oligopoly (game theory) - lancaster university€œkinked” demand curvedemand curve p elastic...
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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