On the way to understanding business training, you will inevitably come across the cost function – an essential aspect that requires in-depth knowledge. In this article you will learn what a cost function is and why its definition is important, how to calculate it and what types it can be divided into. It also shows how you can effectively visualize them and understand their connection to revenue. The article thus offers precise insights and practical instructions for a targeted approach to the cost function in business training.

## Definition of the cost function

One **Cost function** is a mathematical function that represents the total costs (variable and fixed costs) of a production system in relation to the production quantity.\( K(x) = K_{f} + K_{v}(x) \), where \( K_{ f} \) represents the fixed and \( K_{v} \) the variable costs.

There are two types of costs: fixed costs remain constant regardless of the level of production, while variable costs are directly related to the quantity of production.

A practical example of using the cost function might be for a manufacturing company to calculate the cost of producing different quantities of products. With the help of the cost function, it can be decided which production quantity causes the lowest costs per unit and therefore makes the most economic sense.

### Areas of application of the cost function

The application of the cost function is not limited to manufacturing companies. The following list gives some examples where cost functions play a role:

- Manufacturing company
- Service provider
- Agricultural sector
- Healthcare
- public sector

#### Cost function in everyday life and studies

In everyday life or at study, you may not be consciously confronted with a cost function. However, the cost function is implicitly applied to many decisions you make or others make for you that are based on economic considerations.

An everyday example could be deciding on a particular mode of transport. You could see the costs of your car (gas, insurance, maintenance) as fixed costs and the costs of tickets when using public transport as variable costs. When you then compare the total costs for different route lengths using both methods, you are essentially doing an analysis based on the cost function.

## Calculation and preparation of the cost function

Calculating and setting up the cost function is one of the central tasks in commercial training. This process gives you insight into your company’s potential expenses, allowing you to make more efficient decisions.

### Calculation of a cost function

To calculate a cost function, you need information about your company’s fixed and variable costs.

Fixed costs are expenses that remain constant regardless of production volume. Examples include rent, salaries or depreciation.

Variable costs, on the other hand, change depending on the production quantity, such as material or energy costs. The total costs of a company can be determined using the formula \, where \( K_f \) represents the fixed costs and \( K_v \cdot x \) represents the variable costs.

In practice, the formula \( K(x) = K_f + v \cdot x \) is often used, where \( v \) is the variable unit costs. These represent the additional costs incurred by producing an additional unit.

#### Step-by-step instructions for calculating the cost function

Suppose a company has fixed costs of €3,000 and variable unit costs of €5 per unit. The cost function of this company could then be calculated using the formula \( K(x) = €3000 + €5 \cdot x \). For example, if the company produces 500 units, the total cost is \( K(500) = €3000 + €5 \cdot 500 = €5500 \).

### Establishing a cost function

Developing a cost function can be a complex process, especially if the company produces a variety of products. You should take into account all costs that the company incurs through production in the total cost function.

It is particularly important to categorize the costs correctly. Fixed and variable costs should be kept separate and not mixed together as they have different impacts on total costs.

#### Elements of a well-structured cost function

A well-designed cost function should contain the following elements:

- Separate list of fixed and variable costs
- Indication of the production quantity x
- A clear and understandable equation

Example: A company’s fixed costs are €10,000 and the variable costs per unit are €2. Then the cost function is: \( K(x) = 10,000 € + 2 € \cdot x \). This function gives the total costs for each production quantity x.

## Different types of cost functions

Different types of cost functions are used in business administration, which differ in their structure and course. Each type of cost function has its own characteristics and applies in specific contexts. The linear and declining cost functions are explained in detail below.

### Linear cost function

The linear cost function, also known as the proportional cost function, is a basic form of the cost function. It is characterized by a linear progression, which means that costs increase in proportion to the production quantity.

With the linear cost function, the total costs increase by a fixed amount for each additional unit produced. This fixed amount is called marginal cost.

#### Features and calculation of the linear cost function

The linear cost function has the general formula \, where \( v \) is the constant amount by which the total cost increases for each additional unit produced. The features of the linear cost function can be summarized as follows:

- Proportional increase in costs: The total costs increase proportionally with the production quantity.
- Constant marginal cost: The additional cost of producing one more unit remains constant.
- Linear cost curve: A graph that plots costs versus production produces a straight line.

Suppose a company has fixed costs of €1000 and marginal costs of €5 per unit. Then the cost function is \( K(x) = 1000 € + 5 € \cdot x \). When producing 200 units, the total costs are \( K(200) = 1000 € + 5 € \cdot 200 = 2000 € \).

### Declining cost function

The declining balance cost function is another important type of cost function. In contrast to the linear cost function, the marginal costs decrease as production volume increases.

A declining cost function depicts a situation in which the costs per additional unit produced decrease. This phenomenon often occurs when a company benefits from volume discounts or similar effects.

#### Characteristics and application of the degressive cost function

The declining cost function has the general formula \, where \( K_v \) is the variable costs that decrease with increasing production volume. The characteristics of the declining cost function are:

- Declining cost increase: The total costs increase with the production quantity, but to a decreasing extent.
- Falling Marginal Cost: The additional cost of producing one more unit decreases.
- Curvy cost curve: A curved line results in a graph that plots costs against production quantity.

Suppose a company has fixed costs of €5,000 and variable costs of €2,000. Then the cost function is \( K(x) = 5000 € + \frac{2000 €}{x} \). When producing 200 units, the total costs are \( K(200) = €5000 + \frac{2000 €}{200} = €5100 \).

## Visualization of cost functions

Cost functions can be represented particularly well visually. Such a representation not only supports understanding of the topic, but also makes it possible to recognize connections and changes at a glance. No matter whether it is linear, declining or another type of cost function, visualization helps understand the impact of production volume on a company’s overall costs.

### Drawing a cost function

Plotting a cost function is a process that requires a few steps. First, you should prepare an appropriate coordinate system, where the horizontal axis represents the production quantity and the vertical axis represents the cost.

The stages of drawing a cost function include identifying the fixed and variable costs, substituting these costs into the appropriate cost function formula, and then drawing the resulting line or curve in the coordinate system.

It’s important to choose the boundaries of your diagram correctly. If the production quantity or costs are very high, it may make sense to adjust the axes accordingly in order to obtain a clear display. It can also be helpful to choose the scale of the axes so that the expected values can be clearly represented.

#### Techniques for making cost function drawings

Creating an accurate drawing requires accuracy and attention at every stage of the process. Here are some steps to help you plot cost functions accurately:

- First, identify your fixed and variable costs.
- Create your coordinate system with production quantity on the x-axis and costs on the y-axis.
- Enter the fixed costs as a horizontal line in your diagram.
- Calculate the total costs for different production quantities and plot these points on your diagram.
- Connect the points to form a line or curve, depending on the type of cost function.

Suppose a company has fixed costs of €500 and variable costs of €2 per unit. Then the intersection with the Y-axis would be €500, these are the fixed costs. For each additional unit produced, costs would increase by €2. Ie with a production of 200 units the total costs would be €900. These points can now be entered into the diagram and connected to each other. This creates a linear cost function.

No matter what type of cost function you draw, the final diagram should show an accurate representation of the company’s cost structure depending on production volume. Not only does it practice your skills in drawing and using cost functions, but it can also serve as an excellent reference for business decision making and planning.

## Connection between cost function and revenue

A fundamental understanding of the connection between the cost function and revenue is crucial for financial planning and strategy development in a company. The cost function and the revenue are directly linked to each other. They both influence the company’s overall results…