Calculating The Classic Bertrand Model
- MC = constant marginal cost (equals constant unit cost of production).
- p1 = firm 1’s price level
- p2 = firm 2’s price level
- pM = monopoly price level
Firm 1's optimum price depends on where it believes firm 2 will set its prices. Pricing just below the other firm will obtain full market demand (D), though this is not optimal if the other firm is pricing below marginal cost as that would entail negative profits. In general terms, firm 1's best response function is p1’’(p2), this gives firm 1 optimal price for each price set by firm 2.
Diagram 1 shows firm 1’s reaction function p1’’(p2), with each firm's strategy on each axis. It shows that when P2 is less than marginal cost (firm 2 pricing below MC) firm 1 prices at marginal cost, p1=MC. When firm 2 prices above MC but below monopoly prices, then firm 1 prices just below firm 2. When firm 2 prices above monopoly prices (PM) firm 1 prices at monopoly level, p1=pM.
Because firm 2 has the same marginal cost as firm 1, its reaction function is symmetrical with respect to the 45 degree line. Diagram 2 shows both reaction functions.
The result of the firms' strategies is a Nash equilibrium, that is, a pair of strategies (prices in this case) where neither firm can increase profits by unilaterally changing price. This is given by the intersection of the reaction curves, Point N on the diagram. At this point p1=p1’’(p2), and p2=p2’’(p1). As you can see, point N on the diagram is where both firms are pricing at marginal cost.
Another way of thinking about it, a simpler way, is to imagine if both firms set equal prices above marginal cost, firms would get half the market at a higher than MC price. However, by lowering prices just slightly, a firm could gain the whole market, so both firms are tempted to lower prices as much as they can. It would be irrational to price below marginal cost, because the firm would make a loss. Therefore, both firms will lower prices until they reach the MC limit.
If one firm has lower average cost (a superior production technology), it will charge the highest price that is lower than the average cost of the other one (i.e. a price just below the lowest price the other firm can manage) and take all the business. This is known as "limit pricing"
Read more about this topic: Bertrand Competition
Famous quotes containing the words calculating the, calculating, classic and/or model:
“[The] elderly and timid single gentleman in Paris ... never drove down the Champs Elysees without expecting an accident, and commonly witnessing one; or found himself in the neighborhood of an official without calculating the chances of a bomb. So long as the rates of progress held good, these bombs would double in force and number every ten years.”
—Henry Brooks Adams (1838–1918)
“Sin in this country has been always said to be rather calculating than impulsive.”
—Frank Moore Colby (1865–1925)
“There is a distinction to be drawn between true collectors and accumulators. Collectors are discriminating; accumulators act at random. The Collyer brothers, who died among the tons of newspapers and trash with which they filled every cubic foot of their house so that they could scarcely move, were a classic example of accumulators, but there are many of us whose houses are filled with all manner of things that we “can’t bear to throw away.””
—Russell Lynes (1910–1991)
“... if we look around us in social life and note down who are the faithful wives, the most patient and careful mothers, the most exemplary housekeepers, the model sisters, the wisest philanthropists, and the women of the most social influence, we will have to admit that most frequently they are women of cultivated minds, without which even warm hearts and good intentions are but partial influences.”
—Mrs. H. O. Ward (1824–1899)