Oligopoly (Game Theory)

Oligopoly: AssumptionsOligopoly: Assumptions

Many buyers Very small number of major sellersy j

(actions and reactions are important) Homogeneous product (usually, but not g p ( y,

necessarily) Perfect knowledge (usually, but not g ( y,

necessarily) Restricted entry (usually, but not y ( y,

necessarily)

Oligopoly ModelsOligopoly Models

1. “Kinked” Demand Curve2 Cournot (1838)2. Cournot (1838)3. Bertrand (1883)4. Nash (1950s): Game Theory

“Kinked” Demand CurveKinked Demand CurvePP

ElasticElastic

Inelasticp*

InelasticD or d

Q or qQ* or q*q

“Kinked” Demand CurveKinked Demand CurvePP

ElasticWhere do p* and q* come Elastic a d q co e

from?

Inelasticp*

Inelastic

D or d

Q or qQ* or q*

D or d

Cournot Competitionp Assume two firms with no entry allowed Assume two firms with no entry allowed

and homogeneous product Firms compete in quantities (q1, q2) Firms compete in quantities (q1, q2) q1 = F(q2) and q2 = G(q1) Linear (inverse) demand P = a – bQ where Linear (inverse) demand, P = a – bQ where

Q = q1 + q2 Assume constant marginal costs i e Assume constant marginal costs, i.e.

TCi = cqi for i = 1,2 Aim: Find q and q and hence p i e find Aim: Find q1and q2 and hence p, i.e. find

the equilibrium.

Cournot CompetitionCournot Competition

Firm 1 (w.o.l.o.g.)Profit = TR TCProfit = TR - TC1 = P.q1 - c.q1

[P = a - bQ and Q = q1 + q2, hencehenceP = a - b(q1 + q2)P = a - bq1 - bq2]

Cournot CompetitionCournot Competition

1 = Pq1-cq1

= (a bq bq )q cq1 = (a - bq1 - bq2)q1 - cq1

1 = aq1 - bq12 - bq1q2 - cq1

Cournot CompetitionCournot Competition1 = aq1-bq1

2-bq1q2-cq11 = aq1-bq1 -bq1q2-cq1

To find the profit maximising level of q1 for p g q1firm 1, differentiate profit with respect to q1and set equal to zero.q

02 211

1 cbqbqaq1q

Cournot Competition02 21 cbqbqa 21 qq

acbqbq 212 qq 21

cabqbq 212 qq 21

212 bqcabq

bbqcaq

22

1

Firm 1’s “Reaction”curveb2

Cournot Competitionp

bbqcaq

22

1

Next graph with q1 on theb

q21

Do the same steps to find q

q1 on the horizontal axis and q2 on theDo the same steps to find q2

bqca

and q2 on the vertical axis

bbqcaq

21

2

Note: We have two equations and two qunknowns so we can solve for q1q1and q2

Cournot CompetitionCou o Co pe o

bbqcaq

22

1

q2

CO Ob21COURNOT EQUILIBRIUM

bqca 1b

qq2

12

q1

Cournot CompetitionCournot Competitionbqca 2 bqca 1

bbqcaq

22

1 b

bqcaq2

12

Step 1: Rewrite q1 Step 3: Factor out 1/21

bbq

bc

baq

2222

1

21 2

1 qbc

baq

bbb 222Step 2: Cancel b Step 4: Sub. in for q2

2222

1q

bc

baq

b

bqcabc

baq

221 1

1222 bb bbb 22

Cournot CompetitionCournot Competition

bqcaca1 1

bbqca

bc

baq

221 1

1

Step 5: Multiply across by 2 to get rid of the fraction

b

b

bqcabc

baq

2112 1

1

Step 6: Simplify

2222 1

1q

bc

ba

bc

baq

222 bbbb

Cournot CompetitionCournot Competition2 1qcacaq

2222 1 bbbbq

Step 7: Multiply across by 2 to get rid of the fraction

22

22

22224 1

1q

bc

ba

bc

baq

Step 8: Simplify222 bbbb

2211

224 qbc

ba

bc

baq

Cournot CompetitionCournot Competition22 caca

11224 q

bc

ba

bc

baq

Step 9: Rearrange and bring q1 over to LHS.

22bc

bc

ba

baqq

224 11

Step 10: Simplify

bc

baq 13 Step 11: Simplify

bcaq

31

bb b3

Cournot CompetitionCournot CompetitionStep 12: Repeat above for q2ca

bq

31 caq

b

q32

Step 13: Solve for price (go back to demand curve)

bQaP

bca

bcabaP

33Step 14: Sub. in for q1 and q2 bb 331 2

Cournot CompetitionCournot Competition

caca

bca

bcabaP

33Step 14: Simplify

cacaaP

33

aP

3333cacaaP

3333

Cournot CompetitionCournot Competition

caca

3333cacaaP

ccaaaP31

31

31

31

3333

22 21caaP32

32

caP32

31

2caP

3P

Cournot Competition: Summaryp y

caq1

caq2

bq

31 bq

32

bca

bcaQ

33

ca2

bcaQ

32

Cournot v BertrandCournot v. Bertrand

Cournot Nash (q1, q2): Firms compete in quantities,i.e. Firm 1 chooses the best q1 given

dq2 and Firm 2 chooses the best q2 given q1

Bertrand Nash (p p ): Firms compete in pricesBertrand Nash (p1, p2): Firms compete in prices,i.e. Firm 1 chooses the best p1 given p2 and

Firm 2 chooses the best p given pFirm 2 chooses the best p2 given p1Nash Equilibrium (s1, s2): Player 1 chooses the best

s1 given s2 and Player 2 chooses the best s21 g 2 y 2given s1

Bertrand Competition: Bertrand Paradox

Assume two firms (as before), a linear demand curve, constant marginal costs , gand a homogenous product.

Bertrand equilibrium: p1 = p2 = c(Thi i li fit d i(This implies zero excess profits and is referred to as the Bertand Paradox)

Perfect Competition v. Monopoly v. Cournot Oligopoly

Givenii cqTCandbQaP

Given

Perfect Competitionca

bcaQCPMCP pc

Monopoly CQPQTCTR

Monopoly CQQbQa

CQPQ

CQbQaQ 2

Perfect Competition v. Monopoly v. Cournot Oligopoly

2

CQbQaQ

bQaP 02 CbQaQ CabaP

bQaP

CabQ2 2

Cab

baP

M

bCaQ m

2

2CaP M

Perfect Competition v Monopoly v Cournot Oligopoly

b

caQCO

32

32caPCO

b3 3

PCQm < Qco < QPC

Pm > Pco > PpcPm > Pco > Ppc