Problem Set #4 Solutions

Transcription

1 4.0 Sprng 00 Page Problem Set #4 Solutons Problem : a) The extensve form of the game s as follows: (,) Inc. (-,-) Entrant (0,0) Inc (5,0) Usng backwards nducton, the ncumbent wll always set hgh prces, thus accommodatng entry. Seeng ths, the entrant wll enter. b) No matter what happens n the frst market, the second market wll turn out exactly as descrbed n part a). Rollng back to the frst market, the ncumbent has no ablty to deter future entry, so he s forced to acuesce and set hgh prces n response to entry. The frst entrant, therefore, wll enter. Hence, we always observe entry and hgh prces. c) Uncertanty as to the payoffs for the ncumbent mples that the entrant(s) must be concerned wth low prces. The threat s much more credble than before. Notatonally, E I let ( a) be the entrant s payoff when takng acton a and ( a t) be the ncumbent s payoff when takng acton a and hs type s t. () Suppose the entrant does ndeed enter, then the sane ncumbent wll accommodate and set hgh prces, whle the crazy ncumbent wll fght by settng low prces. Ths s a matter of proft maxmzaton. Rollng back to the entrant s decson, f he stays out, he earns zero. E ( Out) 0 If he enters, then wth probablty 0.80 he faces a sane opponent who accommodates, and wth probablty 0.0 he faces a crazy opponent who fghts. Hence, hs payoff s E 7 ( In) 0.8( ) + 0.( ) 5 The entrant wll enter. () Wth two markets, the sane ncumbent has an ncentve to pretend to be crazy by fghtng n the frst market; ths may deter the entrant from enterng. In the second market, the sane ncumbent wll always accommodate, and the crazy type wll always

2 4.0 Sprng 00 Problem Set 4 Solutons Page fght (see the prevously gven arguments). Even n the frst market, the crazy type wll always fght. Why? What would be the pont of accommodatng? The more ncumbents accommodate, the more lkely t s that the entrant enters, whch s bad for both the sane and crazy types. Nether wants entry, but only the crazy type can credbly, n a statc sense, threaten to fght. The sane type hopes that by sacrfcng some short run proft n the frst market by fghtng he can convnce the entrant that he s crazy, thus deterrng entry. There can be no pure strategy soluton to ths game: If the sane type always fghts (a poolng eulbrum), then the second entrant gans no nformaton by observng the ncumbent fghtng the frst entrant. Hence, n the second market we are back to the same problem as n part (), and the entrant enters. The ncumbent thus regrets fghtng n the frst market; hs effort to deter entry faled. If the sane type always accommodates, then the second entrant can perfectly detect who s who. If the ncumbent fghts, he must be crazy; the entrant stays out. If the ncumbent fghts, he must be sane; the entrant enters. The sane ncumbent thus wants to fght and fool the entrant. The soluton must be n mxed strateges. In any event, f the second entrant observes accommodaton n the frst market, then he knows that the ncumbent s defntely sane. What about when fghtng s observed? Let x be the probablty that the sane type fghts n the frst market, µ be the probablty that the ncumbent s sane gven that he fought n the frst market (ths belef must be consstent wth Bayes rule), and δ be the probablty that the second entrant enters gven that the ncumbent fought n the frst market: x P µ P δ P ( F sane ) ( sane F ) ( In F ) Frst, takng x, the ncumbent s strategy as gven, we shall derve δ. The entrant takes δ as gven, observes the ncumbent fghtng, and updates hs/her belefs as to the probablty that the ncumbent s sane. Usng Bayes rule, the updatng s as follows: µ P sane F P ( ) P( F sane) P( sane) ( F sane) P( sane) + P( F crazy) P( crazy) 0.8x 4x 0.8x + 0.() 4x + Consder the payoffs for the entrant upon observng fghtng n the frst market:

3 4.0 Sprng 00 Problem Set 4 Solutons Page E ( Out F ) 0 E ( In F ) µ ( ) + ( µ )( ) µ x 8x 4x + 4x + Hence, x Optmal Induced δ Strategy x < / 8 Out δ 0 x / 8 Indfferent δ [ 0,] x > / 8 In δ Next, takng δ, the entrant s strategy as gven, we shall derve x. Consder the payoffs for the ncumbent: I A sane + I ( ) 4 ( F sane) + δ ( ) + ( δ )( 0) 9 8δ Hence, δ Optmal Induced x Strategy δ < 5 / 8 F x δ 5/8 Indfferent x [ 0,] δ > 5/8 A x 0 As we suspected, no pure strateges are consstent: Hgh δ mples low x ; but low x mples low δ. The only consstent behavor s the mxed strategy eulbrum n whch the ncumbent fghts wth probablty x / 8, and the entrant enters, n response to fghtng n the frst market, wth probablty δ 5/ 8. Problem : a) The pharmaceutcal company solves the followng problem: max A 00 P P 60 wth FOC and P, A ( )( ) A ( P) : A ( 00 P)( ) + A ( )( P 60) :00 P P 60 : P 60 : P 80 ( A) : A ( 00 P)( P 60) : 00 : A A 00 : A 40,

4 4.0 Sprng 00 Problem Set 4 Solutons Page 4 and The maxmzed output and proft levels are thus Q A 00 P ,000 ( ) ( 0) ( 00 P)( P 60) ( 0)( 0) 40,000 A A 80,000 40,000 40,000 b) The prce elastcty of demand s Q P A P ( 00)( 80) ε P 4 P Q Q 4,000 whle the advertsng elastcty of demand s Q A A ε A A ( 00 P) 0. 5 A Q A ( 00 P) Last, the advertsng/sales raton s A 40,000 PQ ( 80)( 4,000) 8 Notce that we can verfy these results by usng the followng formula (derved n the notes): ε A A ε PQ Problem : a) Dscussed n class. P ( 4) 8 Problem 4: The multperod aspect of the problem only serves to rescale the payoffs. Gven the dscount factor, f sngle perod payoffs are, then the payoff s 5. Snce each perod s ndependent and dentcal, we can solve for the statc outcome and extrapolate to the fve-perod soluton. a) For smplcty, I shall solve for a general soluton to the Cournot problem (ths should be rudmentary by now), then plug n dfferent margnal costs to solve for the dfferent parts. Consder the two frm asymmetrc Cournot game n whch frm has margnal cost c ; the frm thus solves the followng problem: max 00 c wth FOC ( j )

5 4.0 Sprng 00 Problem Set 4 Solutons Page Q ( 00 j c ) 00 Q Q c j c c c c c Q 00 c Q P c ( P c ) perod Because demand s lnear, we can calculate per-perod consumer surplus as the area of a trangle. max ( P P) Q Q max where P s the prce-ntercept of the nverse demand functon. () When both frms are operatng wth margnal and average costs of $5, the Cournot soluton s Q 70 0 P perod Addtonally, ths leads to consumer surplus of 70 CS 5* () Suppose Frm develops the new technology, what s the new outcome?

6 4.0 Sprng 00 Problem Set 4 Solutons Page Q 75 5 P perod perod 80 5 If the frm does not develop the new technology, Frm s profts are but, by developng t, he makes profts of 5 Cost The frms s wllng to develop the technology as long as Cost 5 Hence, the maxmum wllngness to pay s Consumer surplus s 5 85 Cost CS 5* () The socal value of the new technology s merely the change n consumer surplus plus the change n aggregate profts.

7 4.0 Sprng 00 Problem Set 4 Solutons Page 7 SValue CS + Σ (v) If Frm develops the copycat technology, then the Cournot outcome s Q 60 P perod ,500 By the same reasonng as n part (), Frm s maxmum wllngness to pay s , Also, consumer surplus s CS 5* ( 60) 9000 (v) The socal value of the copycat technology s SValue b) For the undfferentated Bertrand game, n eulbrum frms bd the prce down to the second hghest margnal prce. () When both frms are operatng wth margnal and average costs of $5, the Bertrand outcome s P 5 Q perod 0 0 The consumer surplus s 65 CS 5 * ( 00 5)

8 4.0 Sprng 00 Problem Set 4 Solutons Page 8 () If Frm develops the new technology, t has lower margnal costs and can undercut Frm s prce of 5 by an arbtrarly small amount and steal the whole market. Hence, the outcome s P 5 Q perod perod ( 5 0) Frm s maxmum wllngness to pay, or hs ncentve to develop s 5. The consumer surplus s unchanged. () The socal value of developng the new technology s 5. Consumer surplus s unchanged, as s the proft to Frm. The only value comes through the profts to Frm. (v) If Frm develops a copycat technology, then frms bd the prce down to 0, and the outcome s P 0 Q perod 0 0 and consumer surplus s CS 5 * ( 00 0) (v) The socal value of developng the copycat technology s Consumers gan; Frm loses; and Frm s unchanged. c) Incentves to develop the new technology can be summarzed n the followng chart: Compettve Value Socal Value Cournot Bertrand 5 5 Bertrand competton provdes hgher prvate and socal ncentves to nnovate. The ncreased prvate ncentve comes from the fact that lower margnal benefts the nnovator one-for-one. A dollar n margnal cost savngs s a dollar of proft; addtonally, the nnovator can steal the whole market. In Cournot competton, on the other hand, a dollar n margnal cost savngs s not a dollar of proft due to the prce response. The ncreased socal beneft follows from the same argument.

9 4.0 Sprng 00 Problem Set 4 Solutons Page 9 d) Incentves to develop the copycat technology can be summarzed n the followng chart: Compettve Value Socal Value Cournot Bertrand Cournot competton, however, provdes stronger ncentves, both prvately and socally, to develop copycat technology. Bertrand competton provdes no prvate ncentve, zero profts versus zero profts. Problem 5: a) Each frm solves the followng problem: max 0 ( j ) wth FOC 8 j 0 Hence, the frm s reacton functon s j 8 j Usng symmetry, 8 ( ) ( ) 6 Q P 0 8 ( 8 ) 6 6 b) Payng a manager a share of profts wll not change the eulbrum. A manager wll maxmze hs own payoff α. Ths wll have the followng FOC: α 0 whch reduces to 0 whch s the orgnal FOC. c) The manager for Frm now solves the followng problem: max β 0 ( ) wth FOC β ( 0 ) 0 Hence, Frm s reacton functon s ( ) ( 0 ) Frm s reacton functon s unchanged. Solvng ths system of euatons, we get the followng outcome:

10 4.0 Sprng 00 Problem Set 4 Solutons Page 0 Q P By payng the manager accordng to revenue, the frm effectvely commts to overproduce; the manager behaves as though margnal cost were zero, not two. Hence, lke a Stackelberg-leader, the frm does better by commttng. Frm s shareholders wll want to follow sut and swtch to the new compensaton scheme as well; commt to overproduce n order to boost profts. If you run the numbers, Frm can boost ts profts to.. d) If the frms compete n a dfferentated products Bertrand game, the frm swtchng to the revenue-based compensaton scheme wll over-produce, snce the manager wll stll behave as though margnal cost s zero. As you may recall, leaders n prcng games tend to do worse, whle followers do better; ths s n drect contrast to uantty games. Because the compensaton scheme nduces the manager to behave as though margnal cost has been reduced, we can nterpret ths as an over-nvestment n cost-reducng technology. Cost reducng technology makes a frm tough. When frms compete n uanttes, strategc substtutes, over-nvestment s optmal (Top Dog), as we saw n part c). On the other hand, when frms compete n prces, strategc complements, under nvestment s optmal (Puppy Dog).