A theoretical and empirical analysis of family migration and household production: U.S. 1980-1985.

• Author: Shields, Michael P.

1. Introduction

   Migration studies have usually focussed on individual persons or workers. There has, however, been some interest in family migration. While single individuals in the labor market only have to worry about their own income, families, where both the husband and wife may be in the labor market, need to be concerned with the income of both spouses. Hence, double income families may be less likely to move than single income families. Jacob Mincer |26~ develops and explores this hypothesis, to be called the tied-mover hypothesis.(1) The tied-mover hypothesis has recently been extended to include not only considerations of the wife's earnings but also considerations of her household production as factors in the family's migration decisions |38~. In this paper, the research on family migration will be extended in three ways. A formal model of household production and family migration will be developed and used to generate testable hypotheses about family migration. Variables, which otherwise would not have a clear economic interpretation, are incorporated into the model. The model will then be estimated and some policy implications will be discussed. One policy implication concerns the efficiency of the labor market. Labor market efficiency needs to be judged not only in terms of the efficiency of the job search of single individuals but also in terms of the efficiency of the joint job search of double income families.

   This paper is organized into five sections. First, a comparative static household production model of locational choice is developed.(2) Next, this model of locational choice is used to develop a theoretical model of family migration. Third, the data and empirical model are discussed. Fourth, the empirical results are reported and interpreted in terms of three different types of migration. Estimates of the impact of selected variables on the distance of the move are also considered. The final section summarizes the general findings and offers some policy implications.

2. A Model of Locational Choice

   In this section, a comparative static model of family migration is developed using a Becker-Lancaster household production approach |2; 18~. The family will be assumed to live at its optimum location. The technology of the household and market variables, such as prices, the husband's income and the wife's wage rate, will be assumed to depend upon the family's location. The family moves when these location-specific variables change in a way which makes the current location no longer optimal. We will see, in section V, that this approach is capable of explaining the patterns of family migration which are observed.

   In a standard household production model, the household receives utility directly from the commodities which it consumes. Commodities are produced from inputs which consist of goods purchased in the market and the household's own time. A household maximizes its utility from consuming commodities subject to a linear income constraint and the household's technology. This model can be applied to migration by introducing, into the household's production functions, characteristics of family members and of the location of the household. Consider an example where a household, with both a husband and a wife, maximizes

   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~;H;|t.sub.i~/|Thet a~,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 of the wife, T is the total time the wife has at her disposal, Y is the husband's income which includes non-wage 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.(3) Note that some household characteristics such as education will influence both the household production function, and the earnings variables, w and Y.(4)

   Many of the exogenous variables in the above constraints depend upon the family's location and will change if the family moves to another location. Location will influence family utility in two ways. First, market prices, housing costs, the husband's income and the wife's wage rate depend upon location. Second, location will have a direct effect on household production possibilities because elements in L, which can differ from one location to the next, are arguments in the production functions of the household. Included in L are such factors as climate, proximity to friends and relatives, and the availability of cultural, recreational and educational amenities. Note that the impact that location has on Y, w, and production possibilities will be affected by household characteristics, |Theta~, such as marital status, the number of children, the ages of family members, and the education of family members.

   If the production functions are linearly homogeneous and the wife does not completely specialize in household production, then the optimal production of commodities implies that the constraints in (1) can be combined into a single, linear constraint,

   I = |Pi~Z, (2)

   where I = Y + wT is called full income, |Pi~ is a vector of implicit (shadow) prices of commodities and Z is a vector of commodities.(5) The shadow prices depend upon Y, w, h, |Theta~, L, and a vector of market prices, p.

   In comparing locations, families, which are living at their optimal location, receive rent in the sense that there is some amount of money which would be just sufficient to induce them to move |40~.(6) This implicit rent will be called locational rent. To more concretely define locational rent, suppose there are only two locations. Let I and I|prime~ be full income at each location, and let |Pi~ and |Pi~|prime~ be scalars representing the cost of commodity production at each location. Since locational rent would equate real commodity consumption at both locations, locational rent can be written as

   R = I(|Pi~|prime~/|Pi~) - I|prime~, (3)

   where R is locational rent.

   To illustrate the role locational rent plays in locational choice, consider an example where there are two periods of time. At the beginning of the first period, the family is assumed to be living at its optimal location. Hence, initial locational rent is positive, given the vector of exogenous variables governing the family's choice for that period. At the end of the period, the family will face a new vector of exogenous variables and, hence, would receive a new level of locational rent. If this new level of locational rent is negative, the family moves.

   A basic assumption of the migration model developed here is that the larger the locational rent, the less likely any given change in the family's economic situation will be of sufficient magnitude to cause a move. For example, an increase in the income that the husband could earn at another location will reduce locational rent but will not necessarily cause the family to move. The increase in income must exceed the original level of locational rent if the family is to move.(7)

3. The Migration Model

   Since household migration is assumed to be a function of locational rent, where locational rent is a function of market variables, household characteristics, and locational characteristics, the impact that these variables have on migration can be discussed in terms of how they affect locational rent. The basic underlying model is

   M = f(R), (4)

   where M is the probability that the family will move by the end of a given period and R is locational rent at the beginning of the period.

   It is by no means straight forward to estimate equation (4). Three difficulties arise. First, there are more than two locations. Second, the market variables for the family at an alternative location, denoted as Y|prime~, p|prime~, and w|prime~, are not known to the investigator. Third, the model is intrinsically nonlinear. The model does, however, have some clear implications as to the direction of the impact which the known variables have on migration.

   The direction of this impact can be found by differentiating R with respect to the relevant variables. To do so, consider a simple example where there is only one aggregate commodity produced by a Cobb-Douglas production function

   Z = A|H.sup.|Alpha~~|X.sup.|Beta~~|t.sup.1-|Alpha~-|Beta~~ (5)

   with inputs of market goods, the wife's time, and housing.(8) Furthermore, assume there are only two locations; the optimal current location and a suboptimal alternative location. Finally, to simplify the discussion of regional differences in household technology, assume that the only difference in these technologies is the value of A or A|prime~, where A|prime~ is the value of A at the alternative location.(9) Given this Cobb-Douglas technology, the ratio of shadow prices in equation (3) is

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

   The derivation of equation (6) is shown in Appendix A.

   The partial derivatives of R with respect to each relevant variable can now be found. The relevant variables include market variables at both locations such as the husband's income, Y and Y|prime~, the wife's wage rate, w and w|prime~, and housing prices, h and h|prime~; family characteristics such as the husband's age, the wife's age, the education of the wife, and the education of the husband; and regional amenities such as whether there are close...