Subsidy policies in a fertility choice model.(Statistical Data Included)

Introduction

The neoclassical model treats population growth as an exogenous factor. This hypothesis neglects interactions between the economic growth process and demographic trends. However, a genuine understanding of the economic growth process should take into account the extent to which fertility and mortality affect the population growth rate as an endogenous variable.

Following the work of Becker [1960], who analyses the behavior of demographic and economic changes in developed countries and the role of fertility, several authors have studied the feedback between population growth and development. Important works of reference are that of Becker and Barro [1988] and Barro and Becker [1989], where demographic and economic outcomes are jointly and endogenously determined.

The results obtained by the modern endogenous population literature have yielded insights into a wide range of issues, including the effects of tax and social security programs on fertility. This paper will focus on the effects that a subsidy to health expenditure or a subsidy to child-rearing costs has on welfare, population growth, and income. The effects of financing such a subsidy by a capital income tax and a consumption tax are also studied. The analysis is based on a continuous time version of the Becker and Barro [1988] model.

This paper is organized as follows. The second section presents the model, and the third section studies the effects of a government policy that subsidizes either the health expenditure or child-rearing costs that are financed either by a capital income tax or a consumption tax. Concluding remarks are given in the fourth section.

The Model

This paper considers an extension to the fertility choice model of Becker and Barro [1988] and Barro and Becker [1989], and the continuous time version set out in Barro and Sala-i-Martin [1995] and Blackburn and Cipriani [1998]. Time is continuous, all markets are perfectly competitive, and the economy has a large number of identical households that seek to maximize the intertemporal dynastic utility function [Barro and Sala-i-Martin, 1995], shown as:

U = [[integral].sup.[infinity].sub.0][e.sup.-[rho]t]{[psi] ln N + ln(c + [phi]g) + [phi]ln(n - d(g))}dt, (1)

where [rho] is the rate of time preference and represents parental altruism, N is the size of the typical dynasty, n is the family's fertility rate, c is per capita consumption, g is per capita health expenditure, d(g) is the family's mortality rate, and g = g[e.sup.-xt] is per capita health expenditure in terms of effective labor, with x being the (exogenous) growth rate of technological progress. Households are assumed to give utility to their health status and, treating the health expenditure as a proxy to the health status, derive utility from the sequences of effective consumption, z = c + [phi]g. Parameter [phi], where 0 < [phi] < 1, indicates the extent to which each individual values his welfare in terms of health, as compared to consumption.

The mortality rate at time t is assumed to be affected only by instantaneous health expenditure per efficiency unit of labor. Health has characteristics like capital that could be accumulated (for example, investing in health has a positive effect on mortality throughout the lifetime of a person). However, the modeling of this effect probably would be intractable in this context. Certain studies (for example, see Newhouse [1977]) argue that in countries with high expenditure, the marginal utility of medical care is more likely to produce improvements in so-called subjective components of health rather than improvements in morbidity and mortality rates. These works are based on the difference between caring and curing. Parkin et al. [1987] suggest that the "marginal utility of medical care" does produce an improvement in objective health status, but the cost of this marginal utility is greater for higher-income and higher-expenditure countries. Blackburn and Cipriani [1998] argue instead that greater spending in health is a means of mitigating the potential adverse effects on child welfare of greater economic activity. The hypothesis that the mortality rate depends on health expenditure per efficiency unit of labor seeks to capture these ideas: per capita health expenditure should grow more than economic activity, as proxied by the technological progress, to yield a lower mortality rate.

The mortality rate function, d(g), is assumed to verify that d(g)> 0, d'(g) <0, and d"(g)> 0 [for all] g [greater than or equal to] 0. In other words, the mortality rate decreases as g increases, but decreases in the mortality rate become less pronounced as health expenditure rises. Health gains are assumed to be effectively bounded, so that [lim.sub.g[right arrow][infinity]]d(g) = d, with 0 < d < d(0). The size of the family changes continuously according to:

dN/dt = (n - d(g))N. (2)

The model now introduces child-rearing costs [PHI], which would tend to increase as parental income increases or with other measures of the opportunity costs of parental time. Following Barro and Sala-i-Martin [1995], a linear function, [PHI] = bk, is used. Thus, the family's budget constraint in per capita terms can be expressed as:

dk / dt = w + (1 - [[tau].sub.k])rk - (n - d(g)) k - (1 - [s.sub.b])nbk - (1 + [[tau].sub.c])c - (l - [s.sub.g])g - R , (3)

where w is the wage rate, r is the interest rate, [[tau].sub.c] is the consumption tax rate, [[tau].sub.k] is the rate of tax on capital income, [s.sub.g] is the rate of subsidy to health expenditure, [s.sub.b] is the rate of subsidy to child-rearing costs, and R is a lump sum tax (or transfers). The government is assumed to run a balanced budget. Its budget constraint can be expressed as:

[[tau].sub.k]rk + [[tau].sub.c]c + R = [s.sub.b] nbk + [s.sub.g]g . (4)

A labor tax is not introduced since it acts much like a lump sum tax in this model.

The household optimization problem consists of maximizing (1) subject to (2) and (3). By solving the model (see Appendix), the following expressions are obtained:

dz / dt = ((1 - [[tau].sub.k])r - [rho] - (1 + (1 - [s.sub.b]) b) n + d(g))z , (5)

n = d(g) + [phi][rho]z(1 + [[tau].sub.c])/[rho](1 + (1 - [s.sub.b]) b) k - [psi](1 + [[tau].sub.c]) z , (6)

and

k = [phi](1 + [[tau].sub.c]) - (1 - [s.sub.g])/(1 - [s.sub.b])bd'(g) , (7)

where k = k[e.sup.-xt] represents per capita capital in terms of effective labor. Equation (5) links the growth rate of per capita effective consumption with the rate of return on capital. Equation (6) indicates that, ceteris paribus, a higher fertility rate is associated with a higher mortality rate (and therefore a lower health expenditure), a higher [phi] (which raises the marginal utility of the children), a higher [psi] (which raises the marginal utility of the family size), a lower rate of time preference, [rho], a lower b, a higher subsidy to child-rearing costs, [s.sub.b], and a higher consumption tax rate, [[tau].sub.c]. In addition, a positive association exists between n and z/k. Equation (7) shows that, ceteris paribus, a higher per capita capital in terms of effective labor is associated with a lower [phi] (which decreases the weight of the health expenditure in the effective consumption, therefore decreasing the marginal utility of health expenditure), a lower b, a lower [[tau].sub.c], a higher subsidy to child-rearing costs, [s.sub.b], and a lower subsidy to health expenditure, [s.sub.g]. In addition, a positive one-on-one relationship exists between per capita capital and per capita health expenditure in terms of effective labor.

There are many firms in the market that act competitively and take technological progress as given. Output is produced with a Cobb-Douglas production function, y = A[k.sup.[alpha]], where 0 < [alpha] < 1 and y = y[e.sup.-xt] is per capita income in terms of effective labor. Profit maximization implies that firms pay the marginal product of factors:

r = [alpha]A[k.sup.[alpha]-1]- [delta],(8)

and

w =(1 - [alpha])A[k.sup.[alpha]][e.sup.xt],(9)

where [delta] denotes the depreciation rate of physical capital.

By substituting (6), (8), and (9), and using (4), then (3) becomes:

[FORMULA NOT REPRODUCIBLE IN ASCII] (10)

Substituting (6) into (5), and given that dz/dt = dz/dt [e.sup.-xt] - xz, the motion of effective consumption is given by:

[FORMULA NOT REPRODUCIBLE IN ASCII] (11)

and differentiating (7) with respect to t gives:

dg/dt = -(1 -[s.sub.b])bd' [(g).sup.2]/([phi](1+[[tau].sub.c])-(1-[s.sub.g]))d"(g) dk/dt. (12)

Substituting (10) into (12), using (7) to eliminate k, and using c = z -[phi]g to eliminate c, the former system, (11) and (12), can be expressed as a function of g and z.

Policy Analysis

This section examines the effects that a subsidy has on welfare, income, and population growth. The situation in which there is no government intervention is compared with one in which there is alternatively a subsidy to health expenditure and a subsidy to child-rearing costs, which is financed either by a consumption tax or a capital income tax. The government is assumed to claim a fraction of output for its expenditure. Once the ratio of government expenditure to income, [xi] has been fixed, the corresponding subsidy and tax rates that maintain a balanced budget without using a lump sum tax in the steady state are determined. That is, if a subsidy to health expenditure is instituted, then [xi]y = [s.sub.g]g = [[tau].sub.c]c in the consumption tax setting and [xi]y = [s.sub.g]g = [[tau].sub.k]rk in the capital income tax setting at the new steady state. If a subsidy to child-rearing costs is introduced, then [xi]y = [s.sub.b]nbk = [[tau].sub.c]c and [xi]y = [s.sub.b]nbk = [[tau].sub.k]rk, respectively.