Monopoly and Oligopoly in homogeneous product markets. TA session 1 Lorenzo Clementi

Transcription

1 Monooly and Oligooly in hoogeneous roduct arkets. TA session Lorenzo Cleenti

2 Roada: Hoogeneous Products The aggregate deand Static Monooly: Ch. Tirole Oligooly theory in Hoogeneous roduct industries: section II Ch.5 Tirole Non coetitive behavior: Collusion section II Ch.6 Tirole

3 The aggregate deand Microeconoic theory starts fro individual references and through the utility axiization roble, finds the individual walrasian deand function x,w. The roerties of x,w are well known HD0 in,w, Walras law, convexity and uniqueness, Roy s identity,. In IO we usually focus on a few roducts artial analysis and collase all other goods into a generic outside good. Suose we focus on one good, good. Suing u all the individual deands of consuers we obtain the aggregate deand for good. Suose that there are I consuers, each of the with rational reference relations this ust be: D, 0, w,..., wi ; Z 0 = i xi,, wi ; Z This aggregate deand deends not only on rices but also on the secific wealth levels of the various consuer as well as other deand shifters When we estiate deand in IO we usually ake functional for assutions on D. It is iortant to kee in ind that such assutions ust be consistent with individual deand 3

4 The aggregate deand As an exale with differentiated roducts, consider Hotelling s odel of horizontal differentiation artial analysis focusing on two goods Hotelling rooses to consider a linear city of length, with consuer distributed uniforly along the city. Two shos, located at the two ends of the city, sell the sae hysical good. For soe characteristics, the otial choice at equal rices deends on the articular consuer and tastes vary in the oulation. Consuer have unit deands: consue zero or one unit of the good, and they ust ay a transortation cost to go to the sho. The individual deand will deend on the rice difference charged by the two shos and on the transortation cost. If the rice difference between the two shos does not exceed transortation cost along the whole city and if rices are not too high there exist a consuer that has a location that ake is indifferent between buying fro sho or. Then consuers can be aggregated resect to their location, and we can find the deand faced by each sho. 4

5 The aggregate deand 5

6 The aggregate deand The generalized rice of going to sho is +tx. The generalized rice of going to sho is +t-x. If ŝ denotes the surlus enjoyed by each when he is consuing the good, then the utility of the consuer located at x is ŝ - - tx if he buys at sho, ŝ - t-x. If - <t and rices are not too high there exist a consuer with a location x* who is indifferent between buying fro sho and buying fro sho: + tx* = + t x* x*, = + t t Then the deands are D, =Nx*, and D, =N[-x*, ] If - >t, sho has no deand. Sho has deand D, =N, if < ŝ-t and deand D, =Nŝ-, if > ŝ-t The third case is that every sho has a local onooly ower and the arket is not covered. 6

7 The aggregate deand Why do we care about aggregate deand? It allows us to solve the fir s rofit axiization roble and to find equilibriu rices and quantities, and thus erfor all kinds of welfare analysis or counterfactual olicy exercises 7

8 Monooly theory Main fraework Inverse elasticity rule Second order conditions The revenue function Proerties of onooly rice The Deadweight loss 8

9 Monooly theory Main Fraework Assutions:. Goods roduced by onoolist are given. Monoolist charges the sae rice er unit of good for each good roduced. 9

10 Define: Monooly theory Inverse Elasticity Rule q=d, Deand D <0 ; Deand is differentiable and decreasing in rice. =P q, Inverse Deand with P <0 ; Price is decreasing in quantity deanded. C q, Differentiable cost function, C >0 is differentiable and increasing in outut. The Monoolist s Profit Maxiization Proble hence is: Max,q q C q s. t. q = D Plugging in the constraint we get rid of one unknown: Max D C D F.o.c. : D + D c D D =0 0

11 Monooly theory Inverse Elasticity Rule By rearranging and defining as the onoolist rice we get: Rewriting the equation, we isolate the Lerner index left side: Furtherore: The Lerner index easures the arket ower of the fir, i.e. the ability of the fir to aintain the rice above the arginal cost. In erfect coetition the Lerner Index is zero. ' ' D D D c = ' ' D D D c = ' D D = ε

12 Monooly theory Second order conditions Then we coute second order conditions to check that our solution corresond to a axiu: S.o.c. : D + D C D [D ] - D C D< 0, for every Then Foc is sufficient condition.

13 Monooly theory The Revenue Function Equivalently, let q =D be the onoolist outut we can solve the roble w.r.t. onooly quantities: Max q P q q C q F.o.c. q: P q q + P q C q=0 which yields: MR q =P q q + P q = C q 3

14 4 Proerties of Monooly rice Monooly rice is a non decreasing function of arginal cost. Consider two alternative cost functions for a onoolist C, C Assue differentiability and C q> C q for q>0. If C q If C q hence q C q q C q q C q q C q [ ] [ ] 0 q C q C q C q C

15 Proerties of Monooly rice Or since differentiable: q q [ C ' x C ' x ] dx 0 Since by assution C x> C x for all x, hence q >q Monooly rice is a non decreasing function of arginal cost 5

16 Proerties of Monooly rice the elasticity ε is greater than at. In fact: And c ' D D = = ε D ' ε = > c ' D 6

17 Monooly theory the Deadweight loss 7

18 Oligooly theory A refresher on Gae Theory Bertrand Coetition Cournot Coetition Bertrand with caacity constraints Alication: erger in Cournot 8

19 Gae Theory a refresher Before going into odels let s have a brief review of Gae Theory. Consider two firs i=, with π i a i, a j twice continuous and differentiable, a i є R Nash Equilibriu: π i a* i, a* j > π i a i, a* j, for every a i i=,. f.o.c. π ii a* i, a* j =0 s.o.c. π iii a* i, a* j <0 Assue strict concavity π iii a* i, a* j <0 for every a i, a j then f.o.c. is sufficient for Nash Equilibriu. Let s define the reaction function R i a j s.t.: a i =R i a j is unique by strict concavity. π ii R i a* j, a* j =0 9

20 Gae Theory a refresher Another way to define the N.E.: a* i, a* j s.t. a* i = R i a* j Hence by differentiating π iii R i a j, a j R i a j + π iji a i, a j =0 i.e. i π ij Ri aj, aj R ' a = π R a, a i j i ii i j j Therefore sign R i a j = sign π iji a i, a j 0

21 Gae Theory a refresher A useful concet at this stage is distinguish between strategic coleents and strategic substitutes. We can deterine it by looking to the sign of the derivative of the Reaction Function. These two concets are used for coarative statics.

22 Oligooly theory - Bertrand Short run rice coetition Consider two firs roducing identical goods with constant arginal cost, c. D, total deand. Deand for fir i is d i i, j There is a unique equilibriu i = j =c Bertrand Paradox > = < = j i j i i i j i i i j i i if if D if D d 0,

23 Oligooly theory - Cournot duooly This is a one-stage gae in which firs choose their quantities siultaneously. Profit function for fir i can be generally defined as π i q i, q -i = q i Pq i +q -i -C i q i Each fir axiizes its rofit given the quantity chosen by the other fir. Assuing that the rofit function π i is strictly concave in q i and twice differentiable, we can axiize firs rofit: Max qi π i q i, q -i = q i Pq i +q -i -C i q i f.o.c. Pq i +q -i - C i q i + q i P q i +q -i = 0 This FOC ilicity defines q i = R i q* -i, where R i is fir i s reaction curve: π i i R i q* -i, q* -i =0. Assution: fir i s arginal rofit is decreasing in the other firs quantity. This ilies that reaction function are downward sloing strategic substitutes. 3

24 Oligooly theory Cournot duooly Dealing with Monooly we defined the Lerner Index rice-cost argin. Now, let α i be the fir i s arket share: qi α i = Q L i = P C P ' i And the inverse elasticity of deand: P ' Q ε P In the Cournot odel, the Lerner Index is equal to: P C' i qi qi Q qi P' Q αi Li = = P' Q = P' Q = Q= P P P Q Q P ε Furtherore we can define the index of concentration, known as Herfindhal Index: H n Σ i = α i 4

25 Oligooly theory - Cournot 5

26 Oligooly theory Cournot, a duooly exale Quantity coetition Consider two firs roducing identical goods Deand D=- C i q i =c i q i P q i,q j =- q i -q j Individual fir otiization, taking action by other fir as given Max qi π i q j, q j =q i - q i -q j - c i q i f.o.c. q i -q i + -q i -q j c i =0 q i =- q j c i / s.o.c. q i -<0 6

27 Oligooly theory Cournot, a duooly exale Hence we can derive the Best Resonse for Fir and q c q = = BR q And the Cournot equilibriu: q i ci + c = 3 j q c q = = BR q q q c L i = q q i j i Finally the Lerner index: and the Herfindhal: i j H = q j i + q Q Hence for the duooly case: c + c c + c c c Q= q+ q = + = It follows that in the case in which the firs are identical the total quantity roduced is Q= c 3 7

28 Oligooly theory - Cournot n firs n identical firs with cost function Cq i =c q i, i=,...,n Market Deand PQ=-Q, Q=Σ i=,,n q i Profit π i q,, q n = PΣ i=,,n q i q i -Cq i Individual rofit axiization f.o.c. qi P Q+ P Qq i =c Or equivalently -Q- q i = c Hence individual quantity roduced is q i = -Q- c Suing over i, Q=n-Q- c. Equilibriu: n Q* = c n + c q* = n + + nc* * = n + c π = n + 8

29 Oligooly theory - Cournot We can also find the relation between the two indices. In fact the average weighted Lerner index is equal to the Herfindahl index divided by the rice elasticity of deand, i.e. Σ i α i L i =H/ ε. In a onooly, H =. It can also be couted to hold for a general n-fir Cournot oligooly but requires a bit of calculus. q * = c n + In fact and q* i α i = = Q* n Then. n Q* = c n + Hence H tends to zero as n increases, since is the su of the squares of arket shares. 9

30 Oligooly theory Bertrand with caacity constraints Two stage gae. First stage: Each fir siultaneously chooses its caacity. Second stage: Each fir siultaneously chooses its rice. Efficient Rationing Rule. Suose < and q caacity constraint of Fir with q <D Residual deand for Fir is D q if D > q D = 0 otherwise This rationing is called efficient since it axiizes consuer surlus 30

31 Efficient Rationing Rule Fir faces translated a deand curve 3

32 Oligooly theory Bertrand with caacity constraints Model assutions Q=D =- = -q -q In stage, firs choose caacity siultaneously q i Unit cost of caacity acquisition c 0 є [¾, ] In stage, firs siultaneously choose rices Mc of roduction c = 0 c = We aly the efficient rationing rule Solution first focus on second stage and solve for the equilibriu in the second stage gae given the caacity choices in the first stage. We can restrict our attention to q i,q j є [0, ⅓] Clai: *=-q -q is the unique equilibriu if if q i q i > q q i i Let s exlore the two cases: if i <* if i >* 3

33 Oligooly theory Bertrand with caacity constraints Stage : Fro the solution we know that the rofit of firs takes the for of the Cournot rofit π i = q i q j q i and we can roceed with the usual axiization. Fro the Cournot solution we ve derived earlier we get q i = q i =- q j / This odel shows that in this case everything is as if the two firs ut oututs equal to their caacities on arket and an auctioneer equalled suly and deand. The difference is that firs choose the arket-clearing rice theselves. Caacity constraint softens rice coetition. Check also Kres and Scheinkan

34 Alication to erger in Cournot Cournot case At least 3 identical firs with c=c< Linear deand Q=- n Q = Σ q j= j Q i = Q q i Before erger Max qi π i =q i - Q -i +q i -c f.o.c. q j q j = - Q -i -c/ By syetry q j =q for all i Hence q= [-n- q c]/ That yields c q = n + π i c = n + 34

35 35 Alication to erger in Cournot Merger of two firs Suose that F and F erge given q 3,,q n Max q, q π + π =q - Q-c + q - Q-c =q +q - q -q -q 3 - -q n Let q +q =q + Then ax q + -q + -q 3 - -q n -c Merger of two firs it s like reducing the arket of one fir. The erger aradox: with constant arginal cost erger is less rofitable. Coarison Profit of the two firs Profit of the other firs 3 Industry rofits > + n c n c < + n c n c < + n c n n c n

36 Oligooly theory Final Rearks Strategic interactions are key to understanding oligoolistic arkets In Bertrand we obtain a erfect coetitive outcoe. We can ove away fro the coetitive outcoe with the Cournot odel or with the introduction of caacity constraints. There is another ossibility: the ossibility of collude and ake a cartel and axiize the aggregate rofits. Hence we ust analyze collusion and incentives to do it. 36

37 Collusion Roada Collusion: econoic ersective Model Basic results and Market concentration Inforation lag Fluctuating deand Green and Porter

38 Collusion Econoic Persective Collective exercise of arket ower, i.e. a situation where firs raise rice beyond what would be consistent with short ter rofit incentives. Hence, firs have an incentive to take advantage of their coetitors in the short ter. Default ay be avoided if it triggers a eriod of intense coetition during which the benefits of the collusive outcoe are lost retaliation. It is conceivable that collusion could arise fro arket interactions without concertation. Exlicit agreeents or concertation cannot be enforced by law because they are illegal Hence, only self interest can suort collusive outcoes, irresective of whether there is concertation or not. No fundaental distinction between cartels and tacit collusion. But concertation ay hel in suorting collusive outcoes. 38

39 Collusion The odel Two firs roducing erfect substitutes with the sae constant arginal cost, c. Bertrand coetition for T eriod finite/infinite T t i Σ t = it jt 0 δπ, is the resent discounted value of F i s rofit δ = discount factor H t = 0, 0,, t-, t- history at tie t Perfect equilibriu: for any history H t at date t, F i s strategy fro date t on axiizes the resent discounted value of rofits given F j s strategy fro date t on. T finite, finite horizon Bertrand solved T ties: t=t equilibriu it =c for every i t=t- equilibriu it- =c for every I.. Fro now on T is infinite. 39

40 Collusion Basic Results and Market concentration Monooly outcoe Let the onooly rice Trigger strategy: Payoff of the following strategy at t = 0 Fi charges fro t>0 it = if H t = *, *,..., it = c forever π Payoff uon the best deviation π [IC] -δ - π / > π δ> ½ π + δ + δ... = δ π Folk Theore Any air of rofits π,π s.t. π >0, π >0, π +π < π there exists a threshold δ* large enough such that these air of rofits can be sustained as art of a collusive equilibriu. For n> collusion harder [IC] -δ - π /n > π 40

41 Collusion Inforation lag A fir deviation is detected by its rival only two eriods after t t+ t+ F c F -ε -ε c If n= [IC] -δ - π / > π + δ π 4

42 4 Collusion Observed fluctuating Deand A fir decide to collude or not deending fro the trend boo, recession. During boos they deviate ore. Deand iid over tie with D >D At each eriod, each fir learns the current state of the deand before choosing rice. We look for, s.t. Each fir charges s when s=,, is an equilibriu The exected rofit of each fir along the equilibriu ath is not Pareto doinated by other equilibriu ayoffs. Let V=exected resent rofit along the equilibriu ath D high q D low q = = δ δ + = + Σ = = c D c D c D c D V t t

43 Collusion Fluctuating Deand Trigger strategy: if deviation each fir will charge =c forever. Case Full collusion outcoe: they charge onooly rice with Deand s : s, π s Then π + π V = 4 δ [IC s ] ½ π s +δv >π s +0 with state s=,. IC is ore difficult to satisfy than IC [IC s ] ½ π s + δv >π s iff δv >½ π s While IC δ >δ 0 = π /π + 3π Where ⅔> δ 0 >½ 43

44 44 Collusion Observed fluctuating Deand Case : Can we sustain soe level of collusion when δ < δ 0, where δє[½, δ 0 ] IC π + δv >π iff kπ > π, with k=δ/-3δ IC is binding. With low deand they charge onooly rice. With high deand rice falls below onooly rice. We ay observe countercyclical rice oveents.,.. ax IC IC t s V = + δ π π.. ax, k IC k IC s t π π π π π π +

45 Collusion - Unobserved deand Let s consider a silified version of Green-Porter odel. Two firs choose rice every eriod The goods are erfect substitutes and are roduced at constant arginal cost, c, so consuer all buy fro the low-rice fir. Deand is slit in halves if both firs charge the sae rice. Two ossible state of nature: With robability α there is no deand for the roduct sold by the duoolist With robability -α there is a ositive deand D Realization of deand are indeendently and identical distributed. A fir that does not sell at soe date is unable to observe wherther the absence of deand is due to the realization of the low deand state or to its rival s lower rice. 45

46 Collusion - Unobserved deand It s always coon knowledge that at least fir observes zero sales: Either If there is absence of deand no fir akes rofits. Either If one fir undercuts rice, it knows that the other is going to ake no rofits. As usual there are collusion hase and unishent hase that will last T eriods. Punishent hase will last T eriods because zero sales could be triggered by the absence of deand and not by a deviation. In a non reeated gae or finitely reeated gae both fir charge the coetitive rice. Let V + resectively V - be the resent discounted value of a fir s rofit fro date t on, assuing that at date t the gae is in the collusive hase resectively, starts the unishent hase By stationarity V + and V - do not deend on tie, then: V + = -απ /+δv + +αδv - V - =δ T V + 46

47 Collusion - Unobserved deand The IC in this case are V + > -απ /+δv - +αδv - that states that no fir will undercut rice. Here is the tradeoff for each fir since if undercuts rice it gets π > π /, hence we need to deter undercutting that V - will be sufficiently lower than V +. Plugging the IC into the resent discounted value of collusive hase we get: δv + -V - > π /. By rearranging the resent discounted value of both hases we get: + α π / V = T + α δ α δ V = T α δ π / α δ α δ Finally substituting it into δv + -V - > π /, we get <-αδ+α-δ T+. T + Hence starting in the colluding hase the fir ust solve: ax V + s.t. <-αδ+α-δ T+ To axiize rofits we need to choose the sallest T that satisfies the incentive constraints, since longer unishent hase reduce the exected rofit. 47

48 Multiarket contact and collusion Bernhei and Whinston 990 identify a nuber of circustances, tyically ilying asyetries between firs or arkets, in which ultiarket contact facilitates collusion by otiizing the allocation of available enforcing ower between arkets with a rice coetition odel. They show that when firs and arkets are identical and there are constant returns to scale, ultiarket contact does not strengthen firs' ability to collude irrelevance result. In ultiarket case a deviation get unished in several arkets, but the fir can also deviate in all arkets at the sae tie by ooling of the IC across arkets. Three ain conclusions: Multiarket contact can have real effects; in a wide range of circustances, it relaxes the incentive constraints that liit the extent of collusion. Firs gain fro ultiarket contact by behaving in ways that have been noted in revious eirical discussions of ultiarket firs. This suggests that ultiarket contact ay indeed have effects in ractice. Even when ultiarket contact does have real effects, these effects are not necessarily socially undesirable. 48