APPLICATIONS OF DUALITY THEORY TO AGRICULTURE



December 1982


Western Journal of Agricultural Economics

use of duality in measuring agricultural factor
demand and output supply responses during
the last decade is examined. First the most
popular approach (the cost function) is con-
sidered, which is used to estimate Hicksian
input demands as well as to obtain informa-
tion regarding properties of the underlying
production technology. Next, applications of
the profit function approach, which has al-
lowed researchers to estimate Marshallian
factor demands jointly with multioutput sup-
ply responses, are discussed. Other possible
applications of duality are reviewed mainly
concerning the analysis of supply responses
when markets for certain inputs or outputs
do not exist. That is, the study of producers
behaviour when some of the prices motivat-
ing producer responses are unobservable due
to the fact that many of the trade-offs in the
allocation of resources occur within the farm-
household and not between the farm produc-
er and a market. This model may be relevant
mainly to developing economies where a par-
tial absence of markets in the rural sector
often occurs.

The Cost Function Approach

Several studies have used this approach in
measuring factor demand elasticities, elas-
ticities of substitution and technical change
in agriculture. Binswanger [1974a or 1974b]
and Kako specified a translog cost function
which allowed them to estimate factor shares
in log linear form. The cost function specified
in both studies was:

n

(1) In C = Oi0 ÷ cty In Y ÷ Σ vi In pi
i =i

+ ½ ∑ij yij In pi In Pj

+ ∑i yit In pi In t

where C is the minimum cost of production
level, Y is output, pi is the price of factor i
and t is a time trend variable used as a proxy
for technical change. From (1) one can obtain
a specification for factor shares (Si) via loga-
rithmic differentiation using Shephard’s
lemma.

(2) Si = vɪ + ∑j γij In pj + γit In t

where yij = yji             i=l, ∙ . ., N

Using this specification Binswanger and
Kako1 were able to separate the effect of
biased technical change (represented by the
it parameters) on factor shares from the ef-
fect of ordinary factor substitution due to
factor price changes (represented by the γij
parameters in (2)). They both found that fac-
tor augmenting technical change has been
very important and explains a great deal of
the observed changes in factor shares in the
U.S.A, and Japan.

An important assumption made in both
studies is that the production technology is
homothetic. Therefore expansion paths were
assumed linear and thus changes in the scale
of production would not affect factor shares.
This is why the factor shares in (2) are as-
sumed to be independent of output levels.
The implication of this is that all changes in
factor shares are attributed to substitution
and/or factor augmenting technical change. If
the production technology is not homothetic,
however, a risk of overestimating the effect of
factor substitution or, more likely, technical
change exists. This is so because the time
trend variable used as a proxy for technical
change is generally positively correlated with
output levels.

Lopez [1980] used a more general specifi-
cation for the cost function using Canadian
agricultural data. This specification allows for
a non-homothetic production function but
preserves the same degree of flexibility of the
translog. The cost function specification used

1Both Binswanger and Kako considered the following
inputs: land, labour, machinery, fertilizers and other
intermediate inputs.

354




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