Cournot point

The Cournot point is a name that is particularly well-known in German-speaking countries for the point on the price-sales function of a monopoly company at which the company is at its profit maximum . In the quantity-price diagram, the point captures the two coordinates quantity and price ; The profit can be clearly determined from these. Cournot's point is therefore, loosely speaking, the answer to the question of which price-quantity combination is the maximum profit for a monopolist. It is the result of monopoly pricing .

This point is named after the French economist Antoine Augustin Cournot (1801–1877).

Typical of the Cournot point is that it is to the left of the revenue maximum . In other words: in the profit maximum a smaller amount of the good is sold than would be the case in the revenue maximum.

calculation

Cournot point graphically

Calculation of Cournot's point ( ) with maximum profit price ( ) and maximum profit sales quantity ( ): ${\displaystyle C}$ ${\displaystyle p_{c}}$ ${\displaystyle x_{c}}$

In contrast to the company in perfect competition , which has to accept a market price for its product , the monopolist can set the sales price in a way that maximizes profit. It must have a demand function for this , i. H. at what price he can sell how much of the product. Alternatively, he can gradually approach the profit optimum with his pricing policy ( Cobweb theorem ).

${\displaystyle x=x(p)}$,

or as an inverse function, the price-sales function as

${\displaystyle p=p(x)}$.

From this the total revenue (often , here sales ) is determined as price × quantity ${\displaystyle E}$ ${\displaystyle U}$

${\displaystyle U(x)=p(x)x}$.

With the total cost function , the company makes profit as ${\displaystyle K(x)}$ ${\displaystyle G(x)}$

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

To find the maximum profit, the first derivative of is taken (i.e. ) and set equal to zero. The determined zeros (in the case of an S-shaped cost curve or other non-linear profit curves) must now be inserted into the second derivation. The zero at which this second derivative is negative is the maximum yield that defines the Cournot point. In order to get the Cournot point, the associated price is determined from the price-sales function. ${\displaystyle G(x)}$ ${\displaystyle G'(x)}$ ${\displaystyle x_{c}}$ ${\displaystyle x_{c}}$ ${\displaystyle p_{c}}$

Because when you maximize the profit function

${\displaystyle G'(x)=U'(x)-K'(x)=0}$

also

${\displaystyle U'(x)=K'(x)}$

can write, it follows that the cournotsche point also calculate can by directly the marginal cost of the marginal revenue equates. The value of the intersection forms the profit-maximum sales volume . This must be used in the price-sales function in order to determine the maximum profit price . The maximum profit sales volume and the associated price together form the Cournot point. ${\displaystyle K'(x)}$ ${\displaystyle U'(x)}$ ${\displaystyle x}$ ${\displaystyle x_{c}}$ ${\displaystyle p_{c}}$

Numerical example

Absolute values ​​graphically: dark blue curve revenue, pink curve costs and green curve the resulting profit, the dashed line shows the Cournot point

A monopoly operating company produces extra-light trekking shoes. The sales management has determined that the demand [units of units] for these shoes depends on the price [monetary units (MU)] with the demand function ${\displaystyle x}$ ${\displaystyle p}$

${\displaystyle x=x(p)=100-0{,}01p}$.

Conversely, the price-sales function (demand function dependent on ) results as ${\displaystyle x}$

${\displaystyle p=p(x)=10.000-100x}$.

I.e. that the company no longer sells a pair at a price of 10,000 GE ( prohibitive price) and does not sell more than 100 containers even at a price of 0 GE ( saturation quantity ).

If you evaluate the requested quantity with the currently valid price, you get the turnover as a function ${\displaystyle x}$

${\displaystyle U=U(x)=xp(x)=10.000x-100x^{2}}$.

The company incurs total costs for the production of the trekking shoes, which are dependent on the output quantity [units]. The company's costs can be broken down into the cost function ${\displaystyle x}$

${\displaystyle K=K(x)=50.000+3.000x}$

sum up. The profit is then calculated as sales - costs, that is

${\displaystyle G=G(x)=U(x)-K(x)=10.000x-100x^{2}-50.000-3.000x}$,

so that one as a profit function

${\displaystyle G(x)=7.000x-100x^{2}-50.000}$

receives.

In order to get the profit maximum at Cournot's point, one determines the maximum of the profit function by differentiating : ${\displaystyle G(x)}$

${\displaystyle {\frac {dG}{dx}}=7.000-200x}$.

Setting the derivative to zero then gives the solution: and . ${\displaystyle x_{c}=35}$ ${\displaystyle G_{c}=72.500}$

Because the second derivative

${\displaystyle G''=-200}$

is less than zero, the solution is a win maximum.

To the Cournot crowd

${\displaystyle x_{c}=35}$

belongs to the Cournot price

${\displaystyle p(x)=10000-100x}$,

so

${\displaystyle p(35)=10000-100\cdot 35}$,

so

${\displaystyle p_{c}=6500}$.

So 35 bundles of shoes can be sold at a price of 6500 GE. This brings the company 72,500 GE profit. ( ) ${\displaystyle G(35)=7000\cdot 35-100\cdot 35^{2}-50.000}$

As explained above, it is also possible to bet equal . This gives the same results. ${\displaystyle U'=K'}$

The general solution of profit optimization in competition and with limited capacity can be found in [Gudehus 2007]. If investments have to be made to increase capacity, the fixed costs must also be taken into account when calculating the absolute Cournot point.

literature

• T. Gudehus: Dynamic markets, practice, strategies and benefits for economy and society. Springer, Berlin / Heidelberg / New York 2007, ISBN 978-3-540-72597-8 , 12.4 Profit Maximization. and 12.5 Cournot's point.

Web links

• Cournotscher Punkt - Article at mikrooekonomie.de
• monopoly pricing - definition in the Gabler economic dictionary

Individual evidence

1. ^ Artur Woll: Economics. 12th edition. 1996, ISBN 3-8006-2091-X , p. 205.
2. ↑ Edwin Böventer, Gerhard Illing: Introduction to Microeconomics. 8., completely rework. u. exp. Edition. R. Oldenbourg, 1997, ISBN 3-486-23070-0 , p. 300.