Technical Paper Series

Congressional Budget Office

Washington, DC

Analyzing an Aging Population—A Dynamic

General Equilibrium Approach

Shinichi Nishiyama

Congressional Budget Office

Washington, DC

E-mail: shinichi.nishiyama@cbo.gov

February 2004

2004-03

Technical papers in this series are preliminary and are circulated to stimulate discussion and

critical comment. These papers are not subject to CBO’s formal review and editing processes.

The analysis and conclusions expressed in them are those of the authors and should not be interpreted as those of the Congressional Budget Office. References in publications should be cleared

with the authors. Papers in this series can be obtained at www.cbo.gov (select Publications and

then Technical Papers).
Abstract

This paper shows the macroeconomic and welfare implications of an aging population in

the United States, using an overlapping-generations model with heterogeneous households.

The model uses three population projections in Social Security Administration (2003), and

generates economies as equilibrium transition paths from 1961 to 2200. The paper demonstrates how several different population projections and government financing assumptions—

to make the Social Security system sustainable—affect households’ decisions and welfare.

One of the policy experiments shows that an immediate increase in the payroll tax may not

improve the welfare of future generations as much as it reduces the welfare of current generations.

Journal of Economic Literature Classification Numbers: D9, H3, H5, J1.

Key Words: Social Security, Life Cycle, Overlapping Generations, Heterogeneous Agent,

Aging Population.
1 Introduction

Many proposals for reforming the Social Security system in the United States have been

advanced in recent years. For example, U.S. Representatives Jim Kolbe and Charlie Stenholm proposed the 21st Century Retirement Security Act in 1998 and revised it in 1999 and

2001. In addition, the President’s Commission (2001) proposed three possible reform plans

of U.S. Social Security.

Although many have already analyzed how much those proposals would improve the

actuarial balance, the macroeconomic and welfare implications of those reform plans are

still uncertain. However, to evaluate those effects of Social Security reform, a reasonable

baseline economy is also required.

This paper constructs possible baseline economies with an aging population to analyze

Social Security reform plans, using an overlapping generations (OLG) model with heterogeneous households. In this model, households receive idiosyncratic working ability shocks

and mortality shocks. Then, the paper shows the effects of simple reform plans as policy

experiments.

In this process, the following two aspects are stressed:

First, like most other developed countries, the population distribution of the United Sates

is aging and, accordingly, the economy cannot be described as a stationary equilibrium. This

paper solves the model for equilibrium transition paths from 1961 through 2200, using the

actual (and projected) age-population distribution and mortality rates in this period.1

Second, with a realistic population projection, the current-law Social Security system is

not sustainable. To solve the model for an equilibrium transition path, the model needs to

have an additional financing assumption to close the intertemporal budget constraint for the

Social Security system.2

In Social Security Administration (2003), Trustees of Social Security have used three

1

At the beginning of 1961, households in the model realize an aging population and choose their optimal

consumption, labor supply, and savings based on the correct population projection. The model assumes that this

adjustment process to an aging population from the initial steady state in 1961 is completed before 2004, the

starting year of policy analyses. 2

Some examples of government financing assumptions for the Social Security system are whether the payroll

tax is increased or benefits are reduced, or both, and when the government changes the payroll tax rate or benefit

replacement rates.

1
possible population projections—alternative II (intermediate), alternative I (low cost), and

alternative III (high cost)—to evaluate the sustainability of the current Social Security system. This paper uses the same three population projections by extrapolating those projections

beyond 2080.

Regarding the financing assumption, this paper assumes that the payroll tax is increased

and benefits are reduced when the trust funds are depleted, so that each of those policy

changes covers half of the deficit and that the trust funds are kept at zero thereafter.3

Then, the paper shows the effect of alternative financing assumptions—the payroll tax

is increased when the trust funds are depleted but benefits are kept at the current-law level;

alternatively, benefits are reduced when the trust funds are depleted but the payroll tax rate

is kept at the current-law level; and, finally, the payroll tax rate is increased immediately in

2004 by 10 percent.

The rest of the government budget is made as simple as possible. Other spending is

considered government consumption, which is not in the utility function of the model. In

addition, the government budget is balanced by adjusting government consumption, so that

the per-capita government net wealth grows at the same rate of the productivity growth.4

Compared to a balanced growth path, per-capita private wealth increases because of the

improved longevity and larger life cycle savings in this model. Private saving and saving rates

in the aging baseline are above the levels in the balanced growth path in 2004 but decline

throughout the period from 2004 to 2200. Per-capita labor supply increases until 2013, then

decreases monotonically to a level below that in 2004.

The capital-labor ratio rises in the aging baseline economy and, as a result, the interest

rate falls and the wage rate rises significantly as the population ages. How much factor

prices change depends on the population projection and the financing assumption. One of

the interesting findings from the numerical experiments is that an immediate increase in the

payroll tax rate may not improve the welfare of future generations as much as it reduces the

welfare of current generations, although the trust funds would last much longer.

3

This assumption is arbitrary. The trust funds are accounting devices and the government does not have to

change the law in that timing. 4

There are some alternative assumptions. For example, the government consumption is population indexed,

and individual income tax rates are adjusted so that the rest of the government budget is balanced.

2
This paper is not the first one that calibrates a dynamic general equilibrium OLG model

to an aging society. Auerbach and Kotlikoff (1987) analyzed the effects of two stylized aging

populations on the saving rate and Social Security system, using a nonstochastic overlapping

generations model without heterogeneity in agents’ earning ability. More recently, De Nardi,

 ̇Imrohoroglu, and Sargent (1999) analyzed Social Security reform plans by solving their  ̆

model for 1975-2200.5 Ríos-Rull (2000) calibrates his model to the Spanish economy with

a stylized aging population. Kotlikoff, Smetters, and Walliser (2001) and Fehr, Jokisch, and

Kotlikoff (2003) analyze the effect of demographic changes in the United States.

The rest of the paper is laid out as follows: Section 2 describes the model economy; Section 3 explains the calibration of the model; Section 4 shows the baseline economy with an

aging population; Section 5 shows a few simple policy experiments; and Section 6 concludes

the paper.

2 Model

The base model used in this paper is a standard overlapping generations growth model

with uninsurable idiosyncratic working ability shocks and mortality shocks.6 The economy consists of heterogeneous households, a perfectly competitive representative firm, and a

government with a full commitment technology. Time is discrete, and a period of the model

corresponds to a year. Regarding the openness of the economy, this paper assumes two polar

cases—a closed economy and a small open economy.

The Household’s Problem. Households are heterogeneous with respect to their ages, working abilities, asset holdings, and working histories. For simplicity, all households are assumed to be two-earner married couples of the same age, who make their decisions jointly.

Every year, a large number of new households of age 20 enter into the economy. A

household of age i receives idiosyncratic working ability shock, ei, at the beginning of each

5

The main differences between De Nardi et al. (1999) and the present paper are as follows: the former uses

a quadratic utility function so that the household’s decision rules and the laws of motion are expressed by linear

functions; assumes idiosyncratic endowment shocks rather than working ability shocks; and assumes that labor

income tax or consumption tax is adjusted once in 10 years. 6

The base model is similar to those in Aiyagari (1994), Huggett (1996), and many others, although Aiyagari (1994) assumed infinitely-lived agents. The model is also an extension of that in Nishiyama (2002) and

Nishiyama and Smetters (2003).

3
year t, and chooses its optimal consumption ci, working hours hi, and end-of-period wealth

holding ai+1, taking a government policy rule Ψt, a population projection Φt, and a series

of factor prices and policy variables Ωt, as given.7 At the end of each year, a fraction 1−φi,t

of households die. Households are possibly alive until age 120, and the survival rate at the

end of age 120, φ120,t, is assumed to be zero.

Let si denote the individual state of an age i household,

si = (i, ei, ai, bi), (1)

where i ∈ I = {(0, ..., 19,) 20, ..., 120} is the household’s age, ei ∈ E = [emin, emax] is

its working ability (measured by its hourly wage), ai ∈ A = [amin, amax] is its beginningof-period asset holding, and bi ∈ B = [bmin, bmax] is its average historical earnings.8 Let St

denote the state of the economy at the beginning of year t,

St = (xt (si),WS,t,WG,t), (2)

where xt (si) is the measure of households for si ∈ I × E × A × B, WS,t is the beginningof-period Social Security trust funds, and WG,t is the rest of the government net wealth.9 Let

Ψt denote the government policy rule known at the beginning of year t,

10

Ψt = {WS,t+1,WG,s+1, τPO,s (.), τPH,s (.), trSS,s (.), τI,s (.), CG,s}∞

s=t , (3)

where τP O,s (.) is a payroll tax function for the Old-Age, Survivors, and Disability Insurance (OASDI), τPH,s (.) is a payroll tax function for the Hospital Insurance (HI), trSS,s (.)

is an OASDI benefit function, τI,s (.) is a progressive income tax function, and CG,s is gov7

The government does not solve its optimization problem to determine the policy. The government policy

rule Ψt is equivalent with its financing rule regarding the Social Security budget and the rest of the government

budget, which is assumed to be credible. The population projection Φt is deterministic. Because there are no

aggregate shocks in this economy, households can perfectly foresight a series of future factor prices and policy

variables Ωt based on the information currently available. 8

Ages i ≤ 19 are used to calculate the average number of dependent children and the population of age 20

households in each year t. The average historical earnings bi are the approximation of the Average Monthly

Indexed Earnings (AIME) multiplied by 12 and used to calculate the household’s Social Security benefits in the

model. 9

In other words, xt (si) is the joint distribution of households in year t multiplied by the population in year t. 10At least one of the series in Ψt is unknown to the households. In the policy experiments below, the government announces that it will increase the payroll tax and reduce benefits when the trust funds are depleted, so that

the trust funds will not be negative. Although the government does not announce explicitly when and how much

it will change the payroll tax rate and benefits, households in the model have rational expectations of both the

timing and sizes of those changes.

4
ernment’s consumption. In the present model, government consumption, CG,t, is not in the

utility function of a household.

The household’s problem is

v (si, St; Ψt, Φt) = max ci,hi,ai+1

ui (ci, hi) + βφi,tE [v (si+1,St+1; Ψt+1, Φt+1)|ei] (4)

subject to

ai+1 = 1

1 + μ {wteihi + (1 + rt)ai − τI,t (wteihi, rtai, trSS,t (i, bi)) (5)

−τP O,t (wteihi) − τPH,t (wteihi) + trSS,t (i, bi) − ci} ≥ amin,

a20 = 0, a121 ≥ 0,

where ui(.) is a period utility function of an age i household, β is the time-preference factor,

φi,t is the survival rate at the end of age i, wt is the wage rate, and rt is the real rate of return

to capital.11 Individual variables, except for working hours, are normalized by the steadystate per capita growth rate μ. Let πi,i+1 (ei+1|ei) denote the conditional probability for the

age i + 1 working ability being ei+1 when the age i working ability is ei. Then,

E [v (si+1,St+1)|ei] = ]

E

v (si+1, St+1) πi,i+1 (ei+1|ei) dei+1. (6)

At the beginning of the next period, the state of the household and the state of the economy become

si+1 = (i + 1, ei+1, ai+1 + qt, bi+1) with πi,i+1 (ei+1|ei), (7)

St+1 = (xt+1 (.),WS,t+1, WG,t+1), (8)

where qt denotes accidental bequests that a household receives at the end of the period,

and WS,t+1 and WG,t+1 are determined by the government budget constraints. The average

11The variable wt is the wage rate per efficiency unit of labor, which is normalized to unity in 2001, and wtei

denotes the hourly wage of each household of age i with working ability ei in year t. The variable rt is the

pre-tax real market rate of return to capital, which is around 6.25 percent in 2003 in the model. The real rate of

return to the Social Security trust funds is adjusted to 3.0 percent in 2003 so that the present model generates

projections similar to those of the Trustees Report.

5
historical earnings bi follows

bi+1 =







0 if i ≤ 24

1

i−24 {(i − 25)bi wtwt−1 + min (wteihi/2, wehmax

t )} if 25 ≤ i ≤ 59

(1 + μ)−1bi if i ≥ 60,

(9)

where wehmax

t is the OASDI payroll tax cap. Under the current law, the Average Indexed

Monthly Earnings (AIME) is calculated from the highest 35 years of earnings. For simplicity,

the model assumes that the highest 35 years of earnings correspond to those years of age

between 25 and 59.12

The decision rule of households is shown as

d(si, St; Ψt, Φt) = {ci (si,St; Ψt, Φt), hi (si, St; Ψt, Φt), ai+1 (si, St; Ψt, Φt)} ,

for i = 20, ..., 120.

13

The Measure of Households. Let xt (si) denote the measure of households, and let Xt (si)

be the corresponding cumulative measure. The measure of households is adjusted by the

long-run population growth rate ν.

The measure of newborn people in year t + 1 is calculated, if it is not exogenously

determined, from the age-dependent fertility rates, fi,t, and age-population distribution, that

is

xt+1 (s0) = [

49

i=15

]

E×A×B

fi,tdXt (si), (10)

where s0 = (0, −, 0, 0).

14 Then, the measures of people of ages 1, ..., 19 are calculated from

the survival rates at the end of each age,

xt+1 (si+1) = φi,t

1 + ν

]

E×A×B

dXt (si) for i ≤ 19, (11)

12Earnings before age 60 are wage indexed and earnings after age 60 are price indexed. The approximation of

AIME by the average historical earnings follows previous Social Security literature, for example, Huggett and

Ventura (1999) and De Nardi et. al. (1999). 13The departure from the previous literature occurs because the household’s decision depends not only on its

own state, the state of the economy, and the government policy rule, but also the household’s belief (perfect

foresight) on the population transition. 14The measure of newborn people is equal to the new born population divided by 2, because the decision

unit in the model economy is a married couple. In this paper, Equation (10) is used to extrapolate population

projections in Social Security Administration (2003) beyond 2080.

6
where si = (i, −, 0, 0). A household of age 20 is assumed to have no initial wealth and

working history. So, U

E dXt (20, e20, 0, 0) is the population of age 20 households in year t.

Let 1[a=y] be an indicator function that returns 1 if a = y and 0 if a 9= y. The law of motion

of the measure of households is, for i = 20, ..., 120,

xt+1 (si+1) = φi,t

1 + ν

]

E×A×B

1[ai+1=ai+1(si,St;Ψt,Φt)+qt] (12)

×1[bi+1=bi+1(wteihi(si,St;Ψt,Φt),bi)]πi,i+1(ei+1|ei)dXt (si).

For simplicity, accidental bequests, qt, due to uncertain life span are captured by the government and distributed to all surviving working-age households in a lump-sum manner.15

The Firm’s Problem. National wealth Wt is the sum of total private wealth, the Social

Security trust funds WS,t, and the rest of the government net wealth WG,t. Total labor supply

Lt is measured in efficiency units. Then,

Wt = [

120

i=20

]

E×A×B

ai dXt (si) + WS,t + WG,t, (13)

Lt = [

120

i=20

]

E×A×B

ei hi(si,St; Ψt, Φt) dXt (si). (14)

In a closed economy, capital stock Kt is equal to national wealth, that is, Kt = Wt, and

gross national product Yt is determined by a constant-returns-to-scale production function,16

Yt = F(Kt, Lt). (15)

The profit-maximizing condition of the representative firm is

FK(Kt, Lt) = rt + δ, (16)

FL(Kt, Lt) = (1 + τ

P O,t + τPH,t)wt, (17)

where δ is the depreciation rate of capital, and τ

P O,t + τPH,t is the marginal payroll tax rate

15Some of the computationally feasible extensions of the treatment of accidental bequests are, first, assuming

age-dependent accidental bequest receipts based on the average age difference between parents and children;

and, second, making accidental bequest receipts stochastic, i.e., the wealth left by a deceased household is given

to another relatively young household by lottery. 16In a closed economy, gross national product equals gross domestic product.

7
for the representative firm.17

In a small open economy, factor prices r∗

t and w∗t are fixed at the international levels of

those, and domestic capital stock KD,t and labor supply Lt are determined so that the firm’s

profit maximizing condition satisfies, that is,

FK(KD,t, Lt) = r∗

t + δ, (18)

FL(KD,t, Lt) = (1 + τ

PO,t + τP H,t)w∗t (19)

Gross domestic product YD,t and gross national product Yt are calculated as

YD,t = F(KD,t, Lt), (20)

Yt = (r∗

t + δ)Wt + (1 + τP O,t + τPH,t)w∗t Lt, (21)

respectively.

The Government’s Policy Rule. Government tax revenue consists of income tax, TI,t,

payroll tax for OASDI, TP O,t, and payroll tax for HI, TPH,t. Those revenues are calculated

as

TI,t = [

120

i=20

]

E×A×B

τI,t (wteihi(si, St; Ψt, Φt), rtai, trSS,t (i, bi)) dXt (si), (22)

TP O,t = 2 × [

120

i=20

]

E×A×B

τPO,t (wteihi(si, St; Ψt, Φt)) dXt (si), (23)

TPH,t = 2 × [

120

i=20

]

E×A×B

τPH,t (wteihi(si,St; Ψt, Φt)) dXt (si). (24)

For the computational convenience, the payroll tax functions, τPO,t(.) and τPH,t(.), are assumed to show the taxes levied on employees (married couples) only. Because the same

taxes are also levied on employers, the aggregate tax revenues in equations (23) and (24) are

multiplied by 2.

17Here, (1 + τ

P O,t + τPH,t)wt denotes the cost of an efficiency unit of labor for the firm. For high income

households, whose labor income is above the OASDI payroll tax cap, τ 

P O,t +τPH,t is 0.0145 rather than 0.0765

under the current law. In the calibration below, it is assumed to be 0.0594 so that the goods market clear in 2001.

8
Social Security (OASDI) benefit expenditure, T rSS,t, equals

T rSS,t = [

120

i=20

]

E×A×B

trSS,t (i, bi) dXt (si). (25)

The laws of motion of the trust funds and the rest of government net wealth—both of those

are normalized by productivity growth and long-run population growth—are

WS,t+1 = 1

(1 + μ) (1 + ν)

{(1 + rt)WS,t + TP O,t − T rSS,t} , (26)

WG,t+1 = 1

(1 + μ) (1 + ν)

{(1 + rt)WG,t + TI,t + TPH,t − CG,t} . (27)

Definition Recursive Competitive Equilibrium (Equilibrium Transition Path): Let si =

(i, ei, ai, bi) be the individual state of households, let St = (xt(si),WS,t, WG,t) be the aggregate state of the economy, let Ψt be the government policy rule known at the beginning

of year t,

Ψt = {WS,t+1,WG,s+1, τPO,s (.), τPH,s (.), trSS,s (.), τI,s (.), CG,s}∞

s=t,

and let Φt be the perfect-foresight population projection. A series of factor prices, accidental bequests, the government policy variables, and the parameters φs of government policy

functions,

Ωt={rs, ws, qs, WS,s+1,WG,s+1, CG,s, φs}∞

s=t;

the value function of households, {v (si,Ss; Ψs, Φs)}∞

s=t; the decision rule of households,

{d(si, Ss; Ψs, Φs)}∞

s=t = {ci(si,Ss; Ψs, Φs), hi(si, Ss; Ψs, Φs), ai+1(si,Ss; Ψs, Φs)}∞s=t;

and the measure of households, {xs(si)}∞

s=t, are in a recursive competitive equilibrium if, in

every year s = t, ...,∞, each household solves the utility maximization problem (1) – (5)

taking Ψt and Φt as given; the firm solves the profit maximization problem, and the capital

and labor markets clear, that is, (13) – (19) hold; the government policy rules satisfy (22) –

(27); and the goods market clears.

9
3 Calibration

This section explains the procedure, assumption, and parameterization of the model to

construct baseline economies with an aging population as equilibrium transition paths.

3.1 The Procedure

To solve the model for equilibrium transition paths from 2004 through 2200, we need to

set the initial state of the economy, S2004, which includes the joint distribution of households

at the beginning of year 2004. Since the economy in 2004 is not stationary and the stationary

condition cannot be used to generate the distribution of households, this paper constructs

S2004 by solving the model from 1961 through 2003.18 The final year 2200 of the transition

path is assumed to be in a steady state.19

1. Calibrate the model described in Section 2 to the 2001 U.S. economy as if it is in a

steady state, using the actual age-population distribution and mortality rates in 2001.20

Choose the time preference factor β and the share parameter for consumption α in the

utility function so that the steady-state economy is consistent with the 2001 U.S. economy with respect to the capital-output ratio and the average annual working hours of

households. Also choose total factor productivity A so that the wage rate w is normalized to unity.

2. Calibrate the model to the 1961 U.S. economy as if it is in a steady state, using the

actual age-population distribution and mortality rates in 1961 and the parameters β, α,

and A obtained in the previous step.21 Choose the parameters in the OASDI payroll

tax function and OASDI benefit function so that total OASDI tax revenue and benefit

expenditure in the model economy are equal to those in 1961 as percentages of GDP.

18It is also possible to construct the joint distribution of households from survey data, such as the Survey of

Consumer Finances. In which case, however, the initial state of the economy is not consistent with the model. 19In the final version of the paper, I am planning to solve the model through 2300 for more accurate computation of the economy. 20To solve the model for a steady-state equilibrium using an population distribution and mortality rates in 2001,

we have to assume the households in the model falsely believe that the population distribution is stationary. 21Again, to solve the model for a steady-state equilibrium using an population distribution and mortality

rates in 1961, we have to assume the households in the model falsely believe that the population distribution is

stationary. This assumption is probably acceptable because this step is merely a preparation for constructing a

model economy from 2004 to 2200.

10
3. Solve the model for an equilibrium transition path from 1961 through 2200, using the

same parameters β, α, and A, and the population projection through 2200. For 1961-

2002, choose the parameters in the payroll tax function and the benefit function, and

the rate of return to the trust funds so that those are consistent with the U.S. data as

percentages of GDP.22

3.2 The Government Policy Rule Ψt

Regarding the government policy rule,

Ψt = {WS,s+1,WG,s+1, τPO,s (.), τPH,s (.), trSS,s (.), τI,s (.), CG,s}∞

s=t,

in the baseline economy with an aging population, this paper makes the following financing

assumptions.

3.2.1 The Social Security (OASDI) Budget

The OASDI surplus—the difference between the OASDI payroll tax revenue, TPO,t, and

the benefit expenditure, T rSS,t—is added to the Social Security trust funds, WS,t, as long as

the trust funds are positive. That is,

WS,t+1 = 1

(1 + μ) (1 + ν)

max {(1 + rt)WS,t + TP O,t − T rSS,t, 0} ,

for all t if WS,s > 0 for all s ≤ t.

23 Once the trust funds are depleted, the trust funds are kept

at zero thereafter, and either the payroll tax rate is increased, or benefit replacement rates

are reduced proportionally, or both by splitting the deficit evenly, to close the intertemporal

budget constraint of Social Security.24 That is,

WS,t+1 = 0,

T rSS,t − TPO,t = (1 + rt)WS,t,

22For policy experiments, the model is solved for equilibrium transition paths in 2004-2200, using the state of

the economy in 2004. 23All aggregate variables in the model are normalized using the steady-state (long-run) growth rate μ and

population growth rate ν. 24In the baseline economy, this paper assumes that, when the trust funds are depleted, both the payroll tax rate

is increased and benefit replacement rates are reduced to cover the deficit evenly.

11
for all t if there exists WS,s = 0 for s ≤ t+ 1. If the trust funds are not depleted before 2104

(100 years from now), to obtain the final steady-state equilibrium, the trust funds are kept at

the same level (after growth adjustments) thereafter, and either the payroll tax rate or benefit

replacement rates or both of those are changed to close the intertemporal budget constraint.

That is,

WS,t+1 = WS,t,

T rSS,t − TPO,t = {(1 + rt) − (1 + μ) (1 + ν)} WS,t,

for all t ≥ 2104 if WS,s > 0 for all s ≤ 2104.

3.2.2 The Rest of the Government Budget

The rest of the government net wealth WG,t is simply assumed to grow at the same rate

as the long-run growth rate μ and year-by-year population growth rate νt, that is,

WG,t+1 = 1

(1 + μ) (1 + ν)

{(1 + μ) (1 + νt) WG,t} .

To close the rest of the government intertemporal budget constraint, government consumption is determined as the residual,25

CG,t = (1 + rt)WG,t − (1 + μ) (1 + ν)WG,t+1 + TI,t + TPH,t.

For years before 2004, this paper assumes that the Social Security budget is combined with

the rest of the government budget, or the trust funds are not pre-funded. That is,

WS,t+1 + WG,t+1 = 1

(1 + μ) (1 + ν)

(1 + μ) (1 + νt) (WS,t + WG,t),

and

CG,t = (1 + rt) (WS,t + WG,t) − (1 + μ) (1 + ν) (WS,t+1 + WG,t+1)

+ TI,t + TPO,t + TPH,t − T rSS,t.

12
Table 1: The Source of the Population Projection

Year

1941-1960 1961-2080 2080-2200

Population of ages:

0-99 by age The 2003 Trustees Report Extrapolated

100-120 in total Alternatives I, II, and III

100-120 by age Estimated

Table 2: Age-Specific Fertility Rates (Per 1,000 Females, 2003)

Age of Females Year 2003 Year 2080-2200

15-19 57.0 57.0 ×φf,t

20-24 112.6 112.6 ×φf,t

25-29 113.1 113.1 ×φf,t

30-34 85.0 85.0 ×φf,t

35-39 35.6 35.6 ×φf,t

40-44 7.0 7.0 ×φf,t

45-49 0.3 0.3 ×φf,t

Total Fertility Rate 2.07

Source: U.S. Census Bureau (2003). In the model, the adjustment factor φf,t

of fertility rates and corresponding total fertility rate in 2080-2200 are 1.0581

and 2.19, respectively, in Alternative I, 0.9441 and 1.95 in Alternative II, and

0.8294 and 1.72 in Alternative III. In all three projections, the sex ratio at

birth (male per female) is 1.0498 in 2080-2200.

3.3 The Population Projection Φt

This paper uses the population projections—alternative II (intermediate), alternative I

(low cost), and alternative III (high cost)—in Social Security Administration (2003). The

Social Security Administration projected the population distributions through 2080. As Table

1 shows, populations in 2081-2200 are extrapolated using the age-specific fertility rates of

women aged 15-49 in 2003 and an adjustment factor φf,t = φf,2080. (See Table 2.)

The adjustment factor is calculated for each population projection to match the total fertility rate in 2080.26 The age-population distributions in 1961 and 2001, and the projections

25We can alternatively assume that either income tax or the payroll tax for HI are changed to close the budget

constraint. This paper focuses mainly on the Social Security (OASDI) budget and tries to avoid the policy

influence from the rest of the government budget.

26In addition, the populations of people 100-120 years of age in 1961-2080 are estimated using the population

13
Table 3: Parameters

Time preference parameter β 1.013

Share parameter for consumption α 0.686

Coefficient of relative risk aversion γ 4.0

Capital share of output θ 0.300

Depreciation rate of capital stock δ 0.047

Long-term real growth rate μ 0.018

Long-term population growth rate ν 0.00594, 0.00154, -0.00312

Total factor productivity A 0.982

in selected years (every 25 years from 2003, and 2200) are shown in Figure 1. The long-run

annual population growth rates ν of the population projection alternatives I, II, and III are

0.594%, 0.154%, and -0.312%, respectively.27

The survival rates φi,t of households at the end of age i in year t are simply calculated

from the population projection. For people below age 45, φi,t tend to be greater than one

because the projections include immigrations.

Figure 2 shows the population share of elderly people (aged 65 or above) and the ratio of

the working-age (aged 20-64) population to the elderly population for three alternative population projections. The latter ratio rises very slightly until year 2006, then falls monotonically

(except for Alternative I) thereafter.

3.4 The 2001 Steady-State Economy

Table 3 summarizes the parameter choices. For the 2001 steady-state economy, the degree of time preference β is chosen so that the capital-output ratio is 2.74, total factor productivity A is chosen so that the wage rate w equals unity, and the share parameter for

consumption α is chosen so that the average annual working hours of married couples aged

between 20 and 64 are 3,368 hours. The capital share parameter θ of the production function

and depreciation rate δ are calculated from macroeconomic statistics.28

data in 1941-1960 and mortality rates in 1998 with proportionate year adjustments, so that the total populations

of ages 100-120 are equal to the numbers in the Trustees’ projection. 27The program used to calculate the population projections in this paper will be provided upon request.

28The calibration strategy for the 2001 steady-state economy is similar to those in Nishiyama (2002) and

Nishiyama and Smetters (2003), but many parameters are revised.

14
Figure 1: Population Projections in Selected Years: Author’s calculation from the data in

Social Security Administration (2003)

Year 2200

0

2

4

6

8

10

12

14

16

0 10 20 30 40 50 60 70 80 90 100 110 120

Age

Populat oi n ni Mill oi ns

Alternative I Alternative II

Alternative III Year 2001

Year 1961

0

1

2

3

4

5

6

7

8

0 10 20 30 40 50 60 70 80 90 100 110 120

Age

Population in Millions

Year 2103

0

1

2

3

4

5

6

7

8

0 10 20 30 40 50 60 70 80 90 100 110 120

Age

Population in Millions

Year 2078

0

1

2

3

4

5

6

7

8

0 10 20 30 40 50 60 70 80 90 100 110 120

Age

Population in Millions

Year 2028

0

1

2

3

4

5

6

7

8

0 10 20 30 40 50 60 70 80 90 100 110 120

Age

Popu al t oi n in Millions

Year 2053

0

1

2

3

4

5

6

7

8

0 10 20 30 40 50 60 70 80 90 100 110 120

Age

Popu al tion in Millions

15
Figure 2: The Share of Elderly People: Author’s calculation from the data in Social Security

Administration (2003)

The Share of Ages 65+ in Total population

5

10

15

20

25

30

35

1960 1980 2000 2020 2040 2060 2080 2100 2120

Year

Percent

The Ratio of Ages 20-64 to Ages 65+

1

2

3

4

5

6

1960 1980 2000 2020 2040 2060 2080 2100 2120

Year

Ratio

Alternative I Alternative II

Alternative III

The following sections describe the choice of functional forms and parameter values, and

the choice of target variables and values.

Household’s Utility Function. The model has elastic labor supply and uses the following

Cobb-Douglas utility function with constant relative risk aversion (CRRA), which is compatible with the existence of a steady state,

ui(ci, hi) =

q

((1 + ni/2)−ζ ci)α(hmax

i − hi)1−α

r1−γ

1 − γ ,

where γ is the coefficient of relative risk aversion, ni is the number of dependent children,

ζ is the consumption adjustment parameter, and hmax

i is the maximum working hours. The

coefficient of relative risk aversion is assumed to be 4.0, following Auerbach and Kotlikoff

(1987).29 In this setting, the growth-adjusted β becomes β(1 + μ)α(1−γ)

, which is 0.976 in

the calibration. The numbers of dependent children by parents’ age are calculated from the

Panel Study of Income Dynamics (PSID) 1993 Family Data (see Table 4). Multiplying these

numbers by a time-variant adjustment factor φn,t, the model calculates the average number

29There is no strong consensus about the coefficient of relative risk aversion γ. For the estimates of γ in

previous literature, see Auerbach and Kotlikoff (1987) and Prescott (1986). Citing these two papers, Huggett

(1996) used γ of 1.5 and 3.0. More recently, Gourinchas and Parker (2002) estimated γ varying between 0.5 and

1.4, and Laitner (2003) estimated γ at 2.3 or 2.0.

16
Table 4: Number of People Under 18 Years of Age in a Married Household

Age of Number of people Age of Number of people Age of Number of people

parents under age 18 parents under age 18 parents under age 18

20 1.02 35 1.83 50 0.61

21 0.96 36 1.87 51 0.50

22 0.98 37 1.90 52 0.42

23 0.89 38 1.96 53 0.35

24 0.96 39 1.85 54 0.29

25 1.08 40 1.76 55 0.23

26 1.12 41 1.75 56 0.22

27 1.15 42 1.66 57 0.19

28 1.19 43 1.51 58 0.15

29 1.29 44 1.43 59 0.15

30 1.36 45 1.30 60 0.13

31 1.49 46 1.13 61 0.09

32 1.60 47 0.96 62 0.10

33 1.68 48 0.82 63 0.10

34 1.77 49 0.70 64 0.09

Source: Author’s calculations from the Panel Study of Income Dynamics 1993 Family Data.

In the model, these numbers are multiplied by an adjustment factor φn,t to be consistent with

the population of ages 0-19 in each year t.

of children of ages 0-19, ni, in an age i household, which is consistent with each of three

population projections. The consumption adjustment parameter is assumed to be 0.6.30

The annual working hours in the model are the sum of the working hours of a husband

and a wife. The average working hours of married households between ages 20 and 64 are

3,368 hours in the 1998 Survey of Consumer Finances (SCF). The maximum working hours

hmax

i are set to be 5,460 for all households, which is the 95th percentile in the same survey.

In this calibration, the parameter α is chosen to be 0.686 so that average working hours of

age 20 and age 64 become 3,368 hours in the 2001 steady-state economy.31

Working Ability. The working ability in this calibration corresponds to the hourly wage

(labor income per hour) of each household in the 1998 SCF. The average hourly wage of a

30Since 20.6 = 1.516, a married couple with two dependent children consume about 52 percent more than a

married couple with no children if other things are equal. This increase is slightly smaller than the assumption

in Elmendorf and Sheiner (2000) but larger than the estimates in Laitner (2003). 31According to a separate policy experiment not shown in the present paper, the uncompensated wage elasticity

of labor supply is about 0.15 in the short run under this utility parameter setting.

17
Table 5: Working Abilities of a Household (in U.S. Dollars per Hour)

Percentile Age cohorts

20-24 25-29 30-34 35-39 40-44 45-49

e1 0-20th 3.83 5.42 5.42 6.93 6.12 6.59

e2 20-40th 7.07 8.64 9.76 11.28 11.36 12.70

e3 40-60th 8.68 10.91 13.46 15.01 15.59 17.22

e4 60-80th 10.67 14.01 18.08 19.96 22.09 23.22

e5 80-90th 14.05 17.52 27.17 25.27 30.89 31.58

e6 90-95th 18.20 22.48 33.71 33.38 48.59 44.31

e7 95-99th 28.43 32.64 54.11 52.16 76.13 86.50

e8 99-100th 36.81 46.09 167.15 186.47 221.34 301.99

Percentile Age cohorts

50-54 55-59 60-64 65-69 70-74 75-79

e1 0-20th 5.48 3.52 0.00 0.00 0.00 0.00

e2 20-40th 11.53 10.06 4.54 0.00 0.00 0.00

e3 40-60th 16.16 14.26 11.18 2.82 0.00 0.00

e4 60-80th 23.44 21.28 18.16 10.37 1.81 0.00

e5 80-90th 32.14 30.93 28.56 19.48 12.57 0.00

e6 90-95th 43.01 44.10 59.36 27.68 29.03 1.96

e7 95-99th 78.61 85.29 96.22 59.34 64.91 14.25

e8 99-100th 314.59 379.44 421.55 299.25 195.73 146.14

Source: Nishiyama and Smetters (2003). The authors’ calculations from the

1998 SCF data.

married couple (family members #1 and #2 in SCF) used for the calibration is calculated by

Hourly Wage = Regular and Additional Salaries (#1 + #2) + Welfare or Assistance

max {Working Hours (#1 + #2), 520} .

To capture the earnings risk a household is exposed to more precisely, unemployment or

worker’s compensation, Temporary Assistance for Needy Families (TANF), food stamps,

and other forms of welfare or assistance are added to the salaries before calculating the hourly

wage. Table 5 shows the eight discrete levels of working abilities of five-year age cohorts.32

Using a shape-preserving cubic spline interpolation, the working ability of each age from 20

32One observation of the age 20-24 cohort, which has an hourly wage of $193.01, is ignored. To avoid

dividing by zero, the minimum of the denominator is assumed to 520 hours (10 hours a week per couple). Under

this assumption, the hourly wage of a married couple that works only 260 hours is discounted by 50 percent,

assuming implicitly that the couple could not work longer due to, for example, illness or unemployment. In the

real economy, however, some households have fairly high working ability but choose not to work (for example,

because of schooling).

18
to 79 is obtained. The average hourly earnings of production workers have increased by 16.7

percent during the years from 1997 to 2001.33 In the calibration, the numbers in the table are

multiplied by 1.167 to convert the hourly wages in 1997 into those in 2001.

Markov Transition Matrix. The Markov transition matrix, Γ, of working ability is calculated from the hourly wage of people ages 30-39 in 1991 in the PSID individual data. To

make the working ability process more persistent, the matrix is calculated as the transition

from the average of years 1989 and 1990 to the average of years 1990 and 1991.

Γ =





0.7674 0.2049 0.0183 0.0045 0.0049 0.0000 0.0000 0.0000

0.1810 0.6033 0.1844 0.0129 0.0000 0.0086 0.0046 0.0052

0.0388 0.1517 0.6768 0.1220 0.0011 0.0046 0.0050 0.0000

0.0126 0.0361 0.1039 0.7210 0.0980 0.0139 0.0145 0.0000

0.0000 0.0081 0.0332 0.2360 0.6306 0.0676 0.0145 0.0100

0.0000 0.0000 0.0000 0.0582 0.3224 0.5303 0.0891 0.0000

0.0007 0.0000 0.0000 0.0354 0.0000 0.2827 0.6433 0.0379

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.3553 0.6447





,

where Γ(j, k) = π(ei+1 = ek

i+1 | ei = e

j

i).

The Firm’s Production Function. Production takes the Cobb-Douglas form,

F(Kt, Lt) = AtKθ

t L1−θ

t .

To compute GNP, the model uses the sum of working hours in efficiency units as total labor

supply Lt. The capital share of output θ is chosen by

θ = 1 − Compensation of Employees + (1 − θ) × Proprietors’ Income

National Income + Consumption of Fixed Capital .

The number of θ in 2000 is 0.30.34 The annual growth rate μ is assumed to be 1.8 percent.

Total factor productivity A is chosen to be 0.982 so that the wage per unit of efficient labor

is normalized to be unity.

Fixed Capital and Private Wealth. Fixed capital K is the sum of private fixed assets and

government fixed assets. In 2000, private fixed assets are $21,165 billion, government fixed

33Source: Department of Labor, Bureau of Labor Statistics.

34Source: Department of Commerce, Bureau of Economic Analysis. The average of θ in years between 1996

and 2000 is 0.31.

19
Table 6: Marginal Payroll Tax Rates in 2001

Labor Income per Marginal Tax Rate (%)

worker (wteihi/2) OASDI HI

$0 – $80,400 6.2 × φτPO 1.45

$80,400 – 0.0 × φτPO 1.45

The same amount of tax is levied to employers.

Table 7: OASDI Replacement Rates in 2001

AIME (b65/12) Marginal Replacement Rate (%)

$0 – $561 90.0 × φtrSS

$561 – $3,381 32.0 × φtrSS

$3,381 – 15.0 × φtrSS

The adjustment factor also reflects the DI and survivors insurance.

assets are $5,743 billion, and the government debt held by the public is $3,410 billion.35

From these numbers, the government net wealth is set to a 9.5 percent of total private wealth

in the initial steady-state economy. In 2000, the capital-GDP ratio is 2.74. The time preference parameter β is chosen so that the capital-GDP ratio of the steady state economy (a

balanced growth path) is 2.74.

The depreciation rate of fixed capital δ is chosen by the steady-state condition,

δ = Total Gross Investment

Fixed Capital − μ − ν.

In 2000, private gross fixed investment accounted for 17.2 percent of GDP, and government

(federal and state) gross investment accounted for 3.3 percent of GDP.36 When the capitaloutput ratio is 2.74, the ratio of gross investment to fixed capital is 7.5 percent. Subtracting

the productivity and population growth rates, the annual depreciation rate is assumed to be

4.7 percent.

The Current Law Social Security System. The tax rate levied on both employers and

employees for OASDI is 6.2 percent, and the tax rate for HI is 1.45 percent. In 2001, employee wages above $80,400 were not taxable for OASDI. (See Table 6.) So, the firm’s

35ibid. and Congressional Budget Office (2001). 36ibid.

20
profit-maximization problem becomes

w × (1 + Marginal Payroll Tax Rate) = AFL(K, L),

where the marginal payroll tax rate is either 0.0765 or 0.0145 for high-earnings workers.

Because the marginal payroll tax rates are not uniform across households, the calibration

uses the average payroll tax rate (total payroll tax paid by employers divided by total labor

income) instead, so that the Walras’ law holds. Social Security benefits are calculated from

each worker’s Average Indexed Monthly Earnings (AIME), b65/12, and the replacement rate

schedule in the United States. (See Table 7.)

In 2001, the OASDI payroll tax revenue was 5.25 percent of GDP, and OASDI benefit

expenditure was 4.28 percent of GDP. In the model, the ratio of statutory payroll tax revenue

to GDP is higher because the model assume all households are two-earner married couples.

The payroll tax in the model is multiplied by an adjustment factor φτP O , which is equal to

0.812, so that the size of payroll tax revenue as a percentage of GDP equals 5.25 percent.

(See Table 9.) In the model, survivors’ benefits and disability insurance are simply assumed

to be proportional to the benefits for old-age workers. The statutory old age benefits in the

model are multiplied by an adjustment factor φtrSS , which equals 1.199, so that the size of

OASDI benefits equals 4.28 percent of GDP in 2001.

Federal and Local Income Taxes. Federal income tax and state and local taxes are assumed to be at the level in year 2001 before the Bush tax cuts, “Economic Growth and Tax

Relief Reconciliation Act of 2001 (EGTRRA).” Every household in the model is assumed to

be a married couple, which is subject to the standard deduction and exemptions. In 2001 the

standard deduction for a married household was $7,600, and the exemption was $2,900 per

person. The exemptions for dependent children follow Table 4. Table 8 shows the statutory

marginal tax rates before EGTRRA.

All of the tax brackets, standard deduction, and exemption are assumed to be growth

adjusted so that there is no real bracket creep. Because the economic income of a household

is larger than taxable income, the effective tax rates are lower than the statutory rates. In

2000, the ratio of total individual income tax revenue to nominal GDP was 0.102 and the ratio

21
Table 8: Marginal Individual Income Tax Rates in 2001 (Married Household, Filed Jointly)

Taxable Income Marginal Income Tax Rate (%)

$0 – $45,200 15.0 × φτI

$45,200 – $109,250 28.0 × φτI

$109,250 – $166,500 31.0 × φτI

$166,500 – $297,350 36.0 × φτI

$297,350 – 39.6 × φτI

The standard deduction is $7,600 and exemption per person is $2,900.

of corporate income tax to GDP was 0.021. The statutory federal income tax is multiplied

by the adjustment factor φτI , which is equal to 0.775, so that income tax revenue (including

corporate income tax) is 12.3 percent of GDP in the 2001 steady-state equilibrium. Also,

since the effective capital income tax rates are lower than labor income tax rates, the tax

function is adjusted so that the tax rate on capital income (including corporate income) is

about 30 percent lower than that of labor income. State and local income tax in the model

is simply a 4.0 percent flat tax for an income (excluding Social Security benefits) above the

same standard deduction and exemptions.

3.5 Equilibrium Transition Paths in 1961–2200

This paper solves the model for transition paths from 1961 to construct baseline economies

in 2004-2200 under several different assumptions. Rather than implementing all relevant policy changes in 1961-2003, for simplicity, this paper considers only the changes in the OASDI

payroll tax, benefits, and the trust funds in this period. Table 9 shows the sizes of the OASDI

payroll tax revenue and benefit expenditure as percentages of GDP in 1961-2002. In the

model, the adjustment factors, φτP O,t and φtrSS,t, in the payroll tax function and benefit

function, respectively, are changed in each year so that the sizes of payroll tax revenue and

benefit expenditure in the model are consistent with those in the U.S. historical data.

The adjustment factors of the OASDI payroll tax function in 2002 are 0.810 [0.808] in

a closed [small open] economy. These numbers are smaller than 1.0, because the model

assumes all households as two-earner married couples and this assumption pushes up the

payroll tax revenue without any adjustments. The adjustment factors of the OASDI benefit

function in 2002 are 1.231 [1.218] in a closed [small open] economy. These are about 51-52

22
Table 9: OASDI Payroll Tax Revenue and Benefit Expenditure

OASDI Payroll Tax Revenue OASDI Benefits Expenditure

As a Percent- Adjustment Factor φτP O As a Percent- Adjustment Factor φtrSS

age of GDP Closed Small Open age of GDP Closed Small Open

1961 2.26 0.345 0.344 2.34 0.740 0.729

1962 2.23 0.341 0.340 2.47 0.771 0.761

1963 2.53 0.387 0.386 2.49 0.770 0.759

1964 2.54 0.389 0.387 2.44 0.749 0.739

1965 2.39 0.366 0.364 2.54 0.774 0.764

1966 2.87 0.440 0.439 2.54 0.767 0.756

1967 3.06 0.470 0.468 2.57 0.772 0.762

1968 3.01 0.461 0.460 2.74 0.818 0.810

1969 3.25 0.498 0.497 2.72 0.807 0.800

1970 3.39 0.520 0.518 3.07 0.904 0.897

1971 3.45 0.528 0.527 3.30 0.963 0.956

1972 3.50 0.535 0.534 3.36 0.973 0.967

1973 3.78 0.578 0.577 3.72 1.069 1.064

1974 3.96 0.606 0.604 3.90 1.113 1.108

1975 3.96 0.605 0.603 4.10 1.161 1.157

1976 3.96 0.604 0.603 4.15 1.166 1.163

1977 3.91 0.595 0.594 4.17 1.163 1.161

1978 3.90 0.593 0.592 4.05 1.122 1.121

1979 4.04 0.614 0.613 4.07 1.120 1.121

1980 4.20 0.638 0.637 4.31 1.178 1.179

1981 4.48 0.681 0.680 4.50 1.225 1.226

1982 4.50 0.684 0.683 4.79 1.298 1.300

1983 4.61 0.700 0.699 4.72 1.271 1.273

1984 4.66 0.708 0.707 4.47 1.201 1.202

1985 4.77 0.725 0.724 4.42 1.184 1.184

1986 4.78 0.726 0.725 4.42 1.177 1.178

1987 4.76 0.723 0.722 4.30 1.137 1.138

1988 5.00 0.761 0.760 4.25 1.120 1.120

1989 5.04 0.768 0.767 4.21 1.108 1.108

1990 5.14 0.784 0.783 4.27 1.122 1.121

1991 5.14 0.784 0.783 4.48 1.177 1.177

1992 5.02 0.766 0.765 4.53 1.190 1.190

1993 4.93 0.753 0.752 4.55 1.195 1.194

1994 4.96 0.758 0.757 4.49 1.184 1.182

1995 4.93 0.755 0.754 4.49 1.189 1.186

1996 4.94 0.757 0.756 4.44 1.183 1.178

1997 4.98 0.764 0.763 4.35 1.170 1.164

1998 5.01 0.770 0.769 4.27 1.160 1.153

1999 5.08 0.782 0.781 4.16 1.144 1.136

2000 5.13 0.792 0.790 4.15 1.151 1.141

2001 5.25 0.812 0.810 4.28 1.199 1.188

2002 5.23 0.810 0.808 4.34 1.231 1.218

2003-2200 0.793 0.791 1.177 1.167

Author’s calculation from the data in Table 4A3, Social Security Administration (2002), also available at

http://www.ssa.gov/OACT/STATS/t4a3Income.html and http://www.ssa.gov/OACT/STATS/t4a3Outgo.html.

OASDI payroll tax revenue used in this table excludes interest income. The adjustment factors in 1961-2002

are calculated so that the sizes of revenue and expenditure are consistent with the historic data as percentages

of GDP. The adjustment factors in 2003-2200 are the averages of those in 1998-2002.

23
percent larger than those of the payroll tax function, because both survivors’ benefits and

disability insurance are included into the benefits.37 For 2003-2200, the adjustment factors

are simply assumed to be the average of those in 1998-2002.

The Rate of Return to the Trust Funds. Because the historical rates of return to the

Social Security trust funds in 1961-2002 are in general different from those generated in

the model, the sizes of the trust funds are adjusted so that those are also consistent with the

data as percentages of GDP. The nominal rates of return to the Social Security trust funds

that the Social Security Administration assumes in 2003-2041 are between 5.93 percent and

6.09 percent.38 The Social Security Administration also assumes the CPI inflation rate of 3.0

percent from 2007 and 2041.39

The real market rate of return to capital (before capital income and corporate income

taxes) in a small open economy is fixed at about 6.25 percent. The market rate of return in

2003 in a closed economy is 6.22 percent, which is roughly equal to that in a small open

economy. So, the model assumes a risk premium of 3.25 percent between the market rate of

return and the rate of return to the trust funds, so that the real rate of return to the trust funds

in 2003 is around 3.0 percent in the model economy.

4 Baseline Economy

Baseline economies are obtained as equilibrium transition paths with three population

projections—intermediate (alternative II), low cost (alternative I), and high cost (alternative

III)—in Social Security Administration (2003).

Table 10 shows aging baseline economies—a closed economy and a small open economy

—with the intermediate population projection (alternative II), in which the long-run population growth rate ν is 0.154 percent. The Social Security trust funds are depleted in 2053

[2056] in a closed [small open] economy.40 The model assumes that the OASDI payroll tax

37Because there are no health shocks in the model, disabilities insurance benefits are assumed to be entitled to

households 65 years of age and older proportionally to their OASDI benefits. 38These numbers are calculated from the projections of interest income and Social Security assets in Table

VI.F9 in the Trustees Report (available at http://www.ssa.gov/OACT/TR/TR03/lr6F9-2.html). 39The CPI inflation rates in 2003-2006 are 2.35, 2.39, 2.70, and 2.92 percent. See Table VI.F7 (available at

http://www.ssa.gov/OACT/ TR/TR03/lr6F7-2.html). 40The trust funds last longer in the present paper than those in Social Security Administration (2003) with all

24
Table 10: Equilibrium Transition Paths with an Aging Population Alternative II (Changes

from the population-adjusted balanced growth path through 2004)

Year Average over Years

2028 2053 2078 2200 04-28 29-53 54-78 04-78

Closed Economy

%ch(National Wealth) 21.4 22.7 26.1 26.1 13.3 21.6 24.9 19.9

%ch(Labor Supply) -1.0 -2.7 -4.1 -4.9 1.2 -2.2 -3.4 -1.5

%ch(GNP) 5.3 4.3 4.1 3.5 4.6 4.4 4.3 4.4

%ch(Consumption) 6.9 7.3 7.2 7.1 4.2 7.3 7.2 6.2

ch(Private Savings / GDP%) -1.98 -1.81 -2.69 -3.03 -0.90 -1.74 -2.42 -1.69

ch(OASDI Payroll Tax / GDP%) -0.17 -0.21 0.72 0.85 -0.11 -0.16 0.60 0.11

ch(OASDI Benefits / GDP%) 1.33 1.79 1.84 1.97 0.35 1.94 1.73 1.34

ch(Trust Funds / GDP%) 13.1 -13.5 -14.0 -14.0 9.9 0.2 -14.0 -1.3

ch(Interest Rate%) -1.44 -1.63 -1.90 -1.95 -0.80 -1.54 -1.79 -1.38

%ch(Wage Rate) 6.3 6.9 7.9 8.0 3.4 6.8 7.4 5.9

Small Open Economy

%ch(National Wealth) 25.3 29.8 34.4 35.4 14.8 27.7 32.5 25.0

%ch(Labor Supply) -2.9 -4.9 -7.0 -7.9 0.1 -4.4 -6.1 -3.5

%ch(GNP) 5.6 5.6 5.5 5.2 4.6 5.3 5.5 5.1

%ch(Consumption) 6.5 8.2 8.5 8.8 3.7 7.6 8.2 6.5

ch(Private Savings / GDP%) -0.87 -0.01 -1.44 -1.88 -0.23 -0.68 -1.05 -0.65

ch(OASDI Payroll Tax / GDP%) -0.12 -0.13 0.95 1.07 -0.07 -0.12 0.73 0.18

ch(OASDI Benefits / GDP%) 1.64 2.48 2.06 2.18 0.51 2.25 2.02 1.59

ch(Trust Funds / GDP%) 17.3 -8.1 -14.0 -14.0 11.6 5.7 -13.6 1.2

Percent changes or changes in percentage points from the balanced growth path of 1.8% per-capita real

growth through the 2004 economy. Years 2028, 2053, and 2078 are the 25th, 50th, and 75th years,

respectively, from 2004. The trust funds are depleted in 2053 in a closed economy and in 2056 in a small

open economy. When the trust funds are depleted, both the payroll tax rate is raised and benefits are cut

to balance the OASDI thereafter.

rate is increased and benefit replacement rates are reduced when the trust funds are depleted,

so that each of those policy changes finances half of the deficit and that the trust funds are

kept at zero thereafter. The numbers in the table are either percent changes or changes in percentage points from the balanced growth path through the economy in 2004 with population

adjustments.

Table 11 shows aging economies with the low cost population projection (alternative I),

in which the long-run population growth rate ν is 0.594 percent. The trust funds are depleted

in 2082 [2100] in a closed [small open] economy. Finally, Table 12 shows aging economies

population projections, partly because the current Social Security system uses the consumer price index (CPI)

for the cost of living adjustment of benefits, and CPI inflation rates are higher than personal consumption deflator

growth rates. When the cost of living adjustment was modified in the model, the trust funds would be depleted

in 2045 [2047] under the population projection alternative II in a closed [small open] economy.

25
Table 11: Equilibrium Transition Paths with an Aging Population Alternative I (Changes

from the population-adjusted balanced growth path through 2004)

Year Average over Years

2028 2053 2078 2200 04-28 29-53 54-78 04-78

Closed Economy

%ch(National Wealth) 15.4 9.2 6.4 5.9 10.7 11.8 7.5 10.0

%ch(Labor Supply) -3.0 -5.6 -5.8 -6.2 0.3 -4.9 -5.8 -3.5

%ch(GNP) 2.2 -1.3 -2.3 -2.7 3.2 -0.2 -2.0 0.3

%ch(Consumption) 4.8 3.0 2.6 2.4 3.3 3.9 2.7 3.3

ch(Private Savings / GDP%) -2.64 -2.52 -2.05 -2.47 -1.19 -2.73 -2.27 -2.06

ch(OASDI Payroll Tax / GDP%) -0.15 -0.12 -0.11 0.18 -0.09 -0.13 -0.11 -0.11

ch(OASDI Benefits / GDP%) 1.09 1.47 1.49 1.30 0.26 1.45 1.53 1.08

ch(Trust Funds / GDP%) 15.7 1.9 -11.7 -14.0 10.7 8.6 -5.4 4.6

ch(Interest Rate%) -1.25 -1.06 -0.90 -0.89 -0.71 -1.17 -0.97 -0.95

%ch(Wage Rate) 5.4 4.5 3.7 3.5 3.0 5.0 4.1 4.0

Small Open Economy

%ch(National Wealth) 19.0 15.4 12.3 9.4 12.1 17.1 13.7 14.3

%ch(Labor Supply) -4.5 -6.9 -7.0 -7.3 -0.6 -6.4 -7.1 -4.7

%ch(GNP) 2.5 -0.2 -1.2 -2.3 3.2 0.7 -0.9 1.0

%ch(Consumption) 4.4 3.4 3.2 2.8 2.9 4.0 3.2 3.4

ch(Private Savings / GDP%) -1.91 -2.23 -1.99 -2.17 -0.70 -2.27 -2.10 -1.69

ch(OASDI Payroll Tax / GDP%) -0.10 -0.08 -0.08 0.24 -0.07 -0.09 -0.08 -0.08

ch(OASDI Benefits / GDP%) 1.33 1.57 1.56 1.35 0.39 1.61 1.61 1.20

ch(Trust Funds / GDP%) 19.6 10.2 -1.4 -14.1 12.2 14.9 4.2 10.4

See notes of Table 10.

The trust funds are depleted in 2082 in a closed economy and in 2100 in a small open economy.

with the high cost population projection (alternative III), in which the long-run population

growth rate ν is -0.312 percent. With the same government financing assumption, the trust

funds are depleted in 2044 [2046] in a closed [small open] economy.

With all population projections, per-capita national wealth is larger in aging economies

than in the balanced growth path. With population projections alternative II and alternative

III, per-capita national wealth grows faster than per-capita GDP in 2004-2078. Households

need to accumulate larger life-cycle saving for the period after retirement, because they expect lower mortality rates, lower Social Security benefits, and higher payroll tax rates. Contrary to the forecasts calculated from a model with fixed saving rates by age cohort, the

present model predicts that private wealth per household will be higher under more severe

assumption about aging (Compare Table 10 and Tables 11 and 12).41

41This paper does not consider the increase in government debt due to the increase in Medicare spending

caused by an aging population. The results will change if we assume the financing rule for the rest of the

26
Table 12: Equilibrium Transition Paths with an Aging Population Alternative III (Changes

from the population-adjusted balanced growth path through 2004)

Year Average over Years

2028 2053 2078 2200 04-28 29-53 54-78 04-78

Closed Economy

%ch(National Wealth) 29.4 42.8 51.8 50.5 16.7 35.7 49.0 33.8

%ch(Labor Supply) 1.3 0.2 -3.9 -6.0 2.2 0.5 -1.7 0.4

%ch(GNP) 9.0 11.4 10.2 8.3 6.3 10.0 11.4 9.2

%ch(Consumption) 8.7 10.8 10.8 10.0 4.9 10.2 11.0 8.7

ch(Private Savings / GDP%) -0.84 -1.34 -2.91 -3.62 -0.36 -0.77 -2.04 -1.06

ch(OASDI Payroll Tax / GDP%) -0.20 0.77 1.49 1.79 -0.12 0.12 1.14 0.38

ch(OASDI Benefits / GDP%) 1.55 1.89 2.61 2.92 0.43 2.09 2.26 1.60

ch(Trust Funds / GDP%) 10.1 -13.9 -13.9 -13.9 9.0 -5.6 -13.9 -3.5

ch(Interest Rate%) -1.71 -2.38 -2.97 -3.04 -0.93 -2.05 -2.73 -1.90

%ch(Wage Rate) 7.6 10.4 13.2 13.4 4.0 9.1 12.1 8.4

Small Open Economy

%ch(National Wealth) 33.8 52.2 66.5 69.9 18.4 43.0 61.1 40.9

%ch(Labor Supply) -1.2 -4.0 -9.0 -11.4 0.9 -2.8 -6.4 -2.8

%ch(GNP) 9.5 13.1 13.9 13.3 6.2 11.2 14.1 10.5

%ch(Consumption) 8.3 12.3 14.5 15.5 4.4 10.9 13.5 9.6

ch(Private Savings / GDP%) 0.91 1.18 0.21 -0.98 0.58 1.50 0.93 1.00

ch(OASDI Payroll Tax / GDP%) -0.13 1.15 2.74 2.30 -0.08 0.22 1.62 0.59

ch(OASDI Benefits / GDP%) 1.96 2.25 3.14 3.40 0.63 2.59 2.72 1.98

ch(Trust Funds / GDP%) 14.7 -14.0 -14.0 -14.0 10.9 -2.4 -14.0 -1.8

See notes of Table 10.

The trust funds are depleted in 2044 in a closed economy and in 2046 in a small open economy.

Labor supply tends to be smaller in aging economies than in the balanced growth path.

With the population projection alternative II, per-capita labor supply increases relative to the

balanced growth path until 2013 [2012] in a closed [small open] economy, then, it decreases

relative to the balanced growth path thereafter. Although the retirement decision is endogenous in the model, the increase in the population share of aged 65 or older decreases the

average working hours of households in the long run. Under the most severe assumption of

population aging (alternative III), per-capita labor supply increases slightly on average for

the first 75 years in a closed economy, because households have to work more to prepare for

the earlier and larger reduction of OASDI benefits. But, this is not the case in a small open

economy, in which the wage rate is fixed.

There are two effects of an aging population on the capital-labor ratio. First, an aging

government budget differently.

27
population increases life-cycle saving accumulated for longer periods after retirements, and

an aging population reduces per-capita labor supply because a larger share of households are

retired. This effect increases the capital-labor ratio of the economy.

Second, under the financing assumption in which the payroll tax rate is increased to make

the Social Security system sustainable, an aging population increases the lifetime payroll tax

payments and lifetime benefit receipts of households and, accordingly, it reduces private

savings.42 This effect tends to decrease the capital-labor ratio.

When the initial size of the Social Security system—the average replacement rate of

OASI benefits—is relatively small, the first effect is larger than the second one, and the

capital-labor ratio rises as the population ages. When the initial size of Social Security is

relatively large, the capital-labor ratio possibly falls.

In the present model and parameter settings, the first effect on the capital-labor ratio turns

out to be larger than the second one, and the capital-labor ratio goes up in a closed economy

with all three population projections—up by 32.6 percent with alternative II, 12.9 percent

with alternative I, and 60.1 percent with alternative III in the long run. The second effect is

largest when the government is assumed to increase the payroll tax, keeping the benefits at

the current-law level. Even under this financing assumption, however, the capital-labor ratio

rises by 24.2 percent in the long run with alternative II.43

In the model economy, the household saving rate is above the steady-state level in 2004

and, accordingly, national wealth grows faster than both GDP and labor supply. But, the

saving rate declines from this level during most of the years from 2004 to 2200. The saving

rate, measured by the ratio of savings to GDP, declines by 2.69 [1.44] percentage points for

the first 75 years and by 3.03 [1.88] percentage points in the long run in a closed [small

open] economy with the intermediate population projection, returning to the steady-state

saving rate.44

42Under the financing assumption, where the payroll tax rate is kept at the current-law level and only future

benefits are reduced, there are no effects of increasing the lifetime tax payments and benefit receipts. 43Fehr, Jokisch, and Kotlikoff (2003) calculate equilibrium transition paths with an aging population. In their

“Base Case” of the U.S. economy, the capital-labor ratio falls by 53 percent from 2000 to 2100 as the population

ages. Kotlikoff, Smetters, and Walliser (2001) obtain similar results. These authors assumed that the OASDI

benefits are kept at the current-law level and the payroll tax rate is increased to balance the Social Security

account, and that all of the government expenditure, which is population indexed, is financed with a wage tax

rather than an income tax or a consumption tax. 44Unfortunately, the present model does not predict the decline in the private saving rates in 1990s. Gokhale,

28
Figure 3: Baseline Economies with the Population Projection Alternative II (Changes from

the population-adjusted balanced growth path through year 2004; Payroll tax is increased

and benefits are reduced when the trust funds are depleted)

(d) Consumption

-15

-10

-5

0

5

10

-40 -20 0 20 40 60 80 100 120 140 160

Year-2000

%ch from the Balanced

Growth Path

(a) National Wealth

-30

-20

-10

0

10

20

30

40

-40 -20 0 20 40 60 80 100 120 140 160

Year-2000

%ch from the Balanced

Growth Path

(b) Labor Supply

-20

-15

-10

-5

0

5

-40 -20 0 20 40 60 80 100 120 140 160

Year-2000

%ch from the Balanced

Growth Path

(c) GNP

-20

-15

-10

-5

0

5

10

-40 -20 0 20 40 60 80 100 120 140 160

Year-2000

%ch from the Balanced

Growth Path

(e) Private Saving / GDP (%)

-5

-4

-3

-2

-1

0

1

-40 -20 0 20 40 60 80 100 120 140 160

Year-2000

Ch from the Balanced

Growth Path

(f) Trust Funds / GDP (%)

-20

-15

-10

-5

0

5

10

15

20

-40 -20 0 20 40 60 80 100 120 140 160

Year-2000

Ch from the Balanced

Growth Path

(g) OASDI Payroll Tax / GDP (%)

-3

-2

-1

0

1

2

3

-40 -20 0 20 40 60 80 100 120 140 160

Year-2000

Ch from the Balanced

Growth Path

(h) OASDI Benefits / GDP (%)

-3

-2

-1

0

1

2

3

-40 -20 0 20 40 60 80 100 120 140 160

Year-2000

Ch from the Balanced

Growth Path

Closed Economy Small Open Economy

29
Before the Social Security trust funds are depleted, the OASDI payroll tax revenue is

simply determined by household earnings—labor supply multiplied by the wage rate—and

its ratio to GDP declines slightly as labor supply decreases. The OASDI benefit expenditure increases rapidly, but its pace depends on the population projection. Although benefit

replacement rates are reduced when the trust funds are depleted, the ratio of the benefit expenditure to GDP rises further as the aging continues.

With the intermediate population projection, the OASDI payroll tax rate is 19.3 [22.2]

percent higher and benefit replacement rates are 13.9 [15.4] percent lower in 2078 (after 75

years) than the current-law levels in a closed [small open] economy45 (Those numbers are

not in Table 10.) Also, in a closed economy, the interest rate falls by 1.90 percentage points

and the wage rate rises by 7.9 percent during the first 75 years, due to the increase in national

wealth and the decrease in labor supply.

Because the intermediate population projection is used as the baseline economy for the

policy experiments in Section 5, selected variables in the equilibrium transition paths with

the projection alternative II are also shown in Figure 3. The figures before 2004 (the vertical

line) do not necessarily show the forecast of the model, but those indicate the adjustment

process from the 1961 steady-state economy to the 2003 aging economy.

5 Policy Experiments

In Section 4, the financing assumption for the OASDI budget is that the payroll tax rate

and benefit replacement rates are kept at the current-law levels as long as the trust funds

last and that, when the trust funds are depleted, the payroll tax rate is increased and benefit

replacement rates are reduced to keep the trust funds at zero thereafter. Once the trust funds

are depleted, the OASDI becomes the pay-as-you-go system.

In this section, alternative financing assumptions for the OASDI budget are examined.

The first alternative assumption is that, when the trust funds are depleted, benefit replacement rates are kept at the current-law levels and the payroll tax rate is increased to make the

Kotlikoff, and Sabelhaus (1996) show that the government transfers from young and future households to current

old households, such as the increase in Medicare benefits, may explain a large part of this decline. 45When the trust funds are depleted, a half of the OASDI deficit is covered by the increase in the payroll

tax rate and the rest of the deficit is covered by the reduction of benefits. The numbers in percent changes are

different because payroll tax revenue is larger than benefit expenditure in 2004.

30
OASDI budget balanced (and to keep the trust funds at zero) thereafter.46

The second alternative is that, when the trust funds are depleted, the payroll tax rate is

kept at the current-law levels and benefit replacement rates are reduced to make the OASDI

budget balanced thereafter.

The last policy experiment assumes that the payroll tax rate is increased immediately

in 2004 by 10 percent and that, when the trust funds are depleted, the payroll tax rate is

increased further and benefit replacement rates are reduced to make the OASDI budget balanced thereafter.

Due to limits of space, all of those policy experiments use the intermediate population

projection (alternative II), and the results are shown as changes from the baseline economy,

which assumes both the payroll tax and benefits are changed when the trust funds are depleted, with the same population projection.

5.1 Alternative Government Financing Assumptions

The financing assumption for the OASDI budget affects the size of the Social Security

system—the lifetime payroll tax payments and the lifetime benefit receipts per household.

Accordingly, the assumption affects the life-cycle savings and working hours of households

even before the payroll tax rate and benefit replacement rates are changed.

The first alternative financing assumption and the second one are symmetric. The policy experiments under those two assumptions show the possible range of macroeconomic

variables and social welfare in the economy with the population projection alternative II.

5.1.1 The Payroll Tax Rate Is Increased When the Trust Funds Are Depleted

Table 13 shows the result of this policy experiment. The trust funds are depleted in 2053

[2056] in a closed [small open] economy. In the long run, the OASDI payroll tax rate has

to be increased to 18.0 [18.5] percent from the current-law rate—12.4 percent—in a closed

[small open] economy. (The numbers are not shown in Table 13.)

Compared to the aging baseline economy, in the long run, the payroll tax rate is 2.9 [3.0]

46As explained briefly in Section 4, Kotlikoff, Smetters, and Walliser (2001) and Fehr, Jokisch, and Kotlikoff

(2003) use this type of financing assumption. But, those authors assume the pay-as-you-go Social Security

system without any trust funds, and the payroll tax rates are increased immediately.

31
Table 13: The Payroll Tax Rate Is Increased and Benefit Replacement Rates Are Kept at

the Current-Law Level When the Trust Funds Are Depleted (Changes from the Baseline

Economy with Alternative II)

Year Average over Years

2028 2053 2078 2200 04-28 29-53 54-78 04-78

Closed Economy

%ch(National Wealth) -0.8 -4.1 -6.3 -7.2 -0.3 -2.3 -5.4 -2.7

%ch(Labor Supply) -0.3 -0.9 -0.9 -0.9 -0.1 -0.4 -0.8 -0.5

%ch(GNP) -0.4 -1.8 -2.5 -2.8 -0.2 -1.0 -2.2 -1.1

%ch(Consumption) 0.2 0.0 -0.5 -0.9 0.1 0.3 -0.3 0.0

ch(Private Savings / GDP%) -0.28 -1.05 -0.46 -0.30 -0.12 -0.52 -0.53 -0.39

ch(OASDI Payroll Tax / GDP%) 0.00 0.96 1.04 1.16 0.00 0.05 0.91 0.32

ch(OASDI Benefits / GDP%) 0.02 0.47 1.04 1.16 0.01 0.07 0.91 0.33

ch(Trust Funds / GDP%) 0.0 -0.5 0.0 0.0 0.0 -0.1 0.0 0.0

ch(Interest Rate%) 0.04 0.21 0.36 0.42 0.01 0.13 0.30 0.15

%ch(Wage Rate) -0.2 -1.6 -2.3 -2.7 0.0 -0.6 -2.0 -0.9

Small Open Economy

%ch(National Wealth) -0.6 -3.8 -6.7 -8.2 -0.2 -2.0 -5.5 -2.6

%ch(Labor Supply) -0.1 -0.0 -0.2 0.0 -0.1 -0.1 -0.2 -0.1

%ch(GNP) -0.3 -1.4 -2.7 -3.2 -0.1 -0.8 -2.2 -1.0

%ch(Consumption) 0.2 0.4 -0.7 -1.4 0.1 0.3 -0.3 0.0

ch(Private Savings / GDP%) -0.28 -0.98 -0.85 -0.65 -0.11 -0.65 -0.91 -0.56

ch(OASDI Payroll Tax / GDP%) 0.00 0.00 1.09 1.20 0.00 0.00 0.88 0.29

ch(OASDI Benefits / GDP%) 0.01 0.00 1.09 1.20 0.00 0.00 0.88 0.29

ch(Trust Funds / GDP%) 0.0 -0.1 0.0 0.0 0.0 -0.1 0.0 0.0

Percent changes or changes in percentage points from the baseline economy with both payroll tax increases

and benefit cuts. The trust funds are depleted in 2053 [2056] in a closed [small open] economy.

percentage points higher, and the payroll tax revenue and benefit expenditure increase by 1.16

[1.20] percent as a percentage of GDP, in a closed [small open] economy. The increase in the

lifetime payroll tax and benefits reduces private wealth and labor supply, because households

expect larger benefits compared to the baseline economy, and they have to accumulate less

life-cycle savings for the period after retirement. The higher payroll tax rate reduces the

disposable income and savings of households. In the long run, national wealth decreases by

7.2 [8.2] percent in a closed [small open] economy. The saving rate is lower throughout the

transition path. Private consumption increases in the short run because of the lower saving

rate, but it decreases in the log run due to the lower disposable income.

The effect on labor supply is ambiguous in a small open economy. Before the payroll

tax is increased, labor supply is increased by the intertemporal substitution effect and de32
Figure 4: Welfare Gains and Losses from the Baseline Economy—The Payroll Tax Rate Is

Increased When the Trust Funds Are Depleted (Compensating Variation in Wealth as Percentages of Per-Household GDP)

(a) The Average of Each Age

-25

-20

-15

-10

-5

0

5

10

-60 -40 -20 0 20 40 60 80 100 120

Year-2000 When Age 20

Ch as a % of perHousehold GDP

(d) Temporary Ability e8 (Top 1 Percent)

-40

-30

-20

-10

0

10

20

30

40

-60 -40 -20 0 20 40 60 80 100 120

Year-2000 When Age 20

Ch as a % of perHousehold GDP

Closed Economy Small Open Economy

(b) Temporary Ability e1 (1st Quintile)

-4

-3

-2

-1

0

1

2

-60 -40 -20 0 20 40 60 80 100 120

Year-2000 When Age 20

Ch as a % of perHousehold GDP

(c) Temporary Ability e3 (3rd Quintile)

-25

-20

-15

-10

-5

0

5

10

-60 -40 -20 0 20 40 60 80 100 120

Year-2000 When Age 20

Ch as a % of perHousehold GDP

creased by the income effect from the increase in the lifetime benefits. Once the payroll

tax is increased, labor supply could fall from the substitution effect, but this effect would be

partially offset by higher future benefit replacement rates. In a closed economy, labor supply

decreases slightly throughout the transition path because of the lower wage rate. In a closed

economy, the interest rate is 0.42 percentage points higher in the long run than the baseline,

and the wage rate is 2.7 percent lower than the baseline.

The welfare gains and losses from the policy experiment are measured by compensating

variations in wealth. That is, the welfare gain [loss] of a household is calculated as once-in-apolicy-change wealth tax [transfer] (which is made when the policy change is announced for

current households or when the future household becomes age 20) that makes each household

as well off as in the baseline economy.

The welfare gains and losses differ according to the state of each household. Figure 4

33
(a) shows the average gains and losses by age cohort, and Figure 4 (b), (c), and (d) show the

average gains and losses by age cohort for a specific temporary (not lifetime) working ability

class—the bottom 20 percent, the mid 20 percent, and the top 1 percent, respectively.

According to Figure 4 (a), households that enter to the economy before 2030 [2032] in

a closed [small open] economy are on average better off by the policy change, and future

households that enter to the economy after 2030 [2032] are worse off. If the new households

belong to the top 1 percent temporary working ability class, households that enter to the

economy before 2042 [2037] in a closed [small open] economy are still better off.

5.1.2 Benefit Replacement Rates Are Reduced When the Trust Funds Are Depleted

Table 14 shows the result of this policy change. The trust funds are depleted in almost

the same years as before—2053 [2057] in a closed [small open] economy. In the long run,

benefit replacement rates have to be reduced by 30.2 [33.3] percent from the current-law

rates in a closed [small open] economy. (These numbers are not shown in Table 14.)

Compared to the aging baseline economy, in the long run, the payroll tax revenue and

benefit expenditure decrease by 1.10 [1.22] percent as a percentage of GDP in a closed [small

open] economy. Expecting smaller benefits, households accumulate larger life-cycle savings

for their retirements.

In the long run, national wealth increases by 8.3 [9.4] percent in a closed [small open]

economy. The saving rate is higher throughout the transition path, and private consumption

decreases in the short run but increases in the log run. The effect on labor supply is ambiguous in a small open economy, but labor supply will increase in a closed economy because of

the higher wage rate. In a closed economy, the interest rate falls by 0.43 percentage points in

the long run, and the wage rate rises by 2.9 percent.

Figure 5 (a) shows the average gains and losses by age cohort, and Figure 5 (b), (c), and

(d) show the average gains and losses by age cohort and by temporary working ability class.

According to Figure 5, households that enter to the economy before 2035 [2034] in a closed

[small open] economy are on average worse off by the policy change, and households that

enter to the economy after 2035 [2034] are on average better off.

In Figure 4, the welfare loss to future households (the area below 0) is larger than the

34
Table 14: Benefit Replacement Rates Are Reduced and the Payroll Tax Rate Is Kept at

the Current-Law Level When the Trust Funds Are Depleted (Changes from the Baseline

Economy with Alternative II)

Year Average over Years

2028 2053 2078 2200 04-28 29-53 54-78 04-78

Closed Economy

%ch(National Wealth) 0.9 4.6 7.2 8.3 0.3 2.6 6.2 3.0

%ch(Labor Supply) 0.3 0.7 0.9 0.9 0.1 0.4 0.8 0.5

%ch(GNP) 0.5 1.9 2.7 3.1 0.2 1.1 2.4 1.2

%ch(Consumption) -0.2 -0.2 0.4 0.9 -0.1 -0.3 0.2 -0.1

ch(Private Savings / GDP%) 0.31 1.20 0.51 0.34 0.13 0.59 0.59 0.44

ch(OASDI Payroll Tax / GDP%) 0.00 -0.39 -0.96 -1.10 0.00 -0.02 -0.83 -0.29

ch(OASDI Benefits / GDP%) -0.02 0.13 -0.96 -1.10 -0.01 -0.05 -0.83 -0.30

ch(Trust Funds / GDP%) 0.0 0.5 0.0 0.0 0.0 0.1 0.0 0.0

ch(Interest Rate%) -0.04 -0.25 -0.38 -0.43 -0.01 -0.14 -0.32 -0.16

%ch(Wage Rate) 0.2 1.4 2.5 2.9 0.1 0.7 2.1 0.9

Small Open Economy

%ch(National Wealth) 0.8 4.3 7.7 9.4 0.3 2.3 6.3 3.0

%ch(Labor Supply) 0.1 0.0 0.1 -0.1 0.1 0.1 0.2 0.1

%ch(GNP) 0.3 1.6 3.0 3.6 0.1 0.9 2.5 1.2

%ch(Consumption) -0.3 -0.4 0.7 1.5 -0.1 -0.4 0.3 -0.1

ch(Private Savings / GDP%) 0.33 1.11 0.99 0.75 0.13 0.74 1.05 0.64

ch(OASDI Payroll Tax / GDP%) 0.00 0.00 -1.10 -1.22 0.00 0.00 -0.87 -0.29

ch(OASDI Benefits / GDP%) -0.01 0.01 -1.10 -1.22 0.00 0.00 -0.87 -0.29

ch(Trust Funds / GDP%) 0.0 0.1 0.0 0.0 0.0 0.1 0.0 0.0

Percent changes or changes in percentage points from the baseline economy with both payroll tax increases

and benefit cuts. The trust funds are depleted in 2053 [2057] in a closed [small open] economy.

welfare gain to current and near-future households (the area above 0). Figure 5 shows that

the welfare gain to future households is larger than the welfare loss to current and near-future

households. Although both of those two alternative economies are not Pareto superior to the

baseline economy, under a modest assumption of the discount rate, the latter with reducing

future benefits seems to be more efficient than the former with increasing payroll tax rates.

More rigorous evaluations of welfare gains are left for future research.47

5.2 The Payroll Tax Rate Is Increased Immediately by 10 Percent

With a reasonable population projection, such as the intermediate population projection

alternative II, the current-law Social Security system is not sustainable. At some point, the

47Nishiyama and Smetters (2003) introduce a mechanism called the lump-sum redistribution authority, which

was originally developed in Auerbach and Kotlikoff (1987), to their heterogeneous-agent lifecycle model, and

they analyze the efficiency gain or loss of a policy change in an equilibrium transition path.

35
Figure 5: Welfare Gains and Losses from the Baseline Economy—Benefit Replacement

Rates Are Reduced When the Trust Funds Are Depleted (Compensating Variation in Wealth

as Percentages of Per-Household GDP)

(a) The Average of Each Age

-10

-5

0

5

10

15

20

25

-60 -40 -20 0 20 40 60 80 100 120

Year-2000 When Age 20

Ch as a % of perHouseho dl GDP

(d) Temporary Ability e8 (Top 1 Percent)

-40

-30

-20

-10

0

10

20

30

40

50

-60 -40 -20 0 20 40 60 80 100 120

Year-2000 When Age 20

Ch as a % of perHouseho dl GDP

Closed Economy Small Open Economy

(b) Temporary Ability e1 (1st Quintile)

-3

-2

-1

0

1

2

3

4

5

-60 -40 -20 0 20 40 60 80 100 120

Year-2000 When Age 20

Ch as a % of perHousehold GDP

(c) Temporary Ability e3 (3rd Quintile)

-10

-5

0

5

10

15

20

25

-60 -40 -20 0 20 40 60 80 100 120

Year-2000 When Age 20

Ch as a % of perHousehold GDP

government has to increase payroll tax, reduce benefits, or make adjustments elsewhere in

the government budget. One of the simplest reform plans is increasing the payroll tax rate

immediately rather than waiting for the depletion of the trust funds.

In this policy experiment, the payroll tax rate is increased immediately (in 2004) by 10

percent. This action is not sufficient to put the Social Security system on a sustainable path.

When the trust funds are eventually depleted, additional action is needed. In the simulation,

the payroll tax rate is increased further and benefit replacement rates are reduced so that each

of those changes from the current-law (2003) levels covers half of the deficit that would be

generated under the current-law payroll tax rate and benefit replacement rates.

Table 15 shows the result of this policy change. The trust funds last until 2070 [2081] in a

closed [small open] economy. In the long run, this economy returns to the baseline economy

by assumption.

36
Table 15: The Payroll Tax Rate Is Increased Immediately by 10 Percent (Changes from the

Baseline Economy with Alternative II)

Year Average over Years

2028 2053 2078 2200 04-28 29-53 54-78 04-78

Closed Economy

%ch(National Wealth) 1.7 1.6 0.0 0.0 1.0 1.8 0.5 1.1

%ch(Labor Supply) 0.0 0.2 0.0 0.0 0.0 0.0 0.1 0.0

%ch(GNP) 0.5 0.6 0.0 0.0 0.3 0.6 0.3 0.4

%ch(Consumption) -0.4 -0.2 0.0 0.0 -0.5 -0.3 0.1 -0.2

ch(Private Savings / GDP%) -0.41 -0.19 -0.02 0.00 -0.30 -0.59 0.45 -0.15

ch(OASDI Payroll Tax / GDP%) 0.47 0.10 0.00 0.00 0.48 0.46 -0.20 0.24

ch(OASDI Benefits / GDP%) -0.03 0.34 0.00 0.00 -0.01 -0.02 0.50 0.15

ch(Trust Funds / GDP%) 10.2 19.0 0.0 0.0 5.3 14.9 6.8 9.0

ch(Interest Rate%) -0.11 -0.09 0.00 0.00 -0.07 -0.11 -0.02 -0.07

%ch(Wage Rate) 0.2 0.3 0.0 0.0 0.0 0.2 0.2 0.1

Small Open Economy

%ch(National Wealth) 2.2 3.3 0.9 0.0 1.2 3.0 2.2 2.1

%ch(Labor Supply) -0.1 -0.1 0.2 0.0 -0.1 -0.1 0.2 0.0

%ch(GNP) 0.7 1.2 0.5 0.0 0.3 1.0 1.0 0.8

%ch(Consumption) -0.5 -0.3 0.4 0.0 -0.5 -0.4 0.2 -0.2

ch(Private Savings / GDP%) -0.36 -1.13 1.18 0.00 -0.23 -0.70 0.36 -0.19

ch(OASDI Payroll Tax / GDP%) 0.48 0.48 -0.62 0.00 0.49 0.48 -0.39 0.19

ch(OASDI Benefits / GDP%) 0.00 -0.01 1.09 0.00 0.00 -0.01 0.84 0.28

ch(Trust Funds / GDP%) 12.5 29.0 6.5 0.0 6.0 20.7 21.7 16.1

Percent changes or changes in percentage points from the baseline economy with both payroll tax increases

and benefit cuts. The trust funds are depleted in 2070 [2081] in a closed [small open] economy.

Because of the lower after-tax income, private saving as a percentage of GDP is smaller

until 2053 [2056] in a closed [small open] economy, and private wealth decreases from the

baseline economy in this period. National wealth, however, is larger in this period because of

the increase in the trust funds. As explained before, the effect on labor supply in a small open

economy is ambiguous. In a closed economy, labor supply increases slightly in the short run

because of the higher wage rate. For the first 75 years, the interest rate is on average 0.07

percentage points lower than the baseline economy, and the wage rate is on average 0.1

percent higher than the baseline.

Figure 6 (a) shows the average welfare gains and losses from the baseline economy by

age cohort, and Figure 6 (b), (c), and (d) show the average gains and losses by age cohort

for selected temporary working ability classes. According to Figure 6, households that enter

the economy before 2056 [2062] in a closed [small open] economy are on average worse

37
Figure 6: Welfare Gains and Losses from the Baseline Economy—The Payroll Tax Rate Is

Increased Immediately by 10 Percent (Compensating Variation in Wealth as Percentages of

Per-Household GDP)

(a) The Average of Each Age

-10

-8

-6

-4

-2

0

2

4

-60 -40 -20 0 20 40 60 80 100 120

Year-2000 When Age 20

Ch as a % of perHouseho dl GDP

(d) Temporary Ability e8 (Top 1 Percent)

-60

-50

-40

-30

-20

-10

0

10

-60 -40 -20 0 20 40 60 80 100 120

Year-2000 When Age 20

Ch as a % of perHouseho dl GDP

Closed Economy Small Open Economy

(b) Temporary Ability e1 (1st Quintile)

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

-60 -40 -20 0 20 40 60 80 100 120

Year-2000 When Age 20

Ch as a % of perHousehold GDP

(c) Temporary Ability e3 (3rd Quintile)

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

-60 -40 -20 0 20 40 60 80 100 120

Year-2000 When Age 20

Ch as a % of perHousehold GDP

off, although welfare losses by this policy change are modest. Households that enter the

economy in 2056-2070 [2062-2095] are on average better off.

Households of ages between 45 and 60 in 2004 are hurt more than others, because those

households have to pay higher payroll taxes but do not likely gain from the postponement

of benefit cuts. The welfare losses of households that enter the economy in 2005-2010 are

relatively small, because those households are more likely benefited from the higher benefits

relative to the baseline. Finally, households that enter the economy in 2053-2070 [2056-

2081] tend to be better off than other households, because those households have to pay

lower payroll taxes.

Although some households would be better off and the others would be worse off in this

policy experiment, the welfare gains are small and achieved only in the distant future. Under

most reasonable assumptions about the discount rate, those gains are likely to be substantially

38
smaller than the near-term welfare losses under the parameter of this model.48

6 Concluding Remarks

This paper extends a heterogeneous-agent overlapping generations model with idiosyncratic working ability shocks and mortality shocks by introducing an aging population of

the United States. The paper constructs baseline economies as equilibrium transition paths

with three population projections in Social Security Administration (2003). Then, the alternative economies with other government financing assumptions are examined as policy

experiments.

There are two main findings in this paper. First, under a reasonable parameter setting and

population projection, private wealth per household is likely to increase and labor supply per

capita is likely to decrease as the population ages, although the result depends on the initial

size of the Social Security system and the financing assumption for both the Social Security

budget and the rest of the government budget. Second, according to the model, the welfare

gains to future households from an immediate increase in the payroll tax rate could be much

smaller than the welfare losses of current households.

Overall, those numerical exercises show how the economy with an aging population

looks like and how the aging population projection and the government financing assumption

affect macroeconomic and welfare implications.

Although policy experiments performed in this paper are very simple, more complex experiments on Social Security reform can easily be done, since the present model has already

been equipped with a detailed Social Security system. More realistic reform plans, including

those with individual (personal savings) accounts, will be examined in the separate paper,

using one of baseline economies constructed in this paper.

48According to Cutler, Poterba, Sheiner, and Summers (1990), the optimal policy response to an anticipated

demographic change is almost certainly a reduction in the national saving rate. Elmendorf and Sheiner (2000)

also argue that the optimal response to the aging of the U.S. population is to allow future cohorts to bear much

or all that burden. The last policy experiment in the present paper is confirmative with those conclusions.

39
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41