Pareto optimality in a Becker model of time allocation.

Holtman, A.G. ; Idson, Todd L. ; Qian Ding 等

I. Introduction

Though it is widely known that Gary Becker's (1965) classic model on the allocation of time does not result in unambiguous comparative static results (see Holtmann, 1972; DeSerpa, 1975; Pollak and Wachter, 1975), his model has been a major tool of analysis for empirical studies of consumer activities which require significant amounts of time, e.g., health care activities (see Holtmann, 1972; Vertinsky and Uyeno, 1973), church activities (see Azzi and Ehrenberg, 1975), and criminal activities (see Ehrlich, 1973). Nevertheless, it is sometimes suggested that this model does not result in a Pareto efficient allocation of resources (see Borland and Pulsinelli, 1989). In fact, there has been a paucity of studies of the welfare implications of this popular model. The issue of Pareto optimality arises in this model specifically because consumers base their choices on full costs-both time and money costs-with the result that marginal rates of substitution are not equated for different individuals (see Kraus, 1979; Borland and Pulsinelli, 1989).

Although it is true the marginal rates of substitution will tend to differ for each individual in a Becker model, Pareto optimality does not necessarily require that marginal rates of substitution are equal for all consumers. For example, in an economy with Samuelsonian public goods (see Samuelson, 1954), Pareto optimality requires that the sum of the marginal rates of substitution of public for private goods be equal to the marginal rate of transformation of public for private goods, thereby implying that marginal rates of substitution of public for private goods will not be equal among consumers. Nevertheless, the question still remains as to whether the Becker model is consistent with Pareto optimality. In this paper we will show that Becker's model does, in fact, fulfill the conditions for a Pareto optimal allocation.

II. The Model

Consider a traditional Becker household consumption model in which the consumer uses goods to produce commodities that generate utility. In this case, the consumer's problem is to maximize the following:

[Mathematical Expression Omitted] where [U.sup.1] is individual 1's utility function; the Q's are commodities produced according to the input-output technology

[Mathematical Expression Omitted]

[T.sub.1] and [T.sub.2], are the time used to produce commodities one and two respectively; [X.sub.1] and [X.sub.2] are the goods used to produce the commodities one and two respectively; [t.sub.1] and [t.sub.2] are the time input-output coefficients; [b.sub.1] and [b.sub.2] are the good input-output coefficients; T is total time available; w is the wage rate; [p.sub.1] and [p.sub.2] are the goods' prices; [Lambda] is a Lagrange multiplier and [T.sub.L] is work time. The constraint in (1) is derived from the time constraint T = [T.sub.L] + [T.sub.1] + [T.sub.2] and the commodity constraint [T.sub.L]W = [p.sub.1][X.sub.1] + [p.sub.2][X.sub.2]. The first-order conditions for a maximum of (1) are:

[Mathematical Expression Omitted]

Thus,

[Mathematical Expression Omitted]

This is the standard result, showing that the marginal rate of substitution of the commodities must be equal to the full price ratio for each consumer. To investigate the issue of Pareto optimality in a time allocation context, assume a welfare function of the form

[Mathematical Expression Omitted]

and a production possibility function

[Mathematical Expression Omitted].

In this case, superscripts refer to each of two consumers, though the model generalizes to n consumers and commodities. Subscripts refer to the two commodities and their respective inputs of time and goods (where the symbols are as defined earlier), where

[Mathematical Expressions Omitted]

Then, maximizing W subject to f, we have:

[Mathematical Expression Omitted]

From (3) we see that the marginal rate of substitution of commodities is not necessarily equal among consumers when welfare is maximized. Furthermore, when we recognize that some of the commodities being produced are attributes such as good health, this result is not surprising-different consumers are not likely to have either identical time prices or input-output coefficients.

Nevertheless, the central question is whether or not these results are consistent with equation (2) for all consumers: Does this system support a Pareto optimal allocation of resources? To answer this question, move the second term of the first two equations in (3) to the right hand side and divide the first equation in (3) by the second equation, giving:

[Mathematical Expression Omitted]

Now, from the constraints on (1) note that

[Mathematical Expressions Omitted]

Further, from f(. . .) = 0, we know that

[Mathematical Expression Omitted]

and [Delta] X.sub.1]/[Delta] X.sub.2] =

-([Delta]f/[Delta][X.sub.2])/([Delta]f/[Delta] X.sub.1]). Hence, we can rewrite (4) as:

[Mathematical Expression Omitted]

This last result is, of course, the same as we obtained in (2'), which indicates that the consumer's maximization of utility is consistent with Pareto optimality derived from (3). The result will hold for all consumers. In general, the t's and b's will differ among consumers, so that the marginal rates of substitution of Q's will rarely be equal among consumers. However, as we have shown, equality among the marginal rates of substitution is not a requirement for Pareto optimality.

III. Conclusions

This paper has investigated the Pareto optimality properties of Becker's time allocation model, and has found that the model is consistent with a Pareto efficient allocation. The key point in deriving this result is that we must consider all actual constraints when we are deriving the Pareto conditions for the economy. If we cannot substitute one person's time for another's in certain consumption activities, we must then recognize that constraint in deriving our Pareto conditions. However, we should recognize that input goods are not carefully defined in this general model and there may be more substitution possibilities than suggested by the model. For example, I can substitute a physician's time for my time in producing "good health." In our analysis, we have allowed wages and input prices to be equal among consumers, but, if there are differences in productivity among suppliers of labor, these should be reflected in the different wages and different marginal rates of transformation by type of consumer.

Of course, if there are imperfections in the input markets, these would violate the conditions for Pareto optimality. Differences in wages, of course, do not imply market failure when workers are not homogeneous. In the Becker model different wages, for example, would simply mean that marginal rates of transformation of labor for other inputs would vary among workers of different productivities. In any case, the Becker model will generate Pareto optimal results in an economy with efficient input or goods' markets.

References

Azzi, C., and R. Ehrenberg, "Household Allocation of Time and Church Attendance," The Journal of Political Economy, 83, February 1975:27-56.

Becker, G., "A Theory of the Allocation of Time," The Economic Journal, 75, September 1965:493-517.

Borland, M., and R. Pulsinelli, "Household Commodity Production and Social Harassment Costs," The Southern Economic Journal, 56, 2, October 1989:291-301.

DeSerpa, A. C., "On the Comparative Statics of Time Allocation," Canadian Journal of Economics, 8, February 1975:101-11.

Ehrlich, I., "Participation in Illegitimate Activities: A Theoretical and Empirical Investigation," The Journal of Political Economy, 81, May-June 1973:521-565.

Holtmann, A. G., "Prices, Time, and Technology in the Medical Care Market," The Journal of Human Resources, 7, Spring 1972:179-190.

Kraus, M., "On Pareto-Optimal Time Allocation," Economic Inquiry, 17, January 1979:142-45.

Pollak, R. A., and M. L. Wachter, "The Relevance of the Household Production Function and Its Implication for the Allocation of Time," Journal of Political Economy, 83, 1975.

Samuelson, P., "The Pure Theory of Public Expenditures," The Review of Economics and Statistics, 36, November 1954:387-89.

Vertinsky, I. and D. Uyeno, "Demand for Health Services and the Allocation of Time," Applied Economics, 5, 1973:249-60.