Lopez
Duality Applications
The problem with estimating (14) is that
the variable L may be correlated with the
disturbance term and hence the estimates of
α would not be consistent. However, if τr (∙)
is linear in the parameters (as is usually the
case when flexible functional forms are used)
then one can use an instrumental variable
technique and thus obtain consistent esti-
mates of a. Therefore, if an appropriate in-
strumental variable for L exists, then one can
estimate equation (14) and obtain consistent
estimates (ɑ) for the parameters of the condi-
tional variable profit function. Using the es-
timated vector & one can evaluate the func-
.. ârr(q;T,L;â) „ , 1
tɪon----—----which is the true shadow
price of land measured with an error. Thus
the true shadow price of T is equal to the
estimated shadow price plus an error term
assumed to be stochastic. That is:
∕ικ∖ ∂π(q,T,Lsα) ∂π(q,T,I√d)
(ɪɔ) ----ΞT---- = ----ΞT---- + Rτ∙
ɑl ol
Notice that equation (15) is not estimated;
the ɑ parameters are obtained by estimating
equation (14) and substituted in^ɪ. In other
дТ
words, by estimating & in (14) one obtains a
measure of the “true” shadow price of capital
subject to an error, μτ. If (15) is used in (13.c)
then one may interpret equation (13. c) as an
error of measurement of the dependent vari-
able situation which offers no estimation
problems:
∕lc∖ ⅛(q,T.Lα) дС/дТ , _
v ' дТ âG/ду
where ê3 ≡ e3 — μτ
If e3 and μτ are normally distributed and
independent of p,q,T and y in (13 c) and (15)
then ê3 will possess the same properties.
Thus, there is no problem with using the
predicted rather than the actual shadow price
of T in estimating (13 c).
Although an explicit labour equation can-
not be estimated, if the parameters of (14)
and (16) are estimated then one obtains an
implicit representation of L on the left-hand-
side of (16) and hence all the relevant
economic information regarding the factors
determining the equilibrium level of family
labour can be derived.
In summary, it appears that estimation of a
complete system of production and consump-
tion equations for the farm-household is fea-
sible and desirable. This system should be
jointly estimated given the inter-
dependences of the production and con-
sumption sectors emphasized by the fact that
all behavioural equations are derived from
the same indirect utility function. Finally,
the testable implications of the farm-
household model derived above can be im-
posed or tested in the estimating model.
Looking Forward
There are at least three additional poten-
tially fruitful areas of research where duality
may prove to be a useful approach. The use
of duality in the context of dynamic models
and on modelling supply responses under
risk which, I understand, is covered in the
other paper presented in this session, are
certainly important areas of further applica-
tions of duality in agriculture.
The use of duality has also helped to sim-
plify the analysis of competitive market
equilibrium analysis and allows one to use
less restrictive a priori assumptions on the
derivation and characterization of competi-
tive market equilibrium. An example of this
approach is the analysis of the land market
and agricultural supply and demand re-
sponses to exogeneous changes in factor or
output prices in the context of a small open
economy [Lopez 1982a]. I think that further
work in this area appears quite promising.
A third direction of research using duality
may be in the context of the analysis of non-
competitive behaviour mainly at the food
processing, distribution and retailing (PDR)
sector. This sector is, in general, highly con-
centrated in North America and one could
expect that the use of conventional models
based on price taking behaviour might not be
very appropriate. In Agriculture Canada we
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