Bertrand Duopoly

We have seen that Oligopoly is a situation where there are just a few firms. In this situation each firm understands that the outcomes of its actions are also influenced by the actions of the other firms. In a Cournot duopoly the firms competed on quantity. There we saw that the output level and price of the firms fell somewhere between the monopoly level and the competitive level. In the Bertrand model firms will compete on price. It seems to me firms in the computer business compete on price. In other situations firms may compete on features of the product. Let’s turn to the Bertrand model.

Bertrand Duopoly Some assumptions of the model: If the two firms charge the same price each will get half of the market demand at that price. If one firm charges more than the other, even just a little bit, then the one with the higher price will sell nothing and the one with the lower price will have all the demand at that price. Each firm wants to maximize its profit. Let’s say the demand in a market is Q = 100 – 5P And say the marginal cost for each firm is 2.

Let’s see what the result would be if there was only one firm in this market. With Q = 100 – 5P P = 20 - .2Q MR = 20 -.4Q and with MR = MC for profit maximization 20 - .4Q = 2, so Q = 18/.4 = 45 and then P = 11. On the next slide let’s explore the demand from firm 2’s perspective. We will want to remember the assumptions we made on a previous slide.

Firm 2’s perspective on the demand it will have Q2 = 0 if P2 > P1 and profit = 0. Q2 = .5(100 – 5P2) if P2 = P1 and profit = P2Q2 – MCQ2 =(P2 – MC)Q2 =(P2 – MC).5(100 – 5P2) Q2 = (100 – 5P2) if P2 < P1 and profit = P2Q2 – MCQ2 =(P2 – MC)(100 – 5P2) Let’s say both charge the monopoly price of 11. Firm two would then have Q2 = .5(100 – 5(11)) = 22.5 and profit = (11-2)22.5 = 202.5 If firm 2 charged just a little less than 11, say 10.99, when firm 1 charges 11, then firm 2 will make Q2 = 100 – 5(10.99) = 45.05 and profit will be (10.99 – 2)45.05 = 404.9995

Firm two finds it irresistible to not charge a lower price here, when the other firm is charging a monopoly price. Now, imagine that firm 1 charges less than its marginal cost of production. Firm 1 would lose money. If firm 2 charged an even lower price firm 2 would lose money. Firm 2 would be better off not producing at all. So, when firm 1 has price lower than marginal cost, firm 2 does not want to have a lower price. If firm 1 has price at marginal cost then firm 2 doesn’t want to have a lower price because it would lose money and it doesn’t want to have a higher price because it won’t sell anything. On the next slide I will have a summary of firm 2’s reactions given what firm 1 does – we will see the best price response for firm 2.

Firm 2’s best price response If P1 > monopoly price of 11, set P2 = 11 and sell monopoly Q, 2(=mc) < P1 <= 11, set P2 = P1 – small amount, sell all of mkt at P2, P1 = 2(=mc), set P2 = 2(=mc) and sell half of market at P2 = 2, 2(=mc) > P1 >= 0, set P2 > P1 and sell nothing. Firm 1’s best price response is similar P2 > monopoly price of 11, set P1 = 11 and sell monopoly Q, 2(=mc) < P2<= 11, set P1 = P2 – small amount, sell all of mkt at P1, P2 = 2(=mc), set P1= 2(=mc) and sell half of market at P1 = 2, 2(=mc) > P2 >= 0, set P1 > P2 and sell nothing.

Let’s check some possible solutions to see if they qualify as a Nash Equilibrium. Is P1 = 11, P2 = 10.99 a Nash Equilibrium? Firm 2 does not want to change from P2 = 10.99 if firm 1 has P1 = 11, but firm 1 would want to change (what does firm 1’s best response from the previous slide suggest?). Firm 1 would want P1 = 10.98 in this case. But then firm 2 would want 10.97. This would spiral downward. What about P1 = 1.5 and P2 = 1.6 as a Nash equilibrium? Firm 2 would not want to change, but firm 1 would want something like 1.61. But then firm two would want P2=1.62. This would spiral upward.

Is P1=2 and P2=2 a Nash equilibrium Is P1=2 and P2=2 a Nash equilibrium? Both would not want to change so it is a Nash Equilibrium. Wow, here with only two firms if they compete on price the price gets driven to MC. This is the competitive solution!