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3.2. Cournot Model Matilde Machado 3.2. Cournot Model Assumptions: All firms produce an homogenous product The market price is therefore the result of the total supply (same price for all firms) the total supply (same price for


  1. 3.2. Cournot Model Matilde Machado �

  2. 3.2. Cournot Model Assumptions: All firms produce an homogenous product � The market price is therefore the result of � the total supply (same price for all firms) the total supply (same price for all firms) Firms decide simultaneously how much to � produce Quantity is the strategic variable. If OPEC was not a � cartel, then oil extraction would be a good example of Cournot competition. Agricultural products? http://www.iser.osaka-u.ac.jp/library/dp/2010/DP0766.pdf ? The equilibrium concept used is Nash � Equilibrium (Cournot-Nash) ��������������������� ��������������� ������������������ �

  3. 3.2. Cournot Model Graphically: Let’s assume the duopoly case (n=2) � MC=c � Residual demand of firm 1: Residual demand of firm 1: � RD 1 (p,q 2 )=D(p)-q 2 . The problem of the firm with residual demand RD is similar to the monopolist’s. ��������������������� ��������������� ������������������ �

  4. 3.2. Cournot Model Graphically (cont.): P p* MC D(p) RD1(q2) = Residual demand q* 1 = q 2 R 1 (q 2 ) MR ��������������������� ��������������� ������������������ �

  5. 3.2. Cournot Model Graphically (cont.): q* 1 (q 2 )=R 1 (q 2 ) is the optimal quantity as a function of q 2 Let’s take 2 extreme cases q 2 : Let’s take 2 extreme cases q : Case I: q 2 =0 ⇒ RD 1 (p,0)=D(p) whole demand ⇓ Firm 1 should q* 1 (0)=q M produce the Monopolist’ s quantity ��������������������� ��������������� ������������������ �

  6. 3.2. Cournot Model Case 2: q 2 =q c ⇒ RD 1 (p,q c )=D(p)-q c D(p) c Residual Demand q c c MR<MC ⇒ q* 1 =0 q c MR ��������������������� ��������������� ������������������ �

  7. 3.2. Cournot Model Note: If both demand and cost functions are linear, reaction function will be linear as well. q1 q1 Reaction function of q M firm 1 q c q2 ��������������������� ��������������� ������������������ �

  8. 3.2. Cournot Model If firms are symmetric q1 then the equilibrium is in the 45º line, the q c reaction curves are q1=q2 q1=q2 symmetric and symmetric and q M q* 1 =q* 2 q* 1 E 45º q M q c q* 2 q2 ��������������������� ��������������� ������������������ �

  9. 3.2. Cournot Model Comparison between Cournot, Monopoly and Perfect Competition q M <q N <q c q1 q c q 1 +q 2 =q N q M q 1 +q 2 =q c q M q 1 +q 2 =q N q c q2 q 1 +q 2 =q M ��������������������� ��������������� ������������������ �

  10. 3.2. Cournot Model Derivation of the Cournot Equilibrium for n=2 P=a-bQ=a-b(q 1 +q 2 ) Takes the strategy of firm 2 as given, i.e. takes MC 1 =MC 2 =c q 2 as a constant. Note the residual demand For firm 1: For firm 1: here here ( ) ( ) ( ) Π = − = − + − Max q q p c q a b q q c q 1 , ( ) 1 2 1 1 2 1 q 1 ∂Π = 1 ⇔ − − − − = a bq bq c bq FOC: 0 0 ∂ q 1 2 1 1 ⇔ = − − bq a bq c 2 Reaction function of firm 1: 1 2 − optimal quantity firm 1 a c q ⇔ = − q 2 should produce given q2. If b 1 2 2 q2 changes, q1 changes as well. ��������������������� ��������������� ������������������ ��

  11. 3.2. Cournot Model We solve a similar problem for firm 2 and obtain a system of 2 equations and 2 variables. −  a c q = − q 2   b 1 2 2   − − a a c c q q  = − q 1   b 2 2 2 If firms are symmetric, then = = q q q * * * i.e. we impose that the eq. quantity is in the 45º line 1 2 − − a c q a c * ⇒ = − ⇔ = = N = N q q q q * * b b 1 2 2 2 3 Solution of the Symmetric equilibrium ��������������������� ��������������� ������������������ ��

  12. 3.2. Cournot Model Solution of the Symmetric equilibrium = = q q q * * * 1 2 − − a c q a c * ⇒ = − ⇔ = = N = N q q q q * * b b b b 1 1 2 2 2 2 2 2 3 3 Total quantity and the market price are: −  a c  2 N = N + N = Q q q   b 1 2   3 + a c 2 2 ( ) = − = − − = N N p a bQ a a c 3 3 ��������������������� ��������������� ������������������ ��

  13. 3.2. Cournot Model Comparing with Monopoly and Perfect Competition < < c N M p p p � � � + + c a c a c 2 3 2 Where we obtain that: Where we obtain that: ∂ c ∂ N ∂ M In perfect competition p p p > < prices increase 1-to-1 with ∂ ∂ ∂ c c c � � � costs. = 1 2 1 = = 3 2 ��������������������� ��������������� ������������������ ��

  14. 3.2. Cournot Model In the Case of n ≥ 2 firms: ( ) ( ) Π = − + + + − Max q q a b q q q c q ,... ( ... ) N N 1 1 1 2 1 q 1 − − + + + + + + − − − − = = a a b q b q q q q q c c bq bq FOC: FOC: ( ( ... ... ) ) 0 0 N N 1 1 2 2 1 1 − + + − a b q q c ( ... ) ⇔ = q N 2 1 b 2 If all firms are symmetric: = = = = q q q q ... N 1 2 − − − − − a b n q c   a c a c ( 1) 1 = ⇔ + − = ⇔ N = q n q q 1 ( 1)   + b b n b   2 2 2 ( 1) ��������������������� ��������������� ������������������ ��

  15. 3.2. Cournot Model Total quantity and the equilibrium price are: − − n a c a c = =  →∞ → = N N n c Q nq q + n b b 1 − − n n a a c c a a n n →∞ →∞ N N = = − − N N = = − − = = + +   n n → → p p a a bQ bQ a a b b c c c c + + + n b n n 1 1 1 If the number of firms in the oligopoly converges to ∞ , the Nash-Cournot equilibrium converges to perfect competition. The model is, therefore, robust since with n → ∞ the conditions of the model coincide with those of the perfect competition. ��������������������� ��������������� ������������������ ��

  16. 3.2. Cournot Model DWL in the Cournot model = a rea where the willingness to pay is higher than MC p N DWL c c ( )( ) 1 = N − c c − N DWL p p Q Q 2 Q N q c − − n a c n a c    1 1 = + − − a c c    When the number of firms + + + n n b n b    2 1 1 1 converges to infinity, the − 2 DWL converges to zero,  a c  1 =  n →∞ →   0 which is the same as in + b n   2 1 Perfect Competition. The DWL decreases faster than either price or quantity (rate of n 2 ) ��������������������� ��������������� ������������������ ��

  17. 3.2. Cournot Model In the Asymmetric duopoly case with constant marginal costs. + = − + P q q a b q q linear demand ( ) ( ) 1 2 1 2 = c MC of firm 1 1 = c MC of firm 2 2 2 The FOC (from where we derive the reaction functions): ′ + + + − = − + − + − =  q P q q P q q c  bq a b q q c ( ) ( ) 0 ( ) 0 ⇔ 1 1 2 1 2 1 1 1 2 1   ′ + + + − = − + − + − = q P q q P q q c bq a b q q c   ( ) ( ) 0 ( ) 0 2 1 2 1 2 2 2 1 2 2 − − a bq c  = q 2 1   b 1 2 ⇔  Replace q 2 in the reaction function − − a bq c  of firm 1 and solve for q 1 = q 1 2   2 b 2 ��������������������� ��������������� ������������������ ��

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