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Keller's Formula


Definition

Keller's formula: sectoral APE can be written in terms of CES parameters and shares


Suppose the production function is nested


Y = G(VA(K, L), F, M, S)


: the substitution elasticity among intermediate inputs (F, M, S) and value-added (VA)


: the substitution elasticity among components of value-added


Let M, N = TRAD_COMM {F, M, S}
i, j = ENDW_COMM {K, L}


To solve the substitution elasticity between any inputs, we need to:
(1) Solve the conditional demand from profit maximization problem
(2) take total derivative of each demand
(3) divided it by the level of the demand itself, and times 100, to make the equation of percentage change.
(4) The elasticity of substitution can be calculated with the percentage change function


So we have the equation to calculate cross-input substitution elasticity (Allen partial elasticity of substitution) , from CES substitution elasticities , and cost share of each inputs:
Note: the cost share of VA equals to the sum of cost share of F and L


Simplified version of Keller's formula

For all M, N in the set TRAD_COMM, which is the set of output (recall the set in GTAP)


Note: VA refers to the value-added nest.


For all i, j in the set ENDW_COMM


For endowment the change of price will have composite effects. For example, the rise of labor price causes:


From weak separability, we have
The note on specific production function further says


Note:
For large value of relative to , it is possible that < 0
Recall > 0 means the price increase of factor i causes the increase of other factor's (j) quantity, or i and j substitute with each other.
When < 0, the price increase of factor i cause the decrease of factor j's quantity, or i and j are complements. The reason is that when it is very easy to substitute between intermediate input and value added, the price increase in one factor causes the price increase of value added, then the demand of the entire value-added nested decrease, so are all factors.


Mathematically, this condition can be represented by:


General version of Keller's formula





Where:
K: the lowest common level at which a component exists associated with n and m
L: the highest level in the nested production function
: the Allen partial elasticity of substitution associated with the l the level nest in which input n is involved
is a cost share at the l th level, defined by


Note

Keller's formula allows us to calculate between:


Example