Lopez
Duality Applications
hold non-labour income. Similarly, demand
for consumer goods is jointly determined by
the parameters of the production and con-
sumption sides. It is also important to indi-
cate that the net output supply equations
specified by (9i) are unconditional, in the
sense that they are evaluated at the utility
maximizing level of L.
Equation (9iii) provides a specification for
the shadow price of land, дтт/дТ. If the condi-
tions of the implicit function theorem are
satisfied by aπ(tTL,T) then one can also de-
ЭТ
rive from (9iii) a specification for the equilib-
rium utility maximizing level of work, L.
Lopez [1981b] has shown that the farm-
household consumption demand functions,
net output supply functions and the equilib-
rium level of hours of work are homogeneous
of degree zero in consumption good prices,
net output prices and non-labour income. In
particular, unlike the net output supply func-
tions of the conventional firm, the net output
supply functions of the farm-household are
not homogeneous of degree zero in net out-
put prices. Also, unlike the conventional
model, the farm-household consumption de-
cisions (i.e., demand for consumer goods),
production decisions and the equilibrim level
of work are all dependent on parameters of
both the consumption and production sides
of the model.
As indicated earlier, in contrast with the
conventional model of the firm, net output
supply functions are not necessarily upward
sloping. However, I have derived a compen-
sated net output supply expression which can
be shown to be non-negative:
∕1m aQi aQi aL r>
<10> ⅛ ɪ s7Q1--⅛u≡"
i=l, —S
where eq. is the second partial derivative with
respect to qi of an expenditure function
e(p,q,T⅛μ.) defined by
(11) e(p,q,Tjμ) ≡ Min {pχ-ττ (q;T,L):
H-L,X
U(H-L,x)S=μ}
aQi
∂qi
Thus, although the sign of the directly
observed Marshallian output supply effect
) cannot be predicted, the compensated
or “Hicksian” output supply effect (i.e., the
left-hand-side of (IO)) is non-negative. There-
fore, if one can estimate^-1, ⅛ and ʃ then
<9qi ∂L ду
it is possible to empirically verify inequality
(10). This is an additional testable prediction
obtained from the farm-household model.
Another prediction from the model is that,
although the effect of a change in net output
price qi on the equilibrium level of work L is
in general unknown, the utility constant ef-
fect is unambiguously non-negative.
(12) {aL _ ^Q}=-(e +πq.>0
5L ⅛qi ду '~ v ll '1 ɪ`ɪ
i = l,. . .,S
Thus, if the equation system (9) is es-
timated then the left-hand-side of (12) can be
calculated and the non-negative restriction
implied by (12) can be tested. Notice that the
<3L
sign of -— is in general ambiguous and that
aqi
if the weak assumption that -^1>0 is made
i)L
then a testable prediction of the model is that
⅛ Qi≡≡0, for i = l, . . .S.
∂qi ду
Symmetry of the function e allows us to
show that reciprocity conditions between
production and consumption decisions also
hold. Lopez [198lb] showed the following
testable symmetry relationship which is obvi-
ously absent in the conventional models of
the firm and the household:
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