Elasticities of substitution and complementarity: A synthesis

We derive the relationships between the net and gross elasticities of substitution and complementarity (i.e., the elasticities that refer either to the conditional or unconditional, direct or inverse demand system) in the general case of non-homothetic, variable-returns-to-scale technologies. We also show that the so-called Hicks Elasticity of Complementarity (Hicks, 1970) is dual to a full-fledged elasticity of gross input substitution that we call the Hotelling/Lau Elasticity of Substitution (Lau, 1978). The former is, in fact, the proper elasticity of substitution in the case of the inverse, unconditional input demand. Our results should clarify some issues about the input substitutability classification