Problem Set #2 Solutions

Transcription

1 4.0 Sprng 003 Page Proble Set # Solutons Proble : a) A onopolst solves the followng proble: ( Q ) Q C ( )= 00Q Q 0Q ax P Q wth frst-order condton (FOC) b) Gven the results fro part a, Q 90 Q = 0 Q P = 55 π = 4050 * = 45 * P = P = 55 * π = 05 On the other hand, a Cournot copettor, as n the punshent phases, solves the followng proble: ax 00 ( + j ) 0 wth FOC 90 j = 0 whch ples the followng best response functon: j C Due to syetry, we know that C = j ; hence, C C = j = 60 P C = 40 C π = 800 Last, suppose I decde to cheat; what uantty do I produce? I should produce the best response to what the other fr (who s not cheatng) s producng. Reeber, as long as I a cheatng, I ght as well cheat as well as possble. Hence, f I cheat, or devate fro the proposed eulbru, I should d ( 45)= P d = d π = 78.5 If I play along wth the cooperatve strategy, the PDV of y proft strea s

2 4.0 Sprng 003 Proble Set Solutons Page But, f I devate, the PDV s ( + δ + δ +...) = δ ( δ + δ +...) = δ δ The cooperatve strategy s an eulbru f δ δ δ δ ( δ ) = δ 478.5δ δ c) th a perod detecton lag, the devator can cheat for two perods. If I devate, the PDV s 3 800δ δ + 800(δ + δ +...) = 78.5( + δ ) + δ The cooperatve strategy s an eulbru f δ 78.5( + δ ) + δ δ δ )+ 800δ ( = δ δ δ d) Frs copetng n prce n a hoogeneous goods arket ths s a Bertrand gae. Frs bd the prce down to argnal cost. Hence, the only eulbru s for both frs to charge a prce eual to the argnal cost of 0. The total arket uantty s 80. e) hen both frs cooperate by settng prce to be the onopoly prce P = 55, each fr akes a proft of 05 n each perod (See Part a). In the punshent phases, frs are Bertrand copettors and ake zero proft. hat about when one fr cheats? A fr s optal cheatng strategy s to set prce eual to P ε,.e., just undercut the other fr and capture the whole arket. In ths way, the cheatng fr essentally acts as the sole onopolst and akes the entre onopoly proft of The cooperatve strategy s an eulbru f

3 4.0 Sprng 003 Proble Set Solutons Page δ δ δ Prce copetton s ore lkely to sustan a onopoly outcoe because, due to the ferce copetton nspred by prce copetton, the punshent s uch harsher. f) Frs are capacty constraned no one can produce the full arket deand at a prce eual to the argnal cost. Frs keep undercuttng the rval s prce down to argnal cost, at whch pont no fr akes proft and can beneft fro rasng ts prce. Snce nether fr can eet the full arket deand, soe consuers are forced to ether not buy or buy fro the other fr. Frs then begn agan to undercut each other back down n prce. In ths way, prces cycle up and down. Ths phenoenon, known as Edgeworth cycles, actually happens. Proble : a) The jont proft-axzaton outcoe s the sae as n part a of Proble. In the cartel, the total producton uota, allotted soehow aong the fve frs, ust be Q = Q The cartel can actually allocate the 90 unts any way t wants; evenly dvdng t aong the fve frs sees lkely. The prce s the onopoly prce of 55; profts are dstrbuted aong the fve frs n the sae way that the uanttes are. b) In the fve fr cartel, each fr akes proft 80 (=4050/5). Suppose one fr decdes not to enter nto the cartel. Let Q 4 be the total uantty produced by the four frs n the cartel. Then the non-cartel fr solves the followng proble: ax 00 ( Q 4 + ) 0 wth FOC 90 Q 4 = 0 Q 4 Knowng the non-cartel fr wll behave n ths way, the cartel axzes ts jont proft as follows: ax 00 Q 4 Q Q 4 Q 4 0Q 4 = 90 The FOC s 90 Q 4 = 0 Hence, the eulbru s Q 4 Q Q 4 4

4 4.0 Sprng 003 Proble Set Solutons Page 4 Q 4 = 45 P = 3. 5 It thus pays not to enter the cartel. non cartel π = ( P 0) = 0. 5 c) Suppose there are n frs. Let us check f an n-fr cartel s stable; does a fr have ncentve not to enter. Frs n the n-fr cartel ake proft 4050/n. Usng reasonng analogous to part b, a non-cartel fr solves the followng proble: ax 00 ( Q n + ) 0 wth FOC 90 Q n = 0 Q n Knowng the non-cartel fr wll behave n ths way, the cartel axzes ts jont proft as follows: ax 90 Q n Q n Q n The FOC s 90 Q n = 0 Hence, the eulbru s Q n Q n = 45 P = 3. 5 non cartel π = 0.5 Therefore, coparng the proft to frs nsde the n-fr cartel to the proft of the noncartel fr suggests n < 4 Stable cartel n = 4 Indfferent n > 4 Incentve to not jon Due to the syetry of the frs, there s no dfference between the (n-)-fr cartel and a sngle Stackelberg leader; look at the FOC to confr ths. th a large nuber of frs, collusve profts are heavly dluted; thus, despte the fact that a Stackelberg follower does coparatvely worse than the leader, t stll pays to hold out and gan the undsspated Stackelberg follower profts. Proble 3: a) Each fr solves the followng proble:

5 4.0 Sprng 003 Proble Set Solutons Page 5 wth FOC ax P Q C ( Q )= (P 0 )( 80 P + P j ) P 80 4P + P j + 0 = P j P = 4 Due to syetry, we know that P R = P ; hence, P R = P = 00 / 3 wth FOC π = 70 3 = b) ave knows Rah s best response functon s 00 + P = PR 4 because Rah s proft-axzaton proble has not changed; therefore, ave solves the followng proble: ax (P P ) 80 P + P 4 Therefore, 80 4P P = P = 0 P = 35 P = 35 P = R π = π R = 8. 5 ave should take advantage of the cotent strategy because t (slghtly) ncreases hs profts. Proble 4: For splcty, label the ponts at whch the players ove as nodes one through seven, nuberng the up and down, left to rght; player one oves at nodes, 4, 5, and 6, whle player two oves at nodes and 3. The two nodes connected by the dotted lne are n fact a sngle node. The dotted lne sgnfes the fact that player one does not know at whch node he was. To fnd an SPNE, we eploy backwards nducton. At node 6, player one wll choose u, whether player two had chosen U or D at node three. At node fve, player one wll choose d ; at node four, he wll choose u. Steppng back to nodes two and three, we can decde what s optal for player two knowng what player one wll do at the ensung nodes. At node three, player two wll choose U; at node two, he wll also choose U. Last,

6 4.0 Sprng 003 Proble Set Solutons Page 6 player one wll begn the gae by choosng d. e can represent the eulbru strateges for the two players as follows: ( d, u, d, u ) ( U, U ) where the frst vector s the strateges for player one and the second vector s the strateges for player two, strateges are ordered by node. Proble 5: a) Ths s analogous to the GE/estnghouse case when GE began postng all ts transacton prces. Ths s lkely to facltate colluson because t ontors the actons of the frs; colludng frs leave ontorng and detectng cheatng to the Cosson. b) The envronental polces are lkely to nhbt colluson. Colluson s enforced by the threat of future punshent,.e., the attractveness of colludng s the collusve future revenue strea. The fact that the product wll be banned n fve years truncates the possble beneft to colludng; less ncentve to collude s ore ncentve to not collude. c) Uncertanty tends to hnder colluson prarly by akng detecton ore dffcult. For exaple, a low prce ay ndcate low deand or cheatng be a fr, producng less output. d) Multarket contact s lkely to facltate colluson. It akes punshent easer; as referred to n the lecture notes, May relax ncentve constrant; transfer slack n nocheatng condton fro one arket to another to expand scope of cooperatve behavor. Proble 6: a) The ajor reason the two prces are dfferent s probably prce dscrnaton. Suppose soe consuers have uch lower search costs than others; they enjoy surfng the web bargan-huntng or are ore techno-savvy. Slarly, t ay be dscrnaton between dfferent types of consuers; college students and yuppes pay less than older adults. Busness travelers ay not search around for low prces n the sae way that pleasure travelers do. Possbly, the two roos are not really dentcal; there ay be sall dfferences between the products that we do not observe. b) The hotels ay want to prevent prces fro beng posted because t helps attract ore consuers to the lower prces. Fewer consuers pay the hgh prces relshed by the hotels. Prcelne, on the other hand, ay not want prces posted because t akes ther servce less attractve n the future. th prces ade publc, people ay be able to bargan drectly wth the hotels, or prces ay becoe standardzed at soe lower prce, akng Prcelne obsolete.