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Chapter 11: Problem 13

With two inputs, cross-price effects on input demand can be easily calculated using the procedure outlined in Problem 11.12 a. Use steps (b), (d), and (e) from Problem 11.12 to show that \\[ e_{K, w}=s_{L}\left(\sigma+e_{Q, P}\right) \quad \text { and } \quad e_{L, v}=s_{K}\left(\sigma+e_{Q, P}\right) \\] b. Describe intuitively why input shares appear somewhat differently in the demand elasticities in part (e) of Problem 11.12 than they do in part (a) of this problem. c. The expression computed in part (a) can be easily generalized to the many- input case as \(e_{x_{i}, w_{i}}=s_{j}\left(A_{i j}+e_{Q, P}\right),\) where \(A_{i j}\) is the Allen elasticity of substitution defined in Problem 10.12 . For reasons described in Problems 10.11 and 10.12 , this approach to input demand in the multi-input case is generally inferior to using Morishima elasticities. One oddity might be mentioned, however. For the case \(i=j\) this expression seems to say that \(e_{L, w}=s_{L}\left(A_{L L}+e_{Q . P}\right),\) and if we jumped to the conclusion that \(A_{L L}=\sigma\) in the two-input case, then this would contradict the result from Problem \(11.12 .\) You can resolve this paradox by using the definitions from Problem 10.12 to show that, with two inputs, \(A_{L L}=\left(-s_{K} / s_{L}\right) \cdot A_{K L}=\left(-s_{K} / s_{L}\right) \cdot \sigma\) and so there is no disagreement.

Short Answer

Expert verified
Question: Derive the cross-price effects on input demand for a production function with two inputs, explain the intuitive reason behind the differences in input shares regarding demand elasticities, and generalize the derived expression for many-inputs. Answer: The cross-price effects on input demand for a production function with two inputs, K and L, can be found as \(e_{K, w}=s_{L}\left(\sigma+e_{Q, P}\right)\) and \(e_{L,v}=s_{K}\left(\sigma+e_{Q, P}\right)\). The difference in input shares and demand elasticities arises from the fact that input shares represent the static allocation of inputs while demand elasticities describe the dynamic adjustments in input usage as prices change. For the many-input case, the expression can be generalized as \(e_{x_i, w_i}=s_j\left(A_{i j}+e_{Q, P}\right)\) where \(A_{i j}\) is the Allen elasticity of substitution.

Step by step solution

01

Derive the Cross-Price Effects on Input Demand Using Problem 11.12 Steps (b), (d), and (e)

Let's use the steps from Problem 11.12: b. Compute the input share of K and L: \(s_K\) and \(s_L\). d. Calculate the elasticity of output with respect to price: \(e_{Q, P}\). e. Find the elasticity of substitution between K and L: \(\sigma\). Now, using these values, we can find the cross-price effects on input demand e_K_w and e_L_v as: \[ e_{K, w}=s_{L}\left(\sigma+e_{Q, P}\right) \quad \text{ and } \quad e_{L,v}=s_{K}\left(\sigma+e_{Q, P}\right) \]
02

Intuitive Explanation of the Difference in Input Shares and Demand Elasticities

Intuitively, the difference in input shares and demand elasticities in part (e) of Problem 11.12 and part (a) of this problem arise from the fact that input shares emphasize the proportion of each input in production, while demand elasticities emphasize the responsiveness of input demand to changes in input prices. In other words, input shares represent the static allocation of inputs, whereas demand elasticities describe the dynamic adjustments in input usage as prices change.
03

Generalize the Expression for the Many-Input Case

For the many-input case, the expression can be generalized as: \[ e_{x_i, w_i}=s_j\left(A_{i j}+e_{Q, P}\right) \] where \(A_{i j}\) is the Allen elasticity of substitution.
04

Discuss the Oddity and Resolution for the Case \(i=j\)

When \(i=j\), this expression seems to say that \(e_{L, w}=s_{L}\left(A_{L L}+e_{Q . P}\right)\). If we assume \(A_{L L}=\sigma\) in the two-input case, this would contradict the result from Problem 11.12. To resolve the paradox, we can use the definitions from Problem 10.12 to show that, with two inputs, \(A_{L L}=\left(-s_{K} / s_{L}\right) \cdot A_{K L}=\left(-s_{K} / s_{L}\right) \cdot \sigma\). Thus, there is no disagreement between these two results.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Input Demand Cross-Price Effects
In the study of microeconomics, understanding the relationship between different inputs used in production is critical. One of the intriguing dynamics observed is the input demand cross-price effect. This effect measures how the demand for one input, such as capital (K), changes when the price of another input, like labor (L), changes. Little nuances in production theory, such as these cross-price effects, significantly influence how businesses plan their resource allocation.

When we use the equation \(e_{K, w}=s_{L}(\sigma+e_{Q, P})\), it reveals the cross-price elasticity of demand for capital when the wage rate changes. Similarly, \(e_{L, v}=s_{K}(\sigma+e_{Q, P})\) tells us about the labor demand when the rental rate of capital alters. The symbols \(s_{L}\) and \(s_{K}\) represent the share of labor and capital in total production costs, respectively, working as weights in the overall substitution effect. The term \(\sigma\) represents the elasticity of substitution between capital and labor, and \(e_{Q, P}\) is the elasticity of output to price changes. Understanding these cross-price effects equips businesses with the analytical tools to better adjust their input combinations in response to market shifts, contributing to more efficient and economical production strategies.
Allen Elasticity of Substitution
Among the concepts critical to comprehending how businesses adjust to changing market conditions is the Allen Elasticity of Substitution (AES). Named after Sir Roy Allen, the AES is a nuanced measure that captures the rate of substitution between inputs (such as labor and capital) in response to changes in their relative prices. It’s a refinement over the standard elasticity of substitution that takes into account all inputs used in production.

The general formula of AES in the multi-input context is given by \(e_{x_i, w_i}=s_j(A_{ij}+e_{Q, P})\). Here, \(A_{ij}\) denotes the AES between inputs \(i\) and \(j\), and it helps determine how easily one input can be substituted for another. A higher AES implies inputs can substitute each other more easily, contributing to greater flexibility in production planning. This concept importantly influences decision-making, especially when businesses are faced with different combinations of input prices and are looking to optimize production costs.
Input Demand Responsiveness
Input demand responsiveness is a key microeconomic concept that indicates how sensitive the quantity of an input demanded is to changes in its price. It’s crucial for businesses to understand this as it directly affects cost management and production efficiency. The demand elasticities derived earlier, such as \(e_{K, w}\) and \(e_{L, v}\), are measures of this responsiveness.

When the elasticity value is high, we say that the demand for an input is more responsive to price changes. For instance, a high elasticity for labor demand \(e_{L, v}\) means that an increase in the cost of labor would result in a substantial reduction in the amount of labor hired. These elasticities help firms forecast how changes in the market – such as wage increases, the introduction of automation, or fluctuations in interest rates – might alter their input demands. This agility to anticipate and react to price changes is a fundamental component of a successful business strategy in today's ever-changing marketplace.

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Most popular questions from this chapter

With a CES production function of the form \(q=\left(k^{\rho}+l^{\rho}\right)^{\gamma / \rho}\) a whole lot of algebra is needed to compute the profit function as \(\Pi(P, v, w)=K P^{1 /(1-\gamma)}\left(v^{1-\alpha}+w^{1-\sigma}\right)^{\gamma /(1-\sigma)(\gamma-1)},\) where \(\sigma=1 /(1-\rho)\) and \(K\) is a constant a. If you are a glutton for punishment (or if your instructor is), prove that the profit function takes this form. Perhaps the easiest way to do so is to start from the CES cost function in Example 10.2 b. Explain why this profit function provides a reasonable representation of a firm's behavior only for \(0<\gamma<1\) c. Explain the role of the elasticity of substitution ( \(\sigma\) ) in this profit function. What is the supply function in this case? How does \(\sigma\) determine the extent to which that function shifts when input prices change? e. Derive the input demand functions in this case. How are these functions affected by the size of \(\sigma ?\)

The demand for any input depends ultimately on the demand for the goods that input produces. This can be shown most explicitly by deriving an entire industry's demand for inputs. To do so, we assume that an industry produces a homogencous good, \(Q,\) under constant returns to scale using only capital and labor. The demand function for \(Q\) is given by \(Q=D(P)\), where \(P\) is the market price of the good being produced. Because of the constant returns-to- scale assumption, \(P=M C=A C\). Throughout this problem let \(C(v, w, 1)\) be the firm's unit cost function. a. Explain why the total industry demands for capital and labor are given by \(K=Q C_{v}\) and \(L=Q C_{w}\) b. Show that \\[ \frac{\partial K}{\partial v}=Q C_{v v}+D^{\prime} C_{v}^{2} \quad \text { and } \quad \frac{\partial L}{\partial w}=Q C_{w w}+D^{\prime} C_{w}^{2} \\] c. Prove that \\[ C_{w v}=\frac{-w}{v} C_{v w} \quad \text { and } \quad C_{w w}=\frac{-v}{w} C_{N w} \\] d. Use the results from parts (b) and (c) together with the elasticity of substitution defined as \(\sigma=C C_{v n} / C_{\nu} C_{w}\) to show that \\[ \frac{\partial K}{\partial v}=\frac{w L}{Q} \cdot \frac{\sigma K}{v C}+\frac{D^{\prime} K^{2}}{Q^{2}} \text { and } \frac{\partial L}{\partial w}=\frac{v K}{Q} \cdot \frac{\sigma L}{w C}+\frac{D^{\prime} L^{2}}{Q^{2}} \\] e. Convert the derivatives in part (d) into elasticities to show that \\[ e_{K, v}=-s_{L} \sigma+s_{K} e_{Q, p} \quad \text { and } \quad e_{L, w}=-s_{K} \sigma+s_{L} e_{Q, P} \\] where \(e_{Q, P}\) is the price elasticity of demand for the product being produced. f. Discuss the importance of the results in part (e) using the notions of substitution and output effects from Chapter 11 Note: The notion that the elasticity of the derived demand for an input depends on the price elasticity of demand for the output being produced was first suggested by Alfred Marshall. The proof given here follows that in D. Hamermesh, Labor Demand (Princeton, NJ: Princeton University Press, 1993).

How would you expect an increase in output price, \(P\), to affect the demand for capital and labor inputs? a. Explain graphically why, if neither input is inferior, it seems clear that a rise in \(P\) must not reduce the demand for either factor. b. Show that the graphical presumption from part (a) is demonstrated by the input demand functions that can be derived in the Cobb-Douglas case. c. Use the profit function to show how the presence of inferior inputs would lead to ambiguity in the effect of \(P\) on input demand.

This problem has you work through some of the calculations associated with the numerical example in the Extensions. Refer to the Extensions for a discussion of the theory in the case of Fisher Body and General Motors (GM), who we imagine are deciding between remaining as separate firms or having GM acquire Fisher Body and thus become one (larger) firm. Let the total surplus that the units generate together be \(S\left(x_{F}, x_{G}\right)=x_{F}^{1 / 2}+a x_{G}^{1 / 2},\) where \(x_{F}\) and \(x_{G}\) are the investments undertaken by the managers of the two units before negotiating, and where a unit of investment costs \(\$ 1 .\) The parameter \(a\) measures the importance of GM's manager's investment. Show that, according to the property rights model worked out in the Extensions, it is efficient for GM to acquire Fisher Body if and only if GM's manager's investment is important enough, in particular, if \(a>\sqrt{3}\)

Young's theorem can be used in combination with the envelope results in this chapter to derive some useful results. a. Show that \(\partial l(P, v, w) / \partial v=\partial k(P, v, w) / \partial w\). Interpret this result using substitution and output effects. b. Use the result from part (a) to show how a unit tax on labor would be expected to affect capital input. c. Show that \(\partial q / \partial w=-\partial l / \partial P\). Interpret this result. d. Use the result from part (c) to discuss how a unit tax on labor input would affect quantity supplied.
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