<P> The inverse demand function is the same as the average revenue function, since P = AR . </P> <P> To compute the inverse demand function, simply solve for P from the demand function . For example, if the demand function has the form Q = 240 - 2P then the inverse demand function would be P = 120 - 0.5 Q . </P> <P> 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 - 0.5 Q2 . 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 . </P> <P> For example, assume cost, C, equals 420 + 60Q + Q. 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 . </P>

Given the demand function q = ap write down a linear relationship between the variables