We consider the participation and supply decisions in the context of traditional probit and Tobit models applied to household production data. For each household, i, i = 1, 2, ..., N, assume that the decision to participate, gi = 1 (if participation is observed and gi = 0 otherwise) is conditioned by a vector of household-specific covariates, xi, so that the decision rule is to participate when the utility of doing so, say, Ui(xi) exceeds the utility of not doing so, say, Vi(xi), which is also the utility reaped in return for resources xi (a k-vector of household characteristics) allocated to some alternative, feasible, enterprise. Taking Taylor-series expansions of these two utility functions around the point xi = 0, yields the linear model, gi = 1 if xi g ³ xi d, gi = 0 if xi g < xi d, where g and d are k-vectors of first-order effects depicting the impacts on the two utilities of changes in the levels of the various resources. Subtracting the left-hand-side from both sides of the inequalities, equating the result to a latent variable, zi, and permitting the equality to hold with error, ui, we are left with the expression:
zi = xi a + ui, zi ³ 0 if yi = 1, zi < 0, otherwise (1)
Here a º g – d measures the difference in allocating resources to either enterprise. By assuming that the errors are normally distributed, ui ~ N(0, 1), the familiar probit specification evolves as a reasonable, linear representation of the participation decision. Here, the error variance is constrained to equal one to overcome an identification problem arising due to the fact that the latent-variable specification in equation (1) is neither scale nor location independent.
Supply decisions are modelled in a similar way. We assume that the quantity supplied on the market is a linear function of another set of household characteristics, which may be the same as the set represented by the covariates xi, above. Specifically, using vi, i = 1, 2, ..., N to denote quantities, supply is specified as:
qi = xi b + ui (2)
Unlike the latent specification in the probit model, the dependent variable in (2) takes on positive and zero values. When a zero value is observed, we assume this to imply that the household in question, rather than possessing an excess of the marketable product, actually has a demand for the commodity (that is, a negative supply). Hence, sales quantities are left-censored at zero. This simple observation is developed further in Figure 1.
Figure 1. Utility-maximising sales values and distance-to-market latency.
Figure 1 depicts the utility-maximising household-supply decision. Utility (which is latent or unobservable) is depicted on the vertical axis and the potential sales quantity is depicted on the horizontal axis. For two households (households i and j) one household maximises utility by producing a positive sales quantity (qj) whereas the second finds utility maximised in the negative quadrant over the supply quantity (qi). Unlike the first household, the second household's implicit supply quantity is unobserved and latent. The quantity vi, in Figure 1, is used to represent this latent value. This value is very important for policy purposes because it provides a simple and highly intuitive quantity with which to measure a household's distance from market (di). As such, the values, vi = di, for i Î(the censor set) c º {i | qi = 0}, are an important part of the estimation exercise. In the section that follows we show how they can be used to simplify the estimation problems arising due to censoring in the sales data and latency arising in the probit regression.
