Parents' discount rates for child quality.

• Author: Agee, Mark D.

1. Introduction

   Contemporary discussions of economic growth make human capital or knowledge accumulation its primary engine [25]. Human capital as the lead component of the stock of wealth defines a developed economy. Becker and Murphy [4] show that human capital both increases and is increased by the scope of the market. Its rate of return rises with the size of its stock [5]. Human capital also plays significant roles in models of human fertility choice [40], population and environmental degradation [11; 34], capital mobility [3], and endogenous technological progress [33].

   A substantial portion of an adult's human capital is accumulated during childhood. Children as children, however, cannot independently contract for the resources required to support investments in this human capital. Though nearly every society grants parents wide discretion about the nurture they provide their children, little is understood about what motivates investment by one generation in another, the relative importance of different investment modes, or the discount rates which parents apply. Whatever the motivation and the mode, because of the long time span between birth and adulthood, parents' discount rates potentially exert great leverage upon the investment decisions they make for their children. Knowledge of these discount rates is a prerequisite to acquiring an improved understanding of the internal decision processes of the household, a collective decision unit which economics mostly treats as a black box.(1)

   Important policy questions, as well as scientific accuracy and logical consistency, demand knowledge of the discount rates parents apply to investments in their children. Healthier, better educated, and wealthier children have a better chance in life. But there is no evident reason to suppose that intrafamily discount rates are socially optimal.(2) Many of the private reactions and behaviors which reinforce or temper the intended effects of public policies are internal to the household. When these intended effects have a substantial temporal dimension, parental discount rates will be influential. For example, increased public investments in children can be neutralized by compensatory reductions in children's shares of household budgets. A public investment in children which yields relatively quick payoffs may substitute in the budget for parental investments with longer term payoffs for children. Similarly, publicly provided social security for parents may replace parental investment in children as a source of retirement funds [9]. In addition, for given prices of caregiving alternatives, the extent to which expanded female labor supply opportunities outside the household will increase child care time and effort will vary inversely with the discount rate the family applies to child investments. If these investments enhance the child's adult production and consumption levels, expanded female current employment opportunities combined with sufficiently low familial discount rates could expand future societal wealth and welfare. Lastly, those who question certain environmental and depletable resource policies on the ground that the current adult generation might give insufficient weight to posterity's interests must account for the relative strengths of the intergenerational emotional and material dependencies which different parental discount rates register.(3)

   In this paper, we use parents' decisions to treat their children's body burdens of lead to infer the discount rates parents attach to investments in their children's intellectual and behavioral development. Lead is a persistent micropollutant widely acknowledged in the biomedical literature to have long-lasting neurological consequences when allowed to accumulate in the bodies of young children [44].(4) The following two sections develop a simple altruistic model of the impact of the endogenous risks posed by a child's body lead burden upon parental utility from child health capital. Section IV describes the data and the setting used in section V to produce estimates of the discount rates which sample parents applied to an investment in the learning capacities and thus the prospective adult production and consumption levels of their children. The paper concludes with a discussion of the implications of our empirical findings for family economics.

2. Parental Altruism and Endogenous Risk

   Consider the following parsimonious statement of parental altruism and endogenous risk in a one period lifetime setting.(5) We ignore the continuous linkages among the generations expressed in dynastic utility formulations [6] in order to focus myopically on a parent-child relationship in which parents make a single, once-and-for-all, tied, inter vivos transfer which they believe influences the child's lifetime prospects. Our model incorporates limited forms of parental altruism, including simple caring about the child's health [49], or strategic interests in the child's adult ability and willingness to protect the parents from the hazards of old age [26]. Adults in multiple adult households are assumed to have identified a cooperative equilibrium with respect to the shared household good which the child represents.

   Let the parents' utility be a quasi-concave, weakly separable, homothetic function of a vector, X, whose components represent the quantities of non-child health commodities consumed and of parents' perceived risk, R, that their child will develop chronic lead-induced health effects,

   U = U(X, R), (1)

   with [U.sub.X] [greater than] 0 and [U.sub.R] [less than] 0. R is a measure on a real interval that could be either the perceived probability or the expected severity of the child's ill-health. The child's siblings are included in X and their numbers are predetermined. Perceived risk, R, admits that the child's adult intellectual, physical, social and moral endowments are not yet fully revealed to the parents. Risk perceived to be posed by body burdens of lead is endogenous because parents are aware that they can influence it by reducing exposures, A, and by the use of medical treatment, L:

   R = R(A, L; l, [Gamma]), (2)

   where l is the child's body lead burden, and [Gamma] is any predetermined family characteristics such as parents' educations which influence risk perception. Reduced exposures to lead can be achieved by using market goods to prevent or correct hazards (mainly interior paint and plumbing) in the home and increased time spent on child supervision [28]. Chelation therapy is the standard medical procedure for treating elevated body burdens of lead [44].(6) Expenditures on chelation therapy produce nothing of value other than reductions in R.

   Given smooth substitutability between parental work and other activities, the full-income budget constraint faced by households can be written as

   [r.sub.X]X + [r.sub.A]A + [r.sub.L]L = Y, (3)

   with [r.sub.j] = ([p.sub.j] + w[t.sub.j]), Y = I + w(T - [t.sub.s]), and j [element of] X, A, L. [p.sub.j] is the parametric money price of commodity j, w is the parametric opportunity cost of parental time, [t.sub.j] is the time required to consume one unit of commodity j, and [t.sub.s] is time spent away from the workplace. Full income, Y, is the sum of nonlabor income, I, and labor income, w(T - [t.sub.s]). Equation (3) requires that total time available be allocated among all possible uses of time, including child care and chelation activities.(7) Full income must equal total expenditures. Parents allocate their time between work and consumption activities; their income is allocated between child health-related expenditures and expenditures on other goods.

   Under local nonsatiation, we define the parents' indirect utility function, V, as the solution to the above utility maximization problem with wage-price vector r = (w, p):

   V [equivalent to] max {U[X, R(A, L; l, [Gamma])] [where] [r.sub.x]X + [r.sub.A]A + [r.sub.L]L = Y}. (4)

   Utility is a function of full income, Y, full prices, r, and a contingent claim which specifies the conditions under which a given child body lead burden will confront the parents; that is, V = V(Y, l, r). For a given reduction in perceived risk associated with a guaranteed reduction, [Mathematical Expression Omitted], of child body lead burden, the parents' indirect utility function in (4) can be defined in annualized terms as

   [Mathematical Expression Omitted],

   where CS is the Hicksian compensating surplus. The parents' rate of time preference, [Rho], generates a discount term, [Delta], equal to [Rho]/(1 + [Rho]). CS is the maximum wealth the parents are willing to forego to reduce perceived risks from their child's lead burden while maintaining their original utility level, V. Given that parents perceive the effects of the lead burden to be lifelong for their child and thus (roughly) infinite in length for the parents, the compounded sum of instantaneous parental compensating surpluses is

   [summation of] CS[(1 where r = 0 to [infinity] + [Rho]).sup.[Tau]]

   and the present value of this sum is

   CS/[(1 + [Rho])(1 - [(1 + [Rho]).sup.-[Tau]])].(8) (7)

   Since (5) constitutes an exact index of lifetime utility and since the parents' horizon is infinite, expression (7) simplifies to CS/(1 + [Rho]), and annualized compensating surplus can be written as

   [Delta]CS = [[Rho]/(1 + [Rho])]CS. (8)

   A small reduction in the child's body lead burden can thus be valued in terms of the parents' annualized marginal willingness-to-pay (MWT[P.sub.l]). MWT[P.sub.l] is obtained by totally differentiating the indirect utility function in (5) with respect to l and setting the result equal to zero:

   [Delta]MWT[P.sub.l] [equivalent to] [Delta](-dCS/dl) = -([Delta]V/[Delta]l)/([Delta]V/[Delta]Y) [equivalent to] -[V.sub.l]/[V.sub.Y]. (9)

   Expression (9) portrays parents' marginal rate of substitution, -[V.sub.l]/[V.sub.Y], between their child's body lead burden and their wealth. This is the annualized implicit price of risk...