Multiple-output child health production functions: the impact of time-varying and time-invariant inputs.

• Author: Agee, Mark D.

1. Introduction

   In this paper we attempt to broaden and to redirect the standard theoretical and empirical approach in economics to the household production of human health, especially child health. The last decade has witnessed increased concern with the impact of public policies and private investments on child health, perhaps motivated by an increased concern for equity and an enhanced recognition that human capital formation in children contributes significantly to society's future well-being. Children live in households with parents or adults who combine their skills with good as well as bad inputs that produce good as well as bad child health outcomes: physical, behavioral, and cognitive. A good input could be the time spent reading to a child, while a bad input could be parental smoking near a child. A good outcome could be higher reading skills, while a bad outcome could be a child's ill-health.

   Three features distinguish our modeling approach to the household production of child outcomes. First, we argue that multiple good and bad inputs can affect multiple measures of good and bad outcomes for children. Ignoring this by estimating single-equation relationships produces omitted variable bias and fails to model the tradeoffs among inputs and outcomes. To examine jointness among several good and bad outcomes as well as inputs and allow for heterogeneity among households in the productivity of observed child inputs, we specify and estimate an output-based directional distance function (Chambers, Chung, and Fare 1998) for a balanced panel of households with coresident children. This function allows us to estimate the maximum expansion of goods and the contraction of bads subject to a given level of observed inputs for a household using the underlying joint technology.

   Second, we argue that households will produce child health with varying degrees of effectiveness. The "best practice" household is defined as that household that cannot make further increases in all good dimensions of child health and reductions in all bad dimensions by some additive amount, using a given level of observed inputs. This household practices its child care on the production frontier. Households within this frontier make less efficient use of personal and marketed child care inputs. Thus, we estimate the technical efficiency for household units in producing multiple outputs from multiple inputs.

   Finally, although we have panel data and compute a fixed-effects estimator, we recover the effects of time-invariant variables (such as sex, race, and parent attributes) on goods and bads in a second-stage regression. We adjust their estimated standard errors and correct for the bias caused by weak instruments in the first stage using a jackknife technique.

   The next section presents background for our hypotheses of (i) jointness in the household production of child health outcomes and (ii) the presence of technical inefficiency among households in the production process. In section 3, we discuss properties of the directional distance function and the calculation of partial effects. Results are presented in section 4, and conclusions follow in section 5. We find that some time-varying as well as time-invariant inputs are significant determinants of child health outcomes. The latter are nearly always overlooked in a fixed-effects analysis. Further, we find that the good child outcomes can be individually increased on the production frontier only with an increase in a bad outcome, that the average sample household using a given child health production technology falls short of the "best practice" household by approximately 1.5 standard deviations, and that this household inefficiency diminishes over the time range of our panel.

2. Background

   The household production model (Becker 1965; Lancaster 1966) is the theoretical workhorse for economists studying household members' behavior, well-being, or both. When applied to health issues, the model emphasizes that relative prices and incomes, along with biological processes, condition members' health input choices (Rosenzweig and Schultz 1983). Applications often posit a cooperative agreement among adult members about a household utility function to be maximized subject to a full income constraint determined by members' pooled resources (e.g., Jacobson 2000). This household utility function includes parent and child health as outputs of production functions whose endogenous inputs include members' time and purchased goods and services as well as exogenous or predetermined endowment and environmental factors (e.g., Grossman 1972). The literature applying this model has either regressed a single health outcome on a set of observed input choices (e.g., Todd and Wolpin 2003, 2006) and employed instruments to account for endogeneity or estimated "reduced form" production functions employing a common set of presumably predetermined or exogenous inputs (e.g., Blau 1999). We attempt to model systematically the jointness of health outcomes and endogeneity, as well as the differing efficiencies among households.

   Jointness

   The biomedical and the economics literature agree that human health, including child health, is multidimensional. For example, Blau (1999) assesses the impact of parental income on six child outcome measures involving cognitive skills, behavioral problems, and motor and social development using data from the National Longitudinal Survey of Youth. Dawson (1991) assesses 17 outcome measures, many of which are constructed from multiple items referring to children's physical health, emotional well-being, and behaviors inventoried in the 1988 National Health Inventory Survey. A wide variety of children's health outcomes are considered in the Browning (1992), Haveman and Wolfe (1995), and Thornton (2001) reviews of child quality production functions. Many of these child health outcomes are plausibly produced jointly because of technical interdependencies. For example, by giving their children more time and attention, parents might enhance their children's cognitive skills and their good behavior. Playing video games may help a child's hand and eye coordination while impeding his social development, or watching television may aid his verbal memory and acuity while making him obese. Readily noticed examples of this sort would seem to justify estimating child health outcomes jointly rather than independently.

   Heterogeneity

   Operation on the production frontier is a necessary condition for utility maximization in the household production model. However, there is abundant reason to believe that parents do not always have the requisite skills to maximize utility or that a child will be receptive to operating on his health production frontier. Even though firms are continuously subjected to the pressures of pricing and innovation from rivals, they frequently perform at less than the "best practice" production level for their industry. (1) If firms frequently fall short of best practice, it is likely that households will also. This may be due to adult heads of household who optimize individual utility rather than all members' aggregate utilities (Lundberg and Pollak 1993; Fehr and Tyran 2005). (2) Further, parents' efforts to understand the effects of inputs on child health outcomes and the synergies among such outcomes can be costly (Stigler 1976; Heiner 1983). Overcoming a "lack of commitment" to the household stemming from a short time horizon, small assets and thus low gains from intrahousehold cooperation, or weak external enforcement of one's claims on household assets also prevent optimization of joint utility (Lundberg and Pollak 1993; Vagstad 2001). Whatever their source, many of these variables preventing optimization are unobservable.

   However, an extensive empirical economics literature dealing with the effects of parental inputs on child health (e.g., Carlin and Sandy 1991; Agee and Crocker 1996) as well as public investment and safety programs on child health (Currie 2000; Kenkel 2000) universally posits household utility maximization subject to a two-sided random error term designed to reflect random unobservables. A one-sided error term reflecting random failure of inefficient households to reach the household production frontier is not modeled. The value-added specifications (Krueger 2000; Hanushek 2003) assume that by adding a lagged child outcome measure, one has included as a regressor all components of the one-sided error. The remaining error term is strictly two-sided. In explicit recognition of the presence of a one-sided error term, fixed-effect specifications have been employed to difference out time-invariant, family-specific unobservables from panel data (Blau 1999). A time-varying, one-sided error term can also be associated with these regressions.

3. The Directional Distance Function

   Specification

   Consider a household production technology where parents combine multiple good inputs, x = ([x.sub.1], ..., [x.sub.N]) [member of] [R.sup.N.sub.+], to produce multiple good outputs, y = ([y.sub.1], ..., [y.sub.G]) [member of] [R.sup.G.sub.+]. The household's production technology, S(x, y, t), can be written as

   S(x, y, t) = {(x, y) : x can produce y at time t}, (1)

   where t = 1, ..., T is time. The technology must satisfy a set of basic axioms discussed in Fare (1988), including convexity of S(x, y, t) for all x and free disposability of inputs and outputs.

   Production of "bad" outputs (e.g., a child's ill-health or behavioral problems) can be appended to Equation 1 simply by defining a vector of B bads, b = ([b.sub.1], ..., [b.sub.B]) [member of] [R.sup.B.sub.+], which is produced jointly with y. As in Pollak and Wachter (1975), joint production of bads in the production of goods can be a function of both inputs and outputs. Following Chambers, Chung, and Fare (1998), we define the output directional distance function as

   [[??].sub.o] (x, y, b; 0, [[delta].sub.y]...