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

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

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

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

(18) [partial derivative]u/[partial derivative][t.sub.t] = [lambda]'[w.sup.**]

and

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

where [lambda]' denotes the Lagrange multiplier that is associated with budget constraint (13). (6) The internal wage [w.sup.**] satisfies the following relation:

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

We next compare the optimality conditions under the HET and RES hypotheses. First, the endogenous internal wage appears on both production and consumption sides of the conditions. Therefore, the AHM under the two hypotheses are both nonseparable.

Second, the two internal wages are determined in a different manner. Equation (11) shows that [w.sup.*] is determined to equate the value of the marginal product of farm labor (MPL) with the marginal rate of substitution (MRS) that is associated with farm labor. On the other hand (20) shows that [w.sup.**] is determined to equate the same MPL to the MRS that is associated with total labor.

Third, when we specifically examine the production side, (5) and (6) coincide with (15) and (16). Equation (4) includes a different internal wage [w.sup.*] from another internal wage [w.sup.**] in (14), but both are unobservable. Therefore, system (4)-(6) seems to be indistinguishable from system (14)-(16) in empirical analyses.

Finally, when we specifically examine the consumption side, (7)-(10) under the HET hypothesis yield a demand system of three commodities. Equations (17)-(19) under the RES hypothesis yield a demand system of two commodities. In particular the former yield supply functions of farm and nonfarm labor with their respective own wages [w.sup.*] and w, whereas the latter yield a supply function of total labor with its own wage [w.sup.**].

Consequently, we can distinguish AHM under the HET and RES hypotheses only on their consumption side. The remainder of this section describes the importance of their distinction by examining responses of the internal wages and quantity variables to a fall in the price r of the consumption commodity.

Figure 1(a) shows how internal wage [w.sup.*] is determined under the HET hypothesis. Therein, [L.sup.D([tau]).sub.f] and [L.sup.S([tau]).sub.f] , respectively, represent the demand and supply curves of farm labor at time [tau] (= 0, 1); they determine the work hours [L.sup.(tau)].sub.f] and internal wage [w.sup.*([tau])]. In addition, [L.sup.D([tau]).sub.m] and [L.sup.S([tau]).sub.m] represent similar curves of nonfarm labor, and they determine work hours [L.sup.([tau]).sub.m]. The curve [L.sup.D([tau])].sub.m] is a horizontal line at the competitive market wage [w.sup.([tau])], while the curve [L.sup.S([tau]).sub.m] is depicted as a vertical line for ease of visual interpretation. (7)

A fall in the price r only affects the supply curves and causes them to shift leftward if the income effect is greater than the substitution effect. Consequently, the internal wage rises from [w.sup.*(0)] to [w.sup.*(1)]. We follow Sonoda and Maruyama (2000) to express the response of the internal wage [w.sup.*] to a change in an exogenous variable s as:

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therein, [L.sup.S.sub.f] [equivalent to] [T.sub.f]-[t.sub.f] defines the supply of farm labor, and [([partial derivative]Q/[partial derivative]s).sub.const] represents the response of quantity Q with the internal wage fixed. On the right-hand side of (21), terms in the numerator represent shifts in the demand and supply functions of farm labor, whereas terms in the denominator represent their slopes.

Figure 1(b) shows how internal wage [w.sup.**] is determined under the RES hypothesis. The demand curve for total labor comprises the demand curve [L.sup.D([tau]).sub.f] for farm labor and that for nonfarm labor (the horizontal segment of length [[bar.L].sub.m] at w = [w.sup.([tau])]). This composite curve and the supply curve [L.sup.S(tau)].sub.t] determine total work hours [L.sup.([tau]).sub.t] = [L.sup.([tau]).sub.f] + [[bar.L].sub.m] and internal wage [w.sup.**([tau])]. Compared with figure l(a), curve [L.sup.S([tau])].sub.t] has a gentler slope than curve [L.sup.S([tau]).sub.m] but has a steeper slope than curve [L.sup.S([tau]).sub.f]. In addition work hours and internal wages at time 0 have identical levels in figures 1(a) and (b).

A fall in the price r only affects the supply curve. We set its shift equal to the sum of the shifts in farm and nonfarm labor in figure l(a). Then, the internal wage rises sharply from [w.sup.**(0)] to [w.sup.**(1)]. Consequently, [w.sup.**] is expected to respond more than [w.sup.*] under the assumptions described above, which will be verified in the empirical analysis. We express the response of internal wage [w.sup.**] to a change in an exogenous variable s as:

(22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [L.sup.S.sub.t] [equivalent to] [T.sub.t] - [t.sub.t] denotes the total supply of labor.

[FIGURE 1 OMITTED]

Different responses of the internal wages in (21) and (22) engender different responses of quantity variables, which might be expressed as:

(23) [partial derivative]Q/[partial derivative]s = [([partial derivative]Q/[partial derivative]s).sub.const] + ([partial derivative]Q/[partial derivative][w.sup.*])([partial derivative][w.sup.*]/[partial derivative]s)

and

(24) [partial derivative]Q/[partial derivative]s = [([partial derivative]Q/[partial derivative]s).sub.const] + ([partial derivative]Q/[partial derivative][w.sup.*])([partial derivative][w.sup.**]/[partial derivative]s)

On the respective right-hand sides of (23) and (24), [([partial derivative]Q/[partial derivative]s).sub.const] represents a direct effect of a change in an exogenous variable s, whereas ([partial derivative]Q/[partial derivative][w.sup.*(*)])([partial derivative][w.sup.*(*)]/[partial derivative]s) reflects an internal wage effect. The former is identical, but the latter differs under HET and RES. In particular if [w.sup.**] is more responsive than [w.sup.*], as shown in figure 1, the internal wage effect tends to be greater under the RES hypothesis.

A System Comparison Approach to Distinguish AHM under HET and RES

We employ a method of two-step estimation for our nonseparable models, as did Sonoda and Maruyama (1999). When estimating AHM under HET, we first simultaneously estimate the optimality condition (5) for other variable inputs and the production function (6). This conjecture yields estimated parameters of the production function, which in turn yield the estimated internal wage [[??].sup.*] as [w.sup.*] = p([partial derivative]f/[partial derivative][L.sub.f]) and the estimated full income [[??].sup.HET] from its definition. Subsequently, we estimate a system of commodity demand functions that is derived from optimality conditions (7)-(10), given [[??].sup.*] and [[??].sup.HET]. We employ a similar method for estimating AHM under RES.

In this case it is natural to compare production and consumption sides of AHM separately. Based on the previous discussion, we compare only the consumption side of AHM under HET and RES. Before starting the comparison, we specify the production technology and consumption preference and explain their estimation method.

Specification and Estimation Method of the Production Side

Let ALL = {[L.sub.f], F, K} be composed of all production and shift factors. The vector K consists of the real capital stock RK, total area planted A, the intensity rate SAP of the set-aside program (see the next section) and ETT = exp(TT) (TT: time trend). Also, let ALL/{F} be constructed by excluding F from ALL. We specify the production function as:

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In this case the optimality condition for other variable inputs is expressible as:

(26) qF/pX = [b.sub.F] + [summation over (z[member of]ALL])] [b.sub.F,z] ln z.

We estimate (25) and (26) simultaneously using the generalized method of moments (GMM) under both the HET and RES hypotheses. Consequently, we obtain the same estimates of the production function parameters and the same estimated internal wage, [[??].sup.*] = [[??].sup.**] = p([partial derivative]f/[partial derivative][L.sub.f]), under the two hypotheses.

Specification, Estimation Method, and Comparison of the Consumption Side

Under the HET hypothesis we obtain three demand functions for consumption commodities and leisure hours of farm and nonfarm workers. Similarly to Shively and Fisher (2004), we specify an almost ideal demand system (AIDS) and estimate the following two share equations for leisure hours of farm and nonfarm workers:

(27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Under the RES hypothesis we obtain two demand functions for consumption commodities and total leisure hours. We specify an AIDS model and estimate the following single share equation for total leisure hours:

(28) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The two demand systems are estimated using GMM after imposing adding-up, homogeneity and symmetry restrictions.