1 C 6, 3 4, 4. Player S 5, 2 3, 1

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1 University of alifornia, Davis - Deartment of Economics PRING EN / ARE : MIROEONOMI TEORY Professor Giacomo Bonanno ====================================================================== MIDTERM EXAM ANWER A QUETION (total ots). [ ots] There are stars arranged a circle. Player can remove one star or two contiguous (= next to each other) stars. Player then removes one or two stars accordg to the followg rules: () one of the stars that she removes must be contiguous to (= next to) a star removed by layer and () if she removes two stars, they have to be contiguous (= next to each other). Then it is layer s turn, followed by layer, etc. The wner is the layer who ends the game by removg the last star(s) (i.e. the layer after whose move no stars are left). Notice that the rules of the game are such that, after each move, the remag stars always form a contuous arc (i.e. with no gas). (a) [ ots] One of the two layers has a wng strategy. Who is it and what his the wng strategy? (b) [ ots] Answer the same question but assumg that there are stars itially (rather than ; everythg else is the same).. [ ots] onsider the followg two-layer game (the numbers the matrix are von Neumann-Morgenstern ayoffs; the first number is layer s ayoff and the second number is layer s ayoff): Player Player,, 6,, (a) [ ots] Assume first that the game is layed simultaneously. Fd all the Nash equilibria. (b) [ ots] Assume now that layer moves first, layer observes s choice and moves second. Draw the extensive game and fd the backward duction solution. (c) [ ots] Assume aga that layer moves first, but layer observes s choice imerfectly as follows. et < ε <. If layer chooses, with robability ε layer is told that chose and with robability ε layer is told that chose. imilarly, if chooses, with robability ε layer is told that chose and with robability ε layer is told that chose. Draw the extensive game, the corresondg normal-form game and fd all the ure-strategy Nash equilibria. (d) [ ots] What is surrisg ab this examle? (The answer nothg is unaccetable!).

2 . [ ots] [Note: this is based on a true story; the names, however, have been changed] Glide Inc. has been a monooly the market for shoes for many years. The roduction of each air of shoes requires hours of labor. At the moment the wage rate is $ er hour. There is also a fixed cost of $6. The workers the dustry are unionized. The union charter states that the wage rate should be uniform across firms the dustry, that is, the union will demand the same wage rate from all the shoe roducers. Thus any newcomer to the dustry will have to ay whatever wage rate Glide Inc. is ayg at the time of entry. Mecometoo is an entrereneur who has announced that he tends to set u a new shoe-roduction facility. If Mecometoo enters the shoe dustry then there will be cometition ournot style. The demand for shoes is given by: P = Q. Mecometoo will enter only if it exects to make ositive rofits. Followg the announcement by Mecometoo, Glide Inc. set u a meetg with the workers union and offered to double the wage rate. The offer was acceted by the union. Mecometoo sues Glide Inc. for anticometitive ractices claimg that the wage crease was offered with the tention of deterrg entry. You have been hired as an exert witness by Mecometoo and have to convce the judge that the charge is well founded. In articular you have to rove that: (a) [ ots] if there hadn t been a threat of entry, Glide Inc. would not have offered the wage crease [Note: you have to justify your answer by calculatg Glide s rofits], (b) [ ots] if the wage crease had not been offered, entry would have taken lace, (c) [ ots] the higher wage has deterred entry by makg it unrofitable.

3 University of alifornia, Davis -- Deartment of Economics PRING 9 EON. : MIROEONOMI TEORY Giacomo Bonanno Midterm Exam ANWER. (a) When there are stars left, the layer who has to move can be made to lose (at most she can remove ). Thus is a losg osition. ence 6 is a losg osition: after the layer has removed one or two stars, the other can take her to the losg osition of stars. ence 9 is a losg osition. ence is a losg osition. It follows that layer has a wng strategy, as follows. After layer has moved, remove as many stars as necessary to leave 9 ( if layer removed, if layer removed ). Then, after layer has moved, remove as many stars as necessary to leave 6 (aga, if layer removed, if layer removed ). Then, after layer has moved, remove as many stars as necessary to leave (aga, if layer removed, if layer removed ). Then, after layer has moved, remove the last star(s). (b) Alyg the same reasong, the losg ositions are:, 6, 9, ence layer has a wng strategy: remove one star and then take layer to ositions 96, 9, 9,..., 9, 6, by removg at each stage ( x) stars where x is the number of stars removed by layer.. (a) is a strictly domant strategy for layer, hence (,) is the unique Nash equilibrium. (b) The extensive game is as follows. 6 The subgame-erfect equilibrium is (, ( if, if )). The equilibrium ayoffs are (,). Page of

4 (c) The extensive form is as follows. Nature Nature ε ε ε ε 6 corresondg normal form is (note that layer has strategies) 6 The Player Player, ( ε) + ε, ( ε) + ε ( ε) + ε, ( ε) + ε, 6, ( ε) + 6ε, ( ε) + ε Exandg we get ( ε)6 + ε, ( ε) + ε Player, Player, ε, ε + ε, + ε, 6, + ε, ε 6 ε, + ε, No matter what ε is, the only ure-strategy Nash equilibrium is (, (,)) with ayoffs (,). Page of

5 (d) The surrisg thg is that if ε is very small, then we are very close to the game of art (b) where layer gas from beg the first mover. Yet, with even a ty robability that his first move is not erfectly observed, we are back to the come of the simultaneous game.. If the wage rate is $, the cost of roducg one air of shoes is =. Thus the cost function is = 6 + Q. If the wage rate is $6, the cost of roducg one air of shoes is 6 =. Thus the cost function is = 6 + Q. (a) AE : no threat of entry. With w = the rofit function is Π(Q) = Q ( Q) (6 + Q) olvg d Π = gives Q =., P = 66, Π = 69. dq With w = 6 the rofit function is Π(Q) = Q ( Q) (6 + Q). olvg d Π = gives Q =, P = 7, Π = 6. Thus creasg the wage dq rate leads to a reduction rofits. (b) AE : w=, there is entry and ournot cometition. With w = the rofit functions are (Glide is firm and Mecometoo is firm ) Π (q,q ) = q [ (q + q )] (6 + q ) Π (q,q ) = q [ (q + q )] (6 + q ) olvg Π q take lace. = and Π q = gives q = q = 9, P = 8, Π = Π = 6. ce Π >, entry would (c) AE : w= 6, there is entry and ournot cometition. With w = 6 the rofit functions are (Glide is firm and Mecometoo is firm ) Π (q,q ) = q [ (q + q )] (6 + q ) Π (q,q ) = q [ (q + q )] (6 + q ) olvg Π q take lace. = and Π q = gives q = q = 8, P = 6, Π = Π =. ce Π <, entry will not Thus the charge is well founded: by offerg to ay a higher wage rate, Glide Inc is makg entry unrofitable thereby guaranteeg its osition of monooly. Page of

6 University of alifornia, Davis - Deartment of Economics PRING EN / ARE : MIROEONOMI TEORY Professor Giacomo Bonanno ====================================================================== FINA EXAM ANWER A QUETION (total ots). [ ots] onsider the followg game, where the ayoffs are von Neumann- Morgenstern ayoffs. Player M R Player A,,, B,,, 9,,, (a) [ ots]for each layer, fd all the rationalizable ure strategies, that is, the strategies that could be layed a situation where there is common knowledge of rationality. (b) onsider the followg model of this game Player / / / / a b c d Player / / a b c d 's strategy: A A 's strategy: M R et R i (i =,) be the event that layer i is rational (that is, she is maximizg her exected ayoff). (b.) [6 ots] Fd the followg events: R (layer is rational), R (layer is rational), K R (layer knows that layer is rational), K R (layer knows that layer is rational). (b.) For each of the followg statements, determe whether the statement is true or false at state a (i) [ ots] Player knows that layer knows that layer is rational. (ii) [ ots] Player knows that layer knows that layer is rational. (iii) [ ots] It is common knowledge that both layers are rational. Page of

7 . [ ots] There are two tyes of dividuals. They have identical itial wealth of $,, they face a otential loss of $9 and they have a utility-of-money function U(m) = m. For dividuals of tye the robability of loss is = while for dividuals of tye the robability of loss is =. et N be the number of tyes and N the number of tye. The surance market is a monooly. The monoolist knows all of the above data but cannot tell whether any articular customer is of tye or tye. The monoolist is considerg several otions (refer to the followg figure). Assume that () if different between surg and not surg, a consumer would choose to sure and () if different between two contracts, then the consumer would choose the one with the lower deductible. I have not labeled the difference curves with the tye of consumer, because you should be able to figure it yourself. wealth good state (no loss) o le no surance A B D wealth bad state (a) [8 ots] Otion : offer only contract A. alculate the monoolist s rofits this case. (b) [8 ots] Otion : offer only contract B. alculate the monoolist s rofits this case. (c) [ ots] Otion : Offer contracts and D and let consumers choose. The remium for contract is $6. and the remium for contract D is $9. alculate the monoolist s rofits this case. (d) [ ots] If N = and N =,, which of the three otions would the monoolist choose? (e) [ ots] If N = and N =,, which of the three otions would the monoolist choose? Page of

8 . [ ots] A market consists of a sgle consumer who is willg to buy one and only one unit rovided the rice does not exceed $α >. If two firms are resent the market, the consumer will buy from the cheaer firm (hence the roducts are homogeneous). If there are two firms and they charge the same rice, then the consumer will choose randomly between the two firms with equal robability. The firms are risk neutral. onsider the followg two-stage game. In stage one two firms, and, simultaneously decide whether to enter the market or not. Entry costs $β (with <β < α) and the entry cost is sunk (i.e. irrecoverable). At the end of stage one each firm knows if the other firm has entered or not ( fact, this is common knowledge). In stage two, if both firms have entered they simultaneously comete rices, otherwise if only one firm has entered it will behave as a monoolist. If a firm does not enter its rofits are zero. Production costs are zero. (a) Assume first that only two rices are ossible: high,, and low, (thus each firm has to choose whether to charge or ). (a.) [9 ots] Draw the extensive form of this game. (a.) [8 ots] Assume that < < < α, and β <. What are the subgameerfect equilibria of this extensive game if >? (a.) [8 ots] Assume that < < < α, and β <. What are the subgameerfect equilibria of this extensive game if <? (b) [8 ots] Assume now that firms can charge any nonnegative rice (thus dro the assumtion that only two rices are ossible). What are the subgame-erfect equilibria of this game? Page of

9 University of alifornia, Davis -- Deartment of Economics PRING EN / ARE : MIROEONOMI TEORY Giacomo Bonanno Fal Exam ANWER. (a) For layer, B is strictly domated by (½ A, ½ ); for layer R is strictly domated by (½, ½ M). Elimatg B and R we are left with Player M Player A,, 9,, In this game no layer has a strictly domated strategy. Thus for layer both A and are rationalizable and for layer both and M are rationalizable. (b.) R = {a,b}, R = {a,b,c}, K R = {a,b}, K R = {a} (b.) (i) False (sce K K R = ); (ii) true (sce K K R = {a}; (iii) false because of (i) (alternatively, one can note that the common knowledge artition is the trivial artition and it is not the case that both layers are rational at every state).. The two steeer difference curves belong to the tyes. (a) ontract A is a full-surance contract that will be bought only by the tyes. To calculate the remium for contract A solve + 6 = h. The solution is h = 96. Thus the monoolist s rofits are π = 96 9 N = 6N. (b) ontract B is a full-surance contract that will be bought by all tyes. To calculate the remium for contract B solve = h. The solution is h = 99. Thus the monoolist s rofits are π = 99 9 N N = 9N 8N. (c) The tyes would choose the full-surance contract D while the tyes would choose the artial-surance contract. To comute the deductible for contract solve 9, 6. +, 6. d = 99. The solution is d = 88.. Thus the monoolist s rofits are π = N + ( 9 88.) N = N +.9N. [lightly different numbers are obtaed if the remium for contract is comuted usg the difference curve of the tye by solvg, 9 =, 6. h +, 6. ; this case the solution is 88. with corresondg rofits of π = N + ( 9 88.) N = N +.9N.] (d) When N = and N =,, π =,6, π = 9 and π =, 9. Thus this case the third otion is the best. (e) When N = and N =,, π = 8,, π =, and π = 7,8. Thus this case the first otion is the best. Page of

10 . (a.) β β At the sgleton decision nodes the otimal choice is. Thus the game simlifies to: β β (a.) If > the simultaneous subgame on the left has a unique Nash equilibrium: (, ). Thus this case the game simlifies to: Page of

11 The unique Nash equilibrium of this game is (, ). Thus the case where > there is a unique subgame-erfect equilibrium given as follows. s equilibrium strategy: (,, ), s equilibrium strategy: (,, ) (a.) If < the simultaneous subgame on the left has two Nash equilibria: (, ) and (, ). Thus there are two subgame-erfect equilibria: one as before, that is ((,, ), (,, )) the other as follows: ((,, ), (,, )). (b) By Bertrand s theorem, the subgame when both firms have entered (corresondg to the simultaneous subgame on the left the revious figure) has the roerty that each firm makes zero revenue at every Nash equilibrium, hence negative rofits equal to. Thus there are only two subgame-erfect equilibria: () firm s strategy is (,, α) and firm s strategy is (,, α) [the second comonent is the duooly rice and the third comonent is the monooly rice], () firm s strategy is (,, α) and firm s strategy is (,, α). Page of