Question

The relative markup (P-MC)/P is a good measure of profitability for a monopolist. If a monopolist has the demand function Q = 900P^-2 and is maximizing profit, what is this markup? What is it if the monopolist’s demand increases to Q = 1,600P^-2?

Answer 1

Answer

In order to maximize profit a firm produces that quantity at which MR = MC

where MC = Marginal Cost and MR = Marginal Revenue

MR = d(TR)/dQ = d(P*Q)/dQ = P + Q(dP/dQ) = P(1 + (Q/P)(dP/dQ)) = P(1 + 1/e)

where TR = Total Revenue = P*Q, P = Price, Q = quantity, e = elasticity of demand = (dQ/dP)(P/Q)

Thus, MR = MC => P(1 + 1/e) = MC

=> P + P/e = MC

=> (P - MC)/P = -1/e ---------------------Optimal value of relative markup or Profit maximizing level of relative markup

where e = elasticity of demand

Now, Q = 900P^-2

Elasticity of demand(e) = (dQ/dP)(P/Q) = -2*900P^-3(P/(900P^-2)) = -2

Hence e = -2

Thus, (P - MC)/P = -1/e => (P - MC)/P = -1/(-2) =  0.5

Hence, When Q = 900P^-2 and is maximizing profit, this markup = 0.5

Now, Demand increases to Q = 1600P^-2(new demand)

For this new demand, Elasticity of demand(e) = (dQ/dP)(P/Q) = -2*1600P^-3(P/(1600P^-2)) = -2

Hence e = -2

Thus, (P - MC)/P = -1/e => (P - MC)/P = -1/(-2) =  0.5

Hence, When Q = 1600P^-2 and is maximizing profit, this markup = 0.5