Nonparametric Testable Restrictions of Household Behavior.

• Author: Snyder, Susan K.

Susan K. Snyder [*]

This paper uses semialgebraic theory to derive nonparametric testable restrictions of Pareto-efficient bargaining behavior within a household. These tests are analogous in form to Samuelson's Weak Axiom of Revealed Preference (WARP) and are defined over data on household-level consumption and individual labor supplies. Thus, without observing intrahousehold division of consumption, we can nonparametrically test whether there exist nonsatiated utility functions such that household behavior is Pareto efficient. I apply these tests to data from the National Longitudinal Surveys on U.S. households and find that preferences exist that are consistent with Pareto efficiency for each household in the data set.

1. Introduction

   Economists often treat the household as a single, utility-maximizing agent, regardless of the number of members making up the household. There is certainly intuitive appeal to the idea that a household's members have common goals on which they act; however, modeling households this way is often inconsistent with the dominant model of behavior in economics, the model of the individual rational decision maker. One reason for the persistence of the unitary model of the household is that observing intrahousehold decision-making processes, allocations, or income divisions is generally difficult. More often data are available at the household level. Thus, if we want to model households as collections of individually rational agents, we should ask what empirical implications for household behavior result from individual utility maximization.

   This paper uses semialgebraic theory to derive nonparametric testable restrictions of Pareto-efficient bargaining behavior within the household. These tests are in the form of a set of polynomial inequalities defined only over potentially observable variables: household-level data plus individual labor supplies. The derived tests are analogous in form to Samuelson's Weak Axiom of Revealed Preference (WARP), a nonparametric test of individual utility maximization. These testable restrictions are necessary and sufficient conditions, and as such, they are all the testable restrictions of the model, given the data we assume are observable.

   A previous approach to this problem is found in Chiappori (1988), who first presented non-vacuous testable restrictions on household-level data and individual labor supplies of Pareto-efficient behavior within the household. These are necessary and sufficient tests that are in the form of finding whether a set of polynomial inequalities has a solution. If the program has a solution, then the data are consistent with the model; if not, the data are not consistent with the model.

   A related problem is that of determining the empirical implications for aggregate demand data generated by individual utility maximization. Most results relating to this problem are negative, however; for example, well-known results in aggregation theory tell us that the aggregate demand of a group of rational decision makers has the same characteristics as the demand of one rational decision maker only under very strong restrictions (Shafer and Sonnenschein 1982). One positive result relating to this problem is shown by Brown and Matzkin (1996), who use semialgebraic theory to find nonvacuous testable restrictions on discrete observations of the equilibrium manifold of an economy. Thus, there are interesting empirical implications generated by competitive equilibrium behavior on aggregate-level data together with individual endowments or incomes. Another interpretation of their result is that individual utility maximization generates empirical implications on aggregate demand data together with data on indi vidual incomes for members of a group or economy.

   In this paper, I merge these two approaches: Using the semialgebraic theory techniques in Brown and Matzkin (1996), I derive testable restrictions for Pareto-efficient intrahousehold allocation. These tests are of a different form than, but are equivalent to, the linear programming tests derived in Chiappori (1988). Like WARP, these tests will be easy to apply in practice and can be readily compared to and interpreted in terms of the nonparametric testable restrictions of other hypotheses of household behavior.

   Empirical work on household or individual consumption is usually conducted with restrictive assumptions about the preferences of the decision makers. Generally, parametric specifications of the utility functions are used to derive testable propositions of the models. For example, the empirical work testing between the unitary model and models of Pareto-efficient intrahousehold allocation has used parametric methods (for example, see Browning et al. 1994). The validity of this work then depends in part on the validity of these assumptions about preferences.

   The strength of nonparametric tests such as those presented in this paper is that they make yery weak assumptions about preferences, essentially only that utility functions are nonsatiated. Additionally, if we have more than one observation of each household over time, we need make no assumptions about preferences across households. These tests could prove particularly useful as specification tests before one does more traditional econometric work on a data set.

   The outline of the paper is as follows. Section 2 gives a more detailed explanation of the collective rationality model. Section 3 derives the nonparametric testable restrictions of the model. Section 4 discusses how to use these tests to distinguish whether households behave as unitary actors or as a collective of rational individuals, with an application to data on U.S. households. The conclusion follows.

2. Collective Rationality within the Household

   The recognition that the unitary model of the household is often inconsistent with individual rationality, as well as the inability of the unitary model to meaningfully address questions concerning the distribution of income or consumption within the household, has led to the development of models of collective decision making within the household. Using both cooperative and noncooperative bargaining theory, these models describe household behavior as the outcome of an explicit bargaining process between the members of the household (see Manser and Brown 1980; McElroy and Homey 1981; Lundberg and Pollak 1994). These models can be used to derive rich insights into intrahousehold allocation. However, the empirical propositions derived from these models often depend on intrahousehold data that can be difficult to observe. Thus, it can be difficult to test whether these collective models describe household behavior better than the unitary model. Empirical results suggest, however, the unitary model restrictions are often not satisfied regardless of the alternative model specified (Schultz 1990; Thomas 1990).

   Chiappori (1988) develops a more general model of collective rationality in household behavior that can be tested with data on aggregate household behavior and individual labor supplies. The hypothesis is that the individuals within the household reach a Pareto-efficient allocation. Thus, instead of specifying a particular point on the contract curve as a function of individual threat points, as a Nash-bargaining model would, this model specifies only that the household be somewhere on this contact curve.

   The model is as follows. A household consists of two individuals, a and b. For i = a, b, each member can supply some amount of labor, [[ell].sub.i] in a market outside the household. Let T represent the fixed amount of total time available to each a and b; [L.sub.i] T - [[ell].sub.i] defines leisure consumption for each individual (there is no household production). There is also a privately consumed good C; let [c.sub.i], a nonnegative number, denote each member's consumption of the good. The price of the consumption good is normalized to one. Consumption and labor choices are made given wages, [w.sub.a], [w.sub.b], and nonlabor household income, Y.

   Here, we focus on the variant of the collective rationality model that assumes household members each have preferences over only their own personal consumption; in Chiappori's terminology, these are egoistic agents (the model can incorporate more general preferences). [1] Assume each agent has preferences representable by a nonsatiated utility function, [U.sub.i]([L.sub.i], [c.sub.i]).

   Although the exact mechanism for determining household consumption is left unspecified, we can think of a Pareto-efficient allocation as resulting from individual decisions subject to an agreement about the sharing of resources within the household. Let an income-sharing rule be some function, G: G([w.sub.a], [w.sub.b], Y) = ([y.sub.a], [y.sub.b]) such that [y.sub.a] + [y.sub.b] = Y. Then if the consumption choices for each individual i, ([L.sub.i], [c.sub.i]), are the solution of the following problem,

   Max [U.sub.i]([L.sub.i], [c.sub.i]) s.t. [c.sub.i] [leq] [y.sub.i] + [w.sub.i][[ell].sub.i],

   the resulting allocation will be Pareto efficient, given ([U.sub.a], [U.sub.b] G) (Chiappori 1992). The sharing rule is completely unspecified, and it may change over time. Also, [y.sub.i] is allowed to be negative, which would imply one member agrees to transfer some portion of their wage income to the other.

   This model constitutes a quite natural alternative to the basic unitary model of household behavior. Instead of thinking of the household maximizing one utility function subject to a budget constraint, we think of the household as being composed of two individuals, each maximizing their own utility function subject to their own budget constraint, with individual incomes summing up to household income.

   Using parametric methods, Browning et al. (1994) estimate the collective rationality model for data on Canadian family expenditures and do not reject the collective rationality...