Lopez
Duality Applications
is a generalized Ieontief which is also a flex-
ible functional form:
(3) C = Y ∑i ∑j bjj Pi‘/2 pj 1/2 + Y2 ∑i αipi
+ Yt ∑i "yipi
From (3) using Shepard’s lemma one can
obtain the factor demand equations in in-
put∕output ratio forms:
X∙ /n ∖ ½
(4) -2 = ∑bij (g +αi Y + 7it t
γ j ∖p7
where bij = bji i=l, ..., N
Note that specification (4) allows one to
separate the effect of relative factor price
substitution, factor augmenting technical
change and the scale of production on the
input-output ratios (and, hence, on factor
shares). In particular, equation system (4)
allows, as a special case, for homotheticity.
This occurs if αi = о for all i, that is, when the
input-output ratios are independent of out-
put. By estimating a system of four input-
output ratios (labour, capital, land and struc-
tures and other intermediate inputs) Lopez
[1980] showed that the hypotehsis of ho-
motheticity is rejected by a wide margin and
that changes in the scale of production ex-
plain a very important proportion of changes
in the input-output or share equations. The
effect of non-neutral technical change was
found to be insignificant, which was a rather
surprising result. However, a recent more
disaggregated study by Lopez and Tung us-
ing combined cross section and time series
data for Canadian agriculture2 found that the
factor augmenting technical change parame-
ters (γit) were jointly significant. However,
the technical change effect was substantially
less dramatic than those obtained by Bins-
wanger or Kako, while the output scale effect
is very strong and significant.
2The inputs considered were energy, energy-based,
labour, capital, land and other intermediate inputs.
The own price elasticities of factor demand
are quite similar for the four studies, despite
using different data and models (Table 1). An
overall analysis of Table 1 allows one to con-
clude that, in general, factor demands are
inelastic; that land demand elasticity is some-
where between —0.35 and —0.50, that
labour demand elasticity is roughly between
— 0.40 and —0.50 (Binswanger’s result is an
outlier). Demand for fertilizers and chemi-
cals tends to be more elastic at least in the
studies using North American data (roughly
— 0.9) and farm capital demand also exhibits
somewhat lower values than the former. In
general, one can say that the estimated de-
mand elasticities may provide policymakers
with some notion of the various degrees of
price responsiveness of the inputs used in
agricultural production.3
Unfortunately, the studies do not show the
same consistency in the estimation of input
substitution measures. Binswanger found
that land is a substitute for labour, machinery
and fertilizers. Fertilizers and land were
found to be the best substitutes. These re-
sults are consistent with the findings of
Lopez and Tung who found that land and
energy-based inputs (largely fertilizers and
other chemicals) were the best substitutes
among all input pairs. Kako found that land
was a substitute with all other inputs except
machinery. In contrast with Binswanger’s,
Kako’s and Lopez’s results, the study by
Lopez and Tung found that capital and land
are complements. Labour and farm capital
have been consistently found to be substitute
inputs in all studies reviewed. However,
labour and energy-based inputs are strong
substitutes in the study by Lopez and Tung
while they are complements in the studies by
Binswanger and Kako.
In general, one can say that the various
cost function studies have shown that (1)
input demands are moderately responsive to
3It is important to note, however, that these are Hick-
sian elasticities. That is, they measure factor demand
responses for given output levels neglecting the indirect
factor demand effects associated with changes in output
scale due to factor price changes.
355