Inverse Demand Function
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 g of quantity demanded: P = f(Q). To compute the inverse demand equation, simply solve for P from the demand equation. For example, if the demand equation is Q = 240 - 2P then the inverse demand equation would be P = 120 - .5Q, the right side of which is the inverse demand function.
The inverse demand function is useful in deriving the total and marginal revenue functions. Total revenue equals price, P, times quantity, Q, or TR = P×Q. Multiply the inverse demand function by Q to derive the total revenue function: TR = (120 - .5Q) × Q = 120Q - 0.5Q². The marginal revenue function is the first derivative of the total revenue function; here MR = 120 - Q. Note that the MR function has the same y-intercept as the inverse demand function in this linear example; the x-intercept of the MR function is one-half the value of that 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 able to quickly calculate MR is that the profit-maximizing condition for firms regardless of market structure is to produce where marginal revenue equals marginal cost (MC). To derive MC the first derivative of the total cost function is taken. For example assume cost, C, equals 420 + 60Q + Q2. Then MC = 60 + 2Q. 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.
Read more about this topic: Demand
Famous quotes containing the words inverse, demand and/or function:
“The quality of moral behaviour varies in inverse ratio to the number of human beings involved.”
—Aldous Huxley (1894–1963)
“To be sure, a good work of art can and will have moral consequences, but to demand of the artists moral intentions, means ruining their craft.”
—Johann Wolfgang Von Goethe (1749–1832)
“Think of the tools in a tool-box: there is a hammer, pliers, a saw, a screwdriver, a rule, a glue-pot, nails and screws.—The function of words are as diverse as the functions of these objects.”
—Ludwig Wittgenstein (1889–1951)