Microeconomics 2 John Hey

The remaining lectures We have three weeks left this term: This week chapters 31 and 32. Next week chapters 33 and 34. Final week? I propose looking at the second half of the first specimen paper, and general revision (questions in advance please). Next term we have two slots: 13.15 Tuesday the 30th of April and 13.15 Tuesday the 7th of May. I propose to use them for the second specimen paper... ... and for telling you in what way I have strengthened the examination for this year. OK? Comments?

Game Theory In Chapter 29 we talked about games, in which two players have to choose the value of some decision variables, the values of which affect both players. We introduced the idea of a Nash Equilibrium in which each is optimising given the decision of the other. We had our doubts about NE in general but in some cases it seems reasonable. Do note that the NE depends heavily on the ‘rules of the game’, particularly about sequentiality and repetition. This week’s tutorial reinforces these results. Today we will apply game theory in Duopoly.

Duopoly We deliberately keep things simple, but we do not lose anything of interest. Consider a market in which there are two identical sellers – two firms – Firm 1 and Firm 2, selling an identical good. Suppose the demand curve in the market is given by: p = a – b(q1 + q2)

The Cournot Model Each firm chooses independently its output. The price is determined by the demand curve (recall that their output is identical). What outputs do the firms choose? Nash: each firm choosing profit-maximising output given the output of the other firm.

Profits Let us denote the profit of firm 1 by π1. Suppose that its total cost function is given by: C(q1) = cq1 (Note that this is linear and so constant returns to scale.) Hence its profits are given by: π1 = pq1 - cq1 = [a – b(q1 + q2)]q1 - cq1 Assuming the same cost function for firm 2 its profits are given by π2 = pq2 – cq2 = [a – b(q1 + q2)]q2 – cq2

Isoprofit Curves For firm 1 an isoprofit curve is given by: π1.= constant Hence [a – b(q1 + q2)]q1 - cq1 = constant Note that this is quadratic in q1 for given q2. For firm 2 an isoprofit curve is given by: π2.= constant [a – b(q1 + q2)]q2 – cq2 = constant Note that this is quadratic in q2 for given q1.

Parameters in Maple file C(q) = 10q (Note linear so Constant Returns to Scale). and hence the Marginal Cost is 10. Demand is p = 110 – q. So a = 110, b = 1 and c = 10. Hence Revenue = pq = 110q – q2 and so Marginal Revenue = 110 – 2q. Hence the monopoly output (given by MR=MC) is 50, and the monopoly price is 60. Competitive output (given by Price=MC) is 100, and the competitive price) is10.

Reaction Curve (showing how a firm should optimally react to the decision of the other) If Firm 1 chooses its output to maximise its profits given a level of output of Firm 2, we get: q1 = (a-c-bq2 )/2b …the reaction curve of Firm 1. Note that its slope is negative. (We will see this graphically in Maple.) (To find, differentiate π1 = [a – b(q1 + q2)]q1 - cq1 with respect to q1 and put the derivative equal to zero.) Let’s go to Maple...

The Nash Equilibrium with quantity-setting The NE is given by the intersection of the two reaction curves… Total output = 2(a-c)/3b With monopoly (marginal cost – marginal revenue) output = (a-c)/2b With perfect competition (price equal to marginal cost) output = (a-c)/b The output with a duopoly is between the monopoly output and the competitive output. In an oligopoly with n identical firms, the NE has a total output of n(a-c)/(n+1)b. Rises with n.

The Bertrand Model Each firm independently chooses its price. The demand all goes to the firm with the lowest price (recall that the firms produce an identical product). What prices will the firms choose? Nash: each firm choosing optimal price given the price of the other firm.

What happens with a duopoly Is very sensitive to the rules of the game…

Summary With quantity-setting rules in the Nash Equilibrium the total output with a duopoly is between the monopoly output and the competitive output. A collusive outcome is better for both firms – but is unstable. For a firm it is better to be the leader. With price-setting the Nash Equilibrium has price equal to marginal cost (and therefore like competition). Surplus is lost with quantity-setting but not with price-setting. After this week’s tutorial you might like to ask what happens if the ‘game’ is repeated many times. Might some kind of (illicit?) agreement be reached? Who would enforce it? Legality of such agreements? Government regulation of duopolies/oligopolies?

Chapter 31 Goodbye!