Inverse demand function: Difference between revisions

Revision as of 01:15, 30 September 2009

Revision as of 01:15, 30 September 2009

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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 = 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<small>2</small><ref>Perloff, Microeconomics, Theory & Applications with Calculus (Pearon 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 becuase 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 clcullating elasticity. The formula for calcularing elasticity is PED =(∆Q/∆P) x (P/Q).