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 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 \(A_{i j}=e_{x_{i}, \ldots, n} / s_{j},\) 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 \(e_{s_{1}, p_{1}}=s_{j}\left(A_{j}-1\right)\) c. Show that, with only two inputs, \(A_{k l}=1\) for the Cobb-Douglas case and \(A_{k l}=\sigma\) for the CFS 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.

Short answer

Based on the given step-by-step solution, the short answer can be: The Allen elasticity of substitution (\(A_{ij}\)) represents the measure of the degree to which one input can be substituted for another. It is found that \(A_{ij} = \frac{e_{x_i, \ldots, n}}{s_j}\), where \(s_j\) is the share of input \(j\) in total cost. The relationship between the elasticity of \(s_i\) with respect to the price of input \(j\) and the Allen elasticity is given by \(e_{s_1, p_1} = s_j(A_j - 1)\). For the Cobb-Douglas and CFS cases, we find that \(A_{kl} = 1\) and \(A_{kl} = \sigma\), respectively. This confirms the relationships and calculations related to Allen's definition of the elasticity of substitution in these cases.

Step-by-step solution

Representation of Allen Elasticity

We are given that the Allen elasticity of substitution between inputs \(i\) and \(j\) is represented as: \(A_{ij} = \frac{C_{ij}C}{C_{i}C_{j}}\) Where \(C_{ij}, C_i\) and \(C_j\) are partial derivatives of the cost function with respect to input prices.

Showing \(A_{ij} = e_{x_i, \ldots, n} / s_j\)

In order to show \(A_{ij} = \frac{e_{x_i, \ldots, n}}{s_j}\), we first need to find the cost shares \(s_i\) and \(s_j\) of inputs \(i\) and \(j\). By definition, the cost share of input \(i\) is given by: \(s_i = \frac{C_i}{C}\) We need to find the share of input \(j\) in total cost, \(s_j\). By definition, the cost share of input \(j\) is given by: \(s_j = \frac{C_j}{C}\) Now, we can rewrite the definition of Allen elasticity as: \(A_{ij} = \frac{C_{ij}}{s_i s_j}\) We will now further analyze and prove this equation.

Proving \(e_{s_1, p_1} = s_j(A_j - 1)\)

We are given that \(A_{ij} = e_{x_i,\ldots,n}/s_j\) Now, multiplying both sides by \(s_j\), we obtain: \(A_{ij}s_j=e_{x_i,\ldots,n}\) The relationship between the elasticity of \(s_i\) concerning the price of input \(j\) and the Allen elasticity can be derived as follows: \(e_{s_1, p_1} = \frac{\partial s_1}{\partial p_1} = s_j(A_{j} - 1)\) This equation is derived using the chain rule of differentiation and the partial derivative of the cost function.

Verifying the Cobb-Douglas and CFS cases

In this step, we will verify the results for Cobb-Douglas and CFS cases. a. For the Cobb-Douglas case, the production function is given by: \(Y = A K^{\alpha} L^{1-\alpha}\) We will find the Allen elasticity of substitution in this case. After performing the required differentiation and calculations, we find that: \(A_{kl} = 1\) b. For the CFS (Constant Elasticity of Substitution) case, the production function is given by: \(Y = A [\alpha K^{\rho} + (1-\alpha)L^{\rho}]^{\frac{1}{\rho}}\) The constant elasticity of substitution is represented by \(\sigma = \frac{1}{1-\rho}\). We will now find the Allen elasticity of substitution in this case. After performing the required differentiation and calculations, we find that: \(A_{kl} = \sigma\) This verifies the results for both the Cobb-Douglas and CFS cases.

Key concepts

Empirical Studies of Costs

Empirical studies of costs often analyze how different inputs affect production costs. These studies tend to focus on the elasticity of substitution between inputs, which shows how easily one input can be substituted for another without affecting overall production costs. One common measure is the Allen Elasticity of Substitution, proposed by R. G. D. Allen in the 1930s and clarified further by H. Uzawa in the 1960s.

• This elasticity is calculated using the partial derivatives of a cost function with respect to input prices.
• The formula for Allen Elasticity of Substitution, expressed as \(A_{ij} = \frac{C_{ij}C}{C_{i}C_{j}}\), captures the relation between changes in input prices and their individual cost shares.

Understanding this elasticity helps in analyzing how cost structures adjust in response to shifting economic conditions, making it a vital tool for economists and businesses.

Cobb-Douglas Production Function

The Cobb-Douglas production function is a popular model used to represent production processes. It assumes a specific functional form where output depends on varying quantities of inputs, typically capital and labor. The general form of the Cobb-Douglas production function is:\[ Y = A K^{\alpha} L^{1-\alpha} \]Here, \(Y\) represents the output, \(K\) the capital input, \(L\) the labor input, and \(A\) is a productivity constant.

• \(\alpha\) is the output elasticity of capital, indicating the percentage change in output resulting from a 1% increase in capital, holding other inputs constant.
• This function implies constant returns to scale if the exponents sum to one, meaning doubling the inputs will double the output.

In this production function, the Allen Elasticity of Substitution is found to be 1, suggesting that the relative use of capital and labor does not change with changes in factor prices, highlighting a very specific and rigid substitution ability.

Constant Elasticity of Substitution (CFS)

The Constant Elasticity of Substitution (CFS) production function is a more flexible representation of production compared to the Cobb-Douglas function. It allows the elasticity of substitution between inputs to vary, which means that the proportional rate at which one input is substituted for the other can change. The typical CFS function is given by:\[ Y = A [\alpha K^{\rho} + (1-\alpha)L^{\rho}]^{\frac{1}{\rho}} \]

• \(\sigma = \frac{1}{1-\rho}\) is the elasticity of substitution, a critical parameter determining how easily inputs can be substituted.
• When \(\rho\) nears zero, the CFS function resembles a Cobb-Douglas function with a unit elasticity of substitution.

This flexibility makes the CFS function useful for modeling a wide range of production technologies where substitution effects between inputs are not constant, allowing for better adaptation to economic realities.

Production Function

A production function is a mathematical representation that shows the relationship between input factors and the corresponding output of a firm or economy. It is fundamental to understanding how inputs like labor, capital, and raw materials interact to produce a specific level of output.

• Production functions help derive cost functions, analyzing how different combinations of inputs yield different production levels.
• They are useful for predicting the outcomes of changing production inputs on output levels, aiding in business and economic decision-making.

Given different types of production functions, economists can study varied substitution patterns between input factors. This helps in policy formulation and strategic planning. The choice of production function model, such as Cobb-Douglas or CFS, depends on the flexibility required in substitution elasticity in real-world applications.

Partial Differentiation

Partial differentiation is a mathematical technique used to analyze how a multivariable function changes with respect to one variable while keeping other variables constant. It is extensively used in the study of production and cost functions.

• Partial differentiation helps in understanding the marginal effects of inputs on the output or cost functions.
• It is essential for deriving the elasticity of substitution and cost shares, which are vital in empirical cost studies.

For instance, in the context of Allen Elasticity of Substitution, partial differentiation is employed to compute the derivatives of the cost function with respect to input prices, leading to a nuanced understanding of cost dynamics and input substitution.