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Restrictions on the production function


Overview

Restrictions: to limit the possible number of parameters used in the model to manageable level


Restrictions are applied in terms of Allen partial elasticities of substitution (APEs)
Advantage of using APE: APE can be calculated with parameters of CES function using Keller's formula, so we can further reduce the number of parameters.


Without restrictions, if we have N inputs, we need to estimate N * N APEs


But we can apply some general restrictions from cost minimization:
(1) Symmetry of APEs
(2) Homogeneity: conditional demand for production is homogeneous with zero degree of price
(3) Adding up: row sum of conditional demand elasticities () equals zero
As a result, now we need to estimate free parameters, but that number can be still large.


So, we need to further apply particular restrictions:
(4) Weak separability: which allows to break the cost minimization to two steps (VA level and top level)
(5) Choose CES function as the production functional form. So we only need to estimate one APE for each CES function.


Note: particular restrictions may vary by models and applications (that's why they are particular!)
We can use other functional form, but it will change the number of parameters needed.


Theory-based geneal explanation of restrictions


Setting-up the problem

For simplicity, denote:

where:

(Recall the definition of sets)


Note: there we take the sector j as an example, so we ignore the index j


Let to be the minimum cost function of one industry in GTAP. It is important that:
(1) C is positive linear homogeneous (PLH) in input price P
(2) C is PLH in output Q via constant return to scale
(3) is the derived demand for input i for this industry
(4) is negative semi-definite


Note: (1) - (4) are the required feature of the minimum cost function in GTAP. We do not set a specific functional form here, but the selected functional form must satisfy these conditions.


General restriction

From (1) to (3), we know that the derived demand of input i is:


Then we have the percentage form of as :
(5)
where:


Procedure to solve (5) is available here.


The row sum of conditional demand elasticities equals zero:
(6)


This is the general restriction (3)


The proof of (6) is given in (R1) below.


To reduce the number of parameters needed, we define the Allen partial elasticity of substitution (APE) as:
(7)


(8)


where:


From (6) and (8), we have
(R1)


For explanation of (R1), check the note of general restrictions on the production function


Because , the APE defined in 7) is symmetric, or
(R2)
This is the general restriction (1)


Combine (R1) and (R2), we have:
(R3)
(R3) gives us the column restriction on the matrix of APE


Since is negative semi-definite, then based on (4) and (7), we know:
(R4) is negative semi-definite.


Rewrite the linearized demand function 5) with APE and symmetry (R2), we have:
(9)
With the combination of (R1) - (R3), for N demanded commodities, we only need free parameters in (9) instead of N*N elasticities in (5).


Particular restriction

Follow Berndt and Christensen (Review of Economic Studies, 1973), we assume two inputs i and j are weakly separable from a third input k, or
(R5)
This is the particular restriction (4).


This restriction implies that the marginal rate of substitution () between and is independent of the level .


Note that firm's optimal mix of and hinges on comparing to the relative price of inputs and


With the weak separability of primary factors and intermediate inputs, we can write the production function as:
(10)
where:


Equation (10) gives the general structure of production functions used in GTAP, which separate value-added with the composite intermediate inputs.


Dual to (10), we can have the weakly separable cost function:
(11)


In addition to the separability restriction, we impose a particular functional form on F and G, that:
F takes the form of CES function
C takes the form of price index solved from CES function and cost minimization.


This is the particular restriction (5).


The CES functional form allow us to further reduce the number of parameters to estimate


Calculate APE with restrictions

With the general and particular restrictions, now we can reduce the number of unknown elasticities parameters to only two:


Then we can calculate Allen partial elasticities of substitution with the nested CES production function can be calculated with Keller's formula, with:


Example

How restriction helps to reduce the number of parameters need
For example, consider a production function of two layers (Value-added and intermediate inputs) of 10 sectors. And the value-added input is produced with two primary factors, then: