# How Your Organization’s Production Function Is Related to Its Marginal Product of Labor

If you are running a business or managing an organization, you might be interested in knowing how your production function is related to your marginal product of labor. In this article, we will explain what these two concepts mean, how they are calculated, and how they affect your output and costs.

## What is a Production Function?

A production function is a mathematical equation that describes the relationship between the inputs and the output of a production process. It shows how much output can be produced with a given amount of inputs, such as labor, capital, land, and technology. A production function can be written as:

Q = f(L, K, T)

where Q is the output, L is the labor input, K is the capital input, and T is the technology input. The function f represents how the inputs are combined to produce the output.

A production function can have different shapes and properties depending on the nature of the production process. For example, some production functions exhibit increasing returns to scale, meaning that doubling the inputs will more than double the output. Some production functions exhibit decreasing returns to scale, meaning that doubling the inputs will less than double the output. Some production functions exhibit constant returns to scale, meaning that doubling the inputs will exactly double the output.

## What is Marginal Product of Labor?

Marginal product of labor (MPL) is the change in output that results from employing an additional unit of labor, holding all other inputs constant. It measures how productive a worker is in contributing to the output. Marginal product of labor can be calculated as:

MPL = ∆Q / ∆L

where ∆Q is the change in output and ∆L is the change in labor input. Alternatively, marginal product of labor can be obtained by taking the first derivative of the production function with respect to labor:

MPL = dQ / dL

Marginal product of labor can also be shown graphically as the slope of the production function at a given point.

The production function and the marginal product of labor are closely related because they both describe how output varies with labor input. The shape and properties of the production function determine the behavior and pattern of the marginal product of labor.

For example, if the production function exhibits increasing returns to scale, then the marginal product of labor will be increasing as well. This means that each additional worker will add more to the output than the previous one. This situation can occur when there are increasing returns to specialization or learning by doing.

If the production function exhibits decreasing returns to scale, then the marginal product of labor will be decreasing as well. This means that each additional worker will add less to the output than the previous one. This situation can occur when there are diminishing returns due to congestion or inefficiency.

If the production function exhibits constant returns to scale, then the marginal product of labor will be constant as well. This means that each additional worker will add the same amount to the output as any other one. This situation can occur when there are no externalities or spillovers from production.