Conjectures n Cournot Duopoly under Cost Uncertanty Suyeol Ryu and Iltae Km * Ths paper presents a Cournot duopoly model based on a condton when frms are facng cost uncertanty under rsk neutralty and rsk averson. Each frm conjectures about the rval s output level, and ts cost functon s assumed to be unknown to ts rval. The Cournot model shows that the expected utlty maxmzng frms, under rsk averson, show dfferent behavors from the expected proft maxmzng frms. Ths mples that each frm can ncrease or decrease ts output, whch depends on the nteracton between both frms under cost uncertanty, assumng that both frms are rskaverse. Keywords: Cournot equlbrum, Cost uncertanty, Rsk averson JEL Classfcaton: D8, L3 I. Introducton Economsts have extensvely analyzed the effects of uncertanty on the optmzng behavor of a sngle agent. These economsts nclude Sandmo (970, 97), Leland (97), Cheng et al. (987), Feder (977), Meyer and Ormston (985), and Km et al. (005). Sandmo (97) has specfcally conducted a systematc study of the theory of the frm under prce uncertanty and rsk averson. He has argued that the uncertanty effect on the optmal output level of rsk-averse frm s negatve. Dardanon (988) has ntroduced a unfed framework for the analyss *Assstant Professor, Department of Economcs, Andong Natonal Unversty, Andong 760-749, Korea, (Tel) +8-54-80-609, (Fax) +8-54-83-63, (E-mal) syryu@andong.ac.kr; Correspondng Author, Professor, Department of Economcs, Chonnam Natonal Unversty, Gwangju 500-757, Korea, (Tel) +8-6-530-550, (Fax) +8-6-530-559, (E-mal) kt603@chonnam.ac.kr, respectvely. The authors wsh to thank two anonymous referees for ther useful comments. [Seoul Journal of Economcs 0, Vol. 4, No. ]
74 SEOUL JOURNAL OF ECONOMICS of two-argument utlty functon wth a random budget constrant and presented the effect of uncertanty on an agent under plausble normalty condtons. Hs analyss showed that the optmal level of the choce varable ncreases for mean-preservng ncreases n rsk n the dstrbuton of the random parameter f the absolute rsk averson ncreases. Hs results are appled n dervng the rules for effcent taxaton under uncertanty. Prevous studes on the duopoly model under uncertanty have assumed that each frm s rsk-neutral and can share or exchange ts nformaton on market uncertanty wth ts rval. Examples of such studes nclude those of Bresnahan (98), Clarke (983a, 983b), Gal- Or (986), Krby (988), L (985), Novshek and Sonnenschen (98), Saka (990, 99), Saka and Yamato (989), Shapro (986), and Vves (984). They have nvestgated how market uncertanty wth ether unknown market demand or unknown constant margnal cost affects frms behavor. Ths paper ntroduces cost uncertanty n a smple Cournot duopoly model and extends prevous analyses n two drectons. Frst, we assume that there s no nformaton sharng or exchange between frms. Each frm does not know the other frm s cost functon at the tme the output s produced. In a Cournot duopoly game, Frm s optmal output choce depends on ts own cost functon and Frm j s output choce, whch also depends on ts cost functon and Frm s output choce (ths type of relatonshp goes on). Ths mples that, n equlbrum, each frm s optmal output depends on the other frm s cost functon as well as ts own cost functon. Therefore, wth uncertanty, each frm has to use ts belefs on the other frm s cost functon, and these belefs can be summarzed n ts own subjectve probablty dstrbuton. Second, we assume that frms are rsk-averse. We show that under rsk neutralty, the wellknown Cournot equlbrum vares based on the assumpton on the objectve functons of the frms. We also demonstrate that the expected utlty maxmzng frms under rsk averson show dfferent behavors from the expected proft maxmzng frms. We adopt the Sandmo s approach (97) to compare the uncertanty case wth the certanty one. The equlbrum output under cost functon uncertanty s lesser or greater than one under certanty, n whch t depends on each frm s conjecture about the other frm s output level. Ths paper s organzed nto sectons. Secton presents our basc duopoly model and analyzes the expected proft maxmzng behavors n t. Secton 3 nvestgates the expected utlty maxmzng behavors
CONJECTURES IN COURNOT DUOPOLY UNDER COST UNCERTAINTY 75 and compares the results wth the certanty case. Fnally, Secton 4 dscusses the concludng remarks. II. Basc Model We consder a smple Cournot duopoly model n whch two frms producng dentcal products face a market demand. The nverse demand functon s derved through P=a-q -q j, where q and q j denote the amount of output produced by Frms and j, respectvely, the demand ntercept s a, and the slope of market demand s one for smplcty. We assume that technology exhbts constant returns to scale, so that Frm has constant unt cost c wth c >0. We brefly repeat the well-known results under certanty n whch the cost functons are exactly known to each other, for comparson wth later results under cost uncertanty. The proft of Frm s descrbed as π = ( a q qj c) q, for, j=, j. () The frst-order condton for proft maxmzaton through () by Frm s π q = a q qj c = 0. () The second-order condton s always satsfed because π q = < 0. (3) From Equaton (), the optmal reacton functon for Frm s derved through q = ( a c qj ). (4) c Therefore, the unque Cournot equlbrum for Frm, q through: s derved
76 SEOUL JOURNAL OF ECONOMICS c q = ( a c + c j ), for, j=, j. (5) 3 We then ntroduce the uncertanty through randomness n the cost functon. Each frm knows exactly ts own constant margnal cost but has partal nformaton only on the other frm s margnal cost. That s, each frm has ts subjectve probablty dstrbuton on the other frm s constant margnal cost. We consder the symmetrc uncertanty n whch both frms have uncertan nformaton on the other frm s margnal cost. Before proceedng, we defne the random varables to dstngush them from the nonrandom varables n the followng way: q j Ω, c j Ω, and π Ω j, for, j=, j, where Ω s the set of nformaton avalable to Frm. III. Expected Proft Maxmzng Behavor (Symmetrc Uncertanty) Each frm s rsk-neutral and chooses ts output to maxmze ts expected proft based on ts prvate nformaton. Frm s expected proft gven ts prvate nformaton, Ω, s derved through: { } E( π Ω ) = E ( a q q c Ω ) q j { ( )} = a q c E qj Ω q for, j=, j, (6) where E s the expectatons operator. The frst-order condton of (6) s E( π Ω ) q = a q c E( q Ω ) = 0. (7) j The second-order condton s always satsfed because E( π Ω ) q = < 0. (8) The optmal reacton functons from the frst-order condtons for both
CONJECTURES IN COURNOT DUOPOLY UNDER COST UNCERTAINTY 77 frms are derved through: q = { a c E( q Ω ) } (9) and q = { a c E( q Ω ) }. (0) The unque Cournot equlbrum s derved from solvng Equatons (9) and (0) smultaneously. However, the equlbrum can vary, dependng on each frm s expectaton, E(q Ω j ). Frm can only expect Frm s reacton wth ts prvate nformaton through Eq ( Ω ) = E a c Eq ( Ω) Ω { } = ( ) ( ). a E c Ω E E q Ω Ω () If we assume that both frms have ther prvate nformaton, Frm s expectaton on Frm s reacton, E(q Ω ), s located between the upper bound [E(q Ω )] u and the lower bound [E(q Ω )] l. Therefore, Frm s optmal output, q *, s q mn q * q max. Smlarly, Frm s expectaton on Frm s reacton, E(q Ω ), s located between [E(q Ω )] u, and [E(q Ω )] l, and ts optmal output, q *, s q mn q * q max. Therefore, the possble Cournot equlbrum for ths uncertanty game s shown n Fgure. The pont A represents an equlbrum pont wth (q c, q c ) beng the par of equlbrum output strateges f both frms have perfect nformaton on the constant margnal cost of ther rvals. The equlbrum s shown n the box of dashed lnes f both frms face cost uncertanty. There are multple equlbrum output strateges for
78 SEOUL JOURNAL OF ECONOMICS FIGURE COURNOT DUOPOLY EQUILIBRIA UNDER SYMMETRIC UNCERTAINTY two frms n ths case. Let us then look at how the Cournot equlbrum can be derved. Consder Equatons (9) and (0), whch have four unknowns, q, q, E(q Ω ), and E(q Ω ). We can solve the problem through an teratve substtuton method. q = + ( ) 3 a c + E E c 3 Ωj Ω + + Ec ( ) ( ). j Ω + E E Ec 4 j Ω Ωj Ω + () We assume that the nformaton set of each frm s exclusve to each other to smplfy the expresson: Assumpton : Let Frm j s expectaton on c be wrtten as, E(c Ω j )=c +b, where b s assumed to be a small (relatve to c ) non-systematc bas. Ths mples that Frm s expectaton on Frm j s expectatonal bas, b, s zero: E(b Ω )=0. Then
CONJECTURES IN COURNOT DUOPOLY UNDER COST UNCERTAINTY 79 EEc [ ( Ω ) Ω ] = E[( c + b) Ω ] = c + Eb ( Ω ) = c j E{ E[ E( c Ω ) Ω ] Ω } = E[ E( c Ω ) Ω ] = c j j E{ EEc [ ( Ω ) Ω ] Ω } = Ec ( Ω ) = c + b. j j j The equlbrum output, usng Assumpton and Equaton (), s derved through q* = a c + ( c j + bj). (3) 3 Let the maxmum and mnmum values for the bas, b j be denoted by b j u and b j l, respectvely, so the equlbrum output for frm, q *, s resolved as q mn q * q max, where q mn =/3[a-c +(c j +b j l )]=q c +(/3)b j l, q max =q c +(/3)b j u, and q c means the Cournot equlbrum output of Frm wthout cost uncertanty. Ths mples that f Frm s expectaton on c j s greater (smaller) than real c j, then Frm s equlbrum output, q *, s greater (smaller) than the certanty equlbrum output, q c. Frm j s expectaton on Frm s reacton s also bounded as Eq ( ) j Eq ( j) Eq ( j) Ω Ω Ω, u (4) where Eq ( Ω j) = a ( c + b ) q j and u u Eq ( ) j a ( c b ) q Ω = + j. IV. Expected Utlty Maxmzng Behavor (Symmetrc Uncertanty) We brefly ntroduce the Sandmo s approach before proceedng. Sandmo (97) has studed the theory of the compettve frm under prce uncertanty and rsk averson. The expected utlty of profts can be wrtten as [ ] [ ] E u( π ) = E u( px C( x) B), (5)
80 SEOUL JOURNAL OF ECONOMICS where p s the prce of output and a random varable wth the expected value E(p)=μ, the varable cost C(x) and the fxed cost B. The frst-order condton of Equaton (5) s [ π ] E u( )( p C ( x)) = 0. (6) E(p)=μ s the prce under certanty to compare wth the level of output under certanty. Equaton (6) can be wrtten as [ π ] [ π ] E u( ) p = E u( ) C ( x). (7) Subtractng E[u (π)μ] from each sde of Equaton (7), we derve [ π μ ] [ π μ ] E u( )( p ) = E u( )( C ( x) ). (8) We derve that π=e[π]+(p-μ)x because E[π]=μ x-c(x)-b. Clearly, [ π] u ( π) u ( E ) f p μ. (9) It mmedately follows that [ ] u ( π)( p μ) u ( E π )( p μ). (0) Ths nequalty holds for all p. The nequalty sgn of Equaton (9) s reversed for p μ. However, multplyng both sdes of the nequalty by (p-μ) wll stll hold the nequalty of (0). Takng expectatons on both sdes of (0) and notng that E[u (π)] s a gven number, we obtan [ ] [ ] E u( π)( p μ) u( E π ) E( p μ). However, the rght-hand sde s equal to zero by defnton; hence, the left-hand sde s negatve. We then know that the rght-hand sde of (8) s also negatve. Meanwhle, Equaton (8) can be wrtten as [ π ] E u ( ) ( C ( x) μ) 0,
CONJECTURES IN COURNOT DUOPOLY UNDER COST UNCERTAINTY 8 and ths mples C ( x) μ () because margnal utlty s always postve. Therefore, output under prce uncertanty s smaller than the certanty output. We now assume that both frms are rsk-averse (u >0 and u <0). The objectve functon of the expected utlty maxmzng Frm s { } E u( π Ω ) = E u ( a q qj c Ω) q. () The frst- and second-order condtons of () can be wrtten as ( ) E u ( π Ω) a q c ( qj Ω ) = 0 (3) and E{ u ( π Ω ) a q c ( q j Ω) u ( π Ω )} < 0 (4) because the second-order condton s always satsfed for the rskaverse frms, u ( )>0 and u <0. E[u (π Ω )(a-q -c )]=E[u (π Ω )(q j Ω )] s derved from the frstorder condton (3). Subtractng E[u (π Ω )E(q j Ω )] from both sdes, we then derve E u ( π )[ a q c E( qj )] E{ u Ω Ω = ( π Ω)[ qj Ω E( qj Ω)]}. (5) We know that: π Ω = ( a q c qj Ω ) q and E( π Ω ) = a q c E( qj Ω) q. Therefore, π Ω E( π Ω ) = qj E( qj ) Ω Ω q.
8 SEOUL JOURNAL OF ECONOMICS Clearly, t follows that u ( π Ω) u E( π Ω), f qj Ω E( qj Ω). The followng nequalty now holds for all q j Ω, u ( π Ω ) qj E( qj ) u E( π ) qj E( qj ) Ω Ω Ω Ω Ω. Takng expectatons on both sdes, we derve E{ u ( π Ω )[ q Ω E( q Ω )]} E{ u [ E( π Ω )][ q Ω E( q Ω )]} j j j j = u [ E( π Ω )][ E( q Ω ) E( q Ω )] = 0. j j (6) From Equatons (5) and (6), we derve E{ u ( π Ω )[ a q c E( q Ω )]} j = [ a q c E( q Ω )] E[ u ( π Ω )] 0. j (7) Ths mples that the followng nequaltes must be satsfed for optmal decson: q a c E( q Ω) (8) and q a c E( q Ω). (9) The present model shows that each frm does n fact react to the other n a way that depends on conjectures based on ts nformaton. Let r j be Frm s response to the change n Frm j s output level. r j s equal to -(/) under certanty but r j s equal to and less than -(/) under cost uncertanty. Ths means that f Frm j reduces ts output by
CONJECTURES IN COURNOT DUOPOLY UNDER COST UNCERTAINTY 83 unt, Frm ncreases ts output by equal to and less than / unt. Ths makes Frm s reacton functon to move nward by k (=, ) under cost uncertanty (compared wth certanty stuaton). Let k be the amount of output, q whch s reduced under cost uncertanty by Frm. We follow the Sandmo s approach (97) and the certanty equlbrum output, q c, can be compared wth the uncertanty equlbrum output, q *, when b =b j =0. Equatons (8) and (9), f b =b j =0, become q = ( a c q ) k (30) q = ( a c q ) k. (3) Equatons (30) and (3) show that reacton curves of both frms under cost uncertanty are nsde ther certanty reacton curves. Solvng (30) and (3), the equlbrum outputs under cost uncertanty are * c 4 q = q + k k (3) 3 3 * c 4 q = q + k k, (33) 3 3 where q c s the Cournot equlbrum output under certanty. Let k j /k be the relatve sze of the amount of output from both frms (reduced under cost uncertanty). The values of k s measure the dfference of the frms reacton between uncertanty and certanty. Uncertanty n the cost functon s generated by random elements, such as uncertanty delvery and tmng of nputs, uncertan prces of nput factors, and uncertan technologcal relatonshp between nput and output. Therefore, these rsk factors affect the rato. We derve three possble cases, when compared wth the results under certanty stuaton (Fgure ). The followng theorem s derved after solvng (3) and (33). Theorem Supposng that both frms are rsk-averse, there exst three Cournot
84 SEOUL JOURNAL OF ECONOMICS FIGURE COURNOT DUOPOLY EQUILIBRIA UNDER RISK AVERSION equlbrum cases under cost uncertanty: Case : If k k, then q * q c c and q * q k Case : If 0 k Case 3: If (representng Area K n Fgure ). k k >, then q *<q c c and q *>q (representng Area L n Fgure )., then q *>q c c and q *<q (representng Area M n Fgure ). The equlbrum output level under cost uncertanty depends on the relatve sze of k and k, that s, the nteractve behavor of both frms. The followngs are observed f the results are compared wth the results under certanty: () both frms reduce ther optmal level of output n Case ; () Frm ncreases ts optmal output whereas Frm reduces ts optmal output n Case ; and () Frm decreases ts optmal
CONJECTURES IN COURNOT DUOPOLY UNDER COST UNCERTAINTY 85 output whereas Frm ncreases ts optmal output n Case 3. V. Concludng Remarks Ths paper presents a Cournot duopoly model based on a stuaton when frms are facng cost uncertanty under rsk neutralty and rsk averson, and compares the results wth certanty case. The Cournot duopoly model shows that the expected utlty maxmzng frms show dfferent behavors from the expected proft maxmzng frms under cost uncertanty. A comparson wth the results under certanty shows that each rsk-averse frm can ncrease or decrease ts output, whch depends on the nteracton between both frms under cost uncertanty. (Receved 4 February 00; Revsed 6 August 00; Accepted 5 August 00) References Bresnahan, T. F. Duopoly Models wth Consstent Conjectures. Amercan Economc Revew 7 (No. 5 98): 934-45. Cheng, H. C., Magll, M., and Shafer, W. Some Results on Comparatve Statcs under Uncertanty. Internatonal Economc Revew 8 (No. 987): 493-507. Clarke, R. N. Colluson and the Incentves for Informaton Sharng. Bell Journal of Economcs 4 (No. 983a): 383-94.. Duopolst Don t Wsh to Share Informaton. Economcs Letters (Nos. - 983b): 33-6. Dardanon, V. Optmal Choces under Uncertanty: the Case of Two- Argument Utlty Functons. Economc Journal 98 (No. 39 988): We need a strong assumpton (for example, that a rsk-neutral Frm s expectaton on a rsk-neutral Frm j s reacton, E(q j Ω ) s the same as a rskaverse Frm s expectaton on a rsk-averse Frm j s reacton), so we can compare a rsk-neutral frm s behavor wth a rsk-averse frm s behavor under symmetrc uncertanty. That s, b j under rsk neutralty s the same as that under rsk averson. The result n ths case s smlar to that of comparng a rsk-averse frm under certanty wth a rsk-averse frm under uncertanty. Note that we cannot compare a rsk-neutral frm s behavor wth a rsk-averse frm s behavor under symmetrc uncertanty f a rsk-neutral Frm s expectaton on a rsk-neutral Frm j s reacton, E(q j Ω ), s dfferent from a rsk-averse Frm s expectaton on a rsk-averse Frm j s reacton.
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