Inverse demand function: Difference between revisions

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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 = 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 or MR = 120 - Q. Note that in this linear example 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 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.<ref>Perloff, Microeconomics, Theory & Applications with Calculus (Pearson 2008) 240.{{ISBN |0-321-27794-5}}</ref> 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 economists place the independent variable on the y-axis and the dependent variable on the x-axis. The slope of the inverse function is ∆P/∆Q. This fact should be kept in mind when calculating elasticity. The formula for elasticity is (∆Q/∆P) × (P/Q).