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Nonparametric Testable Restrictions of Household Behavior.

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 restrictions. Additionally, they find that the restrictions implied by the unitary model are rejected. Although the validity of these results depends in part on the validity of the parametric assumptions made in estimation, the collective rationality model appears to be a promising alternative to the unitary model.

3. Nonparaetric Testable Restrictions

We will first derive the testable restrictions of this model with the assumption that we can observe data on consumption choices and income at the individual level. We will then use those results to derive restrictions for when we cannot observe individual-level data.

Suppose we could observe a series of observations on the behavior of a household both at the household level and at the individual level; that is, we observe household consumption, household nonlabor income, wages, individual labor supplies, individual consumption, and individual shares of nonlabor income, ([C.sup.r], [Y.sup.r], [[w.sup.r].sub.i], [[[ell].sup.r].sub.i], [[c.sup.r].sub.i], [[y.sup.r].sub.i]) for i = a, b, where i indicates a member of a household, and r = 1,..., R, where r indicates a distinct time period. Can we say whether there exist any nonsatiated utility functions, [U.sub.i], and a sharing rule, G, such that the collective rationality model could have generated this data? In other words, are there restrictions on the data that will tell us whether or not the data could have been generated by the collective rationality model, without making any further parametric specifications of preferences or the sharing rule?

In the collective rationality model, each individual maximizes his or her own utility function subject to a budget constraint. A necessary and sufficient condition for a finite set of data on individual consumption and prices to be consistent with the maximization of some nonsatiated utility function is the satisfaction of the Generalized Axiom of Revealed Preference (GARP), a generalized version of WARP (Varian 1982). It is clear that GARP is a necessary condition of utility maximization. It is the sufficient part of this theorem that is perhaps not obvious. This is not sufficient in the sense that the data can imply that consumers act as utility maximizers--we can never know that for sure--but sufficient in the sense that the data imply that there exists a nonsatiated utility function such that the consumer could be maximizing utility with her choices. [2] Given that we cannot observe utility functions, this is the strongest notion of sufficiency that the data can satisfy.

Thus, the collective rationality model implies that each individual in the household must satisfy GARP. Together with some aggregation conditions, this gives us all the empirical implications of the collective rationality model.

The following theorem and corollary are restatements of Chiappori's (1988) Proposition 2: [3]

THEOREM 1. Let data for one household. ([C.sup.r], Y, [[w.sup.r].sub.i], [[[ell].sup.r].sub.i], [[c.sup.r].sub.i], [[y.sup.r].sub.i]) for i = a, b, and r = 1, 2, be given. Then there exist strictly monotonic, strictly concave utility functions [([U.sub.i]).sub.i=a,b] and income-sharing rule (G) such that the data are consistent with a Pareto-optimal allocation within the household ([U.sub.a]([L.sub.a], [c.sub.a]), [U.sub.b]([L.sub.b], [c.sub.b])) iff

(a) Household aggregation conditions satisfied (for r = 1, 2)

[[C.sup.r].sub.a] + [[C.sup.r].sub.b] = [C.sup.r] [[y.sup.r].sub.a] + [[y.sup.r].sub.a] = [Y.sup.r].

(b) Individual budget constraints satisfied

[[C.sup.r].sub.i] = [[y.sup.r].sub.i] + [[w.sup.r].sub.i][[[ell].sup.r].sub.i] for i = a, b; r = 1, 2.

(c) Individuals satisfy the Strong Axiom of Revealed Preference (SARP): [4]

[[c.sup.2].sub.i] + [[w.sup.1].sub.i][[L.sup.2].sub.i] [greater than] [[c.sup.1].sub.i] + [[w.sup.1].sub.i] [[L.sup.1].sub.i] or [[c.sup.1].sub.i] + [[w.sup.2].sub.i][[L.sup.1].sub.i] [greater than] [[c.sup.2].sub.i] + [[w.sup.2].sub.i][[L.sup.2].sub.i] for i = a, b.

(d) Feasibility satisfied

[[c.sup.r].sub.a] [geq] 0, [[c.sup.r].sub.b] [geq] 0.

This theorem provides a restatement of the collective rationality model in the form of a finite set of polynomial inequalities defined over a finite set of variables. As such, it provides the testable restrictions of the collective rationality model, given we could observe both household- and individual-level behavior. If data from a given household satisfy these inequalities, then there exist utility functions and income-sharing rules such that the data are consistent with collective rationality. If data do not satisfy these inequalities, then there are no nonsatiated utility functions (with single-valued demand) such that the data could have been generated by the collective rationality model.

We do not expect it is generally possible to observe all these variables, however, particularly the intrahousehold income sharing or individual consumption. Corollary 1--which assumes that ([[c.sup.r].sub.i], [[y.sup.r].sub.i]), the individual consumptions and the assigned individual nonlabor incomes, are unobserved, and that ([C.sup.r], [Y.sup.r], [[w.sup.r].sub.i], [[[ell].sup.r].sub.i]), the household-level consumptions, and nonlabor incomes, individual wages, and individual leisure/labor supplies, are observed--follows directly from theorem 1.

COROLLARY 1. Let ([C.sup.r], [Y.sup.r], [[w.sup.r].sub.i], [[[ell].sup.r].sub.i]) for i = a, b, and r = 1, 2 be given. Then there exist strictly monotonic, strictly concave utility functions, [([U.sub.i]).sub.i=a,b], and an income-sharing rule, (G), such that the data are consistent with a Pareto-optimal allocation within the household ([U.sub.a]([L.sub.a], [C.sub.a]), [U.sub.b]([L.sub.b], [C.sub.b])) iff there exist numbers ([[c.sup.r].sub.i], [[y.sup.r].sub.i]), [[c.sup.r].sub.i] [geq] 0, such that the conditions of theorem 1 are satisfied.

This corollary provides nonparametric testable restrictions of the collective rationality model. The tests are of the following form: If there exist real solutions to the linear programs described in corollary 1, then the data are consistent with the model. If there do not exist real solutions to those programs, then the data are not consistent with the model (Chiappori 1988). Nonparametric tests of this form have been developed extensively in Diewert and Parkan (1985), for example.

Note also that theorem 1 and corollary 1 present the collective rationality model as a finite set of polynomial inequalities in observed and unobserved variables. Thus, the structure of the complete set of testable restrictions of this model is semialgebraic. This results not from any restrictions on preferences, it is instead the finiteness of data that limits the testable restrictions of utility maximization to be of polynomial form.

Brown and Matzkin (1996) use semialgebraic theory to examine the empirical implications of the competitive equilibrium model. It can be shown that the complete set of testable propositions of the competitive equilibrium model over finite data can be written as a finite union of sets of polynomial inequalities; that is, they form a semialgebraic set. Thus, we can use the properties of semialgebraic geometry to address many empirical issues of general equilibrium. The complete set of testable propositions of the collective rationality model also form a semialgebraic set, so we can use semialgebraic theory to address empirical implications of this model as well.

Most importantly, semialgebraic theory provides finite-time methods for deriving testable propositions over a particular data structure. These testable propositions will be in the form of a finite union of sets of polynomial inequalities in the data (Mishra 1993). In other words, given the data we believe to be observable, even if these data do not include many of the model's key variables, we have methods of systematically deriving all of the model's empirical propositions. These methods to derive the restrictions may prove infeasible in large problems (Van Den Dries 1988); if so, we at least (i) know the restrictions will be in the form of a semialgebraic set and (ii) can address whether these restrictions are nonvacuous, which will often be particularly important when key variables of the model are unobservable.

The derivation of testable propositions over a subset of a model's variables is accomplished through the technique of quantifier elimination. Quantifier elimination is the process of eliminating quantified variables; in this case, the unobservable variables are quantified within the model in that they appear in conjunction with the existential quantifier ("there exists").

There are three possible outcomes when quantifier elimination is applied to a system. One possibility is the equivalent system reduces to 1 [equiv] 0, meaning it is impossible to observe data consistent with the model; the theory is empirically inconsistent. Another possibility is the system reduces to 1 [equiv] 1, meaning it is impossible to observe data not consistent with the model; the theory imposes no testable restrictions on the set of observable variables. The third possibility is the system reduces to a finite set of polynomial inequalities involving the observable variables. It is possible to observe data that satisfy these inequalities, and it is possible to observe data that do not satisfy these inequalities. In this case, we say testable restrictions of the model exist, given the set of observable variables (Brown and Matzkin 1996).

It is straightforward to show that this system describing the collective rationality model satisfies empirical consistency; that is, we can find data that are consistent with the model. We can also show that there exist data that are not consistent with the model. Chiappori (1988) provides an example of three observations of data such that the revealed preference axioms cannot be satisfied for each individual; below, I show an example of two observations of data such that the conditions are not satisfied. Thus, there do exist nonvacuous testable restrictions of the collective rationality model over household-level data together with individual wages and labor supplies. If we apply quantifier elimination to the system described in corollary 1, we will derive these testable restrictions. These restrictions will be in the form of checking a set of polynomial inequalities defined directly over observable variables.

Because this system is linear, one can use Fourier-Motzkin elimination to eliminate the unobserved variables. [5] I apply this technique for two observations of data:

THEOREM 2. Let ([[w.sup.r].sub.a], [[w.sup.r].sub.b], [[[ell].sup.r].sub.a], [[[ell].sup.r].sub.b], [C.sup.r], [Y.sup.r]) for r = 1, 2 be given. Then there exist strictly monotonic, strictly concave utility functions, [([U.sub.i]).sub.i=a,b], and income-sharing rules, ([G.sup.r]), such that the data are consistent with a Pareto-optimal allocation within the household ([U.sub.a]([L.sub.a],[C.sub.a]), [U.sub.b]([L.sub.b],[C.sub.b])) iff the collective rationality nonparametric restrictions are satisfied.

Collective Rationality Model Nonparametric Restrictions

[forall]r = 1,2 [C.sup.r] = [Y.sup.r] + [[w.sup.r].sub.a][[[ell].sup.r].sub.a] + [[w.sup.r].sub.b][[[ell].sup.r].sub.b] and [exists] r, s = 1,2 r[neq]s s.t.

{([C.sup.s] + [[w.sup.r].sub.a][[L.sup.s].sub.a] [greater than] [[w.sup.r].sub.a][[L.sup.r].sub.a] and [C.sup.r] + [[w.sup.s].sub.b][[L.sup.r].sub.b] [greater than] [[w.sup.s].sub.b][[L.sup.s].sub.b]) or

([C.sup.s] + [[w.sup.r].sub.a][[L.sup.s].sub.a] + [[w.sup.r].sub.b][[L.sup.s].sub.b] [greater than] [C.sup.r] + [[w.sup.r].sub.a][[L.sup.r].sub.a] + [[w.sup.r].sub.b][[L.sup.r].sub.b] and [C.sup.s] + [[w.sup.r].sub.a][[L.sup.s].sub.a] [greater than] [[w.sup.r].sub.a][[L.sup.r].sub.a] and [C.sup.s] + [[w.sup.r].sub.b][[L.sup.s].sub.b] [greater than] [[w.sup.r].sub.b][[L.sup.r].sub.b])}.

PROOF. Fourier-Motzkin elimination is used to rewrite the equilibrium conditions of the model in terms of only the observable variables. See Appendix A.

Compare the collective rationality restrictions to the testable restrictions of the unitary model.

Unitary Model Nonparametric Restrictions

[forall] r = 1,2 [C.sup.r] = [Y.sup.r] + [[w.sup.r].sub.a][[[ell].sup.r].sub.a] + [[w.sup.r].sub.b][[[ell].sup.r].sub.b] and [exists] r, s = 1,2 r[neq]s s.t.

[C.sup.s] + [[w.sup.r].sub.a][[L.sup.s].sub.a] + [[w.sup.r].sub.b][[L.sup.s].sub.b] [greater than] [C.sup.r] + [[w.sup.r].sub.a][[L.sup.r].sub.a] + [[w.sup.r].sub.b][[L.sup.r].sub.b]

The household budget constraint must be satisfied under both models. An additional restriction in the unitary model is that the household consumption choices must satisfy SARP. As one would expect, the collective rationality model does not imply SARP and is not implied by SARP. Thus, it is possible to observe data that are consistent with one model and not the other, both models, or neither model.

The collective rationality restrictions result from the definition of bounds on consumption and labor supply data such that it is possible for each individual in the household to be satisfying the axiom of revealed preference. The nonvacuousness of the restrictions comes essentially from the nonnegativity restrictions on individual consumption. Given this level of observability, it is always possible for each individual to be maximizing utility given any data; it is not always possible, however, for each individual to be maximizing utility given that the other individual in the household is maximizing utility as well.

Figure 1 illustrates a situation in which the restrictions will not be satisfied. Because the consumption good can be traded within the household, it is useful to use an Edgeworth box to model the problem each period, keeping in mind that leisure is not tradeable within the household. The first picture represents the situation for consumer a. Each period, consumer a faces a utility-maximization problem where he chooses between leisure and consumption. The assumptions about observability imply that it is not known how income is divided within the household, but the slope of the budget constraint that a faces is known (because we observe the normalized wages), and it is known that the budget constraint in each period r must pass through some point ([[c.sup.r].sub.a], [[L.sup.r].sub.a]), where [[L.sup.r].sub.a], leisure, is observed. Thus, we know the range of possible budget constraints in each period, from lowest level of income to highest level of income. This range of budget constraints, from [[B.sup.r].sub.L] to [[B.sup.r].sub.H], is represented on the graph each period.

Note that for any possible level of income, consumer a's second period bundle is always revealed preferred to his first period bundle. Thus, to satisfy SARP, income in the first period must be such that his first period consumption is not revealed preferred to his second period consumption. A budget constraint below [[[hat{B}].sup.1].sub.a] is necessary. In other words, a must consume less than [[[hat{c}].sup.1].sub.a] of the consumption good for SARP to be satisfied.

Suppose b faces the situation in the second graph. Leisure for consumer b replaces a's leisure on the x-axis, while the consumption good remains on the y-axis. With these wages, again we have a situation in which consumer b's second period bundle is always revealed preferred to her first period bundle. Thus, to satisfy SARP, income in the first period must be such that her first period is not revealed preferred to the second period bundle. Again, we can find an upper bound on income such that this is possible. But to ensure that a satisfies SARP, it must be the case that b must consume at least [[[hat{c}].sup.1].sub.b] = [C.sup.1] - [[[hat{c}].sup.1].sub.a] of the consumption good. This puts a lower bound on b's budget constraint, [[[hat{B}].sup.1].sub.b]. In the situation here, this lower bound is above the upper bound on income that will satisfy SARP. Thus, although each consumer could satisfy SARP, they cannot both satisfy SARP at the same time. These data fail the collective rationality restrictions.

4. Implementation and Application

The collective rationality testable restrictions derived above are similar in form to revealed preference tests of individual utility maximization. They require little data to implement and are straightforward to apply.

If the data satisfy the restrictions, one could interpret this as the satisfaction of a specification test. [6] That is, if the data satisfy the collective rationality restrictions, then one could be more confident about doing further empirical work, making more restrictive assumptions about functional form, for example, and estimating parameters such as the sharing rule within the household. Note that these restrictions are necessary and sufficient conditions on the data; thus, if these restrictions were satisfied by a particular data set, there exist no other restrictions on that data set that would imply the collective rationality model was not satisfied.

If the data do not satisfy the restrictions, it is not clear what the interpretation should be. As in other revealed preference tests, we have not allowed for any stochastic elements in the model or the data. Varian (1985, 1990) addresses the problem that there is likely to be some measurement error in the data that could cause nonparametric tests to lead to rejection. These papers address the question of how to judge how close the conditions are to being satisfied. Alternatively, one could posit that there is some stochastic randomness in preferences, as in Brown and Matzkin (1998), who discuss the nonparametric approach to testing when utility depends linearly on a random variable e.

Application to Data on U.S. Households

I will apply these tests to data on U.S. households to determine whether these households appear to be behaving consistently with the collective rationality model, the unitary model, both, or neither. The data I use are from the National Longitudinal Surveys (NLS), sponsored by the Bureau of Labor Statistics. These surveys gathered information on the labor market activities and other income sources of selected American men and women over a number of years. Horney and McElroy (1988) use NLS data to estimate a (parametrized) Nash-bargaining model of household behavior and found some evidence that this model performed better than the unitary model.

I use a subsample from the 1969 and 1971 National Longitudinal Survey of Men (the men were between ages 45 and 59 in 1966). [7] The subsample used only observations of families in which both spouses worked more than zero hours each year, there were no children, the family received no farm income, total household income was positive, and there were no missing observations for the variables we used in the tests. We make no claim that these households are representative of U.S. households in general; for example, these households contain workers that are older than average, and they have a higher percentage of women working outside the home than average.

Both the collective rationality nonparametric restrictions and the unitary restrictions are defined over two observations of data on household consumption, household nonlabor income, wages, and individual labor supplies. I do not have data on consumption expenditures, so it is assumed that household budget constraints were satisfied so that household consumption expenditures equaled total income. The NLS provide data on many sources of nonlabor income. Wages for each spouse were computed from data on hours worked and income from wages and salary. I normalized consumption expenditure and wages by each year's CPI-U price level. Simple statistics and more details about the variables used are given in the Appendix B.

The results of the nonparametric tests of the collective rationality model and the unitary model are reported in Table 1. About 98% of the 243 households satisfied the restrictions for both models, and all households satisfied the collective rationality restrictions. Thus, for every household in the sample, there exist individual nonsatiated utility functions and an intrahousehold sharing rule such that household consumption and labor supply behavior is consistent with Pareto efficiency.

The fact that the sample's households are all consistent with collective rationality is perhaps not surprising. I am essentially testing individually rational behavior with mostly aggregate-level (household-level) data. Also, the parametric estimations of the collective rationality model for other data sets have generally led to good fits; when we weaken the maintained hypotheses made in these empirical studies by assuming no particular functional form, we should expect the results to be even more favorable.

A more fundamental problem with revealed preference tests is that sometimes consumers cannot violate revealed preference conditions no matter how they act. For example, if one household member had both his wage and his nonlabor income increase from year 1 to year 2, then he could not possibly violate SARP because his two budget lines do not cross; any bundle he could choose in year 2 would always be revealed preferred to any bundle he could choose in year 1, and any bundle he could choose in year 1 would never be revealed preferred to any bundle he could choose in year 2. In our problem, however, we assume individual nonlabor income is unobservable; thus, there always exists an assignment of nonlabor income within the household such that at least one household member's budget lines cross. Thus, given the exogenous data of wages and household-level nonlabor income, it is always theoretically possible to have behavior inconsistent with collective rationality. [8]

Additionally, the revealed preference conditions used in the collective rationality restrictions are knife-edge conditions; there is no stochastic element built in. Thus, one might expect to see numerous violations in practice. Errors in reporting or recording the data, omitted variables, misspecification of the basic maximization problem faced by individuals (for example, the assumption that there is no in-house production), or random elements of preference or consumption behavior could all lead to violations of the restrictions.

Interestingly, although the collective rationality restrictions are always satisfied, the unitary restrictions are violated for some households. The number of households violating the unitary restrictions is too small to conclude that the collective rationality model fits the data better. [9]

Given that the data satisfy the collective rationality restrictions, one could proceed to doing further work with NLS data such as making functional form assumptions and estimating sharing rules within the household, or one could continue to work in the nonparametric setting and test sharper hypotheses about preferences or the sharing rule within the household. Further, one can recover bounds on utility functions and sharing rules consistent with this data (Afriat 1967). [10]

5. Conclusion

This paper has used semialgebraic theory to derive nonparametric testable restrictions of the collective rationality model of household behavior. These restrictions are necessary and sufficient conditions for household-level data (including individual-level labor supplies) to be consistent with Pareto-efficient intrahousehold allocation. These nonparametric restrictions should provide a useful supplement to econometric tests that use assumptions about parametric forms; for example, they could be used as specification tests.

I have used these restrictions to test whether data from the NLS describe households that act as one utility-maximizing agent or two utility-maximizing agents who arrange a Pareto-efficient intrahousehold allocation. I conclude there do exist preferences that are consistent with Pareto efficiency for all of the households in this data set. I also find 2% of the households are not behaving consistently with the unitary model. Further investigations could perhaps focus on how close these households were to satisfying the unitary restrictions, as suggested in Varian (1985, 1990); if the fit is close, then we might conclude we cannot reject the unitary model. If so, this could be some evidence that the parametric tests in other empirical work are rejecting not the unitary model but rather the particular parametric specification that was used.

(*.) Department of Economics, Virginia Polytechnic Institute & State University, Blacksburg, VA 24061-0316, USA; E-mail sksnyder@vt.edu.

I would like to thank seminar participants at Arizona State University, VPI, the 1997 North American Summer Meeting of the Econometric Society, the 1997 Summer Institute for Theoretical Economics at Stanford, the 1997 Southern Economic Association Meetings, and the 1999 Allied Social Science Associations meetings. I am especially indebted to Don Brown for his suggestions, and thanks also to Michael Kuehlwein, Gerald Granderson, Robert Pollak, Stuart Rosenthal, Dale Thompson, and an anonymous referee for their helpful comments. This is a revised version of VPI Working Paper E-97-03, "Nonparametric tests of household behavior."

(1.) In the nonegoistic model, individual consumption becomes a public good within the household. Testable propositions of the nonegoistic model are derived using semialgebraic methods in Snyder (1999).

(2.) Further, if there exists a nonsatiated utility function that could have generated the data, there exists a concave utility function that could have generated the data (Afriat 1967).

(3.) To eliminate some uninteresting cases of restrictions, we assume wages are nonzero, and we assume (here and in theorem 2) that ([Y.sup.1], [[w.sup.1].sub.a], [[w.sup.1].sub.b]) [neq] ([Y.sup.2], [[w.sup.2].sub.a], [[w.sup.2].sub.b]); that is, there is some change in at least one of the variables exogenous to the household.

(4.) Using the Strong Axiom of Revealed Preference (SARP) instead of the Generalized Axiom greatly simplifies the resulting tests and simply rules out data that could have been generated only by utility functions that generate nonsingle-valued demands. If the data satisfy SARP, we can construct strictly monotone, strictly concave utility functions to generate the data (Matzkin and Richter 1991).

(5.) Fourier-Motzkin elimination (sometimes called Fourier elimination) is a technique similar to Gaussian elimination (see Dantzig and Eaves 1973).

(6.) See Diewert and Parkan (1983).

(7.) I replicated this approach for the 1969 and 1971 National Longitudinal Survey of Women data set also and found very similar results.

(8.) This does not address the power of the collective rationality test, however, or under what conditions it is possible to see a violation of the collective rationality conditions as given See Bronars (1987) and Manser and McDonald (1988) for different approaches to computing the power of revealed preference tests.

(9.) I replicated these tests for the survey of mature women in the National Longitudinal Survey and found very similar results: Of 108 households, all satisfy the collective rationality tests and 4 out of 108 (4%) fail the unitary tests.

(10.) For more discussion on recovering utility functions from revealed preference tests, see Varian (1982). Approximate bounds could be recovered even if the tests were not satisfied for all observations (Afriat 1973).

References

Afriat, S. N. 1967. The construction of a utility function from expenditure data. International Economic Review 8:67-77.

Afriat, S. N. 1973. On a system of inequalities in demand analysis: An extension of the classic method. International Economic Review 14:460-72.

Bronars, Stephen G. 1987. The power of nonparametric tests of preference maximization. Econometrica 55:693-8.

Brown, Donald J., and Rosa L. Matzkin. 1996. Testable restrictions on the equilibrium manifold. Econometrica 64:1249-62.

Brown, Donald J., and Rosa L. Matzkin, 1998. Estimation of nonparametric functions in simultaneous equations models, with an application to consumer demand. Yale Cowles Foundation Discussion Paper 1175.

Browning, Martin, Francois Bourguignon, Pierre-Andr[acute{e}] Chiappori, and Val[acute{e}rie Lechene. 1994. Incomes and outcomes: A structural model of intra-household allocation. Journal of Political Economy 102:1067-96.

Chiappori, Pierre-Andr[acute{e}]. 1988. Rational household labor supply. Econometrica 56:63-89.

Chiappori, Piene-Andr[acute{e}]. 1992. Collective labor supply and welfare. Journal of Political Economy 100:437-67.

Dantzig, George B., and B. Curtis Eaves. 1973. Fourier-Motzkjn elimination and its dual. Journal of Combinatorial Theory 14:288-97.

Diewert, W. E., and Celik Parkan. 1983. Linear programming tests of regularity conditions for production functions. In Quantitative studies on production and prices, edited by W. Eickhom, R. Henn, K. Neumann, and R. W. Shephard. W[ddot{u}rzburg-Wien: Physica-Verlag, pp. 131-58.

Diewert, W. E., and Celik Parkan. 1985. Tests for the consistency of consumer data. Journal of Econometrics 30:127-47.

Horney, Mary Jean, and Marjorie B. McElroy. 1988. The household allocation problem: Empirical results from a bargaining model. In Research in Population Economics, Volume 6, edited by T. Paul Schultz. Greenwich, CT: JAI Press, pp. 15-38.

Lundberg, Shelly, and Robert A. Pollak. 1994. Noncooperative bargaining models of marriage. American Economic Review 84:132-7.

Manser, Marilyn E., and Murray Brown. 1980. Marriage and household decision-making: A bargaining analysis. International Economic Review 21:31-44.

Manser, Marilyn E., and Richard J. McDonald. 1988. An analysis of substitution bias in measuring inflation, 1959-85, Econometrica 56:909-30.

Matzkin, Rosa L., and M. K. Richter. 1991. Testing strictly concave rationality. Journal of Economic Theory 53:287-303.

McElroy, Marjorie B., and Mary Jean Horney. 1981. Nash-bargained household decisions: Toward a generalization of the theory of demand. International Economic Review 22:333-49.

Mishra, Bhubaneswar. 1993. Algorithmic algebra. New York: Springer-Verlag.

Schultz, T. Paul. 1990. Testing the neoclassical model of family labor supply and fertility. Journal of Human Resources 25:599-634.

Shafer, Wayne, and Hugo Sonnenschein. 1982. Market demand and excess demand functions. In Handbook of Mathematical Economics, Volume 2, edited by K. J. Arrow and M. D. Intriligator. New York: North Holland Publishing, pp. 671-93.

Snyder, Susan K. 1999. Testable restrictions of Pareto optimal public good provision. Journal of Public Economics 71:97-119.

Thomas, Duncan. 1990. Intra-household resource allocation. Journal of Human Resources 25:635-64,

Van Den Dries, Lou. 1988. Alfred Tarski's elimination theory for real closed fields. The Journal of Symbolic Logic 53:7-19.

Varian, Hal R. 1982. The nonparametric approach to demand analysis. Econometrica 50:945-73.

Varian, Hal R. 1985. Non-parametric analysis of optimizing behavior with measurement error. Journal of Econometrics 30:445-58.

Varian, Hal R. 1990. Goodness-of-fit in optimizing models. Journal of Econometrics 46:125-40.

Appendix A: Proof of Theorem 2

Equilibrium in the collective rationality model is described by the following quantified set of polynomial equalities and inequalities, [Gamma].

[Gamma]: There exist numbers ([[c.sup.r].sub.i], [[y.sup.r].sub.i]) such that

[[c.sup.r].sub.a] + [[c.sup.r].sub.b] = [C.sup.r] for r = 1,2 (A1)

[[y.sup.r].sub.a] + [[y.sup.r].sub.b] = [Y.sup.r] for r = 1,2 (A2)

[[y.sup.r].sub.i] + [[w.sup.r].sub.i][[[ell].sup.r].sub.i] = [[c.sup.r].sub.i] for r = 1, 2; i = a, b (A3)

[[c.sup.2].sub.a] + [[w.sup.1].sub.a][[L.sup.2].sub.a] [greater than] [[c.sup.1].sub.a] + [[w.sup.1].sub.a][[L.sup.1].sub.a] or [[c.sup.1].sub.a] + [[w.sup.2].sub.a][[L.sup.1].sub.a] [greater than] [[c.sup.2].sub.a] + [[w.sup.2].sub.a][[L.sup.2].sub.a] (A4)

[[c.sup.2].sub.b] + [[w.sup.1].sub.b][[L.sup.2].sub.b] [greater than] [[c.sup.1].sub.b] + [[w.sup.1].sub.b][[L.sup.1].sub.b] or [[c.sup.1].sub.b] + [[w.sup.2].sub.b][[L.sup.1].sub.b] [greater than] [[c.sup.2].sub.b] + [[w.sup.2].sub.b][[L.sup.2].sub.b] (A5)

[[c.sup.r].sub.i] [geq] 0 for r = 1, 2; i = a, b. (A6)

To derive an equivalent nonquantified set of polynomial equalities and inequalities, the variables ([[c.sup.r].sub.i], [[y.sup.r].sub.i]) must be eliminated. First use the equalities to make some obvious substitutions. Equation A1 allows us to solve for [[c.sup.r].sub.b], and Equation A3 allows us to solve for [[y.sup.r].sub.i]. Substituting back into Equation A2, we get

[Y.sup.r] + [[w.sup.r].sub.a][[[ell].sup.r].sub.a] + [[w.sup.r].sub.b][[[ell].sup.r].sub.b] = [C.sup.r] for r = 1, 2.

These are simply the household budget constraint conditions. They are defined over observable variables and so form testable restrictions of the model.

The following system [Theta], together with the household budget constraints, is equivalent to the original system.

[Theta]: There exist numbers ([[c.sup.r].sub.a]) such that

[[c.sup.r].sub.a] + [[w.sup.1].sub.a][[L.sup.2].sub.a] [greater than] [[c.sup.1].sub.a] + [[w.sup.1].sub.a][[L.sup.1].sub.a] or [[c.sup.1].sub.a] + [[w.sup.2].sub.a][[L.sup.1].sub.a] [greater than] [[c.sup.2].sub.a] + [[w.sup.2].sub.a][[L.sup.2].sub.a]

[C.sup.2] - [[c.sup.2].sub.a] + [[w.sup.1].sub.b][[L.sup.2].sub.b] [greater than] [C.sup.1] - [[c.sup.1].sub.a] + [[w.sup.1].sub.b][[L.sup.1].sub.b] or [C.sup.1] - [[c.sup.1].sub.a] + [[w.sup.2].sub.b][[L.sup.1].sub.b] [greater than] [C.sup.2] - [[c.sup.2].sub.a] + [[w.sup.2].sub.b][[L.sup.2].sub.b]

0 [leq] [[c.sup.r].sub.a] [leq] [C.sup.r] for r = 1, 2.

This system is equivalent to [Phi], the disjunction of four sets of linear inequalities:

[Phi]: [[Phi].sub.1] or [[Phi].sub.2] or [[Phi].sub.3] or [[Phi].sub.4] is satisfied.

[[Phi].sub.1]: There exist numbers ([[c.sup.r].sub.a]) such that

[[c.sup.2].sub.a] + [[w.sup.1].sub.a][[L.sup.2].sub.a] [greater than] [[c.sup.1].sub.a] + [[w.sup.1].sub.a][[L.sup.1].sub.a]

[C.sup.2] - [[c.sup.2].sub.a] + [[w.sup.1].sub.b][[L.sup.2].sub.b] [greater than] [C.sup.1] - [[c.sup.1].sub.a] + [[w.sup.1].sub.b][[L.sup.1].sub.b]

0 [leq] [[c.sup.r].sub.a] [leq] [C.sup.r] for r = 1, 2.

[[Phi].sub.2]: There exist numbers ([[c.sup.r].sub.a]) such that

[[c.sup.2].sub.a] + [[w.sup.1].sub.a][[L.sup.2].sub.a] [greater than] [[c.sup.1].sub.a] + [[w.sup.1].sub.a][[L.sup.1].sub.a]

[C.sup.1] - [[c.sup.1].sub.a] + [[w.sup.2].sub.b][[L.sup.1].sub.b] [greater than] [C.sup.2] - [[c.sup.2].sub.a] + [[w.sup.2].sub.b][[L.sup.2].sub.b]

0 [leq] [[c.sup.r].sub.a] [leq] [C.sup.r] for r = 1, 2.

[[Phi].sub.3]: There exist numbers ([[c.sup.r].sub.a]) such that

[[c.sup.1].sub.a] + [[w.sup.2].sub.a][[L.sup.1].sub.a] [greater than] [[c.sup.2].sub.a] + [[w.sup.2].sub.a][[L.sup.2].sub.a]

[C.sup.2] - [[c.sup.2].sub.a] + [[w.sup.1].sub.b][[L.sup.2].sub.b] [greater than] [C.sup.1] - [[c.sup.1].sub.a] + [[w.sup.1].sub.b][[L.sup.1].sub.b]

0 [leq] [[c.sup.r].sub.a] [leq] [C.sup.r] for r = 1, 2.

[[Phi].sub.4]: There exist numbers ([[c.sup.r].sub.a]) such that

[[c.sup.1].sub.a] + [[w.sup.2].sub.a][[L.sup.1].sub.a] [greater than] [[c.sup.2].sub.a] + [[w.sup.2].sub.a][[L.sup.2].sub.a]

[C.sup.1] - [[c.sup.1].sub.a] + [[w.sup.2].sub.b][[L.sup.1].sub.b] [greater than] [C.sup.2] - [[c.sup.2].sub.a] + [[w.sup.2].sub.b][[L.sup.2].sub.b]

0 [leq] [[c.sup.r].sub.a] [leq] [C.sup.r] for r = 1, 2.

Each [[Phi].sub.1] is a set of inequalities that is linear in the quantified variables so we can apply Fourier-Motzkin elimination to each set. If one [[Phi].sub.1] is satisfied, then [Phi] is satisfied. Note that [[Phi].sub.1] and [[Phi].sub.4] are the same sets of inequalities with reversed observations, and [[Phi].sub.2] and [[Phi].sub.3] are the same sets of inequalities with reversed observations.

Fourier-Motzkin Elimination on [[Phi].sub.1]

First, we will eliminate [[c.sup.1].sub.a] from [[Phi].sub.1]. Rewrite all the inequalities that involve [[c.sup.1].sub.a] with [[c.sup.1].sub.a] isolated:

[[c.sup.1].sub.a] [less than] [[c.sup.2].sub.a] + [[w.sup.1].sub.a][[L.sup.2].sub.a] - [[w.sup.1].sub.a][[L.sup.1].sub.a]

[[c.sup.1].sub.a] [greater than] [C.sup.1] - [C.sup.2] + [[w.sup.1].sub.b][[L.sup.1].sub.b] - [[w.sup.1].sub.b][[L.sup.2].sub.b] + [[c.sup.2].sub.a]

[[c.sup.1].sub.a] [geq] 0 [[c.sup.1].sub.a] [leq] [C.sup.1].

There exists a [[c.sup.1].sub.a] such that that system is satisfied if and only if

[C.sup.1] - [C.sup.2] + [[w.sup.1].sub.b][[L.sup.1].sub.b] - [[w.sup.1].sub.b][[L.sup.2].sub.b] + [[c.sup.2].sub.a] [less than] [[c.sup.2].sub.a] + [[w.sup.1].sub.a][[L.sup.2].sub.a] - [[w.sup.1].sub.a][[L.sup.1].sub.a]

0 [less than] [[c.sup.2].sub.a] + [[w.sup.1].sub.a][[L.sup.2].sub.a] - [[w.sup.1].sub.a][[L.sup.1].sub.a]

[C.sup.1] - [C.sup.2] + [[w.sup.1].sub.b][[L.sup.1].sub.b] - [[w.sup.1].sub.b][[L.sup.2].sub.b] + [[c.sup.2].sub.a] [less than] [C.sup.1].

The first inequality can be rewritten

[C.sup.2] + [[w.sup.1].sub.a][[L.sup.2].sub.a] + [[w.sup.1].sub.b][[L.sup.2].sub.b] [greater than] [C.sup.1] + [[w.sup.1].sub.a][[L.sup.1].sub.a] + [[w.sup.1].sub.b][[L.sup.1].sub.b]

There are no unobservables in this inequality, so it is one of the testable restrictions of this system. Now to eliminate [[c.sup.2].sub.a] from the rest of the system, isolate the [[c.sup.2].sub.a]:

[[c.sup.2].sub.a] [greater than] [[w.sup.1].sub.a]([[L.sup.1].sub.a] - [[L.sup.2].sub.a])

[[c.sup.2].sub.a] [less than] [C.sup.2] + [[w.sup.1].sub.b]([[L.sup.2].sub.b] - [[L.sup.1].sub.b])

[[c.sup.2].sub.a] [geq] 0 [[c.sup.2].sub.a] [leq] [C.sup.2].

There exists a [[c.sup.2].sub.a] such that that system is satisfied if and only if

[[w.sup.1].sub.a]([[L.sup.1].sub.a] - [[L.sup.2].sub.a]) [less than] [C.sup.2] + [[w.sup.1].sub.b]([[L.sup.2].sub.b] - [[L.sup.1].sub.b])

[[w.sup.1].sub.a]([[L.sup.1].sub.a] - [[L.sup.2].sub.a]) [less than] [C.sup.2]

0 [less than] [C.sup.2] + [[w.sup.1].sub.b]([[L.sup.2].sub.b] - [[L.sup.1].sub.b]).

Note that the first inequality is implied by the first testable restriction. The system [[Phi].sub.1] can be satisfied if and only if

[C.sup.2] + [[w.sup.1].sub.a][[L.sup.2].sub.a] + [[w.sup.1].sub.b][[L.sup.2].sub.b] [greater than] [C.sup.1] + [[w.sup.1].sub.a][[L.sup.1].sub.a] + [[w.sup.1].sub.b][[L.sup.1].sub.b]

[C.sup.2] + [[w.sup.1].sub.a][[L.sup.2].sub.a] [greater than] [[w.sup.1].sub.a][[L.sup.1].sub.a]

[C.sup.2] + [[w.sup.1].sub.b][[L.sup.2].sub.b] [greater than] [[w.sup.1].sub.b][[L.sup.1].sub.b].

The testable restrictions of system [[Phi].sub.4] are derived in a similar way. They are

[C.sup.1] + [[w.sup.2].sub.a][[L.sup.1].sub.a] + [[w.sup.2].sub.b][[L.sup.1].sub.b] [greater than] [C.sup.2] + [[w.sup.2].sub.a][[L.sup.2].sub.a] + [[w.sup.2].sub.b][[L.sup.2].sub.b]

[C.sup.1] + [[w.sup.2].sub.a][[L.sup.1].sub.a] [greater than] [[w.sup.2].sub.a][[L.sup.2].sub.a]

[C.sup.1] + [[w.sup.2].sub.b][[L.sup.1].sub.b] [greater than] [[w.sup.2].sub.b][[L.sup.2].sub.b].

Fourier-Motzkin Elimination on [[Phi].sub.2]

First, we will eliminate [[c.sup.1].sub.a] from the system. Rewrite all the inequalities that involve [[c.sup.1].sub.a] with [[c.sup.1].sub.a] isolated:

[[c.sup.1].sub.a] [less than] [[c.sup.2].sub.a] + [[w.sup.1].sub.a][[L.sup.2].sub.a] - [[w.sup.1].sub.a][[L.sup.1].sub.a]

[[c.sup.1].sub.a] [less than] [C.sup.1] - [C.sup.2] + [[w.sup.2].sub.b][[L.sup.1].sub.b] - [[w.sup.2].sub.b][[L.sup.2].sub.b] + [[c.sup.2].sub.a]

[[c.sup.1].sub.a] [geq] 0 [[c.sup.1].sub.a] [leq] [C.sup.1].

There exists a [[c.sup.1].sub.a] such that that system is satisfied if and only if

0 [less than] [[c.sup.2].sub.a] + [[w.sup.1].sub.a][[L.sup.2].sub.a] - [[w.sup.1].sub.a][[L.sup.1].sub.a]

0 [less than] [C.sup.1] - [C.sup.2] + [[w.sup.2].sub.b][[L.sup.1].sub.b] - [[w.sup.2].sub.b][[L.sup.2].sub.b] + [[c.sup.2].sub.a].

Now to eliminate [[c.sup.2].sub.a] from the rest of the system, isolate the [[c.sup.2].sub.a];

[[c.sup.2].sub.a] [greater than] [[w.sup.1].sub.a]([[L.sup.1].sub.a] - [[L.sup.2].sub.a])

[[c.sup.2].sub.a] [greater than] [C.sup.2] - [C.sup.1] + [[w.sup.2].sub.b][[L.sup.2].sub.b] - [[w.sup.2].sub.b][[L.sup.1].sub.b]

[[c.sup.2].sub.a] [geq] 0 [[c.sup.2].sub.a] [leq] [C.sup.2].

There exists a [[c.sup.2].sub.a] such that that system is satisfied if and only if

[[w.sup.1].sub.a]([[L.sup.1].sub.a] - [[L.sup.2].sub.a]) [less than] [C.sup.2] [C.sup.2]-[C.sup.1] + [[w.sup.2].sub.b][[L.sup.2].sub.b] - [[w.sup.2].sub.b][[L.sup.1].sub.b] [less than] [C.sup.2].

The system [[Phi].sub.2] can be satisfied if and only if

[C.sup.2] + [[w.sup.1].sub.a][[L.sup.2].sub.a] [greater than] [[w.sup.1].sub.a][[L.sup.1].sub.a] [C.sup.1] + [[w.sup.2].sub.b][[L.sup.1].sub.b] [greater than] [[w.sup.2].sub.b][[L.sup.2].sub.b]

The testable restrictions of [[Phi].sub.3] are derived in a similar way. They are

[C.sup.1] + [[w.sup.2].sub.a][[L.sup.1].sub.a] [greater than] [[w.sup.2].sub.a][[L.sup.2].sub.a] [C.sup.2] + [[w.sup.1].sub.b][[L.sup.2].sub.b] [greater than] [[w.sup.1].sub.b][[L.sup.1].sub.b].
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