Chapter 7: Problem 14

How do we calculate marginal product?

Short Answer

Expert verified

To calculate the marginal product, we first need to identify the production function, which shows the relationship between inputs and outputs of a good or service, typically in the form \( Q = f(L, K) \). Then, we can find the marginal product of labor (MPL) and the marginal product of capital (MPK) by taking the partial derivatives of the production function with respect to labor (L) and capital (K), while keeping the other input constant. The general formulas are: \( MPL = \frac{\partial Q}{\partial L} \) and \( MPK = \frac{\partial Q}{\partial K} \). After finding the expressions for MPL and MPK, we can use specific input values to calculate the respective marginal products.

Step by step solution

Understand the Concept

The first step is understanding what marginal product is. It measures how much additional output (goods or services) an economy or business will produce if it adds one more unit of an input, like labor or capital, keeping all other inputs constant.

Identify the Production Function

A production function is a mathematical relationship that shows the relationship between inputs and outputs in producing a good or service. Generally, it takes the form: \( Q = f(L, K) \) where Q is the quantity produced, L is the labor input, and K is the capital input. We must have a given production function in order to calculate marginal product.

Calculate the Marginal Product of Labor (MPL)

To calculate the marginal product of labor, we need to take the partial derivative of the production function with respect to labor (L) while keeping capital (K) constant. This will give us the additional output from one additional unit of labor. The general formula is: \( MPL = \frac{\partial Q}{\partial L} \)

Calculate the Marginal Product of Capital (MPK)

Similarly, to calculate the marginal product of capital, we need to take the partial derivative of the production function with respect to capital (K) while keeping labor (L) constant. This will give us the additional output from one additional unit of capital. The general formula is: \( MPK = \frac{\partial Q}{\partial K} \)

Example

Let's take an example of a production function in the form of the Cobb-Douglas function: \( Q = 2L^{0.5} K^{0.5} \) To calculate the MPL and MPK, we need to take the partial derivatives with respect to L and K, respectively. For MPL, we take the partial derivative with respect to L: \( MPL = \frac{\partial Q}{\partial L} = 2 \times 0.5L^{-0.5} K^{0.5} = L^{-0.5} K^{0.5} \) Similarly, for MPK, we take the partial derivative with respect to K: \( MPK = \frac{\partial Q}{\partial K} = 2L^{0.5} \times 0.5K^{-0.5} = L^{0.5} K^{-0.5} \) Now, we have the expressions for MPL and MPK, which can be used to find the marginal product of labor and capital given specific values of L and K.