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
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