Importantly, note that prices are not functions of the expenditure on advertising that firm 1 makes during the first period.

Transcription

1 ECONS 44 STRATEGY AND GAME THEORY HOMEWORK #4 ANSWER KEY Exerise - Chapter 6 Watson Solving by bakward indution:. We start from the seond stage of the game where both firms ompete in pries. Sine market demand is Q = a - p, then produts are homogeneous, and in addition, we are told in the exerise that the firm setting the lowest prie gets all the market. Hene, we are in a Bertrand game of prie ompetition, and we know from lass that the equilibrium prie firms set is P = P = 0. Importantly, note that pries are not funtions of the expenditure on advertising that firm makes during the first period.. Sine this is the ase, firm knows that by spending more money on advertising it will not inrease the profits during the seond period. As a onsequene, a = 0 during the first period. Therefore, the subgame perfet equilibrium is a = 0 during the first stage and P = P = 0 during the seond stage.

2 Exerise 8 - Chapter 6 Watson a. Without payoffs, the extensive form is as follows [Note that we are using dashed lines to denote that firm hooses q without observing firm s output q. Similarly, firm hooses q without observing firm and firm s output, q and q, respetively.]: q q q E E D q q D E q q D q Solving by bakward indution, we must first find the output level of every possible entry/no entry senario. By doing so, we will be able to find the profits resulting from every possible entry/no entry senario, and then we will be ready to ompare firms profits from entering and not entering:. We first solve firms output in the subgame that starts after firm and enter. [In the figure, this subgame is the upper part, where firms are seleting q, q and q ] This is just a Cournot game of quantity ompetition with three firms ompeting with eah other by simultaneously seleting output. Hene, qq = qq = qq =. a. PROFITS: In this ase, note that the profits of every firm in this Cournot oligopoly game with three firms are: ( QQ)qq ii = ( qq qq qq )qq ii = ( ) = = 9.

3 b. Note that we must finally subtrat 0 (entry osts) in the profits of firm and firm (You don t have to do so for firm, sine it was already the inumbent in the market). Hene, the payoff vetor would be (9-0, 9-0, 9) = (-, -, 9). Now we solve the subgame indued after firm enters (E) but firm does not (D). Here we have a Cournot oligopoly game played by firms and (duopoly), where they simultaneously selet qq and qq. Hene, qq = qq = 4. a. PROFITS: In this ase, note that the profits of every ative firm in this Cournot oligopoly game with two firms are: ( QQ)qq ii = ( qq qq )qq ii = ( 4 4) 4 = 4 4 = 6. b. Note that we must finally subtrat 0 (entry osts) from the profits of firm (entrant), whih implies that the payoff vetor beomes (6-0, 0, 6) = (6, 0, 6).. Now we solve the subgame that starts after firm deides not to enter (D), but firm deides to enter (E ). Now we have a Cournot oligopoly game played by firms and (duopoly), where they simultaneously selet qq and qq. Hene, qq = qq = 4. a. PROFITS: In this ase, note that the profits of every ative firm in this Cournot oligopoly game with two firms are: ( QQ)qq ii = ( qq qq )qq ii = ( 4 4) 4 = 4 4 = 6. b. Note that we must finally subtrat 0 (entry osts) from the profits of firm (entrant), whih implies that the payoff vetor beomes (0, 6-0, 6) = (0, 6, 6). 4. Now we solve the subgame indued after firm deides not to enter (D) and firm deides not to enter either (D ). Here firm keeps its monopolisti position, and hooses monopoly output, qq = 6. a. PROFITS: In this ase, note that the profits of the only monopoly in the market (firm ), are: ( QQ)qq = ( 6)6 = 6 b. Note that we don t have to subtrat any entry osts from firm s profits, given that it was already the inumbent in the market. Hene, the payoff vetor in this ase is (0, 0, 6).

4 Plugging all the payoff vetors in the appropriate nodes (see figure at the end of the answer key), and solving by bakward indution, we see that:. Firm (last mover in this game): After observing that firm entered the market, firm deides to not enter, sine its profit from not entering (0) are higher than from entering a too rowded market (profits of -). After observing that firm didn t enter the market, firm hooses to enter, sine its profits from doing so (6, now firm would beome the only ompetitor of firm ) are higher than from not entering (0).. Firm (first mover in this game): Firm deides to enter, given that its profits from entering (and induing firm to stay out afterwards) are 6, while those from not entering (and induing firm to enter the market afterwards) are only 0. Hene, firm enters. Hene, at the subgame perfet equilibrium:. firm selets Enter,. firm hooses not to enter after observing that firm entered, but hooses to enter after observing that firm didn t enter.. Equilibrium output levels at every subgame of this game are: qq = qq = qq = qq = qq = 4 qq = qq = 4 qq = 6 b. In the subgame perfet equilibrium only firm enters, induing firm to stay out of the market. 4

5 Exerise 9 - Chapter 6 Watson a. The government solves: MMMMMM 0 + (pp WW ) pp pp 0 = pp WW Taking first order onditions with respet to pp, we obtain. Sine this result does not depend on pp, it is indiating that the solution to the problem is a orner solution. In partiular, given that > 0, we an onlude that the solution is the upper orner, i.e., the government sets pp as high as possible, regardless of the level of WW. So pp = 0. Knowing how the government will behave, the ASE solves: MMMMMM (WW 0) ww where we have already replaed pp = 0. The first order ondition implies: So in equilibrium y = 0. pp = WW = 0 b. If the government ould ommit ahead of time, it would solve: MMMMMM pp 0 + (pp WW) pp 0 and using the fat that pp = WW, we an rearrange the above expression to obtain MMMMMM ww pp Taking first-order onditions with respet to pp yields < 0, indiating that the solution to this maximization problem is the lower orner, i.e., the government ommits pp = 0 and the ASE would set WW = 0. In (a) u = 0 and v = -5. Now, when ommitment is possible, u = 0 and v = 0.. One way is to have a separate entral bank that does not have a politially eleted head that states its goals. 5

6 BONUS EXERCISE (Exessive entry in an industry) (a) Sine the equilibrium has to be found by bakward indution, first solve the last stage of the game, where firms hoose quantities given the number of firms, n, that have entered the market in the previous stage. In partiular, the n th firm entering produes an output level of n ( ) +, thus obtaining profits of π ( n) ( n + ) = F. Sine n is a real number, the equilibrium number of firms in the industry will be given by solving for n in, given that the n th entrant must be indifferent between entering and staying out. Solving for n we obtain n =. F (b) The soial planner will hoose n to maximize total welfare (the sum of onsumer and produer surplus). Let us first find onsumer surplus, CS. Notie that the CS is given by the area of the triangle between the vertial interept of the demand urve, + n, and the equilibrium prie p= n =. Hene, CS is given by n + n + ( ) CS + n = n = n. Produer surplus is simply given by the n+ n+ ( n+ ) ( ) ( ) aggregate profits all firms make in the industry, i.e., nπ ( n) = n nf. n + Therefore, the sum of onsumer and produer surplus yields a total welfare of n( ) ( n+ ) W = CS + PS = nf. Taking first-order onditions with respet to ( n + ) * n, and solving for n, we obtain ( n = ). F By omparing the optimal number of firms n* we just found with the number of firms entering at the free entry equilibrium from part (a), lear that there exists exess of entry in the industry., it is The following figure illustrates this result. In partiular, the figure depits n and n* as a funtion of F in the horizontal axis, and evaluating both of them 6

7 at a marginal ost of =0.5. (You an obtain similar figures using another value for firms marginal osts of prodution, ). Two important features of the figure are noteworthy. First, note that both n and n* derease in the entry osts, F. Seond, the equilibrium number of firms entering the industry, n, lies above the soially optimal number of firms, n*, for any given level of F; refleting an exessive entry in the industry when entry is unregulated. 7