Producer behavior

Overview

Production tree for output p: a two-layer nested CES functions

Assumptions:

• separability between layers
• constant return to scale production function on each layer

ESUBT: substitution elasticity between all intermediates, and also between intermediates and value added. It is the full elasticity of substitution (σ in CES function).

ESUBVA: substitution elasticity

QO final product

• QF: intermediae input
• QVA: value added
  • QFE: factor inputs
    • Land
    • Labor
    • Capital

Value-added nest

Production function

Where:

• p: the index of output
• e: the index of primary factor
• : elasticity of substitution of the CES function for value added nest.

Solve the producer's cost minimization problem, we have the demand function of each input (QFE) and the price index (price for QVA).

Demand function

Level form

Input demand in levels:

Linearized form

Total differentiating for X1, Y, P1, and P, and dividing by X1, to get the percentage change term:

Price index function

Price index is the price of nested input, here is VA

Level form

Price index

Linearized form

Total differentiating for P, P1 and P2, rewrite the equation with cost share of inputs S:

Technical change variables

ao(p) output augmenting variable
afe(e,p) primary factor augmenting variable
af(t,p) composite intermediate commodity augmenting variable
ava(p) value-added augmented variable

Where:

• p: index of produced commodities
• t: index of traded commodities
• e: index endowment commodities

(For definition of commodities, refers to sets in GTAP)

Now, let us add technological change variables in the production function of value-added layer, then the level equation becomes :
Demand function:

CES price equation:

Where ESUBVA is the substitution parameter above.

I can also refer to the more detailed notes on linearization of CES function

Final product nest

In the top layer, we use value-added inputs (QVA) and Intermediate inputs (QF) to produce outputs (QO). In the similar manner as above, we have the linearized form of demand function for QVA and QF below:

Demand function

Linearized form

Value added (QVA): Equation VADEMAND

qva(j) = qo(j) - ESUBT(j) (pva(j) - ps(j))
where:

• j: index of PROD_COMM

Intermediate inputs (QF): Equation: INTDEMAND
qf(i,j) = qo(j) - ESUBT(j) (pf(i,j) - ps(j))
Where:

• i: index of TRAD_COMM
• j: index of PROD_COMM

(To review commodities in each set, refers to sets in GTAP)

Note

Note 1:

In this section, it seems that we solve the linearized form for demand and price index directly, which does not require APE. The reason is that results shown on this slide is already the outcome after applying a series of restrictions on production function, where we use APE in general restrictions and further use CES function in particular restrictions.

If we start with the generalized functional form for production (see the theory-based explanation in restrictions on production function), in order to linearized the derived demand for a model with N inputs, we need to have N*N output constant price elasticities of demand .

To reduce the number of parameters needed, we define APE , so instead of using N*N parameters for , now we only need parameters for .

To further reduce the number of parameters needed we apply the particular restrictions on Separability and the CES functional form, which allows use to calculate APE from elasticity of substitution parameters (ESUBT, ESUBVA) of CES function from Keller's formula.

So in this note, we jump over all intermediate procedures, depict the producer's behavior with nested CES function and solve the linearized form with ESUBT, ESUBVA

It seems that APEs are not necessary in the final solution, but they are important to understand the entire series of procedures.

Note 2:

It is possible that two primary factors (k and l) care substitutes in VA, but complements overall
that is to say, , or the increase of one factor's price decrease the demand of the other factor.

That occurs when
ESUBT > ESUBVA / (1 - VASHARE)

Example