# Cost Functions

## Purpose of cost functions

For a given firm, pricing as well as production planning require thorough knowledge of the firm's costs at different activity levels

For instance the owner of a pizzeria needs to know about costs incurred to the firm at different activity levels (number of pizzas produced) when pricing in connection with a brochure that is to be distributed door-to-door. Costs vary with activity level, as purchases of raw materials, hiring of employees, choice of oven, and maybe even size of the location depend on the chosen level of activity. Which costs vary with the activity level depend on the owner's decision-making horizon, e.g. if there are big differences between planning the daily operations and strategic considerations regarding the pizzeria's future.

With the purpose of assisting the decision maker (e.g. the owner of the pizzeria) cost functions expressing cost as a function of the activity level are devised. Depending on the functions of the decision-making situation, the following cost functions are applied:

o Total costs (TC, TVC, and TFC)

o Average costs (ATC, AVC, and AFC)

o Marginal costs (MC)

Concerning cost functions, it is important to find a suitable measurement unit for the activity level. The measurement unit varies with business sector and industry. Examples of measurement units are:

o Produced units (number of baked pizzas, printed magazines, manufactured cars)

o Turnover (expresses the activity in restaurants and clothing stores)

o Working hours (expresses the activity for architectural firms and law firms)

## Cost Functions in the Short-Term

### Total functions

In the following, the cost functions are explained within the frame of the firm's short-term time horizon, which is why both fixed costs and variable costs are involved. The basis of the firm's cost functions are the total-functions as described below.

*o "Total fixed costs" (TFC):* The fixed costs do not fluctuate with the activity level, within the chosen time horizon. For instance the pizzeria's rent, interest, oven installation, etc., do not vary based on the number of pizzas baked over a short-term time horizon.

o *"Total variable costs" (TVC):* The variable costs change based on the activity level within the chosen time horizon. For instance the cost of flour, meat, and workers' salaries, vary in conjunction with the number of pizzas baked, even within the short-term time horizon.

o *"Total costs" (TC):* The TC-function is found through vertical adding the TFC and TVC curves, i.e. by compiling the fixed and variable costs.

### Average functions

As is the case when finding any average, the average-functions are calculated on the basis of the firm's total-functions, which are then divided by the relevant factor (in this case quantity). The average-functions are found as shown below:

o *"Average fixed costs" (AFC):* Expresses the fixed costs divided by the output, i.e. TFC/Q (Q = quantity, e.g. number of pizzas produced in a pizzeria).

o *"Average variable costs" (AVC):* Expresses the variable costs divided by firm output, i.e. TVC/Q.

o *"Average total costs" (ATC):* Expresses the total costs divided by firm output, i.e. TC/Q

### Marginal functions

When the firm plans its production level, it is crucial to know how much the total costs change as related to a change in quantity. In this case marginal costs are applied.

*o "Marginal costs" (MC):* Expresses the change that appears in the total costs as a result of the firm producing one unit more. For instance, the marginal cost function for the pizzeria expresses the change in the total costs which appear when one additional pizza is produced.

The firm's marginal cost function is thus found as the change in the TVC function (or the TC function), resulting from a change in the quantity produced. For this reason the MC function is found by differentiating the TVC-function (or the TC function, as the constant is neutralized in any case):

### The cost development

Costs can develop in the following ways:

o Proportionally, i.e. if the activity increases 1% then the total costs also increase 1%.

o Digressively, i.e. if the activity increases 1% then the total costs increase less than 1%.

This phenomenon is called increasing returns to scale, or economies of scale. In modern production this is the norm, as many actions support this: The more you buy the cheaper it gets. Administrative systems are cheaper for each unit, and marketing as well as production lines are utilized more efficiently.

o Progressively, i.e. if the activity increases 1% then the costs increase more than 1%

This phenomenon is called decreasing returns to scale or diseconomies of scale.

In modern production this is not normal. If existing it is probably found in service sectors which are heavily knowledge-based.

The connection is shown in figure 2:

Figure 2: **The connection between MC and TC**

It must be emphasized that cost development is dependent on both activity level and decision-making situation. A firm may, for example, have a digressive cost development (economies of scale) at a low level of activity because of increasing discounts on purchasing, and at the same time experience a progressive cost development because of waste, discarding, less efficient employees etc. Often these circumstances are present simultaneously.

### The connection between total costs, average costs, and marginal costs.

The connection between total costs, average costs, and marginal costs is shown in model below:

The triangle model depicts: **that when you have a mathematical term for the cost function **- moving from one cost function to another is possible in the following ways:

o You move from a total function to an average function by dividing with Q.

o You move from an average function to a total function by multiplying with Q

For instance, to move from TC to ATC you divide TC with Q - and from ATC to TC by multiplying ATC with Q.

Furthermore, the triangle model shows that movement from a total function (TC or TVC) to a marginal function (MC) is possible by differentiating the total function. In this way the marginal function reflects the inclination on the total costs curve. This relationship is logical as the marginal costs show the costs of the production of one additional unit, which is exactly the increase in the total costs.

In the opposite direction, the triangle model also shows that you move from a marginal function (MC) to a total function (TVC) by integrating the marginal function. In this way the total function (TVC) reflects the area below the marginal costs curve. This relationship is equally logical as the total variable costs (TVC) are precisely the sum of all the marginal costs.

Moreover, it is shown that you move from an average function (ATC or AVC) to a marginal function (MC) by multiplying Q and thereafter differentiating the function; i.e. you first find the total function (TC or TVC) and thereafter find the marginal function.