Lerner index

The Lerner index, formalized in 1934 by Abba Lerner, is a measure of a firm's market power. It is defined by:

$L={\frac {P-MC}{P}}$

where P is the market price set by the firm and MC is the firm's marginal cost. The index ranges from 0 to 1. A perfectly competitive firm charges P = MC, L = 0; such a firm has no market power. An oligopolist or monopolist charges P > MC, so its index is L > 0, but the extent of its markup depends on the elasticity (the price-sensitivity) of demand and strategic interaction with competing firms. The index rises to 1 if the firm has MC = 0.

The Lerner Rule or Lerner Condition is that if it is to maximize its profits, the firm must choose its price so that the Lerner Index equals -1 over the elasticity of demand facing the firm (note that this is not necessarily the same as the market elasticity of demand):

${\frac {P-MC}{P}}={\frac {-1}{E_{d}}}$

A drawback of the Lerner Index is that while it is relatively easy to observe a firm's prices, it is quite difficult to measure its marginal costs. In practice, the average cost is often used as an approximation.

The Lerner index can never be greater than one. As a result, if the firm is maximizing profit, the elasticity of demand facing it can never be less than one in magnitude (|E|<1). If it were, the firm could increase its profits by raising its price, because inelastic demand means that a price increase of 1% would reduce quantity by less than 1%, so revenue would rise, and since lower quantity means lower costs, profits would rise. Put another way, a monopolist never operates along the inelastic part of its demand curve.

Derivation

The Lerner Rule comes from the firm's profit maximization problem. A firm choosing quantity Q facing inverse demand curve P(Q) and incurring costs C(Q) has profit equalling revenue (where R = PQ) minus costs:

$Profit=P(Q)Q-C(Q)$

Under suitable conditions (that this is a convex maximization problem, e.g. P(Q) and C(Q) are linear functions), we can find the maximum by taking the derivative of profit with respect to Q and getting the first-order-condition:

${\frac {dProfit}{dQ}}=({\frac {dP}{dQ}}Q+P)-{\frac {dC}{dQ}}=0$

which gives the standard rule of MR = MC. To get the Lerner Rule, switch to the notation dC/dQ = MC and rewrite as

$P-MC=-{\frac {dP}{dQ}}Q$

Divide by P to get

${\begin{aligned}{\frac {P-MC}{P}}&=-{\frac {dP}{dQ}}{\frac {Q}{P}}\\&\\&=-1/\left({\frac {dQ}{dP}}{\frac {P}{Q}}\right)\\&\\&=-{\frac {1}{E_{d}}},\end{aligned}}$

using the derivative definition of elasticity.

See also

• Ramsey problem
• Amoroso–Robinson relation
• Herfindahl Index
• Market segmentation index

References

• Lerner, A. P. (1934). "The Concept of Monopoly and the Measurement of Monopoly Power". The Review of Economic Studies. 1 (3): 157–175. doi:10.2307/2967480. JSTOR 2967480.