## The Cost Minimization Problem

Firms perform cost minimization in order to determine the most efficient way of producing goods and services from at least two inputs. For example, if a firm invests too much on its workers and not enough on its capital, or vice versa, the cost for both will not be efficient.

**Cost minimization** is the rule in which producers seek to calculate the right balance between two inputs in order to have the most cost-effective productivity.

Before we discuss cost minimization in depth, it is important to go over the concepts of substitutes and complements in a factor market. These are types of factors of production whose price affects the demand of similar or related factors of production, which will in turn, affect how to calculate cost minimization.

In general, **substitutes** are goods that can be replaced with similar goods. For example, a consumer might prefer buying e-books to physical books. However, if the price of a desired e-book rises, or is more expensive than the hard copy, then that consumer may buy the physical book instead. The same applies to the factors of production.

**Substitutes in a factor market** are factors of production that can be replaced with similar factors of production.

Two goods are **complements** if consumers tend to buy those two goods together. Think of the relationship between video games and a game console, such as a Playstation or Xbox. If there is an increase in the price of a game console, then the demand to buy video games will decrease. The same applies to the factors of production.

**Complements in a factor market** are factors of production that are jointly utilized with other factors of production.

To learn more check our article on - Factor Markets!

### Cost Minimization and Profit Maximization

Reflecting on substitute and complementary factors of production can help the way we think about cost minimization and profit maximization. As you can imagine, there are different combinations of inputs a firm can calculate to minimize the cost of its capital and labor, in turn maximizing the firm's profits. Let's look at a specific example and see how it might be solved in the real world.

Think of a modern office warehouse with machine operators and automated forklifts that pick up products and move them from one place to another without a driver. In order to run most efficiently, the business needs to figure out how many machine operators (labor) and forklifts (capital) they require. If there are more forklifts than operators, there might be unused machines in the warehouse, not maximizing the total productivity. On the other hand, if there are more operators than forklifts, there may be slower movement of products across the warehouse floor. In both situations, the cost is not minimized.

## Cost Minimization Examples

Figuring out the best cost-effective approach for a firm can be challenging. However, when a firm needs to decide which combination of labor and capital inputs to use, calculating cost minimization may be relatively straightforward.

Let's refer to the example of a modern office warehouse with machine operators and automated forklifts.

Assume the following: if the firm with the office warehouse rents 40 forklifts, they will need to hire 1 machine operator to oversee 5 forklifts, requiring a total of 8 operators. In another scenario, more experienced operators may be more efficient by using other tools to help them move large quantities of product across the warehouse floor. The warehouse can move around the same amount of products with 20 automated forklifts and 20 machine operators.

Let's suppose the cost to rent a forklift is $2,000 per month and hiring a machine operator costs $1,500 per month. The cost of these inputs would look like the following:

Scenario 1 | Scenario 2 |

Cost of capital: 40 x $2,000 = $80,000 | Cost of capital: 20 x $2,000 = $40,000 |

Cost of labor: 8 x $1,500 = $12,000 | Cost of labor: 20 x $1,500 = $30,000 |

Total cost: $92,000 | Total cost: $70,000 |

Table 1. Scenarios of cost minimization - StudySmarter

We see that the second scenario, where the warehouse hires 20 machine operators and rents 20 automated forklifts, is the more cost-effective combination. But these are only two scenarios. How do we know there isn't another *better* scenario that finds the *most* cost-effective combination of capital and labor? We use the cost minimization formula to check for the most cost-effective combination in such cases.

## Cost Minimization Formula

It can be difficult for firms with thousands of employees and even more units of capital to use the cost minimization approach among seemingly infinite scenarios. That is why there is a formula to find the optimal inputs of labor and capital. This formula illustrates the marginal product of labor over the wage rate to equal the marginal product of capital over the rental price of capital.

$\frac{M{P}_{L}}{w}=\frac{M{P}_{K}}{r}$or

$\frac{Marginalproductoflabor}{wagerate}=\frac{Marginalproductofcapital}{rentalprice}$

The marginal product of an input refers to the additional output that results from additional input.

Let's apply this formula to an example. Similar to the relationship between machine operators and automated forklifts, the relationship between bank tellers and ATMs (Automated Teller Machines) may be observed to find the most cost-effective combination between the two.

Suppose the marginal product of labor from bank tellers is 100 units, and the marginal product of capital from ATMs is 200 units. What happens when the wage is $10, and the rental rate of capital is $5?

We use the formula:

$\frac{M{P}_{L}}{w}=\frac{M{P}_{K}}{r}$

$\frac{100}{\$10}<\frac{200}{\$5}$

$10unitsofoutput\left(labor\right)<40unitsofoutput\left(capital\right)$

The output per dollar of labor would be 100/$10 = 10 units, while the output per dollar of capital would be 200/$5 = 40 units. In other words, the business receives 10 units of output that is spent on labor, and 40 units of output spent on capital.

According to these inputs, the bank should invest more in ATMs than in bank tellers. However, the law of diminishing returns reminds us that as the bank rents more ATMs, the marginal product of capital declines. Additionally, as the bank hires less tellers, the marginal product of labor rises.

The bank therefore will change the amount of labor and capital until the marginal product per dollar spent on each input is the same.

## Cost Minimization Approach: The Two Conditions

When representing the cost-minimizing value on a graph, the value must uphold two conditions:

- The value is on the
*y*-isoquant - No other value on the
*y*-isoquant is on a lower isocost line

These two conditions are essential in illustrating a value whose cost is minimized. This puts into perspective what it means to have the most advantageous combination of labor and capital.

Let's take what we learned above and put it into a graph to see how it translates.

In Figure 1, we see the graph drawn with respect to labor (L) and capital (K). The two dark blue lines are isocost lines, which represent all combinations of labor and capital that cost the same total amount. The curved line, labeled as the *y*-isoquant shows the optimal combination of labor and capital that produces the maximum output at minimum cost.

Now focus on point A, which touches the *y*-isoquant and the first isocost line. This point shows the value of cost minimization. If you were to go to point B, there would be a higher input of capital than labor. Point C similarly illustrates a scenario where there is too much of one input. In this case, there is a higher input of labor and not enough capital for it to be considered a cost-effective alternative.

## Cost Minimization Analysis - Key takeaways

**Cost minimization**is the rule in which producers seek to calculate the right balance between two inputs in order to have the most cost-effective productivity.- Substitutes in a factor market are factors of production that can be replaced with similar factors of production. Complements in a factor market are factors of production that are jointly utilized with other factors of production.
- In the cost minimization formula, the marginal product of labor divided by the wage rate equals the marginal product of capital divided by the rental price of capital.
- Depending on the output per dollar of the marginal product of capital or labor, a firm would need to increase capital and decrease labor, or vice versa, until the marginal product per dollar on each input is equal.

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##### Frequently Asked Questions about Cost Minimization

What is cost minimization?

Cost minimization is the rule in which producers seek to calculate the right balance between two inputs in order to have the most cost-effective productivity.

What is an example of cost minimization?

A supermarket wants to implement the right balance of self-checkout machines and cashiers. Using the cost minimization formula, one can find the optimal level of the marginal product of labor with respect to wage until it equals the marginal product of capital with respect to rental rate.

What is the cost minimization formula?

What approaches can be used to minimize costs?

To determine cost minimization between two scenarios, add the products of the cost of capital and the cost of labor. For most cases, use the cost minimization formula to calculate the most cost-effective productivity.

What are the two conditions for cost minimization?

(1) The value is on the y-isoquant, and (2) no other value on the y-isoquant is on a lower isocost line.

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