Demographics, productivity growth and the macroeconomic equilibrium.

Sheng-Cheng Hu

I. INTRODUCTION

The demographics in the United States are, as in other industrial countries, experiencing drastic changes. The birth rate is falling while life expectancy is rising. As a consequence, the ratio of the elderly to the working-age population is projected to rise sharply in the next fifty years.

The aging of the population imposes both real and imagined burdens on the economy (Aaron [1990]). Most importantly, it is predicted to cause substantial increases in the tax rates needed to sustain existing social security benefits, thereby bringing about an increase in conflict between the working and the elderly generations (Wildasin [1991], Von Weizsacker [1990] and Verbon [1988]). Population aging also means it is more likely that the pay-as-you-go system of financing social security will be dynamically inefficient (Hu [1993]). In this case, the economy is better served by switching to another financing system. Indeed, a number of countries, most notably Chile, have chosen to privatize their state pension systems.

More broadly, there is evidence that the changing age distribution in the U.S. population has significantly affected consumption, housing investment, money demand and the labor-force participation rate (Fair and Dominguez [1991]), and is potentially very important for explaining the U.S. rate of national saving for the next fifty years (Auerbach and Kotlikoff [1990]). It also affects capital accumulation by causing changes in family insurance and intergenerational trade (Ehrlich and Lui [1991]).

Existing studies of the economic effects of demographics assume either that both work hours and retirement behavior are exogenous, or that work hours are endogenous but retirement behavior is exogenous. For example, Cutler, Poterba, Sheiner and Summers [1990] model the labor-market effects of population aging by imposing the projected age distributions on the current labor-force participation rates across age cohorts, rather than on those corresponding to the projected demographics. As such, they implicitly assume exogenous work hours and concentrate on the age-distribution effect of population aging. Auerbach and Kotlikoff [1990] set an exogenous retirement age of sixty-five although they allow endogenous determination of work hours. Feldstein [1980] provides evidence that incorporation of induced retirement improves the prediction of how social security affects capital accumulation. Likewise, studies of the effects of demographic changes and their interactions with social security will be more fruitful if retirement decisions are explicitly recognized. This is particularly true since the 1983 Amendment to the Social Security Act would gradually relax the retirement test, thus removing the distortions that prevent retirement decisions from adjusting to changing demographics.

This paper provides a computational study of the effects of demographics on the macroeconomic equilibrium and welfare within the framework of an intertemporal optimizing model with age heterogeneity and endogenous productivity growth. The model extends Tobin [1967] and Cass and Yaari [1967] to allow for endogenous retirement decisions. It differs from Hu [1978; 1993] in that it incorporates endogenous productivity growth. A key feature that differentiates this analysis from previous studies is that retirement decisions are characterized by a binary choice of whether to work or not to work, but work hours are exogenous during working years.(1) To facilitate comparisons, we also provide a simulation which shows what would happen if social security provisions were such as to prevent the retirement age from adjusting to the demographic changes.

To study the effects of demographics on the macroeconomic equilibrium involves explaining their effects on the labor-force participation rate and aggregate consumption, their interactions with social security, and their implications for the welfare of the economy. In our computational analysis, we first calibrate the model with current U.S. data, and then consider three alternative scenarios. The first two scenarios correspond to the demographics projected for 2015 and 2040, respectively. While these two scenarios depict an "aging population," the third scenario incorporates an increase in the growth rate of labor productivity. However, since demographic projections are highly speculative, our purpose is to illustrate, not to predict what might happen in the future.

The paper is organized as follows. Section II briefly describes the model and individual intertemporal optimizing behavior. Section III derives aggregation of the endogenous variables. Section IV provides a computational analysis of the short-run and long-run economic effects of demographics and productivity growth. Section V considers their implications for the government budget. Section VII extends the analysis to the case where the growth rate of the economy is endogenous. The final section summarizes the results.

II. THE MODEL

Assume that the population grows at the rate of g. Individuals have a maximum life span of T years and face a constant conditional probability of death [Pi] each year until age T. They work full time for the first N years and retire at age N if they survive until that time. During each working year t, they earn a wage income of w(t), paying a payroll tax at the rate of [[Tau].sub.s] and a wage-income tax at the rate of [[Tau].sub.w]; thus their after-tax income is (1-[[Tau].sub.w]-[[Tau].sub.s])w(t). Upon retirement, they receive each year social security benefits in the amount of y(t), which are nontaxable and indexed with respect to real wages. They also pay a tax at the rate of [[Tau].sub.c] on their capital income. Individuals are intertemporal optimizers and their behavior is characterized by the life-cycle/permanent income hypothesis of consumption extended to allow for endogenous retirement. A description of the extended model appears in the appendix.

The structure of the social security system is summarized by the income-replacement ratio, [Psi] = y(t)/[(1-[[Tau].sub.s]-[[Tau].sub.w])w(t)]. Undoubtedly, how the social-security system is calibrated will affect the computational results. Until 1983 the U.S. Social Security system was financed by a payroll tax under a pay-as-you-go system, and it imposed an earnings test that amounts to a heavy "implicit tax" on incomes earned by social security recipients. Although the 1983 Amendment to the Social Security Act shifted the financing system from a pay-as-you-go to a partially funded system, the accumulated funds are expected to be exhausted by around 2020. The 1983 Act also contained a number of other changes that would greatly reduce the distortions on retirement decisions imposed by the earnings test. For example, the normal retirement age would be increased from sixty-five to sixty-six by 2009 and further to sixty-seven by 2027. In addition to the increases in the normal retirement age, there would be a gradual rise in the penalty for early retirement and in the delayed retirement credit, as well as a gradual liberalization of the earnings test. On the basis of these pending changes in social security provisions, we assume that the social security system is financed by a pay-as-you-go system but does not impose an earnings test. To facilitate comparisons, we also provide computational results under the assumption of an exogenous retirement age, N. This pertains to the case where social security imposes a stringent retirement test which prevents workers from adjusting their retirement decisions to the new demographics.

Tables I and II report simulation results illustrating how retirement and consumption decisions respond to changes in the economic and demographic variables. In these simulations, the model is calibrated with the U.S. data. In the benchmark case, the values of the three demographic parameters are as follows: the maximum life span (T) is eighty-five years; the conditional probability of death ([Pi]) is 0.85 percent; and the population growth rate is g = 1.85 percent per annum, which is taken to be the sum of the live birth rate and the net immigration rate.(2) We also assume that initially the wage growth rate ([Theta]) is 1.5 percent, and the before-tax interest rate, r, is 5.04 percent per annum.(3) The subjective discount rate ([Rho]) is 4 percent per annum. There is considerable controversy about the elasticity of instantaneous utility with respect to consumption, [Beta]. To accommodate the various estimates of the elasticity, we set the benchmark value of [Beta] equal to [Beta] = 0.15 but allow for the alternative cases where [Beta] = 0.5 and where [Beta] = -1.(4) We also set the utility cost of work so that the benchmark retirement age is sixty-two.(5) The payroll tax rate is 14 percent, while [TABULAR DATA FOR TABLE I OMITTED] [TABULAR DATA FOR TABLE II OMITTED] the wage-income tax rate is 13.5 percent and the capital-income tax rate is 33.1 percent.(6) Given the benchmark demographic parameters, the income-replacement ratio ([Psi]) is 88 percent of after-tax earnings, or 66 percent of before-tax earnings ((1-[[Tau].sub.w] - [[Tau].sub.s])[Psi]).(7)

As shown in the appendix, retirement takes place at the point where the marginal cost equals the marginal benefits of earlier retirement. The former is the net income forgone due to earlier retirement, while the latter is the marginal valuation of leisure and is an increasing function of total wealth. Line 1, Table I shows that a permanent increase in the real wage rate does not affect the retirement decision in the steady state when physical wealth is endogenously determined, because a wage-rate increase affects both the marginal cost and marginal benefit of retirement equally. However, in the short run when physical wealth is exogenously given by past history, the wage-rate increase affects the marginal cost of retirement more than it does the marginal benefit, and thus it causes a delay in retirement. Unlike a once-and-for-all increase in w, a rise in its growth rate [Theta] affects the retirement decision both in the short run and in the long run. The long-run effect of a rise in [Theta] from 1.5 percent to 2.5 percent is to cause a delay in retirement by 0.31 years (see line 2, Table I). The life-cycle/permanent income hypothesis literature has extensively discussed how a change in the interest rate induces intertemporal substitution in consumption (see, for example, Hall [1988]). Line 3, Table I shows that it also induces intertemporal substitution in labor supply. A rise in r from 5 to 6 percent lowers the retirement age by 0.35 years from age 62 to 61.65.

Although a change in the level of wage income, w, does not have a long-run effect on the retirement decision, any tax change that alters the income-replacement ratio has a long-run effect on the retirement decision. For example, lines 4 and 5 in Table I show that the retirement age decreases by 0.2 years from age 62 to 61.80 if there is a one-percentage point increase in the wage-income tax rate, while it decreases by slightly more than 1.2 years from age 62 to 60.77 if there is a one-percentage point increase in the payroll tax rate. The reason that each percentage-point increase in the payroll tax rate has a larger effect on the retirement decision than the corresponding increase in the wage-income tax rate is that it raises the income-replacement ratio [Psi], by more (7.6 percentage points) than the latter does (1.22 percentage points).

An extension of the maximum life span, T, or a fall in the death rate, [Pi], leads to a postponement of retirement by bringing about a negative wealth effect on the marginal benefits of retirement, while a lower population growth rate, g, affects the retirement decision only under a pay-as-you-go social security system. In this case, a fall in g shifts the age distribution of the population toward older cohorts, and thereby lowers the income-replacement ratio [Psi] that can be supported by the existing payroll tax rate. A less generous [Psi] in turn reduces the marginal benefits relative to the marginal cost of retirement and thereby delays retirement. Table I shows that, other things being equal, the retirement age increases by six years from age 62 to 68.08 if the maximum life span increases from 85 to 100 (the value of T assumed in scenario 1, Table III); in other words, it is delayed by 0.4 years per year extension of the maximum life span. The retirement age also increases by 0.75 years from age 62 to 62.75 if the death rate falls from 0.85 percent to 0.7 percent (the value of [Pi] assumed in scenario 1, Table III). Finally, a fall in the population growth rate from 1.85 percent to 1.47 percent (the value of g assumed in scenario 1, Table III) increases the retirement age by nearly two years from age 62 to 63.93 in a steady state.

Let c(s, t) be consumption at time t of a representative individual born at time s. As shown in the appendix, c(s, t) is proportionate to the sum of physical (a(s, t)) and human (h(s, t)) wealth, the latter being the present value of future earnings. In a perfect-foresight steady state equilibrium, individual consumption so determined rises or falls with age (n = t - s) at the rate of [Mu] = ([r.sub.a] - [Rho])/(1- [Rho]). Moreover, initial consumption (c(s, s)) rises with the birth date, s, at the rate of [Theta], as does initial human wealth (h(s, s)). In column 1, Table II, we normalize initial human wealth and consumption for the youngest cohort so that they are equal to 100 in the benchmark equilibrium. By assumption, physical wealth equals zero on the date of birth. The marginal propensity to consume refers to the ratio of initial consumption to initial wealth. We see that an increase in life span from age eighty-five to one hundred lowers the marginal propensity to consume slightly by about 0.1 percentage points from 5.15 percent to 5.04 percent. Human wealth is left unchanged if retirement is exogenous, but it rises by about 7 percent to 107.11 if retirement is endogenous. As a result, initial consumption falls by slightly more than 2 percent to 97.78 if retirement is exogenous, but rises by nearly 5 percent to 104.88 if retirement is allowed to respond to the increased life span. Similar results also hold for a fall in the probability of death. A slowdown in population growth affects individual consumption only to the extent that the social security system is pay-as-you-go. In this case, the fall in population growth from 1.85 percent to 1.47 percent reduces initial consumption by 2 percent to 98.00 if retirement is exogenous, but leaves it unchanged if retirement is endogenous, because the induced change in retirement fully accommodates the decline in the income-replacement ratio that accompanies the fall in population growth. At the benchmark equilibrium, consumption rises with age at the rate of [Mu] = -0.74 percent. This rate is invariant to changes in wages or demographics, which only affect consumption through the wealth effect, but leave the intertemporal substitution in consumption unchanged. However, a rise in interest rates not only affects human wealth but induces substitution of future consumption for current consumption. As a result, regardless of whether retirement is endogenous, a one percentage-point increase in the before-tax interest rate increases the rate of consumption by 0.8 percentage points from -0.74 to 0.08 percent, while it lowers initial consumption by 15.3 percent.

III. THE ECONOMY

In this economy, the production side is characterized by a Cobb-Douglas technology that displays constant returns to scale in labor and capital. The elasticity of output with respect to capital ([Alpha]) is 30 percent and, as before, labor productivity grows at the rate of [Theta] = 1.5 percent. Following Mankiw, Romer and Weil [1992], we set the depreciation rate of capital equal to [Delta] = 4 percent. Finally, factor markets are perfectly competitive, so that the price of each factor of production is equal to the marginal product of that factor.

Since workers retire at age N, the labor force consists of all workers age N and below. Therefore, the labor-force participation rate, l, at any time t is

[TABULAR DATA FOR TABLE III OMITTED]

[Mathematical Expression Omitted],

where

f(n)=(g+[Pi])exp[-(g+[Pi])n]/(1-exp[-(g+[Pi])T])

is the density function of age cohort n = t - s. Given the benchmark parameter values, the labor-force participation rate, l, is 82 percent.(8) The implied dependency ratio, d, is 21.92 percent.(9) (See line 0, Table I.) A demographic change affects labor-force participation not only directly through the age-distribution effect but also by inducing retirement. The former effect refers to the change in labor-force participation brought about by the demographic change while retirement age is held constant (column 2a, Table I). A longer maximum life span, a lower probability of death or a lower population growth rate all contribute to a fall in the labor-force participation rate by bringing about a rise in the ratio of the retired to the total population. This effect plays a key role in the pessimistic view about the economic consequences of the projected population aging as described by Wildasin [1991]. However, the support for this view is weakened if the induced-retirement effect is also taken into account, because, as illustrated in column 2b, Table I, the induced-retirement effect more or less offsets the age-distribution effect. For example, if life span increases from eighty-five to one hundred years, the age-distribution effect is to lower the labor-force participation rate by 5.23 percentage points, while the induced-retirement effect is to increase it by 5.40 percentage points. Consequently, the total effect of the increase in life span is to cause a slight increase in the labor-force participation rate by 0.17 percentage points from 82 to 82.17 percent. Likewise, a fall in the population growth rate from 1.85 to 1.47 percent raises the labor-force participation rate slightly to 82.08 percent but would have lowered it by 2.24 percentage points to 79.96 percent if it did not induce later retirement. A fall in the death rate from 0.85 to 0.7 percent raises the labor-force participation rate to 82.02 but would have lowered it to 80.22 percent without inducing later retirement. (See lines 7 and 8, Table I.)

As can be seen from equation (1), changes in the wage growth rate, the real interest rate and the tax rates affect labor-force participation only by inducing retirement. Substituting the numbers in column 1, Table I into equation (1), we find that the labor-force participation rate rises by 0.33 percentage points if the wage growth rate increases by one percentage point. It falls by 0.36 percentage points if the interest rate increases by one percentage point. It falls by 0.21 percentage points if the wage income tax rate rises by one percentage point, and by 1.3 percentage points if the payroll tax rate rises by one percentage point. (See column 2c.)

Aggregate consumption per capita, [Mathematical Expression Omitted], is the sum of individual consumption by all age cohorts (t - s), weighted by their densities f(t - s):

[Mathematical Expression Omitted].

The effect of a demographic change on aggregate consumption is twofold. First is the age-distribution effect. This refers to the change in aggregate consumption brought about by the shift in the age distribution of the population, represented by f(n), due to the demographic change. If individual consumption falls (rises) with age,(10) any shift in the age distribution of the population toward older cohorts reduces (increases) aggregate consumption. As can be seen from column 2b, Table II, at the benchmark equilibrium, [r.sub.a] = (1-[[Tau].sub.c])r=(1-0.331) x 5.04 percent = 3.37 percent [less than] 4 percent = [Rho]; therefore, the age-distribution effect is negative. Second is the induced-consumption effect. This effect refers to the sum of the changes in initial consumption induced by the demographic change. When retirement decisions are exogenous, the induced-consumption effect of an increase in life span or a fall in the death rate or a fall in population growth under the pay-as-you-go social security system is negative and reinforces the negative age-distribution effect, giving rise to a negative total effect. When retirement is delayed by the demographic change, the induced-consumption effect is positive and offsets the age-distribution effect, leading to a small increase in aggregate consumption. The upshot of this illustration is that it underscores the importance of recognizing age-based heterogeneity (Fair and Dominguez [1991]) and induced retirement in the study of demographic effects on consumption. (See columns 2a and 2b, Table II.)

A rise in productivity growth does not affect the age distribution of the population based on head counts, but it does affect the age distribution of human capital and thereby aggregate consumption. Suppose there is an increase in the productivity growth rate from 1.5 percent to 2.5 percent. The initial endowment of human wealth of the youngest age cohort, a(t, t), rises by 28 percent, and so does their consumption. However, consumption of an older age cohort relative to that of the youngest cohort, c(s,t)/c(t,t), falls with n = t - s at the higher rate of [Theta] = 2.5 percent. Thus, the percentage increase in aggregate consumption (8.59 percent) is less than that in initial consumption of the youngest age cohort (28.11 percent).

IV. SHORT-RUN AND LONG-RUN EFFECTS OF DEMOGRAPHIC CHANGES

We now consider the short-run and the long-run equilibrium effects of the demographic changes. The main differences between the short-run and the long-run steady state are that (1) the capital stock (or aggregate physical wealth) per person rises at the constant rate of [Theta] in the long run, but it can rise at either a higher or a lower rate in the short run as output net of depreciation and growth requirements is larger or smaller than aggregate consumption plus government spending per person; and (2) the distribution of aggregate physical wealth among age cohorts is endogenous in the steady state but is exogenously given by past history in the short run.

As noted above, in the short run, since aggregate wealth and its distribution among age cohorts are exogenous, a once-and-for-all rise in the wage rate increases the marginal cost of earlier retirement more than it does the marginal benefits; therefore it leads to a delay in retirement and a higher labor-force participation rate. In the long run, when aggregate wealth and its distribution among age cohorts are endogenous, the wage increase leaves retirement decisions and the labor-force participation rate unchanged. Likewise, because the capital stock is historically given in the short run, the aggregate demand for labor is less elastic in the short run than in the steady state. This is shown in Figure 1 by the fact that the aggregate labor-supply curve is upward sloped in the short run (labeled [l.sub.s]) but is vertical in the steady state (labeled [l.sub.L]), and the aggregate labor-demand curve is steeper in the short run (labeled [Mathematical Expression Omitted]) than in the steady state (labeled [Mathematical Expression Omitted]). Comparing the short-run ([e.sub.s]) and the steady-state ([e.sub.L]) equilibria, we see that a demographic change has a larger impact on the labor-force participation rate but, depending upon the relative slopes of the short-run demand and supply curves of labor, its effect on the wage rate can be either larger or smaller in the short run than in the long run.

Table III illustrates long-run equilibrium effects of demographics for three scenarios. The first scenario depicts the demographics projected for 2015: the maximum life span is 100 years, the population growth rate is 1.47 percent and the conditional probability of death is 0.70 percent. The second scenario pertains to the demographics projected for 2040: the maximum life span is 110 years, the population growth rate is 1.47 percent and the conditional probability of death is 0.61 percent.(11) Although the productivity growth rate is held unchanged at 1.5 percent in these two scenarios, there are theoretical grounds and empirical evidence that a change in demographics may induce a corresponding change in the growth rate of the economy. (See Ehrlich and Lui [1991] and Cutler, Poterba, Sheiner and Summers [1990].) While section VI considers the growth effects of the demographic changes, the third scenario assumes that the demographic parameters remain at their benchmark values but the growth rate of labor productivity rises exogenously from 1.5 percent to 2.5 percent. Ignoring the second-order terms, we can take the total effect of demographic and productivity changes as the sum of the two effects (line 1 or 2 plus line 3, Table III).

As mentioned above, the benchmark equilibrium values of r, the before-tax interest rate, and N, the retirement age, are, r = 5.04 percent and N = 62 years. In the first scenario (see line 1a, panel A), the new equilibrium retirement age is 72.43 years. This implies a labor-force participation rate of 82.48 percent, and a dependency ratio (d) of 21.24 percent, which is below, albeit only slightly, the benchmark ratio of 21.94 percent despite the aging of the population. Cutler, Poterba, Sheiner and Summers [1990] measure the welfare implications of the demographic changes by their effects on the standard of living (consumption). They show that the projected demographic changes will improve the American standard of living in the short run but lower it slightly over the very long run. We measure the welfare implications of the demographic changes by their effects on the steady-state equilibrium level of individual lifetime utility As such, we take into account the tradeoff between consumption and leisure. Using this measure, we show that the economy is better off as a result of the projected demographic changes. Line 1b, panel A shows what the predicted equilibrium values would be if the induced-retirement effects of the demographic transition to scenario I are ignored. With the retirement age staying at sixty-two, the labor-force participation rate falls to 72.6 percent and the dependency ratio rises to 37.75 percent. The model now predicts that each retiree will be supported by 2.6, rather than 5, workers. Ignoring induced retirement leads to less optimistic predictions of the economic consequences of "population aging."

While the recognition of induced retirement significantly alters the predicted effects of the demographic changes on the dependency ratio and the labor-force participation rate, it has only a minor impact on the predicted equilibrium wage and interest rates. This is because the increases in output and aggregate consumption that accompany the increased labor-force participation rate roughly offset each other, although the former is slightly larger than the latter. Therefore, the predicted real wage rate is only slightly higher (102.68 vs. 102.55) and the predicted interest rate is only slightly lower (4.50 vs. 4.52 percent) when retirement is endogenous than when it is exogenous. These results are shown to be robust with respect to the specification of the elasticity of utility with respect to consumption, [Beta], in panels B and C, Table III.

A higher growth rate of labor productivity implies a larger labor supply in efficiency units. It is expected to cause a fall in the wage rate per efficient unit of labor, denoted [Mathematical Expression Omitted], and a rise in the real interest rate (see Baily [1981]). Line 3a, panel A, Table III confirms that a rise in labor productivity growth ([Theta]) from 1.5 percent to 2.5 percent lowers the real wage rate per efficiency unit of labor by 5.86 percent, while raising the real rate of interest by almost 1.4 percentage points from 5.04 to 6.41 percent. However, the implications of the rise in labor productivity growth for retirement decisions depend on the elasticity of utility with respect to consumption ([Beta]): it leads to earlier retirement if [Beta] = 0.15 (from 62 to 61.95 years) and if [Beta] = 0.5 (from 62 to 61.56 years), but to later retirement if [Beta] = -1 (from 62 to 62.23 years).

Columns 4 and 9 in Table III show, respectively, consumption per effective person ([Mathematical Expression Omitted]) and lifetime utility of the representative individual (U). We see that productivity growth improves welfare while it lowers consumption per effective person. For positive values of [Beta], the welfare gains from productivity growth are less when retirement is endogenous than when it is exogenous. The reason is that under the pay-as-you-go social security system, a higher rate of productivity growth increases the divergence of the real rate of interest from the natural rate of growth (g + [Theta]), which in turn leads to a greater distortion of labor supply and capital accumulation, offsetting the gains from choosing the retirement age optimally. This distortional effect is not present in the exogenous retirement case.

Table IV illustrates the short-run effects of a transition in demographics from the benchmark case to scenario 1. As mentioned above, the short-run effects of the demographic changes are highly dependent on past history that affects the level of aggregate wealth and its distribution among age cohorts. We assume that the demographic transition is immediate in order to dramatize the differences between its short-run and long-run effects. The table shows that if retirement is endogenous, the adjustments in the real wage and interest rates are monotonic. The labor-force participation rate, on the other hand, falls initially to 77.82 percent before it rises back to the new steady-state equilibrium level of 82.48 percent. As a consequence, there is also a short-run worsening in aggregate consumption, the dependency ratio and the income-replacement ratio. If retirement is exogenous, the adjustments in aggregate consumption and the income-replacement ratio are monotonic, but there is overshooting in the wage and interest rates. The real wage rate rises to 103.72 before it falls back to the new steady-state equilibrium level of 102.55. The interest rate falls initially to 4.30 percent before it rises back to 4.52 percent in the new steady state.

In sum, the overshooting responses to the demographic changes are likely to take place in quantity variables such as consumption and the labor-force participation rate if retirement is endogenous, but in price variables such as the real wage and interest rates if retirement is exogenous.

V. FISCAL IMPLICATIONS

The demographic changes affect not only the government's social security budget but also its operating (non-social security) budget. The distinction between the two budgets is important; under a pay-as-you-go system the social-security budget is balanced each year, but the government's operating budget can have a permanent deficit or surplus.

Since, by assumption, social security is pay-as-you-go, its budget is balanced each year. Using the balanced-budget condition, we can calculate the income-replacement ratio to be [Psi] [equivalent to] y/[(1 - [[Tau].sub.s] - [[Tau].sub.w])w] = [[Tau].sub.s] l/[(1 - [[Tau].sub.s] - [[Tau].sub.w]) (1- l)], or 88 percent in the benchmark equilibrium (see line 0, panel A, Table III). With exogenous retirement, the income-replacement ratio sustainable (with respect to after-tax earnings) by the current payroll tax rate would fall to 51.2 percent in scenario 1, and to 43.7 percent in scenario 2. To put it another way, if the income-replacement ratio is to be maintained at the benchmark equilibrium level, the payroll tax rate must be raised from 14 percent to 24 percent and to 29 percent, respectively, in the two scenarios. This result is compatible with that obtained by Wildasin [1991]. However, if retirement is allowed to change, the dependency ratio and thereby the benchmark income-replacement ratio can be sustained without having to raise the payroll tax rate despite "population aging." For example, in scenario 1, the dependency ratio would actually fall slightly from 21.94 percent to 21.24 [TABULAR DATA FOR TABLE IV OMITTED] percent and, consequently, the income-replacement ratio would rise slightly to 90.92 percent despite the aging of the population. Thus, failure to take into account the induced-retirement effect may lead to an overstatement of the burdens of social security on workers.

Turning to the government's operating budget, we assume that government spending ([Mathematical Expression Omitted]) is financed by a tax on wage income at the rate of [[Tau].sub.w] as well as a tax on capital income at the rate of [[Tau].sub.c]. The operating-budget surplus as a fraction of net output is

[Mathematical Expression Omitted].

where [Mathematical Expression Omitted] is the aggregate capital stock, [Mathematical Expression Omitted] is government spending, and [Mathematical Expression Omitted] is net output per person. A constant [Xi] means that in the steady state, the government's operating budget surplus per capita rises at the rate of productivity growth, but the ratio of government debt to aggregate output stays constant. Column 8, Table III indicates that in the benchmark equilibrium the government's operating budget sustains a deficit equal to 0.18 percent of net output. In terms of the absolute amount, it rises at the rate of 1.5 percent per annum.

A change in demographics or productivity growth affects [Xi] both directly and by causing changes in r, the interest rate, and w, wage income. However, under the assumed spending rule and the constant-returns-to-scale technology, any induced change in the labor-force participation rate affects both government revenue and expenditure nearly proportionately, and thus its effect on [Xi], if any, is insignificant. The recognition of the induced-retirement effect does not alter the predicted effects of the demographic and productivity changes on the government operating budget. For example, the government operating budget deficit as a fraction of net output rises from the benchmark value of 0.18 percent to 0.49 percent in the endogenous-retirement case, and to 0.50 percent in the exogenous-retirement case in scenarios 1 and 2. However, a rise in productivity growth from 1.5 to 2.5 percent per annum turns the government operating budget from a deficit of 0.18 percent into a surplus of 0.5 percent of net output, because a higher productivity growth rate increases the ratio of capital to net output. In sum, regardless of whether retirement is induced, changes in productivity growth and demographics affect the social security budget more than they do the government's operating budget. The reason is that the spending rule already takes into account the induced-retirement effect of the demographic changes. This result still holds if the tax and expenditure rules are based on gross output.(12) That is to say, even though fiscal policy is passive, the rule that stipulates spending as a constant fraction of output provides the feedback of the demographic changes on output and the budget deficit.

VI. DEMOGRAPHICS AND THE GROWTH RATE

We have assumed that the technology displays constant returns to scale with respect to both capital and labor. As a result, the steady-state equilibrium growth rate of output per person is determined by the exogenous growth rate of labor productivity, [Theta]. Demographic changes affect only the steady-state equilibrium level of output per effective person. However, a large current literature suggests that the growth rate of the economy may be affected by the spillover from learning by doing and other forms of externalities due to physical and human capital accumulation. (See, for example, Romer [1989] and Stokey and Rebelo [1993].) Following this argument, demographics affects the growth rate of the economy to the extent that it affects intertemporal substitution in consumption.

This section considers the demographic effects on the rate of economic growth in a slightly different direction than that taken by Ehrlich and Lui [1991]. We focus on how induced retirement affects the economy's ability to capture the externalities generated by capital accumulation within the framework of the so-called "Ak" model (see Barro and Sala-i-Martin [1992]). As shown in the appendix, within this framework, although the production function is subject to diminishing returns at the micro (firm) level, it displays constant returns to scale at the aggregate level, with respect to capital. Consequently, the growth rate of labor productivity [Theta] and thus the growth rates of output and consumption per person are no longer exogenous.

As shown in Table V, the projected demographic changes would raise the growth rate of output by around 0.4 percentage points regardless of whether induced retirement is taken into account. That the demographic changes have a positive effect on the growth rate of the economy is consistent with the findings of Ehrlich and Lui [1991]. Here, population aging increases saving, and thereby capital accumulation, as well as the accompanying learning by doing. The reason that induced retirement does not significantly alter the growth effect of the projected demographic changes is because induced retirement affects both aggregate consumption and output by roughly the same proportion, thus leaving the aggregate savings ratio roughly unchanged. In other words, although induced retirement affects significantly the composition of consumption (leisure vs. real consumption), it has only an insignificant effect on the predicted growth rate of productivity.

VII. CONCLUDING REMARKS

We have studied how changes in demographics and productivity growth affect the steady-state equilibrium. These effects can be decomposed into an age-distribution effect and an induced-retirement effect. Our numerical analysis reveals that (1) not only aggregate consumption and output but retirement decisions are highly sensitive to changes in demographics and productivity growth, thus induced retirement cannot be ignored in studies of the economic effects of the projected population aging; (2) whether retirement is endogenous [TABULAR DATA FOR TABLE V OMITTED] affects primarily the predicted effects of the changes in demographics and productivity growth on the quantity variables (such as labor supply, aggregate consumption and output), but not their effects on the price variables (such as wage and interest rates); (3) depending on whether retirement is endogenous, the economy tends to display different patterns of dynamic adjustment in the quantity and in the price variables; and (4) the fiscal implications of the changes in demographics and productivity growth fall primarily on the social-security budget rather than the government's operating budget.

We find that the projected demographic changes improve the welfare of the economy even as they bring about population aging. More importantly, the welfare improvement tends to be larger if there are no distortions (such as an earnings test in social security) which prevent individuals from adjusting their retirement decisions to the changing demographics, although induced retirement does not affect the projected increase in the growth rate.

A caveat to this analysis is that it has not taken into account the demographic effects on health-care expenditures. It is implicit here that the population as a whole is as healthy, if older, in scenarios 1 and 2 as it is in the benchmark equilibrium. Thus, health-care expenditures are not affected by population aging. Unfortunately, the relationship between the projected population aging and health-care expenditures is not yet well understood. On the one hand, there is empirical evidence that with or without adjustment for health, an older person has a greater demand for health-care. (See Burner, Waldo and McKusick [1992].) If the aging of the population is not accompanied by an improvement in health, it potentially can cause a sharp increase in health-care expenditures. On the other hand, one expects future cohorts to have a longer life span only because they will be healthier than the current cohorts are at the same age. Therefore, the net effect of the projected population aging on health-care expenditures is ambiguous. Insofar as population aging leads to an increase in health-care expenditures, its implications for the economy depend on the substitutability between health-care and non-health-care consumption.

APPENDIX

Derivation of Optimal Consumption and Retirement Decisions

This section describes the model that is used in the computational analysis. Consider a representative individual who was born at time s. If he retires at age N, then his life-time allocation problem at time s [less than] t [less than] s + N is given by

[Mathematical Expression Omitted]

subject to

(A2) [Delta]a(s, v)/[Delta]v = ([r.sub.a] + [Pi])a(s, v), + z(v) - c(s, v),

where 1 [greater than] [Phi] [greater than] 0 is the parameter representing the disutility of work, 1 [greater than] [Beta] [greater than] -[infinity] is the parameter representing the elasticity of intertemporal substitution, a(s, v) is the individual's physical wealth (consisting of both corporate capital and government debt), c(s,v) is his consumption, and z(v) is his after-tax income at calendar time v, with z(v) = (1 - [[Tau].sub.s] - [[Tau].sub.w]) w(v) if v - s [less than or equal to] N, and = y(v) if v - s [greater than] N. Implicit in equation (A2) is that there exists a perfect annuity market to insure against uncertain life span at the actuarially-fair rate of [Pi]. (See Blanchard [1985].) Since the after-tax rate of interest is [r.sub.a], the total after-tax rate of return on physical assets entered in equation (A2) is [r.sub.a] + [Pi].

Holding the retirement age (N) exogenous, the Euler equations for the optimal consumption allocation require that [Mathematical Expression Omitted], and that u[prime](c) = [u.sup.+[prime]] (c) at t = s + N. Using these conditions, we can solve for (s, t):

[Mathematical Expression Omitted].

Here h(s, t) is the individual's human wealth at time t, namely the present value of after-tax wage earnings and social security benefits that he expects to receive over his life, and [Gamma](s, t) is the marginal propensity to consume out of total wealth. The marginal propensity to consume in the text refers to [Gamma](s, s).

Substituting equation (A3) in (A1) and differentiating the resulting expression with respect to N yields the optimality condition for N:

(A4) [Gamma](s, s + N)(1 - [(1 - v)

/([Beta][Gamma] v)]([Rho] + [Pi] - [Mu][Beta])

/{1 - exp[-([Rho] + [Pi] - [Mu][Beta])(T - N)]})

(a[s, s + N] + h[s, s + N])

= (1 - [Psi] - [Psi]N{1 - exp[-([r.sub.a] + [Pi])

(T - N)]} / ([r.sub.a] + [Pi]))

(1 - [[Tau].sub.w] - [[Tau].sub.s])w(s + N),

where the right-hand side is the marginal cost and the left-hand side the marginal benefit of earlier retirement.

In a perfect-foresight steady-state equilibrium, expectations at t[prime] regarding a(s, t) for t [greater than] t[prime] must be realized, which implies that

(A5) a(s, t) = [integral of] [w(v) between limits t and s

- c(s, v)]exp[([r.sub.a] + [Pi])(t - v)]dv

= w(t) (exp[[r.sub.a] + [Pi] - g)T] - 1)/([r.sub.a] + [Pi] - g)

- [Gamma](t, t)h(t, t) (exp[([r.sub.a] + [Pi] - [Mu])T]-1)

/([r.sub.a] + [Pi] - [Mu]).

The short-run effects of demographics are analyzed under the assumption that a(s, t) is exogenous, while the long-run effects are studied under the assumption that a(s, t) is determined by the above equation.

Equilibrium Conditions

The accumulation of the capital stock per capita, [Mathematical Expression Omitted], can be written as

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted] is gross output, [Mathematical Expression Omitted] is net output, [Mathematical Expression Omitted] is aggregate consumption, and [Mathematical Expression Omitted] is government spending, all in per capita terms. We set

[Mathematical Expression Omitted]

exogenous in the short run, but allow it to be determined in the long run by equation (A6) under the condition that [Mathematical Expression Omitted]. Sections IV and V assume that dlogA/dt = [Theta] = exogenous. Section VI assumes that A = [A.sub.O][Kappa], where [A.sub.O] is a constant, and that [Kappa] is exogenous to individual firms, but [Mathematical Expression Omitted] at the aggregate level, due to the learning by doing induced by capital accumulation.

1. Although both the labor-force participation rate and retirement decisions are responsive to economic incentives as well as other non-economic influences, existing literature seems to suggest that social security affects primarily the retirement behavior, not the work hours of male workers. See, for example, Hurd and Boskin [1984], Krueger and Pischke [1989]), and Leonesio [1991].

2. The live birth rate and the net immigration rate were, respectively, 15.8 per thousand population and 0.27 percent in 1989. See U.S. Department of Commerce [1991].

3. The growth rate of labor productivity is based on the average growth rate of output per hour in the U.S. for 1979-91, compiled by the U.S. Bureau of Labor Statistics [1992]. The depreciation rate follows Cutler, Poterba, Sheiner and Summers [1990] and is consistent with those used by Mankiw, Romer and Weil [1992], who assume the depreciation rate, [Delta] equals 3 percent, and by Romer [1989], who suggests that [Delta] = 3-4 percent. Cutler, Poterba, Sheiner and Summers [1990] assume the real interest rate (net marginal product of capital) to be 10.3 percent, while the U.S. Social Security Administration (Yang and Goss [1992]) assumes the real interest rate to be 2 percent for 1987-90.

4. See, for example, Eichenbaum, Hansen and Singleton [1988], Zeldes [1989], and Mankiw, Rotemberg and Summers [1985].

5. Calculations based on the estimates by Leonesio [1991] indicate that the mean retirement age for male workers is 63.6, with the exit rate being highest at 65 and 62. According to the estimates of Sueyoshi [1989], the mean age of full retirement is 64.75 and the mean age of partial retirement is 62.79. The mean married female retirement age estimated by McCarty [1990] is 58.01.

6. The total tax rate is 13.85 percent for Old-Age and Survivors Insurance and is 1.45 percent for health insurance. The tax rate on wage income is taken to be the marginal income tax rate on modified income, table 3.4, Statistics of Income, 1989. The capital-income tax rate is based on the estimate made by Fullerton and Henderson [1989].

7. A study of the Retirement History Survey for 1979 by Boskin and Shoven [1987] indicates that after adjusting for taxes, children and certain bonuses, social security fully replaces average earnings of the elderly poor and replaces over half for middle-income families. Specifically, the replacement rates are 152.3 percent for married couples with career average annual indexed earning less than $7,500, 99.6 percent for $7,500-12,500, 77.3 percent for $20,000-30,000, and 45.5 percent for $30,000-50,000.

8. The labor-force participation rate so calculated is consistent with the male labor-force participation rate excluding the sixteen to nineteen-year-old age group, but is higher than the overall labor participation rate. See Yang and Goss [1992].

9. The dependency ratio defined here is different from the commonly used "aged dependency ratio." The latter is the ratio of the population ages sixty-five and above to the population ages twenty to sixty-four. The latter ratio accurately reflects the dependency of the elderly population on the economy only when workers actually retire at age sixty-five.

10. This occurs if the growth rate of consumption, [Mu] [less than] ([greater than])0, namely if the after-tax interest rate [r.sub.a] [less than] ([greater than])[Rho], the subjective discount rate.

11. There is considerable controversy about whether the maximum life span can be extended. The U.S. Social Security Administration projections assume that the maximum life span cannot be extended substantially beyond 85, while others argue that it can be extended to 120. We consider here the "worst-case" scenario in which the maximum life span is 100 for 2015 and 110 for 2040. The live birth rate is projected to be 12 per 1000 population by 2015 but to remain steady thereafter. Assume no change in immigration policy, so that the rate of immigration remains 0.27 percent. The death rate was .84 percent and is projected to fall to 0.70 percent in 2015 and further to .61 percent in 2040. (See Bell, Wade and Goss [1992] and Wade [1992].) My calculation is based on a projection by the U.S. Social Security Administration (Wade [1992]). The projections are provided for males and females separately under three alternatives. I use the most pessimistic projection in the study. The life expectancy used here is the weighted average of the male and female life expectancies.

12. Replacing net output with gross output, we can show that this conclusion requires that the capital-income tax be based on gross income, without allowance for depreciation.

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