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