Cournot Duopoly Here we study a special type of market structure - two firms.
Say we have a market where the demand in the market is P = 100 - 2Q (graph on next slide). If we had only one firm in the market - a monopoly - and if the firm had a constant MC = 10, then we know the profit maximizing firm would find the Q where MR = MC and set the Q back into the demand curve to get the price. Let’s do this. MR = 100 - 4Q = 10 = MC, or Q = 22.5, and then P = 55. If MC is constant at 10, then the cost of each unit added is 10. Part of total cost then is MC times Q. For what we do here we will only have this part of TC and thus profit here is 55(22.5) - 10(22.5) = 1012.5
MR=100-4Q Pm=55 P=100-2Q Pc=10 MC=10 Qm = 22.5 Qc = 45
Now, since price is driven to MC in a competitive market with many firms, Pc = MC = 10 and the market output would be (100 - 2Q = 10) Q = 45. The overall profit across firms would be 10(45) - 10(45) = 0. Now, let’s consider a model thought up by a guy named Cournot (rhymes with tour go). Cournot said consider only two firms selling in a market. The way that each firm understands the market is that each will pick its own output level. Then when the other firm’s output level is added in the price will determined from the demand curve.
Now, if there are two firms in the market, the market demand would be served by both firms and we might think of the demand as P = 100 - 2Q = 100 - 2(q1 + q2), where q1 & q2 represent the output of each firm. Each firm wants to maximize its own profit. Again, each firm will pick a level of output and when added together will determine the market price. The profit for each firm will be firm 1 Pq1 - mcq1 firm 2 Pq2 - mcq2. As an example, if both make 1 unit (and MC = 10 for both), then P = 100 - 2(1+1) = 96, firm 1 profit = 96(1)-10(1) = 86 and firm 2 profit = 96(1)-10(1)=86. On the next slide I put a game theory matrix with this outcome and you will work out other outcomes.
In the table we have profit amounts and each firm can choose a level of output. firm 2 1 2 3 15 16 24 25 1 86, 86 2 firm 1 3 15 16 24 25
Firm 1 looks at the market demand and sees P = 100 - 2q2 - 2q1, where q2 is listed first because firm 1 can not directly pick q2 and thus treats it as a given much like the number 100. The MR for firm 1 is MR = 100 - 2q2 - 4q1, and with a MC = 10 we see the MR = MC rule give us 100 - 2q2 - 4q1 = 10, or q1 = (90/4) - (2/4)q2,
The result for firm 1, q1 = (90/4) - (2/4)q2, has typically been called a reaction function because firm 1’s profit maximizing level of output depends on what amount firm 2 makes. So, firm 1 reacts to what firm 2 does. The authors suggest we call the function the best response function for firm 1 Firm 2 has a similar best response function: q2 = (90/4) - (2/4)q1. Now, think for a minute what firm 1 would do if firm 2 made nothing. Firm 1 would have Q = 90/4 or 22.5, and we remember this is the monopoly solution. But, if firm 2 saw firm 1 make 22.5 it would want to make Q = 11.25. But if firm 1 saw firm 2 make 11.25..... This goes on.
Imagine we place all these possibilities in a big game theory matrix. What is each firm to do? It has been shown that the solution to the Cournot duopoly problem is actually a Nash equilibrium. This means each firm will not have an incentive to change its output, given the output of the other firm. Here is how we arrive at the Cournot solution: Firm 1 best response function Firm 2 best response function q1 = (90/4) - (2/4)q2, q2 = (90/4) - (2/4)q1, substitute this over into that, like on the next page
q1 = (90/4) - (2/4)q2, = (90/4) - (2/4)[(90/4) - (2/4)q1] I am going to forget the subscript on the q for a minute because it is easier to type. So, we have (when we multiply through by 4) 4q = 90 - 2[(90/4) - (2/4)q) = 90 - (90/2) + q, so 3q = 45, or q1 = 15. Similarly firm 2 would have q2 = 15. The market output would be 30 and the price in the market from the demand curve would be P = 100 - 2Q, or P = 40.
Summary: P Q Monopoly 55 22.5 P. Comp 10 45 Duopoly 40 30 So, the duopoly solution is between the monopoly and perfect competition solutions.
Cournot’s analysis suggests that as the number of firms increases – as the market structure becomes less concentrated – the mark-up of price over marginal cost shrinks. Thus, structure influences performance. A section of the text also suggests that if firms do not have the same MC, the one with the higher cost will have a smaller market share and lower profits. From a game theory point of view, the Cournot model assumes firms act simultaneously.