We mentioned that the Morishima elasticity of substitution is, in general, asymmetric. Many empirical studies

of cost report an alternative but symmetric de nition of the elasticity of substitution between inputs. This is first proposed by R.G.D.Allen in the 1930s and further clari ed by H.Uzawa in the 1960s. The Allen elasticity of

substitution is given by Aij = (CijC)/(CiCj), where C is the cost function, Ci, Cj are the partial derivatives with

respect to various inputs, and Cij is the second-order cross partial derivative of the cost function.

a) Show that Aij = exci, wj/sj where xci,wj are the contingent demand for input i and price for input j, exci, wj is the cross-price contingent demand elasticity and sj is the cost share of the input j