Question

The following formula shows that the firm’s “markup” over marginal cost depends inversely on the elasticity of market demand, which is called "Lerner Index". Prove this formula step by step from a monopoly's profit-maximization problem.

The following formula shows that the firm's “markup” over marginal cost depends inversely on the elasticity of market demand, which is called "Lerner Index". Prove this formula step by step from a monopoly's profit-maximization problem. Pm – MC 1 1 Pm CDP

Answer 1

Ans. Inverse demand curve, P = F(Q)

Total revenue, TR = P*Q = Q*F(Q)

=> Marginal revenue, MR = dTR/dQ = F(Q) - Q*F'(Q)

Now, the price elasticity of demand, e = %Change in Quantity demanded / %Change in Price

=> e = dQ/dP * P/Q
=> e = -1/F'(Q) * F(Q)/Q --> Eq1 (Because elasticity of demand is always negative due to law of demand)

At profit maximizing level of output for a monopolist,

MR = MC

=> F(Q) - Q*F'(Q) = MC

=> F(Q) - MC = Q*F'(Q)

=> (F(Q) - MC)/F(Q) = Q/F(Q) * (F'(Q))

=> (P - MC)/P = -1/e [From Eq1 and inverse demand function]

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