Lerner Index Monopolist PDF

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Monopolist Lerner Index Foundations of Strategy September 4, 2017

1

Note on simple Lerner index

1.1

Purpose

Derivation of relation to demand elasticity for Lerner index of a monopolist.

1.2

Assumptions

• Start from a single monopolist and assume profit maximization (MR=MC). • The monopolist is price-maker.

1.3

Definitions

Definition of the Lerner index, a proxy for market power: L≡

P (Q) − M C(Q) P (Q)

(1)

The non-decreasing (under the assumption of efficiency) total cost T C(Q) determines the marginal cost: M C(Q) =

dT C dQ

(2)

The total revenue T R(Q) = P (Q)Q similarly determines the marginal revenue: M R(Q) =

dP (Q) dT R = P (Q) + Q dQ dQ

(3)

Elasticity of demand (how the relative price drops by a small relative increase in supply): 1 dP (QM ) QM =− η dQ P M

(4)

The profit function: π(Q) = T R(Q) − T C(Q) = P (Q)Q − T C(Q)

1

(5)

1

NOTE ON SIMPLE LERNER INDEX

1.4

2

Derivation

Decision of QM at optimal profit P (QM ) +

dπ dQ

= 0 ⇔ M R = M C:

dP (QM ) M Q = M C(QM ) dQ

Simplify notation and write P (QM ) as P M . Then, factor out P M :   dP (QM ) QM M P 1+ = M C(QM ) dQ P M Substituting with inverted elasticity of demand:   1 M = M C(QM ) P 1− η

(6)

(7)

(8)

Reshuffle and we have the Lerner index of the monopolist: LM =

1 P M − M C(QM ) = η PM

(9)...