A theoretical and empirical analysis of family migration and household production: comment.

• Author: Roe, Brian
• Position: Response to Michael P. Shields and Gail M. Shields, Southern Economic Journal, p. 768, April 1993

In a recent paper in this Journal [2], Michael P. Shields and Gall M. Shields (SS) model migration in a utility maximization framework in which the household derives utility from commodities produced via a household production function. The SS approach considers income contributions of both the husband and wife in a traditional household structure and thus is well-suited to study how labor market conditions for both spouses affect migration decisions. However, SS present two key comparative static results linking changes in the wife's wage to the family's migration decision which are incorrect. These mathematical mistakes have two important implications. First, they may change the sign of the SS result on the decision to migrate. Second, they cause SS to overlook competing income and substitution effects of a change in the wife's wage. In this comment I correct the comparative static error, and then identify and interpret the income and substitution effects. Finally, I discuss the importance of the wife's time in household production and the size of the wife's full income in the migration decision.

1. The SS Household Migration Model

   Following SS's notation and equation numbering, the household's utility maximization problem is:

   maximize U = U([Z.sub.1], [Z.sub.2],..., [Z.sub.n]) (1a)

   subject to

   B = Y + w(T - [summation of] ([t.sub.i])) where i=1 to n = [summation of] ([p.sub.j][X.sub.j]) where j=1 to m + hH (1b)

   and

   [Z.sub.i] = [f.sub.i]([X.sub.1i],[X.sub.2i],..., [X.sub.mi]; [t.sub.i][where][Theta], L), for I = 1, 2,..., n, (1c)

   where

   [X.sub.j] = [summation of] ([X.sub.ji]) where i=1 to n, for j = 1, 2,..., m. (1d)

   U is family utility, [Z.sub.i] is the ith commodity, [f.sub.i] is the household production function for the ith commodity, [X.sub.ji] is the quantity of good j used to produce commodity i, [p.sub.j] is the price of good j, [t.sub.i] is the wife's time used to produce commodity i, w is the market wage rate for the wife, T is the wife's time endowment, Y is the husband's income including any exogenous family income, B is the family budget, h is the price of housing services, H is housing services, [Theta] is a vector of household characteristics, and L is a vector of relevant attributes of the family's current location.

   Assuming that the wife does not specialize in household production and that the household production functions are linearly homogeneous, SS collapse the constraints (1b) and (1c) into a single, linear constraint,

   I...