The present section deals in succinctly deals with basic model for estimating producer supply and demand functions. It is based on Diewert and Wales (1992) and Diewert Chapter 9 on index number theory. For greater detail in to the modelling technique and methodology involved here, please refer to the aforesaid references. First we define the variable profit function V(k,p) as V(p,k) max y {pT y :k = F(y)}. The function V(p,k) must be linearly homogeneous and convex in p for fixed k. The economy's system of profit maximizing supply and demand functions y(p,k) can be obtained by differentiating V(p,k) with respect to the components of p: (Hotelling’s (1932) Lemma). Then we define the unit profit function v(p) as V(p,1), which is basically the gross return to capital we can achieve using one unit of capital. Based on this Diewert etal show that we can develop a Translog unit profit function, v(p) with CRS:
Which satisfy the following properties as part of being a flexible functional form (Diewert Chapter 9, Section 2):
However economy becomes more efficient over time because of technical progress. Thus we generalize the Translog unit profit function defined above to include time trends to try and capture the effects of technical progress. Thus we now define the period t unit profit function v(p,t) as follows:
39 For greater detail on the derivation please review Chapter 9 on Flexible forms by Diewert, which is available in his website11. The aforesaid equation is adapted for PMOD 1 described in the following section. As for PMOD2 which is Translog variable profit functions with CRS and linear splines to model technical progress. In that case the aforesaid system of equations is modified by introducing more than one linear time trend; for instance instead of β0t, we get β01t1... β0ntn. Similar adjustments are made in
other areas of the equation system.
Thus given econometric estimates for the α i , β i and γ ij , which are denoted by α i* , βi* and
γij* , the estimated or fitted shares in period t, s it* is developed and is given by:
Based on this Diewert etal develop the period t cross elasticities of net supply, e ijt :
Similarly using econometric estimates one can obtain the following formula for the period t own elasticities of net supply, e iit , :
In measuring the Technical Progress they define V(k,p,t) kv(p,t), and then differentiate V(k,p,t) with respect to time t and evaluate the resulting expression at the period t data, which yields:
Where Tt is the desired measure of technical progress.
PMOD3, as defined in the following section, is a basic Leontief with CRS functions with no substitution between inputs and outputs and linear splines to model technical progress. The rationale for CRS is that when one uses fixed costs or nonconstant returns to scale then one gets absurd result like technical progress are usually way too big while on the other hand, estimates of returns to scale are way too small.
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40 Next Diewert etal develop the normalized quadratic profit function with CRS and linear splines to model technical progress. First they define the production unit’s period t variable profit function V(k,p,t) as follows:
where b T [b 1 ,...,b N ] and c T [c 1 ,...,c N ] are parameter vectors and B [b ij ] is a matrix of
parameters. The matrix B needs to satisfy the following restrictions:
I. Matrix B has to be symmetric II. Bp* = 0 N for some p* >> 0 N .
Next they define vector of period t normalized prices is defined as v t (α T p t ) −1 p t. Finally they develop a system of equations:
Where and
Finally on the basis of this Diewert etal develop measure for elasticity and technical progress pertinent for this functional form. The price elasticity matrices are given by:
While period t, Technical Progress is measured by :
where .
If the estimated B matrix turns out to be not positive definite, then we can rerun the aforesaid model by replacing B by B = AA T , where A is a lower triangular matrix and satisfies A T p* = 0 N . These aforesaid
models can have splines incorporated in them and this is discussed in greater detail in section 17 chapter 9 Flexible Functional Forms of Diewert.
In the aforesaid formulation we have left the substitution matrix B unchanged over time. Diewert etal showed that, as a result of this the previously discussed functional forms have a built in trend in
41 elasticities. This can be solved by allowing B to change with time. Thus in accordance with Diewert, the author set the matrix B equal to a weighted average of a matrix C (which characterizes substitution possibilities at the beginning of the sample period) and a matrix D (which characterizes substitution possibilities at the end of the sample period). Thus B is defined as follows in terms of C and D and the time variable t:
B t = (1 − [t/T])C + [t/T]D ; t = 0,1,2,…,T.
Also the correct curvature conditions can be imposed globally by setting C and D equal to the product of UUT and VVT respectively, where U and V are lower triangular matrices; i.e. C = UU T and D = VV T; where U and V are lower triangular matrices. We can also impose the following normalizations on the matrices U and V: U Tp* = 0 N ; V Tp* = 0 N . In the following section we implement these modelling
techniques based on Diewert etal, for the Swiss economy. In first part there is a succinct description of the specific types of model used. For greater detail please review the aforementioned references.