Question

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

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

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: \[ e_{K, w}=s_{L}\left(\sigma+e_{Q, P}\right) \quad \text{ and } \quad e_{L,v}=s_{K}\left(\sigma+e_{Q, P}\right) \]

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.

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.

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.

Key concepts

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.