The Lerner index, formalized in 1934 by British economist of Russian origin Abba Lerner, is a measure of a firm's market power.

Definition

The Lerner index is defined by:

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

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 following factors affect the value of the Lerner index:

• the price elasticity of demand for goods produced by the company — the smaller the fluctuations in demand under the influence of prices, the smaller the elasticity and the greater the value of L;
• the interaction with competitors — the more of them and the larger their size, the less the company's ability to maximize profits and the smaller the L;
• the degree of regulation — the more actively the state conducts an antitrust policy, the lower the value of L.

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):

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

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:

${\displaystyle Profit=P(Q)Q-C(Q)}$

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:

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

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

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

Divide by P to get

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

using the derivative definition of elasticity.

Examples

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.
• https://journal.open-broker.ru/economy/indeks-lernera/
• https://www.cjournal.cz/files/91.pdf
• https://cyberleninka.ru/article/n/primenenie-indeksa-lernera-k-otsenke-konkurentsii-na-rynke-kreditovaniya-malogo-i-srednego-biznesa-v-rossii/viewer
• ru:Коэффициент Лернера