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

Question: Prove that if a production function f(x1, x2, ..., xn) is homogeneous of degree k, then: $$ \sum_{j=1}^{n} \sum_{i=1}^{n} x_{i} x_{j} f_{ij} = k(k-1) f $$ where \(f_{ij}\) is the second derivative of f with respect to \(x_i\) and \(x_j\). Additionally, discuss the case when n = 2 and k = 1. Finally, analyze the implications for the multivariable Cobb-Douglas production function.

Step-by-step solution

Deriving the Partial Derivatives

To prove the statement in part (a), we first need to find the mixed partial derivatives for the function \(f\). The second derivative of \(f\) with respect to \(x_i\) and \(x_j\) is denoted by \(f_{ij}\). This can be calculated as follows: $$ f_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j} $$

Using Euler's Theorem

Now, let's use Euler's theorem \(\sum_{i} x_{i}f_{i}=k f\) for the homogeneous function \(f\) of degree \(k\). We will differentiate both sides of the theorem with respect to \(x_j\) to get: $$ \sum_{i=1}^{n} x_{i} f_{ij} + \sum_{i=1}^{n} f_{i} \delta_{ij} = k f_j, \text{ where } \delta_{ij} = \begin{cases} 1 & i=j \\ 0 & i\neq j. \end{cases} $$

Multiplying by \(x_j\)

Next, we will multiply both sides of the equation by \(x_j\) and sum over \(j = 1,...,n\): $$ \sum_{j=1}^{n} \sum_{i=1}^{n} x_{i} x_{j} f_{ij} + \sum_{j=1}^{n} x_{j} f_{j} = \sum_{j=1}^{n} k f_j x_j $$

Applying Euler's Theorem

Now, we can apply Euler's theorem \(\sum_{i} x_{i}f_{i}=k f\) again to rewrite the last term in the equation: $$ \sum_{j=1}^{n} \sum_{i=1}^{n} x_{i} x_{j} f_{ij} = k(k-1) f $$

Case n = 2 and k = 1

Now let's discuss the case where there are only two inputs,\( n=2\), and the production function is homogeneous of degree 1, \(k=1\). In this situation, the statement in part (a) implies: $$ x_1x_2f_{12} + x_2x_1f_{21} = 0 $$ Since the second-order partial derivatives are symmetric, \(f_{12} = f_{21}\), and we can rewrite the equation as: $$ 2x_1x_2f_{12} = 0 $$ This implies that \(f_{12}\) must be zero, meaning that the two inputs are independent of each other. If \(k > 1\), there will be no restriction on the second-order partial derivative \(f_{12}\), and if \(k < 1\), \(f_{12}\) must be negative because of the negative value on the right side.

Generalization for any number of inputs

Generalizing the result in part (b) for a production function with any number of inputs, we can conclude that the second-order partial derivatives must be zero if the function is homogeneous of degree 1. For \(k > 1\), there will be no restrictions, and for \(k < 1\), the second-order partial derivatives must be negative.

Implication for Cobb-Douglas production function

The multivariable Cobb-Douglas production function can be written as: $$ f(x_{1}, x_{2}, \ldots, x_{n}) = \prod_{i=1}^{n} x_{i}^{\alpha_{i}} $$ The degree of homogeneity, \(k\) for this function is the sum of the parameters \(\alpha_i\). If the sum is 1, or \(k=1\), all second-order partial derivatives will be zero, implying the inputs are independent of each other. If the sum \(\alpha_i\), or \(k\) is greater than 1, there will be no restrictions on the second-order partial derivatives. If the sum \(\alpha_i\), or \(k\) is less than 1, the second-order partial derivatives must be negative.

Key concepts

Homogeneous Functions

A function is said to be homogeneous if it satisfies certain scaling properties. In the realm of microeconomics, this relates to how output responds to proportional increases in all inputs. For a function \( f(x_1, x_2, ..., x_n) \) to be homogeneous of degree \( k \), it must satisfy the condition: \[ f(t imes x_1, t imes x_2, ..., t imes x_n) = t^k imes f(x_1, x_2, ..., x_n) \] where \( t \) is a positive scalar. Key Points:

• If \( k = 1 \), the function is linearly homogeneous, meaning a doubling of inputs results in a doubling of output.
• If \( k > 1 \), the function exhibits increasing returns to scale.
• If \( k < 1 \), the function demonstrates decreasing returns to scale.

Understanding the degree of homogeneity is fundamental because it helps us analyze the behavior of production functions within an economy.

Partial Derivatives

Partial derivatives are crucial in understanding the behavior of functions when multiple variables are involved. In simple terms, a partial derivative measures how a function changes as only one of the variables changes while keeping the others constant. For a function \( f(x_1, x_2, ..., x_n) \), the partial derivative with respect to \( x_i \) is denoted as: \[ f_i = \frac{\partial f}{\partial x_i} \] When discussing second-order partial derivatives, denoted as \( f_{ij} \), these measure how the rate of change of one partial derivative with respect to another variable changes. In the context of the problem, these second-order derivatives help us understand interaction effects between different inputs. Applications:

• Determine how the change in one input affects the output, holding all other inputs constant.
• Identify input interaction effects, crucial in understanding complementary or substitution relationships.

Cobb-Douglas Production Function

The Cobb-Douglas production function is one of the most widely used functional forms in economics, particularly useful for modeling how input factors like labor and capital allocate resources to production. It's expressed as: \[ f(x_{1}, x_{2}, ..., x_{n}) = \prod_{i=1}^{n} x_{i}^{\alpha_{i}} \] where \( \alpha_i \geq 0 \) are the output elasticities or share parameters. Properties:

• If the sum of the \( \alpha_i \) equals 1, the production function has constant returns to scale.
• When the sum exceeds 1, it signifies increasing returns to scale.
• If the sum is less than 1, it indicates decreasing returns to scale.

This model captures the relationship between inputs like labor and capital and the output, providing insights into efficiency and productivity of firms.

Degree of Homogeneity

The degree of homogeneity is a pivotal concept in the study of production functions, directly impacting the understanding of scale economies. In essence, it describes how the scale of production changes when all inputs are changed by a constant factor. For example, Euler's theorem in economics states that for a homogeneous function of degree \( k \), the function can be expressed as: \[ \sum_{i=1}^{n} x_i f_i = k f(x_1, x_2, ..., x_n) \] This helps us determine if a production function exhibits increasing, constant, or decreasing returns to scale:

• \( k = 1 \): Constant returns to scale.
• \( k > 1 \): Increasing returns to scale, signaling more output from proportional input hikes.
• \( k < 1 \): Decreasing returns to scale, where inputs have a diminishing marginal return.

Understanding the degree of homogeneity allows businesses and economists to predict production capabilities and optimize resource allocation.