In economics, an inverse demand function is a function that maps the quantity of output demanded to the market price (dependent variable) for that output. Quantity demanded, Q, is a function of price; the inverse demand function treats price as a function of quantity demanded, and is also called the price function.
In mathematical terms, if the demand function is f(x), then the inverse demand function is f -1(x). This is to say that the inverse demand function is the demand function with the axes switched. This is useful because economists typically place Price (P) on the vertical axis and Quantity (Q) on the horizontal axis.[1] The Inverse Demand Function therefore measures what price a good has to be in order for the consumer to choose that level of consumption.[2]
To compute the inverse demand function simply solve for P in the demand function. For example, if the demand function has the form Q = 10 - 2P then the inverse demand function would be P = 5 - Q/2.
The inverse demand function can be used to derive the total and marginal revenue functions. Total revenue equals price, P, times quantity, Q, or TR = PQ. Multiply the inverse demand function by Q to derive the total revenue function: TR = (5 - 0.5Q) x Q = 5Q - 0.5Q². The marginal revenue function is the first derivative of the total revenue function or MR = 5 - Q. Note that the MR function has the same y-intercept as the inverse demand function, the x-intercept of the MR function is one-half the value of the inverse demand function and the slope of the MR function is twice that of the inverse demand function. This relationship holds true for all linear demand equations. The importance of being ablie to quickly calculate MR is that the profit maximizing conditions for firms regardless of market structure is to produce where marginal revenue equals marginal cost. To derive MC you take the first derivative of the total cost function.
For example assume cost, C, equals 10 + 10q = 4q2+q3. then MC = 10 - 8q + 3q2[3]
References
- ^ Mathematics and every other scientific disci[line [laces the independent variable on the y axis and the dependent variable on the x axis. Economist were the first science to use graphical analysis extensively. At that time there was accordingly no convention so economist were free to place the variables as they pleased. Once convention was established to the contrary economist refused to change. The problem is that the demand equation is expressed in terms of Q being a function of price. Yet the inverse or price equation is depicted as if price were a function of Q. This flip flopping can cause trouble for the unwary. For example, price elasticity of demand is (dQ/dP) x P/Q. That is PED is the slope of the demand equation times the ratio of price to quantity. If you use the slope of the price equation your calculation wil be wrong unless you remember to invert the price/quantity ratio.
- ^ Varian, H.R (2006) Intermediate Microeconomics, Seventh Edition, W.W Norton & Company: London
- ^ Perloff, Microeconomics, Theory & Applications with Calculus (Pearon 2008) 240.ISBN 0-321-27794-5
See also
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