3.2. Cournot Model Cournot Model

Transcription

1 Matlde Machado Assumptons: All frms produce an homogenous product The market prce s therefore the result of the total supply (same prce for all frms) Frms decde smultaneously how much to produce Quantty s the strategc varable. If OPEC was not a cartel, then ol extracton would be a good example of Cournot competton. Agrcultural products? The equlbrum concept used s Nash Equlbrum (Cournot-Nash) Industral Economcs- Matlde Machado

2 Graphcally: Let s assume the duopoly case (n) MCc Resdual demand of frm : RD (p,q )D(p)-q. The problem of the frm wth resdual demand RD s smlar to the monopolst s. Industral Economcs- Matlde Machado 3 Graphcally (cont.): P p* MC D(p) q* R (q ) MR q RD(q) Resdual demand Industral Economcs- Matlde Machado 4

3 Graphcally (cont.): q* (q )R (q ) s the optmal quantty as a functon of q Let s take extreme cases q : Case I: q 0 RD (p,0)d(p) whole demand q* (0)q M Frm should produce the Monopolst s quantty Industral Economcs- Matlde Machado 5 Case : q RD (p, )D(p)- D(p) Resdual Demand c c MR<MC q* 0 MR Industral Economcs- Matlde Machado 6 3

4 Note: If both demand and cost functons are lnear, reacton functon wll be lnear as well. q q M Reacton functon of frm q Industral Economcs- Matlde Machado 7 q q M qq If frms are symmetrc then the equlbrum s n the 45º lne, the reacton curves are symmetrc and q* q* q* E 45º q* q M q Industral Economcs- Matlde Machado 8 4

5 Comparson between Cournot, Monopoly and Perfect Competton q q M <q N < q +q q N q M q +q q +q q M q M q +q q N q Industral Economcs- Matlde Machado 9 Dervaton of the Cournot Equlbrum for n Pa-bQa-b(q +q ) MC MC c For frm : (, ) ( ) ( ( ) ) Π + Max q q p c q a b q q q Takes the strategy of frm as gven,.e. takes q as a constant. Note the resdual demand here Π a bq bq c bq q FOC: 0 0 bq a bq c a c q q b Reacton functon of frm : optmal quantty frm should produce gven q. If hanges, hanges as well. Industral Economcs- Matlde Machado 0 5

6 We solve a smlar problem for frm and obtan a system of equatons and varables. a c q q b a c q q b If frms are symmetrc, then * * * q q q.e. we mpose that the eq. quantty s n the 45º lne * * a c q * a c q q q q b 3b N N Soluton of the Symmetrc equlbrum Industral Economcs- Matlde Machado Soluton of the Symmetrc equlbrum q q q * * * * * a c q * a c q q q q b 3b Total quantty and the market prce are: N N N a c Q q + q 3 b N N a + c p a bq a ( a c) 3 3 N N Industral Economcs- Matlde Machado 6

7 Comparng wth Monopoly and Perfect Competton Where we obtan that: c N M { p < { p < { p c a+ c a+ c 3 c N M p p p > < { c { c { c 3 In perfect competton prces ncrease -to- wth costs. Industral Economcs- Matlde Machado 3 In the Case of n frms: q If all frms are symmetrc: (,... ) ( (... ) ) Max Π q q a b q + q + + q c q N N FOC: a b( q + q q ) c bq 0 N a b( q qn ) c q b q q... qn q a b( n ) q c a c N a c q ( n ) q q b + b ( n + ) b Industral Economcs- Matlde Machado 4 7

8 Total quantty and the equlbrum prce are: N N n a c n a c c Q nq q n + b b N N n a c a n n p a bq a b + c c n + b n + n + If the number of frms n the olgopoly converges to, the Nash-Cournot equlbrum converges to perfect competton. The model s, therefore, robust snce wth n the condtons of the model concde wth those of the perfect competton. Industral Economcs- Matlde Machado 5 DWL n the Cournot model area where the wllngness to pay s hgher than MC N c c N DWL ( p p )( Q Q ) n a c n a c a + c c n + n + b n + b a c n 0 b n + Industral Economcs- Matlde Machado 6 p N c Q N DWL When the number of frms converges to nfnty, the DWL converges to zero, whch s the same as n Perfect Competton. The DWL decreases faster than ether prce or quantty (rate of n ) 8

9 In the Asymmetrc duopoly case wth constant margnal costs. lnear demand P( q + q ) a b( q + q ) c MC of frm c MC of frm The FOC (from where we derve the reacton functons): qp ( q + q) + P( q + q) c 0 bq + a b( q + q) c 0 qp ( q + q) + P( q + q) c 0 bq + a b( q + q) c 0 a bq c q b Replace q n the reacton functon a bq c q of frm and solve for q b Industral Economcs- Matlde Machado 7 In the Asymmetrc duopoly case wth constant margnal costs. a c a bq c 3 a c c q q + b b 4 4b 4b b * a + c c q 3b Whch we replace back n q : * * a bq c a a + c c c a c + c q b b 3b b 3b a + c c a c + c a c c Q q + q + 3b 3b 3b * * * a c c a + c + c p a b( q + q) a 3 3 * * * Industral Economcs- Matlde Machado 8 9

10 From the equlbrum quanttes we may conclude that: q a + c c ; a q c + c 3b 3b * * If c <c (.e. frm s more effcent): q a c c a c c c c q + + > 0 3b 3b 3b 3b 3b 3b b * * q > q * * In Cournot, the frm wth the largest market share s the most effcent Industral Economcs- Matlde Machado 9 From the prevous result, the more effcent frm s also the one wth a larger prce-mcost margn: L p c p c > L { p p 3 s s ε ε Industral Economcs- Matlde Machado 0 0

11 Comparatve Statcs: The output of a frm when: a + c * j c q 3b own costs costs of rval q c Shfts the reacton curve of frm to the left E E q* and q* q Industral Economcs- Matlde Machado Profts are: ( ) ( ( ) ) Π p c q a b q + q c q * * * * * * ( a + c c ) a c c a + c c a b c 3b 3b 9b Increase wth rval s costs Decrease wth own costs Π > c Π < c 0 0 Symmetrc to frm. Industral Economcs- Matlde Machado

12 More generally for any demand and cost functon. There s a negatve externalty between Cournot frms. Frms do not nternalze the effect that an ncrease n the quantty they produce has on the other frms. That s when q the frm lowers the prce to every frm n the market (note that the good s homogenous). From the pont of vew of the ndustry (.e. of max the total proft) there wll be excessve producton. Max Π ( q, q ) q P( Q) C ( q ) q j Π CPO: 0 qp ( Q) + P( Q) C ( q ) 0 q effect of the ncrease n quantty on the nframargnal unts Externalty: frms only take nto account the effect of the prce change n ther own output. Then ther output s hgher than what would be optmal from the ndustry s pont of vew. proftablty of the margnal unt Industral Economcs- Matlde Machado 3 If we defne the Lerner ndex of the market as: L s L we obtan: s H s L s s ε ε ε Is the Herfndhal Concentraton Index Industral Economcs- Matlde Machado 4

13 The postve relatonshp between proftablty and the Herfndhal Concentraton Index under Cournot: Remember the FOC for each frm n that ndustry can be wrtten as: p c s p ε The Industry-wde profts are then: n n n n ( ) ( ) p c s pq s p q Π p c q pq Q p ε ε Q n n s p pq pq Q s H κ H ε ε ε The concentraton ndex s up to a constant an exact measure of ndustry proftablty. Industral Economcs- Matlde Machado 5 Note: The Cournot model s often tmes crtczed because n realty frms tend to choose prces not quanttes. The answer to ths crtcsm s that when the cournot model s modfed to ncorporate two perods, the frst where frms choose capacty and the second where frms compete n prces. Ths two perod model gves the same outcome as the smple Cournot model. Industral Economcs- Matlde Machado 6 3