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.**]
(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.
Next, we compare systems (27) and (28). They differ not only in
their dependent and independent variables but also in the number of
equations. Consequently, they are complicatedly nonnested, as stated in
the first section. Nonnested models have often been tested using the J
test of Davidson and MacKinnon (1981) or the Cox test of Pesaran (1974).
The J test is usually used to compare systems that have the same
number of equations and the same dependent variables. Using that test,
Lopez (1984) attempted to compare separable and nonseparable models
which have different numbers of equations. His method requires a
complicated transformation of the models to apply the J test (see [22]
of Lopez). Moreover, this transformation might not maintain properties
of the original models, which might engender inappropriate comparisons
of the models. (8)
In contrast the Cox test does not require this type of strict
correspondence between two systems; instead, it compares their
likelihoods. As Mizon and Richard (1986) indicate, it can be interpreted
as an encompassing test that evaluates the extent to which the null
model explains an important characteristic (likelihood, parameters,
etc.) of the alternative. In our case we can use a Cox-type test of
Smith (1992), which compares their GMM objective functions instead of
their respective likelihoods.
We evaluate the GMM objective function for model (28) in two ways
when we test the null model (27) against the alternative (28). (9) One
is to evaluate it on the assumption that model (28) is correct. The
other is to evaluate its probability limit, assuming that model (27) is
correct. We interpret that model (27) can explain model (28) if the
difference between the two values is sufficiently small. Smith (1992)
shows that the difference is asymptotically normal. The definition and
computation of the Cox-type statistic are complicated. For that reason
they are explained in the Appendix.
Empirical Analysis
Although the use of micro data is preferable in estimating AHM,
this study uses aggregate data for the following reasons. Lopez (1984)
and Sonoda and Maruyama (1999) use aggregate data to estimate AHM under
the HET and RES hypotheses. Furthermore, most Japanese studies have used
aggregate data to estimate AHM (e.g., Kuroda and Yotopoulos (1980) and
Arayama (1986)). For that reason, estimating AHM under the two
hypotheses using Japanese aggregate data at least makes practical sense.
Use of aggregate data renders the discrete choices of farm
households unobservable, which might cause a major problem. In our case
the problem is related to whether Japanese farm households employ
nonfamily workers in rice farming or not, and whether their members work
at nonfarm firms or not. As inferred from the hours of hired farm labor
and the number of nonfarm workers shown in table 1, only a very limited
number of farm households use significant hours of nonfamily labor and
supply none of their members' labor to nonfarm firms. Consequently,
we are not likely to observe a serious difference between the two AHMs
estimated from our aggregate (but not so highly aggregated) data and
those estimated from the original micro-data.
Data
We use aggregate data for rice-farming households in Japan, which
are mainly adapted from the Survey of Farm Household Economy by Types of
Farm Households (FHET) (Japan, Ministry of Agriculture, Forestry and
Fisheries 1982-1991). We also use data from the Statistics of Prices and
Wages in Rural Areas (PWRA) (Japan, Ministry of Agriculture, Forestry
and Fisheries 1982-1991) and the Annual Report on the Consumer Price
Index (RCPI) (Japan, Management and Coordination Agency 1982-1991).
Annual aggregate data are available for seven paddy-scale classes
in eight regions for 1982-1991. (10) The paddy-scale classes include
0.5-1.0, 1.0-1.5, 1.5-2.0, 2.0-2.5, 2.5-3.0, 3.0-5.0, and 5.0 ha and
larger. The regions include Tohoku, Hokuriku, Kanto-Tosan, Tokai, Kinki,
Chugoku, Shikoku, and Kyushu, which cover almost the entire area of
Japan. We would have 10 (years) x 7 (scale classes) x 8 (regions) = 560
observations if data were to constitute a complete time-series
cross-section. However, our incomplete data include only 283
observations, reflecting the actual situation: a small number of farm
households operate large paddy areas. (11)
Table 1 shows the means and standard deviations of variables used
in empirical analyses. The price of consumption commodities is obtained
from the general index by region in RCPI. The Divisia price indexes of
other variable inputs and capital goods are constructed using
expenditures for relevant items in FHET and the price indexes for the
same items in PWRA. Using FHET data, the price of rice is obtained by
dividing the gross revenue derived from rice sales by the amount of rice
produced.
The nonfarm wage rate is obtained by dividing the salaries and
wages in nonagricultural gross income by the total hours of nonfarm
work. Empirical studies employing micro-data commonly replace the wage
rate, when thus constructed, with the predicted one in estimating their
labor supply function to allow for measurement errors in work hours
(e.g., Sahn and Alderman 1988). Instead, we choose a standard solution
to the measurement error problem in estimating AIDS models: we use a set
of instrumental variables that includes exogenous factors affecting the
nonfarm wage. One reason for our choice is that variations in a
predicted wage are smaller than those in the original wage, particularly
when we use aggregate data, which often makes estimated parameters less
precise. Also, the FHET surveys work hours of farm households by asking
them to record their daily work hours throughout the year, from which we
infer that those work hours do not include overly large errors.
The amount of rice produced, hours of family farm labor and total
area planted are obtained from FHET. Quantities of other variable
inputs, capital stock and purchased commodities are obtained by dividing
their expenditures or value in FHET by their respective price indexes.
The endowed time of farm, nonfarm and total workers is estimated as
[T.sub.i] = 16 x 365 x [N.sub.i] (i = f, m, t), where [N.sub.f] and
[N.sub.m], respectively, represent the numbers of farm and nonfarm
workers; also, [N.sub.t] [equivalent to] [N.sub.f] + [N.sub.m]. Other
production costs OC aside from the cost qF of other variable inputs and
the number NF of household members are available from FHET for
subsequent analysis.
Finally, a set-aside program in Japan was introduced to assuage the
mounting surplus of rice in the late 1960s. Since that time, the
government has set annual targets for fallow paddy areas, and
rice-farming households have complied with that program to receive
compensatory payments and avoid a sharp decline in the price of rice.
The intensity rate SAP of the set-aside program is defined by the farm
household's fallow paddy area as a fraction of its total paddy
area. Its data are obtained from FHET.
Estimation of Production Side and Variations in the Internal and
Non farm Wages
To deal with our incomplete time-series cross-section data, we
simply assume fixed effects and express the coefficients [b.sub.x] and
[b.sub.F] in (25) and (26) as:
(29) [b.sub.z] = [b.sub.z,0] + [summation over (r)]
[[pi].sub.z,r][RD.sub.r] + [summation over (s)][[rho].sub.z,s][SD.sub.s]
(z = X, F;r = 1, ..., 7;s = 1, ..., 6)
where [RD.sub.r] and [SD.sub.s], respectively, denote regional and
paddy-scale dummies. After substituting (29) into (25) and (26), we
estimate them using GMM.
In (25), (26) and (29), exogenous variables suitable for
instrumental variables are a constant term, lnRK, lnA, lnSAP, TT, p, q,
[RD.sub.r], and [SD.sub.s]. Substitution of (29) into (25) produces
cross terms between dummies and endogenous regressor lnF. Therefore, we
construct a basic set of instruments by adding [RD.sub.r] x TT and
[SD.sub.s] x TT to the exogenous variables described above. (12)
Furthermore, to allow for cross and quadratic terms between endogenous
regressors lnF and ln[L.sub.f] in (25), we construct cross and quadratic
terms among lnRK, lnA, lnSAP, TT, p, and q and combine them with the
basic set in various ways. We eventually choose plnRK, plnA, plnSAP,
[(In A).sup.2], [p.sup.2], qlnRK, and qlnA to combine them with the
basic set. (13) The combined set yields the estimation results shown in
table 2, (14) which: satisfies monotonicity (in all inputs) and
concavity (in variable inputs) of the production function; passes the
overidentifying restrictions (OIR) test of Hansen (1982) at the 5%
level; and yields the most stable t-values of the parameters.
We can estimate the internal wage from this result, table 1 shows
the mean of the internal wage (687 yen/hour) at about one-half that of
the nonfarm wage (1302 yen/hour). The remainder of this subsection
describes these wages as exhibiting different and sufficient variations
for our purpose of identifying the two demand systems. Detailed
variations in the internal wage are examined systematically in the final
subsection.
The sources of variation in the nonfarm wage w are attributed to
education, age, sex, region, and macroeconomic factors, whereas those in
internal wage [[??].sup.*] are attributed to all factors affecting the
farm household's behavior. Consequently, the latter is expected to
vary in a more complicated manner. Actually, the coefficients of
variation of w and [[??].sup.*] are 0.22 and 0.26, respectively, and
their correlation coefficient is 0.21.
Next, we examine variations in nonfarm wages in greater detail. The
FHET does not provide data for education, sex, and age. Therefore, we
attempt to explain those variations using our available information.
First, the FHET provides data showing the shares of white-collar
permanent workers among all nonfarm workers. According to Fukui (1993),
this type of variable is useful as a proxy for the educational level
because farm household members with a higher education level tend to
engage in white-collar jobs. Second, the Survey of the Farm Household
Economy (Japan, Ministry of Agriculture, Forestry and Fisheries
1982-1991) (15) shows that the share of males among nonfarm workers is
quite stable during 1982-1991, at about 60% for the eight regions
examined here. Therefore, a gender-related variable does not vary so
much and might not strongly affect variations in the nonfarm wage.
Finally, it might be difficult to identify the effects of age for our
aggregate data because we often expect two nonfarm workers in each farm
household, as shown in table 1.
With this information in mind, the logarithm of the nonfarm wage is
regressed on a constant term, regional dummies, time trend and the
shares of white-collar and blue-collar workers. The result shows that
all coefficients are significant at the 5% level, and the adjusted
coefficient of determination is 0.79. All regional dummies have
significant coefficients because the eight regions cover almost the
entire area of Japan, and regional differences in the nonfarm wage have
expanded along with economic development. The time trend has a positive
coefficient, implying an upward trend in nonfarm wages. Finally, both
shares of white-collar and blue-collar workers have positive
coefficients; the coefficient for the former is 2.7 times larger than
that for the latter, verifying our previous inference related to
educational level.
Estimation and Comparison of Consumption Systems
When we allow for fixed effects, the farm household's
characteristics and its taste changes, we specify coefficient
[[alpha].sub.i] (i = f, m, t) in (27) and (28) as:
(30) [[alpha].sub.i] = [[alpha].sub.i,0] + [[alpha].sub.i,2]FSHARE
+ [[alpha].sub.i,3]TT + [summation over (r)]
[[pi].sup.i.sub.r][RD.sub.r] + [summation over
(s)][[rho].sup.i.sub.s][SD.sub.s]
where NF and FSHARE, respectively, represent the number of
household members and the share of farm workers among all workers. After
substituting (30) into (27) and (28) and imposing the theoretical
restrictions, we estimate each system using nonlinear GMM with the
[[alpha].sub.0] parameter in ln [P.sup.k] (k = HET, RES) set equal to
zero.
In (27), (28) and (30), exogenous variables suitable for
instruments are: a constant term, r, w, NF, [N.sub.f], [N.sub.m], TT,
[RD.sub.r], and [SD.sub.s], where [N.sub.f] and [N.sub.m] are used in
place of FSHARE = [N.sub.f]([N.sub.f] + [N.sub.m]) to obtain precise
estimates of parameters. These variables are combined with exogenous
variables included in the full income (ln RK, lnA, ln SAP, p, q, and OC)
to construct a set IV-1 of instruments. (16) Hours [L.sub.m] of nonfarm
work are exogenous only under the RES hypothesis. Therefore, this
variable is not included in view of the criterion stated in the
Appendix. In addition we use another set IV-2 to take care of
measurement errors in the nonfarm wage. This set is constructed by
replacing the wage w in set IV-1 with the shares of white-collar and
blue-collar workers in nonfarm workers. (17)
The estimation results satisfy concavity in all the cases when we
estimate the demand systems (27) and (28) using sets IV-1 and IV-2.
Three of the four results pass the OIR test at the 5% level, and the
remaining one (system [28] for IV-l) passes it at the 1% level.
Furthermore, most coefficients are significant at the 5% level.
Using these results, we can compute the Cox-type statistic in (A.1)
of the Appendix. In testing the HET hypothesis against RES, we noted
that the statistic [[lambda].sub.HET,RES] is equal to 1.17 [0.24] for
set IV-1 and 0.87 [0.39] for set IV-2, where the p-value is shown in
brackets. Conversely, in testing the RES hypothesis against HET, we
found that the statistic [[lambda].sub.RES,HET] is equal to 4.47 [0.00]
for set IV-1 and 2.98 [0.00] for set IV-2. Therefore, the RES hypothesis
is rejected, but the HET hypothesis is not rejected for both sets of
instruments. We conclude that the HET hypothesis is more appropriate in
our statistical tests.
Finally, our Cox-type test used "generated regressors"
(the estimated internal wage and full incomes), which might engender
biases in nonnested tests, as McAleer and McKenzie (1991) suggest. One
way to examine this possibility might be to use a similar method to
Newey's (1984) and compare uncorrected and corrected standard
errors of the parameters. table 3 compares two sets of t-values of the
AIDS parameters under the HET hypothesis, where the set IV-2 of
instruments is used for estimation. (18) The two sets of t-values differ
greatly for about one-third of the parameters, but they do not give
different results in tests of significance at the 5 % level, suggesting
that biases in our nonnested test are not serious. Similar results are
verified under the RES hypothesis.
Comparative Statics Analysis
We now compare price elasticities of internal wages and quantity
variables under the HET and RES hypotheses. We can use (21)-(24) for
this purpose. The responses of factor demand and output supply on the
right-hand sides of these equations are obtained by differentiating the
optimality conditions (4)-(6) (or [14]-[16]) and using estimated
parameters of the production function. These responses are identical
under the HET and RES hypotheses because we use the same estimation
method on the production side.
Similar responses of labor supply are obtained in the following
way. For our AIDS models we can express the full income elasticity of
leisure demand: [t.sub.i] (i = f, m, t), under hypothesis k (= HET,
RES), as [[eta].sup.k.sub.i] [equivalent to] [partial derivative] ln
[t.sub.i]/[partial derivative]ln[M.sup.k] = 1 +
([[beta].sub.i]/[S.sup.k.sub.i]). Under the HET hypothesis price
elasticities of leisure demand: [t.sub.i] (i = f, m) are then
expressible as: (19)
(31) [([partial derivative] ln [t.sub.i]/[partial derivative] ln
y).sub.const] = [([partial derivative] ln [t.sub.i]/[partial derivative]
ln y).sub.M] + y [H.sub.y] [[eta].sup.HET.sub.i] / [M.sup.HET] (y = p,
q, r, w, [w.sup.*]).
The first term on the right-hand side represents the price
elasticity of leisure [t.sub.i] with full income fixed, whereas the
second term represents the effect of changes in full income, with
[H.sub.p] = X, [H.sub.q] = -F, [H.sub.r] = O, [H.sub.w] = [T.sub.m], and
[H.sub.w*]. = [T.sub.f] - [L.sub.f]. For example, [([partial derivative]
ln [t.sub.f]/[partial derivative] ln [w.sup.*]).sub.M] = -1 +
{[[gamma].sub.f,f] - [[beta].sub.f]([[alpha].sub.f] + [[gamma].sub.f,f]
ln [w.sup.*] + [[gamma].sub.m,f] ln w + [[gamma].sub.c,f] ln
r)}/[S.sub.f].
Under the RES hypothesis, price elasticities of leisure demand are
expressed as:
(32) [([partial derivative] ln [t.sub.t]/[partial derivative] ln
y).sub.const] = [([partial derivative] ln [t.sub.t]/[partial derivative]
ln y).sub.M] + y[R.sub.y][[eta].sup.RES.sub.t]/[M.sup.RES] (y = p, q, r,
w, [w.sup.**])
where [R.sub.p] = X, [R.sub.q] = - F, [R.sub.r] = O, [R.sub.w] =
[L.sub.m], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Price elasticities of labor supplies are obtained by multiplying
those of leisure demands by - [t.sub.f]/[L.sup.S.sub.f] or
-[t.sub.m]/[L.sub.m] under HET and by -[t.sub.t]/[L.sup.S.sub.t] under
RES.
The price elasticities of quantity variables thus evaluated, along
with their nonlabor income elasticities, are shown in panel (i) of table
4. Elasticities in this panel are substituted into (21) or (22) to yield
elasticities of the internal wages, which are in turn substituted into
(23) or (24) to yield elasticities of quantity variables, including
internal wage effects. Their computed values are shown in panel (ii) for
the HET hypothesis and in panel (iii) for the RES hypothesis.
First, panel (i) of table 4 shows that the internal wage elasticity
of farm labor supply, 0.38, is much greater than that of the total labor
supply, 0.14. Therefore, the supply function of total labor is actually
steeper than that of farm labor, as depicted in figure 1. This result
reflects the fact that household members work much longer hours at
nonfarm firms than on their farm. It also reflects the fact that the
supply of nonfarm labor responds less flexibly than that of farm labor.
One rationale for the latter is that it seems more difficult for
self-employed farmers to adjust their time at nonfarm firms than on
their own farm. Another explanation suggests that principal (nonfarm)
workers are likely to supply their labor less flexibly than nonprincipal
(farm) workers, reflecting a commonly observed pattern of labor division
mentioned in the first section.
Next, the market price elasticities of the total labor supply are
greater in absolute value than that of the farm labor supply when the
internal wages are fixed, as shown in panel (i). These relations imply
that a shift in the supply curve [L.sup.S.sub.t] of total labor is
greater than the corresponding shift in the supply curve [L.sup.S.sub.f]
of farm labor in figure 1. We note two points to interpret that
difference. First, panel (i) of table 4 shows that the difference in
shifts among the supply functions of farm, nonfarm and total labor
clearly reflect the size of their respective income elasticities. (20)
Second, the estimated income elasticity of the total labor supply -0.35
is inferred to be biased downward because it is greater in absolute
value than both the income elasticities of farm and nonfarm labor supply
-0.26 and -0.21. These points suggest that the greater shifts of the
supply function of total labor result from the biased income elasticity
of the total labor supply.
In view of (21) and (22) as well as figure 1, the steeper slope and
the greater shifts of the supply function of total labor imply that the
internal wage [w.sup.**] responds more sensitively to changes in market
prices than the other internal wage [w.sup.*] does. Panels (ii) and
(iii) of table 4 verify this relation.
In view of (23) and (24), such greater elasticities of internal
wage [w.sup.**] cause greater internal wage effects on quantity
variables. Actually, most elasticities of demand for farm labor and
other variable inputs include much greater internal wage effects in
panel (iii) than in panel (ii). The elasticity of output supply includes
moderately greater internal wage effects in panel (iii) because internal
wage effects on the demands for farm labor and other variable inputs are
mutually offsetting to some degree.
Conclusion
This paper describes a system comparison approach to distinguish
nonseparable AHM under the HET and RES hypotheses. Using a two-step
estimation, we found their consumption side to be distinguished because
they yield demand systems that not only have different dependent
variables but also different numbers of equations. We proposed an
empirical procedure to apply the Cox-type test of Smith (1992) to make
an appropriate comparison of the nonnested systems.
Our specific comp