Health, education, and life cycle savings in the development process.

Tang, Kam Ki ; Zhang, Jie

I. INTRODUCTION

Between countries with low and high life expectancy, there are striking differences in their school enrollments, investment to gross domestic product (GDP) ratios, health spending, and growth rates of per capita income. According to the data set in Barro and Lee (1994), in countries with life expectancy in 1960 below 50 (with a mean of 43.4 years), the average ratio of private investment to GDP was 14%, the average secondary school enrollment ratio was 17.6%, and the average growth rate of per capita GDP was 1.4%, for the period 1960-1989. By contrast, in countries with life expectancy in 1960 above 65, the corresponding average figures were 22%, 71%, and 2.96%, respectively. With no exception, countries with life expectancy in 1960 below 50 (28 in total) are poor and their very low life expectancy is a consequence of little health investment. On the other hand, those with life expectancy in 1960 above 65 are developed countries [24 in total, mostly the members of the Organisation for Economic Cooperation and Development (OECD)] and have much higher health spending than countries at the other end of the spectrum. (1)

Similar to the cross-country comparison, there were upward trends in the ratios of health and education spending to GDP and in life expectancy in the time series data of the United States, in Table 1, for the period 1870-2000. The postwar average growth rate of per capita GDP also appeared to be higher than the prewar average growth rate in the United States, as in many other developed countries according to Maddison (1991). Though the long-term saving rate did not have a discernable trend in the United States, it typically had an upward trend in other developed countries as documented by Maddison (1992). It is, thus, important to explore the interaction between life expectancy and growth by investigating household decisions on health investment, human capital investment, and life cycle savings.

Moreover, in many developed countries, health and education expenditures are heavily subsidized or publicly provided through distortionary taxes. To a lesser extent, health and education expenditures are also subsidized in some less developed countries. Thus, it is also important to investigate the impacts of these subsidies on household decisions about health spending, education spending, and life cycle savings. Through this investigation, we can learn how these subsidies affect life expectancy, output growth, and welfare.

Recently, the relationship between longevity and household decisions on savings and human capital investment has received a great deal of attention. The typical view is that rising longevity or declining mortality encourages savings and human capital investment and hence promotes economic growth [see, e.g., Barro and Sala-i-Martin (1995), Boucekkine, de la Croix, and Licandro (2002, 2003), de la Croix and Licandro (1999), Ehrlich and Lui (1991), Skinner (1985), Zhang and Zhang (2005), Zhang, Zhang, and Lee (2001, 2003)]. However, the rate of survival or death is usually treated as exogenous in these papers. Though some studies have considered health investment, for example, Ehrlich and Chuma (1990), Leung, Zhang, and Zhang (2004), and Philipson and Becker (1998), they have not considered human capital investment at the same time and therefore their models do not permit sustainable growth in the long run. A recent exception is the study of Aisa and Pueyo (2006) that explores how government health spending affects sustainable output growth nonmonotonically, assuming an AK technology in final production without human capital. Intuitively, economic growth promises more resource available for future improvements in health care and life expectancy, while rising life expectancy may in turn motivate savings and human capital investment.

Some recent studies have also considered health and education expenditures together in a life cycle model. Among them, Chakraborty and Das (2005) focus on how the distribution of wealth interacts with health investment and human capital investment in accounting for the high intergenerational correlation of economic status and persistent disparities in health status between the rich and the poor. Also, Corrigan, Glomm, and Mendez (2005) find large growth effects of an acquired immunodeficiency syndrome (AIDS) epidemic and relatively small effects of policies such as the subsidization of AIDS medication.

In this paper, we investigate health investment, human capital investment, and life cycle savings in an endogenous growth model. Health investment improves survival to old age that has a lower bound supported by an endowment of health to each young individual. Unlike Chakraborty and Das (2005) and Corrigan, Glomm, and Mendez (2005), however, we focus on whether the equilibrium solution can differ significantly in different stages of development in a way that resembles what we observe in the real world. Also, we explore how subsidies on education spending or health spending influence capital accumulation, health investment, and welfare in different stages of development.

Our model predicts two distinctive phases of development. When income is sufficiently low, there is no health investment because the marginal utility of consumption would then exceed the marginal utility of health investment, given the lower bound on the expected rate of survival. When the rate of survival is at its minimum, the saving rate is at its minimum as well, leading to very slow growth. When income grows, health investment will become positive and the saving rate will rise substantially, leading to higher life expectancy and faster growth. These results capture some of the stylized facts mentioned earlier.

Interestingly, a health subsidy can move the economy from the no-health-investment phase to the next, a transition that brings about higher life expectancy, greater savings, and faster growth. A growing economy in this model converges to a unique balanced growth path. A health subsidy does not affect the balanced growth rate, while an education subsidy increases it under plausible conditions. These subsidies through a wage income tax can also improve welfare with an externality from average health spending in the determination of health status, particularly in the short run. Numerically, with high income, the ideal subsidy rates are found to be around 60% for both education and health expenditures. The findings support recent reform in some developed countries converting a public health and education system into one with shared financial responsibility between the state and households.

The rest of the paper proceeds as follows. Section II describes the model. Sections III and IV provide analytical and numerical results, respectively. Section V summarizes our findings and discusses further extensions. The last section concludes.

II. THE MODEL

The economy consists of overlapping generations of agents who live for three periods (one period in childhood and two in adulthood). Children learn to embody human capital through education, young adults work, and old adults live in retirement. Each worker gives birth to one child, and each working generation has a mass L. Survival from childhood to young adulthood is certain, while survival to old age is uncertain at a rate of P([h.sub.t]) that is increasing and concave in health status [h.sub.t]. We abstract from child mortality and the choice of the number of children to keep the model tractable. Workers make decisions on life cycle savings s, investment in health m, and investment in the human capital of their children q.

The health status of a worker is determined according to

(1) [h.sub.t] = [A.sub.h][m.sup.[alpha].sub.t][([[bar.m].sub.t]).sup.1-[alpha]] + [bar.h]

where [m.sub.t] is the worker's investment in health, [[bar.m].sub.t] refers to average investment in health, [bar.h] refers to an endowment of health to each young adult, [A.sub.h] > 0 indicates the effectiveness of the health technology, and 0 < [alpha] < 1 indicates the relative importance of an individual's own health spending versus average health spending in the economy. One rationale for the inclusion of average health investment in the determination of health status is that with little health spending on average there would be lack of health professionals and health care facilities as in many poor countries. In this scenario, health spending by a single agent can hardly enhance his health status. The endowed component of health reflects the fact that there is still a chance expected to survive to old age even without health spending. As we will see, this endowment of health can lead to realistic transitional dynamics in this model.

The human capital of a child is determined by

(2) [e.sub.t] + 1 = [A.sub.e][q.sup.[eta].sub.t][e.sup.1-[eta].sub.t],

where [e.sub.t + 1] and [e.sub.t] stand for the human capital of the child and his parent, respectively, and [q.sub.t] is the amount of education spending. In addition, [A.sub.e] > 0 is an efficiency parameter in education and 0 < [eta] < 1 measures the relative importance of education spending versus parental human capital in education.

The rate of survival to old age is an exponential function of health status:

(3) [P.sub.t = 1 -/exp([h.sub.t]),

which is clearly increasing and concave in health status. This assumption in Equation (3) implies a strong interaction between health expenditures and life expectancy and highlights the important role of health investment today. Consequently, health subsidies can be particularly effective. If medicine were ineffective, an increase in health expenditures need not raise life expectancy and the mechanisms highlighted in the paper would be at best attenuated. Indeed, there has long been skepticism on the effect of medicine on longevity since Joseph Dietl's criticism on certain medical practices as therapeutic nihilism back in 1845 [see, e.g., Lindemann (1999), and a counterargument in Johansson (1999)]. With the lack of consensus on the effectiveness of medicine prior to the twentieth century, one should be cautious when interpreting health investment in the early stage of development.

Final production uses physical capital, K, and effective labor, Le:

(4) [Y.sub.t] = [A.sub.y][K.sup.[phi].sub.t][([Le.sub.t]).sup.1-[phi]],

where [A.sub.y] > 0 and 0 [phi] < 1 are the total factor productivity parameter and the share parameter of physical capital, respectively. Production inputs are compensated according to their marginal products:

(5) 1 + [r.sub.t] [phi][A.sub.y][([e.sub.t]/[k.sub.t]).sup.1-[phi]],

(6) [w.sub.t = (1 - [phi])[A.sub.y][([k.sub.t]/[e.sub.t]).sup.[phi]].

where k is physical capital per worker; w, the real wage rate; and r, the real interest rate.

We assume a perfect annuity market through which workers invest their savings in exchange for income for retirement conditional on survival. Under this assumption, savings left by savers who die at the end of working age will be shared by the rest of savers who survive to old age. This assumption implies that the rate of return on savings is equal to (1 + [r.sub.t+1])/[[bar.P].sub.t], where [bar.P] is the average rate of survival (i.e., the portion of the working population surviving to old age). The household budget constraints are given by:

(7) [w.sub.t][e.sub.t](l - [[tau].sub.w]) = [m.sub.t](1 - [[pi].sub.m]) + [q.sub.t](1- [[pi].sub.e] + [s.sub.t] + [c.sub.t],

(8) [z.sub.t] + 1 = [s.sub.t](1 + [r.sub.t] + 1)/[[bar.P].sub.t].

Here, health and education expenditures are subsidized at respective rates [[pi].sub.m] and [[pi].sub.e] (funded by a wage income tax [[tau].sub.w]), c refers to working-age consumption, s refers to life cycle savings, and z stands for old-age consumption. Also, we assume inelastic supply of one unit of labor per worker to keep things simple.

The government budget constraint is balanced in every period:

(9) [[tau].sub.w][w.sub.t][e.sub.t] = [[pi].sub.m][m.sub.t] [[pi].sub.e][q.sub.t].

We treat the tax rate [[tau].sub.w] as a variable, and the subsidy rates, [[pi].sub.e] and [[pi].sub.m], as policy parameters.

The preferences are assumed to be:

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

That is, an agent derives utility from working-age consumption, old-age consumption (conditional on survival), and the human capital of his child. The parameters in Equation (10) include two discounting factors, [delta] and [zeta], and one constant coefficient of relative risk aversion, [theta]. The presence of the product [z.sup.1-[theta]] P/(1 - [theta]) for [theta] [not equal to] 1, or Plnz for [theta] = 1 introduces nonconcavity in the utility function. To ensure the concavity for any interior solution to be optimal, we set 0 < [theta] [less than or equal to] 1. The constant elasticity of substitution utility function is widely used in the literature on growth.

The capital market clears when

(11) [k.sub.t + 1] = [s.sub.t],

where k = K/L is physical capital per worker. Correspondingly, output per worker is [y.sub.t] = [A.sub.y][k.sub.[alpha].sub.t][e.sup.1-[alpha].sub.t]. In equilibrium, we expect [e.sub.t] = [[bar.e].sub.t], [m.sub.t] = [[bar.m].sub.t], and [P.sub.t] = [[bar.P].sub.t] by symmetry because workers in the same generation are identical ex ante in this model.

III. EQUILIBRIUM AND RESULTS

Taking ([r.sub.t] + 1, [w.sub.t], [e.sub.t], [[bar.m].sub.t], [[bar.P].sub.t]) as given, the household problem in period t may be formulated as

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

by choice of nonnegative variables ([c.sub.t], [z.sub.t + 1], [m.sub.t], [q.sub.t]), where [lambda] is the Lagrange multiplier.

The first-order conditions of the household problem are given below for [theta] [member of] (0, 1):

13) [partial derivative][L.sub.t]/[partial derivative][c.sub.t] = [c.sup.-0.sub.t] - [[lambda].sub.t] [less than or equal to] 0, [c.sub.t] [partial derivative][L.sub.t]/[partial derivative][c.sub.t] = 0,

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For the log utility, we set [theta] = 1 in Equations (13), (14), and (16), and replace [z.sup.1-[theta].sub.t+1]/(1 - [theta]) with [lnz.sub.t+1] in Equation (15). The first-order conditions indicate that the net marginal benefits of all the choice variables are nonpositive. If the net marginal benefit of a household variable is strictly negative, then this variable must equal zero. Obviously, when consumption and investment in human capital approach zero, their marginal benefits will approach infinity. However, when health investment approaches zero, its marginal benefit does not approach infinity because [h.sub.t] [greater than or equal to] [[bar.h] provides a lower bound on exp([h.sub.t]) or an upper bound on exp(-[h.sub.t]) in Equation (15). In other words, there may be a corner solution for health investment.

We give the condition for a corner solution for health investment below and relegate the proof to the Appendix:

PROPOSITION 1. For small enough initial capital stocks ([k.sub.0], [e.sub.0]) relative to a given [bar.h], we have [m.sub.t] = 0 for some t [greater than or equal to] 0. Subsidizing health investment can avoid such a corner solution.

Proposition 1 is consistent with the fact that many poor countries suffer from lack of health services with little health spending. The intuition is that when households are so poor that their marginal utility of consumption exceeds their marginal utility of health investment, they are unwilling to divert their small income for health investment. This is mainly because the expected rate of survival is bounded below by the endowed component of health such that the marginal utility of health investment is bounded above. Another interesting result in Proposition 1 is that governments can induce positive health investment by subsidizing it.

With a corner solution for health investment, Equations [m.sub.t] = 0, (13), (14), and (16) and the budget constraints lead to the evolution of human and physical capital below:

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with

[THETA] = [[([phi][A.sub.y]).sup.0-1] [A.sup.1/[eta].sub.e] ([eta][zeta]/1 - [[pi].sub.e])1/ [delta]].sup.[eta]/(1-[eta])] > 0.

For some parameterizations (e.g., a large [A.sub.y]), there exists sustainable growth in this corner solution without health investment, as in endogenous growth models with both human and physical capital in the literature. In a growing economy, health investment will eventually become positive.

With an interior solution for health investment, the first-order conditions of the household problem imply

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(20) [zeta][e.sup.-0.sub.t] + 1 [eta][A.sub.e][([e.sub.t]/[q.sub.t]).sup.1-[eta]] = (1 - [[pi].sub.e])[c.sup.-0.sub.t],

(21) [s.sub.t][alpha][A.sub.h] = (1 - [[pi].sub.m])[exp([h.sub.t] - 1](1 - [theta].

for 0 < [theta] < 1. Further, the budget constraints (7) to (9) imply [w.sub.t][e.sub.t] = [m.sub.t] + [q.sub.t] + [s.sub.t] + [c.sub.t].

From these equations, plus [k.sub.t]+ l = [s.sub.t] and the technologies in production and education, we can determine the evolution of ([e.sub.t], [k.sub.t) for 0 < [theta] < 1 with positive health investment as

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

where

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

In the log utility case, the evolution of ([e.sub.t], [k.sub.t]) is a special case of (22) and (23) with [theta] = 1.

According to Equation (21), in a growing economy in which [k.sub.t + 1] = [s.sub.t] grows over time, exp([h.sub.t]) will also grow over time and therefore the rate of survival P([h.sub.t]) will converge to 1. We now establish the convergence of a growing economy to its unique balanced growth path below and relegate the proof to the Appendix.

PROPOSITION 2. If 1 - [phi](1 - [theta]) - [phi](1 - [eta]) > 0, a growing economy starting with either m = 0 or m > 0 converges to a unique balanced growth path with [lim.sub.t][right arrow]x] P(h.sub.t]) = 1. The balanced growth rate is positive if [A.sub.y] is large enough.

The condition in Proposition 2 for a growing economy to converge to its unique balanced growth path is satisfied if the share parameter associated with effective labor exceeds that associated with physical capital in final production, as is widely accepted in the literature, that is if 1 - [phi] > [phi]. The condition is also satisfied if the utility function is logarithmic, that is [theta] = 1, for all [phi] [member of] (0, 1). As is typical in a model with physical and human capital, the convergence refers to the ratio of physical to human capital [k.sub.t][le.sub.t] in the proof of Proposition 2 in the Appendix. The economic meaning of the condition for convergence is that for the ratio of physical to human capital to approach its steady-state value on the balanced growth path, physical capital investment cannot be too responsive relative to human capital investment. There are three sources for the responsiveness of physical capital investment in this model. First, a large share parameter [phi] in final production indicates an important role of physical capital relative to effective labor. Second, a high elasticity of intertemporal substitution (a small [theta]) means that consumers do not mind large swings in consumption over their lifetime, implying responsive savings (or physical capital investment). Third, a small share parameter q in the education technology means a small role of human capital investment in the adjustment process. The convergence condition may be violated by a combination of a large value of [phi] and small values of [theta] and [eta] such that physical capital investment dominates the adjustment in the capital ratio. However, the violation would need the share parameter of physical capital in final production to exceed that of labor. Finally, for high enough total factor productivity ([A.sub.y]), the balanced growth rate is positive.

We now investigate how subsidies on health and education spending affect the economy. The proof is relegated to the Appendix.

PROPOSITION 3. With O < [theta] < 1 and an initial state ([e.sub.t], [k.sub.t]), a health subsidy raises health spending but has ambiguous effects on savings and education spending; it has no effect on the balanced growth rate. An education subsidy increases education spending but has ambiguous effects on savings and health spending; it also increases the balanced growth rate if [A.sub.y] is sujficiently large under 1 - [phi](1 - [theta]) - [phi](1 - [eta]) > 0.

The results in Proposition 3 are intuitive. Concerning the spending that is directly subsidized, the positive net effect indicates that the substitution effect of each of the subsidies dominates its income effect. For other types of spending that are not directly subsidized, the net effects of the subsidies will depend on parameterizations and are likely to be negative because the substitution effects may be negative. On the balanced growth path, the balanced growth rate is dependent on the ratio of physical to human capital. Because the rate of survival on the balanced growth path is equal to 1, the health subsidy can no longer increase the survival rate. In this case, the health subsidy may have proportionate effects on physical and human capital accumulation, as it does in this model. Therefore, the health subsidy has no effect on the capital ratio and the growth rate on the balanced growth path in our model. This result differs from that of Aisa and Pueyo (2006), whereby government health spending raises (reduces) the growth rate when income is low (high). The difference arises from the difference in the framework: human capital investment and private health investment are present in our model but absent in their model.

On the other hand, by promoting investment in human capital, the education subsidy can increase the balanced growth rate as long as total factor productivity A,. is high enough and the share parameter of labor exceeds the share parameter of capital in production.

In the case of the log utility with [theta] = 1, the equilibrium solution with positive health spending can be determined by the following equations:

(24) [w.sub.t][.sub.t] = [m.sub.t] + {1 + [1 + [delta](1 - exp(- [h.sub.t]([m.sub.t])))] 1 - [[pi].sub.e]/[zeta][eta]}[q.sub.t],

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the large expression inside the bracket {...} on the LHS of Equation- (25) is equal to [lnz.sub.t+l] and [h.sub.t]([m.sub.t]) = [A.sub.h][m.sub.t] + [bar.h]. We establish the following result and relegate the proof to the Appendix.

PROPOSITION 4. With the log utility and an initial state ([e.sub.t] [k.sub.t]), a health subsidy raises health spending, reduces education spending, and has an ambiguous effect on savings; it has no effect on the balanced growth rate. An education subsidy raises education spending but has ambiguous effects on savings and health spending, If 1 - [phi] > [phi], the education subsidy raises the balanced growth rate unless the subsidy rate is too high.

The results with the log utility are similar to those with [theta] [member of] (0, 1) in most aspects. One difference is that in the case of the log utility, a health subsidy reduces education spending. Another difference is that the education subsidy can increase the balanced growth rate unless the subsidy rate is too high under 1 [phi] > [phi].

IV. NUMERICAL RESULTS

We now explore the quantitative implications of the model using a numerical approach. We first calibrate the model to the observed subsidies and some key variables in the U.S. economy and then make counterfactual exercises for different subsidy rates. To this end, we numerically solve the evolution equations of human and physical capital from ([k.sub.t], [e.sub.t]) to ([k.sub.t+l], [e.sub.t+l]) and determine the other variables. In doing so, we will verify whether the condition for a corner solution for health investment is satisfied in each period.

A. Calibration Results

We consider 20 periods (25 years per period), which are enough to illustrate the convergence process. The observed information in Table 1 from 1870 to 2000 can help us in calibrating the model to the U.S. economy for some periods. Suppose that we start from 1825 when per capita income in the United States was about $1,400-$1,500 (the 1985 dollar). Then, the first three periods of our calibration correspond to the last three quarters of the nineteenth century, and the next four periods correspond to the twentieth century. Implied by Column 6 of Table 1, health subsidy rates in the United States were 20% in 1930, 29% in 1950, 41% in 1970, 44% in 1990, and 46% in 2000. From these figures, we assume that in Periods 5-7 (the last three quarters of the twentieth century) the average health subsidy rates are 24.5%, 35%, and 43.5%, respectively. For later periods in the simulation, the health subsidy rate is assumed to be 46% as in 2000. Without available information for the health subsidy prior to 1930, we assume that it is zero in the first three periods of the simulation (for the nineteenth century), and that it is 7% in the fourth period (the first quarter of the twentieth century).

Also, we assume that the education subsidy rates were 10% in the first four periods (from the second quarter of nineteenth century to the first quarter of the 20th), given the very low ratio of total education spending to GDP in the early twentieth century (1.5% in 1910 in Table 1). For Periods 5-7, the education subsidy rates are assumed to be 20%, 30%, and 40%, respectively, and remain at 40% in later periods. These values of the education subsidy rate are lower than the reported figures on public and private spending on formal schooling (nearly 70% of formal education spending was publicly provided in the United States), as education spending in Table 1 refers to formal education. In fact, however, families and firms also spend on education in some important ways outside formal schooling, for example, after-school programs and labor training, which may not be easily observed by the government [see Davies, Zeng, and Zhang (2000)]. If the additional private spending is taken into consideration, the actual subsidy rate tends to be lower than that on formal schooling alone.

We choose the values of parameters in preferences and technologies such that the proportional allocations of income to savings, health, and education, the growth rate, and life expectancy are close to the observed values in Table 1. The parameterization is given below

[alpha] = 1/4, [theta] = 0.85, [delta] = 0.5, [zeta] = 0.3, [bar.h] = 0.1, [A.sub.h] = 0.5, [A.sub.e] = 3.3, [A.sub.y] = 4, [eta] = [phi] 1/3.

In this parameterization, the share parameter of physical capital at 1/3 is widely accepted. Also, when one period refers to 25 years here, [delta] = 1/(1 + [rho]) = 0.5 corresponds to an annual rate of time preference p = 0.028, which is within its usual range used in the literature. In addition, we assume initial capital stocks as [e.sub.0] = 0.9506 and [k.sub.0] = 0.0481.

It is interesting to ask whether the calibration results can predict the observed trends in Table 1. The calibration results are reported in Figure 1 with four panels (a-d), where the rate of survival from middle age to old age, rather than life expectancy, is reported. The rate of survival from age 20 to 65 was 79% for males and 87% for females in 2000 in the United States, when life expectancy across gender was 77 in Table 1. (2) We report proportional allocations of output to savings, health, and education in Panel a, the growth rate of output and the rate of survival in Panel b, health spending and log output per worker in Panel c, and welfare in Panel d.

In Figure 1, there is no health investment in the first three periods (the three quarters of the nineteenth century), as output per worker is initially low at 1.42 (or $1,420 in the 1820s). When income becomes high enough in Period 4 (or 1900-1925), health investment becomes positive and rises over time. The ratio of health investment to output rises in Period 4 through to 7, from nearly 5% to 15%, and then falls gradually in the long run. Reflecting the near-zero health investment in the first few periods and the rising trend afterward, the rate of survival is very low initially (below 0.1) and then rises toward its long-run level (equal to 1). At a low rate of survival, the saving rate and the growth rate of output per worker are very low initially, below 10% and around 1.3%, respectively. The ratio of education spending to output is relatively smooth throughout (5%-7%). When income grows, the saving rate and the growth rate of output converge to their long-run levels (20.9% and 2.594%, respectively) that are higher than in the first three periods without health investment.

[FIGURE 1 OMITTED]

The patterns of movements of the variables in the several periods with a rising ratio of health spending to output in Figure 1 capture some key features in the time series data in the United States in Table 1 and in the other developed countries. The remarkable rise in health investment relative to output in Table 1 is well echoed in Periods 4-7 in Figure 1. Also, the postwar average growth rate appeared to be higher than the prewar average growth rate in the United States and some other developed countries from 1870 to 1990 according to Maddison (1991). This overall rise in the long-term average growth rate is reflected in Figure lb.

As mentioned earlier, according to Maddison (1992), there was a discernable upward trend in the long-term saving rates of 11 developed countries for the period 1870-1987 (except the United States), as captured in Periods from 4 through to 7 in our Figure la. For example, the average saving rate of Canada rose from 9.1% in 1870-1889 to 14.4% in 1914-1938, and further to 23.4% in 19601973. Since the mid-1970s, it had declined slightly to 20.4% in 1981-1987. It appears that Figure 1a has a better match with the saving rate in Canada than in the United States. This is perhaps because Canada started with much lower per capita GDP in 1870 than the United States: $1,330 versus $2,244 (the 1985 dollar). With such a low level of per capita GDP, Canada was in an early stage of development in the 1870s, which fits better into the first few periods in our Figure 1.

B. Counterfactual Exercises

We now depart from the calibration based' on the U.S. data by changing the subsidy rates: no subsidies to health and education; a 60% health subsidy; and 60% subsidies to both health and education. (3) We report the numerical results in these cases in Figures 2-4.

In Figure 2, without any subsidies, the trends in the key variables are similar to those in Figure 1. However, the pace of the convergence toward the balanced growth path is slower without subsidies than with subsidies, and the balanced growth rate is lower as well.

In Figure 3, with a 60% health subsidy, the economy jumps to the phase with positive health investment immediately in the first period as predicted in Propositions 1 and 2, raising the rate of survival and the saving rate substantially on the transitional path. Consequently, the growth rate is higher in the first few periods on the transitional path with the health subsidy than without, as in Aisa and Pueyo (2006). Also greater is the short-run level of welfare in Figure 3d with the health subsidy (8.99656 in Period 1) than in Figure ld with a 10% education subsidy (8.99062 in Period 1) and than in Figure 2d without subsidies (8.98961). The reason for this welfare improvement arises from a positive externality from average health spending in the determination of workers' health status. Intuitively, the externality causes underinvestment in health, leaving room for welfare improvements.

In Figure 4 with 60% subsidies to both education and health, the economy initially has just one period with zero health investment and moves to the phase with positive health investment in the second period. Compared to Figures 1-3, the comprehensive 60% subsidies to health and education reduce welfare initially (8.94953 in the initial period), but raise it later by accelerating human capital accumulation and output growth.

This welfare comparison suggests that poor countries may benefit more from subsidizing health spending than from subsidizing both health and education spending in the short run. In addition to these subsidies, poor countries may also benefit from subsidizing investment in physical capital or savings. The reason lies in the first-order condition with respect to health investment (15). That is, when the externality in health investment causes underinvestment in health, subsidizing savings can help raise the marginal benefit of health investment through increasing expected old-age consumption and hence encouraging more health investment. Also, a saving subsidy can promote capital accumulation and growth.

To focus on the welfare ranking with different combinations of subsidies in rich countries, we select a new initial condition with [e.sub.0] = 11.936 and [k.sub.0] = 2.425. Without any subsidy, this new initial state gives a level of output per worker at 28.22, which is almost 20 times as much as the output level 1.42 with the previous initial condition. This resembles a comparison between a high income level $20,000 in developed countries and a low income level $1,000 in poor countries. Also, the corresponding rate of survival is equal to 0.79, which is close to the. current rate of survival from age 20-65 for males in the United States.

Comparing different subsidies (20%, 40%, 60% and 80% to either or both of education and health), we find numerically that the optimal subsidy policy in rich countries is around a 60% subsidy to both education and health (details available upon request).

V. SUMMARIES AND EXTENSIONS

Our model captures some stylized facts over different stages of development. When the initial income level is very low, health investment and savings are low as well relative to income. As a consequence, the rate of survival and the rate of output growth are low. When the income level rises over time, both health investment and savings will increase rapidly relative to income for some periods, leading to higher life expectancy and faster growth. Eventually, the ratio of health investment to output will fall when survival to old age becomes almost certain, whereas the saving rate and the rate of survival converge to steady-state levels on the balanced growth path. In our numerical results, the balanced growth rate is much higher than in the corner solution without health investment. The contrasting patterns of these variables at low and high levels of income resemble what we observe between poor and rich countries or across different stages of development in time series data in some developed countries.

[FIGURE 2 OMITTED]

Regarding public policies, we find that a health subsidy increases health investment and may raise or reduce the growth rate of output on the transitional path. However, it has no effect on the balanced growth rate in the long run. In terms of its welfare effect, economies with little health investment and low income may benefit more from subsidizing health investment than from subsidizing both health and human capital investment comprehensively. On the contrary, economies with high income may benefit more from subsidizing both than from subsidizing just one of them. This also captures the fact that rich countries have more public spending on education and health as fractions of GDP than poor countries.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Our model can be extended in several directions. First, one can assume that health may contribute to utility directly in addition to its role in enhancing survival as in Corrigan, Glomm, and Mendes (2005). Similarly, one can assume that health can enhance productivity as in Chakraborty and Das (2005) and Corrigan, Glomm, and Mendez (2005). These additional motives for health investment may induce more health spending. However, since the level of health status is bounded below by its endowed component [h.sub.t] [greater than or equal to] [bar.h], a corner solution for health investment may still occur in this extended version when income is sufficiently low such that the desire for consumption dominates the desire for health investment. Even with positive health investment at low income levels, the ratio of health investment to output is likely to be low because of the endowed component of health.

Second, one may assume idiosyncratic health shocks in the form of terminal diseases like AIDS or cancer that may make survival to old age impossible as in the work of Corrigan, Glomm, and Mendez (2005). Correspondingly, one may assume another component of health expenditure that contributes to utility by easing the suffering from such terminal diseases but does not contribute to the rate of survival to old age. In this second extension, average health spending is expected to be always positive due to the new component of health expenditure. However, the component of health spending aiming at enhancing survival to old age is expected to behave in the same way as in the original version of the model.

VI. CONCLUSION

In this paper, we have investigated how health investment interacts with human capital investment and life cycle savings in an endogenous growth model. We have found that the equilibrium solutions for some key variables depend critically on the initial level of income per capita. When initial income is sufficiently low, the desire for consumption is stronger than the desire for health, resulting in zero health investment and hence a low rate of survival. The low rate of survival in turn leads to little savings for old age and slow growth in output per worker. When income becomes high enough in a growing economy, households will be willing to strike a balance among health, human capital investment, and savings, leading to higher life expectancy and faster growth than in the early stage without health investment. The findings capture some stylized facts in cross-country comparisons between poor and rich countries as well as in time series data in the United States.

Interestingly, subsidizing health spending can move an economy from the no-health-spending equilibrium to the other, a transition that brings about higher life expectancy, greater savings, higher welfare, and perhaps faster growth on the transitional path. Subsidizing both education and health spending may reduce welfare in the short run for poor countries but will lead to higher life expectancy, faster growth, and higher welfare in the future. An example of this transition in recent history is the development in the last several decades in China compared to the rest of the developing world. Starting with one of the lowest levels of income and life expectancy but with substantial state funding for education and health services, China has achieved not only phenomenal economic growth but also one of the highest levels of life expectancy in the developing world. (4)

Starting with high income and positive health investment, we have also found that the initial generation of workers are better off from subsidizing both health and education expenditures at realistic rates around 60%. This result is consistent with the practice of substantial government spending on health and education in many developed countries. The welfare gain is attributed to the externality of average health spending in the determination of health status and the shortsightedness of agents in a typical overlapping generations model concerning human capital investment. However, further rises in the subsidy rate on both education and health expenditures, say 80% or over, are found to reduce welfare in our numerical results, although the higher subsidy rates may yield higher welfare compared to cases without any subsidies. This result supports recent reforms in public funding for education and health, from one with almost free public access to education and health services to one with some sort of shared financial responsibility between the state and households.

ABBREVIATIONS

AIDS: Acquired Immunodeficiency Syndrome

GDP: Gross Domestic Product

NIPA: National Income and Product Accounts

OECD: Organisation for Economic Cooperation and Development

doi:10.1111/j.1465-7295.2007.00020.x

APPENDIX

Proof of Proposition 1. From the first-order conditions. we must have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In particular, if the initial stocks ([k.sub.0], [e.sub.0]) and hence [y.sub.0] = [A.sub.y][K.sup.[phi].sub.0][e.sup.1-[phi].sub.0], are so low that

[[delta][z.sup.1-[theta].sub.1]/1 - [theta]] [c.sup.[theta].sub.0][alpha][A.sub.h] < (1 - [[pi].sub.m])exp([bar.h]),

we must have

[[delta][z.sup.1-[theta].sub.1]/1 - [theta]] [c.sup.[theta].sub.0][alpha][A.sub.h] < (1 - [[pi].sub.m])exp([h.sub.0]), since [h.sub.0] [greater than or equal to] [bar.h].

Together with [m.sub.t][[delta][z.sup.1-[theta].sub.t + 1][c.sup.[theta].sub.t]a[A.sub.h]/(1-[theta]) - (1 - [pi].sub.m]) exp([h.sub.t]))] = 0 we must have [m.sub.t] = 0 for some t [greater than or equal to] 0. Obviously, one can always increase the health subsidy rate [[pi].sub.m] such that the user cost of health investment (the RHS of the above inequalities) is equal to or below the marginal benefit (the LHS) in order to induce positive health investment. Finally. it is easy to verify that the result holds for the log utility case when setting [theta] = 1 and replacing [z.sup.1-0.sub.t + 1]/(1 - [theta]) with 1n[z.sub.t + 1] in the above inequalities.

Proof of Proposition 2. Consider first that the economy starts from a corner solution with m = 0. The convergence of this economy is based on Equation (18). For convenience, let [[GAMMA].sub.i] be the coefficient on the ratio k/e in Equation (18). denote x = k/e and rewrite Equation (18) as

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

Differentiating it with respect to [x.sub.t] yields

[dx.sub.t + 1] / [x.sub.t + 1] / [dx.sub.t]/[x.sub.t] = F([x.sub.t + 1])

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and therefore F < B. We thus have

0 < [dx.sub.t + 1]/[x.sub.t + 1] / [dx.sub.t]/[x.sub.t] = F([x.sub.t + 1]) / B([x.sub.t + 1]) < 1.

which implies the convergence of [x.sub.t], that is, [lim.sub.1[right arrow][infinity]] [x.sub.t] = [x.sub.[infinity]] Taking t [right arrow] [infinity] in Equation (A-1) and dividing it by [x.sub.[infinity]], [x.sub.[infinity]] is determined by

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

Here, the LHS is a positive constant, while the RHS is increasing monotonically with [x.sub.[infinity]] (starting below [[GAMMA].sub.1]) because all the exponents of [x.sub.[infinity]] in Equation (A-2) are positive under 1 -[phi])(l -[theta]) - [phi](1 -[eta]) > 0. Thus, [x.sub.[infinity]] is unique under this condition. Corresponding to the unique [x.sub.[infinity]] = [k.sub.[infinity]]/[e.sub.[infinity]], there is a unique balanced growth rate from Equation (17) with m = 0. From Proposition 1, since a growing economy will eventually have [m.sub.t] > 0. it will not converge to the balanced growth path with m = 0.

Now we consider the case with m > 0. According to Equation (21), when [k.sub.t + 1] = [s.sub.t] grows over time, exp([h.sub.t] will also grow over time and therefore the rate of survival P will converge to 1. Thus, when t [right arrow] [infinity], Equations (19) and (20) plus the education technology imply

(A-3) [lim.sub.t[right arrow][infinity]] [e.sub.t + 1]/[e.sub.t] = [[theta].sub.0][([k.sub.[infinity]/[e.sub.[infinity]).sup.[eta] [1-[phi](1-(phi))]/(1-[eta])],

where

[[THETA].sub.0] = [[([phi][A.sub.y]).sup.0-1] [A.sup.1/[eta].sub.e] ([eta][xi]/1 - [[pi].sub.e])1/[delta]].sup.[eta]/(1-[eta]) > 0.

This equation links the long-run growth rate of human capital to the long-run ratio of physical to human capital. Given [e.sub.t] [member of] (0. [infinity]) and [k.sub.t] [member of] (0,[infinity]) in period t, [[e.sub.t + 1]) must be bounded according to the education technology and the household constraint on education spending [q.sub.t]. That is. the growth rate of human capital is bounded in every period. By Equation (A-3), the capital ratio [k.sub.[infinity]/[e.sub.[infinity]] must be bounded as well. Rewrite Equation (22) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Divide both sides of this version of Equation (22) by e,. let t [right arrow] [infinity] and [e.sub.t] [right arrow] [infinity]. and use Equation (A-3) to replace the growth rate by the ratio of physical to human capital. In doing so. the ratios of all the constant terms in Equation (22) to [e.sub.t] will converge to zero as [e.sub.t] [right arrow] [infinity] in the long run. Because we have noted that the long-run growth rate and the long-run capital ratio are all bounded, the last term on the LHS and the first term on the RHS of Equation (22) will also be driven to zero when they are divided by a rising [e.sub.t]. As grows. (1n[e.sub.t])/[e.sub.t] also converges to zero since [lim.sub.x [right arrow] [infinity]] (log x)/x = [lim.sub.x [right arrow] [infinity]] 1/x = 0. In the long run. the resultant equation containing the remaining terms in Equation (22) governs the evolution of the capital ratio:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As in Equation (A-1), the capital ratio in the above equation will converge to [k.sub.[infinity]/[e.sub.x]:

(A-4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that all the terms in Equation (A-4) are positive. The LHS is a constant, while the RHS depends on [k.sub.[infinity]/e[infinity], which will have a unique solution if the RHS is increasing with it monotonically. To this end, we only need to show that all the exponents of [k.sub.[infinity]/e[?] are positive. The last one is positive because [1 - [phi])(1 - [theta])][1 - [eta](1 - [theta])] - [theta][phi](1 - [eta]) [equivalent to] F([theta]) > 0 as

F(0)=(1-[phi])(1-[eta]) > 0,

F'([theta]) = [phi][1 - [eta](1 + [theta]) + [eta][1 - [theta](1 - [theta])] - [phi](1 - [eta]) = [phi][eta][theta] + [eta][1 - [phi](1 - [theta])] > 0,

under 0 < 0 [less than or equal to] 1. The other exponents of [k.sub.[infinity]] / [e.sub.[infinity]] are positive under 1 - [phi](1 - [theta]) - [phi](l - [eta]) > 0. Given this condition, there is a unique finite solution for [k.sub.[infinity]]/[e.sub.[infinity]], implying that physical and human capital (hence also output) must share the same balanced growth rate in the long run. Combining this with Equation (A-3), the balanced growth rate must also be finite and unique.

In order to see whether the balanced growth rate can be positive, we set [[pi].sub.e] = 0 and substitute Equation (A-3) into Equation (A-4) to replace [k.sub.[infinity]]/[e.sub.[infinity]] by the balanced growth rate [g.sub.[infinity]]:

(A-5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The LHS of Equation (A-5) is increasing with [A.sub.y], while the RHS is decreasing directly with [A.sub.y]. Also, the RHS is increasing with [g.sub.[infinity]] because all the exponents of I + [g.sub.[infinity]] are positive. Thus, we have d[g.sub.[infinity]] / d[A.sub.y] > 0. That is, if [A.sub.y] is large enough, then [g.sub.[infinity]] > 0.

In the log utility case with [theta] = 1, the analysis of convergence to a unique balanced growth path is similar by imposing [theta] = 1 in Equations (A-3) and (A-4). Also. if [A.sub.y], is large enough, then [g.sub.[infinity]] > 0, which can be easily verified by setting [theta] = 1 in Equation (A-5).

Proof of Proposition 3. Taking logs on Equations (19)-(21) and making substitutions, we have

(A-6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [[DELTA].sub.1] = [([phi][A.sub.y]).sup.([theta]-1)/[theta] [[DELTA].sup.1/[theta]] > 0, [[DELTA].sub.2] = [[[xi][eta][A.sup.1-[theta].sub.e].sup.1/[theta] > 0, and [[DELTA].sub.3] = (1 - [theta]) / {[alpha][A.sub.h]) > 0,

(A-7) [c.sub.t] = [[DELTA].sub.2][(1 - [[pi].sub.e]).sup.1/[theta]] [q.sup.1 [eta](1 [theta]) / [theta].sup.t] [e.sup.(1 p[eta])(1 [theta])/[theta].sub.t],

(A-8) [S.sub.t] = [[DELTA].sub.3](1 - [[pi].sub.m])[exp([h.sub.t]) - 1].

Substituting Equations (A-7) and (A-8) into the constraint [w.sub.t][e.sub.t] = [m.sub.t] + [q.sub.t] + [s.sub.t] + [c.sub.t] leads to

(A-9) [w.sub.t][e.sub.t] = [m.sub.t] + [q.sub.t] + [[DELTA].sub.3](1 - [[pi].sub.m])[exp ([h.sub.t]) - 1] + [[DELTA].sub.2][(1 -[[pi].sub.e]).sup.1/[theta][q.sup.1-[eta](1-theta) / [theta].sub.t] [e.sup.-(1-[eta])(1-[theta])/[theta].sub.t].

Note that Equations (A-6) and (A-9) contain implicit solutions for [m.sub.t] and [q.sub.t] via [h.sub.t] = [A.sub.h][m.sub.t] + [bar.h] it and Equation (5). Taking total differentiation, we get

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

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

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

While the signing is obvious for [a.sub.2], [a.sub.3], and [b.sub.j] for all j, the signing of [a.sub.1] is more involved. The sign of [a.sub.1] depends on [1 - (1 - [theta])[phi] - [theta]exp(-[h.sub.t]] or [1 - (1 - [theta])[phi]]exp([h.sub.t]) - [theta]. Since 1 > [theta] + (1 - [theta])[phi], or 1 > [theta]/[1 -(1 - [theta][phi])], for 0 < [theta] < 1 and 0 < [phi] < 1, it follows

exp([h.sub.t) [greater than or equal to] 1 > [theta] / 1 - (1 - [theta])[phi],

which implies [a.sub.1] > 0. Now, it is obvious that [partial derivative][m.sub.t]/[partial derivative][[pi].sub.m] > 0 and [partial derivative][q.sub.t]/[partial derivative][[pi].sub.e] > 0. The effects of the health subsidy on savings and education spending are ambiguous, and so are the effects of the education subsidy on savings and health spending.

According to Equations (A-3) and (A-4), the health subsidy has no effect on the balanced growth rate since it does not appear in these two equations. Differentiating Equations (A-3) and (A-4) with respect to 1/(1 - [[pi].sub.e]) yields the following condition for [dg.sub.[infinity]/d[[pi].sub.e] > 0:

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

where [g.sub.[infinity]] is the balanced growth rate. For [theta] [member of] (0, 1), a larger [A.sub.y] means a greater LHS, both directly and indirectly through raising [g.sub.[inifity]] under 1 - [phi](1 - [theta]) - [phi](1 - [eta]) > 0, as shown below Equation (A-5). Thus, the education subsidy increases the balanced growth rate if A,. is sufficiently large and if 1 - [phi](1 - [theta]) - [phi](1 - [eta]) > 0.

Proof of Proposition 4. Totally differentiating Equations (24) and (25) with respect to [[pi].sub.m] and collecting terms, we have:

[dm.sub.t]/d[[pi].sub.m] = [xi][eta][c.sub.t][P.sub.t]exp([h.sub.t])[G.sub.3]/[G.sub.1][G.sub.2] + [c.sub.t][A.sub.h][G.sub.3][G.sub.4] > 0, [dq.sub.t] / d[[pi].sub.m] = - ([G.sub.1]/[G.sub.3])([dm.sub.t] /d[[pi].sub.m] < 0,

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In so doing, we used 1n[z.sub.t+1] = (1 - [[pi.sub.m]exp(h.sub.t)/([c.sub.t][delta][alpha][A.sub.h]) and [c.sub.t] = (1 - [[pi].sub.e])[q.sub.t]/[xi][eta]). The sign of the effect on savings is ambiguous as follows:

[ds.sub.t]/d[[pi].sub.m] = exp(-[h.sub.t])[delta][c.sub.t][A.sub.h] d[m.sub.t]/ d[[pi].sub.m] - [P.sub.t][xi][eta][delta](1 - [[pi].sub.e])([G.sub.1]/[G.sub.3])d[m.sub.t]/d[pi].sub.m]

where the first term on the RHS is positive but the second one is negative.

Similarly, we also have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The effects of the education subsidy on health spending and savings are ambiguous.

Again, the health subsidy has no effect on the balanced growth rate as in Proposition 3. Setting [theta] = I in Equation (A-12), the condition for d[g.sub.[infinity]/d[[pi].sub.e]>0 becomes:

[[pi].sub.e] < (1 - [phi])(1 + [delta]) - [phi][eta][xi] / (1 - [phi])(1 + [delta]).

The RHS of this inequality is positive but less than 1 under 1 - [phi] > [phi].

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Corrigan, P., G. Glomm, and F. Mendez. "AIDS Crisis and Growth." Journal of Development Economies, 77(1), 20(15, 107-24.

Davies, J. B., J. Zeng, and J. Zhang. "Consumption vs. Income Taxes When Private Human Capital Investments Are Imperfectly Observable." Journal of Public Economics, 77, 2000, 1-28.

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The World Bank. The World Development Indicators 2001, Washington, DC, 2002.

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--. Statistical Abstract of the United States. Washington, DC, 2004 and other years.

Zhang. J., and J. Zhang. "The Effect of Life Expectancy on Fertility, Saving, Schooling and Economic Growth: Theory and Evidence." Scandinavian Journal of Economics, 107(1), 2005, 45 66.

Zhang, J., J. Zhang, and R. Lee. "Mortality Decline and Long-Run Economic Growth." Journal of Public Economics, 80(3), 2001, 485-507.

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(1.) For example, health spending accounted for 13% of GDP during 1990-1998 in the United States, and nearly 9% of GDP in other advanced countries according to the World Bank (2002), amounting to a per capita health expenditure at 2,000 U.S. dollars or more per year. On the other hand, in countries with very low life expectancy, health expenditures per capita were mostly below 50 U.S. dollars per year, for example,, merely 1 U.S. dollar in Liberia and below 10 U.S. dollars in other eight such countries in 1997.

(2.) The Life Tables give death rates for each five-year age gap, for example, [d.sub.20.25], ... [d.sub.60.65]. The rate of survival from age 20 to 65 is computed as the product (1 - [d.sub.20.25])(1 - [d.sub.25.30]) ... (1 - [d.sub.60.65).

(3.) The 20% saving rate is close to the 22% ratio of private investment to GDP in countries with life expectancy in 1960 above 65 for the period 1960-1989. In these countries the ratio of public education spending to GDP is about 6%. With a 60% education subsidy, the corresponding figure for the ratio of total education spending to GDP is thus 10% to meet the 6% ratio of public education spending to GDP. Since there are substantial government subsidies on education and health expenditures in many OECD countries, the 60% subsidy rates are plausible figures for these countries on average.

KAM KI TANG and JIE ZHANG *

* We are grateful to very insightful and useful comments by two anonymous referees. All remaining errors are our own. The research was supported by a grant from the National University of Singapore.

Tang: Assistant Professor, School of Economics, University of Queensland, Brisbane, Qld 4072, Australia. Phone 61-7-3365 9796, Fax 61-7-3365 7299, Email kk.tang@uq.edu.au

Zhang: Professor, Department of Economics, National University of Singapore, Singapore, 11907 and School of Economics, University of Queensland, Brisbane, Qld 4072 Australia. Fax: 61-7-3365 7299. Email: j.zhang@uq.edu.au

TABLE 1
Selected Statistics of the United States from 1870 to 2000

 Life Expectancy GDP per Capita
Year At birth (a) (1985 $U.S.) (b)

1870 41.4 2,244
1890 43.5 3,101
1910 51.9 4,538
1930 59.7 5,642
1950 68.2 8,605
1970 70.8 12,815
1990 75.4 18,258
2000 77.0 23,190

 Average Annual Average Health
 GDP per Capita Expenditure (% GDP)
Year Growth Rate (%) (c) (Private + Public) (d)

1870 -- --
1890 1.62 --
1910 1.90 --
1930 1.09 4.0
1950 2.11 5.3
1970 1.99 8.9
1990 1.77 13.1
2000 2.39 14.5

 Ratio of Public Average Education
 to Private Health Expenditure (% GDP)
Year Expenditure (c) (Private + Public) (f)

1870 -- --
1890 -- --
1910 -- 1.5
1930 0.25 3.2
1950 0.40 4.4
1970 0.70 7.5
1990 0.80 7.4
2000 0.84 7.6

 Average Saving
 Rate (% GDP)
Year (Year, Rate) (g)

1870 1870-1889, 19.1
1890 1890-1913, 18.3
1910 1914-1938, 17.0
1930 1939-1949, 15.2
1950 1950-1973, 19.7
1970 1974-1987, 18.0
1990 1990-2000, 16.3
2000 2000-2004, 15.1

Notes: Sources of data are as follows, a and f: U.S. Census Bureau
(1975, 2004). b and c: table 12.10, Barro and Sala-i-Martin (1995);
figures for 2000 are calculated using tables 1.1.6 and CA1-3, National
Income and Product Accounts (NIPA) Tables, 2005, U.S. Department of
Commerce: d and e: NIPA Tables, and table B236-247 in U.S. Census
Bureau (1975), g: figures prior to 1987 are from Maddison (1992);
figures for 1990 and 2000 are from NIPA Tables.