Oligopoly Models - University of Washington

Oligopoly Models

Patrick BajariEcon 4631

Patrick Bajari Econ 4631 () Oligopoly Models 1 / 55

Motivation

In chapter 6 we will discuss game theoretic models of competition

In many markets, there are a small number of dominant �rms thatinteract

Examples:

1 National TV Networks- NBC, ABC, CBS, Fox2 Automobiles- Toyota, Ford, GM, Honda,3 Micro Processors- Intel and AMD4 Upper Midwest Research Universities- Minnesota, Wisconsin,Michigan, Michigan State, Chicago, Northwestern, Urbana, Iowa,Iowa Stata


Motivation

When making prices, output or other strategic decisions, these �rmsknow that their decisions are interrelated

This violates a major assumption of the basic competitive model

In that model, �rms are small and prices are taken as given

How do we analyze these games?


Nash Equilibrium

There are i = 1, ...,N players in the game

Let ai be a strategy for player i

Let πi (ai , a�i ) be the payo¤s of player i as function of i�s strategy aiand the strategy a�i of all the other players


Nash Equilibrium

An equilibrium a�1 , ..., a�N is a collection of strategies that are

maximizing, holding the strategies of the other agents �xed

That is, for all i

a�i maximizes πi (ai , a��i )

In words, Nash equilibrium has two assumptions:

1 Maximization- that is, agents choose the strategy that is in their bestinterests

2 Rational Expectations- You anticipate other players equilibriumactions a��i


Example

Battle of the sexes

Pat/Pat�s wife Ashton Kutcher Movie Ice Fishing

Ashton Kutcher Movie (1,3) (0,0)

Ice Fishing (0,0) (3,1)


Prisoner�s Dilemma

Fink/Don�t Fink

Each player serves 1 year if they don�t �nk

If one player �nks, he gets no penalties while the player that doesn�tgets 10 years

If both players cooperate, they both get 3 years

1/2 Don�t Fink Fink

Don�t Fink (-.5,-.5) (-10,0)

Fink (0,-10) (-3,-3)


Rationality assumptions

Is Nash equilibrium a reasonable assumption?

Human stupidity is certainly present in some market situations

Why do we use models of rationality?



1 In some markets, people are smart and they make decisions well.

If they can�t make a decision well, they will hire someone or acquiresomeone to assist them with a decision

Calculus is a metaphor for optimization

Just because market participants can�t solve our models, doesn�tmean they aren�t useful



2. Rationality is the outcome of simpler and less demanding forms ofreinforcement.

Maynard Smith has noted that Nash equilibrium can be the outcomeof evolution

He imagines strategies replicating based on their success, analogousto sexual reproduction



Sexual reproduction does not require an understanding of calculus

The reason why it leads to equilibrium is pretty clear.

Since there is "survival of the �ttest", strategies with poor payo¤s willbe weeded out

If the dynamics of survival of the �ttest settle down, it must be anequilibrium

Otherwise, there is always a "mutant" strategy that can pro�tablyenter with higher payo¤ and have proportionally more o¤spring

Is a trout rational or a �y? Probably not, but it acts as if it wasrational.


Rationality Assumptions

In evolutionary model, strategies with higher payo¤s survive becausethese players "die" and are less likely to reproduce

Markets can have strong reinforcement e¤ects as well

Instead of dying, you go broke

Firms and other economic actors are often punished if they makedecisions that lead to lower pro�tability



Just like in our evolutionary models, if bad strategies are punished,the only thing that you can converge to is equilibrium

Economists have studied "evolutionary models"

To be candid, the academic economics literature has not fullyappreciated the power of Darwin�s ideas

Learning is another adjustment process

For example, you maximize forming your beliefs given what you haveseen in the past



4. It can be hard to beat the market

A lack of rationality implies that an academic economist can make abetter decision than a �rm in the marketplace

In general, this is much harder than it seems from the outside



5. The economic models of irrationality just aren�t that good

Much of the work in behavioral economics documents various"anamolies" in decision making

It is often disputed whether the anamolies actually exist or are afunction of measurement problems (e.g. biased forecasts by analysts)

There is no general and logically coherent framework that can beapplied as broadly as Nash equilibrium models

Economics is distinguished by having formal and logically coherentanalysis based on math

While there are competing traditions in economics, they have notmade nearly the progress in the past 100 years



None of this is meant to say that we won�t eventually �gure out howto build better models that perturb the rationality assumptions

Neuro economics seems to be doing some quite precise measurementand interesting work

Some behavioral work in experimental economics is also quite good


Cournot

In the Cournot model, N �rms compete by producing a homogenousgood

The strategy of each �rm is qi the amount of good i to produce

Let the market quantity Q be de�ned by:

Q = q1 + ...+ qN


Cournot

Let P(Q) denote the inverse demand function, that is, the price as afunction of the aggregate quantity Q

Let c denote the unit cost to �rm i of producing the good

For simplicity assume constant marginal costs


Cournot

Then the pro�ts of �rm i can be written as:

πi (qi , q�i ) = (P(q1 + ...+ qN )� c) qi

A Nash equilibrium is a vector of quantities q�1 , ..., q�N such that:

q�i maximizes πi (qi , q��i )

Let�s study the properties of the maximization problem


Residual Demand

It is useful to talk about the "residual demand curve"

When i solves her maximization problem, she takes Q��i as given,where

Q��i = q�1 + ...+ q

�i�1 + q

�i + ...+ q

�N

It is useful to write the equilibrium price as P(qi +Q��i )

We call this the residual demand (or more properly, residual inversedemand), because you express the price as a function of i�s quantity,holding every one else�s quantity �xed


Residual Demand

For illustration, assume that P(qi +Q��i ) is linear, i.e.

p = a� b(qi +Q��i )

Note that the elasticity of the residual demand can be written as:

%∆qi%∆p

=dqidp

pqi

=1b

�a� b(qi +Q��i )

�qi

=abqi

� b�Q��iqi

Thus residual demand becomes more elastic as Q��i increasesA percentage increase in price requires an larger percentage decreasein quantityIf Q��i is near in�nity, this means that a 1 percent increase in pricewould require a nearly in�nite percentage increase in quantity


Residual Demand

Thus, even if the aggregate demand curve P(Q) is quite inelastic, itmay be the case that residual demand in very elastic

From the perspective of the �rm, the important thing in determiningoutput is the elasticity of residual demand, not aggregate demand!


Equilibrium

First order conditions:

πi (qi , q��i ) = (P(qi +Q��i )� c) qiddqi

πi (qi , q��i ) =dPdQqi + P � c

At a maximum, the derivative must be zero:

dPdQqi � P � c = 0

P � c = � dPdQqi

P � cP

= � dPdQ

qiP


Equilibrium

Let assume that the �rms choose a symmetric strategy

That is, since they all have the same cost, they all produce the sameoutput

Then qi = q = QN and hence

P � cP

= � dPdQ

QP1N

= � 1dQdP

PQN

= � 1εN

where ε is the market elasticity of demand


Comparative Statics

Our model implies that markups increase as ε gets more inelastic

This seems reasonable- the less abilty consumers have to substitute,the higher the price cost margin or Lerner index P�c

P should be

The model also implies that as N becomes larger, the price costmargins decrease

This also seems reasonable intuitively- as there is more competition,�rms have less market power

If we send N �! ∞, we get that prices converge to marginal costThat is, Cournot converges to perfect competition


Hirschman-Her�ndal Index

Next we wish to derive the Hirschman-Her�ndal Index or HHIH

This is a famous index used to measure market power and is used inthe merger guidelines

The HHI is motivated in part by the Cournot model



Suppose that we have asymmetric �rms, i.e. the marginal costs are ciwhich need not be equal across �rmsThe �rst order conditions in this case become:

πi (qi , q��i ) = (P(qi +Q��i )� ci ) qiddqi

πi (qi , q��i ) =dPdQqi + P � ci

At a maximum, the derivative must be zero:

dPdQqi � P � ci = 0

P � ciP

= � dPdQ

QPqiQ

= � siε

Where ε is the elasticity of demand for �rm i and si is i�s share



The HHI is the market share weighted price-cost or Lerner index(when the elasticity of demand is one):

HHI =N

∑i=1

P � ciP

si

=1ε

N

∑i=1�s2i

The HHI tells us that the square of market shares is related to pricecost margins (holding aggregate demand elasticity constant)

The bigger P�ciP the larger the ine¢ ciency from market power

Hence, the use of HHI in merger analysis



Note that the HHI only measures market power under theassumptions of the Cournot model

If the market involves di¤erentiated products, then the HHI is amisleading measure

This was certainly the case in Wild Oats/Whole Foods

Their share was small (depending on how you de�ned the market),but pricing pressures and competitive e¤ects could have been large

That is why antitrust guidelines may move towards HHI forhomogenous goods markets and diversion ratios (discussed below) fordi¤erentiated product markets


Di¤erentiated product market

Many of the markets that we encounter in economics aredi¤erentiated product markets

That is, �rms sell similar, but not identical goods

Examples: Autos, Cell Phones, Computers, Universities

In di¤erentiated product models, we usually model competition asBertrand

That is, �rms set prices as their strategic variable


The Hedonic Approach

In di¤erentiated product markets, it is typical to model consumerbehavior using a hedonic approach

In this approach, we model a good as a bundle of characteristics

For example, a laptop could be viewed as a bundle of:

1 Screen size2 CPU speed3 Weight4 Brand (e.g. PC or Mac)



Suppose that there are j = 1, ..., J goods and i = 1, ...,N consumersin a marketLet uij denote the utility of consumer i for good jA classic empirical model of demand is due to McFadden and is calledthe conditional logit (he won the Nobel prize in large part for his workrelated to this model)We write uij as:

uij = β0 + screenj � β1 + CPUj � β2 +Weightjβ3 � αpj + εij

In the above, a consumer�s utility for good j is a function of itscharacteristics.The parameters β represent the "marginal utility" for thesecharacteristicsα represents the disutility of paying a higher price (for the formallyinclined, this is actually an indirect utility function)εij is an iid shock to agents preferences



Suppose that there are j = 1, ..., J goods and i = 1, ...,N consumersin a market

Here consumers make discreet choices, e.g. choices between mutuallyexclusive goods

Let uij denote the utility of consumer i for good j

A classic empirical model of demand is due to McFadden and is calledthe conditional logit (he won the Nobel prize in large part for his workrelated to this model)

We write uij as:

uij = β0 + screenj � β1 + CPUj � β2 +Weightj � β3 � αpj + εij

Households are assumed to be utility maximizers, e.g.

i chooses j if and only if uij > uij 0 for all j0 6= i



In the above, a consumer�s utility for good j is a function of itscharacteristics.

The parameters β represent the "marginal utility" for thesecharacteristics

α represents the disutility of paying a higher price (for the formallyinclined, this is actually an indirect utility function)

εij is a random iid shock to agents preferences



The random preference shock captures preference heterogeneity

This is very important in understanding these markets

People have very distinct preferences even after conditioning onobvious demographic variables: (e.g. location, age, income, etc...)

We use econometrics to measure β- actually, it�s a pretty simpleexample of maximum likelihood estimation


The Conditional Logit

McFadden noted that if you let εij be an "extreme value" randomvariable, we get some very pretty formulas

In particular, the probabiltiy that i chooses good j is equal to:

Pr(i chooses j) =

exp (β0 + screenj � β1 + CPUj � β2 +Weightj � β3 � αpj )

∑j 0 exp (β0 + screenj 0 � β1 + CPUj 0 � β2 +Weightj 0 � β3 � αpj 0)

Note that as pj " the probability that j is chosen goes downAs desirable characteristics are added, the probability that j is chosengoes up


The Conditional Logit

The market share and total quantity of j is equal to:

sj (pj , p�j ) =

exp (β0 + screenj � β1 + CPUj � β2 +Weightj � β3 � αpj )

∑j 0 exp (β0 + screenj 0 � β1 + CPUj 0 � β2 +Weightj 0 � β3 � αpj 0)

Qj = N � sj (pj , p�j )

It can be veri�ed that the elasticity of the residual demand curvefacing j is:

%∆sj%∆pj

= �αpj sj


Bertrand Equilibrium

A Bertrand equilibrium is a Nash equilibrium in pricesThe pro�t of j is:

(pj � cj )Nsj (pj , p�j )

The �rst order condtions are:

sj + (pj � cj )∂sj∂pj

= 0

(pj � cj ) = �sj∆pj∆sj

(pj � cj )pj

= � sjpj

∆pj∆sj

= �%∆pj%∆sj

= � 1εj

Where εj is the residual demand elasticity for good j


Measuring Market Power

The formula below is called the inverse elasticity rule:

(pj � cj )pj

= � 1εj

This rule states that price cost margins are higher (and market powergenerates more ine¢ ciency), if the residual demand is more inelastic

IO economists have been obsessed with the measurement of marketpower

For example, they will go to the data and attempt to estimate εj


Measuring Market Power

In the logit model, for example, εj = �αsjpjIf we know α, we can infer the Lerner index

This is useful in practice since we can identify the most distortedmarkets

Also, we can perhaps identify markets in which mergers might beparticularly bad or compute the e¤ects of collusion

This all boils down to determining how mergers or collusion changedresidual demand elasticities


Calibration of the logit model

It is possible for us to calibrate the logit model in order to ask someinteresting economic questions

As an example, let�s consider the e¤ects of the proposed GoogleYahoo! ad deal that was blocked by the Department of Justice

Google- 65%

Bing-10%

Yahoo!-25%



Search engines make money by selling advertisements

Advertisers bid for "sponsored links" on these sites

They pay their bid times each click that is generated by a search



Suppose that the utility of search engine j takes the form:

uij = δj � αpj + εij

Where δj is the "mean" utility that an advertiser has for searchengine j

pj is the advertising price of search engine j

εij is the preference shock for engine j



To simplify things, let�s suppose that prices of a click are equalizedacross sites

Normalize prices to $1.

Assume that Google�s price cost margin is 27% (this is fromaccounting data, not economic costs)

This implies that:

p � cp

=1

αsjpj

.27 =1

α.65

α =1

.27 � .65 = 5.6980



Next, let�s solve for the δj

sGoog = .65 =exp(δGoog � 5.6980)

D

sBing = .1 =exp(δBing � 5.6980)

D

sYahoo = .25 =exp(δYahoo � 5.6980)

DD = exp(δGoog � 5.6980) + exp(δBing � 5.6980)

+ exp(δYahoo � 5.6980)

Since we know all the shares, this is 3 equations in 3 unknowns



Without loss of generality, we can normalize δBing = 0

Verify this by adding a constant to all of the δj

Then we can solve for δGoog and δYahoo by:

sGoogsBing

=.65.1exp(δGoog � 5.6980)exp(�5.6980)

δGoog = log(.65)� log(.1) = 1.8718δYahoo = log(.25)� log(.1) = .9163


Post Merger Outcome

If we remove Yahoo! from the market, the shares of Bing and Google are:

sGoog =exp(1.8718� 5.6980pGoog )

exp(1.8718� 5.6980pGoog ) + exp(�5.6980pBing )

sGoog =exp(�5.6980pBing )


Let�s give both Bing and Google costs of .83 cents (let�s ad in 10cents of "economic costs" to the price cost margins)


Post Merger Outcome

In a Nash equilibrium, the �rms would maximize:

(pGoog � .83) �exp(1.8718� 5.6980pGoog )


(pBing � .83) �exp(�5.6980pBing )



Post Merger Outcome

The Nash equilibrium would be the following two equations in the twounknowns pGoog , pBing

(pGoog � .83)pGoog

=1

5.6980 exp(1.8718�5.6980pGoog )exp(1.8718�5.6980pGoog )+exp(�5.6980pBing )pGoog

(pBing � .83)pBing

=1

5.6980 exp(�5.6980pBing )exp(1.8718�5.6980pGoog )+exp(�5.6980pBing )pBing

This can�t be solved for analytically. Instead, we do it numerically onthe computer



The post merger prices are pGoog = 1.4509, pBing = 1.2091

This is a very large price increase relative to the baseline prices of 1

This back of the envelope calculation suggests this acquisition couldhave been very harmful to advertisers


Extensions of the Logit

Remark: What happens after this point of the lecture is a bitadvanced and will not be on the exam

While the logit model is widely used, there has been important workon extending this model

In particular, researchers have found it important to include "randomcoe¢ cients in the logit"

In the logit, the only reason why di¤erent people make di¤erentchoices is because they draw di¤erent εij

In the random coe¢ cient model, di¤erent people have di¤erentmarginal utilities

In practice, this turns out to be a very important extension



You will not be tested on this section

One formulation of this model is to suppose that there arer = 1, ...,R types of consumers

These types have utilities characterized by the parameters β(r ), α(r )

This leads to the utility function

uij = β(r )0 + screenj � β

(r )1 + CPUj � β

(r )2 +Weightj � β

(r )3 � α(r )pj + εij

Households are assumed to be utility maximizers, e.g.

i chooses j if and only if uij > uij 0 for all j0 6= i

The probability that type r chooses good j is:



The type r�s choice probability is:

Pr(i chooses j jr) =

exp�

β(r )0 + screenj � β

(r )1 + CPUj � β

(r )2 � α(r )pj

�∑j 0 exp

�β(r )0 + screenj 0 � β

(r )1 + CPUj 0 � β

(r )2 � α(r )pj 0

�Let p(r) denote the probability of type r

The choice probability for j is found by summing over r

Pr( j chosen) =

R

∑r=1

p(r)exp

�β(r )0 + screenj � β

(r )1 + CPUj � β

(r )2 � α(r )pj

�∑j 0 exp

�β(r )0 + screenj 0 � β

(r )1 + CPUj 0 � β

(r )2 � α(r )pj 0

�Patrick Bajari Econ 4631 () Oligopoly Models 54 / 55


Next lecture, we will use this framework to study music piracy


Oligopoly Models - University of Washington

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