Lecture 5: Oligopoly Competition...Introduction Game Theory Bertrand Model Cournot Model General...

Introduction Game Theory Bertrand Model Cournot Model General Case Comparative Statics Cournot vs Bertrand

Lecture 5: Oligopoly CompetitionCabral Chapter 8

Yuta Toyama

Last updated: October 6, 2019

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Oligopoly Competition

I We have studied the extreme cases of market structures so far.I monopolyI perfect competition.

I Most real-world markets are somewhere between the extremes.

I We now study oligopolistic competition.I Situation in which there are few competitors.I Special case: Duopoly (when the number is two)

I OverviewI Quick review of game theoryI Bertrand model (price competition)I Cournot model (quantity competition)

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What is Game Theory

I Game theory studies decision-making when decision makers interactwith each other.I Your payoff (profit) depends on what others do.I To achieve your most favorable outcome, you need to understand how

other players behave and how they interact.

I Why important in IO? Imperfectly-competitive markets!I Imperfect competition (oligopoly): a few firmsI Perfect competition: Many price-taking firms.I Monopoly: only one firm.

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Motivating Example

I Consider the following news article: The wall Street Journal, November16, 1999.

I “In a strategic shift in the United States and Canada, Coca-Cola Co. is . .. gearing up to raise the prices it charges its customers for soft drinks byabout 5% . . . The price changes could help boost Coke’s profit . . .

I Important to the success of Coke and its bottlers is how Pepsi-Cola . . .responds. The No.2 soft-drink company could well sacrifice some marginsto pick up market share on Coke”, some analysts said.

I This article highlights the main points in duopoly.I Firm 1 (Coke) is likely to influence Firm 2 (Pepsi)’s profits, and vice versa.I Coke’s decision process should consider what Pepsi is expected to do.

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Setup

I There are N players index by i ∈ {1, · · · ,N}.

I Player i choose the strategy si from the strategy set Si

I Player i ’s payoff (utility) is given by

ui (s1, · · · , si , si+1, · · · , sN)

I Here, the utility depends on both her strategy and others’ strategy.

I We write the payoff asui (si , s−i )

where s−i is the others’ strategy

s−i ≡ {s1, · · · , si−1, si+1, · · · , sN}

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Nash Equilibrium

I The most standard equilibrium concept.

I Definition: The strategy profile s∗ = (s∗1 , s∗2 , . . . , s

∗n) is a Nash

equilibrium if

ui (s∗i , s∗−i ) ≥ ui (s

′i , s∗−i ) ∀s ′i ∈ Si and all players i .

I Intuition: No player has an incentive to deviate from their strategy s∗i ,given all other players strategies s∗−i .

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Nash equilibrium = mutual best response

I Strategy si is the best response strategy (BR) to s−i if

ui (si , s−i ) ≥ ui (s′, s−i )

for all s ′ ∈ Si

I In other words, si is the BR to s−i if si maximizes your payoff:

si = argmaxs∈Siui (s, s−i )

.

I Players are mutually playing best response strategy to each other inNash equilibrium.

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Example of Normal-Form Games

I Game 1: Prisoner’s Dilemma

Quiet Fink

Quiet −1,−1 −3, 0Fink 0,−3 −2,−2

I Nash equilibrium: (Fink, Fink)

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I Game 2: Coordination GameWorkers

Acquirecomputer skills

Do not acquirecomputer skills

FirmInvest incomputer line 10, 5 −25, 0

Maintainold line

0,−10 0, 0

I (Invest, Acquire) is a NE.

I (Maintain old line, Do not acquire) is another NE.

I We could have multiple equilibria.

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Bertrand Model

I A model where firms set the prices for a good simultaneously.

I Number of firms: j = 1, 2

I Price competition: Each firm sets price pjI Assume symmetric and constant marginal cost: MC (q) = c

I The goods are homogeneous. Market demand D(p)

I Consumers buy the good from the firm with the lowest price.

I Static (one shot) game

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I Denote the quantity firm i sells by Di (p1, p2) as a function of (p1, p2)

I Since the goods are homogenoeus, the buyer chooses the cheapest seller.

I The sales for firm i

Di (p1, p2) =

D(pi ) if pi < pjD(pi )

2 if pi = pj0 if pi > pj

I Therefore, the profit of firm i can be expressed as

πi (p1, p2) = (pi − c)Di (p1, p2).

I Question: What is the equilibrium price?I Use Nash equilibrium to predict the outcome.

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Step 1: Find the best response given the competitor’s price

I Consider firm i ’s best response given competitor j ’s price pj .

I We consider 3 cases, depending on the price pj .

I Case 1: pj > cI firm i ’s profit

πi =

D(pi )(pi − c) if pi < pjD(pi )

2

(pi − c

)if pi = pj

0 if pi > pj

I Firm i wants to undercut the price by just a small amount.

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I Case 2: pj = cI firm i ’s profit

πi =

D(pi )(pi − c) < 0 if pi < pjD(pi )

2

(pi − c) = 0 if pi = pj

0 if pi > pj

I Firm i wants to set pi = c .

I Case 3: pj < cI Firm i earns either 0 or negative profit.I Therefore, it wants to set pi = c .

I The best response function for firm i is

p∗i (pj) =

pj − ε if pj > c

c if pj = c

c if pj < c

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Step 2: Nash Equilibrium as Mutual Best Responce

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I Nash equilibrium is given by p1 = p2 = c .

I To examine this, we should see whether both firms have no incentive todeviate from this situation.

1. pi < c cannot be NE b.c. firm i can weekly increasing profit by rasingprices.

2. pi > pj and pj ≥ c cannot be NE b.c. firm i has a profitable deviation.3. pi = pj > c cannot be NE since both firms have a profitable deviation.4. When p1 = p2 = c , both of them has no profitable deviation.

I In Nash equilibrium, both have zero profit!

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I The model provide is very extreme result.I Equilibrium price is p1 = p2 = c regardless of the number of firms.I Both firms obtain zero profits.

I This is called Bertrand Paradox.

I How can we make the model more realistic?

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Solution to Bertrand Paradox

1. Product DifferentiationI We assumed homonogeous products: Completely same products.I The goods are not exactly the same in reality.

I Example: Pepsi and cokeI Differentiation by charcteristics, brand images, etc...

2. Dynamic Competition (tacit collusion)I Firms compete in one period only, a situation like “Prisoners’ Dilemma”.I In reality, firms compete over some certain period of time.

3. Capacity Constraints and/or increasing Marginal Cost.I We assumed the marginal cost is constant and no capacity constraint.I Capacity constraint matters in practice!

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Cournot Model

I Firms compete with quantity (production) decision.

I SetupI Number of firms: j = 1, 2I Firms set quantity simultaneously. q1 and q2.I Price is determined to clear the demand.I Constant marginal cost: MC (q) = c (same for both firms)I The goods are homogeneous: Market demand P(Q)I Static (one shot) game

I The profit of firm i is

πi (q1, q2) = P(q1 + q2)qi − cqi .

I Use Nash equilibrium to find the equilibrium quantity.18 / 38


Nash Equilibrium (or Cournot Equilibrium)I The conditions for (q∗1 , q

∗2) to be a Nash equilibrium:

π1(q∗1 , q∗2) ≥ π1(q1, q

∗2) for any q1

π2(q∗1 , q∗2) ≥ π2(q∗1 , q2) for any q2

I NE can be found using best-response functions.I Firm i ’s best-response function gives the profit-maximizing choice of

output for firm i as a function of output produced by firm j

I Firm i ’s best-response function is denoted by Ri

qi = Ri (qj).

The NE simultaneously satisfy the best-response functions for both firms:

q∗1 = R1(q∗2)

q∗2 = R2(q∗1)

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I Consider firm 1’s problem, given q2

maxq1

P(q1 + q2)q1 − cq1

I The first order condition for q1:

P +∂P(q1 + q2)

∂Qq1 − c = 0

I This gives the optimal choice for q1 given q2:

q1 = R1(q2).

I Similarly, we can get the best responce for firm 2:

q2 = R2(q1)

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Exercise

I SetupI The market demand is P(Q) = 100− Q (where Q = q1 + q2),I Costs of production are the same among firms 1 and 2: C (q) = cq.

I What is the Cournot equilibrium?

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Exercise cont’d

I Let’s first find the best-response function of firm 1.

I The profit function of firm 1 is

π1(q1, q2) = (100− q2 − q1) q1 − cq1.

I The FOC:∂π1∂q1

= 100− q2 − 2q1 − c = 0.

I The best-response function of firm 1:

R1(q2) =100− q2 − c

2.

I Similarly, the best-response for firm 2

R2(q1) =100− q1 − c

2.

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Best Responses in Figure

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I Nash Equilibrium requires,

q∗1 = R1(q∗2) =100− q∗2 − c

2

q∗2 = R2(q∗1) =100− q∗1 − c

2.

I This yields

q∗1 =100− c

3and q∗2 =

100− c

3.

I The price is

P(Q) = 100−(q1 + q2) =

100 + 2c

3,

and the profit of each firm is

πi (q∗1 , q∗2) =

(100− c)2

9

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General Case with N firms

I We consider the case with N firms.

I SetupI Number of firms: i = 1, 2, · · · ,NI Cournot competition: Firms set quantities, qi .I Cost Functions: Ci (q) (can be different among firms.)I The goods are homogeneous.I (Inverse) market demand P(Q) where Q =

∑i qi .

I Static (one shot) game

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I The objective function for firm i is

πi (qi , q−i ) = P(Q)qi − Ci (qi ).

I q−i : a vector of quantity produced by firms other than firm i

I FOC for firm i is

P +dP

dQqi −MCi (qi ) = 0

P −MCi = −dP

dQqi

P −MCi

P= −dP

dQ

qiP

= −dP

dQ

Q

P

qiQ

I FOC becomesP −MCi

P=

si|ε|.

where ε is the demand elasticity dQdP

PQ and si ≡ qi/Q is firm i ’s market

share.26 / 38


I FOC:P −MCi

P=

si|ε|.

I More efficient firms are larger:I firms with lower MCi have larger market share (higher si ).

I Marker power (or markup) is limited by the demand elasticityI more elastic demand lowers markup

I Cournot markup is always less than monopoly markupI Monopoly when si = 1.

I Suppose that firms are symmetric (MCi is same across firms, so that qiis the same and si = 1

N ). If N →∞, then p → MCi .

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Relation with Herfindahl-Hirschman Index (HHI )

I A common measure of market concentration:

HHI =N∑i=1

s2i .

I HHI varies between 0 (perfect competition) and 1 (monopoly).I HHI is higher when there are fewer firms or when variations in market

share are larger.

I Numerical examples:I (s1, s2, s3) = (40%, 40%, 10%)

I (s1, s2, s3) = (33%, 33%, 33%)

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HHI as a measure of Market Power (Markup)

I If we multiply the both sides of FOC by si and take summation, then

N∑i=1

si

(P −MCi

P

)︸︷︷︸

weighted average markup

(industry-wide Lerner index)

=HHI

|ε|

I HHI as an “incomplete” measure of market power!

I Markup depends on both HHI and |ε|.

I Higher HHI (or concentration) does not always lead to higher markup.

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HHI and Antitrust Policy

I HHI as a screening devise in merger review by antitrust authorities.

I Ex: Japanese-FTC does not review a proposed merger if it satisfies

1. HHI ≤ 0.15 before merger2. HHI > 0.15 & HHI ≤ 0.25 before merger and ∆HHI ≤ 0.0253. HHI > 0.25 before merger and ∆HHI < 0.015.

I We will study horizontal mergers later this course in more detail.

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Comparative Statics

I Comparative Statics: a exercise to see the equilibrium effect ofchanges in exogenous conditions.

I Example 1: If the price of chicken falls, how does the price of friedchicken change?

I Example 2: If you have a opportunity to invest in a new cost-reductiontechnology, how should you decide?

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Setup

I Number of firms: j = 1, 2

I Constant marginal cost: MCi (q) = ci (different for each firm)

I The goods are homogeneous: Market demand P(Q) = a− bQ

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I The best-response function of firm 1 and 2 are,

R1(q2) =a− bq2 − c1

2b,

R2(q1) =a− bq1 − c2

2b.

I And the equilibrium quantities, price and profits are

q∗1 =a− 2c1 + c2

3band q∗2 =

a− 2c2 + c13b

Q∗ =2a− c1 − c2

3band P∗ =

a + c1 + c23

π∗1 =1

b

(a− 2c1 + c2

3

)2

and π∗2 =1

b

(a + c1 − 2c2

3

)2

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Example 1: Increase in Input Cost

I Let’s first assume c1 = c2 = c . What happens if c increases?

∂Q∗

∂c=

∂

∂c

(2a− c1 − c2

3b

)= − 2

3band

∂P∗

∂c=

∂

∂c

(a + c1 + c2

3

)=

2

3

I If marginal cost increases by 40%, equilibrium price will increase by2340% = 26.6%.I Only 2/3 of the increase in MC will translate into the increase in price.I In perfect competition, 100% of the increase in marginal cost will

translate into the increase in price.I Note: Here MC is constant, so that the pass-through is 100% under

perfect competition. If MC is not constant (i.e., increasing marginal cost),the pass-through is less than 100%.

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Benefit of Cost Reduction

I Thought experiment:I Suppose that you currently produce 30.I You have an investment opportunity to reduce your marginal cost by 5.I An idea: “By the investment, I can reduce the cost by 150. So, I should

invest if the investment cost is less than 150”I Does this make sense? No in general!

I Difficult question becauseI By reducing the cost, the firm can increase the output.I The rival rationally expect that the firm increases output and best-respond

to the change in output.I Without solving the model, it is impossible to foresee the final outcome.

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Benefit of Cost Reduction

I Note that

∂q∗1∂c1

= − 2

3b,∂q∗2∂c1

=1

3b∂P∗

∂c1=

1

3b,∂Q∗

∂c1= − 1

3b

∂π∗1∂c1

=1

b2

(a− 2c1 + c2

3

)−2

3= −4

3q∗1

I The decrease in firm 1’s marginal cost increases the output.I Firm 2 rationally expect that and reduces its output.I Total output increases and price decreases.I In equilibrium, the profit increases by 4

3q∗1 .

I If current output level is 30 and you can reduce your cost by 5, the valueof such investment is more than 150 (approximately 200).

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Bertrand Competition with Capacity Decisions (Kreps and Scheinkman1983)

I Consider the following two-stage model.I Two firms: i = 1, 2I Two periods: t = 1, 2

Period 1 Firms decide their capacity, ki simultaneously.Period 2 Firms set the price simultaneously.

I To build their capacity, firms need to pay cost, c(k) = ck .I The production in period 2 is costless. However, firms can’t produce more

than their capacity.I The goods are homogeneous: Market demand P(Q)I Consumers buy the good with lower price.

I In the subgame perfect Nash equilibrium, the solution to this game isthe same as the Nash equilibrium in Cournot model with marginalcost c.

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Cournot vs Bertrand

I Which model provides a better description depends on the feature ofindustries.

I Case 1: If firms must make capacity decision which is hard to adjust inthe short run, Cournot is a better approximation.I The two stage model of capacity and price decisions provides the same

prediction as Cournot (see in the previous slide).I ex. Cement, steel, etc..

I Case 2: Firms can adjust the output easily, Bertrand is a betterapproximation.I In a Bertrand model, firms set prices and receives demand based on those

prices.I This implicitly assumes that firms can produce an output exactly equal to

the quantity demanded.I ex. software, retail gasoline station.

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Lecture 5: Oligopoly Competition...Introduction Game Theory Bertrand Model Cournot Model General...

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