The Allen elasticity of substitution 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 defined in footnote 6 of Chapter 9:\( A_{i j}=C_{i j} C / C_{i} C_{j} \)where the subscripts indicate partial differentiation with respect to various input prices. Clearly, the Allen definition is symmetric. a. Show\( A_{i j}=e_{x_{i}^{c}, w_{j}} / s_{j} \), where \( s_{j} \)is the share of input j in total cost. b. Show that the elasticity of si with respect to the price of input j is related to the Allen elasticity by\( e_{s_{i}, p_{i}}=s_{j}\left(A_{i j}-1 ight) \) c. Show that, with only two inputs,\( A_{k l}=1 \)for the Cobb-Douglas case and\( A_{k l}=\sigma \)for the CES case. d. Read Blackorby and Russell (1989: ''Will the Real Elasticity of Substitution Please Stand Up?'') to see why the Morishima definition is preferred for most purposes.