Constant Best-Response Functions: Interpreting Cournot

Transcription

1 Internatonal Journal of Busness and Economcs, 009, Vol. 8, No., -6 Constant Best-Response Functons: Interpretng Cournot Zvan Forshner Department of Economcs, Unversty of Hafa, Israel Oz Shy * Research Department, Federal Reserve Bank of Boston, U.S.A. Abstract Followng Amr and Grlo (999), we characterze a class of demand functons that generate constant quantty best-response functons. We examne mplcatons of constant best-response functons for the nvarance of equlbrum outcomes wth respect to the assumed market structure of quantty games. We argue that, unlke the class of lnear demand functons, ths class of demand functons supports the pure nterpretaton of Cournot conjectures. Key words: constant best-response functons; Cournot equlbrum; Cournot conjectures; Stackelberg equlbrum JEL classfcaton: C7; D43. Introducton Perhaps the major dffculty wth emprcal research related to olgopoly theory s that equlbrum market outcomes tend to be hghly senstve to changng the order of decson makng by frms n an ndustry and/or changng the equlbrum concept (.e., market structure). A natural queston to ask s whether there exst demand functons n whch the resultng market equlbrum outcomes are nvarant wth respect to the assumed market structure. Amr and Grlo (999, p. 9) demonstrated the exstence of a class of demand functons for whch the resultng best-response functons are constant n quantty olgopoly games. The desred nvarance to the mpact of the order of decsons n quantty olgopoly games on market outcomes mmedately follows. From a theoretcal pont of vew, the Cournot market structure may have two nterpretatons assocated wth two dfferent assumptons concernng frms expectatons. The wdely-used (weak) nterpretaton s that a Cournot equlbrum s Receved May 5, 009, revsed May 7, 009, accepted May 8, 009. * Correspondence to: Research Department, Federal Reserve Bank Boston, 600 Atlantc Avenue, Boston, MA 00, U.S.A. E-mal: Oz.Shy@bos.frb.org. We thank Edtor Jong-Shn We for most valuable suggestons and comments on earler drafts.

2 Internatonal Journal of Busness and Economcs a Nash equlbrum (Nash, 950) n the sense that each frm expects the other frm to mantan a constant output level. Clearly, these expectatons are consstent only n equlbrum, at least for lnear demand functons. However, a stronger nterpretaton of Cournot equlbrum s to assume that frms expect rval frms to have constant best-response functons. In contrast wth lnear demand functons, the class of demand functons dentfed n ths paper s consstent wth the strong nterpretaton of Cournot conjectures. In fact, strategc games may have several nterpretatons; see for example Osborne and Rubnsten (994, Secton..). The paper s organzed as follows. Secton dentfes the class of demand functons yeldng constant best-response functons. Secton 3 demonstrates that the realzaton of output levels s nvarant wth respect to changng from Cournot to Stackelberg quantty-settng market structures. Secton 4 concludes wth a general dscusson and demonstrates how, n contrast wth lnear demand functons, ths class of demand functons supports the pure nterpretaton of Cournot conjectures.. Demand Functons Yeldng Constant Best-Response Functons Consder a market for a homogeneous product wth an aggregate downward slopng nverse demand functon denoted by p (Q), where Q denotes aggregate quantty demanded and p the market prce. We assume that p (Q) s twce contnuously dfferentable wth respect to Q. There are N frms ndexed by =, K, N ( N ) whch can costlessly produce ths product. Let q ( q 0 ) denote the output of frm, so N Q = q. = Also, defne q = ( q, K, q N ) as the N -dmensonal vector of frms output levels and q() = ( q, K, q, q+, K, qn) as the ( N )-dmensonal vector of output N levels of all frms except frm. Fnally, let π ( q) = p( q ) j j q = denote the proft functon of frm, =, K, N. For a gven q (), the set of maxmzers s gven by: { ( j ) ( j ) }. ψ () + j j ( q ) = q R : p q + q q p x + q x for all x [0, ) N We assume that ψ ( q ) empty ( for all ) q() R +. We call ψ the best-response correspondence. A best-response functon R( q () ) s a selecton out of ψ ( q() ); N.e., R( q() ): R+ ψ ( q() ). A best-response functon s sad to be constant f there exsts a constant k 0 for whch R( q() ) = k for all q (). Followng Amr and Grlo (999, p. 9), we formalze the exstence of ths class of functons n the followng proposton. Proposton. There exsts a class of downward-slopng market demand functons for whch the correspondng best-response functons are constant. Formally, ths class of demand functons s gven by: N q / Q = pq ( ) = e = e ()

3 Zvan Forshner and Oz Shy 3 for any > 0. Proof. We derve the best-response functon of frm, whch chooses q that solves: () ( q qj ) j + max π ( q, q ) qe =. q The frst-order condton s gven by: π q N ( q ) q j j = ( q, q() ) = e = 0 Hence, q = s an extremum pont. Snce the local second-order condton satsfes: ( q ) N ( ) q j j= π (, q q() ) = e < 0 q = and upon checkng the graph of π (as a functon of q ) we see ths local maxmum happens to be the global maxmum over (0, ) at q =. Observe that: + N. ( + q ) j j π ( q, K, q,, q, K, q ) = e > 0. Fnally, snce π (0, q() ) = 0, π attans a global maxmum over [0, ) at q =. It follows that ψ ( q( ) ) empty, and n fact ψ( q() ) = { }. Therefore, the best-response correspondence s a functon. We have shown that producng output level consttutes a domnant strategy of any frm., 3. Equlbrum Olgopoly Market Outcomes We now solve for the equlbrum market outcomes under Cournot and Stackelberg market structures. A Cournot-Nash equlbrum (Cournot, 838) s an outcome q c R N + such c c that R( q() ) = q for all =, K, N. Clearly, n ths envronment, the unque c Nash-Cournot equlbrum s q = and the aggregate output level s Q = N. A Stackelberg equlbrum n whch frm s a leader (Stackelberg, 934) s the S N outcome R, whch s obtaned by solvng: q + max π ( q, q ) q () s.t. q = R ( q ) for all j. j j ( j) ()

4 4 Internatonal Journal of Busness and Economcs That s, the leader solves for the followers best-response functons and chooses ts output level accordngly. Followers behave n a Cournot fashon by smply followng ther best-response functons. Note that once we know that q = s the unque domnant strategy of every frm, t mmedately follows that equlbra n any of the followng three scenaros yeld dentcal output decsons. (a) The leader moves frst, whle the other N frms move smultaneously as followers. (b) All N frms move sequentally and one at a tme. (c) All N frms move smultaneously. Hence, the order of output determnaton does not nfluence output level decsons. Thus, the order-nvarance result derved for ths specfc market demand functon consttutes the major strength of the present model. Moreover, for ths reason, we delberately refraned from defnng the Stackelberg market structure as a specfc (ad hoc) extensve-form game. Ths s because our result s more general n the sense that t may apply to sequental games whch cannot be represented by a conventonal tree but stll have some normal-form representatons. We can now state the followng proposton. Proposton. The realzaton of output levels s nvarant wth respect to the choce of Cournot and Stackelberg market structures. It s nterestng to pont out that for ths partcular nverse-demand functon, each frm can consder tself to be a Stackelberg leader n the sense that t solves the maxmzaton problem (), whle frms conjectures mantan mutual consstency. Ths nvaldates a common percepton that, n a Stackelberg market structure when there are two frms, there s no equlbrum when both frms vew themselves as leaders (e.g., Kreps, p. 330). Moreover, our result s related to the endogenous-tmng papers, such as Hamlton and Slutsky (990) and Amr and Grlo (999), whch obtan n a more general settng the possblty that both frms wll choose ther output level at the same tme (.e., a smultaneous game). Here, we demonstrate the possblty of havng a complete nvarance wth respect to the order of moves n the absence of producton costs. In the presence of producton costs, t s clear that for suffcently-hgh producton costs at least one frm wll choose not to produce n whch case the order of moves may affect market outcome. 4. Interpretng Cournot: A Dscusson From a theoretcal pont of vew, we would lke to pont out the followng. (a) Cournot (97) never suggested the wdely used class of lnear demand functons as an example for hs model. The class of lnear demand functons belongs to a modern nterpretaton of hs model. In fact, t seems as f Cournot left t vague whch class of demand functons s consstent wth hs model. (b) Unlke the class of lnear demand functons, the class of demand functons dentfed n ths note s consstent wth the strong nterpretaton of Cournot conjectures n the sense that frms correctly beleve that rval frms wll not

5 Zvan Forshner and Oz Shy 5 devate from ther output level regardless of ther actons. Ths nterpretaton does not hold for the class of lnear demand functons. (c) One cannot argue that the wdely-used class of lnear demand functons s more general than the class dentfed n ths note. In fact, as we argue below, we beleve that the present class of demand functons better suts econometrc modelng. (d) It s ndeed possble that Cournot hmself thought about constant best-response functons. Our nterpretaton of Cournot s based on Cournot s system of two equatons gven n Cournot (97, Ch. 7, p. 66), whch can be wrtten usng our notaton and for N = as: π ( q, q ) dp( Q) Q = 0 p( + q ) + q dq q π ( q, q ) dp( Q) Q = 0 p( q + ) + q dq q 0 0, for all q (3), for all q (4) Equaton (3) reflects what frm conjectures about frm, namely that ts optmzaton always yelds q = regardless of the level of output set by frm. Thus, frm conjectures that q = s a domnant acton of frm. Smlarly, (4) reflects what frm conjectures about frm, namely that ts optmzaton always yelds q = regardless of the level of output set by frm. Thus, frm conjectures that q = s a domnant acton of frm. Therefore, our nterpretaton of Cournot s conjecture s that (3) and (4) hold as denttes ndependently of each other. In contrast, the common nterpretaton s to requre that ths system of equatons holds n equlbrum, n whch case the Cournot conjecture about rval frms holdng ther output constant s nconsstent wth frms havng downward-slopng best-response functons. Fnally, varous authors (e.g., Bergstrom and Varan, 985) have attempted to facltate emprcal research n olgopoly markets usng smple Cournot market structures. The class of demand functons characterzed n ths note has sgnfcant mplcatons for emprcal research. Ths s because econometrc models based on a demand functon n the class dentfed by () can have strong predctve power snce the market outcome s nvarant wth respect to the assumed market structure. Ths means that, n these markets, appled economsts do not have to search across sequental dynamc games n order to fnd a partcular ad hoc order of moves whch best fts the data. References Amr, R. and I. Grlo, (999), Stackelberg versus Cournot Equlbrum, Games and Economc Behavor, 6, -.

6 6 Internatonal Journal of Busness and Economcs Bergstrom, T. and H. Varan, (985), When Are Nash Equlbra Independent of the Dstrbuton of Agents Characterstcs? Revew of Economcs Studes, 5, Cournot, A., (97), Researches nto the Mathematcal Prncples of the Theory of Wealth, Translated by Nathanel T. Bacon, New York: Macmllan. Hamlton, J. and S. Slutsky, (990), Endogenous Tmng n Duopoly Games: Stackelberg or Cournot Equlbra, Games and Economc Behavor,, Kreps, D., (990), A Course n Mcroeconomc Theory, Prnceton: Prnceton Unversty Press. Nash, J., (950), Equlbrum Ponts n N-Person Games, Proceedngs of the Natonal Academy of Scences of the Unted States of Amerca, 36, Osborne, M. and A. Rubnsten, (994), A Course n Game Theory, Cambrdge, MA: The MIT Press. von Stackelberg, H., (934), Marktform und Glechgewcht, Venna: Sprnger-Verlag. Englsh Translaton: The Theory of the Market Economy, Oxford: Oxford Unversty Press.