Static Oligopoly Models

Industrial Organization (UG)

Li Zhao, SJTU

Fall, 2016

Outline

Game Theory

Quantity Competition

Price Competition

Paper - Scissors - Rock

Key Elements of Games

I Players: You and I.

I Rules: Simultaneous move, information available at eachmove.

I Actions: {P , S ,R} × {P , S ,R}.I Payo�: R

beats→ S , Sbeats→ P and P

beats→ R .

You \ I P S RP (0,0) (0,1) (1,0)S (1,0) (0,0) (0,1)R (0,1) (1,0) (0,0)

Why Game Theory?

I Game theory is the study of strategic decision making.

I Most industries are dominated by not one, but a smallnumber of relative large �rms.

I These decisions should incorporate strategic reasoning,because payo�s between �rms are interdependent.

I For example, when Coca-Cola raises price, what shouldPepsi response?

I We will not only discuss price competition, but alsostrategic decision on quantity, capacity, product variety,and many others.

Types of Games

I Cooperative and non-cooperative game.

I The timing of moves: Static / dynamic game.

I Uncertainty about the payo�s of rivals: Complete /incomplete information game.

I In this class we will discuss:I Static game with complete info (this class),I Dynamic game with complete info (after 10/1),I Static game with incomplete info (auction).

Static Games of Complete Information

I There are i = 1, 2, ...N �rms in the market.

I Let ai be a strategy of �rm i .

I It could be price (Bertrand), quantity (Cournot), producttype, entry, location, etc.

I Let πi(ai , a−i) be the payo�s of �rm i as function of i 'sstrategy ai and the strategy a−i of all the other players.

I A Nash equilibrium a∗1, ..., a∗N is a collection of strategies

that are maximizing, holding the strategies of the otheragents �xed

a∗i maximizes πi(ai , a∗−i).

Dominant Strategy

I A strictly dominant strategy for player i is one whichmaximizes player i 's payo� regardless of the strategieschosen by i 's rivals.

I Formally, ai is a strictly dominant strategy for player i iffor all a′i ∈ Ai and a−i ∈ A−i .

πi(ai , a−i) > πi(a′i , a−i).

Example of Dominant Strategy

I Consider the case of two airlines, China Eastern Airlineand Tibet Airline, both o�er a daily �ight from Shanghaito Lhasa.

I 70 percent of the potential clientele for the �ight wouldprefer to leave Shanghai in the afternoon.

TVMorning Afternoon

MUMorning (15,15) (30,70)Afternoon (70,30) (35,35)

I This game has a dominated strategy.

Example of Dominant Stategy

C1 C2 C3R1 (4,3) (5,1) (6,4)R2 (2,1) (3,4) (3,6)R3 (3,0) (4,6) (2,8)

I R1 is a dominant strategy for play 1.

I C3 is a dominant strategy for play 2.

Best Response

I The strategy ai is a best response for player i to a−i if

πi(ai , a−i) ≥ πi(a′i , a−i),

for all a′i ∈ Ai .

I Or equivalently, we say ai is a best response for player ito a−i if

ai maximizes πi(a′

i , a−i),

for all a′i ∈ Ai .

I We will denote best response as ai = BRi(a−i).

I Example, BRi(Rockj) = Paper , BRi(Paperj) = Scissors.

Nash EquilibriumI Nash equilibrium is the most common equilibrium conceptused in industrial organization and we will make extensiveuse of it.

I A Nash equilibrium is a strategy pro�le such that everyplayer's strategy is a best response to the strategies of allthe other players.

I The strategy pro�le a∗ to be a Nash equilibrium if

a∗i = BRi(a∗−i).

I From the de�nition of a best response,a∗ to be a Nashequilibrium if

a∗i maximizes πi(ai , a∗−i)

for all ai ∈ Ai .I There is a notion called �mixed strategy� which we don'tconsider for now.

Nash Equilibrium - Example

I The prisoner's dilemma is a standard example of a gameanalyzed in game theory that shows why two completely"rational" individuals might not cooperate, even if itappears that it is in their best interests to do so.

I Let's look at its payo� function

Prisoner A \ B Deny ConfessDeny (-1,-1) (-3,0)Confess (0,3) (-2,-2)

.

I How to we �nd Nash Equilibrium?I We �nd BR1(a2) and BR2(a1) for each a1 ∈ A1 and

a2 ∈ A2.I Then we collect (a1, a2) such that a1 = BR1(a2) and

a2 = BR2(a1).

Nash Equilibrium - Example

I Consider the case of two airlines, China Eastern Airlineand Tibet Airline, but with di�erent payo� matrix:

TVMorning Afternoon

MUMorning (25,25) (50,50)Afternoon (50,50) (25,25)

.

I Can you solve the Nash Equilibrium of the model?

I Is Nash equilibrium a reasonable assumption?

Outline

Game Theory

Quantity Competition

Price Competition

Quantity Competition

I There are i = 1, 2, ...N �rms in the market.

I Let ai be a strategy of �rm i ., say quantity.

I Let πi(ai , a−i) be the payo�s of �rm i as function of i 'sstrategy ai and the strategy a−i of all the other players.

I Pro�t πi (qi ; qj) = P(qi , q−i ) ∗ qi − C (qi ).

I A Nash equilibrium a∗1, ..., a∗N is a collection of strategies

that are maximizing, holding the strategies of the otheragents �xed

a∗i maximizes πi(ai , a∗−i).

I We look for q∗such that q∗i maximizes πi (qi , q∗−i ), or

equivalently, we look for q∗i = BRi (q∗−i ).

Cournot Quantity-Setting Model

I In the Cournot model, N �rms compete by producing ahomogenous good.

I The strategy of each �rm is qi the amount of good i toproduce.

I Let the market quantity Q be denoted by:Q = q1 + q2 + ...qN .

I Let P(Q) denote the inverse demand function. AssumeP = a − bQ.

I For simplicity, assume constant marginal cost c .

Cournot - Residual Demand

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

I When i solves her maximization problem, she takes Q∗−ias given, where

Q∗−i = q∗1 + ...+ q∗i−1 + q∗i+1 + ...+ q∗N .

I A Nash equilibrium is a vector of quantities q∗1 , ...q∗N such

that:

q∗i maximizes πi(qi ,Q∗−i) = (P(qi + Q∗−i)− c)qi .

2-Firm Cournot Competition - Best Response

π1(q1; q2) = (a − b(q1 + q2)− c1)q1;

π2(q2; q1) = (a − b(q1 + q2)− c2)q2;

I To maximize �rms' pro�ts, we take FOC

d

dq1(π1(q1; q2)) = −bq1 + (a − b(q1 + q2)− c1) = 0 :

d

dq2(π2(q2; q1)) = −bq2 + (a − b(q1 + q2)− c2) = 0 :

I Two best response functions are

q1 = BR1(q2) =a − bq2 − c1

2b,

q2 = BR2(q1) =a − bq1 − c2

2b.

Quantity: Strategic Substitutes

Nash Equilibrium

I Firms' best responses are

q1 = BR1(q2) =a − bq2 − c1

2b,

q2 = BR2(q1) =a − bq1 − c2

2b.

I We get qi =a+cj−2ci

3b. For symmetric cost, we get

q1 = q2 =a−c3b

.

I P? CS? Pro�t?

I Compare with Monopoly and Perfect Competition?

Cournot - N Firms

I We look for symmetric equilibrium thereforeQ∗−i = (n − 1)q∗and q = q∗i .

I We then get qCournot = a−c(n+1)b and QCournot = (a−c)

bn

n+1.

I Pro�t for individual �rm πCournoti = (a−c)2(n+1)2b

.

I Equilibrium price is pCournot = a+ncn+1

.

I When N goes up ....Think about S-C-P.

E�ect of N on Prices

source: Church textbook, table 8.1 on page 244.

Cournot - Market Power

πi(qi ,Q∗−i) = (P(qi + Q∗−i)− ci)qi .

I First order condition

d

dqi(πi(qi ,Q

∗−i)) =

dP

dQqi + (P − ci) = 0.

I Cournot markup (or Lerner Index) is

P − ciP

= − 1dQdP

PQ

qiQ

=siε.

I Lerner index is proportional to the �rm's market shareand inversely proportional to the elasticity of demand.

I Irrelevant to functional form of demand curve.

Entry - Number of Firms in Market

I So far we treat number of �rms �xed, as we assume no�xed cost.

I If we have �xed cost FC > 0, how many �rms will be inthe market?

I Previous we talked about

Rev − VC =(a − c)2

(n + 1)2b,

which is decreasing in N .

I A �rm will enter as long as FC < (a−c)2(n+1)2b

.

I The number of �rms in the market is [ a−c√b·FC − 1].

Entry - Comparative Statics

I N ' a−c√b·FC − 1.

I High demand?

I High marginal cost?

I Steeper demand (more inelastic demand)?

I High �xed cost?

I Ine�ciency of entry.

Outline

Game Theory

Quantity Competition

Price Competition

Bertrand Price-Setting Model

I In the short run, changing prices is much easier thanchanging quantities.

I Two identical �rms, homogeneous products. Firm 1 setsp1, �rm 2 sets p2.

I Firm 1's pro�t is

π1 =

(p1 − c)( a

b− 1

bp1) if p1 < p2

1

2(p1 − c)( a

b− 1

bp1) if p1 = p2

0 if p1>p2

.

I Firm 1's best response:

BR1(p2) =

{p2 − ε if p2 − ε > cc otherwise

.

I �Bertrand Paradox�, the unique equilibrium is pBertrand

= c .

Di�erentiated Good

I Now two goods, e.g. �rm 1 produces blue jeans and �rm2 produces black jeans.

I Demands for these two goods are

q1 = a − bp1 + dp2;

q2 = a − bp2 + dp1.

I Assume b > d , demand more responsive to own-price.

I Assume zero marginal cost (for simplicity).

Bertrand with Di�erentiated Products (1)

I Firm 1 maximizes

maxp1

π1 = p1(a − b · p1 + d · p2).

I Firm 2 maximizes

maxp2

π2 = p2(a − b · p2 + d · p1).

I Therefore BR1(p2) =a+d ·p2

2band BR2(p1) =

a+d ·p12b

.

I Strategic complements.

Strategic Complements

Bertrand with Di�erentiated Products (2)

The Model

q1 = a − b · p1 + d · p2;q2 = a − b · p2 + d · p1.

Best Response BRi(pj) =a+d ·pj2b

Equil. Price p = a2b−d

Equil. Quantity qi =ab

2b−dEquil. Pro�t πi =

a2b(2b−d)2

Summary

I We discussed game-theoretic approach in oligopolymodels.

I Nash equilibrium:

I A strategy prole in which each player's NE strategy is abest-response to opponents' best-response strategies2-player case:

I a1 = BR1(a2); a2 = BR2(a1).

I We looked at quantity (Cournot) competition and price(Bertrand) competition.

I All models we discussed so far are static completeinformation game.

I The game is complete because rival's costs are known.I We will look at dynamic game in later.