# What Is Cost Minimization?

Cost minimization is a basic rule used by producers to determine what mix of labor and capital produces output at the lowest cost. In other words, what the most cost-effective method of delivering goods and services would be while maintaining a desired level of quality.

An essential financial strategy it is important to understand why cost minimization is important and how it works.

## The Flexibility of the Production Function

In the long run, a producer has the flexibility over all aspects of production—how many workers to hire, how big of a factory to have, what technology to use, and so on. In more specific economic terms, a producer can vary both the amount of capital and the amount of labor it uses in the long run.

Therefore, the long-run production function has 2 inputs: capital (K) and labor (L). In the table provided here, q represents the quantity of output that is created.

## Choices of Production Process

In many businesses, there are a number of ways in which a particular quantity of output can get created. If your business is making sweaters, for example, you could produce sweaters either by hiring people and buying knitting needles or by buying or renting some automated knitting machinery.

In economic terms, the first process uses a small quantity of capital and a large quantity of labor (i.e., is "labor intensive"), whereas the second process uses a large quantity of capital and a small quantity of labor (i.e., is "capital intensive"). You could even choose a process that is in between these 2 extremes.

Given that there are often a number of different ways to produce a given quantity of output, how can a company decide what mix of capital and labor to use? Not surprisingly, companies are generally going to want to choose the combination that produces a given quantity of output at the lowest cost.

## Deciding the Cheapest Production

How can a company decide what combination is the cheapest?

One option would be to map out all of the combinations of labor and capital that would yield the desired quantity of output, calculate the cost of each of these options, and then choose the option with the lowest cost. Unfortunately, this can get pretty tedious and is in some cases not even feasible.

Luckily, there is a simple condition that companies can use to determine whether their mix of capital and labor is cost minimizing.

## The Cost-Minimization Rule

Cost is minimized at the levels of capital and labor such that the marginal product of labor divided by the wage (w) is equal to the marginal product of capital divided by the rental price of capital (r).

More intuitively, you can think of cost being minimized and, by extension, production being most efficient when the additional output per dollar spent on each of the inputs is the same. In less formal terms, you get the same "bang for your buck" from each input. This formula can even be extended to apply to production processes that have more than 2 inputs.

To understand why this rule works, let's consider a situation that is not cost minimizing and think about why this is the case.

## When Inputs Are Not in Balance

Let's consider a production scenario, as shown here, where the marginal product of labor divided by the wage is greater than the marginal product of capital divided by the rental price of capital.

In this situation, each dollar spent on labor creates more output than each dollar spent on capital. If you were this company, wouldn't you want to shift resources away from capital and towards labor? This would allow you to produce more output for the same cost, or, equivalently, produce the same quantity of output at a lower cost.

Of course, the concept of diminishing marginal product implies that it's generally not worthwhile to keep shifting from capital to labor forever, since increasing the quantity of labor used will decrease the marginal product of labor, and decreasing the quantity of capital used will increase the marginal product of capital. This phenomenon implies that shifting towards the input with more marginal product per dollar will eventually bring the inputs into cost-minimization balance.

It's worth noting that input doesn't have to have a higher marginal product to have a higher marginal product per dollar, and it may be the case that it could be worthwhile to shift to less productive inputs to production if those inputs are significantly cheaper.

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