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Economic Theory

by: April Jerde

Economic Theory ECON 711

April Jerde
GPA 3.6


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This 9 page Class Notes was uploaded by April Jerde on Thursday September 17, 2015. The Class Notes belongs to ECON 711 at University of Wisconsin - Madison taught by Serrano-Padial in Fall. Since its upload, it has received 37 views. For similar materials see /class/205146/econ-711-university-of-wisconsin-madison in Economcs at University of Wisconsin - Madison.


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Date Created: 09/17/15
Econ 711 Notes on Market Powerquot Ricardo Serrano Padial November 3 2008 This notes cover three different models of imperfect competition monopoly Bertrand and Cournot oligopolies 1 Monopoly There is a single rm the monopolist with strictly increasing cost function Cq facing a demand function x foojl a R1 which is strictly decreasing Let the inverse demand function be pq x 1q The monopolist solves mix pQq 7 C 1 Note that the objective function incorporates both the effect of the mo nopolist choice on market prices and the market clearing condition In ad dition it implicitly assumes that consumers are anonymous so that the mo nopolist cannot charge different prices We make the following assumptions i and are continuous and twice differentiable at all q gt 0 we can use rst and second order conditions ii p0 gt c 0 the monopolist wants to produce a positive quantity iii quo 6 000 st pq0 c q0 and p q0 lt c q0 unique socially optimal outcome1 This notes are based on different sources MWG chapter 12 David Millerls lecture notes for UCSD lst year micro and some papers mentioned in the notes Comments are welcome especially those regarding errors and typos 1The last condition implies that the inverse demand and marginal cost functions actu ally cross and only once at 40 rather than touching tangentially in which case qo would not be the social optimum iv p qq 2p q lt c q for all q gt 0 second order condition is satis ed uniqueness of maximum Proposition 1 If i iii are satis ed there is a solution ifquot gt 0 to 1 which satis es p qmqm pm C qm 2 Moreover this solution is unique if iv is also satis ed Proof Existence part By assumption iii it must be that pq lt c q for all q gt go since p and 0 cannot cross more than once But this implies that any q gt go pro ts will yield lower pro ts than go Therefore the monopolists solution if it exists must lie in 0q0 Since 0q0 is a compact interval and the objective function is continuous a solution exists By assumption qm needs to satisfy the FONC p qmqmpqm S c qm with equality if q gt 0 Finally assumption ii guarantees that q gt 0 Uniqueness part The second order locally suf cient condition is p qmqm 2pqm S c qm Assumption iv guarantees that it is satis ed with strict inequality globally which means there is a unique solution I The necessary condition for an optimum is the familiar requirement that marginal revenue equals marginal cost In the typical case of p qm lt 0 we have that pqm gt c qm and therefore q lt 10 since is strictly decreasing This implies that since pq gt c q for all q E qmq0 by iii aggregate surplus is strictly smaller than the socially optimum The welfare loss deadweight loss of monopoly is given by O pq e c qldq The intuition for this result is the following the rst order condition shows that at go a small reduction in the quantity produced causes an increase in the price on all the units the monopolist sells the p qq term while it causes only a small decrease in the amount of sales the pq term The former is a rst order effect since it is a small change in price multiplied by a large quantity while the latter is second order a small difference between price and cost multiplied by a small change in quantity 2 Oligopoly In this section we review two canonical models of oligopoly the Bertrand oligopoly7 in which two or more rms compete for the market demand by choosing prices and the Cournot oligopoly7 in which they compete by choos ing quantities The modern treatment ofthese and other models of imperfect competition relies on the use of game theory to analyze rms7 strategic in teraction The treatment given here follows a decision theoretic approach and it is intended to give you an understanding of the connection between perfect and imperfect competition Price competition Suppose that there are J 2 2 rms with access to the same constant returns to scale technology7 facing a market demand satis es the usual assumptions continuous7 strictly decreasing in 7005 and with xc gt 07 where 0 represents constant unit costs Each rm j chooses his own price 10 taking other rms7 prices as given The rms with the lowest price attract all the demand7 splitting it evenly when two or more choose the same lowest price That is7 rm7s j sales are given by if pj min010J7 W W m I 17 quot397 p1p 71 pl 0 ifpj gt minp1pJ The maximization problem for each rm is then given by maxp 7 cxjp1pj 17120 Proposition 2 In my equilibrium there are at least two rmsj audl such that 19 pl minp1pJ c This remarkable result implies that with just two rms with access to the same constant returns to scale technology7 price competition leads to the competitive outcome Proof Consider two cases a there is a rm j such that c S 19 lt ph for all h 31 j b there are at least two rms such that j and l such that pj pl minp1pJ gt c In case a if 10 0 rm j can increase pro ts by choosing a price 10 E cmin101 101101 10J On the other hand if10 gt c any rm h 79739 can earn strictly positive pro ts by choosing a price 10 E c10 Therefore a cannot be an equilibrium in which all rms are maximizing pro ts given the price decisions of the other rms In case b any rm h 1 J can earn higher pro ts by choosing a price 10 6 010 To see why consider without loss of generality rm l which C WU 2 wlt along with rm j enjoys the highest pro ts 10 7 If rm l charges 10 76 its pro ts would be 10 7 87 cx10 78 gt 10 7 c if for small enough 8 gt 0 It remains to be seen that the above price vector is in fact an equilib rium When there are at least two rms charging a price of c all rms get zero pro ts and none can do better by choosing unilaterally a different price Therefore all of them are maximizing pro ts I The above result deals with the special case of constant returns to scale In general when we have strictly convex costs there are multiple symmetric equilibria ranging from prices above the competitive outcome to having too much77 competition leading to prices below the competitive outcome This result is due to Dastidar 1995 see also a recent paper by Weibull 2006 In a symmetric equilibrium all rms charge the same price 10 In this context individual pro ts are given by Let also 7T7 10 10x10 7 cz10 be the monopoly pro ts We make the following assumptions a is continuous twice differentiable and strictly convex with 00 0 b 7T and are continuous and quasiconcave c 10 gt c 0 condition analogous to assumption iv ofthe monopoly model Assumption a implies that the industry pro ts when all rms charge the same price 10 are bigger than monopoly pro ts at 10 To prove it we use Jensen7s inequality which states that a strictly convex function f satis es fozx 17 00y lt 04fz17 afy for all 04 6 01 and all my in the domain of Since is strictly convex we can use Jensen7s inequality to show that C95P 1 C95P 1 C0 7 cilt J which requires 7T 10 lt Jig10 for all 10 gt 0 Let 10 be the socially optimal competitive price ie 10 c and 10m gt 10 the price at which monopoly pro ts are maximized Proposition 3 There eists a non degenerate interval of symmetric equilib rium prices giveri by 101102 where 101 arid 102 satisfy respectively 7T7101 0 arid 7T7102 max07rm102 Moreover 101 lt 10 lt 102 lt 10 Under strict convexity of costs this striking result tells us that price competition can lead both to excessive competition 10 6 10110 and to too little competition 10 E 10102 The intuition behind this result is the following Imagine a situation in which all rms choose the same price 10 A given rm j has two alternatives either charge a higher price so that it gets zero pro ts its sales are zero or charge a slightly lower price than 10 in which case rm j gets close to monopoly pro ts 7T7quot10 So for 10 to be a symmetric equilibrium it has to happen that individual pro ts satisfy 7T710 gt 0 otherwise rm j will charge a higher price to avoid a loss and 7rj10 2 7T7 10 otherwise rm j will do better by charging a price slightly lower than 10 Due to the convexity of costs and strict concavity of pro ts it turns out that 7T710 2 max07rm10 in an interval of prices 101102 that includes the socially optimal price 10 and 7T710 lt max07rm10 outside that interval The proof of this result is an optional reading beyond the scope of the class so you can skip the rest of this section if you wish and go directly to Quantity competition Before proving the proposition we state a useful lemma which establishes that monopoly pro ts rise faster than oligopoly pro ts for prices below 10m2 It is due to the strict convexity of 2This type of property is called singlecrossing pro t curves cross at most once in 0 10m and you will see many economic models in which it plays a key role especially in game theoretic models Lemma 1 single crossing For all p E 0pm and all p lt p 791 7 7T1 19 lt 7MP 7 WWW Proof Let AWp p 7T7p 7 7Tjp and A mp710 Wmlt10 MCDl First notice that by the quasiconcavity of 7T and the fact that p lt p lt pm we have that Awmp p gt 0 and thus pxp 7p p gt 7cxp 7czp Also notice that zp gt p which implies that cp 7 gt ME 7 CLJPN given that is increasing and strictly convex3 Using both inequalities we can show that M Wp p 7 Malp p 7 1 7 gtlpzp 7 p zp l 60600 7 CWM 9609 961 7 c J gt 7 7 gt lcltzltpvgt 7 cltzltpgtgtl 7 W gt 7 4 gt 0 Proof of Proposition 3 First we show that 7T7p lt 7T p for all p 2 pm Since p is strictly decreasing and continuous there exists p gt p such that zp In addition by assumption b Wmpm gt 7T p Therefore a m m a mp7quot gt mm 7 22 7 a gt p 7 a 7 mp This inequality implies that p gt pm cannot be a symmetric equilibrium Lemma 1 in Weibull 2006 since rm 239 can increase its pro ts by offering pm and getting all the demand Next notice that the necessary and suf cient condition for p 3 pm to be a symmetric equilibrium is 791 2 max0 7T p Proposition 1 in Weibull 2006 Fix a price p 3 pm satisfying this inequality Consider rms 239 alternatives to charging p It can charge a price p gt p which implies 7Tjp 0 or a price p lt p which yields monopoly pro ts 7T p Since i increasing at all p lt pm by strict concavity we have that supplt17 Wmp in 3To see why notice that being increasing and strictly convex implies that for all 12 gt Caa7cab b a 6 01 and all a gt b gt 0 we have that 6 7T p So the best alternative to charging p will yield max07 7rmp7 which is lower than what rm 239 gets in a symmetric equilibrium with price p Now we have to show that there exists a price p1 lt plt pm such that 7T7p1 0 satis es the above condition We have that 790 74 lt 0 ln addition7 notice that 7Tjp f0mpJp 7 c qdq 7 00 gt 0 given that 00 07 c is increasing and 13 c By the continuity of 7T there exists a price pl 6 0710 such that 7T7p1 0 For p1 to be a symmetric equilibrium we need 7T7p1 2 max07 7T 101 But we know that Wmp1 lt J73 p1 07 so the equilibrium condition reduces to 7T7p1 2 07 which is satis ed with equality Since 7Tj is quasiconcave then it is increasing at p1 7T7p lt 0 for allp lt p1 so p cannot be a symmetric equilibrium The next step is to show that there exists a price pg 6 pipm such that 7T7p2 max07rmp2 As we have shown above7 731 lt 7rmpm7 and thus pm cannot be a symmetric equilibrium At the same time7 we have that 7Tp gt Wmp given that 10 lt c q for all q gt M and thus 107 WW 7 MW 19 7 C Qldq lt 0 1PJ Both inequalities and the continuity of 7T and imply that p2 exists lf 7Tjp2 0 then 7rjp lt 0 for all p gt p2 by 7T being quasiconcave given that is decreasing after reaching a maximum7 which so happens at prices below 1024 On the other hand7 if 7T7p2 Wmp2 then 7rjp lt 7T7 p for all p 6 10210 by the single crossing property Thus7 p gt p2 cannot be a symmetric equilibrium Finally7 the single crossing property and the quasiconcavity of 7T guar antee that any p 6 101102 satis es the condition for symmetric equilibrium I Quantity competition In this section we look at the case in which rms choose output rather than prices There are J 2 2 identical rms with cost function c7 facing the inverse demand function We make the following assumptions 47rji reaches a maximum in 101172 given that pijp gt 0 and 7Tj p1 7Tj p2 0i 7 is continuous7 twice differentiable and strictly convex at all q gt 0 is continuous7 twice differentiable and strictly concave at all q gt 0 i C M ii p0 gt c 0 the monopolist wants to produce a positive quantity iii there exists a unique Q0 E 000 st pQ0 C Q70 unique social optimum Let q Q1qJ and Q 271 qj Firm j7s pro t function is given by MW PQQj Cqj m Qh Qjq739 Cqjl 3 hie Our assumptions imply that the FONC is also suf cient and that there exists a unique interior maximum M Qh QM m Qh q CM h 39 h 39 Consider Q satisfying Q Q 7 7 pQJ pQ CJL then a quantity vector in which q QVJ for all j is a symmetric equilib rium Such quantity Q exists because the pro t function is strictly concave It automatically follows that q satis es the FONC given 2 q Q 7 q h The next result shows that the supply in the Cournot oligopoly lies between the monopoly output and the market supply under perfect competition Proposition 4 Q0 gt Q gt g Proof At Q q the LHS of 2 is greater than the LHS of 37 given that q 7 and p lt 0 Thus7 by the strict concavity of and the strict convexity of c7 q gt At Q Q0 the LHS of 3 is lower than pQ0 Thus7 01 lt 01 so Q lt Q0 I The intuition for this result is the following By unilaterally reducing out put from the social optimum there is a rst order gain in the price of all the inframarginal units sold while a second order loss due to the lower marginal quantity sold However this incentive is lower than for the monopolist since its quantity is only 1J times the market supply This implies as shown in the next result that as J a 00 the incentive to reduce output goes to zero Proposition 5 Cournot convergence Jlim Q Jlim Q0 Proof From the FONC of the Cournot rm we have that 7 Q7 7 Q M J M cltJgt As J a 00 the LHS goes to zero But the RHS is zero only at the social optimum assumption iii I References DASTIDAR K G 1995 On the Existence of Pure Strategy Bertrand Equilibrium7 Economic Theory 5 19732 WEIBULL J W 2006 Price Competition and Convex costs77 SSEEFI Working Paper Series in Economics and Finance no 622


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