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Projected U. S. Demographics and Social Security
Mariacristina De Nardi
University of Chicago and Federal Reserve Bank of Chicago
Selahattin İmrohoroğlu
University of Southern California
Thomas J. Sargent
Stanford University and Hoover Institution
November 1, 1998
For comments, we thank Marco Bassetto, Randall P. Mariger, an anonymous referee for this
journal, and the audiences for presentation of drafts of this paper at the Federal Reserve
Macro System Committee Meeting, the University of Minnesota, the London School of
Economics, the Universitat Pompeu Fabra and the 1998 Latin American and Caribbean
Economic Association meeting in Buenos Aires. We also thank He Huang for writing early
versions of some of the computer programs. Neither the Federal Reserve Bank of Chicago
nor the Federal Reserve System is responsible for views expressed in this paper.
Projected U. S. Demographics and Social Security
1. Introduction
Even with two scheduled increases in the normal retirement age in 2008 and 2026, the
Social Security Administration projects that the dependency ratio (the ratio of workers
entitled to social security retirement benefits to those paying payroll taxes) will more than
double between 1997 and 2050. Figure 11 shows four projected paths of the dependency
ratio, corresponding to alternative eligibility rules: perpetuating the current 65 age qualification, adhering to the two legislated postponements to age 66 in 2008 and to age 67 in
2026, adding two additional postponements beyond those two, or with 11 postponements
– eventually leaving the retirement eligibility age at 76. The demographic transition will
require fiscal adjustments to finance our unfunded social security system, with one possibility being further increases in the normal retirement age. Although the demographic
projections contained in Figure 1 have inspired public discussion of social security reforms,
rarely have they been used in general equilibrium computations designed to inform that
discussion.
Dependency Ratio over Time
0.55
retiring at 65
0.5
0.45
current legislation
0.4
current legislation+2
0.35
0.3
0.25
11 postponements
0.2
1980
2000
2020
2040
2060
2080
2100
2120
Figure 1: Projected dependency ratios.
Besides the issue of financing our unfunded social security system, the aging population contributes to what, according to the President’s Council of Economic Advisors
(1997), is an even larger cause for fiscal adjustment: it is projected that medicare and
medicaid spending will increase from 2.7% and 1.2% of GDP in 1996 to 8.1% and 4.9% of
GDP in 2050, respectively. This paper uses projected increases in the dependency ratio
(associated with the current legislation), and medicaid and medicare to create a bench
1
In appendix B we describe how we construct this graph.
1
mark, and then studies the economic consequences of eight alternative fiscal adjustment
packages. These packages either (1) throw all of the fiscal burden onto the labor income tax
rate; (2) raise a consumption tax rate; (3), (4) and (5) reduce benefits in various ways while
also adjusting taxes; (6) and (7) increase the linkage of benefits to cumulative earnings
while also adjusting either the labor income or consumption tax rate; or (8) implement a
privatization by gradually phasing out benefits while adjusting the labor income tax rate.
Except for (8), the experiments abstain from privatization and leave the social security
system unfunded.
In the tradition of Auerbach and Kotlikoff (1987), we use a general equilibrium
model of overlapping generations of long-lived people. As in İmrohoroğlu, İmrohoroğlu, and
Joines (1995), our agents face uncertain life-times and endowments. Huang, İmrohoroğlu
and Sargent (1997) extended the İmrohoroğlu, İmrohoroğlu, and Joines framework to handle the aggregate time-variation occurring during transitions across steady states. We in
turn extend Huang, İmrohoroğlu and Sargent’s (1997) work in four ways: (1) we modify
the technology to incorporate labor-augmenting technical progress; (2) we assume timevarying survival probabilities and demographic patterns; (3) we change the household’s
preferences by activating a life-long bequest motive; (4) we let labor supply choices respond to how retirement benefits are related to past earnings. Innovation (1) introduces
the growth rate as a key parameter affecting the efficiency of an unfunded retirement
arrangement. Innovation (2) lets us study transitions induced by demographic changes.
Innovation (3) allows us to boost savings above what would be produced by pure life-cycle
households, and thereby helps us calibrate the model to realistic capital–output ratios and
age–savings profiles. Innovation (4) not only allows labor supply to respond to policy and
price changes, but also incorporates Auerbach and Kotlikoff’s (1992a, 1992b) and Kotlikoff’s (1997) stress on earnings-relatedness as a key parameter governing the distortions
generated by the social security retirement system.
Our main findings are these:
• In the face of projected demographics, it will be costly to maintain benefits at levels
now promised. Large increases in distorting taxes will arrest capital accumulation
and labor supply. Our work indicates that back-of-the-envelope accounting calculations made outside a general equilibrium model are prone to be overly optimistic.
The Social Security Administration states that a 2.2 percentage point addition to
the 12.4% OASDI payroll tax will restore the financial balance in the social security
trust fund over the 75 year horizon, given intermediate projections of demographics
and other key variables. According to Goss (1998), Deputy Chief Actuary of the
Social Security Administration, a 4.7% immediate increase of the existing OASDI
payroll tax is necessary to finance the existing social security system in perpetuity.
Injecting the same projections of demographics into our calibrated general equilibrium model gives results that diverge from that official assessment. We compute
an additional 17.1 percentage points in the payroll tax rate and large welfare losses
associated with maintaining our current unfunded system. The projected increase
in medicaid and medicare payments adds a further 12.7 percentage points to the
2
required tax on labor income and increases distortions even more.
• Reducing retirement benefits through taxation of benefits and consumption or
through postponing the retirement eligibility age results in a significant reduction
of the fiscal adjustment required to cope with the aging of the population.
• Policies with similar long-run outcomes can have vastly different transient intergenerational distributional implications. With one exception, all our experiments
impose welfare losses on transitional generations. Policies that partially reduce
retirement benefits (by taxing benefits, postponing retirement or taxing consumption) or gradually phase them out without compensation yield welfare gains for
future generations but make most of the current generations worse off. The only
experiment that raises the welfare of all current and future cohorts switches from
the current system to a defined contribution system. Evidently, eliminating the
distortion associated with the social security payroll tax by linking benefits to contributions is very important, confirming arguments by Kotlikoff (1998).
A sustainable social security reform seems to require reduced distortions in labor/leisure and consumption/saving choices and some transition policies that compensate
current generations.
Besides the papers we mention above, many others have studied the U.S. social
security system. Among those, the following seem closest to our work. Kotlikoff, Smetters
and Walliser (1997) use a general equilibrium, long–lived overlapping generations model
to study the consequences of various ways of privatizing the U.S. social security system.
They focus on both inter–generational and intra–generational heterogeneity (the individuals belong to different exogenous earnings–ability classes) and devote particular attention
to matching current U.S. fiscal institutions. They incorporate deductions, exemptions,
and progressive benefits schedules. Altig, Kotlikoff, Walliser and Smetters (1997) use the
model of Kotlikoff, Smetters and Walliser (1997) to study the consequences of different tax
reforms. Their model does not incorporate uncertainty. They assume constant population
growth.
Cooley and Soares (1996) study the design and implementation of a pay–as–you–go
social insurance system as a problem in political economy. They are particularly interested
about the sustainability of such a system in a world with stochastic population growth.
They consider a model with four period–lived agents, no lifespan uncertainty, and exogenous labor supply. They calibrate their population shares up to 1995. They do not study
the substantial aging of the population after that date.
3
2. The model
The model economy consists of overlapping generations of individuals who live no longer
than T + 1 years, and an infinitely lived government. During the first tR + 1 periods of
life, a consumer supplies labor in exchange for wages that she allocates among consumption, taxes, and asset accumulation. During the final T − tR periods of life, the consumer
receives social security benefits. In addition to life span risk, agents face different income
shocks that they cannot insure. They can smooth consumption by accumulating two risk–
free assets: physical capital and government bonds. The government taxes consumption
and income from capital and labor, issues and services debt, purchases goods, and pays
retirement benefits. There is a constant returns to scale Cobb–Douglas aggregate production function, constant labor-augmenting technical progress, and no aggregate uncertainty.
Equilibrium factor prices are time–varying but deterministic.
Cast of characters
For easy reference, we summarize our notation in Table 1. For any variable z, a
subscript t denotes age, and an argument s in parentheses denotes calendar time. There is
an exogenous gross rate ρ > 1 of labor-augmenting technical progress. We let t (s) = t ρs
be an exogenous time-dependent age–efficiency index. The number
PT of people of age t at
time s is Nt (s); the total population alive at time s is N (s) = t=0 Nt (s); kt (s − 1)ρs−1
is physical capital held by an age-t person at the end of time s − 1; K(s − 1)ρs−1 =
PT
s−1
Nt (s − 1) is total physical capital at the end of period s − 1. Where δ
t=0 kt (s − 1)ρ
is the physical rate of depreciation of capital, we let R(s − 1) = 1 + r(s − 1) − δ be the rate
of return on asset holding; r(s − 1) is the gross-of-depreciation rate of return on physical
capital from time s − 1 to time s; g(s)ρs is per capita government purchases of goods at
time s; w(s) is a base wage rate at time s; ct (s)ρs and `t (s) are consumption and labor
supply at time s for someone of age t; Υt (s)ρs denotes total tax payments, St (s)ρs social
security payments, and et (s)ρs the cumulative labor earnings of a household of age t at
time s. We let at−1 (s − 1)ρs−1 be a consumer’s asset holdings at the beginning of age t at
time s; dt is a random component of a household’s endowment, described by dt = Ud,t zt ,
where zt is an exogenous first-order vector stochastic process used to model the flow of
information, and Ud,t is an age-dependent selection vector.
Factor prices
We assume a constant returns
Cobb-Douglas aggregate production function with
PtR
s
s−1
labor and capital arguments ρ
, respectively. From the
t=0 t `t (s) Nt (s) and K(s−1)ρ
firm’s problem in a competitive equilibrium, the rentals r(s − 1) and w(s) are determined
from marginal productivity conditions:2
r (s − 1) = r
2
K (s − 1)
ρL (s)
= α̃A
K (s − 1)
ρL (s)
α̃−1
The presence of ρ in the denominator is due to our timing convention.
4
Demography
Government
g(s)ρs
Nt (s)
population, age t time s
gov’t purchases
αt (s)
one-period survival probability τa (s)
tax rate on assets
λt (s)
survival prob to age t
τ` (s)
labor tax rate
ft (s)
fraction of age t people
τb (s)
tax rate on bequests
T + 1, tR + 1 max life-span, work life-span
Household
Production and Information
ct (s)ρs
consumption at age t
w(s)
wage
`t (s)
labor at age t
t (s)
age-efficiency index
at (s)ρs
asset holdings at age t
r(s)
rate of return
bt (s)ρs
bonds at age t
K(s)ρs
aggregate physical capital
kt (s)ρs
physical capital at t
L(s)
aggregate labor
et (s)ρs
cumulative labor earnings
zt
information
St (s)ρs
retirement benefits
dt
endowment shock
Υt (s)ρs
total tax payments
xt
household state vector
Beq(s)ρs
aggregate bequests
Table 1: Cast of Characters.
w (s) = w
K (s − 1)
ρL (s)
= (1 − α̃) A
K (s − 1)
ρL (s)
α̃
PtR
s
where L(s) =
t=0 t `t (s) Nt (s). The wage of an age-t worker at time s is t ρ w(s).
α̃ ∈ (0, 1) is the income share of capital and A is total factor productivity.
Economy–wide physical resource constraint
Using the firm’s first order conditions and constant returns to scale, we can write the
5
economy–wide physical resource constraint at time s as
T
X
tR
X
R (s − 1)
g (s) N (s) +
K (s − 1) + w (s)
ct (s) Nt (s) + K (s) =
t `t (s) Nt (s) .
ρ
t=0
t=0
Demographics
At date s, a cohort of workers of measure N0 (s) arrives. The luckiest live during
s, s + 1, ...., s + T + 1, but many die before age T + 1. As a cohort ages, mortality is
described by αt (s), the conditional probability of surviving from age t to age t + 1 at time
s. Let Nt (s) be the number of age t people alive at time s. It moves according to
Nt+1 (s + 1) = αt (s) Nt (s) .
(1)
Nt (s) = αt−1 (s − 1) αt−2 (s − 2) · · · α0 (s − t) N0 (s − t) .
(2)
Iterating on (1) gives
The probability that a person born at s − t survives to age t is given by
λt (s) ≡
t
Y
αt−h (s − h) .
(3)
h=1
We assume a path n(s) of
rate of growth of new workers,
Qthe
Qs so that N0 (s) = n(s)N0 (s−1),
s
which implies N0 (s) = h=1 n(h)N0 (0). Let ν(s) = h=1 n(h). Then the fraction ft (s)
of age t people at time s is
ft (s) = PT
λt (s) ν (s)
i=0
λi (s) ν (s − i)
.
(4)
The total population alive at time s is
N (s) =
T
X
Nt (s) .
t=0
We take the paths n(s) and αt (s) for s = 1970, . . . , 2060 + 3T as parameters.
The people that enter the model at t = 0 are 21 years old workers. New retirees are
65 years old and agents can live up to age 90.
Households
We assume the one-period utility function for an age t person
i
h
2
2
u (ct (s) , `t (s)) = −1/2 (ct (s) − γ) + (π2 `t (s))
6
where π2 and γ are preference parameters. Conditional on being alive, the household
discounts future utilities by a constant β.
We adopt ‘warm glow’ altruism, which was first introduced by Andreoni (1989,
1990). It asserts that the agent derives utility from leaving a bequest, independently of
the prospective consumption stream of the beneficiary. We adopt this formulation mainly
for computational manageability. In our setup, agents are long–lived and face a large
state-space. We compute long transitions. Considering a model in which one agent’s utility
depends on the other agent’s state variables would substantially increase the computational
burden. However, Andreoni (1989, 1990) argues that there is empirical evidence against
‘pure altruism models’ that make the consumptions of parent and heir independent of the
distribution of income among them (Barro, 1974). Becker (1974) suggests that ‘warm–
glow’ preferences may arise because perhaps people have a taste for giving: they receive
status or acclaim, or simply experience utility from having done their bit.
We use this device not only to get more capital accumulation than in a pure life–cycle
model (see Jones and Manuelli, 1992 for a discussion of the difficulties in matching capital
accumulation in a pure life–cycle model), but also to reconcile our model with Kotlikoff and
Summers’ (1981) computations, according to which intergenerational transfers account for
70–130% of the current value of the U.S. capital stock. The fact that we do not allow for
inter–vivos transfers in our model is not a restrictive assumption: since we do not have
borrowing constraints, the timing of bequests or inter–vivos transfers is not relevant.
Let the state of an age t person at the start of time s be denoted xt (s) =
0
[ at−1 (s − 1), et−1 (s − 1), zt0 ] . We formulate preferences recursively. We impute to an
age t − 1 person a particular type of bequest motive via a ‘terminal value’ function
Vt (xt (s)| dead at t) = VT +1 (xt (s)) = xt (s)0 PT +1 xt (s), where PT +1 is a negative semidefinite matrix with parameters that determine the bequest motive.3 Our formulation
gradually activates the bequest motive, intensifying it with age as the mortality table
makes the household think more about the hereafter. For t = 0, . . . , T , let Vt (xt (s)) be the
optimal value function for an age t person. The household’s Bellman equations are
Vt (at−1 (s − 1) , et−1 (s − 1) , zt )
=
max
u (ct (s) , `t (s)) + β αt (s) E [Vt+1 (at (s) , et (s) , zt+1 ) |Jt (s)]
(5)
{ct (s),`t (s),at (s)}
+ β (1 − αt (s)) E [VT +1 (at (s) , et (s) , zt+1 ) |Jt (s)] ,
where Jt (s) is the information set of an age-t agent at time s and the maximization is
subject to the constraints:
R (s − 1)
at−1 (s − 1) + w (s) t `t (s) + St (s) − Υt (s) + dt
ρ
Υt (s) = τ` (s) [w (s) t `t (s) + dt ]
ct (s) + at (s) =
3
See the appendix for the details about PT +1 .
7
(6)
R (s − 1)
− 1 at−1 (s − 1) + τc (s) ct (s)
+ τa (s)
ρ
(7)
et−1 (s − 1) + w (s) t `t (s) for t ≤ tR + 1
for t > tR + 1
ρ−1 et−1 (s − 1)
(8)
0
for t ≤ tR + 1
f ixbent (s) + rratet (s) · ρ−1 · et−1 (s − 1) for t > tR + 1
(9)
et (s) =
St (s) =
zt+1 = A22 zt + C2 w̃t+1
Ud,t
dt
=
zt .
γt
Uγ
(10)
(11)
The right side of (6) is the household’s after-tax income, the sum of wages, earnings on
assets, a possibly serially correlated idiosyncratic mean-zero endowment shock dt , retirement benefits (if any), minus tax payments. Equation (7) decomposes total tax payments
Υt into taxes on labor income, assets, and consumption. Equation (8) updates et (s), the
cumulated, wage–indexed, labor earnings of the household that, depending on the parameter rrate in (9), affects the household’s eventual entitlement to retirement benefits.
The worker’s past contributions are indexed to wage productivity growth; the pension she
receives during retirement is not; as in the U.S. Social Security System.
Formula (9) tells how retirement benefits are related to past earnings. Part of
social security payments (f ixbent (s)) is independent of past earnings, and part (rratet (s) ·
et−1 (s)) responds to past earnings.
We compute f ixben as follows. For people living within a steady state,
f ixbent (s) = ρtR +1−t · f ixrate · AV (s) ,
(12)
where AV records the average earnings of a worker who has survived to retirement age:
R
1 X
AV (s) =
t `t (s) w (s) .
tR + 1 t=0
t
(13)
To mimic current U.S. benefits, equation (12) computes average earnings to account for
changes in the average wages since the year the earnings were received; but once a worker
retires, her pension is no longer indexed to productivity growth.
For people living during the transition, we make f ixben a linear combination of the
contribution in the initial steady state and that in the final steady state. This simplifies
the computations.
Bequests are distributed only to newborn workers: each agent born at time s begins
life with assets ρs−1 a−1 (s − 1), which we set equal to a per capita share of total bequests
from people who died at the end of period s − 1. This distribution scheme implies that
within a steady state, per capita initial assets equal per capita bequests adjusted for population and productivity growth. However, during either policy or demographic transitions
8
between steady states, this distribution scheme implies that what a generation receives in
bequests no longer equals what it leaves behind.
In (10), w̃t+1 is a martingale difference process, relative to the history of zτ ’s up
to age t, driving the information flow zt , and Uγ , Udt are selector vectors determining
the preference shock process γt and the endowment shock process dt . In the experiments
reported in this paper, we set the preference shock to a constant but specify dt to be
random process with mean zero: dt = ψdt−1 + w̃1t , with ψ = .8. The martingale difference
0
|Jt ) = I.
sequence w̃t+1 is adapted to Jt = (w̃0t , x0 ), with E(w̃t+1 |Jt ) = 0, E(w̃t+1 w̃t+1
Aggregates and distributions across people
In addition to life span risk, individuals face different sequences of random labor
income shocks, which they cannot insure. People smooth consumption across time and
labor income states only by accumulating two risk-free assets – physical capital and government bonds; they use these together with social security retirement benefits to provide
for old-age consumption.
0
Let Dt (s) = [ ct (s) `t (s) at (s) ] be the vector of decisions made by an age t
worker at time s. Our specification makes Dt (s) a linear time-and-age dependent function
of xt (s)
Dt (s) = Lt (s) xt (s) ,
and makes the state vector follow the linear law of motion xt+1 (s + 1) = At (s)xt (s) +
Ct (s)wt+1 . Our model imposes restrictions on the matrices Lt (s), At(s) and Ct (s). Individuals have rational expectations, and make Lt (s), At (s) and Ct (s) depend on the sequence
of prices and government fiscal policies over their potential life-span s, s + 1, . . . , s + T + 1.
We can compute probability distributions across workers for the state and decision
vectors. Let µt (s) = Ext (s), Σt (s) = E(xt (s) − µt (s))(xt(s) − µt (s))0 . Given a mean and
covariance for the state vector of the new workers µ0 (s), Σ0 (s) , the moments follow the
laws of motion µt+1 (s + 1) = At (s)µt (s) and Σt+1 (s + 1) = At (s)Σt (s)At (s)0 + Ct (s)Ct (s)0 .
Aggregate quantities of interest such as aggregate per capita consumption and aggregate per capita physical capital can be easily computed by obtaining weighted averages
of features of the distributions of quantities across individuals alive at a point in time.
Aggregate quantities are deterministic functions of time because all randomness averages
out across a large number of individuals. Only these aggregate quantities appear in the
government budget constraint and the model’s market clearing conditions.
The government
An age-t person divides his time s asset holdings at (s) between government bonds
and private capital: at (s) = bt (s)+kt (s), where bt (s) is government debt. The government’s
9
budget constraint at s is:
g (s)N (s) +
T
X
t=tR
R (s − 1) X
St (s) Nt (s) +
bt (s − 1) Nt (s)
ρ
+1
t=0
T
T
X
R (s − 1)
= τb
bt (s) Nt (s)
Beq (s) +
ρ
t=0
(
T
X
R (s − 1)
+
− 1 at−1 (s − 1) +
Nt (s) τa (s)
ρ
t=0
)
(14)
τ` (s) w (s) t `t (s) + τc (s) ct (s)
where
Beq (s) =
T
X
(1 − αt (s)) at (s − 1) Nt (s − 1)
(15)
Beq (s) · (1 − τb )
.
N0 (s)
(16)
t=0
and
a−1 (s − 1) =
The amount a−1 (s−1) of assets is inherited by each new worker at time s. We assume that
in administering the bequest tax, the government acquires capital and government bonds
in the same proportions that they are held in the aggregate portfolio. Consistent with this
specification, the per new worker inheritance a−1 (s −1) is divided between physical capital
and government bonds as follows:
k−1 (s − 1) =
b−1 (s − 1) =
PT
t=0
PT
t=0
(1 − αt (s)) kt (s − 1) Nt (s − 1)
N0 (s)
(1 − αt (s)) bt (s − 1) Nt (s − 1)
.
N0 (s)
10
(17)
Production:
α̃ = .33
{t } Hansen (1993) δ = .055
γ = 11
π2 = −1.7
β = .994
JB = 60
PT see text
dt see text
A=2
ρ = 1.016
Household:
JG = .032
Demography:
n(s) assumed path αt (s) from life tables
T = 69
tR = 43
Table 2: Parameters.
The algorithm
We first compute the initial steady state. We use backward induction to compute an
agent’s value functions and policy functions, taking as given government policy, bequests
and prices. We then iterate until convergence on:
(i) the social security benefits, to match the desired replacement rate;
(ii) bequests, so that planned bequests coincide with received ones;
(iii) the labor income or consumption tax to satisfy the government budget constraint;
(iv) factor prices, to match the firms’ first order conditions.
To compute the final steady state we use the same procedure described for the initial
steady state, plus we iterate on the government debt level to match the debt to GDP ratio
we have in the initial steady state. In the initial steady state the debt to gdp ratio was
calibrated, in the second steady state we fix it.
Lastly, we compute the transition dynamics by solving backward the sequence of
value functions and policy functions, taking as given the time–varying transition policies,
prices and bequests. We iterate until convergence on:
(i) a parameterized path for the tax rate to match the final debt to GDP ratio;
(ii) factor prices.
Prices are allowed to adjust for a phase–out period after the changes in the demographics
and policies have ended. Though the model economy would converge to a new steady state
11
only asymptotically (because prices are endogenous) we ‘truncate’ this process and impose
convergence in 2T periods.
Calibrated Transition Demographics
We calibrate and compute an initial steady state, associated with constant pre-1975 values
of the demographic parameters αt , n. We then take time-varying αt (s), n(s) parameters
from 1975-2060, so that
o
if s ≤ 1974;
αt
αt (s) = α̂t (s) if 1975 ≤ s ≤ 2060;
1
αt
if s > 2060,
where αot = αt (1970) from the mortality table and α1t = αt (2060 + t), the SSA numbers
for the cohort to be born in 2060; the α̂t (s) are taken from the SSA.4 We calibrated the
growth rate n(s) to match SSA’s forecasts of the dependency ratio. According to the SSA,
the dependency ratio was about 18% in 1974 and will increase to about 50% in 2060. The
population of new workers continues to grow at its initial steady state value of 1.3% until
1984. After 1984, we gradually diminish n(s) to .8% per year so that the dependency ratio
becomes roughly 50% in 2060, and then stabilizes. Our calculations begin by assuming
that prior to 1975, the economy was in a steady state and that people behaved as if they
expected their survival probabilities to be those experienced by people alive in 1970; but
in 1975, people suddenly realize that the survival probability tables are changing over time
and switch to using the ‘correct’ ones. After the conditional survival probabilities attain a
steady state in 2060, the demographic structure changes for another T + 1 years, until it
reaches a new steady state in 2060+(T +1). The departure of the demographic parameters
αt (s), n(s) from their values at the initial steady state requires fiscal adjustments.
Initial Steady State
All of our experiments start from a common initial steady state. We set T = 69, tR =
43. Since our age 0 people work immediately, we think of them as twenty one year olds, of
new-retirees as 65 year olds, and of age T + 1 workers as 90 year olds. We calibrated the
parameters A, γt, β, π2 , JG 5 , JB so that in this initial steady state the capital-GDP ratio is
3.0, the government purchases to GDP ratio is .21, the debt to GDP ratio is .46, and the
mean age-consumption profile resembles the observed data. For the initial steady state,
we set the tax rate on income from capital to be 30%, the tax rate on bequests at 10%,
and the tax rate on consumption at 5.5%. Given government purchases, steady state debt,
and these tax rates, the steady-state equilibrium tax rate on labor income turns out to be
29.7%. In the initial steady state, the interest rate is 5.9% and the marginal productivity
of labor (w) is 3.2. Each new worker receives a bequest worth about 52% of the average
per capita capital in the economy. Throughout the paper, the rate of technical progress is
kept constant at its initial steady state level of ρ = 1.016.
4
The life tables are taken from Bell, Wade and Goss (1992). See Appendix B for a brief discussion
of how these projections are calculated.
5 In appendix C we perform some sensitivity analysis on JG.
12
Alternative Fiscal Responses
We computed eight equilibrium transition paths associated with alternative government responses to the demographic parameters αt (s), n(s), s = 1975, . . . , 2060. In addition
to the change in these demographic variables, there are two changes that are common to
all eight computations. First, to reflect projected increases in medicare and medicaid, we
gradually increase government purchases so that they eventually become 25% higher than
their initial steady state level. Second, current legislation on the postponement of the
retirement age is implemented, raising the mandatory retirement age by one year in 2008,
and by another year in 2026, for the cohort that qualifies for retirement then.
We name the computations 1, 2, 3, 4, 5, 6, 7 and 8. For easy reference, they are
summarized in table 3. In computations 1 and 2, social security benefits are kept at their
levels in the first steady state (i.e., the benefit rate parameters f ixben is left intact); the
entire burden of adjusting to the demographic changes is absorbed by scheduled increases
in the tax on labor income alone (in experiment 1) or in the tax on consumption alone
(in computation 2). Computations 3 and 4 impose reductions on benefits in the form of
announced increases of tR + 2, the mandatory retirement age, by two additional years, one
in 2032 and the other one in 2036, to eventually raise it to 69. The remaining burden of
adjustment is absorbed by scheduled increases in the labor income tax rate (experiment
3) or the consumption tax rate (experiment 4). Computation 5 also imposes a reduction
in benefits, not by increasing retirement age, but by exposing all social security retirement
benefits fully to the labor income tax rate τ` ; it schedules increases in the labor income
tax rate to complete the fiscal adjustments. Computations 6 and 7 schedule adjustments
in the formula for benefits, fully linking them to past earnings for people retiring in year
2000 or later. Thus, while in the first five experiments and experiment 8 rrate = 0 and
f ixrate = .6, in experiment 6 and 7 , rrate = tR.6+1 , f ixrate = 0 for people retiring in year
2000 and after.6 In experiment 6, the labor income tax is raised to pick up the residual
tax burden, while in experiment 7, the consumption tax is increased. Finally, computation
8 is an uncompensated phase out of the current system, in which benefits are phased out
to zero over a 50–year horizon, starting in the year 2000.
Government tax policy during transitions
In steps, the government increases one tax rate (either τ` or τc ) during a transition,
leaving all other tax rates constant. These tax changes are scheduled and announced as
follows. In 1975 the government announces that starting in year 2000, it will increase the
tax on labor income (in experiments 1, 3, 5, 6 and 8) or on consumption (in experiments 2,
4 and 7) every ten years in order to reach the terminal steady state with the desired debt
to GDP ratio. Starting in 2060, that tax rate is held constant at its new steady state level,
but the wage rate and interest rate continue to vary for another 2(T + 1) periods, after
which time we fix them evermore. We then enter a new phase of T + 1 periods, during
6
We also run a modified version of this ‘linkage’ experiment in which the formula for cumulated
earnings includes the agent’s income shock. This is done to explore the importance of insurance against
income shocks implied in our previous formulation.
13
Experiment
Benefits
1
benchmark:
Tax Adjustment
postpone in 2008, 2026
gradually raise τ` (s)
2
benchmark
gradually raise τc (s)
3
also postpone in 2032, 2038 gradually raise τ` (s)
4
also postpone in 2032, 2038 gradually raise τc (s)
5
tax benefits
gradually raise τ` (s)
6
link benefits to earnings
gradually raise τ` (s)
7
link benefits to earnings
gradually raise τc (s)
8
gradual privatization
gradually raise τ` (s)
Table 3: Eight Experiments.
which the wage rate and interest rate are pegged at their terminal steady state values. As
cohorts born during the transition period die, new ones are born into the terminal steady
state.
14
Experiment
1
2
3
4
5
6
7
8
τ`
29.7% ↑ 59.5%
29.7%
↑ 52.9%
29.7%
↑ 42.8%
↑ 51.3%
29.7%
↓ 26.0%
τc
5.5%
↑ 36.9%
5.5%
↑ 31.2%
5.5%
5.5%
↑ 30.5%
5.5%
interest rate
5.9% ↓ 5.0%
↓ 4.1%
↓ 4.9%
↓ 4.2%
↓ 4.2%
↓ 4.9%
↓ 4.2%
↓ 3.0%
wage
+4.2%
+8.8%
+4.8%
+8.5%
+8.2%
+4.6%
+8.5%
+15.2%
GDP
−17.4%
−4.6%
−10.7%
−2.25%
−3.4%
−2.4%
+3.7%
+8.7%
mean asset holdings
−12.5%
+9.2%
−4.4%
+11.4%
+9.4%
+4.0%
+18.1%
+38.0%
mean capital
−11.7%
+11.3%
−3.4%
+13.5%
+11.3%
+5.0%
+20.3%
+42.5%
mean consumption
−28.4%
−13.4%
−19.1%
−9.8%
−11.1%
−7.0%
−1.4%
−0.2%
mean eff. labor
−20.8%
−12.4%
−14.8%
−10.0%
−10.6%
−6.6%
−4.4%
−5.6%
bequests
+35%
+105%
+57%
+108%
+64%
+100%
+137%
+63%
K/GDP
3.0 ↑ 3.2
↑ 3.5
↑ 3.2
↑ 3.4
↑ 3.4
↑ 3.2
↑ 3.4
↑ 3.9
G / GDP
20.6% ↑ 31.2%
↑ 27.1%
↑ 28.9%
↑ 26.4%
↑ 26.7%
↑ 26.4%
↑ 24.9%
↑ 23.7%
Variable
Table 4: Alterations of Steady States in Eight Experiments.
Numerical Results
Table 4 compares outcomes across steady states for the eight experiments. Comparing the
steady states we only see the positive aspects of taxing or reducing pensions and increasing
savings (the savings and capital increase is also linked to the increased life–span). When
we do welfare comparisons, it will become evident that the various policies affect differently
members of different generations in the transition.
Table 4 refers to variables that are normalized by the exogenous productivity
growth. Therefore, in column 1 for example, GDP −17.4% mean that in the final steady
state for experiment 1, GDP is 17.4% lower than it would have been, should the economy
have grown at the constant, exogenous, productivity growth rate. This is the convention
that we have in mind when discussing the results.
We can summarize our main results as follows:
• When the government uses the labor income tax rate to finance the fiscal burden
(Experiment 1), the tax rate goes from 29.7% to eventually 59.5%, the labor supply
falls by 20.8%, the capital stock decreases by 11.7% and output falls by 17.4%.7
7
For the initial steady state, we computed a ‘long run’ labor elasticity associated with our calibrated
15
The decline in the aggregate labor input owes much to the projected demographics
and the increased distortionary taxation of labor income. Experiment 6 involves a
similar computation except that now retirement benefits are linked to past earnings
which removes a distortion in the leisure/labor choice as far as the social security
contributions are concerned. However, these contributions deliver a rate of return
equal to the growth rate of output in the economy, which is less than the return
on private capital. Overall, the results from Experiment 6 are far better than those
in Experiment 1. τ` rises to 51.3%, the labor input falls only by 6.6%, the capital
stock rises by 5.0%, and GDP decreases by only 2.4%.
• When the government uses the consumption tax to finance the fiscal burden created
by the retirement of the baby boom generation (Experiment 2), the consumption
tax rate rises from 5.5% in the initial steady-state to 36.9% in the final steady-state.
Aggregate labor input falls by 12.4%, capital rises by 11.3% and GDP falls by 4.6%.
Experiment 7 links the retirement benefits to the agent’s past average earnings (as
in Experiment 6), and uses the consumption tax increase to finance government
expenses. With the linkage of benefits and contributions and the use of the consumption tax, the labor supply distortion is the smallest among all experiments.
The tax rate on consumption raises to 29.7%; GDP and mean capital raise 3.7%,
20.3% respectively, and consumption decreases by 1.4%. Aggregate labor input falls
by 4.4%.
• Taxing benefits at the labor income tax rate and using a higher labor income tax
rate to finance the residual burden (Experiment 5) delivers results that are similar
to those of the second experiment in many respects, since taxing benefits is like
taxing the consumption of the old. For example, aggregate labor input falls by
10.6%, capital rises by 11.3% and GDP falls by 3.4%.
• Postponing the retirement age by two additional years, to age 69, and then using
either τ` (Experiment 3) or τc (Experiment 4) to finance the remaining burden significantly reduces the size of the fiscal burden and therefore the size of the additional
tax required to finance it. When the labor income tax is used, it rises to 52.9% in
the final steady-state compared to the 59.5% in Experiment 1. Aggregate labor
input falls only by 14.8% (20.8% in Experiment 1), capital stock falls by 3.4% and
GDP falls by 10.7%. When the consumption tax is used, it rises to 30.5% compared
with the 36.9% in Experiment 2; labor supply falls by 4.4%, capital stock rises by
20.3% and GDP increases by 3.7%.
• When we compare Experiments 3-5 to Experiment 1, it should be noted that the key
difference in the former is the reduction in social security benefits through using
parameters as follows. We fixed all policy parameters at their initial steady state values, and increased
the real wage while leaving the interest rate and level of retirement benefits fixed. We computed the
new labor-supply, then averaged the age-specific labor supply by cohort percentages, to get an average
percentage life-time response in labor supply. This calculation yielded a 1.36% decrease of the agent’s
lifetime labor supply in response to a 10% wage increase, indicating the income-effect’s domination of the
substitution effect for our calibration.
16
the consumption tax instead of the labor income tax, postponing the retirement
age, or taxing social security benefits. All three alternative fiscal policies yield a
higher work effort, higher consumption and larger saving and capital, relative to
the Experiment 1 policy of using the labor income tax to finance the fiscal burden.
As a result, the economy achieves a softer landing to a final steady-state after the
demographic transition.
• The gradual (and uncompensated) phase-out of the unfunded social security system
(Experiment 8) delivers a final steady state in which consumers can only invest in
capital or government debt to provide for retirement. This yields a substantial
increase of 38% of mean asset holding and and raise of 42.5% for capital. As a
result the interest rate decreases from 5.9% to 3.0%. The labor supply falls by 5.6%,
wage increases by 15.2% and the labor income tax rate falls to 26%. Consumption
decreases by 0.2% and GDP rises by 8.73%.
Steady-state profiles
The discipline of using an applied general equilibrium model manifests itself by
generating several forces that act on individuals’ choices over the life cycle and along
the transition to a final steady-state. There are income and intertemporal substitution
effects from changes in the real interest rate; there are incentive effects stemming from
changes in tax rates, reductions in benefits and retirement age postponements; and there
are demographic changes prompting individuals to save more (for both precautionary and
life cycle reasons) as they face an increased life expectancy. In our discussion of steadystate profiles below, we will highlight those factors that we think are most responsible for
the outcomes.
Age-Labor Supply Profiles
Figures 3 and 4 show the age-labor supply profile in our experiments (the graphs
depict both the cross-section and the life–cycle profiles in the steady state: labor efficiency
increases exogenously as a result of the technological progress, but labor supply does not).
The profile labeled ‘0’ belongs to the initial steady-state. Labor supply rises with age,
peaks around age 40, then falls and drops to zero at the mandatory retirement age of 65.
Experiments 1, 2, 5, 6, 7 and 8 have the same mandatory retirement age of 67. Experiments
3 and 4 postpone retirement to age 69. Apart from this difference, all experiments tilt
the age-labor supply profile in the final steady-state in the counterclockwise direction.
Although there are several forces at work, two in particular seem to be responsible for
the reallocation of work effort over the life cycle. First, the real interest rate in all of the
final steady-states is lower than that in the initial steady-state. Second, the postponement
of retirement by at least two years provides an incentive to postpone work effort since
efficiency in these ‘later’ years is still higher than efficiency in the ‘very young years’.
Age-Wealth Profiles
Figures 5 and 6 display the cross–section age asset–holding profiles: individuals in
all the final steady-states inherit higher wealth and decumulate faster early on in the life
cycle. In some of the experiments (2, 4, 5, 7), they accumulate wealth for a longer period of
17
time or decumulate slower later in the life–cycle (1, 3, 6), and leave larger bequests. This
behavior is consistent with a lower real interest rate in the final steady-states combined
with an increase in the incentive to save brought on by the increase in life expectancy
and/or a reduction in benefits.
In the final steady state of experiment 8, the government no longer provides social
security payments (nor taxes to finance them) and consumers lose the life–span insurance
provided by pensions, which are paid as long as they live. Private saving becomes the only
source of consumption during retirement. In this world, consumers between age 50 and 67
save much more than in the other experiments and capital accumulation is much larger.
After retirement, they run down their assets much faster to consume. Since there are no
annuities markets in the model, asset accumulation also serves the purpose of self–insuring
against life–span risk. Should the consumer live long enough, she will run down most of
her assets and leave almost no bequest: her heirs will share the “longevity risk”.
Age-Consumption Profiles
Figures 7 and 8 plot the cross-section consumption profile. In the initial steady state
consumption declines rather steeply for people past retirement age. This happens because
older people retired in periods during which the technological progress was lower and social
security benefits are not productivity–indexed after retirement. Therefore retired, older
consumers, tend to be poorer than retired younger ones and can consume less.
The cross–section age-consumption profiles in the final steady-states of experiments
1–7 are flatter than the one in the initial steady-state. The decline in the real interest
rate, and the enhanced desire to save due to the aging of the population are powerful
forces in shaping these profiles. In experiment 8 an even lower interest rate, the necessity
of financing retirement consumption out of accumulated assets and the wealth effect we
discussed above, combine with an increased life–expectancy to produce an even sharper
decline for cross–sectional consumption past retirement age
Figures 9 and 10 depict the life–cycle consumption profile: consumption over time
from the point of view of an individual born in the initial or final steady state, with
exogenous technological progress increasing the worker’s productivity.
Transition Paths
Figures 11 and 12 show the time path of labor income and consumption tax rates.
As described in the previous section, the government is required to announce and raise the
appropriate tax rate is five steps, each lasting for ten years, and keep them unchanged at
the new steady state levels. Note that in experiment 8 where social security is gradually
phased out the labor income tax rate needs eventually to fall for the government to maintain
the target debt to GDP ratio.
The structure of preferences and our calibration combine to produce consumers who
do not mind substituting intertemporally consumption and leisure: in Figures 15–18 we
can see how average consumption or average labor supply are not smooth over time.
Figures 17 and 18 show the time path of aggregate labor along the transition in our
experiments. For example, in experiment 1, the five sharp drops of aggregate labor coincide
with the implementation of the announced increases in the labor income tax rate. The
18
two spikes that are smaller in size correspond to the scheduled increases in the mandatory
retirement age (years 2008 and 2026 for all experiments, plus years 2032 and 2038 for
experiments 3 and 4 only). Aggregate labor input declines much less under experiments
6 and 7 because labor income taxation is now less distortionary because of the linkage
between benefits and contributions.
Figures 19 and 20 show the time path of aggregate capital. In experiment 8, where
social security is gradually phased out, the capital stock rises near-monotonically to a much
larger value than in all of the other experiments.
Welfare implications
In this subsection we report our findings on the intergenerational redistribution
of welfare. Figure 2 uses the value function of people in experiment 1 as a base from
which to evaluate the other seven experiments. It measures one-time awards of assets to
those people already working or retired in 1975 (the date that the transition from the
initial demographics begins) and to those new workers arriving after 1975. The awards
are designed to make people as well off under the policy parameters of experiment 1
(with compensation) as they would be under the parameters of experiment j (without the
compensation). The awards are made as follows. To people already working or retired
in 1975, we use the appropriate age-indexed value function of a person born in the year
indicated, evaluate it at the mean assets of a surviving person of the relevant age, and
express the award of assets as a ratio to the mean assets owed by people of that age at
birth. To people entering the work force after 1975, we use the value function of a new
entrant, and express the award as a ratio of the assets inherited by a new entrant at
that date. Thus, a positive number indicates that a positive award would be needed to
compensate a person of the indicated birthdate living in experiment 1 to leave him/her as
well off as in experiment j. The figure reveals the different interests served by the different
transition measures. For example, consider an average member of the cohort born in
1940. This individual would rather give up some wealth and stay under the Experiment 1
fiscal policy of rising labor income taxation than accept the Experiment 5 policy of taxing
benefits.
Essentially all future generations are better off under Experiments 2–8 relative to
Experiment 1. In fact, when we compute an overall welfare measure by properly taking
into account the welfare gains and losses of all generations, weighing them by their (timevarying) population shares and discounting the future gains and losses by the after–tax
real interest rate, all of the experiments deliver a welfare gain. Experiment 2 produces a
welfare improvement of 54% of GDP (at the initial steady-state) relative to Experiment
1. Experiments 3, 4, 5, yield overall welfare gains of 49%, 84% and 56% respectively.
Experiments 6, 7 and 8 produce an overall welfare gain of 197%, 189% and 10.9% of GDP,
respectively, relative to Experiment 1.
Despite the fact that different fiscal policies have similar long-run and overall welfare
consequences, existing generations fare quite differently under these policies. The only
fiscal policy that benefits the existing generations in addition to future generations is the
policy of switching from the current defined benefit system to a defined contribution system
19
Compensation for each generation normalized by their mean assets at birth
2.5
8
2
7
4
2
1.5
6
1
3
0.5
6
0
−0.5
5
−1
−1.5
1900
1950
2000
2050
2100
2150
Figure 2: Compensation in terms of fraction of assets to be given to a person born in year
t living under experiment 1 to make her indifferent between experiment j and experiment
1.
and using a higher labor income tax rate to finance the residual fiscal burden (experiment
6). When a link is established between what an agent contributes to the system and what
the agent eventually receives as benefits, much of the labor income tax no longer distorts
labor supply decisions. Evidently, this particular reform of the (still) unfunded system
goes a long way to produce economic benefits even for the generations that are currently
alive. The experiment 3 policy of postponing retirement for two additional years and using
the labor income tax to finance the remaining burden seems to benefit almost all of the
existing generations; only the youngest generations, those that are 21 years old between
1970 and 1980, appear to experience small welfare costs under this policy. In general,
the use of a higher consumption tax hurts existing generations, as experiments 2, 4 and
7 indicate. The magnitude of the welfare cost on the existing generations also depends
on other components of the fiscal package. For example, use of a higher consumption
tax and introducing a linkage of benefits to contributions yield a smaller welfare cost
for the existing generations compared to those produced by experiments 2 and 4. The
largest welfare costs on the existing generations are generated under experiments 5 and
20
8. Experiment 5 makes retirement benefits taxable and uses a higher labor income tax
rate to finance the residual fiscal burden. This policy simultaneously worsens the labor
supply distortion and imposes a large cost on the retirees. A gradual and uncompensated
privatization of the social security system makes all existing generations worse off relative
to maintaining the unfunded system and relying on a higher labor income tax rate to
provide for larger aggregate benefits.
These findings point to the importance of compensation schemes that will cushion the transition to a funded system and underline the significance of the distortionary
taxation inherent in a defined contribution system.
Comparing a labor income tax versus a consumption tax (experiment 1 vs. 2 and 3
vs. 4), we see that the consumption tax significantly reduces distortions. This is partly due
to the well known public finance result that switching to a consumption tax is equivalent to
taxing the initial capital. In our setup, this also derives from the fact that a consumption
tax is also a tax on social security benefits, which are lump–sum, and hence acts as a lump–
sum tax on retirees. The consumption tax has also important redistributional aspects
because a labor tax hits only the workers while the consumption tax hits both workers and
retired agents.
As we have seen comparing the steady states of experiments 1 vs. 2, the drop in
GDP in experiment 2 is much less pronounced and savings, capital and consumption are
much higher. This is linked to several forces: the lump sum component in the consumption
tax, a smaller labor/leisure distortion (the labor tax rate is lower in experiment 2) and
the fact that since the consumption tax reduces pensions, people save more for retirement.
The resulting effect is that the interest rate decreases and real wage increases. Figure 2
shows that people born between 1920 and 1980 are those who lose in experiment 2: they
are hit by the consumption tax while not benefiting from the reduction in the burden
of pensions. As time passes, the second effect becomes stronger than the first one. In
particular baby–boomers (born in 1947–1960) are those who lose a lot in experiment 2:
they retire when the consumption tax starts to hit. They worked and paid to finance the
social security system and, when they retire, they pay additional taxes on consumption.
Postponing the retirement age also reduces distortions, allowing people to work an
additional two periods. Comparing experiment 1 vs. experiment 3 we see that while labor
increases because people work longer, savings increase even more because agents have to
work for two more years in a region where their efficiency is quite low, and this could be a
negative shock to their income in that period. The effect on savings is much smaller than
the one we get with a consumption tax. Postponing the mandatory retirement age seems
to leave most generations unhurt or better off relative to experiment 1. As benefits are
reduced there is less taxation and this offsets the welfare loss associated with having to work
two extra years until retirement. This explains why experiment 4 dominates experiment 2,
and experiment 3 nearly-dominates experiment 1. Experiments 2 and 4 roughly generate
the same winners and losers.
Taxing social security benefits at the same rate as labor income (experiment 5) is a
way of reducing benefits and making the retirees to share the burden of an increased labor
21
tax with the workers. Excluding privatization, experiment 5 is the policy that redistributes
more across generations: it hits old people alive in year 2000 (older than baby–boomers)
hard, but asymptotically it is similar to experiments 2, where the consumption tax is
raised. Again, this finding highlights the similarities in the economic incentives generated
by reducing benefits through retirement age postponement, taxing benefits at the labor
income tax rate, and taxing consumption.
Privatization through a gradual, uncompensated phase-out is the most welfare enhancing policy in the long run. However, the transitional cohorts stand to suffer a great
deal in the absence of any intertemporal redistribution of benefits and losses.8 This policy
especially hurts the younger baby-boomers and the children of the older baby-boomers.
These transitional generations not only see their benefits phased out, but they share in the
burden of financing the retirement of a succession of larger than before cohorts.
One policy which is unambiguously beneficial to all generations and one with a
sizable welfare gain for the future generations is a switch from the current system to a
defined contribution system, namely experiment 6. Figure 2 reveals that even the transitional generations are quite better off under experiment 6 relative to going along with the
transition path under experiment 1. Evidently the reduction of the distortion in the labor
income tax is economically quite important. 9
8
This is one of the points raised by Huang, İmrohoroğlu and Sargent (1997).
In our computation of cumulated earnings which define contributions and determine benefits, we
left out the individual’s idiosyncratic income shock. As a sensitivity check, we computed an alternative
experiment 6 transition in which we included the income shock in the formula for cumulated earnings.
Note that this eliminates the insurance aspect implied by our previous formulation. We found that the
policy functions of the agents and the aggregates of the economy were the same but the agents’ welfare
was slightly lower with respect to the ‘linkage with insurance against income risk’. The effect on welfare
was smaller than 1%.
9
22
Steady State
1
9
10
τ`
29.7% ↑59.5%
↑38.5%
↑46.8%
interest rate
5.9% ↓ 5.0%
0%
↓ 4.7%
wage
+4.2%
0%
+5.8%
GDP
−17.4%
+17.4% −11.0%
mean asset holdings
−12.5%
+88.5% −3.1%
mean capital
−11.7%
+99.4% −1.9%
mean consumption
−28.4%
+4.5%
mean eff. labor
−20.8%
−24.8% −15.8%
bequests
+35%
+210%
+75%
K/GDP
3.0 ↑ 3.2
↑ 5.0
↑ 3.3
G / GDP
20.6% ↑ 31.2% ↓ 17.6% ↑ 23.2%
Variable
−10.8%
Table 5: Comparing Steady States, in Partial or General Equilibrium, with or without
Medicare and Medicaid Expenditure Increase.
Role of Medicare and Medicaid Increases
The Social Security Administration calculates that a 2.2 percentage point increase
to the 12.4% OASDI payroll tax will restore the financial balance in the social security
trust fund. According to Goss (1998), Deputy Chief Actuary of the Social Security Administration, a 4.7% immediate increase of the existing OASDI payroll tax is necessary
to finance the existing social security system in perpetuity. The 1997 Economic Report
of the President argues that projected increases in medicare and medicaid expenditures
will contribute a heavier burden than financing the social security system. In line with
the perspective of the 1997 Economic Report of the President, all of our calculations up
to now assume substantial increases in medicare and medicaid. Therefore, our computed
fiscal adjustments are designed to fund both higher social security and higher medicare
23
and medicaid expenses. In this section, we briefly describe two calculations designed to
shed light on how much of the fiscal burden comes from our having projected increases in
medicare and medicaid.
Thus, in our benchmark experiment 1 (which includes projected increases for medicaid and medicare expenditure), we computed that, in our general equilibrium setup, the
tax on labor income should increase from 29.7% to 59.5%. We now consider two other
experiments. Experiment 9 is a partial equilibrium version of experiment 1 where factor
prices are held fixed at their values in the initial steady state and in which medicaid and
medicare expenditures do not increase over time. Experiment 10 has factor prices adjusting to factor quantities, as in experiment 1, but keeps medicaid and medicare expenditures
constant over time. Both in experiments 9 and 10 the government gradually increases the
tax on labor income to finance its expenditures.
Table 5 compares outcomes across steady states (the initial steady state is common
to all experiments). The first column reports the results for experiment 1, the second and
third columns describe the final steady state for experiments 9 and 10.
Experiment 9 shows that to finance the increased burden of social security in this
environment, the tax rate on labor income should increase by 8.8 percentage points (from
29.7% to 38.5%). Even this partial equilibrium or ‘small open economy’ environment
produces a large jump in the labor income tax rate, much larger than the 2.2% computed
by the SSA to balance the social security trust fund over the next 75 years and somewhat
larger than the 4.7% projected by Goss (1998). The main difference with the computations
by Goss is probably the timing of the tax increases: Goss assumes that the OASDI payroll
tax is raised at once at the time of the computation (1996), while we assume that is is
raised in six steps, every ten years, staring from year 2000. The discrepancy with the
much lower 2.2% increase projected by the SSA to reestablish equilibrium of the social
security trust fun is obvioulsy due (besides the same assumption as Goss on the timing of
tax increases) to the fact that they only consider a 75 years horizon, starting from 1996.
Experiment 10 acknowledges the fact that the U.S. economy is a very large one,
and that changes in its economic scenario will affect the interest rate and the real wage. In
this environment, the tax rate on labor income increases by 17.1 percentage points (from
29.7% to 46.8%), reflecting the fact that the interest rate declines to 4.7% and the wage
increases by 5.8%. Not surprisingly, the decrease in labor supply is less than that in the
‘small open economy’ (−15.8% instead of −24.8%) and capital accumulation decreases by
almost 2%, instead of jumping up by 99%. The aging of the population and the increases
in the labor income tax to finance the social security system have very large effects.
The 1997 Economic Report of the President argues that financing increased expenditures on medicare and medicaid will have a much larger impact on the economy than
the fiscal burden due to maintaining the current unfunded social security system. A comparison of experiments 1 and 10 reveals, instead, that the distortions due to financing our
unfunded social security system using a labor income tax will be large, given the SSA
forecasts about the aging of the poulation.
24
Experiment 10, in which the distortions stem only from the necessity of financing
the unfunded social security (no increase in government health expenditure here), shows
that the labor income tax rate has to be raised from 29.7% to 46.8% to maintain retirement
benefits at current levelsand that average consumption and labor supply will eventually
decrease by 11% and 16%, respectively.
Not surprisingly, however, our results confirm that adding to this burden the projected increase in health expenditure will make the impact on the economy even heavier.
In experiment 1, in which the tax on labor income is raise to finance both our unfunded social security and the projected increase in government health expenditure, the tax on labor
income eventually raises to 59.5% and average consumption and labor supply respectively
decrease by 28% and 21% in the final steady state.
Concluding Remarks
We have studied some implications of the SSA projected demographic dynamics under alternative fiscal adjustments. Our setup allows for exogenous productivity growth and
the projected increase in medicare and medicaid spending. To the best of our knowledge,
this is the first study to address the issue of the retirement of the baby boom generation
in a setting in which two important features for inducing private saving co-exist:
1. Life span uncertainty: This feature of the model induces the households in our
economy with no private annuity markets to save in order to insure against living
longer than expected. An increase in life expectancy, ceteris paribus, generates
higher private saving.
2. Life-long bequest motive: This motive not only helps us match the observed capital
output ratio but also makes the capital stock more resilient to different ways of
financing the fiscal burden.
Our results indicate that the projected demographic transition will induce a transition to a new stationary equilibrium at which a large fiscal adjustment in the form of a
much higher labor income or consumption tax rate needs to be made. We find that reducing benefits (by taxing them or by postponing the normal retirement age) or imposing a
consumption tax will go far toward reducing the rise in the rate of taxation of labor that
will be required to sustain our unfunded social retirement system but will hurt some generations during the transition. An uncompensated phase-out of benefits towards eventual
privatization delivers the largest welfare gains for future generations but at the same time
imposes the largest welfare costs on current and transitional generations. We also find that
a simplification of the social security structure that makes clear the linkage between the
agent’s past contributions and their future pensions, eliminates a labor/leisure distortion
and improves the welfare of all cohorts.
25
References
Altig, David, Alan Auerbach, Laurence J Kotlikoff, Kent Smetters and Jan Walliser (1997). ‘Simulating U.S. Tax Reform’. Mimeo. NBER Working Paper No.
6246.
Andreoni James (1989). ‘Giving with Impure Altruism: Applications to Charity
and Ricardian Equivalence’. Journal of Political Economy, 97(6): 1447–1458.
Andreoni James (1990). ‘Impure altruism and Donations to Public Goods: a
Theory of Warm–Glow Giving ’. The Economic Journal, 100: 464–477.
Auerbach, Alan J. and Laurence J. Kotlikoff (1987). Dynamic Fiscal Policy.
Cambridge, 1987 Cambridge University Press.
Auerbach, Alan J. and Laurence J. Kotlikoff (1992a). ‘The Impact of the Demographic Transition on Capital Formation’. Scandinavian Journal of Economics,
94(2): 281–295.
Auerbach, Alan J. and Laurence J. Kotlikoff (1992b). ‘Tax Aspects of Policy
Toward Aging Populations’. In John Shoven and John Whalley (eds.), CanadaU.S. tax comparisons. Chicago and London, University of Chicago Press: 255-73.
Auerbach, Alan J., Gokhale, J and Laurence J. Kotlikoff (1992b). ‘Restoring
Balance in U.S. Fiscal Policy: What will it take?’. Federal Reserve Bank of
Cleveland Economic Review, 31(1): 2-12.
Bell, F. C., A. Wade, and S C. Goss (1992). Life Tables for the United States
Social Security Area: 1900–2080, Actuarial Study No. 107, U.S. Department
of Health and Human Services. Social Security Administration, Office of the
Actuary, SSA Pub. No. 11-11536, August.
Cooley, Thomas F. and Jorge Soares (1996). ‘Will Social Security Survive the
Baby Boom?’. Carnegie–Rochester Conference Series on Public Policy, 45: 89–
121.
Council of Economic Advisors (1997). Economic Report of the President. United
States Government Printing Office: Washington, D.C.
Council of Economic Advisors (1976). Economic Report of the President. United
States Government Printing Office: Washington, D.C.
Huang, He, Selahattin İmrohoroğlu and Thomas J. Sargent (1997). ‘Two Computations to Fund Social Security’. Macroeconomic Dynamics, 1(1): 7–44..
İmrohoroğlu, Ayşe, Selahattin İmrohoroğlu and Douglas H. Joines (1995). ‘A
Life Cycle Analysis of Social Security’. Economic Theory, 6(1): 83–114.
Hansen, Gary D. (1993). ‘The Cyclical and Secular Behavior of Labor Input:
Comparing Efficiency Units and Hours Worked’. Journal of Applied Econometrics, 8(1): 71-80.
Jones, L. and R. Manuelli (1992). ‘Finite Lifetimes and Growth’. Journal of
Economic Theory, 58(2): 171-97.
Kotlikoff, Laurence J, Kent Smetters and Jan Walliser (1997). ‘Privatizing Social
Security in the U.S: Comparing the Options’. Mimeo. .
Kotlikoff, Laurence J (1997). ‘Privatizing Social Security in the United States:
26
Why and How’. In Alan J. Auerbach (ed.), Fiscal Policy Cambridge, MIT Press.
Kotlikoff, Laurence J. and Summers. L. (1981). ‘The Role of Intergenerational
Transfers in Aggregate Capital Accumulation’. Journal of Political Economy,
89(4): 706–32.
27
Appendix A: Preferences
For ease of exposition, we suppress the time subscript s, but it should be understood to
be present. A person’s Bellman equations are:
)
(
Vt (xt ) = max
ut ,xt+1
u0t Qt ut + x0t Rt xt + βEt Vt+1 (xt+1 )
where
Et Vt+1 (xt+1 ) = αt (s) Et (Vt+1 (xt+1 ) | alive) + (1 − αt (s)) Et (Vt+1 (xt+1 ) | dead)
Vt (xt | alive) = x0t Pt xt + ξt
Vt (xt | dead) = x0t PT +1 xt
2
x0t PT +1 xt = −JG ((1 − τb ) at−1 − JB)
This last term captures the bequest motive. Here JG is a parameter governing the intensity
of the bequest motive and JB is an inheritance bliss point.
Riccati equations for Pt , Ft and ξt are:
Ft = (Qt + βαt (s) Bt0 Pt+1 Bt + β (1 − αt (s)) Bt0 PT +1 Bt )
−1
(βαt (s) Bt0 Pt+1 At + β (1 − αt (s)) Bt0 PT +1 At )
0
Pt = Rt + Ft0 Qt Ft + βαt (s) (At − Bt Ft ) Pt+1 (At − Bt Ft )
0
+ β (1 − αt (s)) (At − Bt Ft ) PT +1 (At − Bt Ft )
ξt = βαt (s) (tr (Pt+1 C 0 C) + ξt+1 ) + β (1 − αt (s)) trace PT +1 C 0 C.
28
Appendix B: Projected Demographics
Figure 1 is constructed by taking the projections of the conditional survival probabilities
from Bell, Wade and Goss (1992), and assuming a growth rate for entrant workers such
that we match the dependency ratio for 1975 and the forecasted one for 2040. The forecasted dependency ratio we match, is the ‘medium’ projection given by the Social Security
Administration, under the current retirement age legislation. These are also the survival
probabilities and the new workers growth rate that we use in our experiments, in particular, the line corresponding to ‘current legislations’ is the dependency ratio implied in
Experiments 1, 2, 5, 6, 7 and 8; the one marked ‘current legislation +2’, is the dependency
ratio in Experiments 3 and 4.
Starting from early 1900s, each successive cohort has faced successively more favorable
vectors of conditional survival probabilities as a result of improvements in exercise and
nutrition habits, medical techniques, environmental practices, etc., which have especially
become important by 1950s. Combined with a drastic increase in fertility in 1950s, the
demographic dynamics have created the anticipations of future increases in the dependency
ratio even before the time the baby boomers start to retire.
The Social Security Administration arrives at this gloomy picture of the future as follows. First, the SSA takes as its starting population for its projections the Social Security
Area as of January 1, 1989, and its breakdown by age, sex and marital status. Second, the
SSA projects future (a) fertility (taking into account historical trends, future use of birth
control methods, female participation in the labor force, divorce, etc.), (b) mortality (taking into account future development and applications of medical methods, environmental
pollutants, exercise and nutrition trends, drug use, etc.), (c) net immigration, (d) marriage,
and (5) divorce. The final step is to compute projections for future survival probabilities,
fertility, immigration, marriage and divorce, after adjusting the above ‘primitive objects’
for a number of reasons. For example, instead of using ‘death rates’, ‘death probabilities’
are computed as the ratio of the number of deaths occurring to a group in a given year to
the number of persons in this group at the beginning (as opposed to the middle) of the
year.
The outcome is a series of tables that show the Social Security Area population by
year, age, sex and marital status under three alternative projections (optimistic, medium,
pessimistic). Since we abstract from immigration, marriage, divorce, etc. in our model,
we approximate the time variation in the cohort shares and therefore the time path of the
dependency ratio by choosing a time path for the fertility rate {n(s)} which is ratio of
newly borns (model age 0 but real time age 21) at time s to those at time s − 1, and using
the cohort-specific conditional survival probabilities given in Bell, Wade and Goss (1992).
29
Appendix C: Changing the intensity of the bequest motive (JG)
To analyze the sensitivity of our results to the strength of the bequest motive, we
compare the results of experiment 1, calibrated as described in the paper, with its results
when either JG (the bequest motive intensity) or JG and β vary.
In the first sensitivity check, we lower the value of JG by 10% (from .0320 to .0288).
In the second one, we decrease JG by 25% (from .0320 to .024) and increase β to obtain
the same capital to GDP ratio as in the initial steady state of experiment 1 (β increases
from .994 to .996). The latter parameter configuration is such that, should the consumer
live up to ninety years of age, she would die with few assets.
Columns 1 and 2 in table 5 refer to the initial and final steady state of experiment 1 in
the original calibration; columns 3 and 4 to its initial and final steady state with a lower
JG and columns 5 and 6 refer to its steady states when a much lower JG and a higher β
are assumed.
In our original calibration, the effective average discount factor of an individual over
her life–time (taking the mean of her β times her conditional survival probability) is .9679
in the initial steady state and .9838 in the final one, reflecting the increase in the life–
expectancy. In the second sensitivity check we run, it is .9702 and .9859, respectively.
Contrasting columns 1 and 2 with columns 3 and 4 in table 5, we see that a 10% decrease
of JG does not change the results substantially. Labor supply stays the same both in the
initial and final steady states. In the run with a lower JG, assets holdings and consumption
are slightly lower in both steady states. The interest rate and the tax rate on labor income
are a little bit higher. The capital to GDP ratio is also pretty much unchanged.
Comparison of the first two columns with columns 5 and 6, it appears clear that, from
the aggregate point of view, considering an environment in which people care less about
leaving bequests but are more patient, does not change much the aggregates and even the
behavior of the economy over time. Not surprisingly, the variable most affected is the
amount of bequests in the economy.
We choose to adopt the model with a stronger bequest motive, rather than the one
with more patient agents, because on the age–asset accumulation profile, the model with
a stronger bequest motive is more consistent with the empirical evidence. In fact, in the
run with more patient agents and lower ‘altruism’, people run down their assets faster in
the second part of their life, much faster then observed in the data. This is due to the fact
that their conditional survival probability is decreasing over time and the increase in their
joy–of–giving, should they die, is lower. Moreover, we feel that the calibration we adopt
in the paper is consistent with Kotlikoff and Summers’ findings on the proportion of the
present value of wealth which is transmitted from one generation to the next.
30
Steady State
1
2
3
4
5
6
Variable
τ`
29.66% 59.48% 29.81% 61.19% 29.66% 59.92%
interest rate
5.93%
5.01%
6.02%
5.17%
5.92%
5.17%
wage
3.18
3.31
3.17
3.29
3.18
3.29
GDP
12.11
10.00
12.06
9.87
12.13
9.94
mean asset holdings
41.55
36.35
41.08
35.42
41.62
35.68
mean capital
36.00
31.77
35.53
30.87
36.07
31.13
mean consumption
6.67
4.77
6.65
4.70
6.68
4.76
mean eff. labor
2.55
2.02
2.55
2.01
2.56
2.02
bequests
18.75
25.32
17.83
21.46
16.65
16.66
K/GDP
2.97
3.17
2.95
3.13
2.97
3.13
G / GDP
.20
.31
.20
.31
.20
.31
Table 6: Steady States Comparisons.
Appendix D: Figures
31
Age Labor−Supply Profiles
4
3.5
1
3
0
4
2.5
2
2
3
1.5
1
0.5
0
20
30
40
50
60
70
80
90
Figure 3: Age-labor supply profiles in steady states; 0 denotes the initial
steady state.
Age Labor−Supply Profiles
4
8
3.5
7
6
3
5
2.5
0
2
1.5
1
0.5
0
20
30
40
50
60
70
80
90
Figure 4: Age-labor supply profiles in alternative steady states; 0 denotes the
initial steady state.
32
Cross Section Age Asset−Holdings Profiles
140
120
100
80
2 and 4
60
3
40
20
0
20
1
0
30
40
50
60
70
80
90
Figure 5: Cross section age-asset holding profiles in alternative steady states.
Cross Section Age Asset−Holdings Profiles
140
8
120
100
80
7
5
60
6
0
40
20
0
20
30
40
50
60
70
80
90
Figure 6: Cross section age-asset holding profiles in steady states.
33
Cross Section Age Consumption Profiles
8
7
0
4
6
2
5
3
4
1
3
2
20
30
40
50
60
70
80
90
Figure 7: Cross section age–consumption profiles in steady states.
Cross Section Age Consumption Profiles
8
8
0
7
6
5
7
5
6
4
3
2
20
30
40
50
60
70
80
90
Figure 8: Cross section age–consumption profiles in steady states.
34
Life−Cycle Age Consumption Profiles
20
0
18
3
16
2
14
4
1
12
10
8
6
4
2
20
30
40
50
60
70
80
90
Figure 9: Life cycle consumption profiles in steady states.
Life−Cycle Age Consumption Profiles
20
0
6
18
7
16
5
14
8
12
10
8
6
4
2
20
30
40
50
60
70
80
90
Figure 10: Life cycle consumption profiles in steady states.
35
Labor Tax Rate over Time
0.6
1
0.55
3
6
0.5
0.45
5
0.4
0.35
2, 4 and 7
0.3
8
0.25
1980
1990
2000
2010
2020
2030
2040
2050
2060
2070
2080
Figure 11: Labor tax rate τ` during transitions.
Consumption Tax Rate over Time
0.4
2
0.35
4
0.3
7
0.25
0.2
0.15
0.1
1,3,5,6 and 8
0.05
1980
1990
2000
2010
2020
2030
2040
2050
2060
2070
2080
Figure 12: Consumption tax rate τc during transitions.
36
Interest Rate over Time
0.06
0.055
1
0.05
3
0.045
4
2
0.04
0.035
0.03
1980
2000
2020
2040
2060
2080
2100
2120
2140
Figure 13: Interest rate during transitions.
Interest Rate Over Time
0.06
0.055
1
0.05
6
0.045
5
7
0.04
0.035
8
0.03
1980
2000
2020
2040
2060
2080
2100
2120
2140
Figure 14: Interest rate during transitions.
37
Average Consumption over Time
7
6.5
4
6
2
3
5.5
5
1
4.5
1980
2000
2020
2040
2060
2080
2100
2120
Figure 15: Average consumption during transitions.
Average Consumption over Time
7
8
7
6.5
6
5
6
5.5
5
1
4.5
1980
2000
2020
2040
2060
2080
2100
2120
Figure 16: Average consumption during transitions.
38
Average Efficient Work over Time
2.7
2.6
2.5
2.4
4
2.3
2
2.2
3
2.1
1
2
1.9
1980
2000
2020
2040
2060
2080
2100
2120
Figure 17: Average labor supply during transition, in efficiency units.
Average Efficient Work over Time
2.7
2.6
2.5
7
2.4
6
8
2.3
5
2.2
2.1
1
2
1.9
1980
2000
2020
2040
2060
2080
2100
2120
Figure 18: Average labor supply during transition, in efficiency units.
39
Average Capital over Time
52
50
48
46
44
42
4
40
2
38
36
3
34
1
32
30
2000
2050
2100
2150
2200
Figure 19: Average capital holdings during transitions.
Average Capital over Time
52
8
50
48
46
44
42
5
40
6 and 7
38
36
34
1
32
30
2000
2050
2100
2150
2200
Figure 20: Average capital holdings during transitions.
40
@FedFRASER