Chapter 3: Oligopoly

Notes on

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.

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.

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+ ).

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:

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 ∂Π i =++p()()()0xx xpxxCx'' +−= (1)() → fx ij i ij ii 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 ⎭⎪

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.

∂Π (,xxˆˆ ) ii j>→0fx (ˆˆ ) > x → ( xx ˆˆ , ) is not a Cournot-Nash equilibrium. ∂x ij i ij i ∂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

Microeconomic Theory IV Oligopoly

Graphic representation

x 2

e 45º x f12()x

m x Cournot-Nash equilibrium

* x 2

f21()x

* m e

x1 x x x1

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).

(iii) Notion of equilibrium. Best response functions. Cournot-Nash equilibrium

In the Cournot oligopoly game we say that

** * ** “ (x12 ,xx ,..,nii )≡ ( xx ,− ) is a Cournot-Nash equilibrium if:

** * Π≥Π∀≥∀=ii(x ,xxxxiin−− i ) ii ( , i ) i 0, ,1,2,..., .”.

Microeconomic Theory IV Oligopoly

In terms of best response functions the definition is:

** * ** ** “ (x12 ,xx ,..,nii )≡ ( xx ,− ) is a Cournot-Nash equilibrium if xiii=∀=fx(− ), ii, 1,2,.., n .”,

where fii(x− ) is firm i’s best response function to all those combinations of strategies whose

total output is x−i .

We next obtain the best response of firm i to all those combinations of strategies (of the

other firms) whose total output is x−i . Firm I’s best response is to choose a strategy xi such that:

maxΠ≡+−ii (xx ,−− i ) px ( i x ii ) x Cx ii ( ) xi ≥0 ∂Π i =++p()()()0xx xpxx'' +− Cx = (1)() → fx iiiii−− ii ii − ∂xi 2 ∂Πi ''''' 2 =+++−<2(pxiiiii x−− ) xpx ( x ) Cx ii ()0 ∂xi

Taking into account the non-negativity constraint, xi ≥ 0, or in terms of game theory that the best response must belong to the player’s strategy space, the best response function is:

fxii()max(),0.−−= { fx ii}

** * ** The Cournot-Nash equilibrium is a strategy profile (x12 ,xx ,..,nii )≡ ( xx ,− ) such that

** xiii=∀=fx(− ), ii, 1,2,.., n .”.

Let us now forget the non-negativity constraint and assume that the best response function is broadly characterized by condition (1) (interior solution). By definition, the best response

∂Π ((fx ),) x must satisfy the first order condition: ii−− i i= 0 → firm i’s best response to ∂xi

Microeconomic Theory IV Oligopoly

** ∂Πii(,xx− i ) x−i ≥ 0 is fii (x− ). The Cournot-Nash equilibrium satisfies = 0 given that ∂xi

** xiii==fx(− ), i1,2,..., n . Here again there is a simple way of checking whether a combination of strategies is a Nash equilibrium: calculating each firm’s marginal profit corresponding to that strategy profile. If any one is other than zero the equilibrium condition is not met.

∂Π (,xxˆˆ ) ii− i>→0fx (ˆˆˆˆ ) > x → ( xx , ) is not a Cournot-Nash equilibrium. ∂x ii−− i ii i ∂Πii(,xxˆˆ− i ) <→0fxii (ˆˆˆˆ−− ) < x i → ( xx ii , ) is not a Cournot-Nash equilibrium. ∂xi

(iii) Lerner index

By assuming an interior solution we are going to transform condition (1) to obtain the Lerner index of market power.

px()()()0++ x xp'' x +− x C x = iiiii−− ii x

px' () px()[1+−= x ] C' ( x ) 0 iiipx()

' xi xp() x ' px()[1+−= ] Cii ( x ) 0 xpx () 1 − ε ()x

x Defining firm i’s share as s = i we get: i x

s px()[1− i ]−= C' ( x ) 0 ε ()x ii

Then the Lerner index of market power for firm i is

p()xCx− ' ( ) s ii= i p()xxε ()

Microeconomic Theory IV Oligopoly

Then the Cournot model is located between the case of pure monopoly (si = 1) and perfect

pC− ' competition ( lim= 0 ). si →0 p

(iv) Special cases. Constant marginal costs

a) Constant marginal cost: cii >=0, 1,.., n .

At equilibrium the first order condition for each firm (interior solution) must be satisfied:

p()()0xx**++ xpxxc *'** +−== i1,2..,. n iiiiii−− x*

By adding up the n first order conditions:

nn **'* np() x+−=∑∑ xii p () x c 0 Nii==11 x*

That is

n **'* np() x+= x p () x∑ ci i=1

Then the total output in Cournot-Nash equilibrium depends exclusively on the sum of the marginal costs, not on their distribution across firms (in an interior solution with all n firms producing positive quantities).

b) Common constant marginal cost: cci = >=0, i1,.., n .

The Lerner index is:

px()− c s = i p()xxε ()

Microeconomic Theory IV Oligopoly

Taking into account that if the product is homogeneous and the marginal cost is the same for all firms then the Cournot-Nash equilibrium should be symmetric:

x* x * 1 sin==i =, = 1,.., . i xnxn**

If the price-elasticity is constant then:

px()− c 1 = p()xnε

Therefore, when the number of firms increases the relative price-marginal cost margin (the

Lerner index) decreases and at the limit when n →∞, p → c.

Microeconomic Theory IV Oligopoly

1.1.3. Welfare analysis

We consider the simplest case where marginal cost is constant and common to all firms.

p()()0xx**++ xpxxc *'** +−== i1,2..,. n iiiii−− x*

By adding the n first order conditions:

np() x**'*+ x p () x−= nc 0

We follow a similar approach to that in the chapter on monopolies to compare the Cournot output with the efficient output.

(Review the obtaining of the welfare function and the problem of maximizing social welfare) maxWx ( )≡− max ux ( ) Cx ( ) xx≥≥00

WuC'''(0)=− (0) (0) >⇒> 0 pC(0) ' (0)

Wx'''()=− ux () Cx () =⇔= 0 Wx '(e ) 0 First order condition.

Wx''()=− ux '' () Cx '' () < 0 Strictly concave welfare function.

⎧⎫Wx' ()0e = ⎪⎪'* ⎨⎬Wx()? ⎪⎪'' ⎩⎭Wx()< 0

ux''() * * P x Wx'*()=− ux '* () Cx '* () =−> px '* ()0 N n N px()* <0

By definition of Cournot output.

⎧⎫Wx' ()0e = ⎪⎪'* 'ee '* * ⎨⎬Wx()0>→ Wx () < Wx () →> x x ⎪⎪'' ⎩⎭Wx()< 0 dW' () x Wx''()<⇔ 0 <→↑↓ 0 x Wx ' () dx

Microeconomic Theory IV Oligopoly

' e Wx()0= '* Wx()0>

* e x x x

Microeconomic Theory IV Oligopoly

1.2. The Bertrand model

1.2.1. Homogeneous product

(i) Context.

(ii) Residual demand.

(iii) Representation of the game in normal form. Notion of equilibrium.

(iv) The Bertrand Paradox. Characterization of equilibrium and uniqueness.

(i) Context

The Bertrand model is characterized by the following elements

1) We consider a market with 2 firms.

2) Firms sell a homogeneous product.

3) Price competition.

4) Simultaneous choice. Each firm has to choose its price with no knowledge of the rival’s decision. Again, simultaneous choice does not mean that choices are made at the same instant in time; the relevant point is that although one firm may play first the other does not observe its decision.

5) Constant marginal cost and common: ccc12= =>0.

(ii) Residual demand

Firms produce a homogeneous product and compete on price. Then from the consumers’ point of view the only relevant point is the relationship between the prices of the two firms; consumers buy the product from the firm that sets the lower price. That is, if one firm charges a lower price than the other, it captures the entire market and the second firm sells nothing. If both firms give the same price then consumers are indifferent between buying

Microeconomic Theory IV Oligopoly from one or the other. For the sake of simplicity we assume that in the case of equal prices each firm sells to the half of the market.

Firm i’s residual demand, ij ,=≠ 1,2, j i , is:

⎧Dp(iij ) p< p ⎪ pi ⎪1 Dppii(, j )==⎨ Dppp () i i j ⎪ 2 ⎩⎪0 pij> p Dii(,pp j )

p j

D()p

1 Dp() xi 2 i

(iii) Representation of the game in normal form. Notion of equilibrium

The game in normal form is:

1) i =1,2. (Players)

2) pi ≥ 0. Any non negative price serves as a strategy for player i. Equivalently, we can

represent the player i’s strategy space as pii ∈ [0,∞= ), 1,2.

3) The firms profits corresponding to the strategy profile (,p12p ) are:

Π=−112(,pp )( p 1 cDpp )(, 112 )⎫ ⎪ ⎬ ≡Πii(,p ppcDppijji j )( = i − )(, ii j ), ,1,2, = ≠ , ⎪ Π=−212(,pp )( p 2 cDpp ) 212 (, )⎭ where the residual demand for firm i, ij,= 1,2, j≠ i , is:

⎧Dp(iij ) p< p ⎪ ⎪1 Dppii(, j )==⎨ Dppp () i i j . ⎪ 2 ⎩⎪0 pij> p

Microeconomic Theory IV Oligopoly

In the Bertrand game we say that

** “ (,)p12p is a Bertrand-Nash equilibrium if:

** * Π≥Π∀≥=≠ii(,p ppppijji j ) ii (,) j i0, ,1,2, ”.

To simplify the analysis we use this definition exclusively because the fact that the residual demand of each firm is a discontinuos function of its own price means that we cannot use standard optimization techniques (in fact instead of having best response functions we would have best response correspondences and the analysis would be more complex).

(iv) The Bertrand Paradox. Characterization of the equilibrium and uniqueness

We demonstrate here that the unique Bertrand-Nash equilibrium is:

** p12= pc=

This result is known as the Bertrand Paradox:

“Two firms competing on prices suffice to obtain a competitive outcome”

Demonstration

** We demonstrate that the strategy profile p12= pc= : a) is a Nash equilibrium. b) is the unique Nash equilibrium.

1 a) The profit of each firm under strategy profile (,)cc is: Π=−(,)cc ( c c ) Dc (), i =1,2. i 2

If firm i unilaterally deviates by charging a price pi > c then its profit will be zero because

it will make no sales. By reducing its price below marginal cost pi < c it would capture the entire market but incur losses. Therefore,

Πiiii(,)cc≥Π ( p ,) c ∀ p ≥0, i,1,2, j = j ≠ i

Microeconomic Theory IV Oligopoly b) We demonstrate that no other combination of strategies can be a Nash equilibrium.The graph below shows the different types of strategy profile. We check whether a strategy profile is an equilibrium or not by calculating the profit of each player corresponding to this combination of strategies and we wonder if any of the players has an incentive to unilaterally deviate. To eliminate a strategy profile as an equilibrium it suffices to show that at least one player can improve by deviating unilaterally.

pp21= p 2 pp21> pm

pp21<

m c p p1

1) Equal prices: pij= p

a) ¿ pij=>pc NE? NO. In a strategy profile like this the profit of each firm would be:

1 Π=−=−(,p ppcDpppcDp )( )(, )( ) (). Firm i =1, 2 would have an incentive to ii j i ii j i2 i

' deviate unilaterally. For example, we can choose a price ppii= −ε (where ε is an arbitrary positive amount as small as required):

1 (p''''−=−=Π>Π=−=−cDp )()( p cDpp )(,) (,) pp ' (,)( pp p cDpp )(,)( p c ) Dp (). i i i iij iij iij i iij i2 i In fact there would be inifinite deviations such that firm i improves with a unilateral deviation.

Microeconomic Theory IV Oligopoly

b) ¿ pij=

1 Π=−=−<(,pp )( p cDpp ) (, )( p c ) Dp ()0. Firm i would have an incentive to ii j i ii j i2 i <0

' deviate unilaterally. For example, any price pii> p :

1 0(=−p''cDpp )(,) =Π (,) pp ' >Π (,)( pp =− p cDpp )(,)( =− p c ) Dp (). i iij iij iij i iij i2 i =0

2) Different prices: pij≠ p

c) ¿ pij>>pc NE? NO. Firm i’s profit in such strategy profile would be zero:

Π=−=ii(,pp j )( p i cDpp )(, ii j )0 and the profit of the other firm, firm j, would be

' Π=−=−>ji(,pp j )( p j cDpp ) ji (, j )( p j cDp )()0. j For firm i any unilateral deviation pi

' such that cp<≤ij p increases profits:

()()()(,)(,)(,)()(,)()00.pcDp''''− =−=Π>Π=−=−= pcDpp pp ' pp pcDpp pc iN i i iij iij iij i iij i ' si ppij<

Although we have already proved that the strategy profile (pij ,pppc ), with i>> j cannot be an equilibrium we can also show that in many cases firm j would also have an incentive

m to deviate unilaterally. (For example, if p ≥>ppcij >any unilateral deviation

' m pijj>>pp increases the profits of firm j. For the cases pij>>ppc > and

m pij>>>ppc it is also straightforward to find deviations such the profits of firm j increase. The only situation where firm j would have no incentive to deviate would be one

m such that pij>=>ppc). d) Other cases:

- ¿ pij>≥cp NE? NO. Firm i would have no incentive to deviate unilaterally while for

' firm j when pij>>cp any p jj> p increases profits and when pij>=cp firm j

Microeconomic Theory IV Oligopoly

m increases profits by conveniently increasing the price. For example, if p ≥>=pcpij any

' m price pij>>pc increases the profits of firm j. When pij>>=pcp any price

m ' p >>pcj (and others) increases the profits of firm j.

- ¿cp≥>ij p NE? NO. Firm i would have no incentive to deviate while for firm j any

' price p jj> p increases profits.

1.2.2. Heterogeneous products (differentiated products)

(i) Heterogeneous product. Residual demand.

(ii) Representation of the game in normal form.

(iii) Notion of equilibrium. Best response function. Bertrand-Nash equilibrium.

(i) Heterogeneous product.Residual demand

We maintain the rest of the assumptions of the Bertrand model (two firms, simultaneous choice, constant and common marginal cost, price competition) but now we consider that the two firms sell heterogeneous products. That is, firms sell products that are close but imperfect substitutes.

The demand for the product of firm i, the residual demand, is given by Dppii ( , j ). Assume

∂D ∂D ∂∂DD that i < 0, i > 0 and ii> ; that is, the demand for product i is a decreasing ∂pi ∂p j ∂∂pijp function of its own price, products are substitutes and the own effect is larger than the cross effect.

Microeconomic Theory IV Oligopoly

(iii)Representation of the game in normal form. Notion of equilibrium

The game in normal form is:

1) i =1,2. (Players)

2) pi ≥ 0 or equivalently pii ∈∞ [0, ), = 1,2.

3) The profit of each firm corresponding to (,p12p )is:

Π=−112(,pp )( p 1 cDpp )(, 112 )⎫ ⎪ ⎬ ≡Πii(,p ppcDppijji j )( = i − )(, ii j ), ,1,2, = ≠ ⎪ Π=−212(,pp )( p 2 cDpp ) 212 (, )⎭

Now the residual demand of each firm is a continous function of its price.

(iv) Notion of equilibrium. Best response function. Bertrand-Nash equilibrium

In terms of best responses the definition of the Bertrand-Nash equilibrium is:

** ** “ (,)p12p is a Bertrand-Nash equilibrium if piij= gp(), ∀= ij,1,2,. j ≠ i”,

where gpij( ) is firm i’s best response to the price p j of its rival.

The best response of firm i consists of choosing pi such that:

maxΠ≡−ii (pp , j ) ( p i cDpp ) ii ( , j ) pi ≥0 ∂Π ∂D ii=+−=→Dpp(, )( p c ) 0 (1) gp ( ) ii j i i j ∂∂ppii 22 ∂Πii ∂DD ∂ i 22=+−<2()pci 0. ∂∂ppii ∂ p i

Microeconomic Theory IV Oligopoly

1.3. Leadership in the choice of output. The Stackelberg model

(i) Context.

(ii) Two-stage game. Perfect information. Notion of strategy.

(iii) Backward induction. Subgame perfect equilibrium.

(iv) Example: linear demand and constant marginal cost.

(v) Other Nash equilibria which are not subgame perfect.

(i) Context

The Stackelberg duopoly model has four basic characteristics: a) We consider a market with 2 firms. b) Homogeneous product. That is, from the consumers’ point of view the two firms produce products which are perfect substitutes.

c) Quantity competition. Let x12 and x be the outputs of firm 1 and 2, respectively. d) Sequential choice. One of the firms (the leader), firm 1, chooses its output level first.

Next the other firm (the follower), firm 2, chooses its output after observing the output chosen by firm 1. From a game theory perspective this is a perfect information game.

(ii) A two-stage game. Perfect information. Notion of strategy

Firms play a two-stage game:

Stage 1: firm 1 chooses its output x1 ≥ 0.

Stage 2: firm 2 chooses its output x2 ≥ 0 after observing the output chosen by firm 1.

Given that both players must have the same perception of the game not only does player 2 observes the choice made by player 1 but also player 1 knows that player 2 observes its choice. That is, there is perfect information and both players have the same perception of the game.

Microeconomic Theory IV Oligopoly

(Note: a game with two stages, i.e. a sequential game, in which the second mover does not observe the output chosen by the first mover –i.e. an imperfect information game– would be equivalent to a simultaneous game, such as the Cournot game).

The strategy spaces of the players are as follows:

- x1 ≥ 0: any non-negative quantity serves as a strategy for player 1; equivalently

x1 ∈∞[0, ) .

- The description of the strategies for player 2 is more complex. Recall that a strategy is a complete description of what a player would do if he/she were called on to play at each one of his/her decision nodes, independently of whether they are attaniable or not given the current behavior of the other(s) player(s). In the Stackelberg game, each possible output of firm 1 generates a different decision node for firm 2. Therefore, firm 2’s strategy is a

function x21()x which tells us how much firm 2 is going to produce for each possible production of firm 1.

Microeconomic Theory IV Oligopoly

(iii) Backward induction. Subgame perfect equilibrium

Although the game seems too complex to be solved, we know that in perfect information games without ties the backward induction criterion proposes a unique solution which coincides with the unique subgame perfect equilibrium. The proceedure is similar to that used with finite games in the previous chapter.

We start from the last subgames, that is, at stage 2.

Stage 2

We eliminate the non credible threats or dominated actions in each subgame. Given an

output of firm 1 (a subgame) x1 the only credible threat is for firm 2 to choose a profit maximizing output level:

maxΠ≡+−212 (xx , ) px ( 1 xx 22 ) Cx 22 ( ) x2 ≥0 ∂Π 2 =++p()()()0xx xpxxCx'' +− = (1)() → fx 12 2 12 22 21 ∂x2 2 ∂Π2 ''''' 2 =+++−<2(px12 x ) xpx 2 ( 12 x ) C 22 ()0 x ∂x2

Taking into account the non-negativity constraint, x2 ≥ 0, we have:

fx21()=→ max(),0{ fx 21 } Firm 2’s strategy in the subgame perfect equilibrium.

In finite games, the proceedure continues by eliminating all the non credible threats and computing the reduced game. In the Stackelberg game eliminating all the incredible threats

is equivalent to eliminating player 1’s strategies other than fx21()= max(),0.{ fx 21 }

Stage 1

Player 1 anticipates that firm 2 will behave at each subgame according to the strategy

fx21()= max(),0.{ fx 21 } Firm 1’s profit function in reduced form is:

Microeconomic Theory IV Oligopoly

Π≡+−11(,x fx 21 ())( px 1 fx 21 ()) x 1 Cx 11 ().

Therefore, the problem for firm 1 becomes:

maxΠ≡+−11 (xfx , 21 ( )) px ( 1 fx 21 ( )) x 1 Cx 11 ( ). x1 ≥0 x

dΠ1 '' ' L =+++p()[1()]()()0xx12 x 1 fxpxxCx 21 12 +−= 11 (2) → x 1 dx1 2 d Π1 2 < 0 dx1

Therefore, the subgame perfect equilibrium is the strategy profile

L (x121 ,fx ( )).

(iv) Example: linear demand and constant marginal cost.

Stage 2

We eliminate the non credible threats or dominated actions in each subgame. Given an

output of firm 1 (a subgame) x1 the only credible threat is for firm 2 to choose a profit maximizing output level:

maxΠ≡+−≡−+−212 (xx , ) px ( 1 xx 22 ) Cx 22 ( ) [ a bx ( 1 x 2 )] x 2 cx 2 x2 ≥0 ∂Πacbx − − 2 =++px()()()0 x xp'' x +− x C x = (1)() → f x = 1 12 2 12 22 21 ∂xb2 2 2 ∂Π2 ''''' 2 =+++−=−<2(px12 x ) xpx 2 ( 12 x ) C 22 () x 2 b 0 ∂x2

Taking into account the non-negativity constraint, x2 ≥ 0, we get:

⎧⎫acbx−− 1 fx21()== max(),0max{} fx 21 ⎨⎬ ,0 Firm 2’s strategy in the SPE. ⎩⎭2b

Microeconomic Theory IV Oligopoly

Stage 1

Player 1 anticipates that firm 2 will behave at each subgame according to the strategy

⎧⎫acbx−− 1 fx21()== max(),0max{} fx 21 ⎨⎬ ,0. Firm 1’s profit function in reduced form is: ⎩⎭2b

Π≡+−11(,x fx 21 ())( px 1 fx 21 ()) x 1 Cx 11 (). Then firm 1’s problem is:

a− c−−− bx11 a c bx maxΠ≡−−+≡−−+≡11 (x ,fx 21 ( )) [ acbx ( 1 fx 21 ( ))] x 1 [ acbx ( 1 )] x 1 [ ] x 1 x1 ≥0 22b dΠ1 '' ' L ac− =+++px(12 x ) x 1 [1 f 21 ( x )] p ( x 12 +−=−−= x ) C 11 ( x ) a c 2 bx 1 0 (2) →= x 1 dx1 2 b 2 d Π1 2 < 0 dx1

L Therefore, the subgame perfect equilibrium is (x121 ,fx ( )).

xe 45º

m L x SPE: (,())x121fx

* x2 s x2 f21()x

x* xL xe 1 N1 x1 xm

In order to obtain the profits of the firms we have to play the game.

Microeconomic Theory IV Oligopoly

⎛⎞ac− L acb−−⎜⎟ acbx−− 2b ac− xfxsL==() 1 =⎝⎠ = 221 224bbb

ac−− ac3( ac − ) xxxsLs=+= + = 1224bb 4 b

3(ac−+ ) a 3 c ac− ppxabxabss==−=−=() s ; pcs −= 44b 4

()()()acac−− ac −22 ()()() acac −− ac − Π=Ls()pcx − L = = = ; Π= ss() pcx − s = = . 1142bb 8 22 44 b 16 b

(v) Other Nash equilibria which are not subgame perfect

e 45º x

* x21()xxx=∀{ 2 1}

m * x EN: (,())xxx121← NO SPE

x* 2 xs 2 f ()x 21 L SPE: (,())x121fx

* L e x x x x 1 N1 1 xm

Microeconomic Theory IV Oligopoly

1.4. Collusion and stability of agreements

1.4.1. Short-term collusion

(i) Cournot model. The collusion agreement is not a short-term equilibrium.

(ii) Bertrand model. The collusion agreement is not a short-term equilibrium.

(i) Cournot model. The collusion agreement is not a short-term equilibrium

If firms colluded they would be interested in maximizing aggregate profits.

maxΠ+Π≡+−++−112 (x , x ) 212 ( x , x ) px ( 1 x 21 ) x C 11 ( x ) px ( 1 x 2 ) x 2 C 2 ( x 2 ) xx12,

∂Π mm mm'' mm m ⎫ =px()()()()012 +++ x x 12 x p x 12 +− x C 11 x = (1)⎪ ∂x1 ⎪ '' ⎬ MRI == C12 C ∂Π =px()()()()0mm +++ x x mm x p'' x mm +− x C x m = (2)⎪ 12 12 12 22 ⎪ ∂x2 ⎭

When marginal costs are constant and equal across firms conditions (1) and (2) are identical. The two-equation system would have infinite solutions: any pair of outputs such

m that x12+=xx would maximize the industry profit. For these cases we also refer to the symmetric collusion agreement where each firm produces a half of the monopoly output:

xm xim ==, 1,2. i 2

We now show that the collusion agreement (implicit of course) cannot be supported as an equilibrium when the game is played once. In other words, we demonstrate that the strategy

mm profile (,)x12x is not a Nash equilibrium in the Cournot game.

Given the strategy x j ≥ 0 the best response of firm i is to choose a strategy xi such that:

maxΠ≡+−ii (xx , j ) px ( i x ji ) x Cx ii ( ) xi ≥0 ∂Π i =++p()()()0xx xpxxCx'' +−= (1)() → fx ij i ij ii ij ∂xi 2 ∂Πi ''''' 2 =+++−<2(pxij x ) xpx i ( ij x ) Cx ii ()0 ∂xi

Microeconomic Theory IV Oligopoly

So the best response function is: fxij()max(),0.= { fx ij }

mm To check that the combination of strategies (,)x12x is not a Nash equilibrium we calculate the marginal profit for each firm:

mm ∂Πii(,)xx j mm m''' mm m m mm =++px()()()()>0ij x x i p x ij +−=−+ x C ii x x j p x ij x ∂xi <0

By definition of collusion agreement.

Then starting from the collusion agreement an increase in output leads to an increase in the firm i’s profit and, therefore, firm i would have an incentive to break the collusion agreement. Put differently, given the definition of best response function

mm mm ∂Πii((fx j ),) x j ∂Πii(,)xx j mm = 0 and as >0 then fij()xx> i . ∂xi ∂xi

As we know what the optimal deviation for firm i is if it decides to break the collusion

m agreement, fij(),x we denote by Πi the profit that i would obtain if it deviates optimally and the other firms do not deviate. That is,

mm Π=Πiiijj(f (xx ), ).

Microeconomic Theory IV Oligopoly

Graphic analysis: linear demand and constant marginal cost

xe 45º

f12()x

m Collusion line: x12+ xx= m x Symmetric collusion * m x C-N m x 2 agreement: xi==, 1,2 m i 2 x2

f21()x

m * m e x1 x1 x x x1

Oligopoly

It is easy to generalize the above result to the case of n firms. The condition which defines the collusion agreement (the strategy profile maximizing the aggregate profit) is:

mm mm'' mm m p()()()()0xxiiiiiiii+++−−− xxp xx +− Cx == i1,..,. n

mm To show that the strategy profile (x1 ,..,xn ) is not a Nash equilibrium we obtain the marginal profit of each firm:

∂Π (,xxmm ) ii− i=++px()()()()>0mm x xpx m''' mm +−=−+ x Cx m x m px mm x ∂x iiiiiiiiii−−−− i <0

By definition of the collusion agreement.

Microeconomic Theory IV Oligopoly

Then starting from the collusion agreement an increase in the output of firm i also increases its profit and, therefore, firm I would have an incentive to break the collusion agreement. In

∂Π ((fxmm ),) x other words, given the definition of best response function ii−− i i= 0 and as ∂xi

mm ∂Πii(,xx− i ) mm >0 then fii(xx− )> i . ∂xi

As we know what the optimal deviation for firm i is if it decides to break the collusion

m agreement, fii(),x− we denote by Πi the profit that i would obtain if it deviates optimally and the other firms do not deviate. That is,

mm Π=Πiiiii(f (xx−− ), ).

Microeconomic Theory IV Oligopoly

(ii) Bertrand model. The collusion agreement is not a short-term equilibrium

Consider the Bertrand model with homogeneous product and constant (and common) marginal cost. The strategy profile which represents the symmetric collusion agreement is

(,)pmmp . The profit of each firm is:

11 Π=Πmmmm(,)(pp = p − c ) Dp () mm = Π ii 22

We know (it has been demonstrated) that a combination of strategies of the type

pij=>pc is not a Nash equilibrium. Any firm would have an incentive to deviate

' m unilaterally. For example, we can choose ppi = −ε (where ε is an arbitrary positive amount as small as necessary). Of course, there are infinite deviations such that firm i is better off.

Finding the optimal deviation for firm i is more problematical. The best that it can do is to undercut its rival’s price by the lowest amount possible, ε >→ 0,ε 0. Although we do not have that optimal deviation well-defined we can be as near as we wish to monopoly price.

Let Πi be firm i’s profit when it optimally breaks the collusion agreement and the rival keeps it. That is,

mmm m m mm Π=Πii(,)()()()()p −εεε p = p − − cDp −N p − cDp =Π ε →0

Microeconomic Theory IV Oligopoly

1.4.2. Stability of agreements. Finite temporal horizon and infinite temporal horizon

We have shown that in the short term the collusion (cooperation) agreement cannot hold as

an equilibrium in either the Cournot game or the Bertrand game. In this section, we study

the possibilities of collusion or cooperation when the game is repeated.

(i) Finite temporal horizon

Backward induction argument → collusion (cooperation) cannot be supported as an

equilibrium (at each stage firms behave as in the one-shot game). The reasoning is similar

to that in the Prisoner’s Dilemma.

(ii) Infinite temporal horizon

There are two ways of interpreting an infinite temporal horizon:

(i) Literal interpretation: the game is repeated an infinite number of times. In this context,

to compare two alternative strategies a player must compare the discounted present value of

the respective gains. Let δ be the discount factor, 0 < δ < 1, and let r be the discount rate

1 ( 0 <<∞r ) where δ = . 1 + r

(ii) Informational interpretation: the game is repeated a finite but unkonwn number of

times. At each stage, there is a probability 0 < δ < 1 of the game continuing. In this setting,

each player must compare the expected value (which might be also discounted) of the

different strategies.

We shall see that the existence of implicit punishment threats may serve to maintain

collusion as an equilibrium of the repeated game.

Microeconomic Theory IV Oligopoly

Note first that there is a subgame perfect equilibrium of the infinitely repeated game where

each player plays his/her short-term Nash equilibrium strategy in each period. In the

Cournot model such a strategy would consist of “producing in each period the Cournot

quantity independently of past history”. In the Bertrand model that strategy would consist

of “charging a price equals to marginal cost independently of past history”.

We next study the possibility that there may be another subgame perfect equilibrium where

players cooperate with each other. Consider the following combination of long term

c ∞ strategies: si ≡ {sit (Ht−1 )}t=1 , i =1,2.

where,

⎧ to collude c ⎪ "cooperate" if all elements of Ht are equal to ("cooperate","cooperate") or = 1 sHit() t−1 = ⎨ t−1 ⎩⎪"not cooperate"(the short-term NE strategy) otherwise

(in Cournot:

⎧ xHxxtmmm if all elements of are equal to ( , ) or = 1 c itii−−1 ) sHit() t−1 = ⎨ * ⎩ xi otherwise

(in Bertrand:

mmm c ⎧ pHppt if all elements of t−1 are equal to ( , ) or = 1 sHit() t−1 = ⎨ ) ⎩ c otherwise

Note that these long term strategies incorporate “implicit punishment threats” in case of

breach of the (implicit) cooperation agreement. The threat is credible because “confess” in

each period (independently of the past history) is a Nash equilibrium of the repeated game.

Microeconomic Theory IV Oligopoly

To check whether it is possible in this context to maintain cooperation as an equilibrium, we

must check that players have no incentive to deviate; that is, we must check that the

c c combination of strategies (s1 ,s2 ) constitutes a Nash equilibrium of the repeated game.

Notation

m Π→i Firm i’s profit under collusion at each stage of the game.

* Π→i Firm i’s profit in the short-term Nash equilibrium at each stage of the game.

Π→i Firm i’s profit if the other firms cooperate and it optimally deviates.

m * Πii>Π >Π i

c c The discounted present value for firm i in the strategy profile (s1 ,s2 ) is given by:

Πm πδδδδ(sscc , )=Π m + Π m +22 Π m + .... =Π m (1 + + + ...) = i ii j i i i i 1−δ

If firm i deviates in the first period its gains are:

Π* πδδδδδδ(ss ,c )=Π + Π*2* + Π + .... =Π + (1 + + 2 + ...) Π * =Π + i ii j i i i i i i 1−δ

Cooperation is supported as a Nash equilibrium if no player has any incentive to deviate;

c c c that is, when π i (si ,s j ) ≥ π i (si , sj ). It is straightforward to check that if δ ≥ δ no firm

has an incentive to break the collusion agreement, where

m Πii−Π δ = * . Πii−Π

Microeconomic Theory IV Oligopoly

Basic Bibliography

Varian, H. R., 1992, Microeconomic Analysis, 3th edition, Norton. Chapter 14, sections:

16.1, 16.3, 16.4, 16.5, 16.6, 16.10 and 16.11.

Complementary Bibliography

Kreps, D. M., 1994, A course in microeconomic theory, Harvester Wheatsheaf.

Tirole, J., 1990, The Theory of Industrial Organization, MIT Press.

Varian, H. R., 1998, Intermediate Microeconomics: A Modern Approach, Norton.