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Linearization of CES function


Key: linearization of CES function, which represents the producer's behavior in GTAP model


Economic model: cost minimization problem
Economic solution: conditional factor demand z(w, q) and unit cost function c(w, q) / q


Set up producer's problem

The producer's production function is CES function, and the cost minimization problem is:


Where:


Property of the CES function:


Solve producer's problem

Full steps to solve the producer's problem:


Write the problem in the standard form of optimization with constraint:


Write Lagrange of the problem:


Calculate FOC with respect to
(1)
(2)
(3)


Divide (1) and (2):


(4)
Note: here equation (4) gives the relative relationship between x1 and x2, given the price p1 and p2. Regardless of the total output y, as long as x1 and x2 are positive, they will satisfy this relationship. However, the exact value of x1 and x2 depend on the condition , which is equivalent with the utility constraint in hicksian demand of expenditure minimization problem of consumer.


That is to say:


If we solve the problem with profit maximization problem, we can still get the relative relationship between x1 and x2, but not the exact value. However, we can transform certain part of the solution into the form of output y, and get exact solution as x(p, y).


substitute (4) into (3):









(5)
Substitute (5) into (4), we have:
(6)
(5) and (6) are the conditional demand of input x(p, y) of the problem. When p1, p2 and y are given, we can have exact value for x1 and x2.


Properties of conditional input demand x1 and x2:


To further simplify the conditional demand, consider the unit cost , where


Property of the unit cost function:


Property of total cost function


Note: the unit cost function c(p1, p2) is not related with specific input use. Instead, it is an optimal variable, a feature of the specific production function.


To further relate conditional demand with unit cost, we can:.
Start with unit cost function

Denote


Note: the notion of A will be used in the linearization of unit cost function below.


Recall that the conditional demand of inputs is:

Then we have:

Recall in CES function, we have the constant elasticity of substitution as
Similarly, we have:


Implication: here δ, α are parameters. When the unit cost relative to input i's own price () increases, the conditional demand of input i would also increase.


Question: Still, I am not sure why to convert the conditional demand in the form of unit cost function. The original conditional demand function is already a function of exogenous variables and parameters, but when converted in the form of unit cost function, we actually incorporate additional endogenous function . Think about this question and keep reading.


Answer to question above: for the producer we have price equals to marginal cost, which further equals unit cost. So by putting into conditional demand function can model the impact of output price change on input use. So there the is not the endogenous variable anymore, it equals to the exogenous variable of output price.


Linearization of conditional demand

Here I use A and B to refer to common variables (note: here A is not the notion above )
Consider the proportional change of A as ,


Operation of proportional changed term:




(Refer to Linearization in Economic Models)


Use rules here to linearize x1(p1, y, cy) and x2(p2, y, cy)


By doing this, it decompose the proportional change of input demand xi into:


Here I have:


Recall the original CES production function:

The original demand function

And the unit cost function:


Linearization of unit cost function

Two ways to linearize the unit cost function:


1) "Brute force" method

From the step in solving producer's problem:

We have:


Total differentiation of the equation above, regarding to α, , and (they are variables in the decomposition of conditional demand function, while ρ or σ are parameters)



Recall that for variable A, we have , then we can simplify the equation above as:








Recall we have:


Then we can get

And

Then we have:

So we can write the function above as:


Denote cost share of input i as
Then we can write the function above as

Implication: the change of unit cost is a function of three technical change (netural technical change and biased technical change ) and two price change .


2) Based on homogeneous of degrees one in input price

Use Euler theorem and Shephard's lemma.


Linearization of cost share

from the cost share equation:


We have the linearized form


Recall we have the linearization of x:


And linearization of cy:

And recall
Substitute these two functions into the linearization of θ (take θ1 as example), we have:



Note here:

So we have:


Denote as the proportional change of the "effective price" of input i.


Then we have
Feature of this expression:


In GTAP model, the CES function is used for producer on


GTAP model



in the form before, we have for the value added CES:


Then the linearized conditional demand of primary factors now becomes
qfe = qva + ESUBVA( (ESUBVA-1) / ESUBVA afe + pva - pfe )


Rearrange it, we get the ENDWDEMAND (endowment demand) equation of GTAP model:
qfe = -afe + qva - ESUBVA(pfe - afe - pva)


Recall the demand function for land input in SIMPLE-G:
E_QLANDgl (all,g,GRID)(all,l,LTYPE)
p_QLANDgl(g,l)

= p_QLANDWTRgl(g,l) - p_AFLAND(g,l)
- EIRRIGgl(g,l) * [p_PLANDgl(g,l) - p_AFLAND(g,l) - p_PLANDWTRgl(g,l) ] ;


Note: note the variable of "p_AFLAND" is the afe in GTAP model. The key here is to understand the definition of afe (biased technical parameter and its relationship with δ: afe = , so we can get the equation in model from the linearization of conditional demand from solving cost minimization problem of CES production function.



unit cost of value added can be obtained from the linearization of unit cost

pva = Σ(sva(pfe - afe))


Recall: afe = :


This is the equation VAPRICE in the GTAP model


Note:
To emphasis it here:
From producer's cost minimization function (given output level and input price), but no information from other functions
We solve for

And linearize them with respect to output level, input price, cost share, substitution elasticity and other parameters in production function.


Similarly, we can have the equation of value added and intermediate as input of qo production:


Note: here ao is the unbiased technical parameter, not the biased technical parameter of qo (it does not exist since qo is already the top-level output in the economy)
qva = qo + ESUBT( ava + ps - pva) + (ESUBT-1)ao
qf = qo + ESUBT( af + ps - pf) + (ESUBT -1)ao


Rearrange them, we have the equation VADEMAND:
qva = -ava + qo - ao - ESUBT(pva - ava - ps -ao)


And equation INTDEMAND
qf = -af + qo - ao - ESUBT(pf - af - ps - ao)


Also, the price = unit cost would be the zero-profit condition in GTAP, which is given by linearization of unit cost function.


ps + ao = θf (pf - af) + θva (pva - ava)


The value of pva is replaced by the unit-cost equation from value-added production function:
pva = Σ(sva(pfe - afe))


Actually this function is the same as the one above it.


Note:
For each production function, it takes:

And solves:


Reference: Gohin, Alexandre, and Thomas Hertel. "A note on the CES functional form and its use in the GTAP model." Center for Global Trade Analysis, Department of Agricultural Economics, Purdue University (2003). https://www.gtap.agecon.purdue.edu/AgEc618/modules/readings/RM2.pdf