Nash Equilibrium A game consists of a set of players a set of strategies for each player A mapping from set of strategies to a set of payoffs, one

Transcription

1 Nsh Euilibrium A gme consists of set of plyers set of strtegies for ech plyer A mpping from set of strtegies to set of pyoffs, one for ech plyer N.E.: A Set of strtegies form NE if, for plyer i, the strtegy chosen by i mximises i s pyoff, given the strtegies chosen by ll other plyers

2 NE is the set of strtegies from which no plyer hs n incentive to unilterlly devite NE is the centrl concept of noncoopertive gme theory I.e. situttions in which binding greements re not possible

3 Exmple Plyer C D Plyer C D (0,0) (0,0) (0,0) (,) This is the gme s pyoff mtrix. Plyer A s pyoff is shown first. Plyer B s pyoff is shown second. NE: (DD) (,)

4 Another Exmple. Plyer B L R Plyer A U D (3,9) (0,0) (,8) (,) Two Nsh euilibri: (U,L) (3,9) (D,R) (,)

5 Applying the NE Concept Modelling Short Run Conduct Bertrnd Competition Cournot Competition [Building blocks in modeling the intensity of competition in n industry in the short run]

6 p p monop P(N))? C N

7 Bertrnd Price Competition Wht if firms compete using only price-setting strtegies,? Gmes in which firms use only price strtegies nd ply simultneously re Bertrnd gmes.

8 Bertrnd Gmes (883). plyers, firms i nd j. Bertrnd Strtegy - All firms simultneously set their prices. 3. Homogenous product 4. Perfect Informtion 5. Ech firm s mrginl production cost is constnt t c.

9 Bertrnd Gmes p i 0 p i ½ (p i c)q p i (p i c)q if p i >p j if p i p j if p i < p j Q: Is there Nsh euilibrium? A: Yes. Exctly one. All firms set their prices eul to the mrginl cost c. Why?

10 Bertrnd Gmes Proof by Contrdiction Suppose one firm sets its price higher thn nother firm s price. Then the higher-priced firm would hve no customers. Hence, t n euilibrium, ll firms must set the sme price.

11 Bertrnd Gmes Suppose the common price set by ll firm is higher thn mrginl cost c. Then one firm cn just slightly lower its price nd sell to ll the buyers, thereby incresing its profit. The only common price which prevents undercutting is c. Hence this is the only Nsh euilibrium.

12 Illustrtion p p p B E C F C A D NE: t point A where p p Cost C p

13 Bertrnd Prdox For n> with firms simultneously setting prices, prices mrginl cost nd profits re zero. Perfectly competitive outcome is replicted Intuitive ssumption..surprising result!

14 This result holds where firms hve identicl costs If firms hve different costs, then there my or my not be pure strtegy euilibrium

15 If firms re cpcity constrined, then mixed strtegy euilibrium results Edgeworth (897) - Cpcity Constrints Neither firm cn meet the entire mrket demnd, but cn meet hlf mrket demnd. Constnt MC to point, then decresing returns Under these conditions, Edgeworth cycle: prices fluctute between high nd low

16 Kreps & Scheinkmn (983) If there is two stge gme, in which firms set cpcity in stge And in stge, given their cpcity, set price Then the Cournot result is observed

17 Differentited Products resolve the Bertrnd Prdox Differentited Products llow price competing oligopolists to mrk up

18 Cournot Competition (838). Plyers (identicl). Cournot strtegy - All firms simultneously set their output 3. Homogenous product 4. Perfect Informtion 5. Liner demnd 6. Constnt MC

19 ) R( c firms Similrily,identicl ) R( c c ) c( ). ( ) c( TC Q P i i 0 Π Π identicl firms, liner demnd, constnt mrginl cost Choose to mx p, given Higher, lower level to mx p

20 An euilibrium is when ech firm s output level is best response to the other firm s output level - then neither wnts to devite from its output level. A pir of output levels ( *, *) is Cournot-Nsh euilibrium if * R( * ) nd * R( * )

21 Firm s rection curve * R( * ) * Cournot-Nsh euilibrium * R ( *) nd * R ( *) * Firm s rection curve * R( * )

22 Solve rection curves to find cournot euilibrium * * * 3 ) ( : 3 ) ( : 3 : 3 : ) ( ) ( ) ( π π + p profit for Solve c Q p price for Solve c untity Cournot Totl euil nsh cournot c m euilibriu in firms identicl c solving c c c R R

23 Q: Are the Cournot-Nsh euilibrium profits the lrgest tht the firms cn ern in totl? A: Firms could ern higher profits if both greed to set hlf the monopoly output (nd thus ern hlf monopoly profit ech)

24 HOWEVER Collusive\Joint profit mx output levels m m not sustinble incentives to unilterlly devite not NE if firm continues to produce m, firm s profit-mximizing response is R ( m )

25 P monop > P cournot > P perfcompbertrnd Q monop < Q cournot < Q perfcompbertrnd p monop > p cournot > p perfcompbertrnd

26 Exmple: P 40 Q; C i 60( i ); firms ply Cournot. Wht re euilibrium outcomes? P Π 40 ( nd solving profit mx Π ) , given

27 * * * : : 53 : 6 : 6 ) ( : π π ofit Solveforpr Q p price for Solve untity Cournot Totl firms identicl nd functions rection solve

28 Monopoly. firm ). (40 40 Π Π Q p P m m m m m m m m m m

29 Perfect Competition (& Bertrnd) P MC 60 P -Q so Q -c Profit p. c. 0 since p c Shows Q m < Q c < Q pc And P m > P c > P pc

30 Thus, Bertrnd fi for N> or, get perfectly competitive outcomes Cn show tht s N, cournot outcome fi perfectly competitive

31 N plyer Cournot. N Plyers (identicl). Cournot strtegy - All firms simultneously set their output 3. Homogenous product 4. Perfect Informtion 5. Liner demnd 6. Zero Cost

32 P - bq ; since N firms re identicl TC i 0+0 so c 0 Firm i : Π Π p * i i * i i i ( b ( N b ( + b ( N i bq ). ) N ) + bq ) b i b. N. Q i i i b 0 Q N i i N ( N i i i ) Q N Note: if y bq. i & Q i + j +.+ n (so dq/d i ) dy/d i bq.+ b i.dq/d i bq+ b i i

33 P / N+ p Nfi p fic 0 N: / N: /3 N3: /4 D Q M Q Duo Q olig Q

34 P(N) function links price cost mrgins to given N p p monop Joint Mx. MC Bertrnd Cournot P for ny given N depends on the intensity of competition (Bertrnd: Most intense) N

35 Entry? Entry s P(N) Stge (Long Run) Stge (Short Run) The entry decision is bckwrd induction procedure

36 We return to modelling entry, where N is endogenous nd depends on P(N) in the ltter prt of the course