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, *E _{d}*. By definition,

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*) = d*P*/d*Q* to assert that so that *E _{d}* = = . Inverting both sides, we see that = -1/

We now have an expression that relates *MR* to the elasticity of demand: *MR* = *P*(1 - 1/*E _{d}*). If demand is inelastic,