A Nonparametric Formula Relating the Elasticity of a Factor Demand to the Elasticity of Substitution

Abstract

It is well known for a Cobb-Douglas production function that the elasticity of a factor demand is the inverse of the share of output going to the other factors. Since Cobb-Douglas has a unit elasticity of substitution, the demand elasticity trivially equals the ratio of the elasticity of substitution to the share of output going to the other factor. I show here that this result can be generalized to any constant returns to scale production function. As a result, if a factor is known to be a substitute for (complement of) other factors, the inverse of the share of output going to other factors will be a lower (upper) bound for the factor’s elasticity of demand.