Many empirical studies of costs report an alternative definition of the elasticity of substitution between

inputs. This alternative definition was first proposed by R. G. D. Allen in the 1930s and further clarified

by H. Uzawa in the 1960s. This definition builds directly on the production function-based elasticity of

substitution given by: A_ij = (C_ijC)/(C_iC_j) , where

• C is the cost function,
• C_i,C_j is the partial derivatives with respect to various input prices.
• C_ij is the second-order cross partial derivative of the cost function

Clearly, the Allen definition is symmetric.

a) Show that (A_ij = e_xi,wj / s_j ) , where x_i is the contingent demand for input i; and w_j is price for input j; s_j is the cost share of input j in total cost.

b) Show that the elasticity of s_i (cost share of input i ) with respect to the price of input j is related to the Allen elasticity by: e_si,wj = s_j (A_ij -1)

c) Show that, Aij = 1 for a Cob-Douglas cost function: C(v,w,q) = q^1/(a+b) Bw^b/(a+b) v^a/(a+b) where B = (a+b)a^-a/(a+b)b^-b/(a+b)

Also show that Aij = ? for a CES function: C(v,w,q) = q^1/? (v^1-? + w^1-? )^1/(1-?) (with v is the price of input i, w is price of input j, )

(The elasticity of substitution is defined as in the attached picture)