3.2. Cournot Model. Matilde Machado


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1 Matilde Machado 1
2 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? The equilibrium concept used is Nash Equilibrium (CournotNash) Industrial Economics Matilde Machado 3.2. Cournot Model 2
3 Graphically: Let s assume the duopoly case (n=2) MC=c 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. Industrial Economics Matilde Machado 3.2. Cournot Model 3
4 Graphically (cont.): P p* MC D(p) q* 1 = R 1 (q 2 ) MR q 2 RD1(q2) = Residual demand Industrial Economics Matilde Machado 3.2. Cournot Model 4
5 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 : Case I: q 2 =0 RD 1 (p,0)=d(p) whole demand q* 1 (0)=q M Firm 1 should produce the Monopolist s quantity Industrial Economics Matilde Machado 3.2. Cournot Model 5
6 Case 2: q 2 =q c RD 1 (p,q c )=D(p)q c D(p) Residual Demand c q c c MR<MC q* 1 =0 MR q c Industrial Economics Matilde Machado 3.2. Cournot Model 6
7 Note: If both demand and cost functions are linear, reaction function will be linear as well. q1 q M Reaction function of firm 1 q c q2 Industrial Economics Matilde Machado 3.2. Cournot Model 7
8 q1 q c q M q1=q2 If firms are symmetric then the equilibrium is in the 45º line, the reaction curves are symmetric and q* 1 =q* 2 q* 1 E 45º q* 2 q M q c q2 Industrial Economics Matilde Machado 3.2. Cournot Model 8
9 Comparison between Cournot, Monopoly and Perfect Competition q q1 M <q N <q c q c q 1 +q 2 =q N q M q 1 +q 2 =q c q 1 +q 2 =q M q M q 1 +q 2 =q N q c q2 Industrial Economics Matilde Machado 3.2. Cournot Model 9
10 Derivation of the Cournot Equilibrium for n=2 P=abQ=ab(q 1 +q 2 ) MC 1 =MC 2 =c For firm 1: 1 (, ) ( ) ( ( ) ) Π = = + 1 Max q1 q2 p c q1 a b q1 q2 c q1 q Takes the strategy of firm 2 as given, i.e. takes q 2 as a constant. Note the residual demand here 1 Π = a bq1 bq2 c bq1 = q1 FOC: 0 0 2bq = a bq c 1 2 a c q2 q1 = 2b 2 Reaction function of firm 1: optimal quantity firm 1 should produce given q2. If q2 changes, q1 changes as well. Industrial Economics Matilde Machado 3.2. Cournot Model 10
11 We solve a similar problem for firm 2 and obtain a system of 2 equations and 2 variables. a c q2 q1 = 2b 2 a c q1 q2 = 2b 2 If firms are symmetric, then q = q = q * * * 1 2 i.e. we impose that the eq. quantity is in the 45º line * * a c q * a c q = q = = q = q 2b 2 3b N N 1 2 Solution of the Symmetric equilibrium Industrial Economics Matilde Machado 3.2. Cournot Model 11
12 Solution of the Symmetric equilibrium q = q = q * * * 1 2 * * a c q * a c q = q = = q = q 2 b 2 3 b Total quantity and the market price are: N N N 2 a c Q = q1 + q2 = 3 b N N 2 a + 2c p = a bq = a ( a c) = 3 3 N N 1 2 Industrial Economics Matilde Machado 3.2. Cournot Model 12
13 Comparing with Monopoly and Perfect Competition Where we obtain that: c N M p < p < p c a+ 2c a+ c 3 2 c N M p p p > < c c c = = = 3 2 In perfect competition prices increase 1to1 with costs. Industrial Economics Matilde Machado 3.2. Cournot Model 13
14 In the Case of n 2 firms: q 1 (,... ) ( (... ) ) Max Π q q = a b q + q + + q c q 1 1 N 1 2 N 1 FOC: a b ( q + q q ) c bq = 0 q = If all firms are symmetric: q1 = q2 =... = qn = q N 1 a b( q qn ) c 2b a b( n 1) q c 1 a c N a c q = 1 ( n 1) q q 2b + = = 2 2 b ( n + 1) b Industrial Economics Matilde Machado 3.2. Cournot Model 14
15 Total quantity and the equilibrium price are: N N n a c n a c c Q = nq = = q n + 1 b b N N n a c a n n p = a bq = a b = + c c n + 1 b n + 1 n + 1 If the number of firms in the oligopoly converges to, the NashCournot 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
16 DWL in the Cournot model = area where the willingness to pay is higher than MC p N 1 N c c N DWL = ( p p )( Q Q ) n a c n a c = a + c c 2 n + 1 n + 1 b n + 1 b 1 a c n = 0 2b n c DWL Industrial Economics Matilde Machado 3.2. Cournot Model 16 Q N q c When the number of firms converges to infinity, the DWL converges to zero, which is the same as in Perfect Competition. The DWL decreases faster than either price or quantity (rate of n 2 )
17 In the Asymmetric duopoly case with constant marginal costs. linear demand P( q + q ) = a b( q + q ) c c 1 2 = = MC of firm 1 MC of firm The FOC (from where we derive the reaction functions): q1p ( q1 + q2) + P( q1 + q2) c1 = 0 bq1 + a b( q1 + q2) c1 = 0 q2p ( q1 + q2) + P( q1 + q2) c2 = 0 bq2 + a b( q1 + q2) c2 = 0 a bq2 c1 q1 = 2b Replace q 2 in the reaction function a bq1 c2 q2 = of firm 1 and solve for q 1 2b Industrial Economics Matilde Machado 3.2. Cournot Model 17
18 In the Asymmetric duopoly case with constant marginal costs. a c1 1 a bq1 c2 3 a c2 c1 q1 = q1 = + 2b 2 2b 4 4b 4b 2b * a + c2 2c1 q1 = 3bb Which we replace back in q 2 : * * a bq1 c2 a q2 2b 1 a + c 2c c a 2c + c 2b 2 3b 2b 3b = = = a + c 2c a 2c + c 2a c c Q = q + q = + = 3b 3b 3b * * * 2a c2 c1 a + c2 + c1 p = a b( q1 + q2) = a = 3 3 * * * Industrial Economics Matilde Machado 3.2. Cournot Model 18
19 From the equilibrium quantities we may conclude that: q a + c 2 c 2 ; a q c + = = c 3b 3b * 2 1 * If c 1 <c 2 (i.e. firm 1 is more efficient): q a c 2c a 2c c c c q = + + = > 3b 3b 3b 3b 3b 3b b * * q > q * * 1 2 In Cournot, the firm with the largest market share is the most efficient Industrial Economics Matilde Machado 3.2. Cournot Model 19
20 From the previous result, the more efficient firm is also the one with a larger pricemcost margin: L p c1 p c2 = > = L p p 1 2 s1 = s2 = ε ε Industrial Economics Matilde Machado 3.2. Cournot Model 20
21 Comparative Statics: The output of a firm when: a + c 2 * j ci qi = 3b own costs costs of rival q 2 c 1 Shifts the reaction curve of firm 1 to the left E E q* 2 and q* 1 q 1 Industrial Economics Matilde Machado 3.2. Cournot Model 21
22 Profits are: ( ) ( ( ) ) Π = p c q = a b q + q c q = 1* * * * * * ( a + c 2c ) 2a c2 c1 a + c2 2c1 = a b c1 = 3b 3 b 9 b Increase with rival s costs Decrease with own costs 1 Π > c 2 1 Π < c Symmetric to firm 2. Industrial Economics Matilde Machado 3.2. Cournot Model 22
23 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 q i 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. i Max Π ( q, q ) = q P( Q) C ( q ) q i i j i i i effect of the increase in quantity on the inframarginal units 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 from the industry s point of view. Πi CPO: = 0 qip ( Q) + P( Q) C i ( qi ) = 0 q i profitability of the marginal unit Industrial Economics Matilde Machado 3.2. Cournot Model 23
24 If we define the Lerner index of the market as: L i s L i i we obtain: s 1 H s L = s = s = ε ε ε Concentration i 2 i i i i i i i Is the Herfindhal Index Industrial Economics Matilde Machado 3.2. Cournot Model 24
25 The positive relationship between profitability and the Herfindhal Concentration Index under Cournot: Remember the FOC for each firm in that industry can be written as: p ci si = p ε The Industrywide profits are then: ( ) p c s pq s p q Π = p c q = pq = = Q = n n n n i i i i i ( i ) i i i= 1 i= 1 p i= 1 ε i= 1 ε Q n 2 n s 2 = i p pq pq Q = si = H = κ H i= 1 ε ε i= 1 ε The concentration index is up to a constant an exact measure of industry profitability. Industrial Economics Matilde Machado 3.2. Cournot Model 25
26 Note: The Cournot model is often times criticized because in reality firms tend to choose prices not quantities. The answer to this criticism is that when the cournot model is modified to incorporate two periods, the first where firms choose capacity and the second where firms compete in prices. This two period model gives the same outcome as the simple Cournot model. Industrial Economics Matilde Machado 3.2. Cournot Model 26
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