BERTRAND VERSUS COURNOT COMPETITION IN A VERTICAL STRUCTURE: A NOTE

BERTRAND VERSUS COURNOT COMPETITION INA VERTICAL STRUCTURE: A NOTE*manc_2228 1..15

byARIJIT MUKHERJEE

University of York and CESifoUDO BROLL

Dresden University of Technologyand

SOMA MUKHERJEE†

University of Nottingham

In a vertical structure with a profit-maximizing upstream firm, we showthat whether the profits in the downstream market are higher underBertrand competition or under Cournot competition depends on thetechnology differences among the downstream firms and on the pricingstrategy (namely uniform pricing or price discrimination) of the upstreamfirm. The upstream firm’s profit, the profit of the upstream and thedownstream firms taken together, and social welfare are always higherunder Bertrand competition than under Cournot competition.

1 Introduction

What are the effects of product market competition on profits and welfare? Ina seminal work, Singh and Vives (1984) show that firms earn higher profitsunder Cournot competition than under Bertrand competition, while welfareis higher under Bertrand competition. Hence, the firms are better off in a lesscompetitive market, whereas the society prefers a more competitive environ-ment. This work by Singh and Vives has been extended in several directionsto show the implications of factors such as the number of firms (Häckner,2000), endogenous market structure (Cellini et al., 2004; Mukherjee, 2005),the impacts of positive primary outputs1 (Acharyya and Marjit, 1998;Zanchettin, 2006), innovation (Delbono and Denicolò, 1990; Reynolds andIsaac, 1992; Bester and Petrakis, 1993; Qiu, 1997; Bonanno and Haworth,1998; Boone, 2001; Symeonidis, 2003; Mukherjee, 2011) and technologylicensing (Mukherjee, 2010). These works share a common feature. They

* Manuscript received 6.7.09; final version received 21.5.10.† We thank an anonymous referee of this journal for helpful comments and suggestions. Arijit

Mukherjee also benefitted from a discussion with Piercarlo Zanchettin. The authors aresolely responsible for the views presented here but not the Universities. The usual dis-claimer applies.

1Positive primary outputs imply that if all the prices are set at the marginal costs, all firms sellpositive outputs.

The Manchester School Vol •• No. •• ••–•• •• 2012doi: 10.1111/j.1467-9957.2012.02228.x

© 2012 The AuthorsThe Manchester School © 2012 Blackwell Publishing Ltd and The University of ManchesterPublished by Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK, and 350 Main Street, Malden, MA 02148, USA.

1

ignore market powers of the input suppliers. This may be appropriate in somereal world situations, yet the input markets are often imperfectly competitive,thus giving market powers to the input suppliers. For example, as pointed outin Komiya (1975), the industries such as iron and steel, petroleum refining,petrochemicals and certain other chemicals, cement, paper and pulp andsugar refining, which produce inputs for several final goods, are characterizedby imperfect competition.

Although vertical relations between the input suppliers and final goodsproducers are quite common in real world, the literature analyzing the effectsof product market competition on profits and welfare did not pay muchattention to this aspect. Whatever efforts have been devoted in this direction(López and Naylor, 2004; López, 2007) look at symmetric final goods pro-ducers. In a bilateral oligopoly with labor unions, López and Naylor (2004)show that the profits are higher under Bertrand competition if the unions aresufficiently powerful and care enough about wages in their utility function.However, ‘if the upstream agents are profit-maximizing firms, the traditionalresult [i.e. profit is higher under Cournot competition than under Bertrandcompetition] is obtained’ (López and Naylor, 2004, p. 694). Consideringendogenous choice of the strategic variables, López (2007) shows that theprofits of the downstream firms are higher under Cournot competition if theupstream agents are profit-maximizing firms and the products are close sub-stitutes. Hence, the existing works on vertical relations show that the profitsin the downstream market are higher under Cournot competition if theupstream agents are profit-maximizing firms and the final goods are closesubstitutes. However, an important assumption of these works is to considersymmetric downstream firms, while asymmetry between the firms is perhapsempirical regularity.

On the other hand, Zanchettin (2006) compares the profitability of firmsunder Bertrand and Cournot competition in the presence of cost asymmetry.However, he assumes perfectly competitive input markets, thus ignoring therole of vertical relations.

In this paper we bring these two factors, namely vertical relationship andasymmetry between the downstream firms, together. In a simple model withhomogeneous final goods, we show that if the downstream firms differ interms of production technologies, the profits in the downstream market maybe higher under Bertrand competition even if the upstream agents are profit-maximizing firms. We show that the pricing strategy of the upstream firm (i.e.uniform pricing or price discrimination) and the degree of asymmetrybetween the downstream firms play important roles in this respect. Underuniform pricing by the upstream firm, the profit in the downstream marketcan be higher under Bertrand competition if the asymmetry between thedownstream firms is sufficiently large. However, under price discriminationby the upstream firm, the profits in the downstream markets are higher underCournot competition.

The Manchester School2

© 2012 The AuthorsThe Manchester School © 2012 Blackwell Publishing Ltd and The University of Manchester

Thus, we show that the results of López and Naylor (2004) and López(2007) do not hold with asymmetric downstream firms, if the upstreamfirm charges a uniform price to the downstream firms. In the presenceof technological asymmetry between the downstream firms, the moretechnologically efficient downstream firm gets a larger market share underBertrand competition compared with Cournot competition. This largermarket share along with a lower input price under Bertrand competitionmay outweigh the effect of the lower product price under Bertrand com-petition, thus creating higher profit of the more technologically efficientdownstream firm under Bertrand competition compared with Cournotcompetition.

On the other hand, unlike Zanchettin (2006), we show that even if thefinal goods producers differ in production technologies but the marginal costsare determined endogenously because of the endogenous input price deter-mination, price discrimination by the input supplier compensates the tech-nology difference, and make the marginal costs of the final goods producerssame. As a result, the downstream profits are higher under Cournot com-petition even if the downstream firms differ in technologies.

We also show that the profit of the upstream firm, the profit of theupstream and the downstream firms taken together, and social welfare arehigher under Bertrand competition compared with Cournot competition,irrespective of the pricing strategy of the upstream firm. We also discuss theimplications of a vertical merger briefly.

To summarize, the contributions of this paper are twofold. First, itshows that, in a vertical structure, the profits can be higher under Bertrandcompetition than under Cournot competition if the final goods producersdiffer in technologies and the input supplier charges a uniform price. Second,we show that the qualitative result derived under exogenously given marginalcost difference does not hold in a vertical structure determining the marginalcosts endogenously, if the input supplier charges discriminatory prices to thedownstream firms.

The remainder of the paper is organized as follows. Sections 2 and 3consider the situations of Cournot and Bertrand competition respectively.Section 4 compares the outcomes under Cournot and Bertrand competition.Section 5 discusses two extensions. Section 6 concludes. The proofs arerelegated to the Appendix.

2 The Case of Cournot Competition

Our basic set-up is similar to the literature on price discrimination in avertical structure (see, for example, Yoshida, 2000 and the references therein).Consider an economy with two final goods producers, firms 1 and 2. Thesefirms compete with a homogeneous product and require a critical input,which is produced by an upstream firm, called firm U. We normalize the

Bertrand and Cournot Competition 3


marginal cost of input production to zero. We assume that the input produc-tivity in firm 1 is one, while it is in firm 2 is 1/l, where l > 1. The inputproductivity is lower in firm 2 than in firm 1. Hence, the input demand in firm1 for producing one unit of output is one, but the input demand in firm 2 forproducing one unit of output is l, which is considered to be a continuousvariable.

We will consider two types of price setting behavior of the upstreamfirm: (i) uniform pricing and (ii) discriminatory pricing. Under uniformpricing, firm U charges the same price to both firm 1 and firm 2. Underdiscriminatory pricing, firm U can charge different price to firms 1 and 2.While it is well-known from Yoshida (2000) that an upstream agent prefersprice discrimination than uniform pricing in the presence of asymmetricdownstream agents, it may follow from Katz (1987) and DeGraba (1990) thatgovernment regulation may require the upstream agent to charge a uniformprice to the asymmetric downstream agents.

The inverse market demand function for the final product is

P a q= − (1)

where P shows price and q shows the total output.We consider the following game. At stage 1, firm U sets the price for the

inputs. At stage 2, firms 1 and 2 choose the outputs simultaneously, and theprofits are realized. We solve the game through backward induction.

2.1 Uniform Pricing

First consider the situation where firm U charges a uniform price to both firm1 and firm 2.

Given the input price w, firms 1 and 2 maximize the following expres-sions respectively:

Maxq

a q w q1

1( )− − (2)

Maxq

a q w q2

2( )− − λ (3)

The equilibrium outputs can be derived as

qa w w

qa w w

1 22

32

3* and *=

− +=

− +λ λ(4)

The total input demand is

q q qa w w

wa

I = + =+ − + +

≤−1 2

21 2 1 23 2 1

* * forλ λ λ λλ

( ) ( )(5a)

q qa w

wa

I = =−

≥−1

2 2 1* for

λ(5b)



It is clear from (5b) that there is no input demand for w > a.Given the input demands, firm U may not want to serve both firms 1 and

2 if firm 2 is sufficiently cost inefficient than firm 1. That is, firm U may bebetter off by charging an input price greater than a/(2l - 1), which allowsonly firm 1 to purchase the input, than charging an input price lower thana/(2l - 1), which allows both firms 1 and 2 to purchase the input. We show inthe Appendix that firm U wants to supply the input to both firms 1 and 2 ifl is less than 2. In the following analysis, we assume that l ∈[1, 2), whichensures positive output of both firms 1 and 2.

Firm U maximizes the following expression to determine the input price:

Maxw

w a w w( ) ( )1 2 1 2

3

2+ − + +[ ]λ λ λ(6)

The equilibrium input price is a(1 + l )/(4 + 4l2 - 4l), which is less thana/(2l - 1) for l ∈[1, 2), and the profit of firm U is [a2(1 + l)2]/[24(1 + l2 - l)].

Given the equilibrium input price, the total input demand and theprofits of firms 1 and 2 are respectively a(1 + l)/6, [a2(2 + 5l2 - 5l)2]/[9(4 + 4l2 - 4l)2] and [a2(5 + 2l2 - 5l)2]/[9(4 + 4l2 - 4l)2].

Social welfare, which is the sum of consumer surplus and the profits offirm 1, firm 2 and firm U, is

Wa a

a

uc =

+ − + + −+ −

++

+

2 2 2 2 2 2

2 2

2 2

5 2 5 2 5 59 4 4 4

124 1

( ) ( )( )

( )(

λ λ λ λλ λ

λλ22

2 2 2

2 2

7 7 10288 1−

++ −

+ −λλ λλ λ)

( )( )

a(7)

2.2 Price Discrimination

Now consider price discrimination by firm U. If firm U charges w1 and w2 tofirms 1 and 2, respectively, firms 1 and 2 maximize the following expressionsrespectively:

Maxq

a q w q1

1 1( )− − (8)

Maxq

a q w q2

2 2( )− − λ (9)

The equilibrium outputs can be found as

qa w w

qa w w

11 2

22 12

32

3* and *=

− +=

− +λ λ(10)

Firm U chooses w1 and w2 to maximize the following expression:

Maxw w

w a w w w a w w1 2

1 1 2 2 2 12 23,

( ) ( )− + + − +λ λ λ(11)



The equilibrium input prices are

wa

wa

1 22 2

= =andλ

(12)

The equilibrium profits of firms 1 and 2 are respectively a2/36 and a2/36.The profit of firm U is a2/6. Social welfare under price discrimination is

Wa

dc =

518

2

(13)

3 The Case of Bertrand Competition

Now consider the situation where firms 1 and 2 compete in prices. Weconsider a game similar to Section 2 with the exception that now firms 1 and2 choose prices simultaneously.

3.1 Uniform Pricing

Since firm 2 needs l > 1 units of inputs to produce one unit of output, giventhe input price, the marginal cost of production for firm 2 is lw. Hence, firm1, which needs one unit of input to produce one unit of output, charges lw asthe price of its product, and only firm 1 produces in this situation.2

The input demand is

q a wI = − λ (14)

Firm U maximizes the following expression to determine the input price:

Maxw

w a w( )− λ (15)

The equilibrium input price is w = a/2l.3 The profits of firm 1, firm 2 and firmU are a2(l - 1)/4l, 0 and a2/4l, respectively.

Social welfare under price competition in the presence of uniform pricingby firm U is

Wa a a

ub =

−+ +

2 2 214 4 8( )λ

λ λ(16)


Now consider price discrimination by firm U. Under price discrimination,there is input demand by firm 1 (firm 2) if w1 < (>)lw2, where w1 and w2 are the

2We assume that, given the input price, monopoly price of the final good is greater than lw. Thishappens if a/(2l - 1) > w.

3Note that a/(2l - 1) > a/2l. It is also easy to check that firm U has no incentive to chargea/(2l - 1) so that firm 1 can charge its monopoly price for its product.



input prices charged to firms 1 and 2, respectively. It can be checked that theequilibrium input prices must satisfy w1 = lw2, and it will be w1 = lw2 = a/2. Inthis situation, the marginal cost of firm 1, which is w1, is equal to the marginalcost of firm 2, which is lw2, and both firms 1 and 2 produce like symmetric-cost Bertrand duopolists.

The profit of firm U is a2/4, and the equilibrium profits of firms 1 and 2are zero. Social welfare under price discrimination is

W adb = 3 82 (17)

4 Comparing Bertrand and Cournot Competition

4.1 Uniform Pricing

First consider the situation under uniform pricing. The input prices underCournot and Bertrand competition are respectively a(1 + l)/(4 + 4l2 - 4l)and a/2l, and the profits of firm U are respectively [a2 (1 + l)2]/[24(1 + l2 - l)]and a2/4l. Comparison of these values gives the following result.

Proposition 1:

(i) The input price is higher under Cournot competition than under Ber-trand competition for l ∈(1, 2).

(ii) The profit of firm U is higher under Bertrand competition than underCournot competition for l ∈[1, 2).

Since the product market competition is more intense under Bertrandcompetition compared with Cournot competition, a reduction in the finalgoods producers’ marginal costs, which are determined by the input prices,affects the outputs of the final goods producers and therefore, the inputdemand more under Bertrand competition compared with Cournot com-petition. Hence, the input demand becomes more elastic under Bertrandcompetition compared with Cournot competition, thus creating a lower inputprice under the former compared with the latter.

Even if the input prices are lower under Bertrand competition comparedwith Cournot competition, a more intense competition under the formerhelps to create a large input demand compared with the latter. As a result, theupstream firm earns higher profit under Bertrand competition compared withCournot competition.

Now we compare the profits of firm 1 and the industry profit underCournot competition to that of under Bertrand competition.

Proposition 2:

(i) The profit of firm 1 is higher under Bertrand competition than underCournot competition if l is greater than 1.27 (approximately).



(ii) The total profits of firms 1 and 2 are higher under Bertrand competi-tion than under Cournot competition if l is greater than 1.53(approximately).

Proof: See the Appendix. �

Intuition for the above result is as follows. In the case of Bertrandcompetition, only firm 1 produces the final goods. While lower input priceand higher market share tend to increase the profit of firm 1 under Bertrandcompetition, lower product price tends to reduce its profit under Bertrandcompetition. If firms 1 and 2 are sufficiently different in terms of inputcoefficient, the product-price effect is sufficiently small and is dominated bythe effects of lower input price and higher market share. In this situation, firm1 earns higher profit under Bertrand competition.

Although profit of firm 1 can be higher under Bertrand competition, it istrivial that the profit of firm 2 is higher under Cournot competition since firm2 gets positive profit under Cournot competition and zero profit under Ber-trand competition. However, the above result shows that the industry profitis still higher under Bertrand competition if the downstream firms are suffi-ciently asymmetric. Under Bertrand competition, only firm 1 produces in themarket, whereas firm 2 has a positive market share under Cournot competi-tion, which creates production inefficiency under Cournot competition com-pared with Bertrand competition. Since the product-price effect is very smallif firms 1 and 2 are sufficiently asymmetric, the input-price effect and theproduction efficiency effect dominate the product-price effect, thus creatinghigher industry profit under Bertrand competition compared with Cournotcompetition if firms 1 and 2 are sufficiently asymmetric.

Now compare the profits of the downstream and the upstream firms taketogether. The profits of the upstream and the downstream firms takentogether are

a a a2 2

2

2 2 2

2 2

2 2 2124 1

2 5 59 4 4 4

5 2 5( )( )

( )( )

( )++ −

++ −

+ −+

+ −λλ λ

λ λλ λ

λ λ99 4 4 42 2( )+ −λ λ

under Cournot competition and a2/4 under Bertrand competition. Note thatthe total profit under Bertrand competition is the monopoly profit of anindustry producing at the zero marginal cost of production. Hence, it is trivialthat the profits of the upstream and the downstream firms taken together arehigher under Bertrand competition than under Cournot competition.

The following proposition summarizes the above finding.

Proposition 3: The profits of the upstream and the downstream firms takentogether are higher under Bertrand competition than under Cournotcompetition.



Now compare welfare under Bertrand and Cournot competition. Con-sumer surplus under Cournot and Bertrand competition are respectively[a2(7 + 7l2 - 10)2]/[288(1 + l2 - l)2] and a2/8, and it is easy to check that thelatter is greater than the former for l ∈[1, 2). Further, Proposition 3 hasshown that the profits of upstream and the downstream firms taken togetherare higher under Bertrand competition than under Cournot competition.Therefore, the following proposition is immediate.

Proposition 4: In the case of uniform pricing by firm U, social welfare ishigher under Bertrand competition than under Cournot competition forl ∈[1, 2).

Even if the total profit in the downstream market may be higher underCournot competition compared with Bertrand competition, intense com-petition under the latter compared with the former helps to create higherwelfare under Bertrand competition compared with Cournot competition.


We now compare Cournot and Bertrand competition under price discrim-ination by firm U.

The equilibrium profits of firms 1 and 2 are respectively a2/36 and a2/36under Cournot competition, while they are zero under Bertrand competition.Hence, it is immediate that the profits of the downstream firms are higherunder Cournot competition than under Bertrand competition.

The profit of firm U is a2/6 under Cournot competition, while it is a2/4under Bertrand competition. Therefore, the profit of the upstream firm ishigher under Bertrand competition than under Cournot competition.

The profits of the upstream and the downstream firms taken together are2a2/9 under Cournot competition, and a2/4 under Bertrand competition.Again, the total profit of the firms under Bertrand competition is equal to themonopoly profit of an industry producing at the zero marginal cost ofproduction. Hence, the profits of the upstream and the downstream firmstaken together are higher under Bertrand competition than under Cournotcompetition.

Consumer surplus under Cournot competition is a2/18, while it is a2/8under Bertrand competition, which implies that consumer surplus is higherunder Bertrand competition compared with Cournot competition.

Since welfare is the sum of consumer surplus and the total profits of theupstream and the downstream firms, it is immediate from the above discus-sion that welfare is higher under Bertrand competition than under Cournotcompetition.

The following proposition summarizes the above discussion.



Proposition 5: In the case of price discrimination by firm U, (i) the profit ofeach downstream firm is always higher under Cournot competition comparedwith Bertrand competition, and (ii) the profit of the upstream firm, the totalprofit of the upstream and the downstream firms taken together, and socialwelfare are always higher under Bertrand competition compared withCournot competition.

5 Two Extensions

5.1 Optimal Pricing Strategy of the Upstream Firm

So far, we have done our analysis under two different pricing strategies ofthe upstream firm. We have shown that the pricing strategy of the upstreamfirm may play an important role in determining the profitability of thefirms. While the factors such as arbitrage possibility, the cost of adminis-tering different prices and government regulation may induce the upstreamfirm to charge a uniform price to the downstream firms, price discrimina-tion gives the upstream firm more flexibility. Hence, one would expect thatthe profit of the upstream firm would be higher under price discriminationcompared with uniform pricing, irrespective of the type of product marketcompetition. It follows from the above analysis that this is indeed the case.Hence, if the type of pricing strategy is a choice variable of the upstreamfirm, it will prefer price discrimination compared with uniform pricing.Therefore, Proposition 3 will be the relevant result if the upstream firmdecides among uniform pricing and price discrimination. This implies thatif the input prices are endogenously determined and the input supplier candetermine whether to charge a uniform price or discriminatory prices, theresult of Zanchettin (2006) may not hold since the upstream firm will preferprice discrimination, and the equilibrium input price difference will createthe same marginal costs of the downstream firms, even if they have differ-ent productivities.

5.2 A Vertical Merger

We have focused on the market transaction between the upstream firm andthe downstream firms, and ignore vertical merger between the firms. Hence,it is implicit in our analysis that either the cost of merger4 or governmentregulation prevents vertical mergers, since vertical merger will create marketforeclosure in our model. However, if the cost of merger is not very high andgovernment allows vertical merger, it is clear that the firms may have the

4See Hart and Tirole (1990) for a discussion on the cost of vertical merger.



incentive for vertical merger, which will affect the pricing strategy of theupstream firms. However, the type of competition in the final goods marketmay affect the incentive for merger and the market outcome. The followingdiscussion shows this.

Assume that firm 1 and the upstream firm decide whether or not tomerge. If they merge, it is optimal for the upstream firm to supply its inputsonly to the merged firm at the marginal cost, which is assumed to be zero inour analysis. Hence, the vertically merged firm will produce like a monopolistand the profit of the merged firm is π v*1

2 4= a ,5 irrespective of Bertrand andCournot competition, which are relevant under vertical separation.

If the cost of vertical merger is M, under Cournot competition, if theupstream firm charges a uniform price under vertical separation, verticalmerger between firm 1 and the upstream firm occurs if

a a aI M

2 2 2

2

2 2 2

2 2 14

124 1

2 5 59 4 4 4

−+

+ −−

+ −+ −

≡ >( )

( )( )

( )λ

λ λλ λλ λ v

cu (18)

However, under Cournot competition, if the upstream firm charges dis-criminatory prices, vertical merger between firm 1 and the upstream firmoccurs if

a a aI M

2 2 2

14 6 36

− − ≡ >vcd (19)

It is clear from (18) and (19) that I Ivcd

vcu

1 1> for l ∈[1, 2], i.e. the incentive forvertical merger under Cournot competition is higher under discriminatoryprices compared with uniform pricing.

Under Bertrand competition, if the upstream firm charges a uniformprice, vertical merger between firm 1 and the upstream firm occurs if

a a a a aM

2 2 2 2 2

4 41

4 4 4− −

−= − >

λλλ

( )(20)

but if the upstream firm charges discriminatory prices, vertical mergerbetween firm 1 and the upstream firm occurs if

a aM

2 2

4 4− > (21)

5Given our assumption of zero marginal cost of input production, the profit of the verticallymerged firm is the same irrespective of a merger between the upstream firm and firm 1 or amerger between the upstream firm and firm 2. If there is a positive marginal cost of inputproduction, a merger between the upstream firm and firm 1 will create higher profit of themerged firm compared with a merger between the upstream firm and firm 2.



It is clear from (20) and (21) that vertical merger does not occur underBertrand competition, irrespective of the pricing strategy of the upstreamfirm.

Now we want to see whether the incentive for vertical merger is higherunder Bertrand competition or under Cournot competition for a particulartype of pricing strategy of the upstream firm. It follows from the abovediscussion that if I I Mv

cdvcu

1 1 0> > > , there can be the cost of vertical mergersuch that vertical merger occurs under Cournot competition but not underBertrand competition, irrespective of the pricing strategy of the upstreamfirm. In this situation, the profits of the upstream and the downstream firmstaken together are the same under Cournot competition and under Bertrandcompetition, irrespective of uniform and discriminatory pricing strategies ofthe upstream firm.

However, if I M Ivcd

vcu

1 1 0> > > , vertical integration occurs only underCournot competition with discriminatory pricing. Hence, vertical separationoccurs under both Bertrand and Cournot competition with uniform pricing.In this situation, the effects of different types of product market competitionfollow from our analysis under uniform pricing. However, under discrimin-atory pricing, vertical merger occurs under Cournot competition and theprofits of the upstream and the downstream firms taken together are the sameunder Cournot competition and under Bertrand competition.

It should be noted that the above discussion on vertical merger considersan exogenously given type of merger. That is, we assume that firm 1 and theupstream firm decide whether to merge or not, and firm 2 remains passive inthe merger process. Under this simple exogenously given merger process, weshow that the type of product market competition may affect the incentive formerger and the equilibrium outcomes. However, a detailed and a richermerger analysis based on the endogenous merger decisions of all firms andwith multiple upstream and downstream firms deserve attention. We leavethis issue for future research.

6 Conclusion

We compare the effects of different types of product market competition,namely Bertrand and Cournot competition, in a vertical structure where amonopolist upstream firm sets the input price for the downstream firms. Weshow that if the upstream firm charges a uniform price, the profit in thedownstream market can be higher under Bertrand competition if the down-stream firms are sufficiently asymmetric. However, the profits of the down-stream firms are always higher under Cournot competition if there is pricediscrimination by the upstream firm. The profit of the upstream firm, theprofits of the upstream and the downstream firms taken together, and socialwelfare are always higher under Bertrand competition.



Appendix

Condition Required for Firm U to Supply Inputs to Both Firm 1 and Firm 2under Cournot Competition

If firm U wants to supply inputs only to firm 1 (i.e. effectively the downstream marketis monopoly), the input demand is given by (5b) in the text and the optimal input priceis a/2. Since a/2 3 a/(2l - 1) for λ ≥ 3

2 , firm U wanting to supply inputs only to firm 1charges the input price a/2 for λ ∈[ , )3

2 2 and a/(2l - 1) for λ ∈[ , ]1 32 to satisfy the

restriction in (5b). The profit of firm U is a2/8 for λ ∈[ , )32 2 and [a2(l - 1)]/(2l - 1)2 for

λ ∈[ , ]1 32 . If l ∈[1, 2), both these profits are lower than [a2(1 + l)2]/[6(4 + 4l2 - 4l)],

which is the profit of firm U if it wants to supply the input to both firm 1 and firm 2 andsets the optimal input price corresponding to the input demand function (5a).

Proof of Proposition 2

(i) The profits of firm 1 under Cournot and Bertrand competition are respectively[a2(2 + 5l2 - 5l)2]/[9(4 + 4l2 - 4l)2] and a2(l - 1)/4l. Firm 1’s profit is higherunder Bertrand competition than under Cournot competition if and only if

λλ

λ λλ λ

−−

+ −+ −

>1 4 2 5 5

9 10

2 2

2 2

( )( )

(A1)

We plot the left-hand side of (A1) in Fig. A16 and the inspection of Fig. A1 proves theresult.

Fig. A1. Left-hand Side of (A1)

(ii) The industry profits under Cournot and Bertrand competition are respectively

π λ λ λ λλ λe

c =+ − + + −

+ −a a2 2 2 2 2 2

2 2

5 2 5 2 5 59 4 4 4

( ) ( )( )

(A2)

π λλe

b =−a2 1

4( )

(A3)

6We use ‘The Mathematica 4.2’ (see Wolfram, 1999) for the figures of this paper.



We subtract (A2) from (A3) and plot the difference in Fig. A2 and the inspection ofFig. A2 proves the result. �

Fig. A2. Subtracting (A2) from (A3)

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BERTRAND VERSUS COURNOT COMPETITION IN A VERTICAL STRUCTURE: A NOTE