leeper tax given by equation (4.8) leaves open the possibility of a total tax rate greater
than one hundred percent if bonds levels are high or if τ1 is large (i.e., τ0t + τ
1
t (Bt/Ht) > 1,
for large Bt/Ht and/or large τ
1). I avoid this by imposing the restriction τ1 is either the
exogenous parameter value chosen by the economist, or the (smaller) parameter value such that
τ0t + τ
1
t (Bt/Ht) = 1. In the policy experiments that follow, the majority of the tax burden
will come from the baseline tax rate τ0, and the Leeper tax τ1 will be used only to calibrate a
dynamically efficient steady state with positive bonds.
116
where zjt represents the benefit paid to a retiree of age j in time t. The parameter
φt is the replacement rate and shows how much of a workers’ wage-indexed average
period earnings (wt+T−j) are replaced by social security benefits.4
The government is not required to balance the social security budget.
If social security taxes are less than social security benefits, the social security
program runs a deficit. The time t deficit is defined as the time t social security
benefits less the pay-roll tax revenue:
Deficitt =
J∑
j=T
Nt+1−jφtwt+T−j −Ht(τ 0t + τ 1t (Bt/Ht))wt
Here Ht indicating the working age population.
The government issuance of bonds is equal to the gross interest on
outstanding debt plus the social security deficit from the previous period.
Bt+1 = RtBt +
J∑
j=T
Nt+1−jφtwt+T−j −Ht(τ 0t + τ 1t (Bt/Ht))wt (4.10)
Markets
There are four markets: labor, capital, bonds, and goods. Prices adjust in
equilibrium to ensure all markets clear.
Labor market clearing requires the total number of hours worked equal the
labor input of the representative firm:
Ht =
T−1∑
j=1
Nt+1−j (4.11)
4There is no technological growth in the model, so this corresponds to the US system.
117
Asset market clearing requires that aggregate capital and bonds next period
are equal to the savings of each cohort:
Kt+1 +Bt+1 =
J∑
j=1
Nt+1−ja
j
t (4.12)
Bond market clearing is ensured by the government’s flow budget constraint,
reprinted below:
Bt+1 = RtBt +
J∑
j=T
Nt+1−jφtwt+T−j −Ht(τ 0t + τ 1t (Bt/Ht))wt (4.10)
Goods market clearing requires that aggregate output is equal to aggregate
consumption and aggregate investment. The goods market clears by Walras law.
The goods market clearing equation is printed below for reference:
F (Kt, Ht) =
T∑
j=1
Nt+1−jc
j
t +Kt+1 − (1− δ)Kt
Along the balanced growth path, cohort size (Nt), labor hours worked (Ht),
output (Yt), capital (Kt), and bonds (Bt) all grow at rate n. Therefore, it will be
convenient to rewrite the market clearing equations (4.10) and (4.12) in per-hours
terms by defining bt = Bt/Ht, and kt = Kt/Ht. I will refer to variables in per-hours
terms as “efficient.”
The capital and bond market clearing equations can be written in efficient
terms as:
(kt+1 + bt+1)(1 + nt) =
∑J
j=1Nt+1−ja
j
t
Ht
(4.13)
(1 + nt)bt+1 = Rtbt +
∑J
j=T Nt+1−jφtwt+T−j
Ht
− (τ 0t + τ 1t (Bt/Ht))wt (4.14)
Rational Expectations Equilibrium
Given initial conditions k0, b0, a
1
−1, · · · aJ−1−1 , and an initial population∑J
j=1(1 + n)
1−jN0 (where N0 is the the initial cohort of young and n is the
118
population growth rate), a competitive equilibrium is a sequences of functions
for the household savings
{
a1t , a
2
t , · · · , aJt
}∞
t=0
, production plans for the firm,
{kt}∞t=1, government bonds {bt}∞t=1, factor prices {Rt, wt}∞t=0, and government policy
variables {τ 0t , τ 1t , φt}∞t=0, that satisfy the following conditions:
1. Given factor prices and government policy variables, individuals’ decisions
solve the household optimization problem (4.1) and (4.2)
2. Factor prices are derived competitively according to (4.6) and (4.7)
3. All markets clear according to (4.11), (4.13), and (4.14)
There are J + 1 equilibrium equations, which hold for time periods t =
0, · · ·∞. The equilibrium equations include the capital clearing equation (4.13), the
bond market clearing equation (4.14), and J − 1 household first order conditions
(4.15). These are reprinted below:
(kt+1 + bt+1)(1 + nt) =
∑J
j=1Nt+1−ja
j
t
Ht
(4.13)
(1 + nt)bt+1 = Rtbt +
∑J
j=T Nt+1−jφtwt+T−j
Ht
− (τ 0t + τ 1t (Bt/Ht))wt (4.14)
(Rta
j−1
t−1 + y
j
t − ajt)−σ = βEt[Rt+1(Rt+1ajt + yj+1t+1 − aj+1t+1)−σ] (4.15)
for j = 1, · · · , J − 1
Here yj indicates period labor income during working life and social security
income after retirement. Note that a0 = 0, and factor prices are given by (4.6)
and (4.7).
The equilibrium definition above accommodates a time-varying population
growth rate nt. In a steady-state, the growth rate of the population is constant.
When the growth rate is constant, several terms can be written concisely by
defining ν = Nt/Ht = (
∑T−1
j=1 (1 + n)
1−j)−1. The steady state is a collection
119
{k, b, a1, · · · , aJ} that solves:
(k + b)(1 + n) = ν
J∑
j=1
(1 + n)1−jaj (4.16)
(1 + n)b = R(k)b+ ν
J∑
j=T
(1 + n)1−jφtw(k)− (τ 0 + τ 1b)w(k) (4.17)
(R(k)aj−1 + yj − aj)−σ = β [R(k)(R(k)aj + yj+1 − aj+1)−σ] (4.18)
for j = 1, · · · , J − 1
with yj = (1− (τ 0 + τ 1b))w(k) for j < T , yj = φw(k) for j ≥ T , aJ = 0, and factor
prices R(k) and w(k) given by (4.6) and (4.7).
The equilibrium of the model is not unique; for many parameter
combinations there are two steady states. The number of steady states depends
on the model parameters and can be characterized by a saddle-node bifurcation.
Chalk (2000) discusses the saddle-node bifurcation in a similar OLG model with
government debt and physical capital.5
The intuition of the bifurcation in this model is that as a given parameter
changes (like the payroll tax rate τ 0 or the population growth rate n), it impacts
the social security deficit or surplus. Suppose the model is calibrated such that
two steady states exist. As the deficit increases (endogenously in response to a
parameter change), this increases government debt and crowds out capital, pushing
the two steady states closer together. At a critical value of the parameter, only
one steady state exists; beyond that, deficits are too large, government debt is
explosive, and no steady states (and no equilibria) exist.
In the exercises that follow, the model will be calibrated such that two
steady states exist. I analyze the determinacy of equilibria by linearizing the model
5See Azariadis (1993) (Chapter 8 and Chapter 20) for a discussion of saddle-node bifurcations
in planar OLG models.
120
around a steady state and computing the eigenvalues of the linearized system (see
Laitner (1990)). There are three predetermined variables in the model (k, b, and
aJ−1) and J-2 free variables (a1,...,aJ−2). There are three possible cases. If three
eigenvalues of the linearized system are less than one in modulus and the remaining
J − 2 have modulus greater than one, the system is determinate. If more than three
eigenvalues lie inside the unit circle, the system is indeterminate. If more than J−2
eigenvalues lie outside the unit circle, the system is explosive. I have confirmed
numerically that when two steady states exist, the steady state with higher capital
is stable (determinate), and the lower capital steady state is explosive. I will focus
only on the determinate solutions in this paper.6
Life-cycle Horizon Learning
Under the rational expectations hypothesis (RE), households make optimal
savings and consumption choices given their (rational) forecasts of future prices
and policy. Households make decisions at age zero looking forward over their entire
life cycle. These rational agents are fully forward looking, and consider every stage
of the life-cycle when making decisions. Under RE agents know the underlying
equations that govern the economy and form expectations of future variables using
the mathematical expected value.
This paper backs away from RE and proposes a learning model in which
agents combine limited knowledge about the structure of the economy with
adaptive forecasts for future macroeconomic aggregates. As a baseline, I consider
agents who do not have perfect foresight over factor prices. Thus, agents do not
know the future path of wages or interest rates over their life-cycle. Agents use
6Chalk (2000) also finds that the high capital steady state is determinate and the low capital
steady state is explosive in a many-period OLG model with debt and capital.
121
adaptive expectations to forecast future prices, but behave optimally in all other
ways. Agents update their forecasts as more information becomes available. I call
this behavioral assumption life-cycle horizon learning (LCH learning).
Under LCH learning, households are forward looking and make optimal
savings and consumption choices given their (adaptive) forecasts of future
macroeconomic aggregates. Adaptive expectations may be viewed as a special case
of adaptive learning suitable for non-stochastic models. The adaptive expectations
assumption in this paper is analogous to constant gain least squares learning in a
model with random shocks (like a productivity shock). LCH learning is similar to
infinite-horizon learning, developed in Marcet and Sargent (1989) and emphasized
by Preston (2005 and 2006). The key difference is that infinite-horizon learning
models are based on a representative agent who lives for an infinite number of
periods. LCH learning applies the same forward looking behavior to finitely lived
agents in an overlapping generations life-cycle model.7 Throughout this paper, I
assume homogenous expectations across agents.
Agents in the LCH learning model forecast wages and interest rates using
adaptive expectations of the following form:
wet+1 = γwt + (1− γ)wet (4.19)
Ret+1 = γRt + (1− γ)Ret (4.20)
with γ ∈ (0, 1). Here we indicates expected wage, and Ret indicates expected
interest rate. Agents also form expectations at time t of prices in period t + j for
7LCH learning is similar to the model developed in Bullard and Duffy (2001), in which agents
forecast inflation over a finite life-cycle and make optimal choices based on those forecasts.
122
j > 1:
wet+j = w
e
t,t+j = w
e
t+1 (4.21)
Ret+j = R
e
t,t+j = R
e
t+1 (4.22)
When the Leeper tax is non-zero (τ 1 > 0), then agents need to know future
bond levels in order to estimate their after-tax pay. For simplicity, I assume that
agents forecast bonds using the same adaptive learning rule they use to forecast
prices.
bet+1 = γbt + (1− γ)bet (4.23)
bet,t+j = b
e
t,t+1 for j > 1 (4.24)
Agents enter the model with knowledge of the previous period’s prices, bonds,
and expectations. At any moment in time, all agents in the model have the same
expectation for future prices and bonds. This assumption is equivalent to assuming
agents inherit expectations from the previous generation. I consider alternative way
for agents to forecast their income net of taxes in section IV.
LCH learning can be viewed as a small deviation from rational expectations
in the sense that agents are still forward looking and only use a different rule to
forecast future aggregates. Adaptive expectations are plausible in a world in which
the true data generating process is complex. Adaptive expectations are optimal
if agents think that capital follows an IMA(1,1) process. That is, expectations
of the form given in Equation (4.19) are rational if the change in capital has the
following form ∆kt = t + θt−1 for shocks  and parameter θ ∈ (0, 1). The use of
adaptive expectations is equivalent to agents believing that capital has a mixture
of permanent and transitory shocks (See Muth (1960)). The gain parameter, γ, has
a natural interpretation in this context. If all shocks are transitory, the optimal
123
forecasting rule is constant, i.e., γ=0. In contrast, if all shocks are permanent, then
the best forecasting rule is a random walk, i.e., γ = 1. In this particular model, all
shocks are permanent, but agents are not endowed with this knowledge. I choose
the gain parameter used in simulations to minimize the welfare cost of agents
of inaccurately forecasting along the transition path. I explore alternative gain
parameters as robustness checks.8
In the baseline LCH learning model, agents have full knowledge of
government policy. Agents know the future path of social taxes and the benefit
replacement rate if there is no policy uncertainty. If future policy is stochastic,
agents form expectations of future policy parameters using rational expectations.
Agents know the finite set of policy parameters (and/or reform dates) and the
relative probability of each. I relax this assumption and have agents learn about
future government policy adaptively in section IV.
A young agent in the LCH learning model chooses first period savings and
consumption and plans future savings and consumption to satisfy her J − 1 first
order equations and her lifetime budget constraint. The young agent’s time t
plan of second period savings is denoted a2t,t+1. In the following period, her time
t + 1 actual choice of second period savings is given by a2t+1. Her time t + 1 plan
does not have to be consistent with her time t choice. She can update her savings
decision based on the new information she receives in period t+1. Abstracting from
policy uncertainty (assuming the path of taxes and social security benefit rates are
8Evans and Ramey show that by appropriately tuning the free parameters of the forecast rule
agents can obtain the best forecast rule within a given class of underparameterized learning rules
(G. Evans and Ramey (2006)).
124
predetermined), the young agent in time t solves:
(Rt+j−1a
j−1
t+j−2 + y
j
t+j−1 − ajt+j−1)−σ = βRet+j(Ret+jajt+j−1 + ye,j+1t+j − aj+1t,t+j)−σ (4.25)
for j = 1, · · · , J − 1
with ye,j+1 indicating expected period labor income or social security income
ye,j+1t+j = (1− τt+j)wet+j for j < T
ye,j+1t+j = φt+jw
e
t+j+T−j for j ≥ T
Similarly, an agent of age two solve J − 2 first order equations, for the
remaining J − 2 periods of her life-cycle. In total, the J cohorts alive in any period
solve
∑J−1
j=1 j =
(J−1)J
2
first order conditions. Together, the decisions of households
of all ages, asset market clearing (4.13), and government bonds (4.14), and the
expectation equations (4.19), (4.20), and (4.23), create a recursive system that
governs the dynamics of the economy.
Finite Horizon Life-cycle Learning
This paper proposes an additional new learning model for finitely lived
agents who form expectations over a short horizon (less than or equal to the length
of their lifecycle) called finite horizon life-cycle learning (FHL). This learning
model is similar to N-step ahead optimal learning, developed by W. Branch, Evans,
and McGough (2013). In both cases, agents make optimal choices based on their
forecast of future prices (and potentially other macro variables) and their budget
constraint over a finite horizon. Agents using FHL learning look forward for a fixed
number of periods and make decisions based on this short-planning horizon. They
update their choices each period as their planning horizon moves forward and they
receive new information.
125
In order to solve the FHL model model, it is necessary to specify a terminal
condition for asset holdings at the end of the planning horizon. When the planning
horizon equals the end of the life-cycle, the terminal condition is simply that the
agent dies without debt (and thus chooses optimally to die with zero assets).
When the planning horizon is shorter than the life-cycle, it is less obvious what
the terminal condition should be.
One simple alternative is to impose that assets are non-negative at the end
of the planning horizon. Under this assumption, agents will choose to hold zero
assets at the end of the planning horizon. This assumption drives the main results
in the short-planning horizon literature and leads to time-inconsistent behavior (see
Caliendo and Aadland (2007) and Park and Feigenbaum (2017)). This assumption
is somewhat unsatisfactory, because agents treat periods beyond the planning
horizon as if they did not exist. An alternative assumption is to have agents hold
wealth at the end of the planning horizon. As a baseline, I assume that agents plan
to hold the same amount of wealth at the end of the planning horizon that older
cohorts held at the same age. Specifically, agents adaptively forecast the steady-
state value of age-specific wealth at the end of their planning horizon.
As in the LCH learning model, agents using FHL learning forecast future
factor prices and bonds according to (4.19) - (4.23). FHL learners also need to
forecast a terminal wealth holding for the end of the planning horizon. I assume
that agents forecast
aj,et,terminal = γa
j
t−1 + (1− γ)aj,et−1,terminal (4.26)
for j = 1, · · · , J − 1
126
Here, aj,et,terminal is amount of asset an age j − 1 agent expects to hold at the end of
the period when they are age j. This asset amount is based on the observed asset
holding of age j agents from last period, and the forecast from last period.
As an example, suppose the planning horizon length is three. That is, agents
looks forward three periods when making decisions. A young agent chooses first
period savings and consumption, and plans saving and consumption for period two
and three, using three first order equations and a terminal condition:
(y1t − a1t )−σ = βRet+1(Ret+1a1t + ye,2t+1 − a2t,t+1)−σ (4.27)
(ye,2t+1 − a2t,t+1)−σ = βRet+2(Ret+2a2t,t+1 + ye,3t+2 − a3t,t+2)−σ (4.28)
(ye,3t+2 − a3t,t+2)−σ = βRet+3(Ret+3a3t,t+2 + ye,4t+2 − a4t,terminal)−σ (4.29)
Older agents follow a similar pattern. Agents who have three or fewer periods of
life remaining using the standard terminal condition aJ = 0.
The FHL learning model can be written as a recursive system. The system
includes the first order equations for the households, asset market clearing (4.13),
and government bonds (4.14), and the expectation equations (4.19) - (4.23) and
(4.26). For a planning horizon of length H, there will be J −H terminal conditions
and H(J −H) + H(H−1)
2
household first order equations. When the planning horizon
is equal to the life-cycle, the system is identical to the life-cycle horizon learning
model.
The shortest possible planning horizon is one. Agents who only look forward
one period make their savings consumption decision based on their Euler equation,
their budget constraint, and the assets they believe they need to hold at the end
of the next period (the terminal condition). W. Branch et al. (2013) call this type
of learning one-step-ahead optimal learning and compare and contrast it to Euler-
equation-learning, a behavioral primitive in which agents do not explicitly consider
127
Parameter Value
α Capital share of income 1
3
β Discount factor ( 1
1+0.005
)10
σ Inverse elasticity of substitution 1
δ Depreciation 1− (1− 0.10)10
n Population growth rate 0.01
A TFP factor 10
TABLE 14. Baseline parameters for 6 period model
their transversality condition. I plan to explore Euler equation learning in an OLG
framework in future work.
Calibration
Agents enter the model at age 25 and live for six periods (J = 6). Each
period is 10 years, and agents die with certainty at age 85. The capital share of
income, discount factor, inverse elasticity of substitution, and depreciation rate are
all set to standard parameter values9. The total factor productivity parameter A is
arbitrarily set to 10 (results do not depend on this assumption). For the majority
of the exercises in this paper, the population growth rate is 0.01. The parameter
values are listed in Table 14.
The SSA estimates the current ratio of social security beneficiaries to
workers to be 0.35 (as of the 2017 Trustee Report). This ratio is expected to
increase to 0.46 by 2035, and to 0.5 by 2095 under the medium cost assumptions.
The increase in the ratio of beneficiaries to workers is driven by both increasing
lifespans for retirees, and declining birthrates for working generations. I abstract
from these details in the model, and capture all population changes using the
9See, for example, W. Branch et al. (2013). My discount factor β is slightly closer to one and
the depreciation rate is slightly higher. I choose the higher discount factor and depreciation rate
to better match the size of tax increase necessary to ensure long-run solvency of social security
128
1980 2000 2020 2040 2060 2080
0.3
0.4
0.5
0.6
calendar year
beneficiares
to workers
intermediate
high
low
model
FIGURE 23. Ratio of beneficiaries to retirees. The red link in this graph illustrates
the ratio of beneficiaries to retirees in the calibrated, six period model. The
demographic change is modeled as a one-time reduction in the population growth
rate n from 0.1802 to 0.01. The yellow, blue, and green lines show the high,
intermediate, and low cost projections of the ratio from the 2017 SSA Trustees’
Report. The model calibration closely tracks the intermediate assumptions from the
SSA.
growth rate n. For the majority of the exercises in section IV, I will choose n such
that the ratio of retirees to workers is 0.3 and then increases gradually to 0.485, as
illustrated in Figure 23.
In the learning models, I set the gain parameter γ = 0.93. This gain
parameter minimizes the maximum welfare cost to an agent of using adaptive
forecasts in the LCH model along the transition path that includes a demographic
shock and a change to social security. I construct a consumption equivalent
variation that compares the utility of consumption of each cohort of agents in the
LCH model to the utility of consumption for “infinitesimally rational” cohorts of
agents. An infinitesimally rational agent is able to predict future prices with perfect
foresight in the LCH world, but is such a small part of the market that she does
not change prices. I compute the consumption equivalent variation that makes an
infinitesimally rational agent indifferent between her own consumption and the
consumption of the life-cycle horizon learners. I chose the gain parameter γ to
129
minimize the maximum welfare cost. It is unsurprising that the gain parameter
is close to γ = 1, since all shocks in the model are permanent. The simulation
results are qualitatively sensitive to the gain parameter, and I explore alternative
parameterizations as robustness checks.
Applications
Announced Social Security Reform
The aging of society put pressure on the social security system. In the
absence of reform, the government would be required to fund social security
benefits by issuing debt. There is a limit to how much the government can borrow,
even in a dynamically inefficient economy. I illustrate these dynamics in the
following example.10 11
Suppose that the government runs a social security program that generates
a small surplus (government assets); however, as the population ages, the operating
surplus changes to a deficit. The government continues to operate the same social
security system which depletes the government assets and causes the government to
accrue debt. After a number of periods, the government reforms social security by
cutting benefits or raising taxes.
I calibrate this example to correspond to the US system. The initial
population growth rate is n = 0.1802 and falls to n = 0.01 in 1980. This generates
a smooth transition in the ratio of retirees to workers from 0.3 to 0.485 after six
10The government could also fund social security benefits using tax revenue from some other
existing tax. For the sake of this paper, I will argue that diverting other federal tax dollars into
the pension system is isomorphic to directly increasing social security taxes, and is thus reform.
11See Chalk (2000) for a discussion of maximum government deficits in a dynamically inefficient
economy.
130
periods (which is near SSA projections). The initial policy mimics the current US
system: the payroll tax rate is set to τ 0 = 0.124 and the benefit replacement rate is
φ = 0.4. Reform is calibrated as either a tax increase to τ 0 = 0.1516, or a benefit
cut to φ = 0.332. The SSA estimates these reforms would eliminate the funding
shortfall in the social security system. In order to have a dynamically efficient
economy with positive government debt, the Leeper tax rate is set to τ 1 = 0.045
before and after reform.
Figure 24 plots the path of capital and bonds for this tax increase reform.
The plot includes paths for an economy with Rational Expectations (in green),
and for an economy with LCH Learning. Government debt increases before the
reform takes place and then converges to the new higher steady state value. Debt
is increasing since the government is running a deficit in the social security system.
Debt would become explosive, were it not for the reform in the year 2030. The
reform is announced, so agents adjust their behavior before the reform date.
Capital increases initially as saving goes up, and then falls to the steady state.
Agents in the learning model overestimate the interest rate and
underestimate the wage in the first few periods following the change in the
population growth rate (which moves the economy away from the initial steady
state). This is depicted in Figure 25. The expectations adjust quickly because the
gain parameter γ is so large. In this example, the gain is set to 0.93.
The agents in the learning model save more relative to the fully rational
agents in the initial periods, since their estimate of prices and bonds is inaccurate.
The savings choices of agents are depicted in Figure 26. 1980 is the first period
with the lower growth rate. Agents with rational expectations do not make (large)
changes in their savings behavior if they know they will not be alive for the reform.
131
1950 2000 2050 2100 2150
7.5
8.0
8.5
9.0
9.5
time
capital
k
reformdem shock
1950 2000 2050 2100 2150
0.0
0.5
1.0
1.5
time
bonds
b
reformdem shock
Rational Expectations
LCH learning
FIGURE 24. Time paths for capital and bonds for RE (green) and LCH (yellow)
for an announced tax increase. The initial population growth rate n is 0.1802 and
falls to 0.01 in 1980. Demographic changes drive increasing debt until tax increase
in 2030. Initial government policy is τ 0, τ 1, φ equal to (0.124, 0.045, 0.4)’. The
policy change increases τ 0 to 0.1516. The gain parameter γ = 0.93 for this example.
This explains the spikes in savings along the rational expectations paths leading up
to reform. The initial decline in savings along the RE paths are due to the changes
in the capital and bonds.
The agents in the model with learning over-save relative to the rational
model, since they do not know future prices. Because the agents initially over-save,
they are able to save less later in life relative to their plan from the first period.
This is evident in Figure 27 by comparing the young (age 1) agent’s planned asset
holdings for future periods against the actual savings choice she makes when she
reaches those periods. For the first several time periods, the agent’s planned savings
are above the actual savings. The planed savings and actual savings of older agents
follows a similar pattern.
132
1950 2000 2050 2100 2150
1.10
1.15
1.20
time
Interest
rate
Ret+1
reformdem shock
1950 2000 2050 2100 2150
12.8
13.0
13.2
13.4
13.6
13.8
14.0
time
wage
wet+1
reformdem shock
1950 2000 2050 2100 2150
0.0
0.5
1.0
1.5
time
bonds bet+1
reformdem shock
expectation
data
FIGURE 25. Time paths for expected interest rates, wages, and government debt
levels (blue lines), and realized interest rates, wages, and debt (yellow lines) in the
LCH model with an announced tax increase. Agents overestimate the interest rate
and underestimate the wage in the initial periods as they learn that capital and
debt are increasing. The initial population growth rate is 0.1802 and falls to 0.01
in 1980. Demographic changes drive increasing debt until tax increase in 2030.
Initial government policy is τ 0, τ 1, φ equal to (0.124, 0.045, 0.4)’. The policy change
increases τ 0 to 0.1516. The gain parameter γ = 0.93.
133
1950 2000 2050 2100 2150
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
time
age 1
assets
a1
reformdem shock
1950 2000 2050 2100 2150
5.5
6.0
6.5
7.0
7.5
time
age 2
assets
a2
reformdem shock
1950 2000 2050 2100 2150
8.0
8.5
9.0
9.5
10.0
10.5
time
age 3
assets
a3
reformdem shock
1950 2000 2050 2100 2150
10.0
10.5
11.0
11.5
12.0
12.5
time
age 4
assets
a4
reformdem shock
1950 2000 2050 2100 2150
6.5
7.0
7.5
time
age 5
assets
a5
reformdem shock
Rational Expectations
LCH learning
FIGURE 26. Time paths for the savings accounts for RE (green) and LCH (yellow)
models with an announced tax increase. Demographic changes drive increasing debt
until tax increase in 2030. The initial population growth rate is 0.1802 and falls to
0.01 in 1980. Initial government policy is τ 0, τ 1, φ equal to (0.124, 0.045, 0.4)’. The
policy change increases τ 0 to 0.1516. The gain parameter γ = 0.93.
134
1950 2000 2050 2100 2150
6.0
6.5
7.0
time
a2t,t+1
age 2
assets
1950 2000 2050 2100 2150
8.5
9.0
9.5
10.0
10.5
time
a3t,t+2
age 3
assets
1950 2000 2050 2100 2150
10.5
11.0
11.5
12.0
12.5
time
a4t,t+3
age 4
assets
1950 2000 2050 2100 2150
6.0
6.5
7.0
7.5
time
a5t,t+4
age 5
assets
age 1 expectation
data
FIGURE 27. This graph shows planned future savings of the period in blue, and
actual future savings in yellow. The expectations (or plans) are made when the
agent in young (age 1) in period t. The initial population growth rate is 0.1802 and
falls to 0.01 in 1980. Demographic changes drive increasing debt until tax increase
in 2030. Initial government policy is τ 0, τ 1, φ equal to (0.124, 0.045, 0.4)’. The
policy change increases τ 0 to 0.1516. The gain parameter γ = 0.93.
135
Announced Change, Finite Horizon Life-cycle Learning
Agents using FHL learning only make decisions based on their forecasts
over their short planning horizon. Agents forecast prices, bonds, and the assets
the expect to hold at the end of the planning horizon. Over time, agents learn
the steady state values of each of these quantities. If agents’ estimates for the
assets they’ll need at the end of the planning horizon are close to the rational
expectations (steady state) values, they make relatively better decisions using a
shorter planning horizon than using a longer horizon. This is because agents using
a longer horizon forecast over more periods and can make larger mistakes. This
is evident by comparing the transition dynamics of different planning horizon
lengths in the FLH model for the a social security tax increase in response to
a demographic shock. The fluctuations are larger for longer planning horizons
and smaller for shorter planning horizons, as depicted in Figure 28. The figure
depicts the time path for capital and bonds for a tax increase. The time paths for
asset holdings display cyclical behavior, with the largest cycles associated with
the longest planning horizon. As in the previous section, this policy experiment
replicates the tax increase suggested by the SSA. The initial population growth rate
is n = 0.1802 and rate falls to n = 0.01 in the year 1980. Initial government policy
is τ 0 = 0.124, τ 1 = 0.045, and φ = 0.4. The policy change increases τ 0 to 0.1516 in
the year 2030.
Agents in the FHL learning model only respond to policy changes when the
announced policy enters their planning horizon. Agents making one step ahead
forecasts will not respond to announced policy until the period right before the
change. Despite this fact, FHL leads to smaller aggregate fluctuations, since agents
136
1950 2000 2050
7.5
8.0
8.5
9.0
9.5
time
capital
k
reformdem shock
1950 2000 2050
0.0
0.5
1.0
1.5
time
bonds
b
reformdem shock
Rational Expectations
life cycle planning horizon
planning horizion 4
planning horizion 3
planning horizion 2
planning horizion 1
FIGURE 28. Time paths for capital and bonds for RE (blue), LCH (yellow), and
FHL (planning horizon 4: green, 3: red, 2: purple, 1: brown ) for an announced
tax increase. Demographic changes drive increasing debt until tax increase in the
year 2030. The initial population growth rate is 0.1802 and falls to 0.01 in 1980.
Initial government policy is τ 0, τ 1, φ equal to (0.124, 0.045, 0.4)’. The policy change
increases τ 0 to 0.1516. γ=0.93 for all learning rules. The red dashed lines indicate
the steady state following the policy change.
using a short planning horizon do not make inaccurate forecasts many periods into
the future.
Welfare
Welfare Cost of Social Security Reform
The demographic changes and social security reforms proposed in the
previous sections harm some generations and help others. I illustrate these effects
by using a Consumption Equivalent Variation (CEV) measure. This CEV measure
shows the percent of period consumption that would have to be added to an agent
in the initial steady state in order to be indifferent between the steady state and
being born at a particular time along a particular transition path. A negative CEV
indicates that the agent would have higher utility in the initial steady state. The
137
CEV is calculated as:
J∑
j=1
βj−1u(cj,ss(1 + ∆)) =
J∑
j=1
βj−1u(cj,t+j−1) (4.30)
where cj,ss indicates age j consumption in the initial steady state, cj,t+j−1 is
consumption of an agent with age j in period t+j−1 along a given transition path,
u(c) is the period utility function as defined by (4.5), and ∆ is the Consumption
Equivalent Variation.
The initial steady state is used as a baseline for comparison to capture both
the effect of the demographic changes and the social security reform. Note that the
initial steady state consumption is not possible after the initial period because of
demographic changes. It is used only as a reference point to compare the relative
harm of the different learning rules. The welfare cost of an announced social
security benefit cut is presented in Figure 29 and the welfare cost of an announced
tax increase is presented in Figure 30. In both examples the economy begins in
a steady state with a ratio of retirees to workers of 0.3. In 1980, the population
growth rate falls to 0.01, which leads the ratio of retirees to workers to increase
to 0.485 over six periods to correspond with the aging of the US population. The
baseline payroll tax rate is set to τ 0 = 0.124 and the benefit replacement rate is
φ = 0.4. The benefit cut reform lowers the replacement rate to φ = 0.332, while
the tax increase reform increases the payroll tax to τ 0 = 0.1516. Throughout both
examples the Leeper tax rate is set to τ 1 = 0.045 before and after reform, and the
gain parameter γ = 0.93.
The underlying demographic changes in these examples harm agents. The
model is calibrated to be dynamically efficient with a social security system about
the size of the US system, as the norm in this literature. Slowing population
138
1950 2000 2050 2100
-0.08
-0.06
-0.04
-0.02
0.00
cohort birth year
CEV
reformd shock
rational expectations
life cycle planning horizon
planning horizion 4
planning horizion 3
planning horizion 2
planning horizion 1
FIGURE 29. Consumption equivalent variation measure of the welfare cost of an
announced social security benefit cut. he initial population growth rate is 0.1802
and falls to 0.01 in 1980. Initial government policy is τ 0, τ 1, φ equal to (0.124,
0.045, 0.4)’. This drives the need for social security reform. In 2030, the benefit
replacement rate falls to φ=0.332. The gain parameter γ=0.93. The cost of the
reform is highest (lowest CEV) for agents in the LCH learning model born in
2030; agents in the initial steady state would have to give up 8.88% of their period
consumption to be indifferent between living in the initial steady and being born in
2030 in the LCH model.
growth makes the economy even more dynamically efficient, which lowers
consumption (and welfare) for all generations.
The demographic changes combined with the benefit cut lower welfare for
agents in all cohorts with all learning rules (and also in the RE model). The CEV
for the final steady state compared to the initial steady state is -7.29% of period
consumption. That is, agents in the initial steady state would have to give up
7.29% of period consumption to be indifferent between the initial steady state and
being born in the final steady state. The welfare cost is greatest for generations
born right after the social security reform. The CEV reaches a minimum of -8.86%
for the LCH model agents born in 2030 (the minimum is slightly smaller when
comparing the RE model). The welfare cost is decreasing in gain parameter γ; if
the gain is decreased to 0.35, the minimum CEV is -9.63% and occurs in the LCH
model for the cohort born in 2040. The timing of the welfare cost is also sensitive
to the gain parameter, since larger gain parameters producer higher frequency
139
1950 2000 2050 2100
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
cohort birth year
CEV
reformd shock
rational expectations
life cycle planning horizon
planning horizion 4
planning horizion 3
planning horizion 2
planning horizion 1
FIGURE 30. Consumption equivalent variation measure of the welfare cost of an
announced social security tax increase. he initial population growth rate is 0.1802
and falls to 0.01 in 1980. Initial government policy is τ 0, τ 1, φ equal to (0.124,
0.045, 0.4)’. In 2030, the payroll tax rate increases to τ 0=0.1516. γ=0.93 for this
example. The cost of the reform is highest (lowest CEV) for agents in the LCH
learning model in 2030; agents in the initial steady state would have to give up
9.96% of their period consumption to be indifferent between living in the initial
steady and being born in 2030 in the LCH model.
cycles. The faster cycle moves the minimum capital stock a few periods ahead and
thus the most harmed cohort is earlier.
Figure 30 depicts the CEV for the tax increase. The demographic changes
combined with the tax increase lower welfare for all cohorts in the RE and learning
models. The CEV in the final state is -9.49%, indicating agents in the initial steady
state would have to give up 9.49% of their consumption to be indifferent between
the initial steady state and being born in the final steady state. The welfare cost is
largest in the LCH model for the cohort born in 2030. The CEV is -9.96% for this
cohort. The magnitude of CEV for the learning models is decreasing in gamma. If
γ is decreased to 0.35, the min CEV is -11.91% for the most harmed cohort.
Welfare Cost of Learning
Agents using finite or life-cycle horizon learning don’t fully realize the
impact of a policy change or other exogenous shock. It take the agents many
periods to learn the new steady state values. Converge to the new steady state
140
is slower in the learning models than the RE model. This section measures the
welfare cost of the learning models relative to the rational expectations baseline.
This welfare comparison will net out the cost of changing demographics and the
change in terminal steady state values due to the new social security policy. By
comparing consumption in the learning model to consumption in the RE model,
this measures the cost of the learning directly.
As in the previous section, welfare is measured using a Consumption
Equivalent Variation measure. Here, the CEV measure shows the percent of period
income that would have to be added to consumer with rational expectations (in
the RE model) in order to be indifferent between living in the RE world or living
in the world with learning in the same cohort. A negative CEV indicates that the
agent would have higher utility in the world with rational expectations. The CEV
is calculated as:
J∑
j=1
βj−1u(cj,REt+j−1(1 + ∆RE)) =
J∑
j=1
βj−1u(cj,Lt+j−1) (4.31)
where cj,REt+j−1 indicates age j consumption in period t + j − 1 in the model with
rational expectations, cj,Lt+j−1 is consumption in the model with learning, u(c) is the
period utility function as defined by (4.5), and ∆RE is the Consumption Equivalent
Variation.
Figures 31 and 32 show the CEV for the tax increase and benefit cut
examples (parameterized as in the previous sections). In contrast to the previous
section, the CEV is not always negative. That is because here the CEV measures
learning compared to rational expectations. After many periods, the learning
models converge to the RE model, so the CEV converges to zero.
Figure 31 illustrates the welfare cost of the tax increase in the learning
models relative to the RE baseline. For the LCH model, the welfare cost is greatest
141
1950 2000 2050 2100
-0.015
-0.010
-0.005
0.000
0.005
0.010
cohort birth year
CEV
reformd shock
life cycle planning horizon
planning horizion 4
planning horizion 3
planning horizion 2
planning horizion 1
FIGURE 31. Consumption equivalent variation measure of the welfare cost of an
announced social security tax increase: learning models compared to RE. he initial
population growth rate is 0.1802 and falls to 0.01 in 1980. Initial government policy
is τ 0, τ 1, φ equal to (0.124, 0.045, 0.4)’. In 2030, the payroll tax rate increases to
τ 0=0.1516. The gain parameter γ=0.93. The cost of the reform is highest (lowest
CEV) for agents in the LCH learning model born in 2040; agents in the RE model
would have to give up 1.20% of their period consumption to be indifferent between
being born in the RE model in 2040 and being born in 2040 in the LCH model.
for the tax increase in the year 2040; agents in the RE model born in 2040 would
be willing to give up 1.20% of their consumption to avoid being born in 2040 in the
LCH model.12
Figure 32 illustrates the welfare cost or the benefit cut in the learning
models relative to the RE baseline. In the LCH model, the welfare cost is greatest
for the cohort born in the year 2030. Agents in the rational model would be willing
to give up 0.81% of their consumption to avoid living LCH world in the same time
periods.13
Many cohorts benefit from the cyclical dynamics introduced via learning.
Agents benefit when the capital stock is higher wages are higher. Consider 1-step-
ahead FHL learning in the context of the social security benefit cut presented
in Figure 32. Cohorts born prior to the reform and many cohorts born after
the reform have higher utility in the FHL model than in the RE model. Indeed,
12When γ = 0.35 the min CEV is -3.38% and occurs for the cohort born in 2040.
13When γ = 0.35 the min CEV is -2.43% and occurs for the cohort born in 2040.
142
1950 2000 2050 2100 2150
-0.015
-0.010
-0.005
0.000
0.005
0.010
cohort birth year
CEV
reformd shock
life cycle planning horizon
planning horizion 4
planning horizion 3
planning horizion 2
planning horizion 1
FIGURE 32. Consumption equivalent variation measure of the welfare cost of an
announced social security benefit cut: learning models compared to RE. The initial
population growth rate is 0.1802 and falls to 0.01 in 1980. Initial government policy
is τ 0, τ 1, φ equal to (0.124, 0.045, 0.4)’. In 2030, the benefit replacement rate falls
to φ=0.332. The gain parameter γ=0.93. The cost of the reform is highest (lowest
CEV) for agents in the LCH learning model born in 2030; agents in the RE model
would have to give up 0.81% of their period consumption to be indifferent between
being born in the RE model in 2030 and being born in 2030 in the LCH model.
the cumulative welfare gain across cohorts can be positive. At first this may
seem like a violation of the Welfare Theorems, but it is not. Bounded rationality
does not present a Pareto improvement over the RE baseline. The learning
dynamics improve the welfare for some agents who enjoy higher wages and higher
consumption due to the increased capital stock from higher saving.
Markets exhibit a pecuniary externality in general equilibrium. Higher
savings today raises capital and wages tomorrow. Agents can exploit this pecuniary
externality in OLG economies, if they can coordinate behavior across generations.
Feigenbaum, Caliendo, and Gahramanov (2011) introduce the concept of “optimal
irrationality” in OLG economies. They show that an irrational consumption rule
always exists that can weakly improve upon the lifecycle/permanent-income rule
in general equilibrium in a two-period model. They present a calibrated continuous
time example with an escalating rule-of-thumb savings rule that increases welfare
relative to rational expectations (for all but the first generation). They argue
that policies that increase savings in a dynamically efficient economy can improve
143
welfare for many (but not all) generations. The welfare gains to some generations
in my model come from the same externality.
The majority of the welfare cost (or welfare gain) of living in a FHL or LCH
learning model come from the general equilibrium changes in the capital stock, not
from the individual forecast errors made by an agent. One way to measure this cost
is to compute the life-cycle consumption for an infinitesimally rational agent who
lives in a learning model.
Suppose a single rational agent is born in the LCH model. The rational
agent understands that everyone else is using LCH learning, and she is also able to
predict future prices with perfect foresight. She is such a small part of the market
that her individual choices do not changes prices. I compute the CEV that equates
the utility of the infinitesimally rational agent with the utility of the LCH agent.
The minimum CEV for this comparison is -0.17% for the benefit cut reform (-0.30%
for the tax increase reform). The CEV is less than 0.2% in magnitude for the first
six periods (driven mainly by the demographic change), and then quickly falls to
zero. The negative sign indicates that LCH agents are worse off than the rational
agent living in a LCH world. This CEV is presented below in Figure 33.
The welfare cost of learning can also be explored in the context of a
recession. Agents using LCH or FHL learning will fare worse during and after a
recession than fully rational agents. This is because the learning agents will not
anticipate the general equilibrium effects of the initial decline in production. An
example recession is presented in Appendix C.
144
1950 2000 2050 2100
-0.0020
-0.0015
-0.0010
-0.0005
0.0000
time
CEV
reformdem shock
FIGURE 33. Consumption Equivalent Variation (CEV) measure of the welfare
cost of using adaptive learning along the transition path of an announced social
security benefit cut. The CEV compares an infinitesimally rational agent to a life-
cycle horizon learner. The infinitesimally rational agent is effected by the general
equilibrium effects of learning, but it able to predict future prices. The initial
population growth rate is 0.1802 and falls to 0.01 in 1980. Initial government policy
is τ 0, τ 1, φ equal to (0.124, 0.045, 0.4)’. In 2030, the benefit replacement rate falls
to φ=0.332. The gain parameter γ=0.93.
Welfare Costs of Policy Uncertainty
The policy changes of the previous sections are non-stochastic and
announced. However, the future of the U.S. social security system seems difficult
to anticipate. A growing body of research focuses on the welfare cost of this type of
uncertainty.14 I calculate the welfare cost of policy uncertainty under RE and under
LCH learning. To do this, I model government policy as an exogenous, stochastic
process.
Let ωt = (τ
0
t , τ
1
t , φt)
′ describe the government policy parameters at time t.
Suppose that the social security program will be reformed at some future date. The
reform date falls within a known, finite set. Suppose that the realization of ωt+1
depends on ωt, and is contained in the finite set Ω. Suppose also that each possible
reform converges to a steady state (the path is non-explosive).
14I follow Caliendo, Casanova, Gorry, and Slavov (2016) and use the words “uncertainty” and
“risk” interchangeably in this paper. All of the examples I will consider have known probabilities,
and thus might be called “risky.”
145
Let the probability of realizing a particular value of ωt+1 given ωt be
described by pi(ωt+1|ωt). Using this notation, the expected value in the household
first order equations (4.4) can be written as:
u′(cjt+j−1) = β
∑
ωt+j∈ Ω
pi(ωt+j|ωt+j−1)Rt+ju′(cj+1t+j ) for j = 1, · · · , J − 1 (4.32)
Introducing aggregate uncertainty to the model raises some concerns; a
steady state wealth distribution will not exist in general in an OLG model with
aggregate uncertainty (see Krueger and Kubler (2004) for a discussion). I overcome
this problem by modeling aggregate policy uncertainty in a simple, stylized way.
Policy uncertainty only exists for a small number of periods and policy reform can
only be one of a small number of policy options. These simplifications, combined
with the calibration of the life-cycle as six periods, allow me to calculate the
equilibrium paths for the economy with policy uncertainty.
Suppose that policy uncertainty take the following form: reform is possible
in either date S, or in date S + 1. Within each period two reforms are possible,
either a benefit cut or a tax increase. Thus, there are four possible paths for the
economy. The probability of each path is p = 0.25. All agents in the economy know
the four possible reforms and their relatively probability. I calibrate this example
to correspond to the US system. The economy starts in a steady with a ratio of
retirees to workers of 0.3 and with government policy: τ 0 = 0.124, τ 1 = 0.045, and
φ = 0.4. In 1980, the growth rate of the population falls to 0.01 leading the ratio
of retirees to workers to grow, eventually reaching 0.485. The demographic change
causes the social security system to run a deficit, which increases government debt.
Reform is calibrated as either a tax increase to τ 0 = 0.1516, or a benefit cut
to φ = 0.32. The reform takes place in either date 2030 or 2040. The SSA estimates
these reforms would eliminate the funding shortfall in the social security system. In
146
order to have a dynamically efficient economy with positive government debt, the
Leeper tax rate is set to τ 1 = 0.045 before and after reform. The gain parameter for
the learning model is set to γ = 0.93.
Before considering the welfare cost of reform, first consider the transition
dynamics of the four possible equilibrium paths in the RE model. The four possible
paths for the economy are identical up to 2020. Agents in (or before) period five
face the same uncertainty about future social security policy. This uncertainty is
either fully or partially resolved in 2030. Along the first two paths of the economy,
agents observe that policy was reformed in 2030 (either as a tax increase or benefit
cut). Uncertainty is resolved for these two paths. Along the other two paths, agents
observe that policy was not reformed in 2030, so they know reform will take place
in 2040, but they do not which reform until the policy is realized. In 2040, all
uncertainty is resolved, and all four paths are (potentially) different. There are
two possible steady states for this example, the tax increase steady state and the
benefit cut steady state. Both tax increase paths converge to the tax increase
steady state; similarly, both benefit cut paths converge to the benefit cut steady
state. The asset paths for this example are presented in Figure 34.
This same policy uncertainty experiment is possible in the learning models;
I will present results for the LCH model. Agents in the learning model do not know
future prices, they make decisions based on their adaptive expectations. In the
initial few periods, following the change in the population growth rate, the LCH
agents over-save compared to the rational agents because they are overestimating
the interest rate. This drives up capital in the LCH model compared to the RE
model. The LCH agents benefit from this higher capital stock (since the economy
is dynamically efficient) and don’t have to save as much in the periods right
147
1940 1960 1980 2000 2020 2040 2060 2080
2.8
3.0
3.2
3.4
3.6
3.8
time
age 1
assets
a1
reformd shock
1940 1960 1980 2000 2020 2040 2060 2080
6.0
6.2
6.4
6.6
6.8
7.0
7.2
7.4
time
age 2
assets
a2
reformd shock
1940 1960 1980 2000 2020 2040 2060 2080
9.0
9.5
10.0
10.5
time
age 3
assets
a3
reformd shock
1940 1960 1980 2000 2020 2040 2060 2080
11.0
11.5
12.0
12.5
time
age 4
assets
a4
reformd shock
1940 1960 1980 2000 2020 2040 2060 2080
6.4
6.6
6.8
7.0
7.2
7.4
7.6
7.8
time
age 5
assets
a5
reformd shock
tax increase, 2030
benefit cut, 2030
tax increase, 2040
benefit cut, 2040
FIGURE 34. Transition paths for asset holdings in the RE model with policy
uncertainty. Four reforms are possible. Either taxes are increased in 2030 (purple
line), benefits are cut in 2030 (red/brown line), taxes are increased in 2040 (light
blue line), or benefits are cut in 2040 (gold line). All four paths are identical until
period 2020. This is because all agents face the same policy uncertainty leading up
to the first possible reform. The two possible reform dates are indicated with red
and pink dots. The beginning of the demographic transition is indicated with a
blue dot.
148
1950 2000 2050 2100 2150 2200
7.0
7.5
8.0
8.5
9.0
9.5
10.0
time
capital
k
reformdem shock
1950 2000 2050 2100 2150 2200
-0.5
0.0
0.5
1.0
1.5
time
bonds
b
reformdem shock
LCH tax increase 2030
LCH benefit cut 2030
LCH tax increase 2040
LCH benefit cut 2040
RE tax increase 2030
RE benefit cut 2030
RE tax increase 2040
RE benefit cut 2040
FIGURE 35. Transition paths for capital and bonds in the LCH model and RE
model with policy uncertainty. Four reforms are possible in each model. Either
taxes are increased in 2030 (LCH: blue, RE: purple line), benefits are cut in 2030
(LCH: yellow, RE: red/brown line), taxes are increased in 2040 (LCH:green, RE:
light blue line), or benefits are cut in 2040 (LCH: red, RE: gold line). The LCH
paths are cyclical, while the RE paths converge to the new steady states quickly.
The dynamics before the reform are driven by demographic change in 1980 and
the forward looking behavior of agents anticipating reform. The benefit cut paths
converge to a steady state with higher capital.
before the policy reform. Following a policy reform, the paths of savings, capital
and bonds are cyclical in the LCH model since agents are slowly updating their
forecasts of prices and bonds.
Note, that all of the agents in the RE and LCH model know the policy
process. All agents know the possible dates of reform, the new policy parameters,
and the probability of each reform. All agents are forward looking (to the end of
their life-cycle). The only difference is that the LCH agents forecast future prices
and bonds adaptively, while the rational agents make fully rational forecasts. The
paths of capital and assets for the LCH and RE models are presented together in
Figures 35 and 36.
149
1950 2000 2050 2100 2150 2200
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
time
age 1
assets
a1
reformdem shock
1950 2000 2050 2100 2150 2200
5.5
6.0
6.5
7.0
7.5
time
age 2
assets
a2
reformdem shock
1950 2000 2050 2100 2150 2200
8.0
8.5
9.0
9.5
10.0
10.5
time
age 3
assets
a3
reformdem shock
1950 2000 2050 2100 2150 2200
10.5
11.0
11.5
12.0
12.5
13.0
13.5
time
age 4
assets
a4
reformdem shock
1950 2000 2050 2100 2150 2200
6.5
7.0
7.5
8.0
time
age 5
assets
a5
reformdem shock
LCH tax increase 2030
LCH benefit cut 2030
LCH tax increase 2040
LCH benefit cut 2040
RE tax increase 2030
RE benefit cut 2030
RE tax increase 2040
RE benefit cut 2040
FIGURE 36. Transition paths for asset holdings in the LCH model and RE model
with policy uncertainty. Four reforms are possible in each model. Either taxes are
increased in 2030 (LCH: blue, RE: purple line), benefits are cut in 2030 (LCH:
yellow, RE: red/brown line), taxes are increased in 2040 (LCH:green, RE: light
blue line), or benefits are cut in 2040 (LCH: red, RE: gold line). The benefit cut
paths converge to a steady state with lower age 1 and 2 savings (a1 and a2) and
higher age 3, 4 and 5 savings (a3, a4, a5). The LCH paths are cyclical, while the
RE paths converge to the new steady states quickly. The dynamics before the
reform are driven by demographic change in 1980; the rational agents respond to
the demographic changes while the learning agents do not. Both types of agents are
forward looking and respond to looming reform.
150
Welfare Cost
The welfare cost of policy uncertainty is calculated in this section using
two ex-post Consumption Equivalent Variation techniques: one for the rational
expectations model, and one for the learning model. In each, the utility of an
agent who experiences a given realization of the uncertain policy process (that is,
they experience a vector of policy parameters over their lifecycle ωt, . . . , ωt+J) is
compared to the utility of an agent who experience the same policy realization
with certainty (that is, the policy parameters ωt, . . . , ωt+J are anticipated). Kitao
(2016) constructs a similar welfare measure to examine the welfare cost of policy
uncertainty (particularly when considering uncertainty over the timing and type of
policy reform). Bu¨tler (1999) uses a similar metric (although her model contains
agent-level policy misperceptions rather than aggregate policy uncertainty). The
ex-post CEV for policy uncertainty in the RE model is calculated as:
J∑
j=1
βj−1u(cj,At+j−1(1 + ∆u)) =
J∑
j=1
βj−1u(cj,A,ut+j−1) (4.33)
where cj,At+j−1 indicates age j consumption in period t+j−1 in the model with policy
path A and rational expectations and cj,A,ut+j−1 is consumption in the model with RE
and policy uncertainty where policy path A was realized. As before, u(c) indicates
the period utility function, and ∆u is the CEV for policy uncertainty.
The welfare cost of social security policy uncertainty is illustrated below
in Figure 37. The figure depict the CEV for each of the four possible paths in the
RE model. The calibration of this example is identical to example 35. Recall, the
economy begins in a steady state with a high population growth rate. In 1980,
the growth rate falls and the social security system begins to run a deficit. Bonds
increase until reform is enacted and the economy converges to a new steady state
151
1950 2000 2050 2100
-0.003
-0.002
-0.001
0.000
0.001
0.002
cohort birth year
CEV
reformd shock
tax increase, 2030
benefit cut, 2030
tax increase, 2040
benefit cut, 2040
FIGURE 37. Consumption equivalent variation measure of the welfare cost of
policy uncertainty in the RE model. The welfare cost is constructed by comparing
the utility of realized policy parameters of a given path with the utility of that
same policy change in a perfect foresight, RE model. The CEV is show pictured for
all four possible paths: taxes are increased in 2030 (purple), benefits are cut in 2030
(red/brown), taxes are increased in 2040 (light blue), or benefits are cut in 2040
(gold). A negative CEV indicates that an agent would have higher utility in a RE
model without policy uncertainty. A positive CEV indicates that utility is higher in
the uncertain model.
associated with the particular reform. The four possible reforms are a benefit cut in
2030 or 2040 or a tax increase in period 2030 or 2040.
The welfare cost of social security policy uncertainty for the RE model is
small. The cost is the less than 0.25% of period consumption for cohorts along all
four paths. The initial cohorts benefit if the reform is a benefit cut, since they do
not have to pay higher taxes and do not experience the benefit reduction in their
lifetime. The agents born a few periods before and after the benefit cut are harmed
because they receive lower benefits, but are not alive to experience the feedback
effect of higher wages that result in the new steady state. The most harmed cohort
are agents born in 2030 if the realized policy is the benefit cut in 2030. These
agents would give up 0.25% of consumption to avoid being born in a world with
policy uncertainty (even though the uncertainty is resolved in the period in which
they are born).
The relatively small welfare cost of policy uncertainty in the rational
expectations framework that I find is consistent with work by other authors. Kitao
152
(2016) finds the welfare cost of pension reform uncertainty in a model calibrated
to the Japanese economy to be 0.8%-1.5% of period consumption.15 Caliendo et
al. (2015) find the welfare cost of social security policy uncertainty to be 0.01% of
period consumption for an agent with average earnings.16 Part of the reason Kitao
and I find larger welfare costs than Caliendo et al. is because Kitao and I both use
a general equilibrium framework. Caliendo et al. use a partial equilibrium model,
which misses the feedback effect of savings on future wages and interest rates.
Bu¨tler (1999) finds that the welfare cost of misperceived social security benefits
is between 0-3.46% of period consumption in a partial equilibrium model. Bu¨tler’s
results are not easily comparable to other papers, because agent beliefs are model
consistent in her paper.17
To assess the cost of policy uncertainty in the learning model the ex-
post Consumption Equivalent Variation compares the lifetime utility of an agent
in a LCH model who experiences a given realization of the uncertain policy
process (that is, they experience a vector of policy parameters over their lifecycle
ωt, . . . , ωt+J) to the utility of an agent in a LCH model who experience the same
15Kitao constructs a CEV measure similar to the CEV used in this paper which compares
utility of agents who experience a particular reform under uncertain with utility of agents who
experience the same reform with certainty. She also compares reforms realized in later periods
to the baseline of earlier reform (and finds CEV in the range of 2-3.5% of period consumption).
Finally, she also considers a CEV that compares the utility of agents who experience a particular
reform with the utility of agents who experience the expected reform. Kitao finds similar
magnitudes using both CEV measures.
16 The welfare metric developed in Caliendo et al. (2015) is not exactly the same as the CEV
developed in this paper. Caliendo et al. consider a continuous distribution of policy reform dates
and options. The CEV metric they construct compares the utility of agent endowed with expected
wealth over all possible reforms who experiences the average (or expected) reform to the utility
of an agent who faces uncertainty. The also perform heterogeneity analysis and find the welfare
cost of social security policy uncertainty to be 2.39 times higher for the lowest earners when the
policy is benefit cut, and 1.7 times higher for tax increase. The welfare cost for highest earners is
0.6 times the welfare cost for an average earning when the policy is a benefit cut, and 2.25 times
higher for tax increase. These costs are still quite small, since 2.39× 0.01 = 0.0239.
17See Phelan (1999) for a discussion of Bu¨tler (1999).
153
1950 2000 2050 2100
-0.02
-0.01
0.00
0.01
cohort birth year
CEV
reformd shock
Tax increase in 2030
Benefit cut in 2030
Tax increase in 2040
Benefit cut in 2040
FIGURE 38. Consumption equivalent variation measure of the welfare cost of
policy uncertainty in the LCH model. The welfare cost is constructed by comparing
the utility of realized policy parameters of a given path with the utility of that
same policy change in a LCH model. The CEV is pictured for all four possible
paths: tax increased in 2030 (blue), benefit cut in 2030 (yellow), tax increase in
2040 (green), or benefit cut in 2040 (red). A negative CEV indicates that an agent
would have higher utility in a LCH model with announced policy (no uncertainty).
A positive CEV indicates that utility is higher in the uncertain model. The gain
parameter γ = 0.93 in this example.
announced policy realization (that is, the policy parameters ωt, . . . , ωt+J are
anticipated).
The learning CEV is calculated as:
J∑
j=1
βj−1u(cj,At+j−1(1 + ∆uL)) =
J∑
j=1
βj−1u(cj,A,ut+j−1) (4.34)
where cj,At+j−1 indicates age j consumption in period t+j−1 in the model with policy
path A and LCH learning, and cj,A,ut+j−1 is consumption in the model with learning
and policy uncertainty where policy path A was realized. As before, u(c) indicates
the period utility function, and ∆uL is the CEV for policy uncertainty comparing
learning to learning.
This learning CEV measures the change in consumption that is the result of
policy uncertainty in the LCH framework. The learning CEV is presented in below
in Figure 38.
The learning uncertainty CEV is positive for several cohorts if taxes are
increased in 2040. This is because the possibility of facing reform in 2030 increases
154
agent savings and drives up the capital stock in the model with policy uncertainty,
relative to the model with announced tax increases. The accumulation of extra
capital raises agent welfare, relative to the baseline. The maximum (positive)
CEV is 1.11% and occurs in 2030. The cohort of agents born in 2030 in the
LCH model with a tax increase in 2040 would need to have 1.11% added to their
period consumption in order to be indifferent between their consumption, and the
consumption of agents born in 2030 in a LCH model that had a 25% chance of each
reform (where the tax increase in 2040 was realized).
The learning uncertainty CEV is negative for a few early cohorts who are
harmed by increased precautionary saving, and for a few cohorts following the
reform who experience the bottom of the swing in the capital stock. The swings in
state variables are larger in the uncertain model, and thus the welfare cost is larger.
The min CEV is -1.98% and occurs in 2040 for agents who are born after the tax
increase in 2030. The minimum CEV along the other three paths is between -0.32%
and -1.92%. As in the previous examples, the CEV is depends partially on the gain
parameter. The min CEV is -1.94% when γ = 0.35.
My analysis suggests that rational expectations models may understate
the welfare cost of social security policy reform and the welfare cost of policy
uncertainty. The life-cycle horizon learning model I propose is a small deviation
from rational expectations in which agents maintain model consistent beliefs Agents
in the LCH model are still optimizing and they are still forward looking, yet the
welfare cost of policy uncertainty is much larger. The welfare cost is driven mainly
by the cyclical changes in capital stock that are introduced into the model by
adaptive learning. The most harmed agents in the LCH model would be willing
to give up nearly 2% of period consumption to avoid living in a world of policy
155
uncertainty. In the RE model, the most harmed agents would only be willing to
give up 0.25% of period consumption to avoid the policy uncertainty. The welfare
cost in the learning model is nearly an order of magnitude larger than in the
rational case.
Robustness
Forecast tax burden
In main specification, agents fully understand the Leeper-tax and forecast
government debt levels in order to estimate their upcoming taxes. I back away from
that assumption in this section and assume that agents do not fully understand
the tax system. Agents observe the total tax burden (rate) they face in the current
period, and they use adaptive learning to forecast their future tax burden. They do
not incorporate knowledge about the structure of the tax system. Agents continue
to forecast social security benefits as the replacement rate φ times the expected
wage at the time of retirement.
Agents in this framework understand that a fraction of their wage is taken
by the government every period, but they do not distinguish between the Leeper
tax and the payroll tax. Agents are not forward looking with regard to tax changes.
They are, however, forward looking with regard to social security benefit changes.
This assumption will be relaxed in the next section.
Under this specification, agents forecast
wet+1 = γwt + (1− γ)wet (4.35)
Ret+1 = γRt + (1− γ)Ret (4.36)
τ et+1 = γ(τ
0
t + τ
1
t bt) + (1− γ)τ et (4.37)
156
with γ ∈ (0, 1). Forecasts many periods ahead are equal to the one-step-ahead
forecasts: xet+j = x
e
t,t+j = x
e
t+1, for x = R,w, τ and j > 1.
Agents do not respond to an announced tax change until after the change
has been is implemented. Savings of working-age cohorts increase as the policy is
implemented, instead of before. Thus the welfare cost of learning (compared to
the RE baseline) is larger in this specification than in the baseline learning model.
For example, consider the same set-up as the announced policy change in section
IV. The initial ratio of retirees to workers is 0.3. The ratio begins to increase in
1980, which causes the social security system to run deficits; capital falls and bonds
increase until reform is enacted. Initial policy is τ 0 = 0.124, τ 1 = 0.045 and φ = 0.4.
Reform is calibrated as a payroll tax increase to τ 0 = 0.1516. This policy change is
depicted in Figure 39 and the CEV is presented in Figure 40.
Agents in the learning models do not respond to the tax increase until the
period of the policy change. The swings in capital stock that follow the policy
change are larger than in the baseline learning model when agents anticipate tax
changes. The minimum CEV in the tax-forecasting learning model is in 2040. An
agent born in 2040 in the RE model would be willing to give up 1.39% of period
consumption to avoid being born in the same period in the LCH tax-forecasting
model. The minimum CEV in the bond-forecasting model is -1.19%. As with
pervious examples, the CEV depends on the learning gain and is smaller for larger
gains. The gain for these examples was set to γ = 0.93.
Forecast Policy Adaptively
As a final robustness check, suppose that agents do not anticipate any policy
changes, but rather agents learn all policy adaptively. Agents forecast their tax
157
1950 2000 2050 2100
7.5
8.0
8.5
9.0
9.5
time
ca
pi
ta
l
k
reformdem shock
1950 2000 2050 2100
0.0
0.5
1.0
1.5
time
bo
nd
s
b
reformdem shock
life cycle planning horizon
planning horizion 4
planning horizion 3
planning horizion 2
planning horizion 1
FIGURE 39. Equilibrium paths for the rational expectations economy and learning
economies that forecast the tax burden according to equation 4.37. The economy
begins with a ratio of retirees to workers of 0.3; this falls to 0.485 over six periods
beginning in 1980. Social security deficits lead to an accumulation of government
debt until taxes are raised in 2030. The RE path is shown in blue, LCH learning
in yellow, finite horizon learning in green (for 4 period planning horizon), red (3),
purple (2), and orange/brown (1-step-ahead).
1950 2000 2050 2100
-0.015
-0.010
-0.005
0.000
0.005
0.010
time
CEV
reformd shock
life cycle planning horizon
planning horizion 4
planning horizion 3
planning horizion 2
planning horizion 1
FIGURE 40. Consumption equivalent variation paths learning economies that
forecast the tax burden according to equation 4.37. The CEV is calculated
according to 4.31. The economy begins with a ratio of retirees to workers of 0.3;
this falls to 0.485 over six periods beginning in 1980. Social security deficits lead
to an accumulation of government debt until taxes are raised in 2030. The CEV
for LCH learning is sown in yellow, finite horizon learning in green (for 4 period
planning horizon), red (3), purple (2), and orange/brown (1-step-ahead). The
welfare cost is greatest (smallest CEV) for LCH learning.
158
burden (1− τt)wt and social security benefit, zt adaptively. Agents expect to receive
the same social security benefit in both periods of retirement (which is consistent
with the actual policy process). They forecast social security benefits for the period
in which they retire.
Agents forecast
wet+1 = γwt + (1− γ)wet (4.38)
Ret+1 = γRt + (1− γ)Ret (4.39)
τ et+1 = γ(τ
0
t + τ
1
t bt) + (1− γ)τ et (4.40)
zet+1 = γφtwt + (1− γ)zet (4.41)
with γ ∈ (0, 1). Forecasts many periods ahead are equal to the one-step-ahead
forecasts: xet+j = x
e
t,t+j = x
e
t+1, for x = R,w, τ, z and j > 1.
Under this specification, agents in learning models do not respond to
announced policy changes. They only respond to tax or benefit changes when they
have been implemented. The welfare cost of tax changes is identical to the previous
section. However, the welfare cost of social security benefit changes is larger. Using
the set-up from the previous section, suppose the population growth rate is initial
high, and then falls in 1980, leading to an accumulation of government debt until
benefits are cut in 2030. The CEV is lowest in 2030 and equals -1.63%. This means
an agent born in the rational model in 2030 would be willing to give up 1.63% of
her period consumption in order to avoid being born in the same period in a model
where agents adaptively learn policy with a life-cycle planning horizon. In contrast,
the minimum CEV in the baseline LCH model (where agents anticipate policy
change and adaptively forecast prices and bonds) is -0.81%. The gain parameter
for these examples was γ = 0.93. The welfare costs are larger for a smaller gain.
159
Sensitivity Analysis
The qualitative results of this paper are sensitive to parameter choice.
As a robustness check, I calibrated the model to be dynamically inefficient and
ran similar experiments (raising taxes or lowering social security benefits). I
parameterize a dynamically inefficient model by setting the discount factor β
greater than one. This increases agents desire to consume in old age and drives
up savings, and thus the capital stock. I use the parameterization of Bullard and
Russell (1999): σ = 4.2, annual β = ( 1
1−0.041), α = 0.25, and annual δ = 0.1.
18
Under this specification, agents in learning models still suffer relative to RE agents
when facing demographic changes, policy changes, and policy uncertainty. However,
the welfare cost of learning is much smaller. For example, the welfare cost that
compares LCH to RE of an announced benefit cut in response to the demographic
shock is less than 1% of period consumption.
Conclusion
Demographic changes in the United States make future social security
reform likely, as more beneficiaries are supported by each worker paying taxes.
If the program is left unchanged, benefits will exceed tax receipts and the social
security trust fund will be depleted by around the year 2034. The welfare cost
to agents of social security reform is not limited to agents alive during the policy
change. Using two new models of bounded rationality, I show that the welfare cost
18My model does not compare directly to Bullard and Russell (1999) in two key ways; first
I do not have stochastic survival probability, nor do I have endogenous labor choice. Thus,
this calibration should be viewed as a robustness exercise only, and not an attempt to replicate
Bullard and Russell (1999).
160
of announced policy changes can be quite large if there are general equilibrium
feedback effects along the transition path.
I relax the rational expectations assumption and model agents who forecast
future interest rates and wages adaptively. I model this in two main ways. First,
I develop a model of life-cycle horizon learning, in which agents make optimal
decisions over their lifecycle, given their (potentially imperfect) forecasts of prices.
Second, I model finite horizon life-cycle learning, in which agents plan over a
shorter planning horizon and make optimal choices conditional on their forecasts
and the assets they plan to hold at the end of their planning horizon.
The maximum welfare cost occurs along the transition path for the tax
increase in the LCH learning model; agents born after the reform while the capital
stock is depressed are worse off by up to about 1% of period consumption compared
to agents in a fully rational model. The welfare costs of learning are not always
negative; agents benefit when the capital stock is increased as the result of over-
saving by the learning agents.
The uncertainty regarding the future of social security is also economically
interesting. Social security policy uncertainty is not terribly costly to agents in
a rational expectations framework. Forward looking, rational agents are able
to save and partially self-insure against the aggregate risk of unfavorable policy
change. The greatest welfare cost of policy uncertainty in the rational model is
equivalent to less than 0.25% reduction in period consumption. I find that the
maximum welfare cost to agents in a model with social security policy uncertainty
is larger in a model with adaptive learning. The maximum welfare cost of policy
uncertainty in the LCH model (compared to the LCH model with announced
161
policy) is 1.98%. The CEV are significantly larger in the learning model than the
rational expectations model.
Policy makers who use rational expectations models to predict the welfare
effects of social security policy change will understate the cost of reform. To the
extent that the learning dynamics are a realistic depiction of agent level behavior,
the welfare costs of announced or uncertain social security reforms might be quite
large. If agents are unable to predict the general equilibrium effects of a change in
government policy, they will not be able to respond optimally. The results of this
paper suggest that policy makers can help agents by announcing policy and also by
explaining how the policy will impact wages and interest rates.
The learning models I develop in the paper include demographic changes
and endogenous government debt. Several interesting questions that are beyond
the scope of this paper can be addressed in this framework. I plan to explore the
relationship between delaying social security reform, growing deficits, and explosive
government debt in future work. The learning models developed in this paper
provide an excellent framework to examine explosive debt that is not possible in
a standard, rational expectations framework. The model could also be used to
explore the relationship between recessions, public pensions, and lifecycle savings.
162
CHAPTER V
CONCLUSION
In the three substantive chapters of this dissertation, I have shown how
agents respond to announced and uncertain social security reform in general
equilibrium over-lapping generations models.
In chapter II, I examine social security in an analytically tractable
model with endogenous deficits and debt. The need for social security reform
is motivated by a saddle-node bifurcation. The number and stability of steady
states depends on the underlying parameters. I calculate the minimum reform
needed to ensure convergence to a steady state in various scenarios. The longer
a reform is delayed, the larger the reform needs to be in order to avoid explosive
government debt. Comparative static analysis reveals that households increase their
saving in response to an increased probability of social security benefit reductions.
Comparative static analysis also shows that the savings rate of households is
increasing in the social security reform benefit rate; agents save less if they
anticipate larger social security benefits. I also conduct welfare analysis using a
social welfare function; optimal reform depends critically on the discount factor and
planning horizon of the planner.
In chapter III, I extend the analysis of the previous chapter to a model
setting with fully forward looking behavior. Households live for multiple periods,
and so they respond to announced and potential reforms many periods in the
future. I examine the response of households to policy timing uncertainty (policy
can take place within one of three periods), and combined timing and option
uncertainty (either taxes or benefits are changed). I also examine the long-lasting
163
effects of policy uncertainty and show that the capital stock and can be depressed
for generations following a reform. Finally, in this chapter, I study the interaction
of policy uncertainty and the sustainability of government debt. I show that if the
probability of reform is high enough, that the response from households can be
large enough to ensure convergence to a steady state, even if the reform never takes
place.
In chapter IV, I introduce two new behavioral models: life-cycle horizon
learning, and finite horizon life-cycle learning. I demonstrate that the welfare
cost of social security policy uncertainty is much larger when agents use adaptive
expectations rather than rational expectations. The welfare cost is not uniform
across generations; cohorts born after reform may still be harmed by the cyclical
dynamics that result from agents learning with policy uncertainty.
The results presented in this dissertation represent the core of a broader
research agenda examining bounded rationality, social security, and retirement.
The learning models I developed in chapter IV can be used to explore a variety
of policy questions that impact the lifecycle. I am currently working on a project
to examine the long-term relationship between unfunded pension liabilities and
growing government debt. By using a model of bounded rationality, I am able
to explore explosive dynamics that are not possible in a fully rational model. I
am also exploring the intergenerational effects of fiscal policy and recessions with
boundedly rational agents. Going forward, I plan to explore the relationship
between Euler-equation learning and finite-horizon learning in OLG economies,
and to explore heterogeneous learning rules in OLG models. I may also model the
consumption profile and retirement timing decision of agents under various learning
rules.
164
I am interested in the relationship between stated household beliefs about
social security policy and current savings decisions. Many households do not believe
they will receive social security benefits (are reported in survey data, such as the
American Life Panel), yet aren’t increasing their private savings. I plan to test this
relationship in a heterogeneous agent life-cycle model with rational expectations
and also using a model of bounded rationality. I suspect that bounded rationality
will generate savings behavior that is consistent with the data.
The aging of the U.S. population ensures that questions about social
security, saving, and retirement will continue to be politically salient. I look
forward to contributing research in this policy area.
165
APPENDIX A
TWO-PERIOD MODEL APPENDIX
Derivation of Golden Rule Level of Capital
The golden rule level of capital is defined as the steady state level of
capital which maximizes aggregate consumption (and therefore aggregate utility).
Aggregate consumption is given by Ct = Yt − It were It indicates aggregate capital
investment. It = Kt+1 − (1 − δ)Kt, with full depreciation It = Kt+1. Note
that principle of the invariance of a maximum implies that the max of Ct will be
equal to the max of Ct/LtAt. Thus, the golden rule level of capital can be found by
maximizing Ct/LtAt with the steady state kt = kt+1 = k imposed
Ct
LtAt
=
Yt
Lt
− Kt+1
LtAt
Lt+1
Lt+1
At+1
At+1
ct = f(kt)− (1 + n)(1 + g)kt+1
c = f(k)− (1 + n)(1 + g)k
max
k
c = f(k)− (1 + n)(1 + g)k
FOC: f ′(k)− (1 + n)(1 + g) = 0
f ′(kgr) = (1 + n)(1 + g)
α(kgr)
α−1 = (1 + n)(1 + g)
kgr =
[
α
(1 + n)(1 + g)
] 1
1−α
Balanced Budget
It is possible for the government to run a balanced budget by setting
social security benefits equal to social security payroll taxes in each period. If the
166
government runs a balanced budget (zero deficit) each period, two special cases are
possible. Either the government has an initial stock of debt (which asymptotes to a
steady state value), or the government government doesn’t have any debt (and the
model collapses to the univariate Diamond model with capital and social security).
I will begin by discussing the planar case, when b0 ≥ 0. I will discuss the simple
univariate case in section A.
When the government runs a balanced social security budget, the payroll
taxes exactly cover social security benefits in each period:
φˆ = (1 + n)τˆ (A.1)
where φˆ and τˆ notation is used to emphasize that the government is running a true,
balanced-budget, pay-as-you-go social security program. When this is the case,
the government only has one choice variable (either τˆ or φˆ) which determines the
size of the social security program. Without loss of generality, I will assume the
government chooses τˆ and sets φˆ = (1 + n)τˆ so that the budget is balanced in each
period and there are no deficits.
Plugging the balanced budget identity (A.1) into the transition equations
(2.8) and (2.9) gives the dynamics for the balanced budget economy:
bt+1 =
1
(1 + n)(1 + g)
[
αkα−1t bt +
(
(1 + n)τˆ
1 + n
− τˆ
)
(1− α)kαt
]
=
αkα−1t bt
(1 + n)(1 + g)
(A.2)
167
0.00 0.05 0.10 0.15 0.20
0.00
0.02
0.04
0.06
0.08
k
b
stable ss
saddle ss
Δk=0
Δb=0
saddle
FIGURE A.41. Phase diagram for balanced budget φˆ=(1+n)τˆ . In the case of a
balanced budget, the constant bond loci collapses to horizontal line at b=0. The
model still has two steady states: the stable sink where the constant capital and
constant bond loci intersect, and a steady state governed by the saddle path. In the
balanced budget case, the saddle steady state is at the golden rule level of capital.
kt+1 =
[
(1 + n)(1 + g) +
(1 + n)τˆ(1− α)(1 + g)
(1 + β)α
]−1
[
β
1 + β
(1− τˆ)(1− α)kαt − αkα−1t bt −
(
(1 + n)τˆ
1 + n
− τˆ
)
(1− α)kαt
]
=
[
(1 + n)(1 + g) +
(1 + n)τˆ(1− α)(1 + g)
(1 + β)α
]−1
[
β
1 + β
(1− τˆ)(1− α)kαt − αkα−1t bt
]
(A.3)
In the planar model with balanced budgets, the model has two steady states:
one at the golden rule level of capital and positive bonds (kgr, b(kgr)), and a second
steady state with zero bonds. Under the baseline parameterization of the model,
the zero bond steady state is dynamically inefficient k > kgr. These steady states
are depicted in the phase diagrams in Figure A.41.
168
It is easy to show that (kgr, b(kgr)) is a steady-state of the model, by
examining equations (A.2) and (A.3). Setting kt+1 = kt = k and bt+1 = bt = b
in (A.2), we find:
(1 + n)(1 + g)b = αkα−1b+
(
(1 + n)τˆ
1 + n
− τˆ
)
(1− α)kα
(1 + n)(1 + g)b = αkα−1b
αkα−1 =
(1 + n)(1 + g)
α
k =
[
α
(1 + n)(1 + g)
] 1
1−α
k = kgr
Setting kt+1 = kt = kgr and bt+1 = bt = b in (A.3), we find b = b(kgr).
ψkgr =
β
1 + β
(1− τˆ)(1− α)kαgr − αkα−1gr b
b =
β
1+β
(1− τˆ)(1− α)kαgr − ψkgr
αkα−1gr
b = b(kgr)
Where ψ = (1 + n)(1 + g) + (1+n)τˆ(1−α)(1+g)
(1+β)α
.
It is also easy to show that b = 0 and k = k(0) is a steady state of the
model. Plugging b = 0 into (A.2) we get 0 = 0 + 0. Plugging b = 0 into (A.3) we
find the steady state value of k associated with zero bonds and zero deficits:
k(0) =
(
αβ(1− τˆ)(1− α)
(1 + n)(1 + g)(α(1 + β) + τˆ(1− α))
) 1
1−α
(A.4)
169
Transcritical Bifurcation of the Balanced Budget Model
With a balanced budget, the model has two steady states: the golden rule
steady state (kgr, b(kgr)), and a zero bond steady state (with capital given by
equation A.4). The stability properties of the two steady states depend on the
parameters of the model. This is called a transciritical bifurcation. For illustrative
purposes, let τˆ be the bifurcation parameter. Holding all other parameters
constant, the number and stability properties of the steady states depends on the
value of the parameter τˆ . At a critical value of the parameter τˆ = τˆc, a single
steady state exists (at that steady state k = kgr and b = 0). At values of τˆ
below the critical value, two steady states exist and the zero bond steady state is
stable while the golden rule steady state is a saddle (this is true under the baseline
parameterization of the model as illustrated in Figure A.41). At values of τˆ above
the critical value, the stability properties are reserved; the zero bond steady state is
a saddle and the golden rule steady state is stable.
The transcritical bifurcation is illustrated in figures A.42 and A.43. Figure
A.42 plots the steady state values of capital associated with different levels of social
security τˆ . For low values of τˆ , the zero bond steady state is stable (and has capital
above the golden rule level). At the critical value τˆc, only one steady state exists.
At values of τˆ above the critical value, there are two steady states and the golden
rule level of capital steady state is stable.
Phase diagrams of the balanced budget model are presented in Figure A.43.
In the top panel, the size of the social security system is relatively small (τˆ < τˆc)
and the zero bond steady state is stable. As the size of the system increases, capital
is crowded out, and the capital stock at the stable steady declines. At the critical
value τˆc, only one steady state exists (at the golden rule level of capital, with zero
170
kgr τ ciritcal
steady state b=0
steady state k = kgr
0.2 0.3 0.4 0.5 0.6
τ=(1+n)ϕ0.00
0.05
0.10
0.15
Steady State
capital
FIGURE A.42. Phase diagrams demonstrating the transcritical bifurcation
with a balanced budget. As the size of the social security system τˆ is increased,
capital is crowded out and the stable steady state (with zero bonds) shifts to the
left. At a critical value of τˆ , only one steady state exists, (kgr, 0). As the tax is
further increased, the zero bond steady state moves further to the left to a lower
value of capital. The stability properties of the steady states switch following the
transcirtical bifurcation, and now the golden rule steady state is stable.
bonds). This is illustrated in the middle panel. As the size of the system grows and
τˆ > τˆc, the stability properties of the steady states switch, as shown in the bottom
panel. The golden rule steady state is now stable and the zero bond steady state is
a saddle.
Analytical Transcritical Bifurcation
The transcritical bifurcation can be demonstrated analytically by showing
that the stability properties of the steady states change as the bifurcation
parameter passes through a critical value.1 I will demonstrate the bifurcation using
the parameter τˆ . As discussed in the main text of the paper (II), the stability of
a steady state depends on the eigenvalues of the Jacobian matrix evaluated at the
1See Sander (1990) for a discussion of transcirtical bifurcations. De La Croix and Michael
(2002) point out that transcritical bifurcations are uncommon in economics, often because they
involve a trivial fixed-point.
171
0.05 0.10 0.15 0.20
k
-0.02
0.02
0.04
0.06
b
kgr
Δk=0
saddle ss
stable ssΔb=0
0.05 0.10 0.15 0.20
k
-0.02
0.02
0.04
0.06
b
kgr
Δk=0
saddle ssΔb=0
0.05 0.10 0.15 0.20
k
-0.02
0.02
0.04
0.06
b
kgr
Δk=0
saddle ss
stable ss
Δb=0
FIGURE A.43. Phase diagrams demonstrating the transcritical bifurcation
with a balanced budget. As the size of the social security system τˆ is increased,
capital is crowded out and the stable steady state (with zero bonds) shifts to the
left. At a critical value of τˆ , only one steady state exists, (kgr, 0). As the tax is
further increased, the zero bond steady state moves further to the left to a lower
value of capital. The stability properties of the steady states switch following the
transcritical bifurcation, and now the golden rule steady state is stable.
172
steady state. If both eigenvalues are less than one in modulus, the steady state is
stable; if one eigenvalue is greater than one in modulus and the other is less than
one in modulus, the steady state is a saddle.
Letting γ = (1 + n)(1 + g) and ψ =
(
γ + τˆ(1−α)
(1+β)α
)
, the Jacobian of the
balanced budget model is given by:
J =

ψ−1
(
αβ(1−τˆ)(1−α)
1+β
kα−1t − (α− 1)αkα−2t bt
)
ψ−1αkα−1t
(α−1)α
γ
kα−2t bt
α
γ
kα−1t
 (A.5)
The critical value of τˆ can be found by noting that when only one steady
exists, k = kgr and b = 0 = b(kgr). The second identity can be used to solve for
τˆc =
α[β(1− α)kα − γ(1 + β)k]
(1− α)(αkα + γk) . (A.6)
For values of τˆ < τˆc, the golden rule steady state is a saddle and the zero
bond steady state is stable; for values of τˆ > τˆc, the reserve is true. This can be
shown by finding the eigenvalues of the Jacobian evaluated at each of the steady
states near the critical value τˆ .
Evaluated at the golden rule steady state, the Jacobian A.5 simplifies to
J |kt=kgr,bt=b(kgr) =

−(1−α)((α+β)(1−τˆ)+βα+τˆ)
α(1+β)+(1−α)τˆ
−α(1+β)
α(1+β)+(1−α)τˆ
(1−α)((1+β)(α+(1−α)τˆ)−(1−α)β)
α(1+β)
1
 (A.7)
The eigenvalues of this matrix are(
(1− α)β(1− τˆ)
α(1 + β) + (1− α)τˆ , α
)
(A.8)
The eigenvalues simplify to (1, α) evaluated at the critical value τˆc. The second
eigenvalue is always less than one and does not depend on the bifurcation
parameter. The magnitude of the first eigenvalue depends on the bifurcation
173
parameter, τˆ . To demonstrate this, first note that the critical value evaluated at
the golden rule level of capital is given by:
τˆc(kgr) =
β(1− α)− α(1 + β)
(1− α)(1 + β) (A.9)
Next, evaluate the first eigenvalue (A.8) at τˆc(kgr) +  where  is very small. This
simplifies to:
λ1 =
β(1 + βα)− β(1 + β)(1− α)
β(1 + βα) + (1 + β)(1− α) (A.10)
When  = 0, the λ1 = 1, as expected. When  > 0, the numerator of the fraction
is smaller than the denominator, and λ1 < 1. When  < 0, the reverse is true, and
λ1 > 1. Thus, for tax rates above the critical value, the golden rule steady state
is stable, and for taxes below the critical value, the golden rule steady state is a
saddle.
Evaluated at the zero bond steady state, the Jacobian A.5 simplifies to
J |kt=k,bt=0 =

α − α(1+β)
(1−α)β(1−τˆ)
0 α(1+β)+(1−α)τˆ
(1−α)β(1−τˆ)
 (A.11)
The eigenvalues of this matrix are
(
α(1+β)+(1−α)τˆ
(1−α)β(1−τˆ) , α
)
. Note that the second
eigenvalue is equal to α and does not depend on the bifurcation variable.
Additionally, the first eigenvalue is the inverse of the eigenvalue evaluated at the
golden rule steady state. Thus, when τˆ = τˆc, the eigenvalue is equal to one; for
values of τˆ above the critical value, the eigenvalue will be great than one; and for
τˆ below the critical value, the eigenvalue will be less than one. Thus the stability
properties of the zero bond steady state are the mirror opposite of the golden rule
steady state.
174
Special Case: Univariate Model
If the government runs a balanced budget and the initial stock of
government debt is zero, then the model collapses into the univariate Diamond
model. This can be shown by plugging bt+1 = bt = 0 into equations (A.2) and (A.3).
The dynamic system is then characterized by the single equation:
kt+1 =
[
(1 + n)(1 + g) +
(1 + n)τˆ(1− α)(1 + g)
(1 + β)α
]−1
[
β
1 + β
(1− τˆ)(1− α)kαt
] (A.12)
Equation (A.12) can also be derived from the household problem and the
market clearing conditions, noting that if the government does not issue debt, that
the capital market clearing condition (2.6) simplifies to
(1 + n)(1 + g)kt+1 = a
∗
t/At
The steady state of the univariate model is given by equation (A.4), just as
in the previous section
k =
(
αβ(1− τˆ)((1− α)
(1 + n)(1 + g)(α(1 + β) + τˆ(1− α))
) 1
1−α
As is standard in the Diamond model, a benevolent planner would only
choose a social security system (that is, choose a value of τˆ > 0) if the economy is
dynamically inefficient, and social security can be used to crowd out private savings
and move the economy closer to the golden rule level of capital. It can be shown
that k = kgr in the univariate model with social security if τˆ = τˆc from equation
(A.9). This is the optimal tax level for the univariate model.
In the univariate case of the model (when the government runs a balanced
budget and never borrows), the unique steady state is always stable. This can
175
be shown by noting that the univariate model is stable if the derivative of the
transition equation (A.12) evaluated at the steady state is less than one in absolute
value. This will be true as long as the production parameter α is less than one.
The derivate of the univariate transition equation is given by:
dkt+1
dkt
= α
[
(1 + n)(1 + g) +
(1 + n)τˆ(1− α)(1 + g)
(1 + β)α
]−1
[
β
1 + β
(1− τˆ)(1− α)kα−1t
] (A.13)
Evaluating equation (A.13) at the steady state given by equation (A.4) and
simplifying:
dkt+1
dkt
|kt=kss = α
[
(1 + n)(1 + g) +
(1 + n)τˆ(1− α)(1 + g)
(1 + β)α
]−1
 β
1 + β
(1− τˆ)(1− α)
[(
αβ(1− τˆ)((1− α)
(1 + n)(1 + g)(α(1 + β) + τˆ(1− α))
) 1
1−α
]α−1
= α
[
(1 + n)(1 + g)(α(1 + β) + τˆ(1− α))
(1 + β)α
]−1
[
β(1− τˆ)(1− α)
(1 + β)
(
αβ(1− τˆ)(1− α)
(1 + n)(1 + g)(α(1 + β) + τˆ(1− α))
)α−1
1−α
]
letting  = (1 + n)(1 + g)(α(1 + β) + τˆ(1− α))
dkt+1
dkt
|kt=kss = α
[

(1 + β)α
]−1 [
β(1− τˆ)(1− α)
(1 + β)
(
αβ(1− τˆ)(1− α)

)−1]
= α
(
(1 + β)α

)(
β(1− τˆ)(1− α)
(1 + β)
)(

αβ(1− τˆ)(1− α)
)
= α
(
(1 + β)α

)(

(1 + β)α
)
= α
176
The stability results are not unique to this model. See Azariadis Chapter 7.4
for a discussion of stability in scalar Diamond models.
Endogenous Government Surpluses with Dynamic Inefficiency
In the baseline calibration of the two-period model, government runs an
endogenous deficit in each period. It is also possible for the government to run an
endogenous surplus.2 Recall from equation (2.5) that the time t deficit is defined as
dt =
(
φ
1 + n
− τ
)
wt
If the tax rate is sufficiently high (if τ > φ/(1 + n)), the government will run a
surplus each period. The government can be a net lender in this scenario, if the
initial stock of debt is low enough. Conversely, if the initial stock of debt is high
enough, the government will be a borrower at steady state, even if it runs a surplus
each period. These two possible steady states are depicted graphically in figures
A.44. Note that if the model is calibrated to be dynamically inefficient, then there
are always two steady states when the government runs a surplus. This is because
the feasible parameter combinations that lead to the government running a surplus
at steady state are such that the bifurcation variable (for example, τ) is always on
the side of the threshold where two steady states exist.
Finally, it is possible for the government to run a surplus that is too large
and approaches infinity. For example, if the government taxes the elderly (that
is, φ < 0), and the young (τ > 0), then the government accumulates large asset
holdings, and the capital stock increases. If the government taxes both generations
enough, the economy fails to converge and the governments assets approach infinity
(bonds approach negative infinity). Recall in this model the government doesn’t
2The United States Social Security system ran a surplus most years since 1983.
177
0.00 0.05 0.10 0.15 0.20
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
k
b
stable ss
saddle ssΔk=0
Δb=0
saddle
FIGURE A.44. If the government runs a surplus each period, it is possible that
there are two non-trivial steady states: a saddle with positive bonds (government
borrows) and a stable sink with negative bonds (the government lends).
spend on consumption or goods; the only function of the government is to run
a social security system. So, if the government taxes both generations, it adds
a pile of tax revenue to its balance sheet each period. The government’s assets
grow by receiving interest each period and by adding the additional revenue
(surplus) each period. This leads government assets to approach infinity, which
prevents the economy from reaching the steady state. In practice, this only occurs
if both generations are taxed a high rate. In the analysis that follows, I rule out
government policies that lead to explosively large surpluses.
Special Case with Dynamic Efficiency
Leeper Tax
The baseline 2-period model is calibrated to be dynamically inefficient. A
special case that can give rise to a dynamically efficient level of steady state capital
is a Leeper tax. A Leeper tax responds automatically to increases in government
178
debt. Although actual social security taxes do not change based on government
debt, a Leeper tax can be thought of as a way to capture legislative unease with
increasing debt. I consider a Leeper tax of the following form:
τt = τ0 + τ1bt (A.14)
Where τ0 ∈ [0, 1] is the baseline tax rate when government debt is zero, and
τ1 ∈ [0, 1] is the incremental tax change if the government borrows. When τ1 = 0
the Leeper tax corresponds to the simple, exogenous tax rate of the baseline model.
When τ1 > 0, taxes increase when government debt increases. This slows the
accumulation of debt and increases the basin of attraction of the stable (high
capital) steady state. The responsive tax rate also allows for a dynamically efficient
saddle steady state. The phase diagram for the model with a Leeper tax is depicted
in Figure A.45.
Leeper taxes are generally modeled as responding to the previous period’s
debt τt = f(bt−1) (see Davig and Leeper 2011 as an example). However, in this
simple model, government debt (bt) and capital (kt) are both predetermined. Thus,
a tax rate in time t that responds to bond in time t is reasonable, since agents with
perfect foresight know bt at the beginning of time t. Following the process outlined
in sections II, II and II, and substituting in equation (A.14) yields the following
(modified) transition equations:
bt+1 =
1
(1 + n)(1 + g)
[
αkα−1t bt +
(
φ
1 + n
− τ0 − τ1bt
)
(1− α)kαt
]
kt+1 =
[
(1 + n)(1 + g) +
φ(1− α)(1 + g)
(1 + β)α
]−1
[
β
1 + β
(1− τ0 − τ1bt)(1− α)kαt − αkα−1t bt −
(
φ
1 + n
− τ0 − τ1bt
)
(1− α)kαt
]
179
kgr
0.05 0.10 0.15 0.20
k
-0.01
0.01
0.02
0.03
0.04
0.05
0.06
b
stable ss
saddle ss
Δk=0
Δb=0
Leeper tax with τ1 = 0.1
kgr
0.05 0.10 0.15 0.20
k
-0.01
0.01
0.02
0.03
0.04
0.05
0.06
b
stable ss
saddle ss
Δk=0
Δb=0
Leeper tax with τ1 = 0.336
FIGURE A.45. The Leeper tax becomes more responsive to government debt as
the τ1 parameter increases. When τ1 is larger, the basin of attraction for the stable
steady state is larger, and it is possible that the saddle steady state is dynamically
efficient, as depicted in panel (b).
The dynamics implied by these modified transitions equations are depicted
graphically in Figure A.45. Note finally that under a Leeper tax, the parameters
τ0 and τ1 can both be bifurcation variables. Holding the other parameters of the
model fixed, there are tax response rates τ1 or baseline tax rate τ0 such that two
steady states exist, one (non-unique) steady state exists, or no steady states exist.
Replacement Rate Uncertainty: Time Varying Probability of Reform
The policy experiments of section II highlight the importance of agents’
expectations about future benefit rates. If agents believe the probability of
reform (benefit cut) is high enough, that belief can keep the economy on a stable
trajectory. This captures the intuitive tension between unstable policy and agent
expectations. A simple extension that ties agents beliefs with the current state of
the economy is to assume that the probability of reform is an increasing function of
the government’s outstanding debt. If government debt is very low, agents are less
worried about debt accumulation and do not place a high likelihood on their social
180
-0.1 0.0 0.1 0.2 0.30.0
0.2
0.4
0.6
0.8
1.0
bt
p(b t)
FIGURE A.46. Graph of p(bt) equation (A.15). The perceived probability of
reform is increasing in debt. As the stock of government debt grows, agents believe
reform (benefit cuts) are more likely.
security benefits being cut. If government debt is high, agents are more concerned
about benefit reductions.
Suppose agents believe the probability of benefit cuts is given by the
following logistic equation:
pt =
1
1 + exp(−η(bt − b))
= p(bt) (A.15)
where η > 0, and b is a target debt level. (Note, if b = 0, then agents believe
there is a 0.5 probability of reform when debt is zero, p(0) = 0.5. If b > 0, then
p(0) < 0.5. The parameter b can be chosen to ensure that the probability of reform
when debt is zero is very small.)
Preliminary results using point expectations suggest the effects of agent’s
believing there is a time-varying probability of reform are qualitatively similar
to the simple case of a fixed probability. In both cases, if agents believe reform
is likely and large (enough), then their beliefs can keep the economy on a stable
trajectory. This time-varying probability p(bt) could be incorporated fully in a
computational model in my future work.
181
Derivation of Social Welfare Function
The social welfare function is the (discounted) sum of consumption for all
generations into the future until time T (which is the planning horizon).
W = (young0 ∗ c1,0 + old0 ∗ c2,0)
+ ρ(young1 ∗ c1,1 + old1 ∗ c2,1)
+ ρ2(young2 ∗ c1,2 + old2 ∗ c2,2)
+ · · ·
+ ρT (youngT ∗ c1,T + oldT ∗ c2,T )
which can be rewritten noting that the young in period i are equal to (1 + n)
times the old:
W = ((1 + n)old0 ∗ c1,0 + old0 ∗ c2,0)
+ ρ((1 + n)old1 ∗ c1,1 + old1 ∗ c2,1)
+ · · ·
+ ρT ((1 + n)oldT ∗ c1,T + oldT ∗ c2,T )
The constant population growth rate implies: oldt+1 = (1 + n)oldt
W = ((1 + n)old0 ∗ c1,0 + old0 ∗ c2,0)
+ ρ((1 + n)2old0 ∗ c1,1 + (1 + n)old0 ∗ c2,1)
+ · · ·
+ ρT ((1 + n)T+1old0 ∗ c1,T + (1 + n)Told0 ∗ c2,T )
182
Normalizing the initial number of old to one, old0 = 1, welfare can be
written as:
T∑
t=0
ρt(1 + n)t [(1 + n)c1,t + c2,t]
Finally, noting that the consumption values depend on the policy
parameters, we arrive at the specification given in equation (2.27).
183
APPENDIX B
THREE PERIOD MODEL APPENDIX
Solution Method
In the prefect foresight case, the model can be solved by first finding a
steady state, and then using a Gauss-Sidel algorithm to solve for the equilibrium
path of the variables. The steady state can be found numerically by searching for
the values {k, b, a1, a2} that solve the steady state equations (3.24), (3.25), (3.26),
(3.27). The steady state is then used as an endpoint to simulate the transition path
(or equilibrium dynamics) of the economy using equations (3.6), (3.7), (3.17), and
(3.18).
Time Path Iteration
The model can be solved computationally using a process called Time Path
Iteration (TPI). This approach was pioneered by Auerbach and Kotlikoff in their
1987 book Dynamic Fiscal Policy, chapter 4. R. W. Evans and Phillips (2014) give
detailed instructions on using TPI to solve OLG models.
The key assumption of TPI is that the economy will reach the steady state
within a finite number of periods T < ∞, as long as the initial conditions lie within
the basin of attraction of the steady state. Using the steady state, the TPI method
can be outlined as follows:
– Solve for the (non-stochastic) steady state given by equations (3.24), (3.25),
(3.26), (3.27). Assume that the economy reaches the steady state in period T .
184
– Assume a path for the state variables {kt}Tt=0 and {bt}Tt=0. The path for the
state variables must include the initial conditions k0 and b0 and the steady
state condition kT and bT .
– Solve for the path of choice variables {a1t}Tt=0 and {a2t}Tt=0 given the path of k
and b assumed in the previous step.
– Use the values of {a1t}T0 and {a2t}T0 from the previous step and the initial
conditions {k0, b0} to find the implied values of {kt}T1 and {bt}T1 .
– If {kt}T1 and {bt}T1 from the previous step differs from the initial guesses
(by more than a given small threshold), update the guess (to a convex
combination of the implied value of capital and bonds, and the values from
the previous iteration) and repeat the cycle. If the distance between the
initial guess and the implied paths is arbitrarily close to zero, stop.
Examples of Steady States in the Three-Period Model
I begin with an economy that is dynamically inefficient in the absence of
social security (Figure B.1). That is, the steady state capital stock is above the
golden rule level of capital. Introducing a balanced budget social security system
crowds out capital (since private savings is depressed), and results in a dynamically
efficient stable steady state (Figure B.2). The steady states and the eigenvalues
of the linearized system evaluated at the steady state are presented in a summary
Table B.1.
185
Stable Steady State Unstable Steady State
Figure τ0 φ deficit k b a1 a2 k b a1 a2
B.1 0.00 0.00 0.00 0.13 0.00 0.09 0.23 0.13 0.01 0.09 0.23
3×108 0.98 0.25 0.24 3×108 1.02 0.25 0.24
B.2 0.0193 0.05 0.00 0.13 -0.01 0.08 0.22 0.12 0.00 0.08 0.22
65.11 0.98 0.25 0.24 66.76 1.03 0.25 0.24
11 0.00 0.01 0.0024 0.13 0.00 0.07 0.26 0.11 0.04 0.10 0.27
172.40 0.79 0.20 0.15 219.68 1.27 0.25 0.15
0.00 0.025 0.0059 0.12 0.01 0.05 0.28 0.09 0.08 0.11 0.30
76.64 0.92 0.21 0.15 83.54 1.09 0.23 0.15
12 0.077 0.20 -0.0003 0.10 0.002 0.078 0.18 0.09 0.02 0.087 0.18
0.8 20.63 0.89 0.27 0.25 22.25 1.12 0.28 0.23
TABLE B.1. Steady States for figures in section III and B. Unless otherwise
specified, the Leeper tax was set to zero for these examples (τ 1=0). Figures B.1
and B.2 are based on the same parameter values. The golden rule level of capital in
for these first simulations is k=0.126. The eigenvalue from matrix M1 evaluated at
a particular steady state are presented below each steady state. If three eigenvalues
are less than one in absolute value, the steady state is saddle stable. If more than
one eigenvalue lies outside the unit circle, the steady state is explosive. For all of
the examples considered, the higher capital steady state is saddle stable and the
lower capital steady state is explosive. The bottom section of the table shows the
steady states and eigenvalues for Figure 11. The underlying parameters here are
different, such that the golden rule level of capital is k=0.0881. Note that as the
deficit increases, the steady states move closer together. As this occurs, the second
eigenvalue for each steady state gets closer to one. By definition, an eigenvalue of
the system evaluated at a steady state will equal one at the critical value where the
saddle-node bifurcation switches between two steady states and one steady state.
The parameters for Figure 12 are τ 0=0.077 and τ 1=0.8.
186
FIGURE B.1. No social security
0.08 0.10 0.12 0.14 0.16 0.18
k
-0.010
-0.005
0.005
0.010
b
kgr stable ss
saddle ss
Δk=0
Δb=0
FIGURE B.2. Social security,
φ = 0.05
0.08 0.10 0.12 0.14 0.16 0.18
k
-0.010
-0.005
0.005
0.010
b
kgr
stable ss
saddle ss
Δk=0
Δb=0
Figures B.1 - B.2: Steady state contour graphs of equations (3.32) and (3.33)
in (k, b) space (capital is plotted on the horizontal axis, bonds on the vertical).
This shows dynamically inefficient economy with no social security. As a balanced
budget social security is introduced, it crowds out capital. The size of the social
security system is indicated by the replacement rate φ. When the social security
system is large enough, the steady state level of capital falls below the golden rule.
The higher capital steady state is saddle stable in each case. The lower capital
steady state is explosive.
Additional Types of Uncertainty
Simple three-period timing uncertainty
Building off the simple case of section III, consider policy that will be
reformed (to know parameters) in period S with probability p, in period S + 1 with
probability q. If policy change is not implemented in S or S + 1, it is implemented
with certainty in period S+2. Assume that p+q < 0. Policy parameters are certain
for time periods t < S and t > S + 1; policy is now uncertain in periods t = S and
t = S + 1.
Under this simple type of policy uncertainty, there are three possible paths
for the economy: either reform occurs in period S, in period S + 1, or in period
S + 2. For notational ease, I will distinguish the three paths using tildes, hats,
187
and bars: x˜ for the path of the economy when reform occurs in period S, and xˆ for
the path of the economy when reform happens in S + 1, and x¯ for the path of the
economy when reform happens in S + 2.
This uncertainty impacts up to three generations: the generation who is
young at time S − 2 (they form expectations for φS for their retirement), and the
generation who is young at time S − 1 (expectations for τS in middle age and φS+1
in retirement), and then generation who is young at time S, if policy is not changed
in S. This generation potentially faces uncertainty about the taxes φS+1 they will
face in middle age. They do not face uncertainty regarding retirement benefits,
since policy reform is enacted with certainty by period S + 2.
The generation who is young at time S − 2 chooses a1S−2 using the non-
stochastic Euler equation (B.1) and a2S−1 using a stochastic Euler equation. There
will be a distinct stochastic Euler equation for each of the three paths; only the
tilde path is represented below as (B.2).
((1− τS−2)w(kS−2)− a1S−2)−σ
= β(R(kS−1)(R(kS−1)a1S−2 + (1− τS−1)w(kS−1)− a2S−1)−σ)
(B.1)
For path x˜:
(R(k˜S−1)a˜1S−2 + (1− τ˜S−1)w(k˜S−1)− a˜2S−1)−σ
= β[p(R(k˜S)(R(kS)a˜
2
S−1 + φ˜neww(k˜S))
−σ)
+ q(R(kˆS)(R(kˆS)aˆ
2
S−1 + φˆoldw(kˆS))
−σ)
+ (1− p− q)(R(k¯S)(R(k¯S)a¯2S−1 + φ¯oldw(k¯S))−σ)]
(B.2)
The generation who is young at time S − 1 chooses a1S−1 and a2S using
stochastic Euler equations. There will be distinct stochastic Euler equations for
each of the three paths; only the tilde path is represented below as (B.3) and (B.4).
188
((1− τ˜S−1)w(k˜S−1)− a˜1S−1)−σ
= β[p(R(kS˜)(R(k˜S)a˜
1
S−1 + (1− τ˜S)w(k˜S)− a˜2S)−σ)
+ q(R(kˆS)(R(kˆS)aˆ
1
S−1 + (1− τˆS)w(kˆS)− aˆ2S)−σ)
+ (1− p− q)(R(k¯S)(R(k¯S)a¯1S−1 + (1− τ¯S)w(k¯S)− a¯2S)−σ)]
(B.3)
p(R(k˜S)a˜
1
S−1 + (1− τ˜S)w(k˜S)− a˜2S)−σ)
+ q(R(kˆS)aˆ
1
S−1 + (1− τˆS)w(kˆS)− aˆ2S)−σ)
+ (1− p− q)(R(k¯S)a¯1S−1 + (1− τ¯S)w(k¯S)− a¯2S)−σ)]
= β[p(R(k˜S+1)(R(k˜S+1)a˜
2
S + φ˜neww(k˜S+1))
−σ)
+ q(R(kˆS+1)(R(kˆS+1)aˆ
2
S + φˆneww(kˆS+1))
−σ)
+ (1− p− q)(R(k¯S+1)(R(k¯S+1)a¯2S + φ¯oldw(k¯S+1))−σ)]
(B.4)
The generation who is young at time S chooses a1S and a
2
S+1 using non-
stochastic Euler equations along the tilde path (reform in period S). Along the
other two paths, a1S is chosen using a stochastic Euler equation.
Path x˜, non-stochastic Euler equations (3.6) and (3.7).
Path xˆ (reform in S + 1), stochastic Euler equation:
((1− τˆold,S)w(kˆS)− aˆ1S)−σ
= β[q(R(kˆS+1)(R(kˆS+1)aˆ
1
S + (1− τˆnew,S+1)w(kˆS+1)− aˆ2S+1)−σ)
+ (1− q)(R(k¯S+1)(R(k¯S+1)a¯1S + (1− τ¯old,S+1)w(k¯S+1)− a¯2S+1)−σ)]
(B.5)
q((R(kˆS+1)aˆ
1
S + (1− τˆnew,S+1)w(kˆS+1)− aˆ2S+1)−σ)
+ (1− q)((R(k¯S+1)a¯1S + (1− τ¯old,S+1)w(k¯S+1)− a¯2S+1)−σ)
= βR(kˆS+2)(R(kˆS+2)aˆ
2
S+1 + φˆneww(kˆS+2))
−σ
(B.6)
189
I illustrate the impact of three-period timing uncertainty by using the same
example as Figure 16 extended to three-periods. The economy begins in a steady
state with social security, and taxes are increased in either period t = 3 (with
probability p = 0.5), in period t = 4 (with probability q = 0.25), or taxes are
increased in period t = 5. This example is displayed below in Figure B.3. As
was the case with two-period timing uncertainty, young agents who expect that
they may have to pay higher taxes next period increase their savings relative to the
previous generation. Agents facing a possible tax hike also save more than agents
who know taxes won’t be increased in their life-cycle. Agents facing uncertainty do
not increase their savings as much as agents in the counterfactual world who know
taxes will be raised next period. This is evident by young saving in period t = 2,
a12. Agents who know taxes will be raised in period t = 3 save the most (depicted
in the graph as the open triangle). Agents who know taxes won’t be raised until
period t = 4 or t = 5 save the least (in graph, open square and open circle). Agents
who face uncertainty save an amount in between. Similarly, the savings of the
young in period t = 3, a13 is highest for the agents who face a certain tax increase in
t = 4; savings is lowest for agents who know the tax increase isn’t coming until
t = 5; agents who face uncertainty save in between the two certain amounts.
The initial values, policy parameters, and final steady state for this experiment
are listed in Table B.2.
I also present the time paths for an economy in a steady state that
experiences a large social security benefit reduction in either period t = 3, t = 4, or
t = 5. This policy experiment is presented in Figure B.4. The probability of policy
change in period t = 3 is p = 0.5, probability of change in period t = 4 is q = 0.25,
and if benefits are not decreased in period 3 or 4, benefits are cut in period 5 with
190
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FIGURE B.3. Time paths for an economy is in a steady state and experiences
a large tax increase in period 3, 4, or 5. Time paths are plotted for reform that
happens with certainty in any of the three period. Time paths are plotted in
addition for an economy that faces probability p=0.5 of reform in period 3,
probability q=0.25 that reform takes place in period 4. If reform doesn’t occur
in period 3 or 4, it occurs with certainty in period 5. There are three possible paths
for the economy under this type of stochastic policy, either the reform takes place
in period 3 (these paths are labeled uncertain reform in 3), period 4 (these paths
are labeled uncertain reform in 4), or in period 5 (labeled uncertain reform in 5).
191
Fig Initial policy New policy Initial values Final steady state(s)
τ0 τ1 φ τ0 τ1 φ k b a1 a2 k b a1 a2
B.3 0.076 0.8 0.2 0.1 0.8 0.2 0.098 0.009 0.081 0.18 0.121 -0.031 0.058 0.163
B.4 0.076 0.8 0.2 0.077 0.8 0.18 0.098 0.009 0.081 0.18 0.113 -0.012 0.071 0.176
TABLE B.2. Policy Experiments with three-period timing uncertainty. Experiment
B.3: Economy in steady state with social security, taxes raised in either period
t = 3, t = 4, or t = 5. Experiment B.4: Economy in steady state with social
security, benefits cut in either period t = 3, t = 4, or t = 5.
certainty. The results of this experiment are qualitatively similar to the two-period
example presented in Figure 19. The initial values, policy parameters, and final
steady state for this experiment are listed in Table B.2.
Two Special Cases with Balanced Social Security Budget
In simplified version of the model without a Leeper tax (τ 1 = 0, and τ 1 = τ),
the social security budget is balanced in every time period if the government does
not run a primary deficit or surplus (τt = φt/(1 + n)(2 + n)). The government’s
initial stock of debt b0 is continually rolled over according to (3.17), which simplifies
to bt+1(1 + n) = Rtbt. In this framework, two steady states are possible. In the
first, the initial government stock is zero and the government never borrows. That
is, bt = 0 for all t. When that is true, the model collapses to a three variable,
three equation system (which will be discussed below). In the second case, steady
state capital occurs at the golden rule level, and steady state bonds are at a
corresponding level implied by equation (3.22).
Balanced Budget, No Government Debt
In the simplest case with balanced budgets, the government simply never
issues debt. The model can be written as a system of three equations by replacing
τt with φt/(1 + n)(2 + n) and noting that bt = 0 for all t. This simplified model
192
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FIGURE B.4. Time paths for an economy is in a steady state and experiences
a large benefit reduction in period 3, 4, or 5. Time paths are plotted for reform
that happens with certainty in any of the three period. Time paths are plotted
in addition for an economy that faces probability p=0.5 of reform in period 3,
probability q=0.25 that reform takes place in period 4. If reform doesn’t occur in
period 3 or 4, it occurs with certainty in period 5. There are three possible paths
for the economy under this type of stochastic policy, either the reform takes place
in period 3 (these paths are labeled uncertain reform in 3), period 4 (these paths
are labeled uncertain reform in 4), or in period 5 (labeled uncertain reform in 5).
193
has two predetermined variables (capital and savings of the young), and one free
variable (savings of the middle aged). The model has a unique, stable steady.
The unique steady state of the model is the collection {k, a1, a2} that solves:((
1− φ
(1 + n)(2 + n)
)
w(k)− a1
)−σ
− βR(k)
(
R(k)a1 +
(
1− φ
(1 + n)(2 + n)
)
w(k)− a2
)−σ
= 0 (B.7)
(
R(k)a1 +
(
1− φ
(1 + n)(2 + n)
)
w(k)− a2
)−σ
− βR(k)(R(k)a2 + φw(k))−σ = 0 (B.8)
(1 + n)k − (1− δ)k − a1 1 + n
2 + n
− a2 1
2 + n
= 0 (B.9)
Equation (B.9) reveals the unique steady state value of capital k =
2+n
n+δ
((1 + n)a1 + a2). The steady state values of a1 and a2 are given jointly by
equation (B.7) and (B.8).
Analysis of the linearized model reveals the steady state is stable. Define
Xˆt = (a
1
t−1, a
2
t , kt)
′ and Xˆt+1 = (a1t , a
2
t+1,kt+1)
′. Then the balanced budget, no debt
model can be written as
Gˆ(XˆtXˆt+1) = 0 (B.10)
where Gˆ(XˆtXˆt+1) is the following:((
1− φt
(1 + n)(2 + n)
)
w(kt)− a1t
)−σ
− βR(kt+1)
(
R(kt+1)a
1
t +
(
1− φt+1
(1 + n)(2 + n)
)
w(kt+1)− a2t+1
)−σ
= 0
(B.11)
(
R(kt+1)a
1
t +
(
1− φt+1
(1 + n)(2 + n)
)
w(kt+1)− a2t+1
)−σ
− βR(kt+2)(R(kt+2)a2t+1 + φt+2w(kt+2))−σ = 0 (B.12)
(kt+1)(1 + n)− (1− δ)kt − a1t
1 + n
2 + n
− a2t
1
2 + n
= 0 (B.13)
Total differentiation of Gˆ yields Hˆ1dXˆt+1 + Hˆ2dXˆt + Hˆ3dZt = 0, where Hˆ are
matrices of partial derivatives. Multiplying through by −(Hˆ−12 ), the linearized
model can be written as:
dXˆt+1 = Mˆ1dXˆt + Mˆ3dZt
where Mˆ1 = −(Hˆ−12 )Hˆ1. The stability of the system is governed by the eigenvalues
of Mˆ1 evaluated at a steady state Xˆ
∗. I found a unique steady state for each
194
parameter combination I tested. Additionally, two of the eigenvalues of Mˆ1
evaluated at a steady state were always within in the unit circle, and one was
outside. Thus, for a given set of parameters, the model has a unique, stable steady
state (See Azariadis (1993) page 59).
Balanced Budget, Constant Debt
The second special case that is possible when the government balances
the social security budget each period, is a steady state at the golden rule level
of capital. Recall the government budget equation:
bt+1(1 + n) = R(kt)bt +
1
(2 + n)(1 + n)
φtw(kt)− τtw(kt) (3.17)
When the government balances the social security budget, this simplifies to:
bt+1(1 + n) = R(kt)bt (B.14)
The steady state version of equation (B.14) is only true at the golden rule level of
capital, when R(k) = 1 + n.
b(1 + n) = R(k)b (B.15)
When capital is at the golden rule level and R(k) = 1 + n, any level of bonds
solves equation (B.15). The unique level of bonds associated with the steady state
is jointly determined by the remaining steady state equations, reprinted below:
((1− τ)w(k)− a1)−σ = βR(k)(R(k)a1 + (1− τ)w(k)− a2)−σ(
R(k)a1 + (1− τ)w(k)− a2)−σ = βR(k) (R(k)a2 + φw(k))−σ
b(1 + n) = R(k)b
(k + b)(1 + n) = (1− δ)k + a1 1 + n
2 + n
+ a2
1
2 + n
195
The stability of the golden rule steady state is governed by the eigenvalues
of matrix M1, as described in Section III and equation (3.30).
196
APPENDIX C
LIFE-CYCLE HORIZON LEARNING APPENDIX
Stability of REE under Learning
Determinacy is often used as an equilibrium selection device in rational
expectations models with multiple equilibria. A complementary approach to
selecting equilibria is to conduct stability analysis under learning. If agents’ (non-
rational) expectations and forecasts converge to a rational expectations equilibrium
(REE) in a model with learning, the REE is said to be stable under learning (see
G. W. Evans and Honkapohja (2001)). W. Branch et al. (2013) show that the
unique REE in an infinite-horizon Ramsey model is stable under N-step optimal
learning. Similarly, I numerically verify that determinate REE in my model are
stable under LCH learning and FHL Learning.
Given constant (potentially incorrect) expectations pe =
(Re, we, be, aj,eterminal)
′, the learning dynamics of the FHL model asymptotically
converge to p = (R,w, b, aj)′. Similarly, the dynamics of the LCH model converge
from beliefs pe = (Re, we, be)′ to p = (R,w, b)′. This converge from beliefs to actual
prices is called a T-map.
T : RJ−H+3 → RJ−H+3 (C.1)
A fixed point of the T map is E-stable if it locally stable under the ordinary
differential equation dp
dτ
= T (p)− p. E-stability requires the real parts the eigenvalues
of the derivative matrix dT < 1. I have numerically verified all determinate steady
197
states in the paper are E-stable under LCH learning and FHL learning (at all
horizons).
The stability under learning of the determinate REE in this model can be
illustrated graphically. The dynamics of the learning model converge to the REE
given arbitrary initial conditions near the steady state. Figure C.5 illustrates this
convergence in the LCH and FHL learning models, calibrated with J = 6 (six
period lives) and parameters calibrated as detailed in section IV. The example
begins with capital, bonds, assets, and expectations of all of these things below
the steady state values. Agents update their expectations as they receive more
information and eventually learn the steady state values, which are indicated
with a dashed red line. Agents overshoot the steady state initially. This pattern
is observed in G. W. Evans et al. (2009).
W. Branch et al. (2013) find that shorter planning horizons converge to
the REE more quickly; and that agents make larger errors (and there are larger
aggregate fluctuations) when the planning horizon is longer. I find a similar result;
the fluctuations in the economy are greatest for life-cycle horizon learning and
decrease as the planning horizon decreases.
Recessions and Life-cycle Learning
The examples in the main text illustrate the welfare cost of life-cycle and
finite-horizon learning in the face of social security policy changes. An alternative
framework to view the welfare cost of learning is to simulate a recession and
compare the lifecycle utility of agents in the RE model with the utility of agents
in a learning model. I do this using the same CEV as defined in equation 4.31.
198
0 2 4 6 8 10
3.30
3.35
3.40
3.45
3.50
3.55
3.60
3.65
time
age 1
assets
a1
0 2 4 6 8 10
6.4
6.5
6.6
6.7
6.8
time
age 2
assets
a2
life cycle planning horizon
planning horizion 4
planning horizion 3
planning horizion 2
planning horizion 1
FIGURE C.5. This graphs shows convergence to a steady state for the Finite
Horizon Lifecycle Learning model (FHL) and Life-cycle Horizon Learning model
(LCH) with different planning horizons. The red dashed line indicates the steady-
state. The initial values were chosen to be near the steady-state. The initial
expectations were all set equal to the initial values. The top graph shows the path
of age 1 assets; the bottom graph shows the path of age 2 assets. The converge of
the other asset choices (a3, . . . , a5), capital, and bonds, follow a similar oscillating
pattern.
199
I simulate as recession as a surprise decrease in the TFP factor A. The
recession is depicted in Figure C.6 below. I model the recession as a one period
reduction in the TFP factor from A = 10 to A = 9.5. The recession reduces output
per worker by 2.9% and aggregate consumption by 7.53%.1 Agents in the RE
model know that the recession is only one period. Agents in the learning models
do not know the recession will end; they continue to forecast prices and bonds
adaptively. The recession generates cycles in capital and bonds, as agents overshoot
the steady state before converging. Note that converge back to the steady state
is non-monotonic in the rational and learning models (see Azariadis, Bullard, and
Ohanian (2004) for a discussion of cycles in rational, multi-period OLG economies).
The oscillatory dynamics are larger and last longer in the learning dynamics. The
decrease in output and consumption are also larger in the learning models relative
to the RE baseline.2 The welfare cost of experiencing the recession in a model with
learning compared to the same recession in the RE model is depicted in Figure C.7.
The welfare cost varies by cohort. The cost is greatest (smallest CEV) for agents
born the period after recession in the LCH model. Agents born in period t = 11,
following a recession in period t = 10, are worse off in the learning model by 1.10%
of period consumption. The largest welfare cost occurs after the recession due to
the depressed capital stock in the learning model relative to the RE baseline.
The welfare cost is greatest to the cohorts born the period after the
recession in the LCH model since they experience lower consumption relative to
1During the Great Recession (2007-2009) output fell by 7.2% and consumption fell by 5.4%.
The average post-war recession saw a decline in output and consumption of 4.4% and 2.1%,
respectively. See Christiano (2017).
2Eusepi and Preston (2011) show that adaptive learning in an RBC model creates dynamics
that more closely match that data than a standard, RE model. They also show that the shocks
required to generate a realistic recession are smaller in the model with learning.
200
0 5 10 15 20 25 30
7.1
7.2
7.3
7.4
7.5
7.6
7.7
time
ca
pi
ta
l
k
0 5 10 15 20 25 30
1.30
1.35
1.40
1.45
time
bo
nd
s
b
RE
life cycle planning horizon
planning horizion 4
planning horizion 3
planning horizion 2
planning horizion 1
FIGURE C.6. Equilibrium paths for capital and bonds in an economy with a
recession in period t=10 that lasts for 1 period. The population growth rate is
constant at n=0.01, and social security policy is constant τ 0=0.1516, τ 1=0.045,
φ=0.4. The recession is a surprise reduction in the TFP factor from A=10 to
A=9.5 in period t=10. The TPF factor is depressed for a single period, and the
returns to the pre-recession level. In the learning models, the gain parameter
γ=0.93.
0 5 10 15 20
-0.010
-0.005
0.000
0.005
cohort birth year
C
E
V
life cycle planning horizon
planning horizion 4
planning horizion 3
planning horizion 2
planning horizion 1
FIGURE C.7. Consumption equivalent variation measure of the welfare cost
of a recession. The population growth rate is constant at n=0.01, and social
security policy is constant τ 0=0.1516, τ 1=0.045, φ=0.4. The recession is a surprise
reduction in the TFP factor from A=10 to A=9.5 in period t=10. The TPF factor
is depressed for a single period, and the returns to the pre-recession level. In the
learning models, the gain parameter γ=0.93.
201
8 10 12 14 16 18
6.3
6.4
6.5
6.6
6.7
6.8
6.9
time
c1
8 10 12 14 16 18
7.3
7.4
7.5
7.6
7.7
7.8
7.9
time
c2
8 10 12 14 16 18
8.5
8.6
8.7
8.8
8.9
9.0
9.1
time
c3
8 10 12 14 16 18
9.8
9.9
10.0
10.1
10.2
10.3
10.4
time
c4
8 10 12 14 16 18
11.3
11.4
11.5
11.6
11.7
11.8
11.9
time
c5
8 10 12 14 16 18
12.8
13.0
13.2
13.4
13.6
time
c6
RE
life cycle planning horizon
planning horizion 4
planning horizion 3
planning horizion 2
planning horizion 1
FIGURE C.8. Path of consumption for the rational and learning models for a
recession in period t=10. Consumption falls the most during the recession for
the 1-period ahead finite horizon learners (dark orange line). Consumption is
depressed for many periods in the LCH learning model (yellow line, often very
close to the green line). The population growth rate is constant at n=0.01, and
social security policy is constant τ 0=0.1516, τ 1=0.045, φ=0.4. The recession is a
surprise reduction in the TFP factor from A=10 to A=9.5 in period t=10. The
TPF factor is depressed for a single period, and the returns to the pre-recession
level. In the learning models, the gain parameter γ=0.93.
the other generations. Consumption falls during the recession (period t = 10) and
during the following for rational agents. The dip in consumption after the recession
reflect the decision of agents to save as the economy recovers. Consumption also
falls for learning agents during the recession. Since the learning agents adaptively
forecast prices, they anticipate depressed wages and interest rates the following
period, so they continue to consume less for several periods. This is depicted in
Figure C.8.
202
The welfare cost of the learning depends on both the length and depth
of the recession, as well as on the gain parameter γ. If the recession were three
periods rather than one, the greatest welfare cost would be 1.98% of period
consumption (compared to 1.10% for a one period recession). Similarly, if the
recession is deeper and the TFP factor falls from A = 10 to A = 9.1 instead
of A = 9.5, the largest welfare cost of a one period recession is 2.26% of period
consumption. In all three cases, the largest welfare cost occurs under LCH learning
for the cohort of agents born after the recession. The welfare cost also depends
on the gain parameter γ. When the gain parameter is larger, the welfare cost is
also larger. This is because agents place more weight on recent data when the gain
parameter is high, and so they expect prices to be more depressed following a one
period recession than agents using a smaller gain. The minimum CEV for LCH
agents experiencing a one period recession is 0.7% of period consumption for a gain
parameter of 0.01, and is equal to 1.10% of period consumption for a gain of 1.
Note on Golden Rule
The golden rule level of capital maximizes consumption of all generations
and occurs when R = (1 + n) in this model.
kgr =
(
α
n+ δ
) 1
1−α
The consumption profile is smooth over the lifecycle if R = β−1. This is not
the case in general. R = β−1 if
ksmooth =
(
α
β−1 + δ − 1
) 1
1−α
Thus, the golden rule level of capital corresponds to the level of capital such
that R = β−1 (and the lifecycle consumption profile is constant) when n = 1−β
β
.
203
The social security program crowds out capital and can be set at a level to
ensure k = kgr, or k = ksmooth; it’s less clear why a planner would want the latter.
204
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