A system comparison approach to distinguish two nonseparable and nonnested agricultural household models.

Numerous empirical studies have emphasized the importance of nonseparable agricultural household models (AHM), in which a farm household simultaneously determines its production organization with its consumption choice (Bardhan and Udry 1999). Those studies devoted their efforts exclusively to showing the superiority of a nonseparable model to the corresponding separable one. Furthermore, they were able to relate the separable model to a special case of the nonseparable one. For example, Jacoby (1993) tested a relation by which the shadow wage is equal to the market wage in the separable model. Benjamin (1992) tested a relation by which the demand for farm labor is unresponsive to changes in household composition in the separable model. Using that simple relation, they used a nested test to make a partial comparison of the two models. (1)

However, no studies have compared two nonseparable models to show that one is superior to the other. This consequence arises probably because we cannot expect a simple relation in comparing two nonseparable models. Instead, we will observe a less familiar, nonnested relation, which can be inferred by comparing the following two common hypotheses, with special attention paid to labor supply functions and shadow or "internal wages" (Sonoda and Maruyama 1999).

One hypothesis, corresponding to heterogeneous labor supply, is denoted as "HET." Under this hypothesis, members of a farm household confront different disutilities from working on and off the farm: they supply heterogeneous farm and nonfarm labor. In addition they freely supply their time to the market for nonfarm labor, but the household does not demand nonfamily farm workers and operates its farm by self-employment. Lopez (1984) tested this hypothesis for Canadian farm households and found the associated nonseparable AHM superior to the corresponding separable one.

The other hypothesis corresponds to a restricted labor market and is denoted "RES." Under this hypothesis members of a farm household are indifferent to working on and off the farm; they supply a single type of total labor (the sum of farm and nonfarm labor). In addition the household operates its farm by self-employment just as under the HET hypothesis. But its members face restricted hours of nonfarm work because nonfarm employers offer a higher than equilibrium wage to use the resulting excess supply of labor as a worker discipline device. Sonoda and Maruyama (1999) tested it for farm households in Japan and found a similar result to Lopez (1984).

Under the HET hypothesis, the household has separate supply functions of farm and nonfarm labor with their respective own wages--an internal wage for farm labor and the market wage for nonfarm labor. In this case we find the internal wage to be relevant to the household simply because it operates its farm by self-employment. Under the RES hypothesis, the household has a single supply function of total labor with a single own wage--another internal wage for total labor. Now we find the internal wage to be relevant to the household, partly because it operates its farm by self-employment and partly because it faces restricted hours of nonfarm work. In this way we face serious inconvenience in comparing AHM under the two hypotheses: they include distinct internal wages and have different numbers of labor supply functions.

For appropriate comparison of those models, it is necessary to apply a nonnested test to determine the distinct internal wages in the two models. Furthermore, comparison of AHM under the HET and RES hypotheses is practical for evaluating the behavior of Japanese rice-farming households, as explained below. For these reasons this study demonstrates a method of distinguishing nonseparable AHM under the two hypotheses. In addition this paper provides an economic reason why we need to distinguish between them in terms of their comparative statics analysis.

We address this issue using data from Japanese rice-farming households during 1982-1991. Sonoda and Maruyama (1999) used the same data to find AHM under the RES hypothesis to be a better model than the corresponding separable AHM. However, the following observations suggest further support for AHM under the HET hypothesis.

During Japan's period of high economic growth (from mid 1950s to early 1970s), opportunities for nonfarm employment increased in and near rural areas. In addition mechanized systems in rice farming had been widely adopted by the early 1980s. According to Hayami (1986), rice-farming households have adopted a division of labor within the household in adapting to these situations: adult males primarily work at nonfarm firms, whereas housewives and elderly household members play a major role in farming. (2) In addition their small paddy field (about one hectare on average) allows them to operate their farm through self-employment. (3) This situation seems to be well described by specialization within the household: one member works off the farm for a market wage, and others work on the farm through self-employment. In this case there is no reason for the market wage (of one person) to be equal to the internal wage (of other family members).

The next section compares optimality conditions for AHM under the HET and RES hypotheses and explains the importance of their distinctions. The third section introduces a system comparison approach to distinguish nonseparable AHM under them. The fourth section applies the framework to data of Japanese rice-farming households and reveals the HET hypothesis to be better. That section also presents a comparison of the respective elasticities of the internal wages and quantity variables under the two hypotheses. The final section presents salient conclusions of this study.

Comparison of AHM under the HET and RES Hypotheses

Japanese rice-farming households operate their farms on small paddy fields; many of them depend solely on workers from their own families. For this reason we assume that a farm household does not hire nonfamily workers.

We first introduce AHM under the HET hypothesis. A farm household devotes Lf hours to its own farm and supplies [L.sub.m] hours to market work at nonfarm firms. A distinctive assumption under HET is heterogeneity of the two types of time: (4)

(1) u = u(C, [t.sub.f], [t.sub.m], G), [t.sub.f] [equivalent to] [T.sub.f] - [L.sub.f] > 0, [t.sub.m] = [T.sub.m] - [L.sub.m] >0

where C and G, respectively, denote the amount of purchased consumption commodities and the vector of household characteristics; [T.sub.f] and [t.sub.f] ([T.sub.m] and [t.sub.m]), respectively, represent the endowed time and leisure hours of farm (nonfarm) workers.

Another distinctive assumption under HET is that the farm household can freely supply its time to nonfarm firms at the market wage w. When it also uses competitive markets for consumption and farm commodities and variable production factors other than family labor (seed and seedlings, fertilizers, feed, agricultural chemicals, fuel, light, heat, and processing materials), the household faces the following budget constraint:

(2) rC = pX - qF + w[L.sub.m] + V

where X and F, respectively, denote the amounts of farm commodity and variable inputs other than labor. Also, p, q, and r, respectively, denote the market prices of farm commodity, other inputs and consumption commodities; V denotes nonlabor income.

Furthermore, farm production technology of this household is expressed as:

(3) X = f([L.sub.f], F, K)

where the vector K includes fixed inputs (farm machinery and land) and shift factors (including policy variables) of the function f(x).

When the farm household maximizes its utility function (1) subject to constraints (2) and (3), the optimality conditions are written as:

(4) p([partial derivative]f/[partial derivate][L.sub.f]) = [w.sup.*]

(5) p([partial derivative]f/[partial derivative]F) = q

(6) X = f([L.sub.f], F, K)

(7) [partial derivative]u/[partial derivative]C = [lambda]r

(8) [partial derivative]u/[partial derivative][t.sub.f] = [lambda][w.sup.*]

(9) [partial derivative]u/[partial derivative][t.sub.m] = [lambda]w

and

(10) rC + [w.sup.*][t.sub.f] + [wt.sub.m] = [M.sup.HET] [equivalent to] pX - [w.sup.*][L.sub.f] - qF + [w.sup.*][T.sub.f] + w[T.sub.m] + V

where [lambda] denotes the Lagrange multiplier that is associated with budget constraint (2). (5) The "internal wage" [w.sup.*] satisfies the following relation:

(11) p([partial derivative]f/[partial derivative][L.sub.f]) = [w.sup.*] = r([partial derivative]u/[partial derivative][t.sub.f])/([partial derivative]u/[partial derivative]C).

Next, we introduce AHM under the RES hypothesis. A distinctive assumption under RES is the homogeneity of farm and nonfarm labor, which might be expressed as:

(12) u = u(C, [t.sub.t], G), [t.sub.t] [equivalent to] [T.sub.t] - [L.sub.f] - [L.sub.m] > 0

where [T.sub.t] and [t.sub.t], respectively, denote the endowed time and leisure hours of all workers.

Another distinctive assumption under RES is that the farm household faces a restricted market for nonfarm labor: it supplies fixed hours [[bar.L].sub.m] of nonfarm labor at the going wage w. When it uses competitive markets for consumption and farm commodities and other variable inputs, its budget constraint is expressed as:

(13) rC = pX - qF + w[[bar.L].sub.m] + V.

When the farm household maximizes its utility function (12) subject to constraints (3) and (13), the optimality conditions are written as:

(14) p([partial derivative]f/[partial derivative][L.sub.f]) = [w.sup.**]