Oligopoly Marc Bourreau - Télécom ParisTechses-perso.telecom-paristech.fr/bourreau/files/cours02_oligopoly_eng.pdf · Oligopoly, deﬁnition and examples Cournot model, where ﬁrms

Industrial Organization 02Oligopoly

Marc Bourreau

Telecom ParisTech

Marc Bourreau (TPT) Lecture 02: Oligopoly 1 / 42

Outline

Oligopoly, definition and examplesCournot model, where firms compete in quantitiesBertrand model, where firms compete in pricesBertrand paradoxeFrom Bertrand to Cournot: capacity constraintsCournot competition with n firmsComparison of market powers: monopoly, Cournot, and Bertrand


Introduction

Introduction

Definition of an oligopoly

A market or industry in which a small number of firms compete.

Most markets fit this description: telecommunications, software industry,but also mineral water industry, etc.In an oligopolistic market, a firm cannot ignore the behavior of competi-tors...... and the reaction of competitors to its own decisions.The theory of oligopoly aims at studying these strategic interactions.


Introduction

An example

The oligopoly of mineral water producers in France:Three firms (Danone, Nestlé, and Neptune) share around 80% of the mar-ket.Products are marketed under various brand names (Danone: Evian, Volvic,...; Nestlé: Perrier, Contrex, ... ; Neptune (formerly Castel): Saint-Yorre,Vichy, Cristalline, ...).Emergence of the Castel group in 1993 following a decision of the EuropeanCommission which authorized the acquisition of la Société des eaux minéralesdu bassin de Vichy.Differentiation between mineral waters (health, well-being), different typesof packaging, makes the price comparisons difficult.Interdependancy of strategic decisions: if a firm increases the price ofmineral water, its competitors can chose to do the same or not to react,hoping to capture a part of the customer base.


Cournot model

Cournot model

Cournot model (1838), developed by Augustin Cournot, a French engineer.Two firms produce identical (homogenous) goods ("perfect substitutes")and compete in quantities.The price is determined in the way that the entire quantity of producedgoods is sold. Therefore one sets p = P(Q) where Q = q1 + q2 is the totalquantity produced.The marginal cost of producing each unit of the good is assumed constantand identical for the two firms: cThen, the profit function of the firm i is

Πi = (P(Q) − c)qi.

For simplicity, we assume that the inverse demand function is P(Q) = 1−Q.


Cournot model

Cournot model

Each firm sets its quantity to maximize its profits, given the quantity setby the other firm.The first-order condition of the maximization problem is:

1 − (q1 + q2) − c − qi = 0.

We compute a symmetric equilibrium such that qi = q. Thus:

q? =1 − c

3.

and

Π? =(1 − c)2

9.


Cournot model

Cournot model

The equilibrium profit is:

Π? =(1 − c)2

9.

In Cournot competition, firms make profits!

However, the Cournot model seems somewhat unrealistic:Few examples of markets where firms set quantities rather than prices;We don’t know well how the price is set up on the market.

A model where firms set prices rather than quantities? → Bertrand model.


Bertrand paradox The model

Bertrand model

Bertrand model (1883), developed by Joseph Bertrand, a French engineer.Two firms produce identical goods ("perfect substitutes") and compete inprices.The demand function is q = D(p).The marginal cost of production is constant and identical for both firms: cWe assume (not indispensable) that the market is split evenly if firms offerthe same price. We thus have:

Di(pi, pj) =

D

(pi)

if pi < pj12 D

(pi)

if pi = pj0 if pi > pj

.

The profit function of firm i is:

Πi(pi, pj) = (pi − ci)Di(pi, pj).


Bertrand paradox Bertrand equilibrium

Bertrand equilibrium

We find the Nash equilibrium for this one-step game.

Bertrand paradox

The game has a unique Nash equilibrium such that firms set p∗1 = p∗2 = c. A theequilibrium, we have Π∗1 = Π∗2 = 0 and W = W∗.

A strong result:When the number of firms goes from one (monopoly) to two (duopoly),the price decreases from the monopoly price to the competitive price andstays at the same level as the number of firms increases further.Two firms suffice to reach a perfectly competitive equilibrium.Because this result is extreme and generally inconsistent with reality, it iscalled the "Bertrand paradox".



An example

The price war between Intel and AMD, chip manufacturers, in 2006.AMD ended the year with a loss and Intel’s benefits dropped by 42% in2006.

Interview with Mario Rivas (CEO of AMD, Source:l’Expansion) :

"This price war with Intel was ridiculous. I wish I wasable to match my prices to the market rate (...). Theproblem was that, despite the 2 billion dollars of stockswe had, Intel had 7 factories to run at full capacity. Theycut the price and we have no choice but to follow theirpricing".



Demonstration

If p∗1 > p∗2 > c:

Then firm 1 increases its profit by setting p∗1 = p∗2 − ε.

If p∗1 = p∗2 > c:

Then firm 1 increases its profit by setting p∗1 = p∗2 − ε, as for a small ε,

D(p∗1)(p∗1 − c)/2 < D(p∗1 − ε)(p∗

1 − c − ε)

If p∗1 > p∗2 = c :

Then firm 2 increases its profit by setting p2 = p∗2 + ε.



Bertrand competition with different marginal costs

Let’s assume that c1 < c2.

If costs are close enough: c1 < c2 < pm(c1)

There is a unique Nash equilibrium with p∗1 = c2 − ε et p∗2 = c2.Only firm 1 makes profit: Π∗1 = (c2 − c1)D(c2) et Π∗2 = 0.

If firm 1 is much more efficient than firm 2: c1 < pm(c1) < c2

There is a unique Nash equilibrium with p∗1 = pm(c1) and p∗2 = c2.Only firm 1 makes profit: Π∗1 = Πm

1 and Π∗2 = 0.


Bertrand paradox Solutions to Bertrand paradox

Solutions to Bertrand paradox

Four important solutions to "Bertrand paradox", corresponding to four impor-tant assumptions on which the model relies :

1 Products are homogeneous2 Short-run competition (a static analysis of a one-step game)3 Firms do not have capacity constraints4 Consumers are perfectly informed

Relaxing one of these assumptions solves the paradox:1 Product differentiation2 Dynamic competition (repeated interactions)3 Firms are capacity-constrained4 Imperfect information



Product differentiation

Let’s assume, for instance, a geographic differentiation.

Two icecream vendors, 1 and 2, are located at the two extremities of abeachIf p1 = c, is p2 = c + ε > c possible?Consumers close to vendor 2 can prefer buying a little more expensivefrom 2 than moving to 1!Horizontal and vertical differentiation theory: the Hotelling model...

In the presence of product differentiation...

A situation such as that pi > c can be an equilibrium.

→ See the lecture about differentiation.



Dynamic competition

Bertrand competition model assumes that firms compete only during asingle period.In this situation, as we start from p1 = p2 > c, a firm has strong incentivesto undercut its price.In a more dynamic framework, what can happen?In a dynamic situation, firms are going to take account the effect of its pricecut on its rival’s behavior in future periods.If there is a "punishment" (price war...), it will compare the short-term andlong-term gains.

In a repeated interaction framework...

A situation with pi > c can be an equilibrium.

→ See the lecture about collusion.



Capacity constraints

Bertrand model assumes that firms do not have capacity constraints.If p1 = p2 = c, the two firms share the demand: D(c)/2If firm 2 increases slightly its price, p2 = c + ε, we assumed that firm 1meets the demand, that is D(c).But firm 1 can be unable to meet the whole demand: capacity constraintsIn that case, firm 2 can increase her price and still retain a part of the market

When firms face capacity constraints...




Imperfect information

With imperfect information...


The Diamond paradox

Consumers are not informed about pricesCost ε to move from a store to another (search cost)If p1 = p2 < pm then deviation is possible to p1 + ε/2


Price competition with capacity constraints A model

Price competition with capacity constraints

We are going to study a price competition model when firms have constraintsin production capacity.

We consider the following model:Two firms compete in prices, but are capacity-constrainedThe demand function is linear, D(p) = 1 − pThe inverse demand function is: p = P(q1 + q2) = 1 − (q1 + q2)Firm i cannot produce more than its production capacity qi. Thus, we haveqi ≤ qi

The unit cost of acquiring capacity qi is c0 ∈ [3/4, 1].There is no production cost (c = 0).


Price competition with capacity constraints A model


We need to define a rationing rule.The rationing rule determines "which consumers" are being served whenthe firm cannot meet the entire demand.We apply the "efficient" rationing rule: consumers with higher willingness topay are first served.If p1 < p2 and q1 < D(p1) :

D̃2(p2

)=

{D

(p2

)− q1 if D

(p2

)≥ q1

0 if D(p2

)< q1

.

Other possible rule: the "proportional" rationing rule


Price competition with capacity constraints Results

Results

We obtain the following result:


The unique Nash equilibrium is such that the firms set the same price

p∗ = 1 −(q1 + q2

).

Consequence: at the stage of defining their production capacities, firms have agross profit equal to:

Πgi =

[1 −

(qi + qj

)]qi,

That is? The profit function in the Cournot model.



Proof: preliminaries

First, given that c0 ∈ [3/4, 1], what is the upper bound for qi?

To answer the question, what is the maximum profit of firm i ?

→ It is the monopoly profit!

As D(p) = 1 − p and c = 0, the monopoly price is equal to... ? 1/2

Thus the monopoly profit is... ? 1/4 − c0qi

Therefore qi ≤ 1/3 because we have 1/4 − c0qi ≥ 0.



Proof of existence

Now, let’s demonstrate that p∗ = 1 −(q1 + q2

)is a Nash equilibrium.

Observation: p∗ > c because q1 + q2 < 2/3.

Can firm i lower her price?

No, she would not increase her profit as she is already at the maximum of herproduction capacity.

→ "Bertrand competition" does not work because firms face capacity con-straints.



Proof of existence

If the firm i sets a higher price p > p∗?Firm j captures the entire demand... up to the maximum of its productioncapacity.A fraction of the demand is not satisfied and firm i takes it back: we call itthe residual demandThe residual demand is equal to 1 − p − qj

Thus firm i makes profit:

p(1 − p − qj

)=

(1 − q − qj

)q.

It is the Cournot profit! It is concave in q and its derivative in q = qi is

1 − 2qi − qj ≥ 0 as qi ≤ 1/3,

Therefore, the optimal value of q is q = qi.



Proof of uniqueness

p∗ = 1 −(q1 + q2

)is the unique Nash equilibrium.

p1 = p2 = p > P(q1 + q2

)is not an equilibrium because at least one firm

does not reach its maximum production capacity. Therefore she can lowerher price.

p1 = p2 = p < P(q1 + q2

)is not an equilibrium. With pi = pi + ε, the firm

would sell the same capacity (its production capacity) at a higher price.p1 < p2 not feasible, because firm 1 is encouraged to increase its price.


Price competition with capacity constraints More general result

More general result

We consider a two-stage game in which:First, firms choose simultaneously their production capacity;Firms observe the chosen capacities and choose simultaneously their prices(price competition).

Kreps and Scheinkman (1983): capacity + price = quantities

If the demand function is concave and we use the efficient-rationing rule, thenthe equilibrium f the two-stage game is equivalent to the equilibrium of Cournotcompetition (competition in quantities).


Cournot or Bertrand competition?


Which one is the best-suited model, Bertrand or Cournot?

Informal rule (rule of thumb)

If the production capacity can be easily adjusted, Bertrand competition modelbetter represents the duopoly competition. Otherwise, if it is difficult to adjustthe production capacity, the Cournot model is more appropriate.

Examples of markets where production capacity is not easy to adjust? →markets forphysical goods (car manufacturers, airplanes, cement industry...).

Examples of markets where production capacity is easy to adjust: → markets forservices (bank, insurance...)...



Bertrand or Cournot? The example of the recordedmusic industry

Let’s consider the recorded music industry.

We observe an evolution in the distribution of recorded music:from the sales of CDs in physical stores,to digital files on web platforms such as iTunes,and now to streaming platforms.

What type of competition model (Cournot, Bertrand) represents better the olddistribution model? And the new one?

What are the consequences we can predict from this change in the competitionmodel ?



The effect of the digitization: Britannica vs Encarta

Britannica: a 200-year-old encyclopedia, 1600 euros the full collection.Encarta: a product launched by Microsoft in 1992 (acquisition of Funk andWagnalls) at 49,95 dollars/eurosResponse of Britannica to Microsoft’s entry?Online encyclopedia at 2000 euros per yearDrop in sales of Britannica to 50% between 1990 and 1996Online encyclopedia at 120 euros/anCD for 200 euros, then for less than 100 euros.


Cournot competition with n firms


Let’s assume a linear demand D(p) = 1 − pn firms on the market compete in quantities (Cournot competition)The cost fonction of firm i is Ci(qi) = cqi

What is the equilibrium price?What is the equilibrium profit?




The inverse demand function is P(Q) = 1 −Q, where

Q =

n∑i=1

qi.

The profit function of a firm i is

Πi = (P(Q) − c)qi.

The first order condition of the maximization problem is:

1 −Q − c − qi = 0.

We look for a symmetric equilibrium with qi = q. Thus

q =1 − cn + 1

.

and

Π =(1 − c)2

(n + 1)2.




The equilibrium profit is:

Π =(1 − c)2

(n + 1)2.

The more numerous firms are, the lower Cournot profit is.

If the number of firms is large, firms hardly any profit.


Comparison in terms of market power


Assumptions:

Let’s assume there are n firms,With the same marginal cost of production, c.We define Li = (pi − c)/pi the Lerner index for firm i.

Comparison: monopoly, Bertrand, CournotIn a monopoly, we have?

Li =1ε

.

In Bertrand competition, we have? → Li = 0.In Cournot competition, we have?

Li =αi

ε, where αi is the market share of firm i



Proof

Let P(Q) represent the inverse demand function, with Q = q1 + q2 the totalquantity producedThen, firm i profit function is(

P(qi + qj

)− c

)qi

The FOC is :P(qi + qj

)− c + qiP′

(qi + qj

)= 0,

That is,

P − cP

= −

qiP′(qi + qj

)P

= −qi

Q

P′(qi + qj

)Q

P

= −qi

QQ

PD′,

ie.Li =

αi

ε.


An application: mergers

Merger in the French mobile telephony market

The French mobile telephony market had a client base of 71 million ofsubscribers in March 2015 (according to ARCEP). (That is, a penetrationrate of 107%).There are four principal operators: Orange (35,6%), SFR (28%), BouyguesTélécom (14,2%) and Free (13,2%)Mobile virtual network operators (La Poste, etc.) have low market shares(8,7%).There have been rumors of mergers between Bouygues Télécom and Free.We are going to build a simple competition model to study the incentivesof Bouygues Télécom and Free to merge.




For simplicity, let’s assume that the mobile market presents 4 firms.We thus ignore the competitive pressure coming from MVNOs.We assume that these 4 firms produce homogeneous (identical) goods andcompete in quantities.Let qi firm i’s quantity and Q = q1 + q2 + q3 + q4 the total quantity.The demand function is D(p) = 50 − p.

Questions:1 Determine the Nash equilibrium for a quantity competition game. Com-

pute the quantity and profit in equilibrium for each firm.2 We assume that firms 3 and 4 merge (Bouygues and Free) and become a

new firm, BF. Compute again the equilibrium, the produced quantity andequilibrium profit for firm BF, 1 and 2.

3 Are firms 3 and 4 better off merging? How do firms 1 and 2’s profitsevolve?




1 Determine the Nash equilibrium for a quantity competition game. Com-pute the quantity and profit in equilibrium for each firm.

The quantity in equilibrium for a Cournot model with n firms is:

q∗(n) =50

n + 1.

The Cournot profit with n firms is:

Π∗i (n) =2500

(n + 1)2.

Therefore, for n = 4, we have: q∗(n) = 50/5 = 10 and Π∗i = 2500/25 = 100




1 We assume that firms 3 and 4 merge (Bouygues and Free) and become anew firm, BF. Compute again the equilibrium, the produced quantity andequilibrium profit for firm BF, 1 and 2.

We move from an oligopoly with 4 firms to an oligopoly with 3 firmsThus, for n = 3, we have: q∗(n) = 50/4 and Π∗i = 2500/16 = 156.25




1 Are firms 3 and 4 better off merging? How do firms 1 and 2’s profitsevolve?

Bouygues and Free are not well advised to merge because 156.25 < 100 *2: their global profit decreases!Whereas, Orange and SFR are better off: 156.25 > 100 !



Merger in Cournot competition

Let’s consider a market with n > 1 firmsMarginal cost is constant and identical: cLinear demand: P(Q) = a − bQIn the equilibrium, we have:

Π∗i (n) =1b

(a − c)2

(n + 1)2.

In case of a merger of k firms, n− k + 1 firms will be present on the market.In that case, a merger is profitable if

Π∗i (n − k + 1) ≥ kΠ∗i (n)

Merger in Cournot competition

In a Cournot competition, a merger is profitable only if it involves more than80% of firms in the market.



Merger in Bertrand

Let’s consider a market with n > 1 firmsMarginal costs are constant and identical: cIn the equilibrium, p∗1 = ... = p∗n = c and profits are 0.If k < n firms merge, how the equilibrium is modified?It is unchanged: the price stays equal to c and profits are 0.

Merger in Bertrand competition

In a Bertrand competition, a merger is profitable only if it involves all (100%)of the firms in the market.



Conclusion on mergers

Merger decisions cannot be explained only by the incentives to reduce compe-tition in the market.

Other dimensions to explain merger?

Synergies: cost cutting...Becoming an industry leader (the competition model changes: Stackelbergrather than Cournot)Portfolio strategy: extending product range, economies of scale in salesand marketing, incentives to bundle.


Take-aways

Take-aways

Bertrand competition between identical firms leads to marginal-cost pric-ing ("Bertrand paradox").The Bertrand paradox is not verified anymore if we relax one of the 4principal assumptions...If capacity constraints can be easily adjusted in short term, it is more likelythat firms compete in Bertrand. If production capacity remains fixed inmedium term, firms compete in Cournot.When n firms compete in Cournot, their profit is inversely proportional tothe number of firms playing in the market.A merger is not always profitable in Cournot competition; it needs toinvolve at least 80% of the firms in the market. In Bertrand competition, itrequires 100% of the firms in the market. Mergers cannot be only explainedby an incentive to reduce competition.


Documents

Oligopoly Marc Bourreau - Télécom ParisTechses-perso.telecom-paristech.fr/bourreau/files/cours02_oligopoly_eng.pdf · Oligopoly, deﬁnition and examples Cournot model, where ﬁrms