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[citation needed].
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.
The Inverse Demand Function therefore measures what price a good has to be in order for the consumer to choose that level of consumption.[1]
To compute the inverse demand function simply solve for P in the demand function. For example, if the demand function has the form Q = 240 - 2P then the inverse demand function would be P = 120 - .5Q[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 = (120 - .5Q) x Q = 120Q - 0.5Q². The marginal revenue function is the first derivative of the total revenue function or MR = 120 - 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 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 420 + 60Q + Q2. then MC = 60 + 2Q[3] Equating MR to MC and solving for Q gives Q = 20. So 20 is the profit maximizing quantity - to find the profit maximizing price simply plug the value of Q into the inverse demand equation and solve for P.
The inverse demand function is the form of the demand function that appears in the famous Marshallian Scissors diagram. The function appears in this form because economist place the independent variable on the y axis and the dependent variable on the y axis. The slope of the inverse funtion is ∆P/∆Q. This fact should be kept in mind when calculating elasticity. The formula for calcularing elasticity is PED = (∆Q/∆P) x (P/Q).
References
See also
Supply and demand Marginal Revenue
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