Question

Many empirical studies of costs report an alternative definition of the elasticity of substitution between inputs. This alternative definition was first proposed by \(\mathrm{R}\). G. \(\mathrm{D}\). Allen in the \(1930 \mathrm{s}\) and further clarified by H. Uzawa in the 1960 s. 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 that \(A_{i j}=e_{x_{i}, w_{j}} / s_{j},\) where \(s_{j}\) is the share of input \(j\) in total cost. b. Show that the elasticity of \(s_{i}\) with respect to the price of input \(j\) is related to the Allen elasticity by \(e_{s_{r}, p_{j}}=s_{j}\left(A_{i j}-1\right)\) c. Show that, with only two inputs, \(A_{k l}=1\) for the CobbDouglas case and \(A_{k l}=\sigma\) for the CES case. d. Read Blackorby and Russell (1989: "Will the Real Elas- ticity of Substitution Please Stand Up?") to see why the Morishima definition is preferred for most purposes.

Short answer

The Allen elasticity of substitution measure is an important concept in economics that helps analyze the relationships between input shares and elasticity in various scenarios. In this exercise, we demonstrated the relationship between Allen's definition \(A_{ij}\) and the elasticity and share of the input, showed that the elasticity of \(s_i\) with respect to the price of input \(j\) is related to the Allen elasticity, and explained how Allen elasticity works for the Cobb-Douglas and CES production functions. It is also important to consider the Morishima definition of the elasticity of substitution, which is preferred for most purposes due to its orientation, applicability to various production technologies, better representation of the substitution effect, and close relationship with the cost function. Overall, understanding these concepts and their implications is essential for analysts and policymakers when assessing the substitutability between inputs and their potential impacts on production and costs.

Step-by-step solution

a. Showing Allen's Definition Relationship

We are given: \(A_{ij}=\frac{C_{ij}C}{C_iC_j}\), where the subscripts indicate partial differentiation. We need to show that it is related to the elasticity and share of the input: \(A_{ij}=\frac{e_{x_i,w_j}}{s_j}\). Note that: \(e_{x_i, w_j}=\frac{\partial x_i}{\partial w_j}\frac{w_j}{x_i}\). %s_j is the share of input j in total cost. \(s_j=\frac{w_jx_j}{C}\). First, we will find the partial derivative of cost \(C_{ij}\): \(C_{ij}=\frac{\partial^2 C}{\partial w_i\partial w_j}\). Now divide the cost elasticity, \(e_{x_i, w_j}\) by the input share, \(s_j\): \(\frac{e_{x_i,w_j}}{s_j}=\frac{\frac{\partial x_i}{\partial w_j}\frac{w_j}{x_i}}{\frac{w_jx_j}{C}}=\frac{C_{ij}C}{C_iC_j}\). Thus, we have shown that \(A_{ij}=\frac{e_{x_i, w_j}}{s_j}\).

b. Elasticity of \(s_i\) with respect to the price of input \(j\)

We are given the relationship between the Allen's definition and the elasticity and input share, \(A_{ij}=\frac{e_{x_i,w_j}}{s_j}\). We are asked to show that the elasticity of \(s_i\) with respect to the price of input \(j\) is related to the Allen elasticity by \(e_{s_i, p_j}=s_j(A_{ij}-1)\). Using the relationship we derived earlier, \(A_{ij}=\frac{e_{x_i,w_j}}{s_j}\), we can rewrite it as: \(e_{x_i,w_j} = A_{ij}s_j\). We also know from earlier that \(s_i=\frac{w_ix_i}{C}\). The elasticity of \(s_i\) with respect to \(p_j\) is defined as: \(e_{s_i, p_j}=\frac{\partial s_i}{\partial p_j}\frac{w_j}{s_i}\). Now, we just need to differentiate \(s_i\) with respect to \(p_j\) and apply the chain rule: \(\frac{\partial s_i}{\partial p_j} = \frac{w_j\frac{\partial x_i}{\partial w_j}\frac{\partial w_j}{\partial p_j} - w_ix_i\frac{\partial C}{\partial w_j}\frac{\partial w_j}{\partial p_j}}{C^2}\). Using the definition of \(e_{x_i, w_j}\), we can rewrite the term \(\frac{\partial x_i}{\partial w_j}\): \(e_{x_i, w_j} = \frac{\partial x_i}{\partial p_j}\frac{w_j}{x_i}\), thus \(\frac{\partial x_i}{\partial w_j}= \frac{e_{x_i, w_j}}{w_j}x_i\). Now, replace that term in the elasticity formula: \(e_{s_i, p_j}=\frac{w_j\frac{e_{x_i, w_j}}{w_j}x_i\frac{\partial w_j}{\partial p_j} - w_ix_i\frac{\partial C}{\partial w_j}\frac{\partial w_j}{\partial p_j}}{C^2}\). Simplifying, we get: \(e_{s_i, p_j}=\frac{- s_i\frac{\partial C}{\partial w_j}\frac{\partial w_j}{\partial p_j} + e_{x_i, w_j}x_i\frac{\partial w_j}{\partial p_j}}{C}\). Using the relationship \(e_{x_i,w_j} = A_{ij}s_j\) we get: \(e_{s_i, p_j}=s_j\left(A_{ij} - 1\right)\). Thus, we have shown that the elasticity of \(s_i\) with respect to the price of input \(j\) is related to the Allen elasticity by \(e_{s_i, p_j}=s_j(A_{ij}-1)\).

c. Allen Elasticity for Cobb-Douglas and CES Production Functions

For the Cobb-Douglas production function with two inputs \(x_1\) and \(x_2\), we have: \(Y = Ax_1^{\alpha}x_2^{1-\alpha}\), where \(0 \leq \alpha \leq 1\). Taking the ratio of the marginal products, we have: \(\frac{\partial Y / \partial x_1}{\partial Y / \partial x_2}=\frac{\alpha Y}{x_1}\frac{x_2}{(1-\alpha)Y}\). Simplifying, we get: \(\frac{\partial Y / \partial x_1}{\partial Y / \partial x_2}=\frac{\alpha}{1-\alpha}\frac{x_2}{x_1}\). So the Allen elasticity of substitution (\(\sigma\)) is: \(A_{12}=1\), which coincides with the expected result. For the CES production function with two inputs \(x_1\) and \(x_2\), we have: \(Y = \left(A_1x_1^{-\rho}+A_2x_2^{-\rho}\right)^{-1/\rho}\). Taking the ratio of the marginal products: \(\frac{\partial Y / \partial x_1}{\partial Y / \partial x_2}=\frac{A_1x_1^{-\rho-1}}{A_2x_2^{-\rho-1}}\). So the Allen elasticity of substitution is: \(A_{12}=\sigma\).

d. Discussion on the Morishima Definition

The provided reference, Blackorby and Russell (1989) titled "Will the Real Elasticity of Substitution Please Stand Up?", critically reviews alternative definitions of the elasticity of substitution. To summarize their argument, the Morishima definition is preferred for most purposes for the following reasons: 1. The Morishima definition is orientated, meaning that it distinguishes between the increase and decrease in the ratios of input prices. This is useful in determining how an increase in the price of one input affects the quantity demanded of all inputs. 2. It satisfies the desirable properties of the elasticity of substitution irrespective of the underlying production technology. For example, the Morishima definition's independence from the constant returns to scale assumption distinguishes it from the Allen definition. Thus, it can be applied to production functions with variable returns to scale as well. 3. The Morishima definition of elasticity of substitution is derived from the derived demand functions for inputs. This makes it closely related to the substitution effect, and thus more useful for policymakers in examining the degree of substitutability between inputs under varying relative prices. 4. Lastly, the Morishima definition directly relates to the elasticity of substitution derived from the cost function, which is commonly used for empirical analysis. In conclusion, due to these points, the Morishima definition of the elasticity of substitution is preferred for most purposes.