Definition
Allen partial elasticities of substitution (APEs) are the cost-share weighted conditional demand elasticities (constant-output).
Where:
: output-constant demand elasticity for the quantity. It is the elasticity of input i with respect to the price of input j, holding output constant
: the share of input j in total costs.
APEs appear in derived demand equations of sectors in GE model
Additional explanation from "Markandya, Anil, and Suzette Pedroso-Galinato. "How substitutable is natural capital?." Environmental and Resource Economics 37.1 (2007): 297-312.":
The full elasticity of substitution between inputs i and j is 
, where:
Where:
are levels of input i and j respectively. - The production function is
However, the full elasticity of substitution is not always available. Thus more commonly, we use the Allen partial elasticity of substitution 
Where:
: share of total cost of production represented by input j
: price of input j
Interpretation of : the weighted cross price elasticity of demand for input i with respect to price of input j.
Relationship between full elasticity of substitution and Allen partial elasticity of substitution:
Where:
: a simple unweighted cross price elasticity of demand for input i, 
. It gives the proportional change in the use of input i for a proportional change in the price of input j.
So 
, which is the same results as definition on the top of the page:
That means 
and 
refers to the same elasticity, the output-constant demand elasticity, or the elasticity of input i with respect to the price of input j, holding output constant
Calculate Allen Partial Elasticity
APE can be calculated with CES parameters and shares, using Keller's formula
Note
Here we talk about three elasticities:
- full elasticity of substitution:
: the elasticity of the ratio of two inputs / goods to a production / utility function with respect to the ratio of their marginal products / utilities (at competitive market, the ratio of marginal product equals the ratio of price) - unweighted output-constant cross-price elasticity of demand,
or 
: the elasticity of input i with respect to the price of input j, holding output constant - cost-share weighted constant-output cross-price elasticity of demand, or Allen partial elasticity of substitution,
or 
Relationship between three elasticities:
