# Marginal cost

The marginal costs (also marginal costs ) are in business administration and microeconomics the costs that arise from the production of an additional unit of quantity of a product. Mathematically, the marginal cost function is the first derivative (the slope) of the cost function with respect to the number of units produced.

Example of the possible graphical course of the marginal cost function

## example

The production of a good X incurs fixed costs of 100,000 euros - for example, in the form of rents for production facilities, the costs of providing machines and salaries for employees. With these resources, a maximum of 5,000 units of the property can be manufactured. To produce a unit of X, raw materials costing 5 euros are required ( variable costs ).

In the case of a manufactured unit, the total production costs are 100,005 euros, for two manufactured units 100,010 euros, etc. With each unit produced, the total costs increase by 5 euros - this amount corresponds to the marginal costs.

Assume that the raw material supplier grants a quantity discount of 0.50 euros (per unit) from the 3,000. Unit and one euro (per unit) for the 4,000. and every further unit. Up to a production of 2,999 units, the marginal costs are then 5 euros, between 3,000 and 3,999 units 4.50 euros and from 4000 units 4 euros - the marginal costs fall.

If the demand rises to over 5000 pieces, i.e. if the capacity limit of the production is exceeded, further costs are incurred for the expansion of production. For example, higher (variable) costs for the maintenance of machines and overtime allowances for the staff have to be calculated. At this point the marginal cost increases. In addition, additional fixed costs ( step- fixed costs ) can arise if production is expanded beyond a certain level.

In addition to this connection, the marginal cost function also indicates the price on the perfect market , which is realized for the quantity of good X. As long as the marginal costs are below the average costs , the company does not break even. It will therefore only produce when the marginal cost function intersects with the average cost function.

## Alternative definitions

"Marginal costs are the increase in costs that arises from the additional production of an output unit."

However, the term so-called jump - fixed costs is to be differentiated here . These arise when a capacity level is exceeded in addition to the absolutely fixed costs of the previous capacity level and are incurred in the new capacity level regardless of the level of employment. Consequently, this means for the specified marginal cost definition that it only remains correct as long as the additional production unit can be created within a capacity level. The prerequisite for this is that there is still free capacity in the capacity level in which a company is currently located (e.g. existing machines, systems, production halls). If this requirement is not met, then step-fixed costs must inevitably be incurred, since new capacities must be created in the form of investments for the production of an additional service unit.

"Viewed graphically, the marginal costs are the slope of the tangent of the total cost curve for the examined output volume."

This definition also only covers part of the necessary explanatory components for marginal costs. Step-fixed costs always play an important role in marginal costs, as already explained above, and should therefore not be disregarded. Furthermore, the block of fixed costs of each company should be included in the discussion in order to determine the intersection of the cost function with the ordinate and to be able to imagine the course of the cost and marginal cost function. Technical standards, in particular, lead to an increasing number of fixed cost components in companies, which must be the focus, especially in medium and long-term decisions.

## Marginal cost function

The marginal cost function graphically represents the first derivation of the cost function. This is explained in more detail below, applied to two different function types. The output quantity x (output) is plotted on the abscissa and the associated total costs K on the ordinate.

### Linear cost trend

In order to first illustrate the principle of marginal costs, the linear cost curve is explained using an example, even if in practice this hardly ever occurs in pure form. The total costs result from the sum of fixed costs and variable unit costs , the latter being multiplied by the output quantity . ${\ displaystyle FK}$${\ displaystyle VK}$${\ displaystyle x}$

Graphic 1: Linear cost trend

Graph 1 shows a linear cost function with the general form

${\ displaystyle K (x) = VK \ cdot x + FK}$

In the example, the cost function is This means that the fixed costs are one (e.g. monetary units: €) and the variable costs that change with the output amount correspond to two. This cost function is shown by the orange graph with an intersection with the -axis at the point and a positive increase of two. ${\ displaystyle K (x) = 2x + 1}$${\ displaystyle y}$${\ displaystyle (0; 1)}$

To determine the marginal cost, it is necessary to implement the mathematical definition of marginal costs and virtually the cost function to differentiate . The first derivation thus results : ${\ displaystyle K}$${\ displaystyle x}$ ${\ displaystyle K '}$

${\ displaystyle K '(x) = {\ frac {dK (x)} {dx}}}$
${\ displaystyle K '= 2}$
${\ displaystyle K '' = 0}$

The marginal cost function (shown in green) is independent of the output volume and runs as a straight line parallel to the abscissa . This linear cost curve represents a special case of marginal costs in which the return on scale tends to zero. The marginal costs are constant, the total costs increase proportionally to the variable quantities . If the production in this company increases by one unit of service, the costs increase by two monetary units under the conditions that were explained in the first section. ${\ displaystyle x}$ ${\ displaystyle GK = VK + FK}$${\ displaystyle x}$

### Non-linear cost history

The non-linear cost curve is based on a cost function under the law of income and has far more practical relevance than the linear cost curve, as it represents the operational costs more realistically.

Graphic 2: non-linear cost development. The marginal cost curve (K ', purple) intersects the variable average costs (VDK, blue) and the total / total average costs (TDK, green) in their respective minima.

Graph 2 illustrates the course of the cost function (black graph), which, as can be seen, is not linear, i.e. H. cannot be interpreted as a straight line. In addition, the average variable costs (blue graph), average total costs (green graph) and marginal costs (purple graph) resulting from the cost function , which are further analyzed below, are shown as independent functions. ${\ displaystyle K}$${\ displaystyle VDK}$${\ displaystyle TDK}$${\ displaystyle K '}$

In operations, marginal costs are usually expected to fall, since the production of large units of measure is more profitable for a company than the production of small quantities. The reasons for this are economies of scale and learning curve effects . The first section of the marginal cost curve up to the minimum of the function therefore initially declines, i.e. H. as the output volume increases, the price of the additional output unit produced decreases. Here the marginal costs are lower than the average total costs and the economies of scale increase. It is therefore possible to achieve double the output without incurring double the costs. Then the marginal cost function reaches its minimum at the inflection point of the cost function  and the marginal costs rise again, the marginal costs are now greater than the average total costs and the returns to scale decrease. H. double sales cannot be achieved with double the cost. The reasons for this trend can be traced with the help of the income-related cost or production function . The marginal cost function cuts the average cost functions in their minimum . ${\ displaystyle K '}$${\ displaystyle TDK}$${\ displaystyle K}$${\ displaystyle TDK}$

## Marginal costs and average costs

The graph of marginal costs always intersects the graph of average costs at its extremes. At the beginning, the marginal costs are lower than the (total) average costs, since the marginal costs only depend on the variable costs, whereas the (total) average costs also include the fixed costs and these only decrease with increasing production volume (see: Fixed cost degression ). As long as the marginal costs are below the (total) average costs, an additional unit causes fewer costs than the average. This leads to the (total) average costs falling further and further, and thus the two graphs converge. For the intersection of the two graphs, this means that an additional unit costs exactly as much as the average for that amount. From this amount onwards, the marginal costs continue to rise, as a result of which an additionally produced unit now costs above average and thus also increases the (total) average costs. Due to the previous decrease in the (total) average costs and the subsequent increase, there must inevitably be a local minimum, and since there can only be an intersection with the marginal costs if the costs of an additionally produced unit do not lead to a change in the average (otherwise the marginal costs ensure that the (total) average costs "equalize"), the intersection must be in the extreme of the (total) average costs.

The slope of the cost function at these points, the marginal costs , is / are thus equal to the average costs . ${\ displaystyle K '(x)}$${\ displaystyle {\ tfrac {K (x)} {x}}}$

If you want to determine the extremes of the average costs, you have to set the first derivative of the average cost function equal to zero ( ): ${\ displaystyle x \ neq 0}$

${\ displaystyle \ left ({\ frac {K (x)} {x}} \ right) ^ {\ prime} = 0}$

From this it follows according to the quotient rule :

${\ displaystyle {\ frac {K ^ {\ prime} (x) \ cdot xK (x)} {x ^ {2}}} = 0}$

It follows:

${\ displaystyle K ^ {\ prime} (x) = {\ frac {K (x)} {x}}}$

Mathematically, this in turn corresponds to the intersection of marginal costs and average costs . ${\ displaystyle K '}$${\ displaystyle {\ tfrac {K (x)} {x}}}$

## Basics of profit maximization in the short term

From an economic point of view, the corporate sector always aims to maximize its profits; That is, all companies are assumed to be profit maximizers. The profit is the difference between total revenue and total costs.

Profit = Total Revenue - Total Cost${\ displaystyle G}$${\ displaystyle R}$${\ displaystyle K}$

“In order to maximize profit, a company selects the output where the difference between the revenue and the costs is greatest.” To achieve this goal, an entrepreneur must be well-informed , especially with regard to his cost and the associated price calculation, depending on his market form be.

### Profit maximization in the competitive company

Marginal cost curve and marginal revenue curve with an intersection

The market form of a polypole is assumed, i. that is, there is perfect competition between the companies. All competing companies are equally exposed to the demand of other economic sectors and also of their own, and thus the price for a product is considered to be fixed, and the achievable revenue from an additional unit of marginal revenue sold corresponds to the price that an economic entity has to pay for the product. The short-term profit maximization condition applies to all competitive companies: ${\ displaystyle R '}$${\ displaystyle P}$

Marginal cost = marginal revenue = price${\ displaystyle K '}$${\ displaystyle R '}$${\ displaystyle P}$

Since the price is viewed as constant, a polypolist can only regulate profit maximization via the sales volume and not via the price.

### Profit maximization in a monopoly

Unlike the polypolist, the monopolist can also determine his profit via the price due to his strong market position as the only buyer / seller. It determines the point of intersection between the marginal revenue curve and the marginal cost curve and receives a profit-maximizing sales volume . The monopolist can use the demand function to determine the associated price. If the monopolist produces below the calculated profit-maximizing quantity, he has fewer costs, but the lost proceeds from the additional sales are greater than the saved costs and thus lead to a reduction in profit.

If, on the other hand, the monopolist produces more than the profit-maximizing production volume, he incurs higher revenues on the one hand, and on the other hand the costs for additional production above the equilibrium volume exceed the revenues and also lead to profit shrinkage. The profit maximization condition applies:

Marginal costs = marginal revenue${\ displaystyle K '}$${\ displaystyle R '}$

In a normal monopoly, there is an area in which marginal costs intersect falling marginal revenue (marginal revenue). With a linear demand curve [PQ], the sales curve is characterized by twice the fall rate, but the same starting point as for the demand curve. For the monopolist, this intersection (Cournot point) is the combination of the quantity offered and the price obtained, which maximizes the total revenue. All other things being equal , this price will be higher than that of the quantity adjuster, and the quantity offered will be lower than that of the perfect competition.

In a natural monopoly , the average cost continues to decrease with the quantity. There is then no intersection between marginal costs and average costs, since the marginal costs are always below the average costs. That is why such a natural monopoly cannot cover its costs with the marginal costs, but must at least offer at average costs. Only when the marginal costs are above the average costs can the price be set equal to the marginal costs, with all costs covered.

If the marginal costs are above the average costs without fixed costs , the operating minimum has been reached. The company should accept the next order. However, if it falls below this limit, it is no longer worthwhile to continue producing, as not even the variable costs can be covered.

However, it is better if the marginal costs are higher than the average costs including fixed costs. With this production volume, one moves above the operating optimum .

• LRIC - alternative to marginal costs

## literature

• Accounting and controlling. Accounting modules and their links. Verlag neue Wirtschaftsbriefe, (NWB), Herne / Berlin 1998, ISBN 3-482-48121-0 , p. 272.
• Robert S. Pindyck, Daniel L. Rubinfeld: Microeconomics. 6th edition. Pearson Studium, Munich 2005, ISBN 3-8273-7164-3 , p. 361.
• Adolf E. Luger: General business administration. Volume 1: The structure of the company. 5th edition. Carl Hanser Verlag, Munich / Vienna 2004, ISBN 3-446-22539-0 .