Question

The Cobb-Douglas production function is given by \(f\left(x_{1}, x_{2}\right)=A x_{1}^{a} x_{2}^{b}\) It turns out that the type of returns to scale of this function will depend on the magnitude of \(a+b .\) Which values of \(a+b\) will be associated with the different kinds of returns to scale?

Short answer

Values of \(a + b\) indicate: - \(a + b = 1\): constant returns, - \(a + b > 1\): increasing returns, - \(a + b < 1\): decreasing returns.

Step-by-step solution

Understand Returns to Scale

The returns to scale of a production function describe how the output changes when all input factors are scaled by the same factor. For a function \(f(x_1, x_2) = Ax_1^a x_2^b\), consider scaling inputs \(x_1\) and \(x_2\) by a constant \(k\). We need to determine how this scaling affects the output.

Scale Input Factors

Scale both input factors by a constant \(k\), so the new input becomes \((kx_1, kx_2)\). Substitute these into the production function: \[ f(kx_1, kx_2) = A (kx_1)^a (kx_2)^b = A k^a x_1^a k^b x_2^b = A k^{a+b} x_1^a x_2^b. \]

Analyze the Expression for Different Values of a+b

The expression becomes \(k^{a+b} f(x_1, x_2)\). Now analyze what happens for different values of \(a+b\):- **If \(a + b = 1\):** This implies constant returns to scale, as \(f(kx_1, kx_2) = k f(x_1, x_2)\), meaning output scales proportionally with inputs.- **If \(a + b > 1\):** This results in increasing returns to scale, as \(f(kx_1, kx_2) = k^{a+b} f(x_1, x_2)\) where \(k^{a+b} > k\).- **If \(a + b < 1\):** This indicates decreasing returns to scale, since \(f(kx_1, kx_2) = k^{a+b} f(x_1, x_2)\) where \(k^{a+b} < k\).

Key concepts

Returns to Scale

Returns to scale is a fundamental concept in economics that describes how the output of a production process changes when all input levels are increased by the same proportion. Imagine a production function like the Cobb-Douglas one, given by \(f(x_1, x_2) = Ax_1^a x_2^b\). Here, the parameters \(a\) and \(b\) define how strongly each input contributes to the output. But what if we scale both inputs by a factor \(k\)?
To find out, we substitute \(kx_1\) and \(kx_2\) into the function, resulting in \(f(kx_1, kx_2) = A(kx_1)^a (kx_2)^b = A k^{a+b} x_1^a x_2^b\). The behavior of the output as the inputs scale depends on the sum \(a + b\):

• If \(a+b = 1\), the function shows constant returns to scale; the output increases by the same proportion as the inputs.
• If \(a+b > 1\), we observe increasing returns to scale, where output increases by a larger proportion than the increase in inputs.
• If \(a+b < 1\), the function demonstrates decreasing returns to scale, as the output grows by a smaller proportion than the inputs.

This classification helps economists and businesses understand how changes in input levels affect overall productivity.

Production Function

A production function is a mathematical representation of the relationship between input factors and the resulting output. For the Cobb-Douglas production function, it is defined as \(f(x_1, x_2) = Ax_1^a x_2^b\). Here, \(A\) is a constant that represents technological efficiency, and \(x_1\) and \(x_2\) are inputs, usually labor and capital.
The exponents \(a\) and \(b\) indicate the output elasticity with respect to each input. This means they denote the percentage change in output resulting from a 1% change in a specific input.
Understanding production functions is crucial because they help in determining the optimal allocation of resources in order to maximize output. Different types of production functions, including Cobb-Douglas, allow businesses to model various output behaviors based on technological constraints and input efficiencies.

Input Scaling

Input scaling refers to adjusting all input factors of a production function by the same factor to understand how it affects the output. It’s a practical approach to studying returns to scale within mathematical models like the Cobb-Douglas production function.
For instance, if we have inputs \(x_1\) and \(x_2\), scaling them by a factor \(k\) changes the production function to \(f(kx_1, kx_2) = A k^{a+b} x_1^a x_2^b\). Here, the key observation is how \(k^{a+b}\) impacts the results: if \(a+b = 1\), the outputs adjust proportionally with inputs, and if these conditions differ, we observe different types of returns to scale.
Input scaling is not just a mathematical exercise; it provides insights into how changes in input allocation affect business operations and production efficiency. This helps businesses in strategic decision-making and planning for scaling operations.