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In mathematical terms, if the demand function is = (), then the inverse demand function is = ().The value of the inverse demand function is the highest price that could be charged and still generate the quantity demanded. [3]
The constant b is the slope of the demand curve and shows how the price of the good affects the quantity demanded. [6] The graph of the demand curve uses the inverse demand function in which price is expressed as a function of quantity. The standard form of the demand equation can be converted to the inverse equation by solving for P:
In its standard form a linear demand equation is Q = a - bP. That is, quantity demanded is a function of price. The inverse demand equation, or price equation, treats price as a function f of quantity demanded: P = f(Q). To compute the inverse demand equation, simply solve for P from the demand equation. [12]
Mathematically, a demand curve is represented by a demand function, giving the quantity demanded as a function of its price and as many other variables as desired to better explain quantity demanded. The two most common specifications are linear demand, e.g., the slanted line =
When the price elasticity of demand is unit (or unitary) elastic (E d = −1), the percentage change in quantity demanded is equal to that in price, so a change in price will not affect total revenue. When the price elasticity of demand is relatively elastic (−∞ < E d < −1), the percentage change in quantity demanded is greater than that ...
Consider the function = (;), where is the quantity demanded of good , is the demand function, is the price of the good and is the list of parameters other than the price. The law of demand states that ∂ f ∂ P x < 0 {\displaystyle {\frac {\partial f}{\partial P_{x}}}<0} .
A synonymous term is uncompensated demand function, because when the price rises the consumer is not compensated with higher nominal income for the fall in their real income, unlike in the Hicksian demand function. Thus the change in quantity demanded is a combination of a substitution effect and a wealth effect.
Mathematically, the markup rule can be derived for a firm with price-setting power by maximizing the following expression for profit: = () where Q = quantity sold, P(Q) = inverse demand function, and thereby the price at which Q can be sold given the existing demand C(Q) = total cost of producing Q.