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Chapter 10: Problem 12

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

Expert verified
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

01

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.
02

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.
03

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.
04

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.

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Most popular questions from this chapter

The own-price elasticities of contingent input demand for labor and capital are defined as \\[ e_{Y},_{w}=\frac{\partial l^{c}}{\partial w} \cdot \frac{w}{l^{c}}, \quad e_{k^{\prime}, v}=\frac{\partial k^{c}}{\partial v} \cdot \frac{v}{k^{\ell}} \\] a. Calculate \(e_{\mathbb{R}, \text { w }}\) and \(e_{k_{k}, v}\) for each of the cost functions shown in Example 10.2 b. Show that, in general, \(c_{r}, w+e_{L r, v}=0\) c. Show that the cross-price derivatives of contingent demand functions are equal-that is, show that \(\partial f / \partial v=\partial k^{c} / \partial w\). Use this fact to show that \(s_{1} e_{\gamma, v}=s_{k} e_{k^{\prime}}, w\) where \(s_{b} s_{k}\) are, respectively, the share of labor in total cost \((w l / C)\) and of capital in total cost \((v k / C)\) d. Use the results from parts (b) and (c) to show that \(s_{i} €_{l^{\prime}, s, s}+s_{k} c_{k^{\prime}, w}=0\) e. Interpret these various elasticity relationships in words and discuss their overall relevance to a general theory of input demand.

An enterprising entrepreneur purchases two factories to produce widgets. Each factory produces identical products, and each has a production function given by \\[ q=\sqrt{k_{i} l_{i}}, \quad i=1,2 \\] The factories differ, however, in the amount of capital equipment each has, In particular, factory 1 has \(k_{1}=25,\) whereas factory 2 has \(k_{2}=100 .\) Rental rates for \(k\) and \(l\) are given by \(w=v=\$ 1\) a. If the entrepreneur wishes to minimize short-run total costs of widget production, how should output be allocated between the two factories? b. Given that output is optimally allocated between the two factories, calculate the short-run total, average, and marginal cost curves. What is the marginal cost of the 100 th widget? The 125 th widget? The 200 th widget? c. How should the entrepreneur allocate widget production between the two factories in the long run? Calculate the long-run total, average, and marginal cost curves for widget production. d. How would your answer to part (c) change if both factories exhibited diminishing returns to scale?

Suppose the total-cost function for a firm is given by \\[ C=q(v+2 \sqrt{v w}+w) \\] a. Use Shephard's lemma to compute the (constant output) demand function for cach input, \(k\) and \(l\). b. Use the results from part (a) to compute the underlying production function for \(q\) c. You can check the result by using results from Fxample 10.2 to show that the CES cost function with \(\sigma=0.5, \rho=-1\) generates this total-cost function.

Suppose that a firm produces two different outputs, the quantities of which are represented by \(q_{1}\) and \(q_{2}\). In general, the firm's total costs can be represented by \(C\left(q_{1}, q_{2}\right) .\) This function exhibits economies of scope if \(C\left(q_{1}, 0\right)+C\left(0, q_{2}\right)>C\left(q_{1}, q_{2}\right)\) for all output levels of either good. a. Explain in words why this mathematical formulation implies that costs will be lower in this multiproduct firm than in two single-product firms producing each good separately. b. If the two outputs are actually the same good, we can define total output as \(q=q_{1}+q_{2}\). Suppose that in this case average cost \((=C / q)\) decreases as \(q\) increases. Show that this firm also enjoys economies of scope under the definition provided here.

Suppose the total-cost function for a firm is given by \\[ C=q w^{2 / 3} v^{1 / 3} \\] a. Use Shephard's lemma to compute the (constant output) demand functions for inputs \(l\) and \(k\). b. Use your results from part (a) to calculate the underlying production function for \(q\)
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