# Profit maximization

Profit maximization , also profit maximization , is a corporate goal in economics , according to which entrepreneurs adjust their production volume in a market economy so that a market equilibrium is achieved. The corporate goal of profit maximization is often assumed in the specialist economic literature, but other goals such as cost recovery or profit maximization are also possible. In the maximum profit situation , the marginal cost equals the marginal revenue .

After a company has entered the market , it will usually try to maximize its profit through optimal production planning. The way in which a company can maximize its profit depends on the type of market in which the company operates and on the company's position in the market.

In the following, the mechanism of profit maximization of a monopolist in a market economy will first be explained. Then profit maximization in equilibrium is explained.

## Profit maximization with a monopolist

A characteristic of this situation is that there is a price-sales function that describes the quantity of a product that can be sold at a certain price. One can generally assume that a larger amount of the product can be sold if prices fall.

The company then chooses the price for its product at which the maximum profit can be achieved. The price is therefore not given, as in a market with perfect competition , in which the companies act as price takers or volume adjusters , but is chosen by the monopolist.

The point on the price-sales function at which a monopoly company achieves maximum profit is called the Cournot's point .

## Formulas for maximizing profit for a monopoly

A particularly easy-to-use version of a profit function formulates profit as a function of the output quantity of a certain good, i.e. the following applies to the profit function : ${\ displaystyle x}$${\ displaystyle G}$

${\ displaystyle G (x) = E (x) -K (x)}$

with the revenue and cost function (each depending on the quantity sold ). ${\ displaystyle E (x)}$${\ displaystyle K (x)}$${\ displaystyle x \ geq 0}$

It is assumed that the profit function is twice continuously differentiable . According to the general rules about the maximization of functions, there is a (local) profit maximum at an inner point if, on the one hand, the marginal profit for this set is zero, that is ${\ displaystyle x_ {0}> 0}$

(1) (necessary condition for a maximum),${\ displaystyle G '(x_ {0}) = 0}$

and on the other hand the second derivative of the profit function is negative in the position${\ displaystyle x_ {0}}$

(2) (sufficient condition for a maximum).${\ displaystyle G '' (x_ {0}) <0}$

Note that from (1) with the definition of the profit function it follows immediately that , that is, the marginal revenue corresponds to the marginal costs. This is intuitive: If the marginal revenue exceeds the marginal costs, one could increase the profit with the production of a (marginal) additional unit, because the additional revenue achieved would outweigh the additional costs incurred. Conversely, if marginal costs exceed marginal revenue, one could increase profit by reducing production by one (marginal) unit, because the cost savings achieved would more than compensate for the resulting decline in revenue. ${\ displaystyle E '(x_ {0}) - K' (x_ {0}) = 0}$

### example

Illustration of the example: The profit maximum is at the point .${\ displaystyle x_ {0} = 1200}$
Illustration of the example: In the maximum place of the profit function, marginal revenue and marginal costs coincide.${\ displaystyle x_ {0} = 1200}$

The price-sales function of a monopolist is given

${\ displaystyle p (x) = 150 - {\ frac {x} {20}}}$

as well as a linear cost function

${\ displaystyle K (x) = 20000 + 30x}$.

The revenue function is initially

${\ displaystyle E (x) = p (x) \ cdot x = 150x - {\ frac {x ^ {2}} {20}}}$.

For the profit function follows

${\ displaystyle G (x) = E (x) -K (x) = \ left (150x - {\ frac {x ^ {2}} {20}} \ right) - (20000 + 30x)}$.

The first order condition for a maximum is , and so ${\ displaystyle G '(x_ {0}) = 0}$

${\ displaystyle G '(x) = E' (x) -K '(x) = 150 - {\ frac {x} {10}} - 30 = 120 - {\ frac {x} {10}} \; {\ overset {!} {=}} \; 0 \ implies x_ {0} = 1200}$.

This is because of

${\ displaystyle G '' (x) = - {\ frac {1} {10}} <0}$

also sufficient. The price-sales function shows that the price for this production volume is. ${\ displaystyle p_ {0} = 90}$

## Profit maximization in equilibrium

For a company in a market with perfect competition and in equilibrium, the maximization of profit is very different from that of a monopolist: with perfect competition the profit in equilibrium is zero! The maximum achievable for a company here is that no losses are made.

At first glance, this does not seem to make sense, since it is assumed that no entrepreneur enters the market without being able to make a profit. Doesn't he want to be 'paid' for his work in the company (planning, organization, etc.) and for the risk he takes?

Even in a perfectly competitive market, such as B. is dealt with by Arrow & Debreu , the entrepreneur appears, albeit as a normal consumer who on the one hand makes his labor available and on the other hand receives the most preferred bundle of goods intended for him by the market, just like every other market participant.

The entrepreneur thus receives a virtual salary for his work. There is no risk for him in this market, he is only responsible with his labor. He has raised capital for buildings, machines, etc., for which he has to pay interest, which appears quite normally in cost accounting and is taken into account by the market.

One hypothetical question is how does a fully competitive market prevent companies from making a profit? To do this, you have to keep in mind that in a market with full competition there are theoretically many suppliers for the same product ( homogeneous polypol ) and that all relevant information is known to everyone. First of all, it follows that no consumer would accept a price higher than the lowest price.

Would a company z. B. can produce more cheaply due to innovative production, the other suppliers would also switch to this production process, which would restore the same conditions and all manufacturers would have to offer the same price without profit. That is the optimal price that is 'found' by the market and that every entrepreneur gets - no more and no less. However, there is no perfect competition anywhere, it is a theoretical construct.

## literature

• Friedrich Breyer: Microeconomics. An introduction . 6th edition. Springer, Heidelberg a. a. 2015, ISBN 978-3-662-45360-5 .

## Remarks

1. Sufficient: twice continuously differentiable on the interval .${\ displaystyle (0, \ infty)}$
2. See the article Extremwert .
3. Conditions (1) and (2) ensure a (local) profit maximum. Note that it generally does not follow from this that every (local) maximum digit of the profit function satisfies conditions (1) and (2). In this case, there could also be a (local) profit maximum. In this case, the possibility remains to check the sign behavior of a stationary point in the vicinity of a stationary point determined on the basis of condition (1) : ${\ displaystyle G '(x_ {0}) = G' '(x_ {0}) = 0}$${\ displaystyle G '(x)}$${\ displaystyle x_ {0}}$
(2 ') A stationary digit is a local maximum digit of the profit function if one exists such that for all and one exists such that for all .${\ displaystyle x_ {0}}$${\ displaystyle a ${\ displaystyle G '(x) \ geq 0}$${\ displaystyle x \ in (a, x_ {0})}$${\ displaystyle b> x_ {0}}$${\ displaystyle G '(x) \ leq 0}$${\ displaystyle x \ in (x_ {0}, b)}$
4. See the article Sum rule .
5. See only Breyer 2015, p. 71 f.
6. Lawrence Boland: Foundations of Economic Method: A Popperian Perspective. 2nd edition 2003. pp. 149, 150