# Gronau’s Model

Gronau (1973) assumed that the offered wage W° is given to each housewife independently of hours worked (H), rather than as a schedule W°(H). Given W°, a housewife maximizes her utility function U(C, X) subject to X = W°H + V and C + H = T, where C is time spent at home for childcare, X represents all other goods, T is total available time, and V is other income. Thus a housewife does not work if (10.7.10) and works if the inequality in (10.7.10) is reversed. If she works, the hours of work H and the actual wage rate W must be such that

Gronau called the left-hand side of (10.7.10) the housewife’s value of time or, more commonly, the reservation wage, denoted W1.12

Assuming that both W° and Wx can be written as linear combinations of independent variables plus error terms, his model may be statistically de­scribed as follows: Щ = ХгА + «2/

W] — z|a + Vj

W, = Wf if Wf > W]

= 0 if WJ, /- 1,2,… , л,

where (u2i, vt) are i. i.d. drawings from a bivariate normal distribution with zero mean, variances and a, and covariance am. Thus the model can be written in the form of( 10.7.1) by putting WJ — W] = yftand FFj^yJ-.Note that H (hours worked) is not explained by this statistical model although it is

Tobit Models 3,89*

•t’ »

determined by Gronau’s theoretical model. A statistical model explaining as well as W was developed by Heckman (1974) and will be discussed,’^ Section 10.8.2.

Because the model (10.7.11) can be transformed into the form (10.7.1) ш» such a way that the parameters of (10.7.11) can be determined from the parameters of (10.7.1), all the parameters of the model are identifiable except V( Wf — Wf), which can be set equal to 1 without loss of generality. If, how­ever, at least one element of x2l is not included in zt, all the parameters are identifiable.13 They can be estimated by the MLE or Heckman’s two-step estimator by procedures described in Section 10.7.1. We can also use the probit MLE (the first step of Heckman’s two-step) to estimate a certain subset of the parameters.14