Example 7: Consider the following SAM:

Example 9a: Consider the following SAM, where sectors have symmetric intermediate demand

The production function in the model is a two-level CES function, so called nested function. We use any abbreviation like "VA", "LY", etc. to let MPSGE know that we are going to work with nested function.

- L and K form a Cobb-Douglass aggregate at the bottom level:

f =A * K^d* L^1-^d

- At the top level, Y and f(L,K) have an elasticity of substitution equal to 0.5:

X = B * [ d^(1/^s)* Y⁽^s-1)/^s+ (1-d)^(1/^s)* f(K,L)⁽^s-1)/^s] ^s/(^s-1)

where d - share of Y in production of X

s - elasticity of substitution between Y and (K,L)

A, B - technological parameters

K, L - production factors

Y - production input for X

PX, PY, PK, PL - price of X, Y, K , L

Pf - price of composite f(K,L)

Alternatively we can use dual form of production functions:

1) Cobb-Douglass function:

min (PK * K + PL * L) s.t. f =A * K^d* L^1-^d

c = f * 1/A * PK^d* PL^1-^d * d^-^d * (1-d)^d-1

The dual form is: Pf = 1/A * PK^d* PL^1-^d

2) CES function:

min (PY * Y + Pf * f(K,L)) s.t. X = B * [ d^(1/^s)* Y⁽^s-1)/^s+ (1-d)^(1/^s)* f(K,L)⁽^s-1)/^s] ^s/(^s-1)

c = X * 1/B * [ d* PY^(1-^s⁾+ (1-d)* Pf^(1-^s⁾] ^1/(1-^s⁾

The dual form is: PX = 1/B * [ d* PY^(1-^s)+ (1-d)* Pf^(1-^s)] ^1/(1-^s)