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 CobbDouglas and CES production functions.

Short answer

In this exercise, we proved that the elasticity of substitution for a homothetic production function \(F(k,l) = [f(k,l)]^\gamma\) is the same as the elasticity of substitution for the function \(f(k,l)\), where \(f(k,l)\) is a constant returns-to-scale production function. By calculating the necessary partial derivatives and the elasticity of substitution formula, we arrived at the conclusion that \(\sigma_F = \sigma_f\). We also applied the result to Cobb-Douglas and CES production functions. For the Cobb-Douglas production function, the elasticity of substitution remains unchanged since it is already a constant returns-to-scale production function. For the CES production function, we showed that the result holds true for the elasticity of substitution, considering the transformation with the homothetic production function.

Step-by-step solution

Derive the necessary partial derivatives for the homothetic production function

First, we need to find the necessary partial derivatives for \(F(k,l)\), which are: \(F_k\), \(F_l\), and \(F_{kl}\). Since \(F(k,l) = [f(k,l)]^\gamma\), we have: \\[ F_k = \gamma [f(k,l)]^{\gamma - 1} f_k \\] \\[ F_l = \gamma [f(k,l)]^{\gamma - 1} f_l \\] \\[ F_{kl} = \gamma(\gamma - 1) [f(k,l)]^{\gamma - 2} f_k f_l + \gamma [f(k,l)]^{\gamma - 1} f_{kl} \\]

Calculate the elasticity of substitution for \(F\)

Using the given formula for calculating elasticity of substitution (\(\sigma\)) and the partial derivatives we calculated in step 1, we can find the elasticity of substitution for \(F\). \\[ \sigma_F = \frac{F_k F_l}{F \cdot F_{kl}} \\] Plug in the values of \(F_k\), \(F_l\), and \(F_{kl}\) we found in step 1: \\[ \sigma_F = \frac{\gamma [f(k,l)]^{\gamma - 1} f_k \cdot \gamma [f(k,l)]^{\gamma - 1} f_l}{[f(k,l)]^\gamma \cdot (\gamma(\gamma - 1) [f(k,l)]^{\gamma - 2} f_k f_l + \gamma [f(k,l)]^{\gamma - 1} f_{kl})} \\]

Simplify the expression for \(\sigma_F\)

By simplifying the expression above, we can show that the elasticity of substitution for the homothetic production function \(F\) is the same as the elasticity of substitution for the function \(f\). \\[ \sigma_F = \frac{\gamma^2 [f(k,l)]^{2\gamma - 2} f_k f_l}{\gamma [f(k,l)]^{\gamma - 1} [f(k,l)]^\gamma ( (\gamma - 1) [f(k,l)]^{- 2} f_k f_l + f_{kl})} \\] Cancel out common terms: \\[ \sigma_F = \frac{f_k f_l}{f \cdot f_{kl}} = \sigma_f \\] Thus, the elasticity of substitution for the homothetic production function \(F\) is the same as the elasticity of substitution for the function \(f\).

Apply the result to Cobb-Douglas and CES production functions

Now we will apply the result to the Cobb-Douglas and CES production functions. a) For Cobb-Douglas production function, it is already a constant returns-to-scale production function. Therefore, no transformation is needed, and the elasticity of substitution remains the same. b) For CES production function, let \(f(k,l)\) be the CES production function. Then we find: \\[ F(k,l) = [f(k,l)]^\gamma = [A(k^\rho + B l^\rho)^{\frac{1}{\rho}}]^\gamma = A^{\gamma}(k^\rho + B l^\rho)^{\frac{\gamma}{\rho}} \\] Now, since the elasticity of substitution for \(F\) is the same as that for \(f\), we know that the result holds for the CES production function as well.