# Price-sales function

The inverse demand function ( PAF ), and price-sales curve (PAH) or inverse demand function (engl. Inverse demand function ), is a model function of the operating and Economics ( Mikroökonomie ) indicating what price the offering companies for a good depending on its sales volume . ${\ displaystyle p}$ ${\ displaystyle x}$ The price-sales function is thus the inverse function of the demand function , which indicates the amount of the good the consumers are willing to purchase depending on the price , whereby a left-handed coordinate system with interchanged axes is usually used for the graphical representation of the latter in economics , in which - unlike in mathematics - the independent variable is not represented by the horizontal abscissa but the vertical ordinate axis, while the abscissa axis takes on the representation of the dependent variable . ${\ displaystyle p (x)}$ ${\ displaystyle x (p)}$ ${\ displaystyle x}$ ${\ displaystyle p}$ ${\ displaystyle p}$ ${\ displaystyle x (p)}$ The terms “price-sales function” and “demand function” are often not clearly delimited from one another in the literature, sometimes even used synonymously , all the more so because their graphic representations are different except for the question of whether they are from or vice versa as should be depicted dependent, practically the same. ${\ displaystyle p}$ ${\ displaystyle x}$ ${\ displaystyle x}$ ${\ displaystyle p}$ While the price-sales function is used in economics to map the functional dependency of price and demand, in business administration it is used to solve specific tasks, e.g. B. used in sales planning. The company , however, know the inverse demand function of a market in reality not often - so they do not know exactly how many units they can sell at what price in the market. One empirical way to create a PAF is to use conjoint analysis .

## presentation

So that the axis assignments of the demand function (see above) are adopted in the representation of the price-sales function, the graphs of both functions remain - unlike the inversion of a function in mathematics - except for the question of which of the two variables is used in each case dependent and independent is the same: the demanded or sold quantity is also represented here by the horizontal abscissa, the achievable price also here by the vertical ordinate, only that the latter now - again following the usual convention in mathematics - the values ​​of the dependent Variables, while the abscissa values ​​are those of the independent variables. ${\ displaystyle x}$ ${\ displaystyle p}$ ## Expressions

Depending on the type of market , different forms of the price-sales function can be distinguished. To simplify matters, a linear demand function is generally assumed.

### In the homogeneous polypole

In the homogeneous Polypol one assumes a decreasing demand with increasing price. Suppliers (and consumers) are price takers. The equilibrium price results from the intersection of the supply and demand curves. The PAF has three sections:

• In the absence of transaction costs, no consumer will buy from providers who try to set a higher price; the deductible amount is therefore zero (the PAF runs on the vertical axis).
• All providers who demand exactly the equilibrium price can sell a maximum of the equilibrium amount. The PAF runs horizontally up to the equilibrium quantity at the level of the equilibrium price. In this area, company demand is completely price elastic (even if the demand function is not).
• Anyone setting a price below the equilibrium price could serve the entire amount of demand. This section of the PAF thus follows the demand curve.

### In monopoly

In contrast to the company in perfect competition , which has to accept an equilibrium price for its product, the monopolist can set the sales price. The buyer then responds with his request. The market solution results from the Cournot point .

### In the oligopoly

In the oligopoly case one assumes a simply broken price-sales function.

### In the heterogeneous polypole (Gutenberg function)

In the case of a heterogeneous polypole , a doubly kinked price-sales function (Gutenberg's price-sales function or Gutenberg function according to Erich Gutenberg ) is assumed. This is due to the present imperfect market , which gives the PAF a special market constitution . She is u. a. typical in the case of monopoly competition .

The monopoly area is in the middle section of the function. This represents the entrepreneur's potential for acquisitions . The whole thing can be traced back to the imperfection of the market (lack of market transparency, greater distances between competitors, customer loyalty, ...), so that the entrepreneur can vary prices in the monopoly area without increasing demand changes greatly. Above a certain price level, however, a disproportionately large number of customers migrate to the competition (price high) or a disproportionately large number of customers come from the competition (price low). For example, at a particularly low price, a customer might find it more useful to travel a further distance to the store than to go "around the corner" to their store.

The prohibitive price marks the point at which market demand comes to a complete standstill. No consumer is willing to pay the price asked (or more) anymore. It can also be assumed that even with a price of 0, the demand will not become infinitely great. The maximum sales volume is determined e.g. B. through the lack of market transparency, transaction costs or the finiteness of demand. Dietzel describes this upper limit as the entrepreneur's acquisition range. It is less than or equal to the saturation level , since z. B. due to incomplete information not every interested party gains knowledge of the offer price.