Question

Although much of our discussion of measuring the elasticity of substitution for various production functions has assumed constant returns to scale, often that assumption is not necessary. This problem illustrates some of these cases. a. In footnote 6 we pointed out that, in the constant returns-to-scale case, the elasticity of substitution for a two-input production function is given by $$\sigma=\frac{f_{k} f_{l}}{f \cdot f_{k l}}$$ Suppose now that we define the homothetic production function \(F\) as $$F(k, l)=[f(k, l)]^{\gamma}$$ where \(f(k, l)\) is a constant returns-to-scale production function and \(\gamma\) is a positive exponent. Show that the elasticity of substitution for this production function is the same as the elasticity of substitution for the function \(f\) b. Show how this result can be applied to both the Cobb-Douglas and CES production functions.

Short answer

In summary, we have shown that the elasticity of substitution for a homothetic production function is the same as the elasticity of substitution for its corresponding constant returns-to-scale production function. We calculated the partial derivatives and second-order partial derivatives of both production functions and used the formula for elasticity of substitution to prove this result. Furthermore, we applied this result to both the Cobb-Douglas and CES production functions, demonstrating its relevance for these specific cases.

Step-by-step solution

Calculate partial derivatives and second-order partial derivatives of the production functions

We begin by first differentiating \(F(k, l)\) and \(f(k, l)\) with respect to \(k\) and \(l\): For \(F(k,l)\): \(F_k=\gamma f^{\gamma-1}f_k\) \(F_l=\gamma f^{\gamma-1}f_l\) Next, we find the second-order partial derivatives: \(F_{kl}=\gamma(\gamma-1)f^{\gamma-2}f_kf_l+\gamma f^{\gamma-1}f_{kl}\) For \(f(k,l)\): \(f_{kl} = f_{kl}\) Now that the partial derivatives have been found, we can use the given formula in Step 2 to find the elasticity of substitution for both functions and show that they are equal.

Find the elasticity of substitution for both functions and show that they are equal

Now we use the formula for elasticity of substitution, \(\sigma = \frac{f_{k} f_{l}}{f \cdot f_{k l}}\), and find the elasticity for both production functions, \(F(k,l)\) and \(f(k,l)\). For \(F(k,l)\): \(\sigma_F = \frac{F_k F_l}{F \cdot F_{kl}} =\frac{\gamma f^{\gamma-1}f_k \cdot \gamma f^{\gamma-1}f_l}{[f^\gamma] \cdot (\gamma (\gamma-1) f^{\gamma-2} f_k f_l + \gamma f^{\gamma-1}f_{kl})}\) Simplifying the above equation, we get: \(\sigma_F = \frac{f_k f_l}{f \cdot f_{kl}}\) For \(f(k,l)\): \(\sigma_f = \frac{f_k f_l}{f \cdot f_{kl}}\) Comparing \(\sigma_F\) with \(\sigma_f\), we find that they are equal: \(\sigma_F = \sigma_f\) The elasticity of substitution for the homothetic production function \(F(k, l)\) is the same as the elasticity of substitution for the constant returns-to-scale production function \(f(k, l)\). Now that we have shown that the elasticities of substitution are equal, we can apply this result to the Cobb-Douglas and CES production functions in Step 3.

Apply the result to Cobb-Douglas and CES production functions

Cobb-Douglas production function is in the form of: \(f(k,l) = k^\alpha l^\beta\) Taking the homothetic production function \(F(k, l) = [f(k, l)]^{\gamma}\), we have: \(F(k,l) = (k^\alpha l^\beta)^{\gamma} = k^{\alpha\gamma} l^{\beta\gamma}\) For CES production function, its form is: \(f(k,l) = [A(k^r) + B(l^r)]^{\frac{1}{r}}\) Taking the homothetic production function \(F(k, l) = [f(k, l)]^{\gamma}\), we have: \(F(k,l) = [A(k^r) + B(l^r)]^{\frac{\gamma}{r}}\) From the result obtained in Step 2, we have shown that the elasticity of substitution for the homothetic production function \(F(k, l)\) is the same as the elasticity of substitution for the constant returns-to-scale production function \(f(k, l)\). This result is applicable to both the Cobb-Douglas and CES production functions as shown above.

Key concepts

Homothetic Production Functions

In economics, production functions are used to represent the relationship between inputs (like capital and labor) and the output of a firm. A homothetic production function is a special case where the function can be represented as a monotonic transformation of a function with constant returns to scale. Specifically, if we have a production function represented as f(k, l), we can create a homothetic function F(k, l) by raising f(k, l) to the power of a positive exponent \(\gamma\).
The importance of homothetic functions lies in their simplicity to analyze since they maintain the same elasticity of substitution across different input combinations. This means that the percentage change in the input ratios resulting from a change in their relative prices is consistent, regardless of scale. This property leads to proportional paths of expansion and offers a simpler way to model economic growth and input demand. By working with homothetic functions, students can more easily understand the concepts of factor substitution and scale effects in production without the complications that arise from non-homothetic functions where elasticity could vary at different levels of output.

Cobb-Douglas Production Function

One of the most utilized forms of the production function in economics is the Cobb-Douglas production function. It has a mathematically simple form given by \(f(k,l) = k^\alpha l^\beta\), where \(k\) represents capital, \(l\) represents labor, and \(\alpha\) and \(\beta\) are the output elasticities of capital and labor respectively. The Cobb-Douglas function is especially popular because it's often used to represent the real-world production processes where inputs are substituted for each other at a diminishing rate.
The function is characterized by its constant elasticity of substitution, which is exactly one, meaning that the percentage decrease in the use of one input, holding output constant, will be the same as the percentage increase in the use of another input. This simplicity enables students to grasp the nature of production inputs working together to produce a fixed amount of output.

CES Production Function

Another vital concept in understanding the flexibility of input substitution in production is the Constant Elasticity of Substitution (CES) production function. This function takes the form \(f(k,l) = [A(k^r) + B(l^r)]^{\frac{1}{r}}\), where \(A\) and \(B\) are positive constants, and \(r\) determines the elasticity of substitution between inputs capital and labor. The CES function is more general than the Cobb-Douglas production function and allows for different values of the elasticity of substitution.
The CES function can represent a range of substitution possibilities, including the case where inputs are perfect substitutes (\(r\to\infty\)) or perfect complements (\(r=0\)). Understanding the CES function is crucial for students as it provides a broader framework for analyzing how different levels of input substitutability impact production processes.

Constant Returns to Scale

The concept of constant returns to scale refers to a situation in which the scale of production does not affect the amounts of inputs required to produce a unit of output. In mathematical terms, a production function has constant returns to scale if, when all inputs are scaled by a factor \(t\), output is also scaled by the same factor. The functional form for constant returns to scale is \(f(tk, tl) = t f(k, l)\).
This concept is fundamental in economics because it implies efficiency in production—no matter how large or small the scale, firms can maintain productivity. When working with production functions, assuming constant returns to scale simplifies calculations and helps in the theoretical understanding of economies of scale, which can be a pivotal concept when discussing market structures and firm behavior.