1 A Recursive Dynamic Computable General Equilibrium Model For ...

σ2
σ1
Figure A1: A Two-level nested production structure
It is assumed that the underlying production function (A.1) is weakly separable
implying (using Chambers’ (1988) notation):
∂
∂x
3
⎛ ∂X
/ ∂x
⎜
⎝ ∂X
/ ∂x
1
2
⎞
⎟ = 0
⎠
52
(A.3)
In words, this expression states that the ratio of marginal products (MRTS) of inputs
x1 and x2, belonging to the same input nest X, is not affected by changes in the level of
input usage of x3 which is not in that nest. The family of convenient functions such as CD
and CES exhibit weak seperability, where in the case of a two-level nested CD production
example:
Y = AX X and X = Ax x
(A.4)
α
1
β
2
The MRTS 11,21 can be shown to be:
MRTS
11,
21
MP
γ x
1
11 21
= =
(A.5)
MP21
δ x11
Clearly, changes in the level of X2 in the upper CD nest, has no effect on the MRTS
between inputs x11 and x21 in the lower nest. Mathematically:
∂ ⎛ γ x
⎜
∂X
2 ⎝ δ x
21
11
⎞
⎟ = 0
⎠
Y
X x3
x1 x2
γ
11
δ
21
(A.6)