STATIC OLIGOPOLY MODELS

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1 SØK/ECON 535 Imperfect Competton and Strategc Interacton STATIC OLIGOPOLY MODELS Lecture notes Nash Equlbrum Descrbes strategc nteracton by equlbrum n non-cooperatve games. Strateges: a, =,,..., n. Payoff (proft) functons: ( a; a ) Π. Defnton: a strategy combnaton (a*,...,a* n ) consttutes a Nash equlbrum f, for =,...,n, a : Π ( a, a ) Π ( a, a ) The Cournot model (838) Duopoly (n = ) Strateges are quanttes (supply/output): q, =,,... n. Frms choose smultaneously (.e. wthout knowng ther compettors strateges). Homogeneous products. Π q, q = P Q q C q, Q = q + q, =,. Proft functons: ( ) ( ) ( ) Π s assumed strctly concave and two tmes dfferentable. FOC: P ( Q ) q P( Q) C ( q ) + =0 Negatve external effect on compettor s proft P ( Q) q ; therefore producers choose a larger quantty than that whch would maxmse total profts: ( ) ( ) ( ) = < ( )[ + ] P Q C q P Q q P Q q q d Q > Q Prce s lower and output greater than n monopoly (perfect cartel). m

2 At equlbrum, margnal costs are equal only f frms are symmetrc; therefore, for gven aggregate output, total costs are generally not mnmsed. From the FOC we obtan the Lerner ndex: L P C α = P ε where α = of demand. Olgopoly (n ) q Q s s market share and ε ( ) Reacton functons (from FOC): Π (R (q ),q ) = 0. R Π = < 0 Π = P Q Q P s the prce elastcty f margnal profts decrease as a functon of the compettor s supply (because Π = Pq + P, a suffcent condton for ths result s that the nverse demand functon s convex,.e. P < 0 ). If so, quanttes are strategc substtutes. Fgure (equlbrum descrbed by reacton functons) Example: Lnear demand and cost functons: C ( q) = cq and P( Q ) = Q ( ) R q = q c Note: a reducton n c shfts the reacton curve outwards and ncreases s proft-maxmsng supply. Π q; Q = P Q q C q, Q = q Proft functons: ( ) ( ) ( ) ì Same FOC as above, and same formula for Lerner-ndces. From the ndvdual Lerner-ndces: P C H L = P ε We defne Π = Π q, Π = Π q and so on.

3 n where [ α ] H = = s the Hrschmann-Herfndahl concentraton ndex and n C = α C the (weghted) average margnal cost. = Example: lnear demand and cost functons Symmetrc model: c = c, =,,..., n: Concentraton ndces H = n q = [ c] n + n Q = [ c] n + p = c+ [ c] c as n n + Summary statstcs for market concentraton,.e. dstrbuton of market shares. Ideally, an ndex should treat frms symmetrcally; not decrease n value f part of a frm s market share s transferred to a larger frm; and decrease n value f, n a symmetrc ndustry, the number of frms ncrease. Order frms such that α α... αn : m m-frm concentraton rato: K = = α n Hrschmann-Herfndahl ndex: H [ α ] m = = These ndces satsfy the above condtons: take on values between 0 and ; maxmsed (=) at monopoly; for a gven number of frms n, the ndex reaches ts lowest value when frms are equally large. Hrschmann-Herfndahl ndex puts more weght on smaller frms. 3

4 Concentraton ndces may be useful f there s a relatonshp between market concentraton and competton, or, more specfcally, performance/welfare. Ths s the case n the Cournot model (under certan condtons, see above). I other cases, ths may not necessarly be true (cf. the Bertrand model). The Bertrand model (883) Strateges: prces p, =,,..., n. Frms choose strateges smultaneously. Homogenous duopoly (n = ) Buyers wll shop where the prce s lowest (f frms charge same prces we assume they share the market equally). Demand: ( ) ( ) D p f p < p D ( p, p) = D p = f p p 0 f p > p where D( p) s market demand. Assume margnal costs are constant and dentcal (= c). Nash equlbrum: p = p = c. ( p > p cannot be an equlbrum because can ncrease profts by rasng prce; p = p > c cannot be an equlbrum because ether frm would want to undercut; p = p < c want to rase ts prce.) cannot be an equlbrum because ether frm would Concluson: duopoly equals perfect competton! In the case of asymmetrc costs (c > c obtanng a monopoly at prce equal to the cost of ; that s, assumptons to guarantee exstence. The "Bertrand Paradox" Solutons to the paradox: product dfferentaton. ), the only equlbrum nvolves p = p (techncal 4

5 capacty constrants (Edgeworth); nter-temporal (dynamc) competton (threat of prce wars); Product dfferentaton Demand does not necessarly dsappear even f prce exceeds those of the compettors. Proft functons: π ( ) = ( ) ( ( ) ) p, p pd p, p C D p, p, =,. π assumed strctly concave and twce dfferentable. Reacton functons (from frst-order condtons): ( ( ) ) π r p, p = 0. π r = >0 π f margnal proft s ncreasng n compettor s prce. If so, prces are strategc complements. Fgure (equlbrum descrbed by reacton functons) Prce competton wth capacty constrants Ratonng Example (costly capacty): Demand: D( p) = Unt capacty costs: c 3, 4 p Varable costs are constant and normalsed to 0 Capacty constrants: q q, =,. Assume < p p and ( ) > D p : Who gets to buy from Frm? q Effcent ratonng: resdual demand equals D( p ) q f ( ) > otherwse. Two stage game ) Frms choose capactes smultaneously. D p, 0 q 5

6 ) Frms choose prces smultaneously. Trade takes place and profts are realsed. We are lookng for a Subgame Perfect Equlbrum, n whch, from each stage of the game, the strateges consttute a Nash equlbrum of the contnuaton game. Solves the game by backward nducton. STAGE : Gross monopoly profts equal /4. Therefore, no frm wll ever choose q > 3 because that would lead to negatve profts. Gven q, =, 3, there exsts a unque Nash equlbrum n whch p = p = p* = q q. Proof of exstence: There s no pont n reducng prces because capacty constrants bnd. Profts for frm when > = * profts s STAGE : p p p equals [ ] [ ] p q < q q q = q + q 0. p p q. The margnal Reduced-form proft functons, gven equlbrum play at Stage, are [ ] Π In other words, Cournot profts. ( q, q ) = q q q, =,. Concluson: The two-stage game leads to the same equlbrum outcome as n the Cournot model. The dfference s that here prces are set by frms rather than by some Walrasan auctoneer or other equlbratng mechansm. Caveats: For larger capactes ( q > 3 ), an equlbrum n pure strateges generally does not exst and one may have to consder mxed strateges (Kreps- Shenkman, 983). The equvalence wth the Cournot model does not necessarly hold for other ratonng rules (for example, proportonal ratonng, such that everyone wth wllngness to pay above the lowest prce has equal probablty of gettng to buy at ths prce). 6

7 Insttutonal detal prcng rules and capacty constrants Market outcomes may depend on the fner detals of market nsttutons; n partcular, how frms are remunerated. Assume demand D s perfectly nelastc. There are two (symmetrc) frms wth capactes k and unt costs c. Frms set prces smultaneously: p, cp. A market prce s set equal to the prce of the margnal suppler; that s, Low-demand perods ( D k ) { p p} { } mn, f D k P = max p, p f D > k There s a unque (Bertrand lke) equlbrum n pure strateges n whch p = p = c. Hgh-demand perods ( D > k ) There are many equlbra, all havng outcomes wth one frm prcng at P and the other prcng so low as to avod undercuttng; that s, p = P and p = b c, bk P D k. [ ] Ref: N-H von der Fehr and D Harbord (993): Spot market competton n the UK electrcty ndustry, The Economc Journal, 03, Under the Bertrand assumpton (.e. each frm beng pad ts own prce) we would have the same equlbrum n low-demand perods, but n hgh-demand perods equlbra would be dfferent; n partcular, only mxed-strategy equlbra exsts. 7