Monopoly and Oligopoly

Monopoly and Oligopoly1. Third-degree price discriminationSince the market is separated into two groups, submarket, then we face two demand functions, )(11y p and )(22y p . Thus, profit maximization problem can be described as)()()(:21222111,21y y c y y p y y p Max y y +-⋅+⋅=πHence, FOC is)()(2111y y MC y MR +=, and )()(2122y y MC y MR +=Other situation: combination of these two separated markets)()()(:21212121y y c y y y y p Max y y +-+⋅+=+πHorizontal summation: )()()(1121121p p p p p p y y y ---=+=+= Optimal solution: )()(y MC y MR =2. Oligopoly behaviorA. Cournot model —— Simultaneously quantity settingConditions: market demand )(21y y b a p +-=, products are identical, marginal cost of these two firms are equal and constant,c MC MC ==21, they decide their output decision simultaneously without knowing other ’s decision advanced. Maximization problem:)(:111y c y p Max -⋅=π s.t. )(21y y b a p +-= )(:222y c y p Max -⋅=π s.t. )(12y y b a p +-=Solution:(1) Since 11211)]([y c y y y b a ⋅-⋅+-=π, then, its FOC is shown as021211=---=∂∂c by y b a y π, after rearranging, this yields the reaction function of firm 1, by b c a y 221--=(2) The same, we have the reaction function of firm 2by b c a y 212--=(3) Combine these two reaction functions, we have⎪⎩⎪⎨⎧-=-=b c a y bc a y 33*2*1B. Bertrand model —— Simultaneously price settingC. Stackelberg model (quantity leadership model)Conditions: market demand )(21y y b a p +-=, products are identical, marginal cost of these two firms are equal and constant,c MC MC ==21, however, firm 1 is quantity leader, who makes his output decision first, firm 2 is follower, who makes his decision with knowing firm 1’s decision. Maximization problem:)(:111y c y p Max -⋅=πs.t. )(21y y b a p +-= & firm 2’s reaction function )(122y f y =)(:222y c y p Max -⋅=π s.t. )(12y y b a p +-=Solution: backward induction(1) Since 22122)]([y c y y y b a ⋅-⋅+-=π, thus,022122=---=∂∂c by y b a y π, hence, firm 2’s reaction function is by b c a y 212--= (2) And 11211)]([y c y y y b a ⋅-⋅+-=π, substitute firm 2’s reaction function, then Iso-profit curves can be shown as)(2121111by cy ay --=πthus, optimal choice appears when0)2(21111=--=by c a dy d πSo that ⎪⎩⎪⎨⎧-=-=b c a y bc a y 42*2*1D. Price leadership modelConditions: market demand )(21y y b a p +-=, products are identical, marginal cost of these two firms are 2212 ,y MC c MC ⋅==α, however, firm 1 is price leader, who makes his pricing decision first, firm 2 is follower, who makes his decision with knowing firm 1’s decision.Traditional Method:Taking use of backward induction: Maximization problems of follower)(:2222y c y p Max -⋅=πFOC: 222)(y y MC p α==Thus, the reaction function of follow is α2)(22pp f y ==, in other words, it ’s supply function of the follower.Then we consider the maximization problem of price leader)(:1111y c y p Max -⋅=πs.t. )()(2121y y b a y y D p +-=+= Substituting α2)(22pp f y == into the above constraint, then, p b b a y )211(1α+-=, more generally, )(11p f y =, which is residual demand for firm 1. Then the maximization problem becomes))(()(:1111p f c p f p Max p-⋅=πThus, FOC is 0)(11111=-+dpdf df dc dp df p p f Hence,22*c b a p ++=αα, then 2)211(2*1c b b a y α+-=, and then αα424*2c b a y ++=Comparison: Stackelberg model VS. Price leadership modelHere, we try to make a comparison between stackelberg model and price leadership model, focusing on the essential difference with respond to solution process and outcome.Conditions of economic situation ；market demand )(21y y b a p +-=, products are identical, marginal cost of these two firms are 2211 ,y MC y MC ⋅=⋅=βα, however, firm 1 is decision leader, who makes his decision first, firm 2 is follower, who makes his decision with knowing firm 1’s decision. Case (1): Stackelberg model Solution:a. Follower ’s Maximization problem with knowing firm1’s output quantity decision)(:22202y c y p Max y -⋅=≥π s.t. )(12y y b a p +-=Thus, FOC is0222122=---=∂∂y by y b a y βπ Hence, the respond function of firm 2 is )(122y f y =β+-=b y b a y 212b. Leader ’s Maximization problem with expecting firm2’s responds)(:11101y c y p Max y -⋅=≥πs.t. )(21y y b a p +-=& firm 2’s reaction function )(122y f y =Substituting all constraints into maximization problem, this yields)()2(11111y c y b by a y b a -⋅⎪⎪⎭⎫⎝⎛+-+-=βπThus, FOC is 02222112111=⋅-+++--=∂∂y y b b b ab by a y αββπHence, αββαβ+++-+=)(2222*1b b ab a ab y , ))(22)(2(2222*2αββαβαβαββα+++++-++=b b b ab b a ab ab yc. Equilibrium))(22)(2(322222223*αββαβαβαββαβαβ++++-+++++=b b b b a ab ab ab ab ab pCase (2): Price Leadership model Solution:a. Follower ’s Maximization problem, knowing firm1’s output quantity and price decision)(:22202y c y p Max y -⋅=≥π s.t. )(12y y b a p +-=At this time, there are two rules that we don ’t know whether it is matched. FOC: 22)(y y MC p ⋅==β, or 12y bpa y --=Hence, there are two probable reaction functions.βpp f y ==)(22, or 1122),(y bpa y p f y --== If both of the above reaction functions are equivalent, then1y b p a p--=β, that is firm 1’s residual demand : p b b a p b p a y ⎪⎪⎭⎫⎝⎛+-=--=ββ111 b. Leader ’s Maximization problem with expecting firm2’s responds)(:111y c y p Max p-⋅=πs.t. p b b a y ⎪⎪⎭⎫⎝⎛+-=β111 FOC:011111121=⎪⎪⎭⎫ ⎝⎛+⎪⎪⎭⎫ ⎝⎛⎪⎪⎭⎫ ⎝⎛+-+⎪⎪⎭⎫⎝⎛+-=∂∂ββαβπb p b b a p b b a p Hence, )2)((22*βββαβαββa ab b b a ab ab p +++++=Thus, )2(2*1ββαβa ab b a a a y ++-+=, )2)((*2βββαβαβa ab b b a ab ab y +++++=Also there are two approaches to solving this problem, firstly, taking the place of max p and substituting max y 1, then, we haveαβαββ++=b b a y 2*1, )2)((22*βαββαβαββa b b b a ab ab p +++++=, )2)((*2βαββαβαβa b b b a ab ab y +++++=Case (3) Existing unmarketable goods in price leadershipOn the other hand, as goods in market are identical and over-supplied is reality, unmarketable goods occur inevitably. Therefore, unmarketable goods of market share areequal. Here we denote unmarketable proportion, 2121211)(y y x x y y p f ++=+=-η, hence, quantity of firm1’s unmarketable goods is 111x y y -=η, and that of firm2’s is 222x y y -=η. Now we can describe the maximization problems as follow:For firm2, )( 2222y 2y C x p Max -⋅=πs.t. 21122)(y y p f y x +==-η Thus, maximization problem becomes if constraint is substituted into the original one.)()( 2222112y C y y y p f p Max -⋅+⋅=-πHence, the FOC of the above problem is22112)()(y y y y b p a p +=-βSo that, firm2’s reaction function can be depicted as ),(122y p y ϕ= For firm 1, the leader, its maximization problem is)( 1111)y (p,1y C x p Max -⋅=πs.t. 21111)(y y p f y x +==-η and follower ’s reaction function ),(122y p y ϕ= Thus, we have )()( 1121111)y (p,1y C y y y p fp Max -+⋅⋅=-πs.t.22112)()(y y y y b p a p +=-βSolution: taking use of Lagrange method,⎪⎪⎭⎫ ⎝⎛+----+⋅-⋅=2211211211)()()(y y y y b p a p y C y y y b p a p L βλ Hence,0)()(22112=+--=∂∂y y y y b p a p L βλ Then, this yield22112)()(y y y y b p a p +=-β (1)022211=⎪⎪⎭⎫ ⎝⎛--+⋅-=∂∂βλb p a y y y b pa p L Then, this yield0211=-+βλy y y (2)0)()(21322122121=⎪⎪⎭⎫ ⎝⎛-+⋅-+⋅-=∂∂y y y y y y y b p a p y L λα Substituting (1) & (2), then,We have 21y y ⋅=⋅βα (1*)034)()1()(12221212212=⎪⎪⎭⎫⎝⎛++++-⋅-=∂∂y y y y y y y b p a p y L λ Substituting (1) & (2), then,We have 221213y y y y =+ (2*) Combining (1*) & (2*), thus,βαα+=3*1y , hence, 22*23βαβα+=yE. Cartel case。