3.2. Cournot Model Matilde Machado 1 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) 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 Industrial Economics-Equilibrium Matilde Machado (Cournot-Nash) 3.2. Cournot Model 2 3.2. Cournot Model Graphically: Let’s assume the duopoly case (n=2) MC=c Residual demand of firm 1: RD1(p,q2)=D(p)-q2. The problem of the firm with residual demand RD is similar to the monopolist’s. Industrial Economics- Matilde Machado 3.2. Cournot Model 3 3.2. Cournot Model Graphically (cont.): P p* MC D(p) RD1(q2) = Residual demand q* 1= q2 R1(q 2) MR Industrial Economics- Matilde Machado 3.2. Cournot Model 4 3.2. Cournot Model Graphically (cont.): q* 1(q2)=R1(q2) is the optimal quantity as a function of q2 Let’s take 2 extreme cases q2: Case I: q2=0 ⇒RD1(p,0)=D(p) whole demand ⇓ Firm 1 should M produce the q* 1(0)=q Monopolist’ s quantity Industrial Economics- Matilde Machado 3.2. Cournot Model 5 3.2. Cournot Model c c c D(p) Case 2: q2=q ⇒RD1(p,q )=D(p)-q c Residual Demand qc c MR<MC ⇒q* 1=0 qc MR Industrial Economics- Matilde Machado 3.2. Cournot Model 6 3.2. Cournot Model Note: If both demand and cost functions are linear, reaction function will be linear as well. q1 Reaction function of qM firm 1 c q q2 Industrial Economics- Matilde Machado 3.2. Cournot Model 7 3.2. Cournot Model q1 If firms are symmetric then the equilibrium qc is in the 45º line, the reaction curves are q1=q2 symmetric and M q q* 1=q* 2 q* 1 E 45º qM qc q* 2 q2 Industrial Economics- Matilde Machado 3.2. Cournot Model 8 3.2. Cournot Model Comparison between Cournot, Monopoly and Perfect Competition qM<q N<q c q1 qc N q1+q 2=q qM c q1+q 2=q M N c q q1+q 2=q q M q2 q1+q 2=q Industrial Economics- Matilde Machado 3.2. Cournot Model 9 3.2. Cournot Model Derivation of the Cournot Equilibrium for n=2 Takes the strategy of P=a-bQ=a-b(q 1+q 2) firm 2 as given, i.e. takes MC1=MC2=c q2 as a constant. Note the residual demand For firm 1: here Π1 ( ) =−( ) =−+−( ) Max qq12, pcq 1 abq ( 12 q ) cq 1 q1 ∂Π 1 FOC: =⇔−0a bq − bq −− c bq = 0 ∂ 1 2 1 q1 ⇔ =− − 2 bq1 a bq 2 c Reaction function of firm 1: a− c q optimal quantity firm 1 ⇔ = − 2 q1 should produce given q2. If 2b 2 q2 changes, q1 changes as Industrial Economics- Matilde Machado 3.2. Cournot Model well. 10 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 1 2b 2 a− c q q = − 1 2 2b 2 If firms are symmetric, then *= * = * q1 q 2 q i.e. we impose that the eq. quantity is in the 45º line acq−* ac − ⇒ q*= −⇔= q * == qqN N 2b 2 3 b 1 2 Solution of the Symmetric Industrial Economics- Matilde Machado 3.2. Cournot Model equilibrium 11 3.2. Cournot Model Solution of the Symmetric equilibrium *= * = * q1 q 2 q acq−* ac − ⇒ q*= −⇔= q * == qqN N 2b 2 3 b 1 2 Total quantity and the market price are: 2 a− c QN= q N + q N = 1 2 3 b 2a+ 2 c pN=− abQ N =− a() ac −= 3 3 Industrial Economics- Matilde Machado 3.2. Cournot Model 12 3.2. Cournot Model Comparing with Monopoly and Perfect Competition c< N < M p p p c ac+2 ac + 3 2 Where we obtain that: ∂pc ∂ p N ∂ p M In perfect competition > < prices increase 1-to-1 with ∂ ∂ ∂ c c c costs. = 1= 2 = 1 3 2 Industrial Economics- Matilde Machado 3.2. Cournot Model 13 3.2. Cournot Model In the Case of n ≥2 firms: Π( ) =−( +++− ) Max11 qq,...N abqq ( 12 ... qN ) cq 1 q1 − +++ −− = FOC: abqq(1 2 ... qN ) cbq 1 0 abq−( ++ ... q ) − c ⇔q = 2 N 1 2b If all firms are symmetric: = == = qq1 2 ... qqN abnqc−−−( 1) 1 ac − ac − q= ⇔+−=⇔=1 ( nqq 1) N 2b 2 2 bnb (+ 1) Industrial Economics- Matilde Machado 3.2. Cournot Model 14 3.2. Cournot Model Total quantity and the equilibrium price are: nac−→∞ ac − QnqNN= = n → = q c n+1 b b nac− a n →∞ pabQabN= − N = − = + c n → c n+1 b n + 1 n + 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. Industrial Economics- Matilde Machado 3.2. Cournot Model 15 3.2. Cournot Model DWL in the Cournot model = a rea where the willingness to pay is higher than MC pN DWL 1 c DWL=()() pN − p c Q c − Q N 2 QN qc 1 1 n ac− nac − =a +− c c − + + + When the number of firms 211n n bnb 1 converges to infinity, the 2 DWL converges to zero, 1 a− c →∞ = →n 0 + which is the same as in 2b n 1 Perfect Competition. The DWL decreases faster than either price or quantity (rate of n 2) Industrial Economics- Matilde Machado 3.2. Cournot Model 16 3.2. Cournot Model In the Asymmetric duopoly case with constant marginal costs. + =− + linear demand Pq(12 q ) abq ( 12 q ) = c1 MC of firm 1 = c2 MC of firm 2 The FOC (from where we derive the reaction functions): qPqq′()()0++ Pqq +−= c −+− bqabqq ()0 +−= c 112 121⇔ 1 121 ′ ++ +−= −+− +−= qPqq212()()0 Pqq 122 c bqabqq 2 ()0 122 c a− bq − c q = 2 1 1 2b ⇔ Replace q in the reaction function a− bq − c 2 q = 1 2 of firm 1 and solve for q 1 2 2b Industrial Economics- Matilde Machado 3.2. Cournot Model 17 3.2. Cournot Model In the Asymmetric duopoly case with constant marginal costs. ac−1 abqc − − 3 a cc q=−1 12 ⇔=+− q 21 1 22b 2 b 44421 bbb a+ c − 2 c ⇔q* = 2 1 1 3b Which we replace back in q 2: a− bq* − c a 1 ac+−2 c c acc −+ 2 q* = 1 2 =−21 −= 2 21 2 2b 223b b 2 b 3 b ac+−2 cacc −+ 2 2 acc −− Q*=+= q * q * 21 + 21 = 21 1 2 3b 3 b 3 b 2ac−− c ac ++ c p*=− abq( * +=− q * ) a 21 = 21 1 2 3 3 Industrial Economics- Matilde Machado 3.2. Cournot Model 18 3.2. Cournot Model From the equilibrium quantities we may conclude that: acc+−2 acc −+ 2 q*=21 ; q * = 21 13b 2 3 b If c 1<c 2 (i.e. firm 1 is more efficient): ac2 c a 2 cccc− q*−=+−−+ q * 21 2121 −= > 0 1 2 33bbbbbb 3 3 3 3 b ⇔* > * In Cournot, the firm with the largest market q1 q 2 share is the most efficient Industrial Economics- Matilde Machado 3.2. Cournot Model 19 3.2. Cournot Model From the previous result, the more efficient firm is also the one with a larger price-Mcost margin: pc− pc − L=1 > 2 = L 1p p 2 s = 1 = s2 ε ε Industrial Economics- Matilde Machado 3.2. Cournot Model 20 3.2. Cournot Model Comparative Statics: ↑ own costs The output of a firm ↓ when: a+ c − 2 c ↓ costs of rival q* = j i i 3b q2 Shifts the reaction curve ↑c1 of firm 1 to the left E’ E ↑q* 2 and ↓q* 1 q1 Industrial Economics- Matilde Machado 3.2. Cournot Model 21 3.2. Cournot Model Profits are: Π=−1** * =− ** +− * = ( p cq11) ( abq( 1211 q ) cq) 2 2acc− − acc+ − 2 ()a+ c − 2 c =−21 −× 21 = 2 1 a b c 1 3b 3 b 9 b ∂Π 1 Increase with rival’s costs > 0 ∂ c2 ∂Π 1 Decrease with own costs < 0 ∂ c1 Symmetric to firm 2. Industrial Economics- Matilde Machado 3.2. Cournot Model 22 3.2. Cournot Model More generally… for any demand and cost function. There is a negative externality between Cournot firms. Firms do not internalize the effect that an increase in the quantity they produce has on the other firms. That is when ↑qi the firm lowers the price to every firm in the market (note that the good is homogenous). From the point of view of the industry (i.e. of max the total profit) there will be excessive production . Externality: firms only take into account the effect of the price change in their own output. Then their output is higher than what would be optimal Πi = − from the industry’s point of view. Max(,) qqij qPQ i () Cq ii () qi ∂Π CPO:i =⇔ 0qPQ′ () +−= PQ () Cq ′ ()0 ∂q i i i i effect of the increase in quantity profitability of the on the inframarginal units marginal unit Industrial Economics- Matilde Machado 3.2.