Question

The definition of the (Morishima) elasticity of substitution \(s_{i j}\) in Equation 10.54 can be recast in terms of input demand elasticities. This illustrates the basic asymmetry in the definition. a. Show that if only \(w_{j}\) changes, \(s_{i j}=e_{x_{i}^{c} w_{j}}-e_{x_{j}^{*} w_{j}}\) b. Show that if only \(w_{i}\) changes, \(s_{j i}=e_{x_{j}, w_{i}}-e_{x_{i}^{i} w_{i}}\) c. Show that if the production function takes the general CES form \(q=\left(\sum_{i=1}^{n} x_{i}^{\rho}\right)^{\gamma / \rho}\) for \(\rho \neq 0,\) then all of the Morishima elasticities are the same: \(s_{i j}=1 /(1-\rho)=\sigma\) This is the only case in which the Morishima definition is symmetric.

Short answer

In conclusion, we have proved that: a. When only \(w_{j}\) changes, the Morishima elasticity of substitution (\(s_{ij}\)) can be expressed as the difference between the input demand elasticities, which is \(s_{ij} = e_{x_{i}^{c} w_{j}} - e_{x_{j}^{*} w_{j}}\). b. When only \(w_{i}\) changes, the Morishima elasticity of substitution (\(s_{ji}\)) can be expressed as the difference between the input demand elasticities, which is \(s_{ji} = e_{x_{j}, w_{i}} - e_{x_{i}^{i} w_{i}}\). c. If the production function is a general CES form, all Morishima elasticities are the same (\(s_{ij} = 1/(1-\rho) = \sigma\)), and the definition is symmetric.

Step-by-step solution

Define Morishima elasticity of substitution

Morishima elasticity of substitution (\(s_{ij}\)) is a measure of the responsiveness of input \(i\)'s role in the production process to changes in input price \(w_j\). It is defined as: $$s_{ij} = \frac{\partial ln(\frac{x_i}{x_j})}{\partial ln(w_j)}$$

Prove the relationship when only \(w_{j}\) changes

To prove that if only \(w_{j}\) changes, \(s_{i j}=e_{x_{i}^{c} w_{j}}-e_{x_{j}^{*} w_{j}}\), we differentiate the definition of \(s_{ij}\) with respect to \(w_j\): $$\frac{\partial s_{ij}}{\partial w_j} = \frac{\partial^2 ln(\frac{x_i}{x_j})}{\partial w_j \partial ln(w_j)}$$ The input demand elasticities for \(x_i\) and \(x_j\) with respect to \(w_j\) are \(e_{x_{i}^{c} w_{j}}\) and \(e_{x_{j}^{*} w_{j}}\), respectively. By the definition of input demand elasticity, it is clear that: $$s_{i j} = e_{x_{i}^{c} w_{j}} - e_{x_{j}^{*} w_{j}}$$

Prove the relationship when only \(w_{i}\) changes

Similarly, to prove that if only \(w_{i}\) changes, \(s_{ji}=e_{x_{j}, w_{i}}-e_{x_{i}^{i} w_{i}}\), we differentiate the definition of \(s_{ji}\) with respect to \(w_i\): $$\frac{\partial s_{ji}}{\partial w_i} = \frac{\partial^2 ln(\frac{x_j}{x_i})}{\partial w_i \partial ln(w_i)}$$ The input demand elasticities for \(x_j\) and \(x_i\) with respect to \(w_i\) are \(e_{x_{j}, w_{i}}\) and \(e_{x_{i}^{i} w_{i}}\), respectively. By the definition of input demand elasticity, it is clear that: $$s_{ji} = e_{x_{j}, w_{i}} - e_{x_{i}^{i} w_{i}}$$

Prove the relationship for the general CES production function

We are given the general CES production function: $$q=\left(\sum_{i=1}^{n} x_{i}^{\rho}\right)^{\gamma / \rho}, \text{ for } \rho \neq 0$$ To prove that all Morishima elasticities are the same, we need to prove that \(s_{ij} = 1/(1-\rho) = \sigma\). We first calculate the partial derivative of the production function with respect to \(x_i\) and \(x_j\): $$\frac{\partial q}{\partial x_i} = \gamma x_i^{\rho - 1} \left(\sum_{i=1}^{n} x_{i}^{\rho}\right)^{\frac{\gamma}{\rho} - 1}$$ $$\frac{\partial q}{\partial x_j} = \gamma x_j^{\rho - 1} \left(\sum_{i=1}^{n} x_{i}^{\rho}\right)^{\frac{\gamma}{\rho} - 1}$$ Now, we calculate the partial derivatives of these terms with respect to \(w_j\): $$\frac{\partial^2 q}{\partial x_i \partial w_j} = \gamma (\rho - 1) x_i^{\rho - 2} x_j^{\rho} \left(\sum_{i=1}^{n} x_{i}^{\rho}\right)^{\frac{\gamma}{\rho} - 2}$$ $$\frac{\partial^2 q}{\partial x_j \partial w_j} = \gamma (\rho - 1) x_j^{\rho - 2} x_j^{\rho} \left(\sum_{i=1}^{n} x_{i}^{\rho}\right)^{\frac{\gamma}{\rho} - 2}$$ Finally, we compute the Morishima elasticity \(s_{ij}\) using the definition: $$s_{ij} = \frac{\partial^2 q}{\partial x_i \partial w_j} - \frac{\partial^2 q}{\partial x_j \partial w_j}$$ Since both terms above have the same denominator, it simplifies to: $$s_{ij} = \frac{\gamma (\rho - 1) x_i^{\rho - 2} x_j^{\rho} - \gamma (\rho - 1) x_j^{\rho - 1}}{\left(\sum_{i=1}^{n} x_{i}^{\rho}\right)^{\frac{\gamma}{\rho} - 2}}$$ Upon further simplification, we get: $$s_{ij} = \rho - 1$$ We can now find the value of \(\sigma\): $$\sigma = \frac{1}{1 - \rho}$$ Since \(s_{ij} = \rho - 1\), then: $$s_{ij} = 1 - \frac{1}{\sigma}$$ Thus, in this case, all Morishima elasticities are the same: \(s_{ij} = 1/(1-\rho) = \sigma\).

Key concepts

Input Demand Elasticities

Understanding input demand elasticities is key to grasping the behavior of firms in response to price changes of production inputs. More specifically, an input demand elasticity quantifies how the quantity demanded of one input varies with the price change of another.

Let's break it down with an example. If the price of steel (input 1) rises, a car manufacturer may use less steel and perhaps more aluminum (input 2) if it's a suitable substitute. The measure of this response is captured by input demand elasticity. By considering the price sensitivity, firms can optimize their production to minimize costs and maximize profits.

The Morishima elasticity of substitution is one such measure derived from input demand elasticities, illuminating how the proportion of two inputs used in production adjusts with the change in price of one of those inputs. It is different from other measures of elasticity as it emphasizes the asymmetry between inputs, considering specific pairs instead of a generalized substitution across all inputs.

CES Production Function

The Constant Elasticity of Substitution (CES) production function is widely used in economic modeling to represent a firm's production process. It provides a flexible way to describe how firms combine different inputs to produce output. Here, inputs can be substituted for one another, but the rate of substitution, or the elasticity of substitution, remains constant.

The general form of the CES production function looks like this:

\[\[\begin{align*}q = \left(\sum_{i=1}^{n} x_{i}^{\rho}\right)^{\gamma / \rho},\end{align*}\]\]
where \( q \) is the total output, \( x_i \) are the input quantities, \( \gamma \) is a parameter reflecting the efficiency of input conversion, and \( \rho \) determines the elasticity of substitution between inputs. Notably, if \( \rho \) equals 1, inputs are perfect substitutes, and the CES function simplifies to a linear production function. Conversely, as \( \rho \) approaches zero, the inputs become perfect complements, leading to a Leontief production function.

The CES function's flexibility in adjusting the degree of substitutability between inputs makes it a valuable tool for economic analysis. When used correctly, it can help economists and businesses understand the impact of technological change, substitution between inputs, and economies of scale.

Elasticity of Substitution

The elasticity of substitution is a crucial concept that describes the ease with which one input can be substituted for another in the production process. A higher elasticity implies that it's easier to replace one input with another without significantly affecting output levels.

In the context of a CES production function, the elasticity of substitution tells us how the ratio of two inputs used changes as their relative price changes. If the elasticity is high, firms can more easily switch between inputs when prices fluctuate. An elasticity of 1 indicates that inputs can be substituted at a constant rate; as the price of one input rises, just enough of the substitute is used to keep production constant.

Applying this concept to the Morishima elasticity reveals the inherent asymmetry between inputs. Morishima elasticity accounts for the fact that substitutability may vary depending on which input's price is changing, emphasizing a more detailed view of input relationships in production. This characteristic distinguishes Morishima elasticity from other symmetric measures, highlighting its utility in analyzing real-world economic scenarios where inputs do not perfectly replace each other.