1. The Cournot model, where the strategic variable is a quantity choice. 2. The Bertrand model, where the strategic variable is the price charged.

Transcription

1 16 Oligopoly Oligopoly refers o a siuaion wih a few rms on he marke and he enral assumpion ha di ers from he ompeiive model is ha he rms undersand ha heir aions a e he marke prie. There is an indusry of di eren models, u mos of hese are in some way or anoher variaions on he wo really enral models of oligopoly. These are: 1. The Courno model, where he sraegi variale is a quaniy hoie. 2. The Berrand model, where he sraegi variale is he prie harged. Besides eing imporan in hemselves, hese models also illusraes well how he ools from game heory is used in eonomis. We will ry o keep hings as simple as possile, so we will ypially only onsider duopoly versions of he models (wo rms) The Courno Model We assume ha: There are wo rms, 1 and 2 (will onsider n rms laer) Consan marginal os equal o The rms produe a homogenous good wih inverse demand p(y) = a y Now, we wan o as he model as a game, so we need o desrie: 1. Who he players are. This is no prolem-we ve already said ha he rms are rm 1 and rm Wha he availale sraegies are for eah player. Wih quaniy ompeiion ha is jus 1

2 quaniy 0 for rm 1 quaniy y 2 0 for rm 2 Noe ha we are assuming ha quaniies are hosen simulaneously sine oherwise we would speify he sraegy of one rm o e oningen plan ( a quaniy for eah quaniy hosen y he oher rm) 3. Finally we need o omplee he model and desrie he payo funions, ha is how he payo s of he players depend on he sraegies hosen y he players. ere we have some work remaining. We will assume ha rms are aou heir pro s exaly as in he ompeiive model and he monopoly model, ha is rm 1 ares aou 1 = p owever, wih only wo rms we wan o hink aou rms who undersand how he prie is a eed y quaniies on he marke. Now, wih a homogenous good he equilirium prie depends on he sum of quaniies hosen y he rms, ha is he prie given quaniies and y 2 is p( + y 2 ) = a ( + y 2 ); So, we an wrie rm 1 s pro as a funion of he sraegies ( ha is quaniies) as 1 ( ; y 2 ) = p( + y 2 ) = = (a ( + y 2 ) ) Noe ha he pro is now desried fully in erms of he sraegies (and parameers), so we have desried a game eween rm 1 and rm 2. Sine here is an in nie numer of possile quaniy ominaions we an no wrie i as a payo marix, u we have desried a more elaorae version of a game exaly like he prisoners dilemma or he ale of he sexes. 2

3 16.2 Nash Equilirium in he Courno Model Reall ha a Nash equilirium is a siuaion where eah player does he es he/she an given wha he oher player(s) are doing. ene a pair of quaniies (y 1; y 2) is a Nash equilirium in he Courno model if Bu, ha is jus saying ha 1 (y 1; y 2) 1 ( ; y 2) for all 0 and 2 (y 1; y 2) 2 (y 1; y 2 ) for all y 2 0. y 1 solves max 0 1( ; y 2) and y 2 solves max y 2 0 2(y 1; y 2 ) Wha his means is ha we an jus ake rs order ondiions as usual (for a xed quaniy y he oher rm) and hen solve he 2 rs order ondiions ou for he Nash equilirium. Firm 1 solves he prolem and he rs order ondiion is max 1( ; y 2 ) = max (a ( + y 2 ) ) 0 0 a ( + y 2 ) = 0 Solving for we ge he es response, he opimal hoie of given any y 2 (y 2 ) = y 2 Symmerially, we ge a es response for rm 2 y 2 ( ) = Nash equilirium is a siuaion where eah rm is doing he es hey an given wha he oher rms is doing, so y 1 = (y 2) and y 2 = y 2 (y 1); so a Nash equilirium is a soluion o he sysem = y 2 y 2 = 3

4 y 2 6 Firm 1 s Bes Response 3 Nash Equilirium Firm 2 s Bes response 3 - Figure 1: Bes Responses and Nash Equilirium in Courno Model The es responses are drawn in Figure 1 where you should oserve a few hings: 1. Evaluaing he es response for a rm when he oher rm piks an oupu=0 we have (0) = y 2 (0) = a : If you reall he monopoly analysis his is exaly he same as he soluion o he monopoly prolem (verify), whih should make perfe sense sine a rm who elieves ha he oher rm will produe nohing should ehave as a monopolis. 2. Moreover, if, say, rm 2 would produe enough oupu so ha p(y 2 ) = ;whih would e he ompeiive (reak-even prie) hen i is raher inuiive ha addiional oupu from rm 1 would mean ha he prie would ge elow he marginal os)loss for he rm. Bu q 1 = a is exaly ha prie sine p( a ) = a = ; so he inereps where he es response is zero has exaly he inerpreaion as he poins where he ompeior produes an oupu ha yields he ompeiive prie. Solving he sysem for a Nash equilirium we an aually hea a lile i sine he equaions are symmeri and herefore ough o have a symmeri soluion y1 = y2: Then we 4

5 have only a single equaion o solve, namely y 1 = 3 2 = =, y 1 = y 2 = 3 You may also nd i ovious from he piure ha a he es responses are a a and y 1 2, respeively on oh axis. y oserving ha he inereps of 16.3 Comparison wih Monopoly Model Suppose ha he rms insead of aing independenly would ge ogeher and hink aou if hey ould improve on he siuaion in he Courno equilirium. They would hen simply se y = + y 2 o maximize he monopoly pro, ha is solve max (a y ) y y We ve already solved his prolem and he soluion is (hek!) y m = : Plugging his ino he pro s and omparing wih he pro s under oligopoly you an hek ha he monopoly pro is more han wie he per rm pro in he Courno equilirium so i is possile for oh rms o gain y forming arel. To see his more learly in he graph we noe ha if we plo isopro s (ominaions of and y 2 suh ha pro s are onsan) for rm 1 hen anyone of hese solves 1 = (a ( + y 2 ) ) for some 1 : Now, sine he es response is he pro maximizing hoie given he pariular y 2 hese mus have a zero slope when inerseing he es response. Moreover, pro s are inreasing he less he oher rm produes so we an depi hese isopro s for rm 1 as in Figure 2. Doing he same hing for rm 2 and omining wih he line onsising of poins where + y 2 equals he monopoly oupu we ge a very insruive piure, whih 5

6 y 2 6 Firm 1 s Bes Response 3 Firm 2 s Bes response U igher 3 pro s - Figure 2: Isopro s in Courno Model is he one in Figure 3. In he piure he line eween ; 0 and 0; are he poins where indusry oupu equals he monopoly oupu ( rms aing as arel). In he gure we have also drawn in he isopro s going hrough he Nash equilirium of he Courno model and everyhing in he shaded areonsiss of poins where oh rms are eer o han in he Courno equilirium. Finally, he poin on he inerseion eween he arel line and he 45 0 line is he poin where he rms agree o spli he monopoly oupu in wo. This piure illusraes niely oh he empaion and he prolem wih arel agreemen. The reason he rms wan o ooperae is ha y reduing oupu relaive he equilirium hey oh an gain. The prolem is ha if hey ry o do his, here is always a empaion o defe sine he rms are no playing es responses The Sakelerg Model naural variaion on he Courno model is o ask wha would happen if one rm would make is deision efore he oher rm. Then he players are sill (in he duopoly version) rms 1 and 2 and heir pro funions are he same, u: 6

7 y 2 6 Firm 1 s Bes Response Firm 2 @ 3 - Figure 3: Carel would e Beer for Firms han Courno Compeiion sraegy for rm 1 is some 0 sraegy for rm 2 is oningen plan ha spei es he oupu as a funion of : We wrie r( ); where r is for response. Exaly as in he example wih he Bale of he sexes wih rs mover advanage here will e los of Nash equiliria in his game. owever, he mos ineresing equilirium is he equilirium where rm 2 ehaves opimally afer any hisory of play, ha is he equilirium where rm 2 would hose an opimal quaniy no maer whih quaniy rm 1 piks. This is he redile Nash equilirium whih we refer o as he akwards induion equilirium The Followers Prolem In he akwards induion equilirium he follower ( rm 2) should maximize pro s given any hoie of ; ha is rm 2 solves max y 2 (a ( + y 2 ) ) y 2 7

8 Bu his is jus like he Courno prolem ha deermines he es responses whih we already solved when analyzing he Courno model. Wihou any furher alulaion we hen know ha he (dynami) response of rm 2 mus e in a akwards induion equilirium. r( ) = The Leaders Prolem Now, exaly as Brue would foresee ha if he moved rs and wen o he game raher han he opera, he leader rm in he Sakelerg model an gure ou wha he follower will do given any quaniy hoie y he leader. The leaders prolem hus akes ino aoun ha y 2 = r( ) so he leader solves or, afer susiuing he response of rm 2 max a = max max (a ( + r ( )) ) + a y1 2 1 = max 2 ( ) I urns ou ha his is he monopoly prolem, so he soluion is o se (hek if neessary!) y1 = This oinidene wih he monopoly oupu is no a general feaure of he Sakelerg Model u has o do wih he geomery on he linear demand and os funions, u he ovious onsequene is ha when he leader akes he opimal responses y he follower ino onsideraion, i inreases is oupu relaive o he Courno model. The equilirium is y 1 = y 2 = r (y 1) = = 1 2 = 4 8

9 y 2 6 Firm 1 s Bes Response 3 4 Nash equilirium In Courno Game Bakwards Induion Equilirium in Sakelerg Game + Firm 2 s Bes response 3 - Figure 4: Equilirium in Sakelerg Model The mos useful hing o undersand his is from Figure 4. The key insigh is ha when he leader akes he response funion raher han a given (onjeured) oupu ino onsideraion, hen he rm will pik he es poin on he followers es response. Bu he es poin is jus a angeny eween he isopro and he es response and sine he slope of he isopro going hrough he Nash equilirium in he Courno game is zero i should e lear from he piure ha his means ha he leader- rm will inrease he oupu relaive he simulaneous model, whih in urn implies ha he follower will redue is oupu. Sine he leader ends up on a higher pro level (isopro loser o he monopoly oupu) here is a rs mover advanage in he model Berrand Compeiion In he Berrand model everyhing is as in he Courno model exep ha rms hoose pries insead of quaniies. Noe ha If p 1 < p 2 hen all onsumers go o rm 1 ha will hen sell q(p 1 ) unis while rm 2 sells nohing. 9

10 If p 1 > p 2 hen all onsumers go o rm 2 ha will hen sell q(p 2 ) unis while rm 1 sells nohing. If p 1 = p 2 hen he rms spli he onsumers, geing half of hem eah. Le q(p) = Bp e he (dire) demand. Now noe ha: 1. Neiher rm ould se a prie elow in equilirium sine he lowes prie rm would hen would make a loss. If rm 2 would se a prie p 2 > ; hen he pro of rm one would e (p 2 ) ( Bp 2) 2 0 if p 1 > p 2 > 0 if p 1 = p 2 (p 1 ) ( Bp 1 ) if p 1 < p 2 ene no rm ould harge a prie aove in equilirium eiher sine y underuing he oher rm y a small amoun (a penny) he rm wih he prie a penny lower would ge he whole marke raher han jus half of i. ene he equilirium is for oh rms o harge a prie equal o marginal os, so wih prie ompeiion wo rms are su ien o generae he ompeiive ouome. While i may seem ouner-inuiive ha he rms don have any marke power he siuaion is very muh like an auion where wo agens value an oje equally high. Thinking of i ha way i is no ha surprising ha he equilirium ids have o e he value of he oje sine if he agens would id elow heir values, one of hem ould win for sure and sill ge some surplus ou of he ouome while he oher would ge nohing. 10