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Weekly UpdatesThe shadow wage and shadow income are the key variables in estimating the labor supply function. Thus, having an appropriate measurement of these variables is crucial to the estimation process. Under a perfect market assumption, the observed market wage is identical to the shadow wage, so it can be used as an appropriate measure of the shadow wage. However, this assumption is usually violated and any failure (or imperfection) in the labor, farm input, or credit market can make the shadow wage deviate from the market wage (Singh, Squire, and Strauss 1986; Thorbecke 1993). As a result of market failures, the market wage is no longer an appropriate measure of the shadow wage. Meanwhile, the measurement of the shadow income is complicated by the household output that cannot be measured directly from the data, such as provision of nutritious meals and a clean living environment.
Jacoby (1993) and Skoufias (1994) are the first to propose a method to derive the shadow wage under market failures. They notice a very important point that the shadow wage is identical to the marginal product of labor (MPL) on the farm regardless of market failures, so they first estimate the production function on the farm and then calculate the corresponding MPL. Using this MPL as the shadow wage, the labor supply function can be estimated. Because of its robustness to market failures, the method has become a standard approach in the literature to estimate the shadow wage and the labor supply function (Lambert and Magnac 1994; Sonoda and Maruyama 1999; Abdulai and Regmin 2000; Seshan 2006; Barrett, Sherlund, and Adesina 2008). (1)
However, there are some concerns with this estimation method. Jacoby raises the first one in his discussion of the estimation of the production function on the farm. Due to data limitations, the values instead of quantities of farm inputs and outputs are used in estimating the production function, so any price variation across regions can bias the estimation. This data limitation is not specific to the sample used in Jacoby's paper but also applies to other papers as well. (2) Also, Jacoby mentions the difficulties of finding convincing instruments to address the well-known endogeneity problem during the estimation of the production function.
The second concern is raised by Skoufias with respect to bias in calculating the shadow income, since the method does not include household outputs. As the labor spent in producing household outputs usually accounts for more than 20% of the total labor supply, the value from household outputs can be a
Kien T. Le is researcher at the University of Virginia Weldon Cooper Center for Public Service.
This article is one chapter in the author's Ph.D. dissertation (2008) submitted to the University of Virginia.
The author is grateful to John McLaren, John Pepper, and Sanjay Jain for their mentorship. The author is indebted to Christopher Barrett, Jeffrey H. Dorfman, and three anonymous reviewers for insightful comments and helpful suggestions. The author is responsible for any remaining errors.
(1) Some articles use the method to estimate the shadow wage only.
(2) The need to sum different types of farm inputs and outputs is another reason for using value instead of quantity in the production function.
significant component in shadow income. (3) The final concern is about the calculation of the MPL and shadow income based on the fitted output from the regression of the production function. Because farmers do not observe the final output when they make decisions about labor supply, their decisions should be based on the expected output, which can be different from the fitted output.
This article tries to address these concerns by proposing a different method to derive the shadow wage and shadow income. The method is still based on Jacoby and Skoufias' important observation that the shadow wage is the MPL, regardless of the market failure. (4) In addition, the method uses the following observation: the shadow wage is not any MPL but the one at the optimal point of both farm and household production functions. Accordingly, under certain assumptions on the functional form of the production function, we can derive both the shadow wage and household outputs without estimating the production function. These functional form assumptions are not restrictive since they result in a production function that is more flexible than the Cobb-Douglas function.
The current method from the literature and the one outlined in this article are used to estimate the labor supply function for a sample of farmers in Vietnam, where failures in the labor market are pervasive due to the legacy of central planning. The results are compared in terms of their consistency with the theory, which states that the coefficient on the shadow income is negative to ensure leisure as a normal good and the coefficient on the shadow wage is positive to ensure a forward sloping labor supply function.
Theoretical Model
The theoretical model is based on the standard time allocation model. (5) A farm household with two persons, male and female, maximizes utility function defined over leisure ([l.sub.m], [l.sub.f]), and consumption (C); U(C, [l.sub.m], [l.sub.f]; A), where A is a vector of preference shifters (e.g., number of children, number of adults, unobserved working preferences, etc.), and C includes goods that can be either purchased in the market (c) or produced at home (v): C = c + v. (6) The labor supply for each person ([h.sub.i]) is defined as the total stock of time (T) minus leisure and is allocated to farm labor ([L.sub.i]), market labor ([M.sub.i]), and household labor ([N.sub.i]): [h.sub.i] = [L.sub.i] + [N.sub.i] + [M.sub.i], where i = {m, f}.
The household production function on the farm is Q([L.sub.m], [L.sub.f], z; F), where z is a vector of variable inputs (e.g., hired labor, fertilizer, seed, etc.), and F is a vector of quasi-fixed inputs (e.g., land, farm equipment, etc.). The production function for the goods produced at home is V([N.sub.m], [N.sub.f]; K), where K is a vector of other inputs beside household labor (e.g., having electricity, having a refrigerator).
The market failure is introduced into the model as a market labor constraint: [M.sub.i] [less than or equal to] [H.sub.i], where [H.sub.i] is the maximum number of hours a farmer can work in the labor market. This type of failure is chosen in this article since it is cited in several studies about agricultural households and particularly important in Vietnam due to the legacy of central planning. (7) It is not difficult to show that the model is still applicable in markets with other types of failures such as credit constraint, transactions or search costs, and work location preferences.
The household maximization problem can be summarized as follows:
(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where p is the farm output price, [p.sub.z] is a vector of input prices, [w.sub.m] and [w.sub.f] are the market wages for male and female, Y is exogenous income (e.g., gifts from relatives).
The first order conditions (FOCs) for this problem are
(2) [w.sup.*.sub.i] = p[partial derivative]Q/[partial derivative][L.sub.i] = [partial derivative]V/[partial derivative][N.sub.i] = [w.sub.i] if [M.sub.i] < [H.sub.i]
(3) [w.sup.*.sub.i] = p[partial derivative]Q/[partial derivative][L.sub.i] = [partial derivative]V/[partial derivative][N.sub.i] < [w.sub.i] if [M.sub.i] = [H.sub.i]
where [w.sup.*.sub.i] is called the shadow wage, the marginal rate of substitution between leisure and consumption: [partial derivative]U/[partial derivative][l.sub.i]/[partial derivative]U/[partial derivative]C. In these FOCs, p[partial derivative]Q/[partial derivative][L.sub.i] is the value of the MPL. However, previous studies in the literature normalize p to 1, so this term is called the MPL. Although p is not normalized in this article, this term is still called the MPL to be consistent with the literature.
The FOC in equation (2) states that, under the perfect market assumption (i.e., the labor market constraint is not binding: [M.sub.i] < [H.sub.i]), farmers will allocate labor to three activities (farm, household, and market work) in a way to equalize the marginal return from all activities (i.e., the equalization of the MPL on the farm, MPL of the household work, and the market wage). Meanwhile, under the market failure (i.e., the labor market constraint is binding: [M.sub.i] = [H.sub.i]), the FOC in equation (3) states that farmers can equalize the marginal return from farm and household work but not the return from the market work.
Using the shadow wage, the nonlinear budget constraint can be replaced with an artificial linear constraint which induces the household to arrive at the same optimal choice (Hall 1973; Hausman 1981). Interested readers can look at Skoufias (1994) for a graphic presentation of this idea. The household maximization problem under the linear budget constraint is:
(4) Max U(C, T - [h.sub.m], T - [h.sub.m], T - [h.sub.f]; A)
subject to: C - [w.sup.*.sub.m][h.sub.m] - [w.sup.*.sub.f][h.sub.f] = [y.sup.*]
where [y.sup.*] is shadow income:
(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Solving this maximization problem, we have the optimal labor supply function
(6) [h.sub.i] = [h.sub.i] ([w.sup.*.sub.m], [w.sup.*.sub.f], [y.sup.*]; A).
Empirical Model
The key variables to estimate the labor supply are the shadow wages ([w.sup.*.sub.m], [w.sup.*.sub.f]) and income ([y.sup.*]) and estimation methods differ in terms of how to calculate these variables. Under the perfect market assumption ([M.sub.i] < [H.sub.i]), the shadow wage can be calculated from the observed market wage ([w.sub.i]) as in equation (2), but this is no longer correct in the event of market failure ([M.sub.i] = [H.sub.i]) as in equation (3). Jacoby and Skoufias notice an important point from both equations (2) and (3) that the shadow wage can always be calculated from the MPL (p[partial derivative]Q/[partial derivative] [L.sub.i]) regardless of market failures. This point, although simple, makes the calculation of the shadow wage and the subsequent estimation of the labor supply function robust to market failures. The following will discuss this estimation.
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