Mixed Cournot-Bertrand Competition in N-firm Differentiated ...

whichisthensubstitutedinto(4)toobtainp K= A K − B K ¯K+ B K ¯K³B ¯K ¯K´−1A ¯K −³B ¯K ¯K´−1p ¯K.If we further introduce the following notation,³α ¯K = B ¯K ¯K´−1A¯K,β ¯KK =∙³B KK − B K ¯KB ¯K ¯K´−1B ¯KK³B ¯K ¯K´−1B¯KK, β ¯K ¯K =α K = A K − B K ¯K ³ B ¯K ¯K´−1A¯K,¸q K³B ¯K ¯K´−1,β KK = B KK − B K ¯K ³ B ¯K ¯K´−1B ¯KK and β K ¯K = −B K ¯K ³ B ¯K ¯K´−1,then p K and q ¯K are written as linear functions of strategic vectors q K and p ¯K :p K = α K − β KK q K − β K ¯K p ¯Kandq ¯K = α ¯K − β ¯KK q K − β ¯K ¯K p ¯K.The k th components of p K and q ¯K ,p k and q¯k, are written asp k =andnP(1+(n−m−1)γ)A k −γ A ī −(1−γ)(1+(n−m)γ)q k +γī=m+11+(n−m−1)γnPnPmPp ī −(1−γ) q iī=m+1i=1,i6=k(1+(n−m−1)γ)A¯k−γ Aī−(1+(n−m−2)γ)p¯k+γpī−(1−γ) q kī=m+1ī=m+1,ī6=¯kk=1q¯k =(1−γ)(1+(n−m−1)γ).(7)We now turn our attention to the case of BC competition in which p k is thestrategic variable for firm k and so is q¯k for firm ¯k. Since BC competition is dualto CB competition, interchanging sets K and ¯K generates the k th componentsof q k and p¯k,q k and p¯k, as functions of the strategic variables:q k =mP(1+(m−1)γ)A k −γ A i −(1+(m−2)γ)p k +γ p i −(1−γ) qīi=1i=1,i6=kī=m+1(1−γ)(1+(m−1)γ)(8)its inverse matrix is1+(n − m − 2)γ(1 − γ)(1 + (n − m − 1)γ)and the off-diagonal element isγ−(1 − γ)(1 + (n − m − 1)γ) .The invertibility of the matirx B KK is defined in the same way. This issue is discussed inmore detail in Szidarovszky and Yakowitz (1978).mPnPnPmP(6)4

andmP mPi=1(1+(m−1)γ)A¯k−γ A i −(1−γ)(1+mγ)q¯k+γ p i −(1−γ)qīi=1ī=m+1,ī6=¯kp¯k =(1+(m−1)γ)(9)Firm k chooses optimal strategy to maximize its profit π k =(p k −c k )x k wherec k denotes the constant marginal cost. In a linear demand oligopoly, changes inthe production cost do not affect the optimal behavior of the firms qualitatively.Thus the zero-cost assumption is usually adopted for the sake of analytical simplicity,and we follow this tradition in the following discussions.3 Review: Mixed DuopolyBefore proceeding to the general n-firm mixed oligopoly, we briefly reviewtheresults on the behavioral comparison in a duopoly framework. To this end,we take n =2and m =1. Let firm 1 be quantity-adjusting and firm 2 priceadjustingunder CB competition. The usual procedure for solving the profitmaximization problems leads to the optimal behavior of the firms as follows:q1 CB = 2A 1 − γA 24 − 3γ 2 ,p CB1 =(1− γ 2 )q1 CB and π CB1 =(1− γ 2 ) ¡ q1CB ¢ 2andp CB2 = −γA 1 +(2− γ 2 )A 24 − 3γ 2 ,q2 CB = p CB2 and π CB2 = ¡ q2CB ¢ 2.Subscript "CB" over variables indicates that the corresponding variables areevaluated at a CB equilibrium.Firm 2 is quantity-adjusting under BC competition and its optimal behavioris dual to the optimal behavior of quantity-adjusting firm 1 under CB competition.We get its optimal behavior by replacing A i by A j ,q i by q j and p i byp j for i 6= j, i, j =1, 2:q2 BC = −γA 1 +2A 24 − 3γ 2 ,p BC2 =(1− γ 2 )q2 BC and π BC2 =(1− γ 2 ) ¡ q2BC ¢ 2.In the same way, duality transforms the optimal behavior of firm 2 under CBcompetition into the optimal behavior of firm 1 under BC competition,p BC1 = (2 − γ2 )A 1 − γA 24 − 3γ 2 ,q1 BC = p BC1 and π BC1 = ¡ q1BC ¢ 2.Let us compare the profits of firms 1 and 2 under CB competition with thoseunder BC competition. Ratios of the profits areπ CB1π BC1µ 2 2 − αγ=(1− γ 2 )2 − γ 2 and πCB 2− αγ π BC2nP= 1 µ (2 − γ 2 2)α − γ1 − γ 2 2α − γ5

where α is the ratio of A 2 over A 1 . Differences between the numerator and thedenominator of the profit ratiosare(1 − γ 2 )(2 − αγ) 2 − (2 − γ 2 − αγ) 2 = −γ 3 (α 2 γ − 2α + γ)and¡(2 − γ 2 )α − γ ¢ 2− (1 − γ 2 )(2α − γ) 2 = γ 3 (α 2 γ − 2α + γ).The right hand sides of these two equations are negative and positive, respectively,if γ < 0. Henceπ CB1 < π BC1 and π CB2 > π BC2 if γ < 0. (10)Assume next that γ > 0. Then it can be shown that α 2 γ − 2α + γ < 0 underthe non-negativity conditions for the optimal prices and outputs,2 − γ 2γ≥ α ≥γ2 − γ 2implying thatπ CB1 > π BC1 and π CB2 < π BC2 if γ > 0. (11)The results (10) and (11) in the mixed duopoly agree with those obtained bySingh and Vives (1984) and, as a benchmark for measuring firms’ profitability,they can be summarized in the following way.Theorem 1 In a mixed duopoly, a quantity-adjusting strategy is more profitablethan a price-adjusting strategy if the goods are substitutes whereas the priceadjustingstrategy is more profitable than the quantity-adjusting strategy if thegoods are complements.4 Mixed N-Firm Oligopoly4.1 CB CompetitionWe will begin our discussion on the n-firm mixed oligopoly by considering theoptimal behavior of the firms under CB competition. We first derive the bestresponses and then obtain equilibrium outputs, prices and profits.4.1.1 Best ResponsesSince firm k ∈ K is quantity-adjusting, it chooses quantity of good k so as tomaximize its profit subject to (6), taking the other firms’ quantities and pricesas given. Solving the first-order condition of the profit maximizing problemyields the best responseq k =nP(1+(n−m−1)γ)A k −γ Aī+γī=m+1nPmPpī−(1−γ) q iī=m+1i6=k2(1−γ)(1+(n−m)γ). (12)6

Using the simple fact that P mi=1 q i = q k + P mi6=k q i, the best response can berewritten asnPnPmPq k =(1+(n−m−1)γ)A k −γ Aī+γī=m+1pī−(1−γ) q iī=m+1i=1(1−γ)(2+(2(n−m)−1)γ). (13)The second-order condition (SOC) for the profit maximization is satisfied if∂ 2 π k∂q 2 k(1 − γ)(1 + (n − m)γ)= −2 < 0. (14)1+(n − m − 1)γIf 0 < γ < 1, then the terms on the right hand side of (14) are positive andthus this SOC is always satisfied. If −1 < γ < 0, thesignsof1+(n − m)γ and1+(n−m−1)γ are ambiguous and then the sign of the SOC is also ambiguous,so we need additional conditions to fulfill the SOC. Since1+(n − m)γ < 1+(n − m − 1)γ, (15)the required condition is 1+(n − m)γ > 0 or 1+(n − m − 1)γ < 0.Since firm ¯k ∈ ¯K is price-adjusting, it chooses a price of good ¯k so as tomaximize its profit subject to (7), taking the other firms’ quantities and pricesas given. Solving the first-order condition of the profit maximizing problemyields the best responsenPnPmP(1+(n−m−1)γ)A¯k−γ Aī+γ pī−(1−γ) q iī=m+1ī6=¯ki=1p¯k =2(1+(n−m−2)γ). (16)Using the simple fact that P nī=m+1 p ī = p¯k + P nī6=¯k p ī, the best response can berewritten asnPnPmP(1+(n−m−1)γ)A¯k−γ Aī+γī=m+1p¯k =The SOC for the profit maximization is satisfied ifpī−(1−γ) q iī=m+1i=12+(2(n−m)−3)γ. (17)∂ 2 π¯k∂p 2¯k(1 + (n − m − 2)γ)= −2(1 − γ)(1 + (n − m − 1)γ) < 0.If 0 < γ < 1, thenthisSOCisalwayssatisfied. If −1 < γ < 0, thenbythesame reason above, we need additional conditions to satisfy the SOC. Since1+(n − m − 1)γ < 1+(n − m − 2)γ, (18)the required condition is 1+(n−m−1)γ > 0 or 1+(n−m−2)γ < 0. From (15)and (18), in order to fulfill the SOCs in the optimizing problems under both CBand BC competitions, we assume the following:Assumption 1. If γ < 0, either 1+(n − m)γ > 0 or 1+(n − m − 2)γ < 0holds.7

4.1.2 CB EquilibriumWe will next show that there is a unique CB equilibrium. Adding (13) over allquantity-adjusting firms and (17) over all price-adjusting firms givesandP(1 + (n − m − 1)γ) m nPA k − mγ A¯k + mγmPk=1¯k=m+1q k =(1 − γ)(2 + (2n − m − 1)γ)k=1nPp¯k =¯k=m+1(1 − γ)If we introduce the notation,PQ =(1− γ) m q k ,P=k=1nP¯k=m+1n P¯k=m+1A¯k − (n − m)(1 − γ)γ m P2+(n − m − 3)γnPp¯k¯k=m+1q ii=1Pp¯k, B k = m A k and B¯k =k=1.P n A¯k,¯k=m+1the above two equations can be rewritten as the simultaneous equations for Qand P⎧⎨ (2 + (2n − m − 1)γ)Q − mγP =(1+(n − m − 1)γ)B k − mγB¯k,(19)⎩(n − m)γQ +(2+(n − m − 3)γ)P =(1− γ)B¯k,whereweassume∆ CB =(2+(2n − m − 1)γ)(2 + (n − m − 3)γ)+m(n − m)γ 2 6=0 (20)to have a non-trivial solution. From the solution of (19) we obtainP − Q = 1∆ CB {−(1 + (n − m − 1)γ)(2 + (2(n − m) − 3)γ)B k+[2+(2n − 3)γ +(2n(m − 1) − (2m − 1)(m +1))γ 2 ]B¯k}.P − Q is substituted into (13) and (17) to obtain the Cournot-Bertrand equilibriumoutput and price, 3andq CBk =p CB ¯ka CBkA k −b CBkB k − b CB ¯kB¯k∆ CB (1 − γ)(2 + (2(n − m) − 1)γ)A k −b CBkB k − b CB ¯kB¯k∆ CB (2 + (2(n − m) − 3)γ)= aCB k(21)(22)3 The CB output and price as well as the CB equilibrium values of another variables to beobtained below in the n-firm oligopoly can be reduced to those in the duopoly obtained aboveif we take n =2and m =1.8

whereandb CBkb CB¯ka CBk= ∆ CB (1 + (n − m − 1)γ),= γ(1 + (n − m − 1)γ)(2 + (2(n − m) − 3)γ)= γ(2 + (2(n − m) − 1)γ)(1 + ((n − m) − 2)γ).From the perceived demand (6) of firm k and the best response (12), we canobtain the price at which firm k sells its output as a function of its optimalquantity,p CBk=(1 − γ)(1 + (n − m)γ)qk CB . (23)1+(n − m − 1)γIn the same way, solving (7) and (16) for q¯k yields the output that firm ¯k sellsas a function of its optimal priceq¯k CB 1+(n − m − 2)γ=(1 − γ)(1 + (n − m − 1)γ) pCB ¯k. (24)Using (23) and (24), the Cournot-Bertrand profits of firms k and ¯k are writtenasπ CB (1 − γ)(1 + (n − m)γ) ¡ ¢k = qCB 2k(25)1+(n − m − 1)γandπ CB¯k=4.2 BC Competition(1 − γ)(1 + (n − m − 1)γ)1+(n − m − 2)γ¡ ¢qCB 2¯k(26)In BC competition, the strategies of firms are interchanged. Namely, the firstm firms in K are price-adjusting and the last (n − m) firms in ¯K are quantityadjusting.By duality, we can obtain the optimal behavior under BC competitionfrom the optimal behavior under CB competition. In particular, the bestresponses of firms k and ¯k areÃ!P(1 + (m − 1)γ)A k − γ m mPnPA i + γ p i − (1 − γ)p k =i=1i=1, i6=k2(1 + (m − 2)γ)q¯k¯k=m+1(27)andÃP(1 + (m − 1)γ)A¯k − γ m mPnPA k + γ p k − (1 − γ)q¯k =qīk=1k=1ī=m+1, ī6=¯k2(1 − γ)(1 + mγ)!(28)It is easy to see that the second-order conditions for the profit maximizingproblems are always satisfiedifthegoodsaresubstitutes. Ifthegoodsarecomplements, then the following additional conditions are necessary.9

Assumption 2. If γ < 0, then either 1+mγ > 0 or 1+(m − 2)γ < 0.Duality also gives BC equilibrium price and output from (21) and (22):andwhereandp BCkq BC¯k= aBC k= aBC kA k − b BCkB k − b BC¯kB¯k∆ BC (2 + (2m − 3)γ)A¯k − b BCkB k − b BC ¯kA¯k∆ BC (1 − γ)(2 + (2m − 1)γ)∆ BC =(2+(m − 3)γ)(2 + (n + m − 1)γ)+m(n − m)γ 2b BCkb BC¯ka BCk= ∆ BC (1 + (m − 1)γ)= γ(1 + (m − 2)γ)(2 + (2m − 1)γ)= γ(1 + (m − 1)γ)(2 + (2m − 3)γ)(29)(30)Superscript "BC" over variables has the similar meaning as "CB."Using the demands and the best responses, the output of the price-adjustingfirm k and the price of quantity-adjusting firm ¯k can be expressed as linearfunctions of the strategic variables, p BCkandand q BC¯k:qk BC 1+(m − 2)γ=(1 − γ)(1 + (m − 1)γ) pBC k (31)p BC ¯k=(1 − γ)(1 + mγ)1+(m − 1)γqBC ¯k(32)From (31) and (32), the profits under BC competition have the formsandπ BCk=π BC¯k(1 − γ)(1 + (m − 1)γ)1+(m − 2)γ=(1 − γ)(1 + mγ)1+(m − 1)γ¡ ¢qBC 2k(33)¡ ¢qBC 2k . (34)5 Comparison of the optimal behaviorWe compare the optimal behavior under CB competition with that under BCcompetition to find out which competition is more profitable. It, however, requirestedious calculations to make the comparison under general values of mand n. To simplify the analysis, we focus on the special case in which the followingcondition is satisfied:Assumption 3. n =2m.Notice that Assumptions 1 and 2 become identical under Assumption 3. Inthe following analysis, we focus our attention to the optimal behavior of firmk. This treatment will suffice for our purpose to show that Theorem 1 does notnecessarily hold in the general n-firm mixed oligopoly.10

5.1 Ratios of optimal valuesReplacing n with 2m, we can verify that∆ CB = ∆ BC =(2+(3m − 1)γ)(2 + (m − 3)γ)+m 2 γ 2 ,b CBk= b BC ¯k= γ(1 + (m − 1)γ)(2 + (2m − 3)γ)andb CB ¯k= b BCk = γ(1 + (m − 2)γ)(2 + (2m − 1)γ).Let ∆ CB and ∆ BC be denoted as ∆. Then we can also verify thata CBk= a BCk= ∆(1 + (m − 1)γ).To simplify the notation, we denote k by 1, ¯k by 2, a CBkand a BCkby a 1 ,b CBkand b BC by b ¯k 1 and b CB and b ¯kBCkby b 2 , respectively. With this new notation,CB price (23) and BC price (29) are rewritten asp CB(1 + mγ)mb 2 A 1¡1 =ϕ CB ¢(α 1 ) − α 2 (35)∆(1 + (m − 1)γ)(2 + (2m − 1)γ)andp BC mb 1 A 1¡1 =ϕ BC ¢(α 1 ) − α 2 (36)∆(2 + (2m − 3)γ)where α 1 , respectively α 2 , denotes the ratio between the average quality offeredby the firms in K, respectively ¯K, and the quality offered by firm k,α 1 = B 1/m, respectively α 2 = B 2/m(37)A 1 A 1andϕ CB (α 1 )= a 1− b 1α 1 and ϕ BC (α 1 )= a 1− b 2α 1 . (38)mb 2 b 2 mb 1 b 1Both ϕ CB (α 1 ) and ϕ BC (α 1 ) are linear in α 1 . If b 1 6= b 2 , then they intersect onceat point (α ∗ 1, α ∗ 2) whereα ∗ 1 = α∗ 2 = a 1(b 1 + b 2 )m and ϕCB (α ∗ 1 )=ϕBC (α ∗ 1 )=α∗ 2 . (39)Taking the ratio of (35) over (36), thenwhereandp CB11 − ε 2 ¡p BC − 1=α21ϕ BC − ϕ(α 1 ) − α p (α 1 ) ¢ (40)2ϕ p (α 1 )= ϕBC (α 1 ) − ε 2 ϕ CB (α 1 )1 − ε 2 (41)ε 2 =(1 + mγ)(1 + (m − 2)γ)(1 + (m − 1)γ) 2 .11

It is easy to confirm the following properties of ϕ p (α 1 ). The definition of ϕ p (α 1 )in (41) indicates that ϕ p (α ∗ 1 ) = α∗ 2 , that is, the α 2 = ϕ p (α 1 ) curve passesthrough the point (α ∗ 1, α ∗ 2). Forα 1 =0ϕ p (0) = a 1(b 2 − ε 2 b 1 ) a 1 γ 3 (1 + (m − 2)γ)m(1 − ε 2 =)b 1 b 2 m(1 − ε 2 )b 1 b 2 1+(m − 1)γ(42)and differentiating ϕ p (α 1 ) with respect to α 1 givesdϕ p (α 1 )dα 1= ε2 b 2 1 − b 2 2(1 − ε 2 )b 1 b 2= 2γ5 (1 + (m − 2)γ)(1 − ε 2 )b 1 b 2, (43)where the definitions of b 1 ,b 2 and ε 2 are substituted into the denominators ofthe middle expressions in (42) and (43) to obtain the last expressions.With the new notation, CB quantity (21) and BC quantity (31) are rewrittenasandq CB1 =mb 2 A 1¡ϕ CB ¢(α 1 ) − α 2∆(1 − γ)(2 + (2m − 1)γ)q1 BC(1 + (m − 2)γ)mb 1 A 1¡=ϕ BC ¢(α 1 ) − α 2 .∆(1 − γ)(1 + (m − 1)γ)(2 + (2m − 3)γ)Taking the ratio of q CB1 over q BC1 ,wehaveq1CBq1BC − 1= (b 1 − b 2 )(b 1 + b 2 )(ϕ BC (α ∗ 1 − α 1 ). (44)(α 1 ) − α 2 )b 1 b 2Finally, from the ratio of (25) over (33), we getwhereandπ CB1π BC − 1= (1 + ε)(1 − ε)(ϕ+ π (α 1 ) − α 2 )(α 2 − ϕ − π (α 1 ))1(ϕ BC (α 1 ) − α 2 ) 2 . (45)ϕ + π (α 1 )= ϕBC (α 1 )+εϕ CB (α 1 )1+εϕ − π (α 1 )= ϕBC (α 1 ) − εϕ CB (α 1 )1 − ε(46)(47)withdϕ − π (α 1 )= εb2 1 − b2 2dα 1 (1 − ε 2 . (48))b 1 b 2Subtracting (43) from (48) givesdϕ − π (α 1 )dα 1− dϕ p(α 1 )dα 1= ε(b2 1 − b 2 2)(1 − ε 2 )b 1 b 2. (49)12

5.2 γ > 0: the goods are substitutesWe first assume that the goods are substitutes (i.e., γ > 0). Let us start withthe price comparison. If γ > 0, then ∆ > 0, a 1 > 0, b 1 > 0, b 2 > 0 andb 1 − b 2 = γ 3 > 0. It also implies that the first factor in each of (35) and (36) ispositive. The non-negativity conditions for the prices areα 2 ≤ ϕ CB (α 1 ) and α 2 ≤ ϕ BC (α 1 ).From (38) and (39), it can be examined that the boundary curves, α 2 = ϕ CB (α 1 )and α 2 = ϕ BC (α 1 ), slope downwards, have positive intercepts and intersect onceat point (α ∗ 1, α ∗ 2) with α ∗ 1 = α ∗ 2 > 0. We denote a feasible region of α 1 and α 2by Ω which is defined byΩ = © (α 1 , α 2 ) | ϕ CB (α 1 ) ≥ α 2 , ϕ BC (α 1 ) ≥ α 2 , α 1 > 0 and α 2 > 0 ª .Sinceϕ CB (α 1 ) − ϕ BC (α 1 )= b2 1 − b2 2(α ∗ 1 − α 1 ) and b 2 1 − b 2 2 > 0,b 1 b 2the upper bound of Ω for α 1 < α ∗ 1 is the α 2 = ϕ BC (α 1 ) curve and the upperbound of Ω for α 1 > α ∗ 1 is the α 2 = ϕ CB (α 1 ) curve. Relations (42) and (43)indicate that the α 2 = ϕ p (α 1 ) curve slopes upwards and has a positive intercept.Since the numerator of the first factor in the right hand side of (40) is positiveas ε 2 is positive and less than unity under γ > 0 4 and the denominator isalso positive by the non-negativity condition of p BC1 , we arrive at the followingcondition for the price difference:sign £ p CB1 − p BC ¤ £1 = sign α2 − ϕ p (α 1 ) ¤ . (50)Hence we have the following results on the price difference:Proposition 1. If γ > 0, then the α 2 = ϕ p (α 1 ) curve divides the feasibleregion Ω into two subregions: p CB1 > p BC1 in the subregion above thecurve and p CB1 0 implies q1 BC ≥ 0 due to (31). We then notice thatthe first factor of (44) is positive. In consequence, we arrive at the followingcondition for the quantity difference,sign £ q1 CB − q1BC ¤= sign[α∗1 − α 1 ]. (51)The sign of the quantity difference is determined by the difference between α ∗ 1and α 1 , so we have the following result.Proposition 2. If γ > 0, then q1 CB >q1 BC for α 1 < α ∗ 1 and q1 CB α ∗ 1.4 (1 + mγ)(1 + (m − 2)γ) − (1 + (m − 1)γ) 2 = −γ 2 < 0 implies ε 2 < 1 and ε 2 > 0 is clear.13

Figure 3. 1+mγ > 0 and ∆ > 0 where γ = −0.1 and m =6.We now assume ∆ < 0 that leads to a 1 < 0. Since the first factors in (35)and (36) are positive, the non-negativity conditions of the prices areϕ CB (α 1 ) − α 2 ≥ 0 and ϕ BC (α 1 ) − α 2 ≥ 0which is the same as the conditions in the case in which the goods are substitutes.Inthesameway,weobtainsign[p CB1 − p BC1 ]=sign[α 2 − ϕ p (α 1 )]sign[q1 CB − q1 BC ]=sign[α ∗ 1 − α 1 ]andsign[π CB1 − π BC1 ]=sign[α 2 − ϕ − π (α 1 )],which are the same as (50), (51) and (52). There are, however, differences: Theequal-price curve, ϕ p (α 1 )=α 2 , is negative sloping and the slope of the profitequalcurve ϕ − π (α 1 )=α 2 , can be of either sign. The division of the feasibleregion is depicted in Figure 4 in which we take point a (i.e., γ = −0.095 andm =10). Despite of these differences, the main results is the same as Theorem2, and summarized as follows.Theorem 4 Assume that γ < 0, 1+mγ > 0 and ∆ < 0, then the α 2 = ϕ − π (α 1 )curvedividesthefeasibleregionΩ into two parts: π CB1 > π BC1 in the dividedregion above the curve and π CB1 < π BC1 below.18

Figure 4. 1+mγ > 0 and ∆ < 0 where γ = −0.095 and m =105.3.2 The case of SOC 2If γ < 0 and 1+(m − 2)γ < 0, then1+(m − 1)γ < 0, 2+(2m − 1)γ < 0 and2+(2m−3)γ < 0, which, in turn, imply that b 1 < 0, b 2 < 0 and b 1 −b 2 = γ 3 < 0.Although ∆α 1 < 0, the sign of ∆ is undetermined and so is the sign of α 1 .Weidentity two cases as in the case of SOC 1 : oneisthecasewhere∆ is positiveand the other is the case where ∆ is negative.Let us suppose that ∆ > 0, then a 1 < 0 which leads to α ∗ 1 > 0 and α ∗ 2 > 0.(35) and (36) imply that the non-negativity conditions for the prices and outputsareϕ CB (α 1 ) ≥ α 2 andϕ BC (α 1 ) ≥ α 2 .The feasible region for the price and outputs are Ω. ϕ p (α 1 ) slopes upwards andϕ + π (α 1 ) > α 2 . It is clear that ϕ − π (α 1 ) slopes upwards and is steeper than ϕ p (α 1 ).From (40), (44) and (45), we havesign[p CB1 − p BC1 ]=sign[α 2 − ϕ p (α 1 )],sign[q1 CB − q1 BC ]=sign[α ∗ 1 − α 1 ]andsign[π CB1 − π BC1 ]=sign[α 2 − ϕ − π (α 1 )].Thesethreeconditionsarethesameas (50), (51) and (52), respectively. Inconsequence the results obtained here are the same as the results obtained inthecaseofγ > 0. The profit differences are summarized as follows:Theorem 5 If γ < 0, 1+(m − 2)γ < 0 and ∆ > 0, then the α 2 = ϕ − π (α 1 )curvedividesthefeasibleregionΩ into two parts: π CB1 > π BC1 in the dividedregion above the curve and π CB1 < π BC1 below.19

We take γ = −0.25 and m =6, which correspond to point b in Figure 2 anddepict the division of the feasible region in Figure 4. As far as profit comparisonis our concern, the similarity between Figures 3 and 5 is clear: π CB1 < π BC1 inthe lighter-gray region and π CB1 > π BC1 in the darker-gray region.Figure 6. 1+(m − 2)γ < 0 and ∆ < 0 where γ = −0.25 and m =6.6 Concluding RemarkIn a mixed duopoly with product differentiation, Theorem 1 indicates that aquantity-strategy is more profitable than a price-strategy if the goods are substitutesand the profit dominance is reversed if the goods are complements. Ourmain aim is to reconsider whether these clear-cut results still hold in a general(n-firm) oligopoly framework or not. Theorems 2, 3, 4, 5 and 6 lead us to theconclusion that the results obtained in the duopoly framework are no longervalid in general oligopoly framework. If the firms can precommit to quantity orprice strategy, the dominant strategy depends on the average quality ratio ofthegoodsproducedbythequantity-adjustingfirms and the goods produced bythe price-adjusting firms.21

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