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Oligopoly (Game Theory)

Oligopoly (Game Theory) - Lancaster University€œKinked” Demand CurveDemand Curve P Elastic Where do p* aadqco end q* come from? Inelastic p* Dord Q* or q* Q or q D or d

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Text of Oligopoly (Game Theory) - Lancaster University€œKinked” Demand CurveDemand Curve P Elastic...

  • 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 - cq11 = aq1 - bq12 - bq1q2 - cq1

  • Cournot CompetitionCournot Competition1 = aq1-bq12-bq1q2-cq11 = aq1-bq1 -bq1q2-cq1To 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 21cabqbq 212 qq 21

    212 bqcabq

    bbqcaq

    22

    1

    Firm 1s Reactioncurveb2

  • Cournot Competitionp

    bbqcaq

    22

    1

    Next graph with q1 on thebq

    21

    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 Ob21

    COURNOT 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

    bbqca

    bc

    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

    bbqca

    bc

    baq

    2112 11

    Step 6: Simplify

    2222 11

    qbc

    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 11

    qbc

    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 b

    caq31

    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 3

    P

  • 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

    bcaQCO

    32

    32caPCO

    b3 3

    PCQm < Qco < QPC

    Pm > Pco > PpcPm > Pco > Ppc