SHADOW PRICES 681 where W* is the shadow price, h is the hours of work, or alternatively, the amount of time the wife does not have available for her nonmarket activities, Wm is the wage of the husband, P is a vector of goods prices, A is the asset income of the household, and Z is a vector of constraints which arise from previous economic choices or chance events, such as the number of children, the education of the family members, and the state of household technology. W* is the value the household places on marginal units of the wife's time in production and con-sumption.

The formal derivation of this function is relegated to Appendix 1. There we establish that if the ordinary labor supply function is a positive monotonic function of wave rates, equation (1) may be derived in a straightforward fashion, with the range of that function constituting the domain of the marginal valuation function. We further establish that equation (1) possesses a continuous partial derivative with respect to h at h= 0 if the household preference function is defined for quantities of leisure in excess of the total time currently available to the wife. Historical time-series and cross-section studies suggest that there is a monotonic positive relationship between wage rates and labor supply for married women (Mincer [18], Ashenfelter and Heckman [3]) so that excluding the" backward bending" case is not objectionable, at least for an analysis of this demographic group. However, in the empirical work presented below,'the hypothesis of a positive relationship is tested rather than directly imposed on the data. While there may be strong intuitive feelings about the direction of the relationship between W* and other variables, the assumption of utility maximization-yields no information on these signs. Nonttheless, previous empirical analysis suggests that children tend to increase W* a-nd that this effect is more pronounced the younger their ages. If leisure is a normal good, it is easy to show that increments in net worth raise W*. Michael [17] argues that the education of the wife raises the wife's efficiency in producing domestic services. Thus, education might affect W* but the direction of that effect is uncertain. The determinants of the market wage rate (W) are better known. Education and years of labor force experience are expected to increase the wage (Mincer [19]). The market wage function may thus be written as