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 CobbDouglas production function \(f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\prod_{i=1}^{m} x_{i}^{\alpha_{i}}\) for \(\alpha_{i} \geq 0 ?\)

Short answer

The problem examines scaling through second-order partial derivatives, imposing conditions on how outputs change with inputs given different homogeneity degrees.

Step-by-step solution

Understanding Euler's Theorem

Euler's theorem states that if a function \( f(x_1, x_2, \ldots, x_n) \) is homogeneous of degree \( k \), then the following holds: \[ \sum_{i=1}^{n} x_i f_i = k f \] where \( f_i = \frac{\partial f}{\partial x_i} \) is the partial derivative of \( f \) with respect to \( x_i \). This means that scaling all inputs by a factor \( \lambda \) scales the output by \( \lambda^k \).

Proving Part (a)

To prove \( \sum_{i=1}^{n} \sum_{j=1}^{n} x_i x_j f_{ij} = k(k-1)f \), consider the application of \( f_i \) being homogeneous of degree \( k-1 \). Differentiating the expression \( x_i f_i \) with respect to \( x_j \), we obtain:\[ x_j f_{ij} + f_i = (k-1)x_i f_{ij} \]Summing over both \( i \) and \( j \), and substituting back into Euler's theorem yields:\[ \sum_{i=1}^{n} \sum_{j=1}^{n} x_i x_j f_{ij} = k(k-1) f \] through distribution and collection of terms including utilization of original Euler's equality \( \sum_{i=1}^{n} x_i f_i = k f \).

Exploring Part (b) with n=2 and k=1

For \( n=2 \) and \( k=1 \), substitute the equation \( \sum_{i=1}^{2} \sum_{j=1}^{2} x_i x_j f_{ij} = 0 \) since \( k(k-1) = 0 \). This implies strict restrictions on the cross partials to preserve the homogeneity and linear scaling assumption. Essentially, the mixed partial derivatives (e.g., \( f_{12} \)) need to balance so that their weighted sum equals zero.

Varying the Degree of Homogeneity (Part b with k>1 or k

If \( k > 1 \), then \( k(k-1) > 0 \), suggesting that the second order partials must collectively result in a positive expression when summed, indicating increasing returns to scale. Conversely, \( k < 1 \) suggests a negative sum with decreasing returns.

Generalizing to More Inputs (Part c)

For any number of inputs \( n \), the core structure of the equations remains. Each pair of inputs contributes to the second-order homogeneity condition. Therefore, the sums involving all combinations of partial derivatives across different \( n \) remain critical in constraining behaviour of the functional response subject to the homogeneity degree \( k \).

Cobb-Douglas Production Function Implications (Part d)

For a Cobb-Douglas function \( f(x_1, x_2, \ldots, x_n) = \prod_{i=1}^{n} x_i^{\alpha_i} \), Euler's theorem suggests that \( \sum_{i=1}^{n} \alpha_i = k \). The implications for second partial derivatives is seen through conditions based on returns to scale represented by \( k \); different scenarios (e.g., increasing or decreasing returns to scale) stress different functional elasticities and overall behaviour, influenced transmissively through the \( \alpha_i \) parameters.

Key concepts

Euler's Theorem

When working with production functions, especially those homogeneous of degree \(k\), Euler's Theorem becomes a central concept. It elegantly states that for such functions \(f(x_1, x_2, \ldots, x_n)\), the sum of each input multiplied by its corresponding partial derivative equals the product of \(k\) and the function itself. In symbolic terms: \[ \sum_{i=1}^{n} x_i f_i = k f \]where \(f_i = \frac{\partial f}{\partial x_i}\) is the partial derivative with respect to input \(x_i\). This relationship provides critical insights. The theorem indicates that scaling all inputs by the same factor \(\lambda\) alters the function's output by \(\lambda^k\), pinpointing the nature of returns to scale.
Understanding this theorem helps to prove further properties, such as proving why the second-order partial derivatives entail certain conditions influenced by \(k\). Integrating this into solving production problems is an effective way to harness mathematical insight to real-world economic behaviors, emphasizing the power of theoretical methodologies.

Cobb-Douglas Production Function

The Cobb-Douglas production function is a cornerstone in economics and offers a clear representation of how inputs contribute to output. It is expressed as:\[ f(x_1, x_2, \ldots, x_n) = \prod_{i=1}^{n} x_i^{\alpha_i} \]where each \(x_i\) represents an input like labor or capital, and \(\alpha_i\) are the parameters that signify input significance and substitutability. The sum of all \(\alpha_i\) values reflects the function’s degree of homogeneity \(k\), thereby indicating the returns to scale.

• If \(\sum \alpha_i = 1\), the function experiences constant returns to scale.
• If \(\sum \alpha_i > 1\), the function has increasing returns to scale.
• If \(\sum \alpha_i < 1\), the function showcases decreasing returns to scale.

The Cobb-Douglas model's flexibility lends itself well to demonstrate practical economic scenarios by adjusting the \(\alpha_i\) values, making it a robust tool for economists to predict and analyze output based on changes in input levels.

Returns to Scale

Returns to scale describe how the output responds to scaling of all inputs, crucial for understanding production efficiency and managerial economics. If a production function is homogeneous of degree \(k\), this directly dictates the return to scale properties:

• \(k = 1\): Constant returns to scale – output changes proportionally with inputs.
• \(k > 1\): Increasing returns to scale – output increases by a larger proportion than inputs.
• \(k < 1\): Decreasing returns to scale – output increases by a smaller proportion than inputs.

These distinctions help firms decide on the extent of resource allocation, impacting business strategy. For instance, increasing returns might encourage expansion, while decreasing returns signal potential inefficiencies with larger scale operations.
Understanding and applying these concepts can form the basis for strategic production planning and optimizing resource usage, making returns to scale an indispensable part of economic analysis.

Partial Derivatives

Partial derivatives are integral to analyzing multivariable functions, including production functions. They represent the rate of change of the function concerning changes in each individual input while keeping others constant.Given a function \( f(x_1, x_2, \ldots, x_n) \), its partial derivative with respect to \(x_i\) is denoted \(f_i = \frac{\partial f}{\partial x_i}\). These derivatives help in understanding how one specific input influences the overall output, and are key to applying Euler's Theorem.Second-order partial derivatives, such as \(f_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j}\), provide insights into how combinations of inputs can interact to affect output. These higher-order derivatives are used to explore deeper characteristics of functions, particularly concerning curvature and interactions between inputs.
In the case of homogeneous functions, understanding how partial derivatives align with the degree \(k\) highlights the functional structure and correlations between inputs and output, serving as a foundational tool in economic and mathematical modeling.