APPLICATIONS OF DUALITY THEORY TO AGRICULTURE

Western Journal of Agricultural Economics

дХ* _ aQj(q_iT,L*)

for all i = 1,. . . ,N
j = l,...,S.

дх*
where the compensated demand effect—¹ =
^a<lj

—¹- — —- Q₁  and the compensated output

ðeɪj ∂y

effect of a change in consumption goods price
∂Qj θQ_j ,ðb _j. 0L_{γx τ},

-- = —ⁱ {-- + —X_i}. lhus, symmetry
ðpɪ ∂L ɑpɪ ∂y

relations (13) can be represented in terms of
expressions which can be empirically es-
timated and thus (13) is a testable prediction
of the farm-household model.

Results similar to (10), (12) and (13) can be
derived for the case in which farmers work
off-farm and when the farm-household pro-
duces goods which are entirely consumed
within the farm-household [Lopez 1981b]. If
household members work off-farm then im-
portant questions are whether they regard
on-farm and off-farm work as perfect substi-
tutes in consumption and if there exist bind-
ing restrictions on the number of hours
which they can work off-farm. If they regard
on-farm and off-farm work as identical “com-
modities” and if they face no binding restric-
tions on hours of off-farm work then the
shadow price of labour becomes exogeneous.
If, in addition, all outputs produced by the
farm-household are at least partially traded,
then one can dichotomize production and
consumption decisions. In this case the con-
ventional models of the firm and household
apply and the predictions discussed before
no longer apply. However, if any of the
above conditions are not met, then utility
maximizing and profit maximization deci-
sions are interdependent and our previous
analysis holds.

A Suggested Econometric Specification
for the Farm-Household Model

Using equations (9ii) and (9i) one can ob-
tain the household demand for consumption
goods, X, and the net output supply response
specifications by postulating appropriate
functional forms for G(∙) and ττ(∙). With re-
spect to L it is necessary to indicate that we
cn only obtain an implicit representation of it
using equation (9iii).

A stochastic structure of the household
equations (9i), (9ii) and (9iii) can be specified
by assuming additive disturbances with zero
means and a positive semi-definite variance-
covariance matrix:

j = l,...,N

(13)  (b)Q,= ≤‰_2i,

i=l,. . .,S

3π(q,T,L)    <9G∕<9T

^<c) -fΓ~ ~ 5c∕57 ^+e3

where e_lj, e₂j and eɜ are the disturbance
terms.

It is evident that equation (13c) cannot be
estimated unless the shadow price of the-
fixed factor of production T is observed. Un-
fortunately, the shadow price of T is rarely
observed if a rental market for factor T does
not exist. Although the shadow price of factor
ɪ, ⅛τ , cannot be observed, the variable ττ
ðɪ

(“profit”) can at least be calculated; it is sim-
ply the net farm returns after payments for all
variable inputs (except L) are deducted from
the gross sales. Hence, given that q, T and L
are also observed one could in principle esti-
mate the vector of parameters, a, which
characterizes the conditional variable profit
function by estimating

(14) ττ = ττ (q,T,L;ot) + μ,
where μ is a disturbance term.

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