A course on Industrial Organizationpareto.uab.cat/xmg/Docencia/IO-en/IO4IDEAHomog.pdfA course on Industrial Organization Xavier Martinez-Giralt Universitat Autonoma de Barcelona` [email protected]

A course on Industrial Organization

Xavier Martinez-Giralt

Universitat Aut̀onoma de Barcelona

[email protected]

Fall 2009-2010 – p.1/133

Static Oligopoly Pricing - Homogeneous Product

Quantity competition

Price competition

Price-Quantity competition

Fall 2009-2010 – p.2/133

Homogeneous Product

Definition:Two products are homogeneous if at the eyes of the consumer theyprovide exactly the same service.Illustration:Assume consumers’ preferences are rough enough so that allchairs are perceived exactly alike regardless of whether they havearms, wheels, made of wood, metal, etc. In such a case,consumers will simply demand chairs. Thus, there can only be asingle demand function for chairs.Examples:

Difficult. Sulphuric acid, electricity

Fall 2009-2010 – p.3/133

Oligopoly

Definition:An industry is said to be oligopolistic whenever the decision of onefirm affects and is affected by the decisions of the other firms in theindustry. [STRATEGIC INTERACTION].Features:

Typically, such situation is associated with a limited number offirms in the industry

Decision variables: prices or quantities; entry; R&D, ...

Decision “timing" across firms: simultaneous, sequential(commitment)

Decision “timing" across decisions: simultaneous, sequential

Equilibrium concept: Nash (and some variations)

Starting point: Cournot oligopoly model (modern version);(Original 1838)

Fall 2009-2010 – p.4/133

Cournot model - Assumptions

Structural:

Static model

Technology: cost function Ci(qi)

Aggregate demand function Q = F (p)

Homogeneous product marketLarge number of consumers

There are n firms in the industry, i = 1, 2, . . . , n

No entry, no exit of firms in the market

Strategic variable of the firms: production levels qi

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Cournot model - Assumptions (2)

Behavioral:

Firms choose production levels to maximize profits.Each firm knows that its production decision depends onits expectation over the rivals’ decisions.Also, every rival’s decision depend of what each of themexpects all the other competitors will decide.All firms take simultaneously their respective productiondecisions.

Consumers choose a consumption bundle to maximize utility.

Fall 2009-2010 – p.6/133


On demand Q = F (p) (A1)

1. f : R+ → R+

2. ∃Q s.t. f(Q)

{> 0 if Q < Q,

= 0 if Q ≥ Q

3. ∃p < ∞ s.t. f(0) = p

4. f(Q) is continuous and C2 in [0, Q]

5. f ′(Q) < 0 for Q ∈ (0, Q)

Implication: qi ∈ [0, Q] ∀i

Fall 2009-2010 – p.7/133


On technology Ci(qi) (A2)For all i,

1. Ci : R+ → R+

2. Ci is continuous and continously differentiable ∀qi > 0

3. Ci(qi) > 0 ∀qi > 0

4. Ci(0) ≥ 0

5. C′

i(qi) > 0 ∀qi ≥ 0

Note:

Symmetry across firms means Ci(·) = Cj(·), ∀i, j, i 6= j

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Cournot model - Profits

Definitions:

q = (q1, q2, q3, . . . , qn) is a production plan.

Πi(q) = qif(Q) − Ci(qi) is firm i’s profit function.

Π(q) = (Π1(q),Π2(q),Π3(q), . . . ,Πn(q)) is a distribution ofprofits in the industry.

Assumptions (A3)

1. Πi : R+ → R+

2. Πi is continuous and C2 ∀qi > 0

3. Πi(q) is strictly concave in qi, ∀q s.t. qi > 0, Q < Q.

Fall 2009-2010 – p.9/133

Cournot model - More definitions

Feasibility:

qi is a feasible output for firm i if qi ∈ [0, Q].

The set F ∈ Rn defined as F

def= [0, Q]× n times. . . ×[0, Q], is the

set of all feasible production plans in the industry.

F is a compact set.

Space of outcomes:

The space of outcomes is the set of all possible distribution of

profits in the industry: {Π(q)|q ∈ F}def= Π(F).

Π(F) is also a compact set.

Pareto optimal outcomes:

PO = {Π(q)|q ∈ F s.t. ∀q′ ∈ F , Π(q) > Π(q′)}, where Π(q) > Π(q′)

means Πi(q) ≥ Πi(q′) ∀i, and ∃j s.t. Πj(q) > Πj(q

′).

Fall 2009-2010 – p.10/133

Cournot model - Equilibrium

Definitions: Cournot-Nash equilibrium

A production plan qc is a C-N equilibrium if no firm canunilaterally improve upon its profit level by modifying itsproduction decision.

A production plan qc is a C-N equilibrium if no firm has anyprofitable unilateral deviation.

Let qc−i

def= (qc

1, qc2, . . . , q

ci−1, q

ci+1, . . . , q

cn). We say that a

production plan qc is a C-N equilibrium ifΠi(q

c) = maxqiΠi(qi, q

c−i) ∀i.

A production plan qc is a C-N equilibrium if

6 ∃q̃def= (qc

1, qc2, . . . , q

ci−1, q̃i, q

ci+1, . . . , q

cn) s.t. Πi(q

c) ≤ Πi(q̃) ∀i.

A production plan qc is a C-N equilibrium ifqci = argmaxqi

Πi(qi, qc−i) ∀i.

Fall 2009-2010 – p.11/133

Cournot equilibrium - Illustration

Duopoly: Firms 1 and 2

p

Q

f(Q)

q2q

c

1(q

2)

0

f(Q)−

q2

Firm 1's residual demand

Firm 1's marginal revenue Firm 1's expectation on Firm 2

C′

1

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Cournot model - Game theory

The Cournot model is a one-shot, simultaneous move,non-cooperative game [in pure strategies].Extensive form

F1

q1

0 Q

F2

Qq20 Qq2

0

Π1(Q, 0) . . .Π1(Q,Q)

Π2(Q, 0) . . .Π2(Q,Q)

Π1(Q, 0) . . .Π1(Q,Q)

Π2(Q, 0) . . .Π2(Q,Q)

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Cournot model - Game theory (2)

Normal form: (N,F ,Π)

N = 1, 2, . . . , n (set of firms)

Fdef= [0, Q]× n times. . . ×[0, Q] (strategy space)

Π(q) = (Π1(q), . . . ,Πn(q)) (payoff vector)

Payoff matrix (duopoly).

2/1 0 . . . q1 . . . Q

0 Π1(0, 0), Π2(0, 0) Π1(q1, 0), Π2(q1, 0) Π1(Q, 0), Π2(Q, 0)

......

......

q2 Π1(0, q2), Π2(0, q2) Π1(q1, q2), Π2(q1, q2) Π1(Q, q2), Π2(Q, q2)

......

......

Q Π1(0, Q), Π2(0, Q) Π1(q1, Q), Π2(q2, Q) Π1(Q, Q), Π2(Q, Q)

Fall 2009-2010 – p.14/133

Cournot equilibrium and Pareto optimality

Proposition 1. Let qc ≫ 0 be a Cournot equilibrium production plan.ThenΠ(qc) is not Pareto optimum.

Proof. ShowQc def=

∑i qc

i < Q.

As qc ≫ 0 it satisfies FOCs. Also,∂Πi

∂qj= qif

′(Q) < 0 ∀i, j i 6= j.

Hence a simultaneous reduction of the output levels of any twofirms,qi andqj would improve their profits. Thus,∃q such thatΠi(q) > Πi(q

c) ∀i.

Althoughqi < qci ,∀i has a negative impact on firmi’s profits, it is

second order effect and offset by previous (first order) effect.

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Cournot equilibrium and Pareto optimality (2)

IntuitionFirm i when deciding qi considers adverse effect of price on itsoutput but ignores the effect on aggregate production.

Impact of variation of qi on price f(Q) is given by

∂f

∂qi=

df

dQ

∂Q

∂qi+

∑

j 6=i

df

dQ

∂Q

∂qi

∂qj

∂qi=

df

dQ

1 +

∑

j 6=i

∂qj

∂qi

First term: impact on price of additional unit of qi

Second term: externality on rivals: conjectural variation.Cournot assumes it away.

Thus, in equilibrium firms produce beyond the optimal industry level.

Fall 2009-2010 – p.16/133

Cournot equilibrium - Existence - Illustration

p

Q

D

D′R0 q1 q2 q3 q4 q5 q6

K

K′

J′

J

B

H

I

E

F

G

qa ∈ [q4, q6], qb ∈ [q2, q3] and [q4, q6] ∩ [q2, q3] = ∅

Fall 2009-2010 – p.17/133

Cournot equilibrium - Existence - Illustration (2)

q1

q5

p

D

J

B

J′

FJ′

R D′

Q0

Fall 2009-2010 – p.18/133

Cournot equilibrium - Existence - Formal approach (1)

Concavity of Πi:

{means ∂2Πi

∂q2

i

≤ 0

implies ∃q̃i maximizing profits for any∑

j 6=i qj

∂2Πi(q)

∂q2i

= 2f ′(Q) + qif′′

(Q) − C′′

i (qi)

∂2Πi(q)

∂q2i

< 0 if

f′′

< 0 and C′′

i > 0

f′′

> 0, C′′

i > 0 and |2f ′| > qif′′

− C′′

i

f′′

< 0, C′′

i < 0 and |2f ′ + qif′′

| > |C′′

i |

Sufficient conditions for existence, not necessary. See Vives (1999),

Mas Colell et al. (1995), Friedman (1977), Okuguchi (1976).

Fall 2009-2010 – p.19/133


Theorem 1. Whenever assumptionsS1 to S3 are fulfilled, there is aninterior Cournot equilibrium,

We will divide the proof in three parts. First, we will show

that assumptions S1 to S3 ensure well-defined reaction functions

(lemma 1); next, we will show that these reaction functions are con-

tinuous (lemma 2); finally we will verify that we can apply Brower’s

fixed point theorem.

Fall 2009-2010 – p.20/133


Lemma 1. Whenever assumptionsS1 to S3 are fulfilled, there will be awell-defined reaction function for every firm.

Lemma 2. wi(q−i) is a continuous function.

Theorem 2(Brower). LetX be a convex and compact set inRn. Let

f : X → X be a continuous application associating a pointf(x) in X to

each pointx in X. Then there exists a fixed pointx̂ = f(X̂).

On fixed point theorems see Border (1992).

Fall 2009-2010 – p.21/133


Proof of lemma 1

Proof. LetF−idef= [0, Q] × [0, Q] × [0, Q]× (n-1) times. . . ×[0, Q].

Consider an arbitrary production planq−i ∈ F−i.Given thatπi(qi, q−i) is continuous inqi, qi ∈ [0, Q] and strictly concave,and given that[0, Q] is compact we can write the first order condition ofthe profit maximization problem

∂πi(qi, q−i)

∂qi= f(Q) + qif

′(Q) − C′

i(qi) = 0,

as a functionqi = wi(q−i) calledfirm i’s reaction function. It tells usfirm i’s profit maximizing strategy conditional to its expectation on thebehavior of the(n − 1) rival firms.

Fall 2009-2010 – p.22/133


Proof of lemma 2

Proof. Let us now define a one-to-one continuous mapping,w(q), of thecompact setF on itself,

w(q) =(w1(q−1), w2(q−2), . . . , wn(q−n)

)

Let {q−i}∞τ=1 a sequence of strategy vectors inF−i, such that

limτ→∞ qτ−i = qo

−i.SinceF−i is compact, we know thatqo

−i ∈ F−i.

The sequence{q−i}∞τ=1 allows us to obtain a sequence{wi(q

τ−i)}

∞τ=1

wherewi(qτ−i) ∈ [0, Q].

Let,

qτi = wi(q

τ−i)

qoi = wi(q

o−i).

Fall 2009-2010 – p.23/133


Proof of lemma 2 (cont’d)

Proof. We say thatwi is continuous iflimτ→∞ qτi = qo

i .By definition,

πi

(wi(q

τ−i), q

τ−i

)≥ π(qi, q

τ−i), qi ∈ [0, Q].

Since the profit function is continuous,

limτ→∞

πi

(wi(q

τ−i), q

τ−i

)= πi

(lim

τ→∞wi(q

τ−i), q

o−i

)

limτ→∞

π(qi, qτ−i) = π(qi, q

o−i),

so that we can write,

πi

(lim

τ→∞wi(q

τ−i), q

o−i

)≥ π(qi, q

o−i), qi ∈ [0, Q].

Fall 2009-2010 – p.24/133


Proof of lemma 2 (cont’d)

Proof. Given thatwi(q−i) is a single-valued function,

limτ→∞

wi(qτ−i) = wi(q

o−i)

or equivalently,lim

τ→∞qτi = qo

−i,

so thatwi(q−i) is a continuous function.

Fall 2009-2010 – p.25/133


Brower’s fixed point theorem - illustration for X = [0, 1]

A

B

C

D

01

1

Proof. We can apply Brower’s fixed point theorem, given thatF iscompact andw(q) is continuous. Therefore, there is at least a pointq∗

such thatw(q∗) = q∗, whereq∗ is the Cournot equilibrium.

Fall 2009-2010 – p.26/133

Cournot equilibrium - Uniqueness

Consider a Cournot duopoly with two equilibria (q∗1 , q∗2) and (y∗1 , y

∗2).

Therefore,

If firm 2 would choose q∗2 firm 1’s best reply would be q∗1 ;

If firm 2 would choose y∗2 firm 1’s best reply would be y∗1

But firm 1 does not know firm 2’s decision. It only makesconjectures.It may happen that firm 1 conjectures that firm 2 will choose q∗2when firm 2’s choice is y∗2 .A production plan (q∗1 , y

∗2) generally will not be an equilibrium vector

of strategies.In other words, there may appear a coordination problem.How guarantee that firms will “point at the same equilibriumproduction plan"?

Study the conditions under which there is a unique equilibrium.

Fall 2009-2010 – p.27/133

Cournot equilibrium - Uniqueness - Illustration

P

0

CE

q1 q2 q3 q4

F

F′

G

G′

J

J′

K

K′

D′

E′

Q

D

D′D: demand; FF ′, GG′, JJ ′,KK ′ isoprofit curves of firm i.Firm i conjectures firm j produces q1 → E equilibrium.Firm i conjectures firm j produces q3 → E′ equilibrium.

Multiplicity → lack of concavity of DD′.

Fall 2009-2010 – p.28/133

Cournot equilibrium - Uniqueness - Formal approach

We need to restrict assumption A3, introducingAssumption (A4) The profit function πi(q) is continuous, twicecontinuously differentiable, and ∀q, q ≫ 0, Q < Q satisfies,

∂2πi(q)

∂q2i

+∑

j 6=i

∣∣∣∣∣∂2πi(q)

∂qi∂qj

∣∣∣∣∣ < 0, or

2f ′(Q) + qif′′

(Q)−C′′

i (qi) + (n − 1)|f ′(Q) + qif′′

(Q)| < 0 (1)

equivalent to assume Hessian negative semidefinite

implies F compact

more restrictive than A3

Fall 2009-2010 – p.29/133

Cournot equilibrium - Uniqueness - Formal approach (2)

Consider the case f′′

< 0, so that

|f ′(Q) + qif′′

(Q)| = −(f ′(Q) + qif

′′

(Q))

.

Then, (1) can be rewritten as

−(n − 3)f ′(Q) − (n − 2)qif′′

(Q) < C′′

i (qi). (2)

For n = 2 always true.For n ≥ 3 left hand side positive.This means that increasing values of n require increasing values ofC

′′

i (qi) to verify (2).

Summarizing, assumptions S1, S2, S4 guarantee that w(q) is a con-

traction.

Fall 2009-2010 – p.30/133

Cournot equilibrium - Uniqueness - Formal approach (3)

Definition: ContractionConsider two vectors q

′

−i and q′′

−i, i = 1, 2, . . . , n. It is said that w(q)

is a contraction if∣∣∣wi(q

′

−i) − wi(q′′

−i)∣∣∣ < ‖q

′

−i − q′′

−i‖

That is, when all the competitor firms vary their strategies in acertain amount, firm i’s best reply varies in a smaller amount. In IR2

this means that w′

i < 1.We introduce now a theorem without proof:

Theorem 3. Letf : Rl → Rl be a contraction. Then,f has a unique

fixed point.We can use this theorem to obtain the result we are after:Theorem 4. AssumeS1, S2, S4. Thenw(q) is a contraction andq∗ is theunique Cournot equilibrium.

Fall 2009-2010 – p.31/133

Cournot equilibrium - Existence & Uniqueness

Van Long and Soubeyran (2000)

Szidarovsky and Yakowicz (1977)

Tirole (1988, pp.224-225)

Fall 2009-2010 – p.32/133

Strategic complements and substitutes

Properties of reaction functions. Consider Duopoly.

w1(q2) solution of

∂π1(q1, q2)

∂q1= f(Q) + q1

df

dQ−

dC1(q1)

q1= 0. (3)

The strategic nature of the relation between firms is given bythe slope of the reaction function.

The slope is obtained by differentiating (3):

∂2π1(q1, q2)

∂q21

dq1 +∂2π1(q1, q2)

∂q1∂q2dq2 = 0, or

dq1

dq2

∣∣∣foc

= −

∂2π1(q1,q2)∂q1∂q2

∂2π1(q1,q2)∂q2

1

. (4)

Fall 2009-2010 – p.33/133

Strategic complements and substitutes (2)

−∂2π1(q1, q2)

∂q21

> 0, from the second order condition.

Then, sign of slope = sign of the numerator in (4).

From (3)

∂2π1(q1, q2)

∂q1∂q2=

∂

∂q2

[∂π1(q1, q2)

∂q1

]=

df

dQ+ q1

d2f

dQ2. (5)

Concave demand → sign (5)<0 → w′1 < 0.

Convex demand → sign (5) ambiguous.

Fall 2009-2010 – p.34/133

Strategic complements and substitutes (3)

Definition [Bulow, Geanakoplos and Klemperer (1985)]the actions of the two firms are strategic complements if∂2π1(q1, q2)

∂q1∂q2> 0.

the actions of the two firms are strategic substitutes if∂2π1(q1, q2)

∂q1∂q2< 0

Nature of the strategic relations among competitors to beexamined case by case.

Prices are often strategic complements.Quantities are often strategic substitutes.Martin (2002, pp. 21-27) counterexamples.

Throughout investigation of games with strategiccomplementarities (supermodular games): Amir (1996, 2005)and Vives (1999, 2005a, 2005b).

Fall 2009-2010 – p.35/133

Geometry of the Cournot model

Assume duopoly with linear demand and costs:

p = a − b(q1 + q2),

Ci(qi) = c0 + cqi, i = 1, 2.

Isoprofit curvesFix a level of profits Πi(q) = Π.Then,

Π = qi(a − b(qi + qj)) − c0 − ciqi, that is

qj = −1

b

(Π + c0

qi+ c − a

)− qi

Fall 2009-2010 – p.36/133

Geometry of the Cournot model (2)

Properties of isoprofit functions

Slope: ∂qj

∂qi

∣∣∣∣Π

= 1b

(Π+c0

q2

i

)− 1.

Critical point:

qi =(Π + c0

b

) 1

2

. (6)

Critical point is increasing in Π:

∂qi

∂Π=

1

2b

(Π + c0

b

)− 1

2

> 0. (7)

Isoprofit curve strictly concave in the space (qi, qj):

∂2qj

∂q2i

= −2(Π + c0)

bq3i

< 0.

Fall 2009-2010 – p.37/133


Extreme isoprofit curves

Maximum Π at monopoly (qmi , 0)

qmi =

a − c

2b, and Πm

i =(a − c)2

4b− c0.

Hence, isoprofit curve tangent to qi axis from below.

Minimum Π when (0, q̃j) maximizes firm i’s profits

Compute q̃j such that∂Πi

∂qi= 0 when qi = 0

This is q̃j =a − c

b.

Fall 2009-2010 – p.38/133


Lemma 3. Under linear demand and costs, the function linking allmaxima of the isoprofit family of curves is linear

Proof. Consider an isoprofit curveΠ. Its maximum wrtqi is given by (6).Substituting it in the isoprofit curve we obtain the associated valueqj :

qj =a − c

b− 2(

Π + c0

b)

1

2 . (8)

Compute now,

∂qj

∂Π=

−1

b

(Π + c0

b

)−1

2

. (9)

Comparing (7) and (9) we see,

∂qi

∂Π= −

1

2

∂qj

∂Π.

Fall 2009-2010 – p.39/133


When there is a variation in the level of profits, the effect on qi

is half the effect on qj regardless of the actual value of profits.

This implies a linear relation between the set of maximumpoints of the family of isoprofit curves. This linear function hasslope −1

2 .

Finally, to identify the expression of this linear function wesubstitute (6) in (8) to obtain,

qi =a − c

2b−

1

2qj . (10)

Fall 2009-2010 – p.40/133


q̃jΠ̂i < Πi < Π

m

i

qj

qiqm

i

Π̂i

Πi

Πm

i

qj

qi

Πm

j

qmj

q̃iΠjΠ̂j

Π̂j < Πj < Πmj

qi =a − c

2b−

1

2qj

qj =a − c

2b−

1

2qi

Fall 2009-2010 – p.41/133


Combining both isoprofit maps,

Cournot eq. characterized by intersection of the two functionslinking max profits.

Tangency point between any two isoprofit curves: distributionof profits such that any alternative share of profits cannotmake both firms better off simultaneously.

Loci of tangency points: set of Pareto optimal productionplans.

Set of tangency points:extremes: (qm

i , 0), (0, qmj ).

solution of maxqi,qj(Πi + Πj) = Q(a + bQ) − 2c0 − cQ :

linear function qi =a − c

2b− qj , with slope −1

Fall 2009-2010 – p.42/133


qmj

qmi

Πc

i

Πc

j

qi

qj

PO

C

Note :

C !∈ PO

Fall 2009-2010 – p.43/133


Reaction functions

Locus of profit maximizing production plans conditional to theexpectation on the behavior of the rival firms.

Solution of FOC of its profit maximization program:

Πi(q) = (a − bQ)qi − c0 − cqi,

∂Πi(q)

∂qi= a − c − bqj − 2bqi = 0,

qi =a − c

2b−

1

2qj .

Same as (10). This means that firm i’s reaction function isprecisely the function linking the maximum points of its familyof isoprofit curves.

Fall 2009-2010 – p.44/133

Comparative statics on n

What if n variable?Two questions:

Quasi-competitiveness of Cournot equilibrium:

∂Qc

∂n> 0?

Convergence to competitive equilibrium:

limn→∞

Qc = Qcompet?

Interest of questions:

Approximation effect of oligopolistic markets on welfare

Relevance of oligopoly theory

Fall 2009-2010 – p.45/133

Comparative statics on n (2)

Illustration 1: Martin (2002, pp: 18-19)Symmetric duopoly, C ′ = 0Cournot eq: Qc → qc

i = Qc/2. Then, FOC:

f(Qd) +Qd

2f ′(Q) ≡ 0, or

2f(Qd) = −Qdf ′(Q)

In general, with n symmetric firms we obtain

nf(Q) = −Qf ′(Q).

Intersection of two curves. Generally, price falls as n increases.

limn→∞

f(Qc) = 0

Fall 2009-2010 – p.46/133


f(Q)

2f(Q)3f(Q)

f(Q)

−Qf ′(Q)

Fall 2009-2010 – p.47/133


Illustration 2: Shubik (1959) - efficient pointDefinitionThe efficient point is a production plan resulting from equating priceto marginal cost for all firms simultaneously, i.e. qe ∈ R

n is anefficient point if it solves the equation f(Q) = C

′

i(qi), ∀i.Intuition:

To study “how far" is the Cournot equilibrium from thecompetitive equilibrium, we will assume that all firms behavecompetitively so that they adjust their production levels to thepoint where price equals marginal cost.

Then we will compare the resulting outcome with the Cournotoutcome.

Fall 2009-2010 – p.48/133


Illustration 2 (cont’d)

n-firm industry

Demand and technology:

Ci(qi) = cqi, c > 0, i = 1, 2, . . . , n.

f(Q) = a − bQ, a, b > 0, Q =

n∑

i=1

qi.

Cournot equilibrium:

qci = a−c

b(n+1) , Qc = n(a−c)b(n+1) , Πi(q

c) = (a−c)2

b(n+1)2 .

Efficient point:a − bQ = c, Qe = a−c

b , qei = a−c

nb .

Fall 2009-2010 – p.49/133


Illustration 2 (cont’d)

Comparing both equilibria we should note that,qci ≤ qe

i , Qc ≤ Qe, P c ≥ P e. and, limn→∞ Qc = a−cb = Qe.

can assume competitive behavior on oligopolistic firmswithout losing much? The answer is NO.

Assume Ci(qi) = k + cqi, k, c > 0. Then,

Profits: Πi(qc) = (a−c)2

b(n+1)2 − k ≥ 0 ⇒ n ≤ a−c√bk

− 1

finite number of firmsnonsense for limn→∞

note (a)∂n

∂k< 0, and (b)

∂Πi

∂n< 0.

Fall 2009-2010 – p.50/133


Quasi-competitiveness of Cournot equilibrium (Telser, 1988)

Let Ci(qi) = cqi, i = 1, 2, . . . , n and p = f(Q), f ′ < 0

Eqbm: q01 = · · · = q0

n = q∗, Q = nq∗, and Π∗i = (f(Q) − c)q∗.

How does Π∗i vary with n?

∂Π∗i

∂n= q∗

∂f

∂Q

(∂Q

∂n−

∂q∗

∂n

).

f ′ < 0 ⇒ sgn∂Π∗

i

∂n= −sgn

(∂Q

∂n−

∂q∗

∂n

).

∂Q

∂n−

∂q∗

∂n=

q∗(2f ′ + q∗f′′

)

f ′(n + 1) + Qf ′′.

If f′′

< 0 then∂Q

∂n−

∂q∗

∂n> 0, and

∂Π∗i

∂n< 0.

Fall 2009-2010 – p.51/133


Convergence of Cournot equilibrium (Telser, 1988)

Assumedemand cuts axes at p and Q.

Ci(qi) increasing, differentiable in q : i ∈ [0, Q],∀i.Ci(0) = 0 and Ci(qi) = Cj(qj), ∀i, j; i 6= j

Proposition 2. Consider a homogeneous product industry satisfyingassumptions above. Then the Cournot equilibrium convergestowards the long run competitive equilibrium ifC

′

i(0) = min ACi(qi), ∀i.

Proposition 3. If Ci(qi) is U -shaped, and∃i s.t.C′

i(0) > ACi(qi),then the Cournot equilibrium does not converge towards thecompetitive equilibrium.

Fall 2009-2010 – p.52/133


Illustration

Let Ci(qi) = cqi, c > 0, i = 1, 2, . . . , n.

f(Q) =

{a − b

∑i qi if

∑i qi ≤

ab = Q, a, b > 0,

0 if∑

i qi > Q

Eqbm:

q∗i = a−c(n+1)b , Q∗ = n(a−c)

(n+1)b , f(Q∗) = a+ncn+1 , Π∗

i (q∗) = (a−c)2

(n+1)2b ,∀i

Differentiate these equilibrium values with respect to n:

∂q∗i∂n

< 0,∂Q∗

∂n> 0,

∂P ∗

∂n< 0,

∂Π∗i

∂n< 0.

Thus, the Cournot equilibrium is quasi-competitive.

Fall 2009-2010 – p.53/133


Illustration (cont’d)

Moreover,limn→∞ q∗i = 0, limn→∞ Q∗ = a−c

b (Q), limn→∞ P ∗ =

c, limn→∞ Π∗i = 0.

Note also C′

i(qi) = c = ACi(qi), so that ,

c = limn→∞

P ∗ = limqi→0

C′

i(q∗i ) = lim

qi→0AC

′

i(q∗i ) = min

qi

ACi(qi) ∀i.

That is the Cournot equilibrium converges towards thecompetitive equilibrium.

Finally, note that assumptions above hold:

p = a < ∞, Q =a

b< ∞, Ci(0) = 0, C

′

i(qi) > 0.

Fall 2009-2010 – p.54/133

Cournot vs. Monopoly vs. Competitive solutions

Proposition 4. Consider a symmetric duopoly whereC′

1 = C′

2 = c. Then,the equilibrium Nash-Cournot price,pN is greater than the competitiveprice,c, and smaller than the monopoly pricepm.

Proof. See lecture notes pp. 61-62

In the space of production plans, we can represent the reactionfunctions and the combinations of output volumes that togethergive rise to the monopoly (QM ) and competitive (QC) output levels.

Proposition says that the aggregate Cournot output (qN1 + qN

2 = QN )

is an intermediate value between the competitive and monopoly

equilibrium outputs.

Fall 2009-2010 – p.55/133

Cournot vs. Monopoly vs. Competitive solutions (2)

0

C

qN

1

qN

2

QN

QN

QM

QM

Qc

Qc

q1 + q2 = Qc

q1 + q2 = QM

q1

q2

Fall 2009-2010 – p.56/133

Stability analysis

Stability in static model? Confusing.

Dynamic assumption ad hoc: Hahn (1962), Seade (1977)

Assumption:Fictitious time. In every period t, t = 1, 2, 3, . . . each firmrecalls the decisions taken by itself and its rival in the previousperiod t − 1.In period t, firm j expects that its rival, firm i will maintain thesame output as in the previous period, qe

it = qit−1, i = 1, 2

Now, reaction functions : qit = wi(q−it−1), i = 1, 2, . . . , n where

q−it−1 denotes a n − 1 dimensional production plans of all firms

except firm i in t − 1.

Fall 2009-2010 – p.57/133

Stability analysis (2)

DefinitionLet qc ∈ R

n be a static Cournot equilibrium production plan.Let q0 = (q10, q20, . . . , qn0) be an arbitrary production plan. Wesay that qc is a stable equilibrium production plan if thesequence of production plans {qt}

∞t=1, qt = (q1t, q2t, . . . , qnt)

converges towards qc. In other words, if limt→∞ qt = qc.

A sufficient condition to guarantee the stability of a Cournotequilibrium is that all reaction functions wi, i = 1, 2, . . . , n becontractions.

Example: Lecture notes pp.64-65.

Fall 2009-2010 – p.58/133


DefinitionLet f be a continuous function defined on [a, b]. Consider twoarbitrary points x, y ∈ [a, b]. We say that f is a contraction if

∣∣∣f(x) − f(y)∣∣∣ ≤ c

∣∣∣x − y∣∣∣ ∀x, y ∈ [a, b], c < 1

In words, f is a contraction if given two arbitrary points in thedomain of the function, the distance between their images issmaller than the distance between the points.

If f is linear this simply means that the slope has to besmaller than one.

Fall 2009-2010 – p.59/133


f(y)

f(x)45

◦

abx y

45◦

a bx y

f(y)

f(x)

Examples where f is a contraction

q1

q20

45◦

α2

α1

Stable equilibrium

q1

q20

45◦

α2

α1

Unstable equilibrium

Fall 2009-2010 – p.60/133


Two objections linked with the construction of the system ofreaction functions.

It does not make any sense to assume that firms are somyopic to ignore the flow of future profits when decidingtoday’s production level.It does not make sense to assume that a firm expects thatits rivals will not vary their decisions from yesterday, inparticular when our firm is changing its decision in everyperiod (see example).

Note that this objections refer to the formation of expectations,i.e. to the construction of the reaction functions, but not to theconcept of Cournot equilibrium.

A more general analysis of the stability of the Cournotequilibrium can be found in Okuguchi (1976), pp. 9-17.

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Price competition - Bertrand (1883)

First critique to Cournot model - 45 years later!!

Critique: the obvious outcome of Cournot’s analysis is thatoligopolists will end up colluding in prices, a behavior ruledout by Cournot.

Variation a Cournot’s model with prices as strategic variable.

In a scenario with perfect and complete information,homogeneous product, without transport costs, and constantmarginal costs, every consumer will decide to buy at the outletwith the lowest price.

Fall 2009-2010 – p.62/133

Price competition (2)

Bertrand’s point goes beyond.

If firms choose quantities, it not specified in Cournot’s modelwhat mechanism determines prices.

In a perfectly competitive market, it is irrelevant what variablesis decided upon because Smith’s “invisible hand" makes themarkets clear.

In oligopoly, there is no such device. A different mechanism isneeded to determine the price that, given the production ofthe firms allow the markets to clear.

Accordingly, it may be more reasonable to assume that firmdecide prices and production is either sold in the market orstocked.

Fall 2009-2010 – p.63/133

Price competition (2)

Bertrand’s model solves one institutional difficulty, but risesanother difficulty. In the real world it is difficult to findhomogeneous product markets. Often we observe marketswhere firms sell their products at different prices and all ofthem obtain positive market shares.In these markets slight variations of prices generate slightmodifications of market shares rather than the bankruptcy ofthe firm quoting the highest price.

Oligopoly models of homogeneous product seem to contain adilemma: Cournot’s model behaves in a reasonable way butuses the wrong strategic variable; in Bertrand’s model the“good" strategic variable is chosen but, as we will see below,behaves in a degenerated way.This is the so-called Bertrand paradox.

Fall 2009-2010 – p.64/133

Bertrand model - Assumptions

n firm industry of a homogeneous product

same constant marginal cost technology, Ci(qi) = cqi ∀i.

strategic variable: prices

Consumers behavior described by a (direct) demand function,Q = f(P ) satisfying all the necessary properties.

Fall 2009-2010 – p.65/133

Bertrand model - Assumptions (2)

Sharing rule:

the firm deciding the lowest price, gets all the demand(Pi < P−i =⇒ Dj(Pi, P−i) = 0, j 6= i)

if all firms decide the same price, they share demand evenly(Pi = Pj , ∀j 6= i =⇒ Di(Pi, P−i) = Dj(Pi, P−i), j 6= i);This is a particular sharing rule based on the symmetry of themodel.A possible alternative sharing rule could be to deciderandomly which firm gets all the market (Hoernig, 2007,Vives, 1998, ch. 5).

consumers have reservation prices sufficiently high so thatthey are all served regardless of the prices decided by firms.To ease computations, wlog, we normalize the size of themarket to the unit, that is

∑ni=1 Di(Pi, P−i) = 1.

Fall 2009-2010 – p.66/133

Bertrand model - Demand & profits

Contingent demand functions

Di(Pi, P−i) =

0 if Pi > Pj , ∀j 6= i,1n if Pi = Pj , ∀j 6= i,

1 if Pi < Pj , ∀j 6= i.

Contingent profit functions

Πi(Pi, P−i) =

0 if Pi > Pj , ∀j 6= i,

(Pi − c) 1n if Pi = Pj , ∀j 6= i,

(Pi − c) if Pi < Pj , ∀j 6= i.

Fall 2009-2010 – p.67/133

Bertrand model - Illustration

Pi

P j

Di(Pi, P j)

c

0 11/2

Πi

PiP jc0

P j − c

1

2(P j − c)

45◦

Figure illustrates firm i’s contingent demand and profits for a

duopolistic market, where P−i reduces to P j the expectation on the

behavior of the rival firm.

Fall 2009-2010 – p.68/133

Bertrand model - Equilibrium

Definition 1. A n-dimensional vector of prices(Pi, P−i) is aBertrand (Nash) equilibrium if and only if

∀i,∀Pi Πi(P∗i , P ∗

−i) ≥ Πi(Pi, P∗−i)

Proposition 5. Let us consider an firm industry where firmsproduce a homogeneous product using the same constant marginalcost technology,Ci(qi) = cqi ∀i. Let us normalize the size of themarket to the unit and assume consumers have sufficiently highreservation prices. Then, there is a unique Bertrand equilibriumgiven byP ∗

i = c ∀i.

Fall 2009-2010 – p.69/133

Bertrand model - Equilibrium (2)

Proof. Duopoly. Consider an alternative price vector(P̃i, P̃j).

if P̃i < P̃j ⇒ Dj(P̃i, P̃j) = 0 and,Πj(P̃i, P̃j) = 0. Firm j can

improve profits withPj < P̃i. Therefore,(P̃i, P̃j) not equilibrium.

if P̃i > P̃j ⇒ Di(P̃i, P̃j) = 0 andΠi(P̃i, P̃j) = 0. Firm i can

improve profits withPi < P̃j . Therefore,(P̃i, P̃j) not equilibrium.

Then, in equilibrium,P̃i = P̃j . ConsiderP̃i = P̃j > c.

Now, Di(P̃ ) = Dj(P̃ ) = 12 andΠi(P̃ ) = Πj(P̃ ) = 1

2(P̃i − c).

But Πi(P̃i − ε, P̃j) = P̃i − ε − c > 12(P̃i − c). Same for firmj.

Price war so that̃Pi = P̃j > c not equilibrium.

Let P̃i = P̃j = c. Now,Di(P̃ ) = Dj(P̃ ) = 12 and

Πi(P̃ ) = Πj(P̃ ) = 0. No firm has a profitable unilateral deviation.

A price vectorP̃i = P̃j = c is the only Bertrand equilibrium.

Fall 2009-2010 – p.70/133


Reaction functions. Let Pm denote the monopoly price.

If Pj > Pm, firm i’s best reply is to choose the monopoly priceto obtain monopoly profits.

If Pj < c, firm i’s best reply is to choose a price equal to themarginal cost to obtain zero profits. Actually, any price Pi > Pj

yields zero profit to firm i, so that the reaction functionbecomes a correspondence.

If c < Pj < Pm we have to distinguish three cases.

If Pi > Pj, then Πi = 0;

If Pi = Pj, then Πi = (Pi − c)12 ;

If Pi < Pj, then Πi = (Pi − c). In this case the profitfunction is increasing in Pi, so that firm i’s best reply isthe highest possible price, that is Pi = Pj − ε, for εarbitrarily small.

Fall 2009-2010 – p.71/133


Summarizing, firm i’s reaction function is,

P ∗i (Pj) =

Pm if Pj > Pm

Pj − ε if c < Pj ≤ Pm

c if Pj ≤ c

By symmetry, firm j has a similar reaction function exchanging the

subindices adequately.

Fall 2009-2010 – p.72/133


Pi

Pj

Pm

Pm

P∗

j (Pi)

P∗

i (Pj)

45◦

c

c

Reaction fncts intersect only at Pi = Pj = c: the Bertrand eqbm.

Fall 2009-2010 – p.73/133

Cournot vs. Bertrand

Why different behavior? ⇒ Different residual demand.Illustration: Duopoly with q1 + q2 same aggregate output in C and B.

D−1(q∗2)

P∗

CM

0

P

C

B

B

B

C

A

q1q∗

1 q∗

1+ q

∗

2

IM

Fall 2009-2010 – p.74/133

Bertrand and “proper" behavior

Attempts to obtain “normal" behavior in Bertrand models: 6variations

Capacity constraints

Contestability

Price rigidities

Commitment

Conjectural variations

Dynamic models

Fall 2009-2010 – p.75/133

Variation 1. Capacity constraints

Assumption on technology: decreasing returns to scale/strictly convex costs.

Models:exogenous: Edgeworth (1897),endogenous: Kreps-Scheinkman (1983)

Preliminaries: Rationing rulesEfficient rationing ruleProportional rationing rule

Consider a duopoly with P1 < P2 and q̄1 ≡ S(P1) < D(P1).

Fall 2009-2010 – p.76/133

Variation 1. Capacity constraints. Rationing (1)

Efficient rationing rule

“first come, first served" rule.Firm 1 serves the most eager consumers; firm 2 serves the rest.

D1(P1) = q1

D2(P2) =

{D(P2) − q1 if D(P2) > q1

0 otherwise

Firm 2’s residual demand: shift market demand inwards by q1.

This rule is efficient because it maximizes consumer surplus.

Fall 2009-2010 – p.77/133


Efficient rationing rule

P

q

P2

q1

q1

+ q2

q2

q = D(P )

q = D(P ) − q1

P1

Fall 2009-2010 – p.78/133


Proportional rationing rule

Randomized rationing rule. Any consumer same prob. of rationed.Probability of not being able to buy from firm 1 is

D(P1) − q1

D(P1).

Firm 2’s residual demand rotates inwards: slope of the residualdemand is modified by the probability of buying at firm 2.

D2(P2) = D(P2)(D(P1) − q1

D(P1)

).

This rule is not efficient. Consumers with valuations below P2 may

buy the commodity because they find it at a bargain price P1.

Fall 2009-2010 – p.79/133



P1

P2

P

qq2

q1

q1

+ q2

D(P )

D(P2)(D(P1) − q

1

D(P1)

)

Fall 2009-2010 – p.80/133


Efficient rationing rulevs.


At any price the PRR yields higher residual demand to firm 2.More consumers are served under PRR although consumersurplus is not maximized.

D(P2)(D(P1) − q

1

D(P1)

)

D(P ) − q1

D(P )

P

q

P1

P2

q1

qef2

qpr

2

Fall 2009-2010 – p.81/133

Variation 1. Capacity constraints. Edgeworth (1)

Assumptions

Demand: P = 1 − q

n firms

CRS up to Ki (Ki = K) Note: K exogenous

nK < 1 Thus, K < 1n

K > 12n

Starting point: Full collusion

qm = 12 ; Pm = 1

2 ; Πm = 14

qi = 12n ; Πi = 1

4n

Collusive agreement stable? NO

Fall 2009-2010 – p.82/133


Price war!

Firm i undercutting Pi = Pm − ε, sells K and obtainsΠK

i = (12 − ε)K > 1

212n for ε sufficiently small.

Firm j may undercut firm i’s price to obtain profitsΠK

j = (Pi − ε)K. And so on

Two points to note:How far will this undercutting arrive? Assume duopolyIf undercutting, rival is monopolist over residual demand

Fall 2009-2010 – p.83/133


0 1

1

1/21/4 K

Pm

= 1/2

Pm

− ε

Fall 2009-2010 – p.84/133


Other features

Residual demand left to undercut firm: RD = 1 − K

Monopoly profits over RD : ΠmKj =

(1−K

2

)2

Output level 1−K2 must be feasible, i.e. 1−K

2 < K or K > 13 .

Range of feasible values for the K to be meaningful:K ∈ (1

3 , 12).

Fall 2009-2010 – p.85/133


q

P

0

1 − K

2

1 − K

1 − K1 − K

2

0

K

1

Fall 2009-2010 – p.86/133


Other features (cont’d)

Both firms full capacity, qi = K, q = 2K and minimum feasiblemarket price is Pmin = 1 − 2K

Undercut firm as two options:undercut its rival, ormax profits over residual demand

Define P̂ as the price yielding the same profits in both

situations: P̂K =

(1−K

2

)2

P̂ ∈(1 − 2K, 1−K

2

)

Fall 2009-2010 – p.87/133


Summarizing

1

3

1

2

1

2

1 − K

2

1 − K

2

1

1

1 − 2K

0K 2K

P

q

Fall 2009-2010 – p.88/133


Behavior

Consider a fictitious time span where rivals decide in alternatetime periods

Assume it is firm i’s turn

Compares profits if undercutting or monopoly over RD

If undercutting more profitable price war goes on

At some point in the mutual undercutting process where thecorresponding firm will be indifferent, i.e. firm will hit price P̂

Next firm, undercut on P̂ yield less profit than monopoly overRD. Therefore, will give up price war and will jump up tomonopoly price 1−K

2 .

Fall 2009-2010 – p.89/133


Conclusion

Price cycle non-stop ⇒ No equilibrium.

P

1/2

1 − K

2

1 − 2K

0

P̂

Fall 2009-2010 – p.90/133

Variation 1. Capacity constraints. Kreps-Scheinkman (1)

Assumptions

two-stage game: production-then-prices

duopoly

capacity levels q̄i

technology: constant mg cost (=0) up to q̄i, then ∞.

efficient rationing rule

concave market demand

Conclusion

Pure-strategy equilibrium results in the production and the price that

would have resulted in a one-shot Cournot game.

Fall 2009-2010 – p.91/133


Tirole’s illustration

q1 + q2 = 1 − P

Price game

assume q̄i was bought in the previous stage at a unit costC0 ∈ [34 , 1]

monopoly profit Πm = P (1 − P ); maxP Π = 14

firm i’s total profit is at most 14 − C0q̄i

firm i’s total profit is negative for q̄i > 13 .

Assume q̄i ≤13 (i.e. q̄i not too large)

Note that qi = 13 is the Cournot symmetric equilibrium

Assume qi ∈ [0, 13 ], i = 1, 2

Fall 2009-2010 – p.92/133


Tirole’s illustration ResultLemma 4. In a pure-strategy equilibrium,P ∗

1 = P ∗2 = 1 − (q1 + q2).

That is, firms sell up to capacity.“Proof"

At P ∗ consumers are not rationed

Hence, no incentives to lower price because firms are alreadyselling their full capacity

Incentives to increase price?Πi(Pi, P

∗) = (1 − qi − q̄j)qi, where qi ≤ q̄i becausePi ≥ P ∗

Πi(Pi, P∗) is profit function of a firm deciding qi given

expectation q̄j

Hence, Πi(Pi, P∗) is Cournot profit function.

Fall 2009-2010 – p.93/133


Tirole’s illustration Result (cont’d)

Incentives to increase price? (cont’d)Πi(Pi, P

∗) is concave in qi

∂Πi

∂qi|q̄i

= 1 − 2q̄i − q̄j > 0 because q̄z < 13

Hence, lowering output below q̄i (i.e. raising the priceabove P ∗) is not optimal.

Capacity gameProfits are Πi(q̄i, q̄j) = (1 − q̄i − q̄j − C0)q̄i

But these are the profits that would have obtained should firms haddecided on output levels (q̄i, q̄j) and market clearing would have setthe price.

Conclusion

Cournot equilibrium is the eq. of first stage of the game.

Fall 2009-2010 – p.94/133

Variation 1. Price-Quantity competition

Simultaneous decisions mixed strategies

Dasgupta-Maskin (1986)

Maskin (1986)

Supply-function equilibriaGrossman (1981)

Hart (1982)

Two-stage capacity-price games

Kreps-Scheinkman (1983)

Vives (1993)

Boccard-Wauthy (2000)

Two-stage quantity-price games

durable goods: Friedman (1988)

Maskin (1986)

perishable goods: Judd (1990)

Simultaneous vs. sequential decisions: Chowdhury (2005)

Dynamic models

durable goods: Judd(1990)

alternate decisions: Maskin-Tirole (1988) Fall 2009-2010 – p.95/133

Variation 2. Contestable markets

Bailey, Baumol, Panzar, Willig (1980s). Martin (2002,supplement)

Extend the theory of perfectly competitive markets tosituations where scale economies are relevant.

A market is contestable when entry is free and exit is costless

Potential entrants evaluate the profitability of entry wrt theprices of the incumbents before entry.In other words, potential entrants think that they can undercutincumbents and “steal" all the demand before the incumbentswill react.

Contestable market if vulnerable to “hit-and-run" entry.

In a contestable market the equilibrium production is alwaysefficient regardless of the number of firms, since price alwaysequals marginal cost.

No Nash strategies.Fall 2009-2010 – p.96/133

Variation 3. Sticky prices

Casual empirical observations of price rigidities downwards.

First model: Sweezy (1939)

Sweezy’s idea: oligopolistic firm when lowering its priceshould expect its rivals’ to react in a similar fashion. But whenthe firm increases its price, its rivals’ should be expected notto react.

Sweezy’s construction assumes a more elastic demand forincreases than for decreases in prices.

Modern treatments of these arguments in Bhaskar (1988),Maskin and Tirole (1988b), or Sen (2004).

Fall 2009-2010 – p.97/133

Variation 3. Sticky prices. Sweezy’s model (1)

Assumptions

Duopoly

Constant marginal cost k

Market demand: p = A − (qi + qj)

Assume firms are producing q̂i and q̂j respectively.

ConjecturesFirm i conjectures that firm j will continue producing q̂j aslong as it produces qi ≤ q̂j (i.e. price increases).

Firm i also conjectures that if it changes its production toqi > q̂j (i.e. price decreases), then firm j will increase itsproduction until level with that of firm i.

Fall 2009-2010 – p.98/133


Given these conjectures, the only consistent production plansare vectors of the type q̂i = q̂j.

If q̂i < q̂j then, firm j’s conjectures say that firm i will increaseits production till q̂j .

Mutatis mutandis in the symmetric case q̂i > q̂j .

Restrict analysis to situations where both firms decide thesame production levels q̂i = q̂j = q̂.

Firm i faces a demand function showing a kink at qj = q̂:

p =

{A − qi − qj if qi < qj ,

A − 2qi if qi > qj .

Marginal revenue function is discontinuous at that point qj = q̂:

IMi =

{A − qj − 2qi if qi < qj ,

A − 4qi if qi > qj .

Fall 2009-2010 – p.99/133


qiqj

A − qj

A − 2qj

A − 3qj

A − 4qj

P

0 A/2 qiqj

A − qj

A − 2qj

A − 3qj

A − 4qj

P

0 A/2

k

k

Fall 2009-2010 – p.100/133


Equilibrium

Assume now that firm j produces qj = q̂.

Firm i’s problem is maxqi(A − qi − yj(qi, q̂) − k)qi

where yj(qi, q̂) = max{qi, q̂}.

yj represents firm j’s reply:

to produce q̂ if qi ≤ q̂ andto produce qi if qi > q̂

Reaction function.If qj is large enough (with respect to k), the equalitybetween marginal revenue and marginal cost appears inthe lower segment of the marginal revenue curve;otherwise marginal revenue and marginal cost intersect inthe upper part of the marginal revenue curve

Fall 2009-2010 – p.101/133


Reaction functions

q∗i (qj) =

0 if qj ≥ A − k,A−k−qi

2 if A−k3 ≤ qj ≤ A − k,

qj if A−k4 ≤ qj ≤

A−k3 ,

A−k4 if qj ≤

A−k4 .

Both curves intersect in the interval qi = qj = q̂ ∈ [A−k4 , A−k

3 ].Therefore, there is a continuum of equilibria.

Note though that in all those equilibria the aggregate production lies

in the interval 2q̂ ∈ [A−k2 , 2(A−k)

3 ], that is from the monopoly output to

the Cournot output.

Fall 2009-2010 – p.102/133


A − k

A − k

A − k

2

A − k

2

A − k

3

A − k

3

A − k

4

A − k

4

qi

qj

q∗

j (qi)

q∗

i (qj)

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Variation 4. Commitment (1)

Commitment

Situation of strategic interaction were one agent may restrictin a credible way its choice set to gain an advantage over acompetitor.

Stackelberg (1934) first to propose a model to capturecommitment in oligopoly pricing.

Stackelberg’s model

A firm acts as leader and several other firms (followers)conditional on the behavior of the leader, choose their actions.

Leader aware of the behavior of the followers.

Leader’s profit maximizing decision is conditional on thereaction of the followers, and the followers’ profit maximizingdecisions are conditional on the choice of the leader.

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Assumptions

duopoly

P = a − b(q1 + q2)

Ci(qi) = c0 + cqi

Firm 1: leader; Firm 2: follower

2-stage model. Firm 1 chooses q1 first. Next, firm observes q1

and chooses q2

Commitment: Firm 1 cannot modify q1.

Equilibrium: Subgame perfect eq: profit maximizing production plan

(q∗1, q2(q∗1)).

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Follower’s optimal choice

maxq2Π2(q1, q2) =

(a − b(q1 + q2)

)q2 − c0 − cq2, yielding,

q2 = a−c2b − 1

2q1.

Leader’s optimal choice

Choose a profit maximizing output level anticipating the impact ofthis decision on the follower:

maxq1

Π1(q1, q2) s.t. q2 =a − c

2b−

1

2q1.

Accordingly, q∗1 = a−c2b .

Thus, follower’s optimal decision is q∗2 = a−c4b

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Intuition

Leader chooses a point on a isoprofit curve on firm 2’s reactionfunction.Follower plugs in the leader’s decision q∗1 in reaction function, toobtain q∗2 .

Completing characterization

P ∗ =a + 3c

4,

Π∗1 =

(a − c)2

8b− c0,

Π∗2 =

(a − c)2

16b− c0.

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q2

q1qc

1

qc

2

q∗

2

q∗

1

Π∗

1

Π∗

2

Πc

2

Πc

1

C

S

0

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Variation 4. Stackelberg vs. Cournot

Cournot eqbrm

qc1 = qc

2 =a − c

3b, P c =

a + 2c

3,

Πc1 = Πc

2 =(a − c)2

9b− c0.

Stackelberg vs.Cournot

q∗1 > qc1; q∗2 < qc

2;

Π∗1 > Πc

1; Π∗2 < Πc

2;

Q∗ > Qc; P ∗ < P c.

Leader has a “first-mover advantage" over the follower.

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Variation 4. SGPE vs. Nash eq.

SGPE means that empty (non-credible) threats by thefollower are ruled-out.

SGPE requires the follower’s strategy to be optimal in front ofany decision of the leader q1, and not only against theequilibrium output q∗1 .

In contrast, a Nash equilibrium only requires optimality alongthe equilibrium path.

In our two-stage game, it only imposes production levels forthe leader that do not generate loses.

For C0 = 0, any output in [0, (a − c)/b] is sustainable as aNash equilibrium of the two-stage game.

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Variation 5. Conjectural variations

Definition

Firms aware of the impact of their decisions on rivals

Incorporate that impact in profit maximization decisionprocess

CV: assumptions on how the behavior of a firm impacts onrivals’ decision processes

CV: set of conjectures (expectations) of every firm on thesequence of moves of rivals

Bowley (1924); Boyer and Moreaux (1983), Bresnahan (1981),Perry (1982).

Models:Bowley; Consistent CV; Marschack-Selten

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Variation 5. Conjectural variations - Bowley (1)

Assumptions

Duopoly

Same assumptions on profits as Cournot

FOC on profits are

∂Πi

∂qi+

∂Πi

∂qj

dqj

dqi= 0, i 6= j

wheredqj

dqirepresents firm i’s conjecture on the behavior of firm j,

after a marginal variation of its production level, dqi.

This is the content of the Conjectural variation

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Illustration

linear demand p = a − b(q1 + q2)

common constant marginal cost c

Reaction function:

qi(qj) =a − bqj − c

2b + b(dqj

dqi

)

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Cooperative CV

Assume firms coordinate to adjust the aggregate production to themonopoly level by equally adjusting their individual outputs.

This translates in a conjectural variationdq2

dq1=

dq1

dq2= 1. Then,

q1(q2) =a − bq2 − c

3b,

q2(q1) =a − bq1 − c

3b.

and

q1 + q2 =a − c

2b.

that corresponds to the monopoly solution.

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Competitive CV

Assume that every firm conjectures that if it reduces production inone unit, the rival will increase its production in one unit so that theaggregate output remains constant.


dq1=

dq1

dq2= −1. Then,

q1(q2) =a − bq2 − c

b, q2(q1) =

a − bq1 − c

b.

and

q1 + q2 =a − c

b.

This is precisely the competitive solution.

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Cournot CV

Assume each firm takes the output of the rival as given.


dq1=

dq1

dq2= 0. Then,

q1(q2) =a − bq2 − c

2b, q2(q1) =

a − bq1 − c

2b.

and

q1 + q2 =2(a − c)

3b.

This is the Cournot solution.

CV ∈ [−1, 1] generate the perfect collusion, perfect competition, and

Cournot solutions.

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Other CV

Bertrand: infinite conjectural variations;

Stackelberg:zero conjectural variation for the followerfinite conjectural variation for the leader

Sweezycollusive conjectural variations when output expandsCournot conjectural variations when output contracts.

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Variation 5. Consistent Conjectural variations (1)

So far, values of the conjectural variations yielding the equilibriumof some particular models.

Definition

Find an equilibrium set of conjectural variations.

In this equilibrium no firm would have incentives neither tomodify its behavior nor to change its conjectural variations.

This approach tries to identify a Nash equilibrium inconjectural variations. This equilibrium is called a set ofconsistent conjectural variations.

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Assume duopoly, linear demand, constant mg. cost and let,

k1 ≡dq2

dq1; k2 ≡

dq1

dq2

Then,

q1 =a − bq2 − c

2b + bk1, and q2 =

a − bq1 − c

2b + bk2

Therefore,

dq1

dq2= k2 = −

1

2 + k1, and

dq2

dq1= k1 = −

1

2 + k2.

Solving,k2 = −1, and k1 = −1

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Comments

In example, equilibrium in consistent conjectural variations,both firms use conjectural variations -1 giving rise to thecompetitive equilibrium.

Suggests possibility that the behavior of the firms producing ahomogeneous product may be approximated by thecompetitive behavior even with few firms in the market.

If simultaneous decision model, it does not make sense qi

being a function of qj or viceversa.

Such a situation implies that firm i observes firm j’s decision,and according to its conjectural variation, determines qi.

Therefore, conjectural variations is a device to understand(not to explain) the decision process of firms aware of theirstrategic dependency.

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Variation 5. Conjectural variations - Marschack-Selten (1 )

Static model, 1977

Firms simultaneously announce prices → public info.

Recursive process to revise pricesFirm 1 revises price → (n − 1) rivals adjustFirm 1 re-revises price → (n − 1) rivals adjust... until Firm 1 does not want to revise its price.Firm 2 revises its price → (n − 1) rivals adjust, etc, etc

Consistent conjectural variations

A non-coop equilibrium: price vector satisfyingno profitable deviation for any firmreaction function maximizes profits

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Variation 5. Conjectural variations - Marschack-Selten (2 )

Dynamic model, 1978

Firms after choosing their first price face an adjustment costfor any price change.

If a firm i varies its price, there is a period of time between thenew price is posted and the competitors adjust their prices.

The adjustment cost faced by firm i is sufficiently high tooffset the extra profits firm i may obtain in the interim perioduntil the rivals react.

Accordingly, a price variation is only profitable if it generatesmore profits in the long run, once the rivals have reacted.

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Variation 6. Dynamic models (1)

Two alternative ways of explicitly consider the introduction oftime in a model:

Repeated games or supergames. These replicate a staticCournot (Bertrand) type of game a finite or infinitenumber of times (see Friedman, 1977).dynamic strategies of firms are of Markov type and theobjective of the model is to characterize a Markov perfectequilibrium (see Maskin and Tirole, 1987, 1988a).

An overview of dynamic oligopoly models can be found inFudenberg and Tirole (1986), Kreps and Spence (1984),Shapiro (1989) or Maskin and Tirole (1987, 1988a, 1988b).

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Repeated games

Set-up

Game is played a certain number of times (iterations)

One iteration is independent of another,

players can condition their present or future behavior to thehistory of moves.

room for punishments as (credible) threats to affect players’future decisions.

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Repeated games (cont’d)

Illustration (Tirole, 1988)

Symmetric Bertrand duopolyfirm lowest price, gets all demandboth firms same price, share evenly demand

Game repeated T + 1 times (T finite or infinite)

Πi(pit, pjt), t = 0, 1, . . . , T

Firm i’s objective: max present value flow of profits

T∑

t=0

δtπi(pit, pjt), δ is the discount rate

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Illustration (Tirole, 1988) (cont’d)

In every period t both firms simultaneously choose a price.

Firms have perfect recall of all the history of past decisions.Let, Ht = (p10, p20; p11, p21; p12, p22; . . . ; p1,t−1, p2,t−1) be thehistory of prices chosen by both firms up to period t.

Firm i’s strategy depends on Ht.

Equilibrium

Characterize a perfect equilibrium.

For any history Ht in period t, firm i’s strategy from t on shouldmaximize the present value of the flow of its future profitsconditional on the expectation of firm j’s strategy in period t.

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T finite.

By backward induction, in period T the model is equivalent tothe static version. Accordingly, in equilibrium price equalsmarginal cost.

Decisions in period T are not dependent on T − 1. Therefore,T − 1, as if it would be the last period. Thus, for any HT−1, theequilibrium strategies price = marginal cost.

We can repeat this reasoning until the initial period.

Summarizing, if the number of iterations is finite, the only equilibrium

of the repeated game is simply the iteration in every period of the

equilibrium strategies of the static game.

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Repeated games (cont’d)T infinite.

Multiplicity of equilibria:

Eq. under T finite

Other equilibria:Let p ∈ [pc, pm] and consider symmetric strategies:

at t = 0 both firms choose pIn t = τ if both chosen p in the past, choose p in t = τIf at t = s deviation, choose p = MC forever

ProfitsIf no deviations, 1

2Π(p)(1 + δ + δ2 + δ3 + · · · ).If at t = s one deviates, obtains π(p) in s, and zeroafterwards:12Π(p)(1+ δ + δ2 + δ3 + · · ·+ δs−1)+ δsπ(p) = π(p) δ

2(1−δ)

If δ ≥ 12 deviating is never optimal

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Repeated games (cont’d)T infinite.

Argument true ∀p ∈ [pc, pm]

Any price p can be supported as equilibrium → folk theorem

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Markov games

Interaction among time periods

“Markov perfect equilibria": firms condition their actions to areduced subset of state variables rather than in the full historyof the game.

Consider an infinite duopoly à la Cournot. Letπi(qit, qjt), t = 0, 1, 2, . . . firm i’s profits in period t.

Assume profits concave in qit and decreasing in qjt

Accordingly, reaction functions are well-defined and arenegatively sloped.

Every firm aims at maximizing the present value of the flow offuture profits,

∑∞s=0 δsπi(q1,t+s, q2,t+s), δ discount factor

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Markov games (cont’d)

decision process:in odd periods, firm 1 decides a production volume thatremains fixed until the next odd period, i.e. until t + 2. Inother words, q1,t+1 = q1,t, if t is odd.

in even periods firm 2 decides a production volume thatremains fixed until the next even period, i.e. q2,t+1 = q2,t, ift is even.

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In every period, the relevant state variables are the onesinvolved in the profit functions.

In odd periods when firm 1 decides, the relevantinformation is the production of firm 2, q2,2k+1 = q2,2k.Firm 1’s decision is contingent only on q2,2k, so that itsreaction function is of the type q1,2+1 = w1(q2,2k).

In even periods when firm 2 chooses its output level, itsreaction function is q2,2k+2 = w2(q1,2k+1).

We call these Markov strategies dynamic reaction functions

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The objective of the model is to find a pair (w1, w2) thatconstitutes a perfect equilibrium.

That is, for any period t, the dynamic reaction function of afirm must maximize the present value of the discounted flowof future profits given the dynamic reaction function of the rivalfirm.

This pair (w1, w2) is called a Markov perfect equilibrium.

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A course on Industrial Organizationpareto.uab.cat/xmg/Docencia/IO-en/IO4IDEAHomog.pdfA course on Industrial Organization Xavier Martinez-Giralt Universitat Autonoma de Barcelona` [email protected]