Question

Suppose that a production function \(f\left(x_{1}, x_{2}, \ldots, x_{n}\right)\) is homogeneous of degree \(k\). Euler's theorem shows that \(\sum_{i} x_{i} f_{i}=k f,\) and this fact can be used to show that the partial derivatives of \(f\) are homogeneous of degree \(k-1\) a. Prove that \(\sum_{i=1}^{n} \sum_{j=1}^{n} x_{i} x_{j} f_{i j}=k(k-1) f\) b. In the case of \(n=2\) and \(k=1\), what kind of restrictions does the result of part (a) impose on the second-order partial derivative \(f_{12} ?\) How do your conclusions change when \(k>1\) or \(k<1 ?\) c. How would the results of part (b) be generalized to a production function with any number of inputs? d. What are the implications of this problem for the parameters of the multivariable Cobb-Douglas production function \(f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\prod_{i=1}^{n} x_{i}^{\alpha_{i}}\) for \(\alpha_{i} \geq 0 ?\)

Short answer

#tag_title#Step 3: Interpret implications for Cobb-Douglas parameters#tag_content# The results of this problem imply that the parameters of the Cobb-Douglas production function play a crucial role in determining the properties of second-order partial derivatives. Specifically, the parameters \(\alpha_i\) affect the degree of homogeneity and subsequently influence restrictions on the second-order partial derivatives. For a multivariable Cobb-Douglas production function, the degree of homogeneity determines how second-order partial derivatives behave with respect to input variables and reveals insights into the underlying properties, such as returns to scale and factor substitution possibilities.

Step-by-step solution

Definition of homogeneous functions

A function is homogeneous of degree k if, for any constant c > 0, we have: \(f(c x_1, c x_2, \dots, c x_n) = c^k f(x_1, x_2, \dots, x_n)\).

Euler's theorem for homogeneous functions

Euler's theorem states that, for a homogeneous function f of degree k, the following equality holds: \(\sum_{i=1}^{n} x_i \frac{\partial f}{\partial x_i} = k f\).

Compute the second partial derivatives

To compute \(\sum_{i=1}^{n} \sum_{j=1}^{n} x_i x_j \frac{\partial^2 f}{\partial x_i \partial x_j}\), we first find the second partial derivatives \(\frac{\partial^2 f}{\partial x_i \partial x_j}\) using the chain rule and the fact that the first partial derivatives \(\frac{\partial f}{\partial x_i}\) are homogeneous of degree k-1.

Use Euler's theorem on the second partial derivatives

Now that we have formula for second partial derivatives in terms of first partial derivatives, we apply Euler's theorem on the first partial derivatives: \(\sum_{i=1}^{n} x_i \frac{\partial^2 f}{\partial x_i \partial x_j} = (k-1) \frac{\partial f}{\partial x_j}\).

Apply Euler's theorem once more

We can now find the expression for \(\sum_{i=1}^{n} \sum_{j=1}^{n} x_i x_j \frac{\partial^2 f}{\partial x_i \partial x_j}\) by applying Euler's theorem to the expression obtained in step 4: \(\sum_{i=1}^{n} \sum_{j=1}^{n} x_i x_j \frac{\partial^2 f}{\partial x_i \partial x_j} = k(k-1) f\). #b. In the case of \(n=2\) and \(k=1\), what kind of restrictions does the result of part (a) impose on the second-order partial derivative \(f_{12} ?\) How do your conclusions change when \(k>1\) or \(k<1 ?\)#

Compute expression for \(n=2\) and \(k=1\)

To analyze the restrictions on second-order partial derivatives, we first write down the expression obtained in part a for the case n=2 and k=1: \( x_1 x_2 \frac{\partial^2 f}{\partial x_1 \partial x_2} = 0\).

Interpret the result of Step 1

The condition obtained in step 1 shows that the second-order partial derivative \(f_{12}\) must be zero when the function is homogeneous of degree 1 with 2 input variables.

Consider other cases k>1 and k

In case the degree of homogeneity is different from 1, the restriction would be different, as the value of k(k-1) would not be zero. #c. How would the results of part (b) be generalized to a production function with any number of inputs?#

Write down the general expression for n and k

First, we write down the general expression obtained in part a for any number of inputs n and degree of homogeneity k: \(\sum_{i=1}^{n} \sum_{j=1}^{n} x_i x_j \frac{\partial^2 f}{\partial x_i \partial x_j} = k(k-1)f\).

Interpret the general expression

This expression implies that, if the function f is homogeneous of degree k, the sum of the products of second-order partial derivatives and input variables would be equal to k(k-1)f. #d. What are the implications of this problem for the parameters of the multivariable Cobb-Douglas production function $f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\prod_{i=1}^{n} x_{i}^{\alpha_{i}}\( for \)\alpha_{i} \geq 0 ?$#

Note the case of Cobb-Douglas production function

The given Cobb-Douglas production function is homogeneous of degree k, where k is the sum of all coefficients \(\alpha_i\). Therefore, we can use the results obtained in previous parts.

Appropriate homogeneity condition

The second-order partial derivatives of the Cobb-Douglas production function will be homogeneous of degree k-1, which means that the second-order partial derivatives will be proportional to the product of input variables raised to the power of (k-1).