Question

Consider a monopolist facing a linear (inverse) demand curve given by:p = a – bQ. Show with calculus that the marginal revenue in fact has the same price-axis intercept but twice the slope as the inverse demand curve above.

Answer 1

Demand is given by : p = a - bQ

Marginal revenue(MR) = d(TR)/dQ where TR = p*Q = (a - bQ)*Q = aQ - bQ²

Thus, MR = d(TR)/dQ = a - 2bQ

Hence, MR = a - 2bQ

When Q = 0 then P = a - b*0 = a and MR = a - 2b*0 = a

This means that both have price axis intercept = a and hence price axis intercept of Both inverse demand curve and Marginal Revenue curve is same.

Slope of Inverse demand curve = dp/dQ = -b and slope of MR curve = dMR/dQ = -2b = 2(-b) = 2*Slope of Inverse demand curve

Thus, Slope of MR curve is twice the Slope as Inverse demand curve.