Monopoly and the Elasticity of Demand
Before investigating the relationship between marginal revenue and elasticity of demand, we will need to digress a moment and recall the elasticity coefficient, Ed. By definition, Ed is the (absolute value of the) percentage change in quantity demanded divided by the percentage change in price: Ed = . A simple rearrangement of this formula shows that Ed = .
With this in mind, suppose a monopolist’s demand curve is given by P = f(Q), revenue is R(Q) = QP = Qf(Q) and marginal revenue is MR = R’(Q) = f(Q) + Qf ’(Q). Suppose we now both divide and multiply the right-hand-side of MR by P = f(Q): MR = P() = P(1 + ). At this point we make use of the facts that demand is monotonically decreasing and f ’(Q) = dP/dQ to assert that so that Ed = = . Inverting both sides, we see that = -1/Ed. Substituting this into our equation for MR we obtain the following: MR = P(1 - 1/Ed).
We now have an expression that relates MR to the elasticity of demand: MR = P(1 - 1/Ed). If demand is inelastic, Ed < 1 and MR < 0. As profit maximization requires that MR = MC and MC is always positive, we see that a monopolist must always price in the elastic portion of the demand curve. MR > 0 if and only if Ed > 1.