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Chapter 3: Oligopoly Notes on Chapter 3: Oligopoly Microeconomic Theory IV 3º - LE-: 2008-2009 Iñaki Aguirre Departamento de Fundamentos del Análisis Económico I Universidad del País Vasco Microeconomic Theory IV Oligopoly Introduction 1.1. The Cournot model. 1.1.1. Duopoly. 1.1.2. Oligopoly (n firms). 1.1.3. Welfare analysis. 1.2. The Bertrand model. 1.2.1. Homogeneous product. 1.2.2. Heterogeneous product. 1.3. Leadership in the choice of output. The Stackelberg model. 1.4. Collusion and the stability of agreements. 1.4.1. Short-term collusion. 1.4.2. The stability of agreements under a finite temporal horizon and under an infinite temporal horizon. 2 Microeconomic Theory IV Oligopoly Introduction Non-Cooperative Game Theory is very useful for modelling and understanding the multi- agent economic problems characterized by strategic interdependency, in particular for analyzing competition between firms in a market. Perfect competition and pure monopoly (not threatened by entry) are special non-realistic cases. It is more frequent in the real life to find industries with not many firms or with a lot of firms but with a smaller number of them producing a large proportion of the total production. With few firms, competition is characterized by strategic considerations: each firm takes its decisions (price, output, advertising, etc.) taking into account or conjecturing the behaviour of the others. Competition in an oligopoly can be seen as a game where firms are the players. So we shall adopt a Game Theory perspective to analyze the different models of oligopoly. For each case, we shall wonder what game firms are playing (information, order of playing, strategies, etc.), and what the equilibrium notion is. An important difference between the games of the previous chapter and the games we shall solve in this chapter is that the former were finite games while the latter are infinite games. 3 Microeconomic Theory IV Oligopoly 1.1. The Cournot model 1.1.1. Duopoly (i) Context. (ii) Representation of the game in normal form. (iii) Notion of equilibrium. (iv) Best response function. Characterization of equilibrium. (v) Example. Graphic representation. (i) Context The Cournot model has four basic characteristics: a) We consider a market served by 2 firms. b) Homogeneous product. That is, from the consumers’ point of view, the goods produced by the two firms are perfect substitutes. c) Quantity competition. The variable of choice of each firm is output level. Denote by x12 and x the production levels of firm 1 and firm 2, respectively. d) Simultaneous choice. The two firms have to choose their outputs simultaneously. That is, each firm has to choose its output without knowing the rival’s choice. Simultaneous Choice does not mean that choices are made at the same instant in time. An equivalent context would be one where one firm chooses its output first and tthe other firm chooses its output afterwards but without observing the decision adopted by the first firm. In other words, sequential choice together with imperfect information (the player moving second does not observe what the first mover does) is equivalent to simultaneous choice. The inverse demand function is p(),x where px' ()< 0. As the product is homogeneous, the price at which a firm can sell will depend on the total output: p()xpxx= (12+ ). 4 Microeconomic Theory IV Oligopoly The production cost of firm i is Cxii( ), i=1,2. (ii) Representation of the game in normal form 1) i =1,2. (Players) 2) xi ≥ 0. Any non-negative amount would serve as a strategy for player i. Equivalently, we can represent the set of strategies of player I asxii ∈ [0,∞= ), 1,2. 3) The payoff obtained by each firm at the combination of strategies (,x12x ) is: Π=+−112(,xx ) px ( 1 xx 21 ) Cx 11 ()⎫ ⎪ ⎬ ≡Πii(,x xpxxxCxijji j ) = ( i + ji ) − ii (), , = 1,2, ≠ . ⎪ Π=+−212(,xx ) px ( 1 xx 22 ) Cx 22 ()⎭ (iii) Notion of equilibrium. Cournot-Nash equilibrium It is very easy to adapt the definition of the Nash equilibrium in the previous chapter to this new context. *** * * * “ sss≡ (1 ,..,n ) is a Nash equilibrium if : Πi(si , s−i) ≥Πi(si, s−i) ∀si ∈Si,∀i,i = 1,...,n .” For the Cournot duopoly game we say: ** ** * “ (,)x12x is a Cournot-Nash equilibrium if Πii(,x xxxxijji j )≥Π ii (,) j ∀ i ≥0,, = 1,2, ≠ ”. The second definition based on best responses proves more useful. *** ** “ sss≡ (1 ,..,n ) is a Nash equilibrium if: sMRsiiiii∈∀=()− ,1,.., n where * ' ' * * ' MRi (s−i ) = {}si ∈Si : Πi (si ,s−i ) ≥Πi(si ,s−i ), ∀si ∈Si ,si ≠ si . ”. For the Cournot duopoly game: 5 Microeconomic Theory IV Oligopoly ** ** “ (,)x12x is a Cournot-Nash equilibrium if xiij= fx(), ij,1,2,=≠ j i”, where fij(x ) is the best-response function of firm i to firm j’s output. (iv) Best response function. Characterization of equilibrium The proceedure that we follow to obtain the Nash equilibrium is similar to that used in the previous chapter. First we calculate the best response of each player to the possible strategies of its rival and then we look for combinations of strategies which are mutually best responses to each other. Given a strategy of firm j we look for the strategy that provides most profit for firm i. That is,, given the strategy x j ≥ 0 firm i’s best response is to choose a strategy xi such that: maxΠ≡+−ii (xx , j ) px ( i x ji ) x Cx ii ( ) xi ≥0 ∂Πi '' =++p()()()0xxij xpxxCx i ij +−= ii (1)() → fx ij ∂xi 2 ∂Πi ''''' 2 =+++−<2(pxij x ) xpx i ( ij x ) Cx ii ()0 ∂xi Taking into account the non-negativity constraint, xi ≥ 0, or in terms of game theory taking into account that the best response must belong to the strategy space of the player, the best response function is: fxij()max(),0.= { fx ij } ** The Cournot-Nash equilibrium is a strategy profile (,)x12x such that the strategy of each player is its best response to the rival’s strategy. That is, ** *⎫ xfx112==()max(),0{ fx 12} ⎪ ⎪ ** * ⎬ ↔=xiijfx()max(),0, ={} fx ij ij,1,2,. = j ≠ i ⎪ xfx**==()max(),0 fx * 221{} 21 ⎭⎪ 6 Microeconomic Theory IV Oligopoly Let us now forget the non-negativity constraint and we are going to assume that the best response function is broadly characterized by condition (1) (interior solution). By definition, ∂Π ((),)fx x the best response must satisfy the first order condition: ii j j= 0 → firm i’s best ∂xi ** ∂Πii(,)xx j response to x j ≥ 0 is fij(x ). The Cournot-Nash equilibrium satisfies = 0 given ∂xi ** that xfxiiij==(), 1,2. We have a simple way of checking whether a combination of strategies is a Nash equilibrium: by calculating each firm’s marginal profit corresponding to that strategy profile. If any one of them is other than zero the equilibrium condition is not met. ∂Πii(,xxˆˆ j ) >→0fxij (ˆˆ ) > x i → ( xx ˆˆ ij , ) is not a Cournot-Nash equilibrium. ∂xi ∂Πii(,xxˆˆ j ) <→0fxij (ˆˆ ) < x i → ( xx ˆˆ ij , ) is not a Cournot-Nash equilibrium. ∂xi (v) Example. Graphic representation Consider the case of linear demand and constant marginal cost: p(xabx ) =− and Cxii()== cxi ii , 1,2. Assume for the sake of simplicity that the marginal cost is the same for the two firms: cci =>0, i =1,2.( ac> for the example to make sense). We first obtain the best response function for firm ii, =1,2. maxΠii (xx , j ) ≡ px ( i + xxCx ji ) − ii ( ) ≡− [ abx ( i + xxcxacbx j )] i − i ≡−− [ ( i + xx j )] i xi ≥0 ∂Πi '' acbx−− j =++p( xij x ) x i p ( x ij +−=−−−=→ x ) C ii ( x ) a 2 bx i bx j c 0 f ij ( x ) ∂xbi 2 2 ∂Πi 2 =−20b < ∂xi So the best response function is: ⎧acbx−− j ⎫ fxij()max(),0max=={} fx ij ⎨ ,0.⎬ ⎩⎭2b 7 Microeconomic Theory IV Oligopoly The Cournot-Nash equilibrium satisfies: * ** ⎧⎫acbx−− 2 xfx112==()max⎨⎬ ,0 >N 0 ⎩⎭2b given that ac> * ** ⎧⎫acbx−− 1 xfx221==()max⎨⎬ ,0 >N 0 ⎩⎭2b given that ac> By solving the system: x**==fx() f (()) fx * 112121 * x2 * ⎛⎞ * acbx−− 1 acbx−+ acb−−⎜⎟ 1 acbx−−* 2b acbx −+ * ac− xx* ==2 ⎝⎠ =2 =→=1 * . 1 22243bbbbb1 ⎛⎞ac− * acb−−⎜⎟ acbx−− 3b 2(ac− ) ac− →=x* 1 =⎝⎠ = = . 2 2263bbbb 2(ac− ) The Cournot-Nash total output is: xxx***=+= and the equilibrium price 12 3b 2(ac−+ ) a 2 c ppxxab***=+=−() = . Profits are: 12 33b acacac−−() −2 Π=Π******(,)[(xx = px + x ) − cx ] = = 1112 12 133bb 9 acacac−−() −2 Π=Π******(,)[(xx = px + x ) − cx ] = = . 2212 12 233bb 9 8 Microeconomic Theory IV Oligopoly Graphic representation x 2 xe 45º f12()x m x Cournot-Nash equilibrium * x 2 f ()x 21 * m e x1 x x x1 9 Microeconomic Theory IV Oligopoly 1.1.2. Oligopoly (i) Representation of the game in normal form. (ii) Notion of equilibrium. Best response function. Cournot-Nash equilibrium. (iii) Lerner index. (iv) Special cases. Constant marginal cost. (i) Representation of the game in normal form 1) in=1,2,..., . (Players) 2) xi ≥ 0. Similarly, xi ∈∞[0, ), in =1,2,.., . 3) The profit of each firm corresponding to strategy profile (xii ,x− ) is: Π=+−=(x ,xpxxxCxi ) ( ) ( ), 1,2,..., n . ii−− i i ii ii x The way of representing the game in normal form has changed slightly. Given the strategy profile (x12 ,xx ,...,n ) the relevant point for firm ii, =1,2,..., n , is the total output produced by the other firms, x−ij= ∑ x . Therefore, (xii ,x− ) is not really a strategy profile and ji≠ Πii(,x x− i ) is firm i’s profit associated with every combination of strategies where firm i produces xi and the other firms produce in aggregate x−i (it being irrelevant to firm i how that production x−i is distributed among the n -1 firms).
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