Longevity and public old-age pensions.

Liu, Liqun ; Rettenmaier, Andrew J. ; Saving, Thomas R. 等

I. INTRODUCTION

Conventional economic analysis of intertemporal or intergenerational resource allocation takes longevity as exogenously determined by biological or technological factors. Within this framework, the institution of a mandatory public old-age pensions of a pay-as-you-go type unambiguously reduces capital accumulation. The intuition behind this result is that when the young expect an old-age pension after retirement and at the same time pay an earmarked payroll tax that lowers disposable income, the ability and the incentive to save are reduced. Such a result, however, ignores the possibility that an individual who has a survivorship-contingent title to an old-age pension might use resources to increase the likelihood of survival to acquire the promised pension. The fact that over the past century both longevity and the size of old-age pensions have risen suggests that a more complete model that incorporates longevity as a choice variable may be useful in understanding the simultaneous increase in longevity and old-age pensions.

Ehrlich and Chuma (1990) consider the demand for longevity in a world where the certain length of life is a fixed function of health care expenditures, thus making longevity endogenous. In their model, consumers choose the optimal depreciation rate for human capital by choosing the path of health care expenditures. Their work paves the way for consideration of the effect of longevity contingent property rights on behavior. In a more recent paper, Philipson and Becker (1998) build on the work of Ehrlich and Chuma by modeling longevity-contingent claims in the form of old-age pensions. Among other things, they find that an increase in a longevity-contingent old-age pension will induce individuals to live longer. As a result of this longer life, Philipson and Becker question the traditional result that pay-as-you-go old-age pensions always reduce the capital stock because workers who choose greater longevity may also choose to save more during their working years to finance their longer retirement period.

In this paper we introduce endogenous and uncertain lifetime into a two-period overlapping generations (OLG) model. We depart from the traditional Diamond-type OLG model by measuring longevity as the probability of survival into the second period and by allowing individuals to affect their survival probability through spending on longevity extending activities. Within this framework, we examine the effects of a pay-as-you-go old-age pension on longevity and capital accumulation under two polar assumptions concerning the existence and absence of a fair intragenerational annuity market.

In either case, the longevity effect of an increase in the old-age pension is neutral while the capital stock effect is strictly negative if the interest rate and population growth rate are equal (e.g, if the economy is on a golden rule path). Surprisingly, increasing the old-age pension produces both a negative longevity and savings effect for the more relevant case where the interest rate is larger than the population growth rate.

That an increase in the old-age pension produces a negative capital stock effect is anticipated, but the neutral or negative effect on longevity is counterintuitive. It is counterintuitive because the empirical evidence seems to show lockstep increases in both longevity and government-provided retirement benefits in the form of pensions and heath care benefits. Furthermore, longevity insurance conjures up the moral hazard problem where increased utilization is expected. With government provision, the funding pool expands with use; the cost being borne by workers.

The fact that the workers bear the burden of increasing the old-age pension helps explain the longevity results we obtain. When the population grows at the same rate as the return on capital, an increase in the pension is offset by a dollar-for-dollar reduction in private savings, leaving second period consumption unchanged. Importantly, an increase in the pension does not affect the price of a longer life and consequently does not induce individuals to increase their longevity investment. So, increasing the pension is longevity-neutral when the rate of return on tax payments and private savings are the same. But when the rate of return on capital exceeds the population growth rate the present value of payroll taxes paid exceeds the present value of benefits resulting in negative lifetime transfers. In this case, increasing old-age pensions actually produce a negative lifetime income effect, which reduces health purchases and ultimately longevity. On the other hand, there is no such negative income effect on longevity in Philipson and Becker because both the interest rate and population growth rate are assumed to be zero.

To investigate whether our results are due to the particular form of uncertainty about the time of death, we replicate some of the main results for an alternative model of stochastic endogenous longevity. Specifically, we consider the length of life in the retirement period as a random variable. For this alternative formulation of endogenous longevity, our principal results--that longevity may well be reduced by increases in old-age pensions and that old-age pensions reduce the equilibrium capital stock--continue to hold for most cases considered. On the other hand, we can replicate Philipson and Becker's result of a positive longevity effect from old-age pensions for a special case in which (1) the maximum limit to human life can be increased, (2) there is no annuity market, and (3) the interest rate does not exceed the growth rate.

The results of our analysis suggest that the observed worldwide increase in longevity is not due to increased old-age benefits. Rather, the cause-effect relation underlying the simultaneous increase of old-age pensions and longevity may well be the other way around, that is, factors other than old-age pensions have caused longevity to increase, which in turn is responsible for the increase in the pensions. To investigate this possibility, we consider a price subsidy on longevity extending consumption and longevity extending technological progress and find that the longevity effects of both tend to be positive.

This article is organized as follows. The model is set up in the next section. In section III, the longevity and saving effects of changes in pay-as-you-go old-age pensions are derived for the two alternative assumptions regarding the existence of fair annuity markets. Section IV considers an alternative model of stochastic endogenous longevity, and in section V we extend our results to altruistic agents. Finally, in section VI we analyze the longevity effects of price subsidies and technological progress. Major conclusions are summarized in the final section.

II. THE MODEL

The model we use is a variant of Diamond's (1965) general equilibrium OLG model. To capture uncertainty about time of death and endogenous longevity, two changes are made to the standard two-period OLG model. First, at the beginning of the second period, each individual gives birth to 1 + n children and then either dies or survives through period two. (1) Second, the probability of surviving through the second period for each individual is a function of first period individual health care expenditures. (2) Otherwise, the model is standard with three components: firms, consumers, and government.

The representative firm uses labor and capital to produce a single good that can be used for consumption, capital investment, or health care expenditures. Production technology is characterized by constant returns to scale. Both the product market and the two factor markets are assumed to be perfectly competitive.

Agents are identical within and across generations. They live for up to two periods, working in the first period by supplying one unit of labor and, if they survive, retire in the second period. During their working period, individuals receive wage payments and unintended bequests (if any) and consume, make health care investments, and accumulate capital. During the retirement period, conditional on survival, individuals receive annuity-type insurance benefits (if any), a public old-age pension, and proceeds from first period saving and consume. The accumulated capital of individuals who fail to survive into the second period is either distributed to the remaining members of their generation or to the younger generation according to the following alternative rules. (3) Distribution rule one assumes the existence of a fair intragenerational annuity market in which the accumulated capital of nonsurvivors is divided among survivors according to the level of the latter's saving (in this case, saving is equivalent to an insurance premium). Distribution rule two assumes no private retirement income insurance market so that any saving of nonsurvivors is treated as an unintended bequest to incoming young. To ensure history independence we further assume that unintended bequests are divided equally among the young.

The sole role of government is to collect taxes from the young to finance any pensions to the old. For simplicity, the government is not allowed to borrow; hence, the government budget is balanced for each and every period. (4)

Let [k.sub.t], f([k.sub.t]); f' > 0, f" [less than or equal to] 0, be, respectively, the capital-labor ratio and output per worker in period t. Without loss of generality, we assume that capital fully depreciates after one period so that perfect competition implies

(1) 1 + [r.sub.t] = f' ([k.sub.t]) [w.sub.t] = f([k.sub.t]) - [k.sub.t]f'([k.sub.t]),

where [r.sub.t] and [w.sub.t] are respectively the (net) rate of return on capital and wage rate in period t.

Each generation is identified by the period in which its members are young. Let the general form of the expected utility function for a representative generation t individual be (5)

(2) [V.sub.t] = [E.sub.t]{U([c.sup.t.sub.t]) + [beta]U([c.sup.t+1.sub.t])}

where [V.sub.t] is the expected utility of a generation t individual, [c.sup.t.sub.t]([c.sup.t+1.sub.t]) is consumption while young (old) by a generation t individual, U(*) satisfies U(0) = 0, U' > 0, U" < 0, and [beta] is the time preference discount factor. (6)

The budget constraints that a generation t individual faces depend on whether the capital proceeds of nonsurvivors are distributed within or between generations. These constraints can be expressed in the following general form:

(3) [c.sup.t.sub.t] = [w.sub.t](1 - [[tau].sub.t]) - [s.sub.t] - [H.sub.t] + (1 - I) x [1 - P([H.sub.t-1])](1 + [r.sub.t]) x [s.sub.t-1]/(1 + n)

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where I [member of] {0, 1} represents the existence of fair intragenerational retirement income insurance (I = 1) or the absence of such insurance (I = 0); [s.sub.t] is the level of saving by a generation t individual in the first period of life; [H.sub.t], [[bar.H].sub.t] are respectively the individual and equilibrium level of health care expenditures by a generation t individual in the first period of life; P([H.sub.t]), P' > 0, P" < 0 is the probability of a generation t individual surviving into the second period; n is the population growth rate; [[tau].sub.t] is the payroll tax rate in period t; and T is the survival-contingent transfer individuals receive from the government at time t + 1. The representative generation t individual chooses [s.sub.t] and [H.sub.t] (and hence [c.sup.t.sub.t] and [c.sup.t+1.sub.t] through [3] and [4]) to maximize expected utility, taking all other variables, including [[bar.H].sub.t], which determines the population survival probability, as given.

The distribution of the proceeds from nonsurvivor saving is identified by the last terms in (3) and (4). With fair insurance (I = 1), all saving is placed in the insurance pool and survivors receive a benefit in proportion to their saving (their premium paid). Specifically, for a generation t individual who saves [s.sub.t], the fair insurance benefit in case of survival is [s.sub.t](1 + [r.sub.t+1])/p([[bar.H].sub.t]), which equals the proceeds from his own saving, [s.sub.t](1 + [r.sub.t+1]), plus [s.sub.t](1 + [r.sub.t+1])/[1 - p([[bar.H].sub.t])]/p([[bar.H].sub.t]). (7) In the absence of fair insurance (I = 0), nonsurvivor savings are distributed equally to the new young generation through what is essentially a reverse intergenerational transfer or 100% death tax. The explicit old-age generation transfer is survival-contingent, whereas the reverse transfer is contingent on dying.

In each period, the government pays a lump-sum transfer, T, to each old individual and finances this old-age subsidy with an earmarked payroll tax on the young. The tax rate in period t is chosen to balance the government budget in that period so that

(5) (1 + n)[w.sub.t][[tau].sub.t] = P([H.sub.t-1])T.

This government balanced budget condition implies that, although each individual faces uncertainty about time of death, there is no aggregate lifetime uncertainty. Thus, if the longevity investment is [H.sub.t-1] per person for generation t - 1, then exactly P([H.sub.t-1])[N.sub.t-1], where [N.sub.t-1] is generation t - 1 population size, survive into period t. (8)

III. EFFECTS OF INCREASING PUBLIC OLD-AGE PENSION

This section examines the effect of increasing the old-age pension on longevity-extending expenditures and capital accumulation under each of two assumptions concerning the existence of fair retirement income insurance.

Fair Retirement Income Insurance Exists (I = 1)

Setting I = 1 utility maximization (through choosing [s.sub.t] and [H.sub.t]) for the representative generation t individual implies

(6) -U'[[w.sub.t](1 - [[tau].sub.t]) - [s.sub.t] - [H.sub.t]] + [beta] (1 + [r.sub.t+1])U'[(1 + [r.sub.t+1])[s.sub.t]/P([H.sub.t]) + T] = 0

(7) -U'[[w.sub.t](1 - [[tau].sub.t]) - [s.sub.t] - [H.sub.t]] + [beta] P'([H.sub.t]) x U[(1 + [r.sub.t+1])[s.sub.t]/P([H.sub.t]) + T] = 0.

where for notational simplicity we drop the bar over [H.sub.t] because the distinction between individual and equilibrium values of health care expenditures is no longer--after taking the first-order derivative with respect to [s.sub.t] and [H.sub.t]--relevant. (9) The steady-state values of H and k can be obtained by first substituting (1) and (5) and the capital market equilibrium condition

(8) [s.sub.t] = (1 + n)[k.sub.t+1]

into (6) and (7), (10) and then by letting [H.sub.t-1] = [H.sub.t] = H, [k.sub.t] = [k.sub.t+1] = k. By the standard comparative steady state manipulation, we prove in appendix the following proposition.

PROPOSITION 1. If fair retirement income insurance exists and f" = 0, then sgn (dH/dT) = sgn(n - r) and dk/dT < 0. In particular, if r = n so that the economy is on the golden rule growth path, then dH/dT = 0 and dk/dT = -P[(1 + n).sup.-2].

There is no theoretical reason to believe that the interest rate should be larger than the population growth rate or vice versa, but from experience of developed economies, a larger interest rate seems more likely to be the case. (11) Then, according to Proposition 1, the longevity effect of an old age subsidy of the Social Security type tends to be negative.

Philipson and Becker consider only the golden rule case in which r = n. From Proposition 1 we have for golden rule economies and fair retirement income insurance that increases in the size of a pay-as-you-go scheme are longevity-neutral and have the usual negative effect on the equilibrium capital stock. Our result here stands in sharp contrast to the very intuitive argument made by Philipson and Becker that an increase in a longevity-contingent public old-age pension distorts the choice of living well versus living long and always works to encourage living longer. However, Philipson and Becket treat the time of death as certain. As a result, a balanced-budget increase in the old-age pension (a mortality-contingent claim) constitutes a compensated decrease in the price of longevity consumption when the interest rate and the population growth rate are equal, which always induces the individual to live longer--a standard moral hazard-based argument.

In contrast, our uncertain lifetime model contains a second mortality-contingent claim represented by the proceeds of one's private saving that goes side by side with the public pension. An increase in the public pension, within the parameter setting considered here (r = n), will be fully offset by a decrease in this second mortality-contingent claim through a reduction in saving, with the full price of living longer remaining unchanged. This argument can be demonstrated precisely as follows. From constraint (4), the second period consumption for survivors--the total mortality-contingent claim--is [c.sup.2] = (1 + r)s/ P([bar.H]) + T when fair retirement insurance exists (I = 1). (12) It is easy to see that an increase in T is fully offset so that total mortality-contingent claims remain constant, if ds/dT = -P([bar.H])/(1 + r). This is exactly the case when r = n, because s = (1 + n)k and dk/dT = -P[(1 + n).sup.-2] from Proposition 1.

The two conditions set forth in Proposition 1 warrant further investigation. First, Proposition 1 assumes the existence of a fair retirement income insurance market. In the absence of fair retirement income insurance, would there still exist a perfect substitute for publically provided retirement income insurance (the public pension) so that an increase in the size of the public pension would be fully offset and would not increase the total mortality-contingent claim when r = n? In the next subsection we show that fair retirement income insurance is not critical for the results in Proposition 1. Second, Proposition 1 assumes constant marginal productivity of capital (i.e., f" = 0), an assumption also made in Philipson and Becker. When f" < 0, an increase in the old-age public pension also has an indirect longevity effect through an interest rate change. It is easily shown (see the last part of the proof of Proposition 1 in the appendix) that if individuals are sufficiently risk averse in the sense that

(9) -U"([c.sub.1])/U'([c.sub.1]) > P'(1 + n)/(Pf'),

then the indirect effect on longevity of a public old-age pension through its impact on the interest rate strengthens the negative partial equilibrium effect. If (9) fails to hold, then the indirect longevity effect through interest rate changes could be great enough to offset the negative partial equilibrium effect on longevity. As a result, the net longevity effect may be positive.

The Absence of Retirement Income Insurance (I = 0)

In the absence of retirement income insurance, the young confiscate the wealth of nonsurvivors so that the utility function can be written as

U{[w.sub.t](1 - [[tau].sub.t]) - [s.sub.t] - [H.sub.t] + [1 - P([H.sub.t-1])] (1 + [r.sub.t]) x [s.sub.t-1]/(1 + n)} + [beta]P([H.sub.t])U[(1 + [r.sub.t+1]) [s.sub.t] + T],

and maximization of expected utility requires

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(11) -U'{[w.sub.t](1 - [[tau].sub.t]) - [s.sub.t] - [H.sub.t] + [1 - P ([H.sub.t-1])](1 + [r.sub.t]) x [s.sub.t-1]/(1 + n)} + [beta]P'([H.sub.t]) x U[(1 + [r.sub.t+1][s.sub.t] + T] = 0.

The steady-state values of capital and longevity can be obtained by substituting (1), (5), and (8) into (10) and (11), and then by letting [H.sub.t-1] = [H.sub.t] = H, [k.sub.t] = [k.sub.t+1] = k. The standard comparative steady state manipulation yields the following proposition:

PROPOSITION 2. If private retirement income insurance is nonexistent and f" = 0, then sgn(dH/dT) = sgn(n - r) and dk/dT < 0 whenever r [greater than or equal to] n. In particular, if r = n so that the economy is operated on the golden rule growth path, then dH/dT = 0 and dk/dT = -[(1 + n).sup.-2].

The intuition for the longevity results of Proposition 2 is similar to Proposition 1. In the absence of fair retirement income insurance, the total mortality-contingent claim in terms of second-period consumption conditional on survival is [c.sup.2] = (1 + r)s + T. Thus, an increase in T would be fully offset so that the total mortality-contingent claim stays the same--that is, there is no additional incentive for living long from an increase in T--as long as ds/dT = -1/(1 + r), which is the case when r = n. The negative longevity effect of an increase in old-age pensions when r > n results from an income effect. When r > n, the pay-as-you-go old-age pension system is a worse deal than private investment. Thus, an increase in the size of old-age pensions in this situation has a negative income effect, implying a negative longevity effect as long as longevity is a normal good.

IV. AN ALTERNATIVE MODEL OF STOCHASTIC ENDOGENOUS LONGEVITY

To ensure tractability in introducing stochastic endogenous longevity into a general equilibrium OLG model, we have assumed that individuals live either one or two periods. A striking result obtained for this model is that the longevity effect from an increase in the old-age pension is negative as long as the interest rate is larger than the population growth rate whether or not fair retirement income insurance exists (Propositions 1 and 2). Our results stand in sharp contrast to the positive longevity effect obtained by Philipson and Becker for a deterministic version of endogenous longevity. One wonders, then, whether our results are due to the simple version of stochastic endogenous longevity adopted in the previous sections, rather than general uncertainty about time of death. We answer this question by analyzing the longevity effect of an increase in old-age pensions for a less restrictive treatment of stochastic endogenous longevity.

The assumption that individuals face no further uncertainty about longevity if they survive into the retirement period makes the public pension more like a lump-sum subsidy than an annuity. (13) We introduce an uncertain annuity by modifying a model due to Sheshinski and Weiss (1981). Specifically, we allow survivors to affect their length of life in the retirement period through a health investment. As in the previous model, we assume interest is accumulated between periods, not within periods. However, this model differs from the earlier one in that each period now should be considered as of unit length rather than a time point of zero length. The fraction of the retirement period actually lived by an individual is a random variable [theta], 0 [less than or equal to] [theta] [less than or equal to] [L.sub.max] < 1, where [L.sub.max] is the biological limit to human life. The distribution function of [theta, denoted by F([theta], [H.sub.t]), where [F.sub.H] < 0, and [H.sub.t] is as before the health investment made by a generation t individual in the working period, has a mean of [[bar].[theta]]([H.sub.t]). (14) Whether health expenditures during the first period of life can affect the biological maximum is an open question. In what follows we consider two cases: (1) where [L.sub.max] is a scalar, and (2) where [L.sub.max] = [L.sub.max] ([H.sub.t]), [L'.sub.max] > 0.

Let [c.sup.t.sub.t] and [c.sup.t+1.sub.t] be the consumption flows in the first and second periods, respectively, of a generation t individual. The expected additive utility of the individual is

U([c.sup.t.sub.t]) + [beta][[bar.[theta]]([H.sub.t])U([c.sup.t+1.sub.t]).

The individual's budget constraints depend on availability and efficiency of the annuity market. As in the last sections, we consider two polar cases: the existence and absence of a perfectly competitive annuity market. (15)

Fair Annuities Exist

When fair annuities exist and there is no aggregate uncertainty, the only attribute of the cdf F that matters is the mean. Herein, allowing first period health expenditures to affect the biological lifetime maximum does not affect the result. When fair annuity markets exist the budget constraints are

[c.sup.t.sub.t] = [w.sub.t](1 - [[tau].sub.t]) - [s.sub.t] - [H.sub.t],

[c.sup.t+1.sub.t] = (1 + [r.sub.t+1][s.sub.t]/ [[bar.[theta]]([[bar.H].sub.t]) + T,

where [[bar.H].sub.t] is the equilibrium level of [H.sub.t] and definitions of other variables and parameters are as before, except that ([c.sup.t.sub.t], [c.sup.t+1.sub.t]) should now be viewed as flows. (16) The first-order conditions of the individual's problem are

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(13) -U'[[w.sub.t](1 - [[tau].sub.t]) - [s.sub.t] - [H.sub.t]] + [beta] [bar.[theta]']([H.sub.t]) x U[(1 + [r.sub.t+1])[s.sub.t]/[bar.[theta]] ([[bar.H].sub.t]) + T] = 0.

The fact that the amount of the public old age transfer is now endogenous results in the government budget balance condition being a function of equilibrium health care expenditures so that

(14) (1 + n)[w.sub.t][[tau].sub.t] = [bar.[theta]]([H.sub.t-1]T.

Substituting (14) into (12) and (13) and letting [H.sub.t] = H, [s.sub.t] = s, [w.sub.t] = w, [r.sub.t] = r yields the characterization of steady-state equilibrium. Letting r = n, the standard comparative statics yields

dH/dT = 0

ds/dT = -[bar.[theta]](H)/(1 + n),

which is basically the result given in Proposition 1.

No Annuities Exist

In the case of no annuity market, the individual's budget constraints are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[c.sup.t+1.sub.t] = (1 + [r.sub.t+1])[s.sub.t]/[L.sub.max]([H.sub.t]) + T,

where we have assumed that consumption flows in the second period must be feasible up to the biological lifetime maximum. As before, we assume that any unintended bequests of a generation are divided equally among the next generation. (17)

The first-order conditions are

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Following the same steps as for the last case, we have, when r = n and [L'.sub.max] = 0, that

dH/dT = 0 ds/dT = -[L.sub.max]/(1 + n)

which is essentially the result given in Proposition 2.

The more interesting case occurs when individuals are able to affect the biological limit on life through first-period health expenditures, that is, [L'.sub.max] > 0. In this case we are finally able to generate the positive longevity effect of old-age subsidies achieved by Philipson and Becker. In general, we have the following proposition, the proof of which is given in the appendix:

PROPOSITION 3. When the economy is on the golden rule path and individuals cannot compensate for an intergenerational transfer via a fair annuity market, then dH/ dT > 0 if and only if first period health expenditures increase the biological limit on life.

Our findings from sections III and IV are summarized in Table 1.

V. THE CASE OF ALTRUISM

When children and parents are linked through altruism and lifetime is uncertain, the initial asset position of newborn individuals in any generation may differ as a result of their parents survival status. Because this history dependence effect accumulates over time, there is no well-defined steady state, complicating the formal analysis of old-age subsidies. However, in a world of exogenous (though stochastic) longevity, Sheshinski and Weiss (1981) show that the existence of a fair annuity market is sufficient to avoid history dependence. They further show that with fair annuity markets, individuals purchase annuities exclusively for possible second-period consumption and have ordinary savings exclusively for a bequest that is independent of survival into the second period.

In this section we show even that when longevity is endogenous, the Sheshinski and Weiss results continue to hold. In addition, in this complete intertemporal market setting, we show that longevity is unaffected by the level of publically provided pensions and that a form of Ricardian equivalence holds regarding the capital stock.

Expected utility when agents are altruistic can be written as

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [V.sub.t] is the expected utility of a generation t individual, (18) and

(18) [c.sup.t.sub.t] = [w.sub.t](1 - [[tau].sub.t]) + [I.sub.t-1] - [s.sub.t] - [B.sub.t] - [H.sub.t],

(19) [c.sup.t+1.sub.t] = [s.sub.t](1 + [r.sub.t+1])/P([[bar.H].sub.t] + [B.sub.t](1 + [r.sub.t+1]) + T - (1 + n)[b.sub.t],

(20) [A.sub.t+1] = [w.sub.t+1](1 - [[tau].sub.t+1]) + [b.sub.t]

(21) [[??].sub.t+1] = [w.sub.t+1](1 - [[tau].sub.t+1]) + [B.sub.t](1 + [r.sub.t+1])/(1 +n),

where [I.sub.t-1] is any inheritance received from the preceding generation; (19) [s.sub.t] is the level of saving for the future, which is invested in the retirement income insurance market earning rate of return (1 + [r.sub.t+1])/P([[bar.H].sub.t]) if the individual survives and minus one if the individual is a nonsurvivor; (20) [B.sub.t] is the savings-for the bequest the individual plans for heirs contingent on failure to survive into the second period, which will earn a rate of return of 1 + [r.sub.t+1]; and [b.sub.t] is the per-child bequest made by the generation t individual conditional on survival; and [A.sub.t+1], [[??].sub.t+1], are respectively an heir's disposable income when the parent survives or dies. All other variables and parameters are as previously defined.

Substituting (18)-(21) into (17), we have an unconstrained optimization problem with choice variables [s.sub.t], [B.sub.t], [H.sub.t], and [b.sub.t]. After substitution of (5), the first-order conditions can be written as

(22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(25) -U'{(1 + [r.sub.t+1][s.sub.t]/P([H.sub.t]) + [B.sub.t](1 + [r.sub.t+1] + T - [b.sub.t](1 + n)} + [V'.sub.t+1]{[w.sub.t+1] - P([H.sub.t])T/(1 + n) + [b.sub.t]} = 0.

From (22), (23) and (25), we have

(26) [V'.sub.t+1][[w.sub.t+1] - P([H.sub.t])T/(1 + n) + [b.sub.t]] = [V'.sub.t+1][[w.sub.t+1] - P([H.sub.t])T/(1 + n) + [B.sub.t](1 + [r.sub.t+1])/(1 + n)],

which implies the bequest received by any heir is independent of the parent's survival. Further, from (1) and (8), [w.sub.t+1], [r.sub.t+1], and [s.sub.t] + [B.sub.t] can be expressed in terms of [k.sub.t+1]. Assume that ([k.sub.t+1], [H.sub.t], [s.sub.t], [B.sub.t], [b.sub.t]) is a solution to (22)-(25) when the subsidy per old person in period t + 1 is T. Now assume an increase in the old-age subsidy of [DELTA]T so that the total becomes T + [DELTA]T. It follows then that

([k.sub.t+1], [H.sub.t], [s.sub.t] - P[DELTA]T/(1 + [r.sub.t+1], [B.sub.t] + P[DELTA]T/(1 + [r.sub.t+1]), [b.sub.t] + P[DELTA]T/(1 + n))

is an equilibrium solution to (22) (25) when T is replaced with T + [DELTA]T. Assuming that the solution is unique, we have that a $1 increase in the transfer from the young to the old will result in a P([H.sub.t]) dollar increase in bequests independent of survival with [k.sub.t+1], [H.sub.t], and the after tax wealth of the next generation unchanged. Thus, longevity is independent of the level of the old-age pension and Ricardian equivalence holds even when the time of death is uncertain and longevity is endogenous. Importantly, this result holds regardless of the relative size of r and n.

PROPOSITION 4. If individuals are altruistic and fair retirement income insurance exists, then longevity and the capital stock are neutral with respect to changes' in the level of a pay-as-you-go public pension.

VI. THE DIRECTION OF CAUSALITY: LONGEVITY OR OLD-AGE PENSIONS?

How can the observed simultaneous increase in longevity and old-age benefits be reconciled with our theoretical results? One plausible explanation is that factors other than old-age pensions have caused longevity to increase, which in turn has caused the increase in old-age benefits. There are at least two reasons that an increase in old-age benefits can result from increased longevity. First, even though per capita benefits are fixed, increased longevity implies a larger size of aggregate benefits. Second, from a public choice point of view, per capita old-age benefits are likely to be larger as the median voter becomes older due to increased longevity. (21) In this section we use the basic framework of the fair annuity case in section III and consider two institutional parameters (in addition to the level of old-age pensions): a health care expenditures subsidy and changes in longevity producing technology.

Let [alpha] be the subsidy rate and [delta] be a technology parameter such that survival probability is now P([delta][H.sub.t]). (22) Our budget constraints now become

(3') [c.sup.t.sub.t] = [w.sub.t](1 - [[tau].sub.t]) - [s.sub.t] - (1 - [alpha])[H.sub.t]

(4') [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and utility maximization requires that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We complete our representation of equilibrium with (1), (8), and the new government-balanced budget condition

(5') (1 + n)[w.sub.t][[tau].sub.t] = P([delta][H.sub.t-1])T + (1 + n)[alpha][H.sub.t].

At the steady-state equilibrium, [H.sub.t] = [H.sub.t-1] = H, [k.sub.t] = [k.sub.t+1] = k, and the comparative steady state results concerning the longevity effects of an increase in a and an increase in [delta] are given by the following proposition (proof is in the appendix):

PROPOSITION 5. If f" = 0, longevity is increasing in the price subsidy on longevity-extending expenditures and in longevity-extending technology. (23)

We offer a simple intuition for these comparative steady-state results in terms of the substitution and income effects of a price change. The key to this interpretation is viewing the generation [??] individual's problem with respect to ([H.sub.t], [s.sub.t]) as a choice between longevity consumption in the form of 8/4t, and nonlongevity consumption in the form of residual wealth,

[J.sub.t] = [w.sub.t](1 - [[tau].sub.t]) - (1 - [alpha])[H.sub.t].

More specifically, the utility associated with a certain combination of [delta][H.sub.t] and [J.sub.t] is realized by choosing [s.sub.t] to maximize (2) subject to (3') and (4'), and the budget constraint on longevity consumption [delta][H.sub.t] and nonlongevity consumption [J.sub.t] is

[J.sub.t] = [(1 - [alpha])/[delta]]([delta][H.sub.t]) = [w.sub.t](1 - [[tau].sub.t]).

Because an increase in either [alpha] or [beta] reduces the price of longevity consumption, both work to increase longevity. However, an increase in [delta] represents a pure decrease in the price of longevity while an increase in [alpha] represents a compensated price decrease because, from (5'), the payroll tax (which is really a lump-sum tax here) must be increased to finance the increased subsidy. Thus, an increase in the price subsidy increases longevity through a pure substitution effect, whereas progress in technology increases longevity through both substitution and income effects.

There exists convincing evidence that price subsidies and technological progress have played important roles in the increases in longevity we have observed in the past several decades. As documented in Cutler (1999), longevity increases began to take off in 1940s. Cutler attributed the sharp longevity increases since then to the discovery and popular use of antibiotics and the birth and subsequent growth of Medicare. Obviously, the former is an example of technological progress. However, Medicare can affect individuals' longevity choice in opposite directions. As a government price subsidy on senior citizens' health care consumption, it has a positive longevity effect. On the other hand, it has a negative longevity effect as a pay-as-you-go, mortality-contingent, old-age subsidy as part of Medicare benefits are financed by a payroll tax on current working generations. It is likely that the price subsidy component of Medicare dominates the pay-as-you-go component. However, this net positive longevity effect of Medicare cannot be attributed to the fact that Medicare is partially a mortality-contingent old-age subsidy.

Because technological progress in the longevity-extending industry has a positive longevity effect, a natural question is: Could Social Security-type public old-age pensions have a positive longevity effect through such technological progress? Our answer is "no." The reason is, although an increase in supply-side research and development activities on longevity-extending technology may be caused by an increase in demand for longevity, as we have proved in this article, larger old-age pensions would not generate higher demand for longevity in the most likely case. However, Medicare may have caused the demand for longevity to increase, which in turn has contributed to the advance in longevity-extending technology. But again, it is not Medicare's component of mortality-contingent claims that causes such increase in the demand for longevity.

Finally, if government longevity expenditures are not in the form of excise subsidies (Medicare, tax deduction, and so on, as represented by the parameter [alpha]), but rather in the form of lump-sum in-kind subsidies (immunization, health education, better environment, etc.), it can be demonstrated following the same procedure that as long as the public longevity expenditures are perfect substitutes for private longevity expenditures, and the public sector is as efficient as the private sector in providing longevity consumption, an increase in government longevity expenditures will be fully offset by an equal decrease in private longevity expenditures, with longevity remaining unchanged.

In contrast to this theoretical result, the empirical work of Anand and Ravallion (1993) suggests that public health expenditures (including lump-sum expenditures) play an important role in explaining longevity. We can reconcile their finding with ours for lump-sum health subsidies in two ways. First, we have ignored the possible complementarity between public and private health expenditures. (24) If public and private health care expenditures are complements at the margin, an increase in the public health expenditures, absent of any efficiency effects, will have a positive effect on longevity similar to that of an increase in the price subsidy. Second, if in addition the complementary public health expenditures are more efficient than their private counterparts, both the income and the substitution effects work to increase longevity, just as in the case of technological progress. (25)

VII. CONCLUSION

The historical simultaneous increase in longevity and old-age pensions financed by pay-as-you-go schemes makes one wonder if there is a degree of causality between a guaranteed provision for one's old age and one's choice of longevity-extending expenditures. Put simply, if you think of an old age pension as a pot of gold, then it is natural to think that the bigger the pot of gold, the more effort an individual will put into acquiring it. In reality, however, the problem is not this simple for two reasons: (1) unlike a leprechaun's pot of gold, this one must be paid for through taxation during each individual's early years; and (2) if intergenerational markets exist, any potential wealth in the autumn of one's life can be consumed in the spring. Indeed, these factors when incorporated into an OLG model result in publically financed old-age pensions having a completely neutral effect on longevity.

In this article we incorporate endogenous longevity by making the time of death uncertain but allowing expenditures on health care to increase the expectation of survival into retirement. When agents are not altruistic and the upper limit to human life cannot be increased by longevity-extending health expenditures, we find that, regardless of the existence of a fair retirement income insurance market, an increase in the old-age pension financed by an increase in the payroll tax will have a positive effect on longevity if and only if the rate of growth in population is greater than the interest rate. Moreover, for any golden rule economy changes in the level of public old-age pensions have no effect on longevity, that is, public old-age pensions are longevity-neutral. Further, we obtain the traditional negative effect of an intergenerational transfer on saving for all cases where the interest rate is not less than the population growth rate and even for some cases when population growth exceeds the rate of interest.

Interestingly, even when nonaltruistic individuals can influence the biological maximum on life, increases in old-age pensions remain longevity-neutral for golden rule economies so long as fair retirement income insurance exists. However, if we completely eliminate retirement income insurance, we can generate a result argued by Philipson and Becket that increases in publically financed old-age pensions will increase longevity.

Assuming altruistic agents raises the question of whether an agent's early death results in that agent's heirs first-period constraint being different than those whose parent survives. If so, then there will be no steady state. But even if a steady state does not exist, the effect on longevity of publicly financed old-age pensions may still be determinate. We show that allowing agents to be altruistic does not change our basic results so long as fair retirement income insurance exists. Moreover, the inheritance that an agent plans for an heir should that agent not survive is identical to the inheritance the agent leaves for the heir on survival. Therefore, when fair retirement insurance exists there is a steady state solution where (1) individuals segment their savings between ordinary savings accounts and retirement income insurance with the former exclusively for bequests and the latter exclusively for own second-period consumption, and (2) the Ricardian equivalence result holds where changes in the level of publicly financed old-age pensions are fully absorbed by private intergenerational transfers, with allocation of resources saving and longevity-extending expenditures in particular unchanged. Once again, changes in publicly supplied old-age pensions are neutral with respect to longevity.

If increases in publicly supplied old-age pensions do not affect longevity and may even reduce it, what then explains the upward trend in both old-age pensions and longevity? We pose an alternative explanation for the concurrence of increases in longevity and old-age pensions with rational agents responding to changes in prices and technology. We find that a price subsidy on longevity-extending health investment or progress in longevity-producing technology will cause the increased longevity. General lump-sum public health expenditures that offer complementary longevity benefits to private health expenditures can cause longevity to increase for the same reason as a price subsidy or technological progress.

Finally, a short-run causal relation between longevity gains and the advent of mortality-contingent claims such as Social Security and Medicare is not inconsistent with our theoretical results. The recorded longevity gains have come during the start-up phase of the program where for early beneficiaries, the new program is a pure transfer for which they paid nothing. Our results pertain to the steady state where once the costs are fully taken into account the longevity gains from the transition would disappear.

APPENDIX

Proof of Proposition 1

The steady-state k and H are given by

-U'[f - kf' - PT[(1 + n).sup.-1] - (1 + n)k - H] + [beta]f'U'[f'(1 + n)k[P.sup.-1] + T] = 0

-U'[f - kf' -PT[(1 + n).sup.-1] - (1 + n)k - H] + [beta]P'U x [f'(1 + n)k[P.sup.-1] + T] = 0

Therefore,

dk/dT = [absolute value of B] / [absolute value of A]

dH/dT = [absolute value of C] / [absolute value of A]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and [absolute value of A], [absolute value of B], [absolute value of C] are evaluated at the steady-state values of (k, H, [c.sub.1], [c.sub.2]).

It can be shown under fairly general conditions that [absolute value of A] > 0, making the saving and longevity effects of an increase in the old-age pension hinge on the signs of [absolute value of B] and [absolute value of C], respectively. (26) Through expansion and substitution of condition (6) it can be shown that

[absolute value of B] = U"([c.sub.1])[1 + P'T/(1 + n) + P'kf'/P][beta] [U"([c.sub.2])f' - U'([c.sub.2])P'] - U([c.sub.2])[beta]P"[U"([c.sub.1] P/(1 + n) + U"([c.sub.2])[beta]f'] < 0,

[absolute value of C] = U"([c.sub.1])[beta](1 +n - f')[U'([c.sub.2])P' - U"([c.sub.2])f'] + f"U'([c.sub.1])U'([c.sub.2])[beta]P[(1 + n).sup.-1] [-U"([c.sub.1])/U'([c.sub.1]) - P'(1 + n)/(Pf')].

Substituting f" = 0 into the expression for [absolute value of C] immediately yields that [absolute value of C] > 0 if and only if n > r.

Letting both f" = 0 and n = r yields [absolute value of C] = 0 and [absolute value of B] / [absolute value of A] = -P[(1 + n).sup.-2].

Also note from the expression of [absolute value of C] that when (9) is satisfied, allowing f" < 0 adds to the negative longevity effect (when r > n) from a larger T.

Proof of Proposition 2

Similar to the proof of Proposition 1, we have

dk/dT = [absolute value of B] / [absolute value of A]

dH/dT = [absolute value of C] / [absolute value of A]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and [absolute value of A], [absolute value of B], [absolute value of C] are evaluated at the steady-state values of (k, H, [c.sub.1], [c.sub.2]).

It is shown next that [absolute value of A] > 0 from the stability condition. Thus, the saving and longevity effects of an increase in the old-age pension hinge on the signs of [absolute value of B] and [absolute value of C], respectively. Through expansion and substitution of condition (10) and setting f" = 0, it can be shown that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Because f' = 1 + r, [absolute value of C] has the same sign as n - r. In addition, r [greater than or equal to] n and [absolute value of A] > 0 imply that [absolute value of B] < 0, and if r = n then dH/dT = 0 and dk/dT = -[(1 + n).sup.2]. Q.E.D.

Proof of [absolute value of [??]A[??]] > 0

Through substitution of (1), (5), and (8), conditions (10) and (11) can be turned into the following first-order difference equation system about k and H,

(A-1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(A-2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The steady-state k and H are given by F(k, k, H, H, T) = 0 and G(k, k, H, H, T) = 0 so that,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Linearizing difference equation system (A-1) and (A-2) at its steady state (k, H) and rearranging it into the standard form, we have

(A-3) [k.sub.t+1] - k = a([k.sub.t] - k) + b([H.sub.t-1] - H),

(A-4) [H.sub.t] - H = c([k.sub.t] - k) + d([H.sub.t-1] - H),

where

(A-5) a = [[OMEGA].sup.-1]([G.sub.1][F.sub.4] - [G.sub.4][F.sub.1]),

(A-6) b = [[OMEGA].sup.-1]([G.sub.3][F.sub.4] - [G.sub.4][F.sub.3]),

(A-7) c = [[OMEGA].sup.-1]([G.sub.2][F.sub.1] - [G.sub.1][F.sub.2]),

(A-8) d = [[OMEGA].sup.-1]([G.sub.2][F.sub.3] - [G.sub.3][F.sub.2]),

and

(A-9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

from the second-order condition of the individual's choice problem.

Because [absolute value of A] = (a + d - 1)[OMEGA] to prove a + d < 1, note first that a + d = [[lambda].sub.1] + [[lambda].sub.2], where [[lambda].sub.1] and [[lambda].sub.2] are the two characteristic roots of the difference equation system (A-3) and (A-4). Second,

(A-10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

because [F.sub.1] = [G.sub.1] and [F.sub.3] - [G.sub.3]. So, a + d is the value of the nonzero characteristic root and stability requires that a + d < 1. Q.E.D.

Proof of Proposition 3

Substituting (14) into (15) and (16) (so [[tau].sub.t] is gone) and letting [H.sub.t] = H, [s.sub.t] = s, [w.sub.t] = w, and [r.sub.t] = r gives us the following characterization of the steady-state equilibrium.

(A-11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(A-12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The comparative static result with respect to the longevity effect of the old-age subsidy is given by

dH/dT = [absolute value of C] / [absolute value of A],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and [absolute value of] > 0 from the stability condition, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It is easy to see that when r = n, [absolute value of C] = 0 if [L'.sub.max]=0 and [absolute value of C] > 0 if [L'.sub.max] > 0. Q.E.D.

Proof of Proposition 5

d([delta]H)/d[alpha] = [absolute value of [C'.sub.[alpha]] / [absolute value of A'],

d([delta]H)/d[delta] = [absolute value of [C'.sub.[delta]] / [absolute value of A'],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For the same reason that [absolute value of A] is positive, [absolute value of A'] is positive. It is easy to show that [absolute value of [C'.sub.[alpha]] > 0, [absolute value of [C'.sub.[delta]] > 0 if f" = 0. Q.E.D.

TABLE 1
The Effects of Government Pensions on Health Expenditures and Savings

 Fair Retirement
 Income Insurance

 r = n r > n

[dL.sub.max]/dH = 0 dH/dT 0 < 0
 dk/dT -P[(1 + n).sup.-2] < 0
[dL.sub.max]/dH > 0 dH/dT 0 < 0
 dk/dT < 0 < 0

 No Retirement
 Income Insurance

 r = n r > n

[dL.sub.max]/dH = 0 dH/dT 0 < 0
 dk/dT -[(1 + n).sup.-2] < 0
[dL.sub.max]/dH > 0 dH/dT > 0 ?
 dk/dT ? ?

ABBREVIATIONS

OLG: Overlapping Generations

(1.) This version of lifetime uncertainty is adopted in Abel (1985), and Eckstein et al. (1985). Sheshinski and Weiss (1981) and Blanchard and Fisher (1989) use alternative versions of uncertainty about time of death. The uncertain lifetime in all those studies, however, is exogenous.

(2.) Whether longevity-extending expenditures are all made in the first-period or at the beginning of the second period before survival status is known, they impact first-period consumption in the same way and an agents' trade-off between saving for retirement consumption and investing in longevity remains unchanged. Thus, the timing of longevity investment, for the purposes of analyzing the effects of old-age pensions, does not affect any results obtained in this article.

(3.) There is no interest accumulation within a period because a period because is just a time point of zero length.

(4.) It can be shown that the equilibrium that results from any policy with government borrowing can be replicated by some intergenerational tax-transfer scheme without any borrowing at any date. For example, see McCandless and Wallace (1991).

(5.) We consider the case where parents are altruistic toward their children in section V. Results concerning an absence of altruism/intended bequests are relevant for at least two reasons: (1) There is strong empirical evidence against the altruism hypothesis. For example, Altonji et al. (1997) find that redistributing $1 from a recipient child to donor parents leads to less than a 13-cent increase in the parents' transfer to the child. Also see Gokhale et al. (2001) for a comprehensive survey on empirical evidence on altruism; (2) previous studies incorporating endogenous longevity (for example Philipson and Becker 1998) do not include altruistic bequest motives, so it is more interesting to study a nonaltruism model for comparison purposes.

(6.) As the budget constraints below make clear, the assumption U(0) = 0 amounts to treating death as equivalent to zero consumption. We could treat death as negative consumption to capture the fear of or loss from death without changing any result obtained in the article.

(7.) By the fair insurance arrangement, individuals choose their coverage (the amount received on survival) by paying the appropriate premium (the amount forgone on nonsurvival). The insurer only knows the average survival probability P([[bar.H].sub.t]) and sets the coverage/premium ratio accordingly.

(8.) The assumption of no aggregate lifetime uncertainty has been taken for granted in the literature of uncertain lifetime and can be justified using the law of large numbers.

(9.) However, the distinction between [H.sub.t] and [[bar.H].sub.t] is important before taking the first-order derivative because the latter is not subject to individual choice.

(10.) The purpose of these substitutions is to represent wage rates, interest rates, tax rates, and savings in terms of capital stocks and health expenditures.

(11.) The existence of capital taxes and lifetime uncertainty both work in the direction of increasing the equilibrium interest rate. The condition r > n also underlies much of the discussion about privatizing Social Security with private retirement accounts to take advantage of a high rate of return on capital, as compared to the low implicit rate of return on one's Social Security contributions, which is the population growth rate in the long run. For details of that discussion, see Mariger (1997), Geanakoplos et al. (1998), and Liu et al. (2000).

(12.) We drop the time subscript to consider only the steady-state variables.

(13.) The subsidy in this version is still a mortality contingent claim because one has to be alive after retirement to be eligible for it.

(14.) One could assume that individuals engage in longevity-extending expenditures during the retirement period, however, this alternative timing of H does not affect any of the following results.

(15.) We retain the assumption that the mean length of life can be treated as parametric once the equilibrium level of health expenditures is reached. In effect, those who die younger than the mean subsidize those who live beyond the mean.

(16.) Using this interpretation, total consumption in the first period is [c.sup.t.sub.t] while total consumption in the second period is [theta]([H.sub.t])[c.sup.t+1.sub.t].

(17.) The first assumption effectively ensures no debt at time of death, and the second one is imposed to ensure history independence, hence the existence of a steady state. We appeal to the law of large numbers to eliminate any aggregate uncertainty in unintended bequests to the young.

(18.) Utility specification (17) is standard following the tradition of Barro (1974). According to Bernheim and Bagwell (1988), this "dynastic family"-type utility is restrictive in the sense that it is responsible for too many "equivalence results." Even within this specification, however, Ricardian equivalence is often found invalid with additional endogeneity. See Enders and Lapan (1990) for an example in which the traditional Ricardian equivalence for altruistic agents does not hold after making fertility endogenous. We have introduced endogenous longevity in the model. Hence, the Bernheim-Bagwell critique should be less critical here.

(19.) [I.sub.t-1] may differ from individual to individual depending on one's parent and grandparents' longevity.

(20.) Obviously, the total saving by the individual is [s.sub.t] + [B.sub.t], and therefore [s.sub.t] in the capital market equilibrium condition (8) should equal [s.sub.t] + [B.sub.t].

(21.) Browning (1975) applies this public choice approach to studying the size of Social Security programs.

(22.) This specification of the technology parameter implies that the longevity benefits of a given level of expenditure [H.sub.t] are [delta][H.sub.t]. In other words, an increase in [delta] represents a cost-saving technology progress.

(23.) Note that our emphasis on the role of technological progress is different from the assumption that longevity is directly determined by the level of technology. In our analysis, technology affects longevity through an individual's choice.

(24.) The importance of incorporating complementarity between these two kinds of health expenditures into the analysis of public health policies is emphasized by Dow et al. (1999).

(25.) For example, health-oriented environment protection and government-financed immunization of contagious diseases may have an efficiency advantage based on the public good and externality arguments.

(26.) We impose these conditions and prove a similar signing result. Because doing so requires being more explicit about the dynamics of the model (one of the conditions is the stability of the dynamic process) and involves many tedious derivations, we choose not to repeat it here. Also note that even imposing f" = 0 without the stability condition would not give us [absolute value of A] > 0. For another example in which the stability condition plays an important role in signing comparative steady state results in an OLG setting, see Lin (1998).

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Geanakoplos, J., O. S. Mitchell, and S. P. Zeldes. "Would a Privatized Social Security System Really Pay a Higher Rate of Return?" Pension Research Council Working Paper 98-6, University of Pennsylvania, 1998.

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Lin, S. "Labor Income Taxation and Human Capital Accumulation." Journal of Public Economics, 68(2), 1998, 291-302.

Liu, L., A. J. Rettenmaier, and T. R. Saving. "Constraints on Big-Bang Solutions: The Case of Intergenerational Transfers." Journal of Institutional and Theoretical Economics, 156(1), 2000, 270-91.

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Philipson, T. J., and G. S. Becker. "Old-Age Longevity and Mortality-Contingent Claims." Journal of Political Economy, 106(3), 1998, 551-73.

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LIQUN LIU, ANDREW J. RETTENMAIER, and THOMAS R. SAVING *

* We thank Scott Freeman, Karsten Jeske, Laurence Kotlikoff, Tomas Philipson, Guoqiang Tian, several anonymous referees, seminar participants at Atlanta Federal Reserve Bank and University of Nevada at Las Vegas, and session participants at Loan Star Economics Conference (1999) and the Econometric Society North American Winter Meeting (2000) for helpful comments on earlier versions of this article. We also thank the Lynde and Harry Bradley Foundation for their generous support of the research underlying this study.

Liu: Associate Research Scientist, Private Enterprise Research Center, Texas A&M University, College Station, TX 77843. Phone 1-979-845-7723, Fax 1-979-845-6636, Email lliu@tamu.edu

Rettenmaier: Executive Associate Director, Private Enterprise Research Center, Texas A&M University, 4231 TAMU College Station, TX 77843. Phone 1-979-845-9658, Fax 1-979-845-6636, Email a-rettenmaier@ tamu.edu

Saving: Director and Distinguished Professor of Economics, Private Enterprise Research Center, Texas A&M University, College Station, TX 77843. Phone 1-979-845-7659, Fax 1-979-845-6636, Email t-saving@ tamu.edu