Iso profit line

The lowest iso-profit line touches the production function in one point (tangential point). This indicates the company's maximum profitable production.

In microeconomics and especially in production theory, the iso-profit line, also iso-profit curve or isoprofit curve (from the Greek ἴσος isos, `` equal '') is the geometric location of all quantity combinations of input goods and one output good that bring the manufacturing company the same profit. The concept of the iso-profit line is closely related to that of the iso- cost line and the iso-utility line , which each represent the geometric locations of the same costs and benefits.

Iso-profit lines are shown differently; They are regularly understood as a function that maps the amount of factor input to the amount of the output good in such a way that a certain profit level is maintained (volume-volume diagram), or, in the case of a non-fixed price level, as a function that maps the amount of factor input to the price of the output good shows that a certain profit level is maintained; In the field of oligopoly theory , iso-profit lines are often shown "crossed" due to the nature of the interaction, which means that, for example, the iso-profit line of company A is visualized in a (output amount of A) - (output amount of B) diagram. In the following, isogwinn lines are understood to mean the first-mentioned form for the sake of simplicity (see also the figure).

Formal definition

Let be the profit function of a company, where the revenue function depends on the amount y of the output goods and the cost function depends on the amount of n input goods. A real-valued function is also implicitly given by this profit function . It is then ${\displaystyle \pi (\mathbf {x} ,y)=R(y)-C(\mathbf {x} )}$ ${\displaystyle R(y)}$ ${\displaystyle C(\mathbf {x} )}$ ${\displaystyle y(\mathbf {x} ,\pi )}$

${\displaystyle I_{\bar {\pi }}:=\left\{\left.\left(\mathbf {x} ,y\left(\mathbf {x} ,\pi \right)\right)\right|\pi ={\bar {\pi }}\right\}}$

the iso gain function to the level . For and y linear in x , the function is also known as the iso-profit line. ${\displaystyle {\bar {\pi }}}$ ${\displaystyle \mathbf {x} \in \mathbb {R} ^{1}}$

example

function

The revenue of a company is , where y is the quantity of the good produced and p the (exogenously given) price for a unit of this good. The company produces this good with the help of an input factor, in our case “work” as an example. Let w be the (exogenously given) price for a unit of labor and x the amount of labor employed, then is the company's cost function . So the profit function is . (Due to the exogeneity of prices, one can imagine the case of short-term profit maximization here, for example.) ${\displaystyle R(y)=py}$ ${\displaystyle C(x)=w\cdot x}$ ${\displaystyle \pi (x,y)=py-wx}$

On an iso-profit line, that means that for every input-output combination , the maximum profit that can be achieved with it corresponds to a certain amount . If you now look at an xy diagram, an iso-gain function is a function . In the example, the form of the profit function is given and it can be rearranged accordingly, so that here the iso profit lines follow the equation . ${\displaystyle \pi (x,y)={\bar {\pi }}}$ ${\displaystyle (x,y)}$ ${\displaystyle {\bar {\pi }}}$ ${\displaystyle I_{\bar {\pi }}=y(x,{\bar {\pi }})}$ ${\displaystyle I_{\bar {\pi }}=y(x,{\bar {\pi }})=\left({\bar {\pi }}+wx\right)/p}$

application

Be it concretely, for example , then we get to the example of the Isogewinnlinie the profit level , the function and the Isogewinnlinie to the level of the function - in other words, to make a profit of 100, the company, for example, to use units of labor (then produced she units of the output good) or units (then she produces units of the output good); in both cases the profit is just 100, as can be easily verified by inserting it into the profit function ( ). ${\displaystyle \pi (x,y)=10y-5x}$ ${\displaystyle y(x,\pi )=(\pi +5x)/10}$ ${\displaystyle {\bar {\pi }}=20}$ ${\displaystyle I_{20}=y(x,{\bar {\pi }}=20)=2+0{,}5x}$ ${\displaystyle {\bar {\pi }}=100}$ ${\displaystyle I_{100}=y(x,{\bar {\pi }}=100)=10+0{,}5x}$ ${\displaystyle x=10}$ ${\displaystyle 10+0{,}5\cdot 10=15}$ ${\displaystyle x=20}$ ${\displaystyle 10+0{,}5\cdot 20=20}$ ${\displaystyle \pi (x=10,y=15)=\pi (x=20,y=20)=100}$

properties

In the simple case of the profit function (sketched above), the slope of an iso profit line corresponds to the quotient of the price of the input good and the price of the output good ( ). If one therefore assumes that a company only produces with work, the slope of the iso profit line corresponds exactly to the real wage . ${\displaystyle \pi (x,y)=py-wx}$ ${\displaystyle w/p}$

In the case of an input factor, the profit-maximum production can easily be viewed graphically with the aid of iso-profit lines. If the production function of the company is plotted in an xy diagram , all that is necessary is to find the lowest possible from the (infinite) family of iso-profit lines (the lower an iso-profit line is, the higher the profit level) that just touches the production function (tangential condition) .

Web links

• 2.2.2 Iso profit lines - chapter in the economics textbook at teialehrbuch.de
• Procedure for profit maximization - Article at wiwiweb.de

literature

• Friedrich Breyer: Microeconomics. An introduction. 5th edition. Springer, Heidelberg a. a. 2011, ISBN 978-3-642-22150-7 .
• Geoffrey A. Jehle and Philip J. Reny: Advanced Microeconomic Theory. 3rd ed. Financial Times / Prentice Hall, Harlow 2011, ISBN 978-0-273-73191-7 .
• Werner Lachmann: Economics 1: Basics. 5. revised u. exp. Edition Springer, Heidelberg a. a. 2006, ISBN 3-540-30086-4 (page 113).
• Jochen Schumann, Ulrich Meyer and Wolfgang Ströbele: Basics of the microeconomic theory. 9th edition. Springer, Heidelberg u. a. 2011, ISBN 978-3-642-21225-3 .
• Hal Varian : Intermediate Microeconomics. A modern approach. 8th edition. WW Norton, New York and London 2010, ISBN 978-0-393-93424-3 .

Individual evidence

1. ↑ See Varian 2010, p. 351 f .; Jehle / Reny 2011, p. 230.
2. ↑ So only Schumann / Meyer / Ströbele 2011, p. 318 f.
3. ↑ See Varian 2010, p. 502 ff.
4. ↑ See Varian 2010, p. 352.