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

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.

I. 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 = [Pi]Z (2)

where I = Y + wT. This is full family income. [Pi] is a vector of implicit prices of the vector of household commodities, Z. Assuming the family is currently in their optimal location, we can imagine that there exists a next best location which will provide the family with a full income of I[prime] and a vector of implicit commodity prices of [Pi][prime]. Location rent, R, is then defined as the lump sum addition to I[prime] that would equate the family's commodity consumption at the two locations:

R = I([Pi][prime]/[Pi]) - I[prime] (3)

where [Pi] and [Pi][prime] are now scalars representing implicit commodity prices at the family's current and best alternative locations. SS connect rent to migration by assuming some general relationship:

M = f(R) (4)

where M is migration. They implicitly assume that [Delta]f/[Delta]R [less than] 0, or that as the rent accruing to a family's current location increases they are less likely to migrate.

To find the effect of key parameters on location rent and to provide expected signs for their subsequent empirical analysis, SS provide a simple example where the family produces a single commodity, Z, using housing services, H, a single purchased good, X, and the wife's time, t, in a Cobb-Douglas production function:

Z = A[H.sup.[Alpha]][X.sup.[Beta]][t.sup.1 - [Alpha] - [Beta]] (5)

where A is a technology shifter. After some manipulation they derive the expression:

[Pi][prime]/[Pi] = (A/A[prime])[(h[prime]/h).sup.[Alpha]][(p[prime]/p).sup.[Beta]][( w[prime]/w).sup.1 - [Alpha] - [Beta]] (6)

where characters with ([prime]) represent parameter values at the best alternative. Substituting this expression into (3) and writing out the expression for full income at the current (I) and alternative (I[prime]) locations yields a detailed expression of location rent for this example:

R = [Y + wT](A/A[prime])[(h[prime]/h).sup.[Alpha]][(p[prime]/p).sup.[Beta] ][(w[prime]/w).sup.1 - [Alpha] - [Beta]] - [Y[prime] + w[prime]T]. (6a)

II. The Impact of the Wife's Wage on Location Rent

The SS paper centers on the tied-mover hypothesis [1] which suggests that double income families are less likely to move than single earner families. As SS point out, exploration of this hypothesis should include ". . not only considerations of the wife's earnings but also considerations of her household production as factors in the family's migration decisions" [2, 768]. To analyze the effect of the wife's labor market opportunities on migration, SS derive the partial derivatives of (6a) with respect to w and w[prime]. They present:

[Delta]R/[Delta]w = (T + I(1 - [Alpha] - [Beta]))[Pi][prime]/[Pi] [greater than] 0, (7)

and

[Delta]R/[Delta]w[prime] = -I[prime](1 - [Alpha] - [Beta]))[Pi][prime]/[Pi] - T [less than] 0. (8)

The interpretation is that rent accruing to the family's current location is unambiguously increasing in wife's wage at the current location and unambiguously decreasing in the wife's wage at the alternative location, respectively. SS use the result in (7) as the a priori sign expectation for empirical results.

The derivatives in the paper, however, are incorrect. The correct partial derivatives are:

[Delta]R/[Delta]w = T([Pi][prime]/[Pi]) - (I/w)(1 - [Alpha] - [Beta])([Pi][prime]/[Pi]), (7a)

and

[Delta]R/[Delta]w[prime] = -T + (I/w[prime])(1 - [Alpha] - [Beta])([Pi][prime]/[Pi]), (8a)

which are clearly different from SS's (7) and (8) respectively. SS have made a sign error for one of the terms and have neglected the w and w[prime] in the denominators. While SS get income and substitution effects that reinforce each other, the true income and substitution effect go in opposite directions. That is, the effect of a change in the wage available to the wife at the current and alternative location cannot be signed a priori in this theoretical framework.

The first term in (7a), T([Pi][prime]/[Pi]), is an income effect. An increase in the wage at the current location increases the potential full income of the family by the wife's time endowment. This is the consideration of wife's earnings which SS mention but fail to connect to the theoretical model. The effect is always positive but is tempered if the implicit price of the household commodity is less at the alternative location than at the current location ([Pi][prime]/[Pi] [less than] 1).

The second term, (I/w)(1 - [Alpha] - [Beta])([Pi][prime]/[Pi]), is the substitution effect and is always negative. In the household production framework, the family earns a full income of Y + wT, but must "buy back" the wife's time which is used in household production. As the wife's time becomes more valuable due to an increase in w, it becomes more expensive to buy back this time and, thus, the implicit price of the commodity at the current location increases. SS discuss the importance of the wife's household production in the family migration discussion, but do not derive it.

In order for [Delta]R/[Delta]w [greater than] 0 and, therefore, for probability of migration to decrease with increases in w, the following must hold:

[Delta]R/[Delta]w [greater than] 0 [implies] ([Alpha] + [Beta])wT [greater than] (1 - [Alpha] - [Beta])Y. (7b)

To see how likely it is that the corrected partial derivative is positive and, thus, in agreement with the positive effect assumed by SS, consider three examples. First, assume the wife's time is the major factor in commodity production (i.e. [Alpha] + [Beta] [approaches] 0). Then Y, the husband's income plus exogenous family income, would need to approach infinity in order for increases in the wife's wage to increase the locational rent and thus decrease the probability of moving. In this case the substitution effect dominates and the probability of migration increases with increases in the wife's current wage. This deviates from the SS result. Next, assume the commodity produced requires very little of the wife's time (i.e. [Alpha] + [Beta] [approaches] 1). Then the income effect dominates and [Delta]R/[Delta]w is positive unless the wife's wage approaches infinity. In this case an increase in the wife's current wage will result in a decrease in the probability of migration. Finally, assume all three factors in commodity production contribute equally (i.e. [Alpha] = [Beta] = 1/3). The probability of migration decreases ([Delta]R/[Delta]w [greater than] 0) only if the wife's full income potential, wT, is greater than 1/2Y. This result depends upon the wife's earning potential relative to income from the husband and exogenous sources.

The expected sign derived from the theoretical model is clearly sensitive to several factors including labor's share in commodity production, (1 - [Alpha] - [Beta]), the wife's wage, and income from the husband and exogenous sources. It is difficult to interpret these theoretical results because the household production factor shares are not empirically recoverable. Even if they were recoverable, they are likely to vary significantly across commodities. While it seems intuitively appealing to say that an increase in a wife's wage at the current location will decrease the probability of migration, the framework adopted by SS does not seem appropriate for capturing this intuition unambiguously. It therefore remains an empirical issue. SS's estimation results suggest that an increase in the wife's wage at the current location significantly decreases the probability of family migration. In light of the corrected theoretical result, we may interpret this empirical result as evidence that the full income effect of a change in the wife's wage dominates the substitution effect.

Brian Roe University of Maryland College Park, Maryland 

References

1. Mincer, Jacob, "Family Migration Decisions." Journal of Political Economy, October, 1978, 749-73.

2. Shields, Gail M. and Michael P. Shields, "A Theoretical and Empirical Analysis of Family Migration and Household Production: U.S. 1980-1985." Southern Economic Journal, April 1993, 768-82.