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Symbiotic production and downstream market competition.

Introduction

The problem of bilateral monopoly has received much attention over the last 150 or so years. A bilateral monopoly is defined as a market structure with a single upstream seller and a single downstream buyer. Ever since Bowley (1928), a well-developed body of literature (such as Machlup and Taber (1960), Blair and Kaserman (1987), Blair et al. (1989), and Irmen (1997)) has examined different aspects of a bilateral monopoly. In particular, as mentioned by Machlup and Taber (1960), Tirole (1984), and Young (1991), imperfect downstream competition leads to successive markups and an inefficient result--namely, a double marginalization problem.

A frequently cited proposition in the industrial organization literature is that collusion (or vertical integration) of bilateral monopolists improves economic efficiency by eliminating double marginalization (e.g., Machlup and Taber (1960), Blair and Kaserman (1987), Blair, et al. (1989), and Irmen (1997)). Several variant contract schemes for solving the incentive problem of a bilateral monopoly are also discussed. For example, Blair and Kaserman (1987) derive an ex ante agreement whereby assigned shares of profits may eliminate the need for successive mark-ups. (1)

The history of this problem can in fact be traced back to the classic writing of Cournot (1838), but as Machlup and Taber (1960) point out, Cournot offers no model of a transaction between a single seller and a single buyer. Instead, Cournot (1838) examines the case of complementary monopolists that has been generally considered analogous to the case of bilateral monopoly. Cournot (1838) develops a simple model of complementary goods to consider the merger of two monopolists that produce complementary goods (zinc and copper) into a single (fused) monopolist that produces a combination of the two complementary goods (brass). He demonstrates that joint ownership by a single integrated monopolist reduces the sum of the two prices, relative to the equilibrium prices of the independents. This is because the two independent firms ignore the effect of their individual markups on each other (i.e., double marginalization), while integrated monopolists internalize this externality.

In their study, Carter and Wright (1994) use the concept of symbiotic production to generalize Cournot's model. Symbiotic production is a different market structure from bilateral monopoly and has two characteristics: (i) each firm produces both an intermediate input and a final good, and (ii) each firm must purchase the intermediate input from the other firms. Moreover, Carter and Wright (1994) also suggest that collusion over material prices will lead to lower output prices. (2) Nevertheless, as pointed out by Vogelsang (2003), the collusion outcome is desirable because it characterizes the noncooperative approach, and undesirable because of the monopoly pricing outcome. This simply implies that the inefficiency measured by the deadweight loss arises from double marginalization and downstream market imperfection.

Our model can be applied to the call-back services in international telecommunication. As pointed by Carter and Wright (1994), international calls are an ideal example of symbiotic production, since both local telecommunication firms are required to provide complementary intermediate goods (access of network). In practice there usually exists dispersion (arbitrage opportunities) between international rates in countries. Say the rate for international calls from a caller in the U.S. to a Country A is in general quite different from those for an international caller in Country A to the U.S. This is understandable because the local telecommunication companies set imperfect competitive prices according to the demand condition. Thus, call-back service makes it possible for a caller in a high rates location to initiate calls as if they originated in a nation with much lower rates, either by callback operators having made use of 0800 numbers or by reverse charge calling card services such as those of AT&T. (3, 4)

The vertical structures of bilateral monopoly and symbolic production are described in Fig. 1. The benchmark structure is a typical bilateral monopoly with a single upstream seller and a single downstream buyer as in the first figure. The symbiotic production with two separated markets, as in Carter and Wright (1994), is described as two monopolists that sell each other's essential intermediate goods. The markets of final output are considered as either separate (the middle figure) such as international telephones without call-back, or integrated (the bottom figure), such as the case allowing call-back services.

For the purpose of deriving analytical results, this paper assumes linear demand and constant marginal cost, and addresses the role of output market competition in vertical relations. Specifically, quantity competition in the output market slightly mitigates the effect of double marginalization, with a deadweight loss being 40.5 % of the Pareto optimum. However, double marginalization arises when output markets are independent, with a deadweight loss being 56.25 % of the Pareto optimum. We demonstrate that incorporating competition in the product market under a symbiotic production structure will partially eliminate inefficiency caused by double marginalization. More specifically, this paper suggests that incorporating quantity competition in the final market increases total profits and reduces output prices and deadweight losses, thereby improving social welfare. It suggests that introducing callback services or Internet telephones creates an environment similar to downward market competition in the output market, and hence, it is observed that international tariffs are significantly reduced.

The structure of the paper is as follows. In the next section we present the basic model of symbiotic production. Then the equilibria under different types of downside market competition are solved, and the welfare comparison among those different types of competition is provided.

[FIGURE 1 OMITTED]

The Model

Following Bowley (1928) and Carter and Wright (1994), we present the market structures of a bilateral monopoly and symbiotic production as follows. A typical structure of a bilateral monopoly is described as two stages of a game. In the first stage, the upstream firm sets the prices of the material (intermediate product) as [p.sub.x], with a production cost [phi](x). In the second stage, the downstream firm purchases x units of the material from the upstream firm and transfers each unit of the material into a unit of final product with a cost C(x). It then sells those x units of the final product to the output market. Next is the structure of symbiotic production. Each firm i, i = 1, 2, is an upstream firm in the product market j, j [not equal to] i, and also is a downstream firm in producing product i, with total production cost [C.sup.i]([x.sup.i], [x.sup.j]) and inverse demand function [P.sup.i]([x.sup.i], [x.sup.j]). It is assumed that the upstream firm can dictate the material price [p.sub.x] whereas the downstream firm determines the output x. This assumption is certainly only one particular case in Bowley (1928). Whereas various distributions of the bargaining power of setting up the material price are considered in the literature of bilateral monopoly (see, for instance, Bowley (1928) and Blair and Kaserman (1987)), the symbiotic literature focuses on the case where the downstream firm is a price taker, as in Cournot (1838), Economides and Salop (1992), Carter and Wright (1994), and Economides (1999).

In the second stage, each firm i maximizes its profits:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first-order conditions then become:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

in which implicit reaction functions [x.sup.i] ([p.sup.i.sub.x] [p.sup.j.sub.x]) - i= 1, 2, are readily obtained. Going back to the first stage, each firm i determines the material price, given the reaction functions [x.sup.i] ([p.sup.i.sub.x] [p.sup.j.sub.x]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The optimization yields the equilibrium conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Given the first bracket in the first line being equal to zero by (1), it follows that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

To obtain analytical results, we consider the simple case described with a linear inverse demand P = [alpha] - [[beta].sub.x], a constant marginal cost c for the upstream firm, a constant marginal cost c + [gamma] for the downstream firms, and given the separable conditions. Note that c is the marginal cost of the material and [gamma] is the marginal transfer cost. It is further assumed that [alpha] > 2c + [gamma] to guarantee positive profits in the market. In fact, the case of symbiotic production with independent markets is equivalent to that of two bilateral monopolies.

To further explore how the degree of the elimination of double marginalization depends on the different types of competition in a downstream market, we will first consider a particular case of symbiotic production with independent product markets as the benchmark case. Then, we will discuss several different competition structures in an integrated downstream output market. To obtain analytical results, in the following, we consider the simplest case with linear inverse demand P = [alpha] - [beta]x in the integrated downstream output market. Two identical independent markets are presented with linear demand functions [x.sup.i]([P.sup.i]) = 1/2[[alpha]/[beta] - [P.sup.i]/[beta]] or inverse demand functions [P.sup.i] = [alpha] - 2[beta][x.sup.i] , i=1, 2. In the case of symbiotic production with an integrated market, in which the total demand is the summation of the two markets, the demand function is x(P) = [x.sup.1] (P) + [x.sup.2](p) = [alpha]/[beta] - P/[beta]. (5)

Equilibrium Under Different Types of Competition in Downstream Markets

A Benchmark Case--Symbiotic Production with Two Independent Product Markets

In the second stage, the downstream firm maximizes its profits:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This yields the first-order condition x([p.sub.x]) = 1/4[beta] ([alpha] - c - [gamma] - [p.sub.x]), which also is considered the reaction function of the downstream firm. The second-order condition is obviously satisfied since [[partial derivative].sup.2][[pi].sub.d] / [partial derivative][x.sup.2] = -4[beta] < 0 Next, consider that the upstream firm maximizes its profits:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The above optimization yields the solution [p.sub.x] = [alpha] - [gamma] / 2 and x([p.sub.x]) = 1/8[beta] ([alpha] - 2c - [gamma]).

The second-order condition is obviously satisfied since [[partial derivative].sup.2][[pi].sub.y] / [partial derivative][p.sup.2] < 0 . Further calculations obtain the final product price:

[P.sup.*I] = 3 [alpha] + 2c + [gamma] / 4, (3)

where superscript I represents independent markets. We now compute the profits for upstream and downstream firms:

[[pi].sup.*I.sub.u] = [([alpha] - [gamma] - 2c).sup.2]/ 16[beta], [[pi].sup.*I.sub.d] = [([alpha] - [gamma] - 2c).sup.2]/ 32[beta].

Because markets are independent, under the symmetric case with two identical firms, the market structure is equivalent to that of two separated bilateral monopolies:

[[pi].sup.*I.sub.1] = [[pi].sup.*I.sub.2] = [[pi].sup.*I.sub.u] + [[pi].sup.*I.sub.d] = 3[([alpha] - [gamma] - 2c).sup.2]/ 32[beta]. (4)

The consumer surpluses in the two markets become:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, the consumer surpluses are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] This results in the social welfare:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Notice that the independence of the demand between the two markets is crucial to separate the symbiotic production into two bilateral monopolies. If independence no longer holds, then an arbitrage may appear. In a very extreme case, if we consider homogeneous final products, then those two markets may be considered like an integrated market as in the next section.

Quantity Competition

This subsection analyzes the equilibrium condition under quantity competition. (6) By the same token, in the second stage, Firm 1 maximizes its profits:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [p.sup.Q] is the inverse demand of the integrated market, which is equivalent to the sum of two bilateral markets as previously described, [P.sup.Q] = [alpha] - [beta] ([x.sub.1] + [x.sub.2]). Finn 2 also maximizes its profits:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

leading to the first-order conditions (7):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, the Nash equilibrium solution is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting the above solutions into the first stage, we have the first-order conditions (8):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, we now solve for the Nash equilibrium:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The final product price then becomes:

[P.sup.*Q] = 1/11 (7[alpha] + 4[gamma] + 8c). (6)

This is followed up to compute the profits for firms:

[[pi].sup.*Q.sup.1] = [[pi].sup.*Q.sup.2] = 14[([alpha] - [gamma] - 2c).sup.2] / 121[beta]. (7)

The consumer surplus can be obtained as:

[CS.sup.*Q] = 8[([alpha] - [gamma] - 2c).sup.2] / 121[beta].

This also gives the social welfare:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Cournot's Model of Complementary Goods

In Cournot's model of complementary goods, the product price is assumed to reflect the sum of material prices:

[p.sup.C] = [p.sup.1.sub.x] + [p.sup.2.sub.x] + [gamma],

where superscript C represents Cournot's original setting. Under this assumption, the maximization problem for firm i becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [x.sup.1] ([p.sup.C])+ [x.sup.2] ([p.sup.C]) = [alpha]/[beta] [p.sup.C]/[beta] is the demand function for the integrated market.

We also can compute the equilibrium material prices, output price, profits, consumer surplus, and social welfare. First is the material price:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The final product price then is:

[p.sup.*C] = 1/3 (2[alpha] + [gamma] +2c). (9)

We next calculate the profits for upstream and downstream firms and consumer surplus:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

This also achieves the social welfare:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Welfare Comparison under Different Types of Competition in Downstream Markets

Through previous discussions and further calculations, we now compare the equilibrium material price, final product price, firms' profits, consumer surplus, and social welfare, which are summarized in Table 1.

Note that assuming [alpha] > 2c + [gamma], helps rank our analytical solutions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

With the above results as a basis, we have the following two propositions.

Proposition 1 Symbiotic production under an integrated market has a lower output price and higher consumer surpluses than that under independent markets. Therefore, allowing competition in the output market reduces the inefficiency of double marginalization, that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The economic reasoning behind Proposition 1 is as follows. This proposition means that symbiotic production with an integrated market achieves a higher consumer surplus, and therefore, it dominates the symbiotic production with independent markets. This property arises because the competition effect in the output market eliminates (partly or completely) the inefficiency of double marginalization in symbiotic production. Symbiotic production with an integrated market provides a lower output price, higher consumer surplus, and, hence, higher social welfare. In the cases of quantity competition or Cournot's model of complementary goods, introducing competition increases firms' profits because competition effects mitigate the inefficiency of double marginalization. Therefore, allowing competition in the output market results in a Pareto improvement. Next, we compare different types of output market competition with an integrated product market.

Proposition 2 Concerning different market structures under symbiotic production with an integrated product market, the rankings of firms' profits, consumer surplus, and social welfare are respectively summarized as. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The performance of different market structures depends on the degree of competition effects. Competition in the output market reduces the inefficiency of double marginalization. However, quantity competition allows competition in the output market but reduces only part of the inefficiency of double marginalization. The case under Cournot's model of complementary goods is similar to the one under quantity competition.

It is worth emphasizing that the deadweight loss in symbiotic production with independent markets (two bilateral monopolies) is very significant at 56.25 %, since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Quantity competition helps reduce the inefficiency, and the deadweight loss shrinks to 1 - [W.sup.*Q]/[w.sup.*P] = 40.50%. The case of Cournot's model of complementary goods is a little worse than that of quantity competition.

Conclusions

The contribution of this study is that it provides new insight by identifying some key forces operative in the interaction between vertical integration and downstream market structure for eliminating double marginalization. In the presence of vertically integrated firms, this paper demonstrates that incorporating competition in the product market will partially resolve the inefficiency from double marginalization and market imperfection. It is shown that the worst case under a linear demand setting with constant marginal cost is an independent market, whereas Cournot's model of complementary goods is a better case but still less than quantity competition.

Our results herein lead to interesting implications for competition policy. Allowing competition in the output market significantly improves the social welfare. For instance, introducing callback services or internet telephones creates an environment similar to allowing competition in the output market, and hence, it is observed that international tariffs are significantly reduced. The policy implication from our model is that call-back services should not be prohibited. This is consistent with the conclusion by the Federal Communication Commission that "call-back advances the public interest, convenience, and necessity by promoting competition in international markets and driving down international phone rates, and that it is in the best interests of consumers around the world." It is also interesting in practice that, besides notwithstanding its recognition of this procompetitive policy, the Commission required U.S. carriers to provide call-back service in a manner that is consistent with the laws of countries in which they operate. The Commission also state that U.S.-based carriers may not offer international call-back using uncompleted call signaling in countries that have specifically prohibited this practice. Further research may involve the interaction of the call-back policies and the access fee (settlement rate) between developed and developing countries.

Similarly, for the local telecommunication services, the monopolist controls the local network and interconnects with complementary segments, such as long-distance cellular phone service providers. Whereas the literature suggests necessary regulation on two-way access pricing (Vogelsang (2003)), our results provide a natural way to resolve the inefficiency by allowing the complement segment (such as long-distance services providers) to compete with the local network for local telecommunication services. Therefore, if homogeneous services are provided with the competition, then it is possible to partially eliminate inefficiency caused by double marginalization.

It is believed that this paper has provided a general model of symbiotic production for the analysis of issues of product compatibility and networks, including postal and telecommunications services, ATMs, airline computer reservation systems (CRS), airline code sharing, and non-network markets of compatible components such as computer CPUs and peripherals, hardware, and software. This is an important area for future studies.

References

Blair, R. D., & Kaserman, D. L. (1987). A note on bilateral monopoly and formula price contracts. American Economic Review, 77(3), 460-463.

Blair, R. D., Kaserman, D. L., & Romano, R. E. (1989). A pedagogical treatment of bilateral monopoly. Southern Economic Journal, 55(4), 831-841.

Bowley, A. L. (1928). Bilateral monopoly. The Economic Journal, 38(152), 651-659.

Brueckner, J. K. (2001). The economics of international codesharing: an analysis of airline alliance. International Journal of Industrial Onganization, 19(10), 1475-1498.

Brueckner, J. K. (2004). Network structure and airline scheduling. Tire Journal of Industrial Economics, 52 (2), 291-314.

Carter, M., & Wright, J. (1994). Symbiotic production: the case of telecommunication pricing. Review of Industrial Organization, 9(4), 365-378.

Cournot A. (1838). Of the competition of producers, Chapter 7 in Researches into the Mathematical Principles of the Theory of Wealth, translated by Nathanial T. Bacon, Macmillan, New York, 1897; reprinted with notes by Irving Fisher, Macmillan, New York, 1927; reprinted, Augustus M. Kelley, New York, 1960; originally published as "Recherches sur Its principles mathematiques de la throrie des richesses," L. Hachette, Paris.

Economides, N. (1999). Quality choice and vertical integration, International Journal of Industrial Organization, 17(6), 903-914.

Economides, N., & Salop, S. C. (1992). Competition and integration among complements, and network market structure. The Journal of Industrial Economics, 40(1), 105-123.

Grecnhut, M. L., & Ohta, H. (1979). Vertical integration of successive oligopolists. American Economic Review, 69(1), 137-141.

Irmen, A. (1997). Mark-up pricing and bilateral monopoly. Economics Letters, 54(2), 179-187. Lam, P. L. (1997). Erosion of monopoly power by call-back: lessons from Hong Kong. Telecommunications Policy, 21(8), 693-695.

Liu, C. J., Mai, C. C., Lai, F. C., & Guo, W. C. (2010). Pollution, factor ownerships, and emission taxes. Atlantic Economic Journal, 38(2), 209-216.

Machlup, F., & Taber, M. (1960). Bilateral monopoly, successive monopoly, and vertical integration. Economica, 27(106), 101-119.

Sandbach, J. (1996). International telephone traffic, callback and policy implications. Telecommunications Policy, 20(7), 507-515.

Tirole, J. (1984). The theory of industrial organization. Cambridge: MIT Press.

Vogelsang, I. (2003). Price regulation of access to telecommunications networks. Journal of Economic Literature. 41(3), 830-862.

Wright, J. (2002). Access pricing under competition: an application to cellular networks. Tire Journal of Industrial Economics, 50(3), 289-316.

Young, A. R. (1991). Vertical structure and nash equilibrium: a note. The Journal of Industrial Economies, 39(6), 717-722.

W.-C. Guo ([mail]) * C.-J. Liu

Department of Economics, National Taipei University, 151, University Rd. San Shia, Taipei 23741, Taiwan

e-mail: guowc@ntu.edu.tw

C.-J. Liu

e-mail: liucj@mail.ntpu.edu.tw

F.-C. Lai

Research Center for Humanities and Social Sciences, Academia Sinica, and Department of Public Finance, National Chengchi University, Taipei, Taiwan

e-mail: uiuclai@gate.sinica.edu.tw

C.-C. Mai

Department of Industrial Economics, Tamkang University, and Research Center for Humanities and Social Sciences, Academia Sinica, Taipei, Taiwan

e-mail: ccmai@gate.sinica.edu.tw

(1) Greenhut and Ohta (1979) and Blair et al. (1989) also point out that inefficiency in the vertical relation of a bilateral monopoly can be resolved by collusion (i.e., joint profit maximization). Irmen (1997) shows an improvement in percentage markups. Economides (1999) provides a framework with the choice of quality and discusses the effects of the interaction of double marginalization.

(2) The form of symbiotic production also can be applied to other networks. For instance, Wright (2002) studies the access pricing problem in the case of cellular networks. Brueckner (2001, 2004) discusses international code sharing of airline networks. Economides and Salop (1992) generalize the Cournot duopoly complements model to the case in which there are multiple brands of compatible components. They analyze equilibrium prices for a variety of market structures that differ in their degree of competition and integration. Moreover, the implication on emission regulation is discussed in Liu et al. (2010), which suggest that emission subsidy could be socially optimal.

(3) In the U.S. (and most developed countries), the regulator, the Federal Communications Commission (FCC), allows call-back services. However, the information provided by the International Telecommunication Union (ITU) suggests that many developing countries still prohibit call-back services. Sandbach (1996) and Lam (1997) provide detailed discussions on the practice of call-back services.

(4) Call-back can also be applied to the communication between fixed-line callers and cellular subscribers. Cellular subscribers can call back fixed-line callers when the price for such calls is significantly below the price of fixed-to-mobile calls.

(5) In fact, our results may be extended in a more general framework with different sizes and slopes of the demand function under some reasonable conditions. However, to avoid unnecessary complex and technical details, we consider only the simple symmetric case. The framework of nonlinear demand in a bilateral monopoly, for instance, is considered by Young (1991).

(6) The quantity competition also is studied in Chapter 7 of Cournot (1838).

(7) The second-order condition requires [[partial derivative].sup.2] [[pi].sub.1] / [partial derivative] [x.sup.2.sub.1] / [[partial derivative].sup.2] [[pi].sub.2] / [partial derivative] [x.sup.2.sub.2] and the stability condition requires [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(8) The second-order condition requires [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the stability condition requires [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We wish to express our gratitude to an anonymous referee for the very valuable suggestions provided. All remaining errors arc ours.

Published online: 2 June 2012

DOI 10.1007/s11293-012-9322-6
Table 1 Comparison among different market structures

                               Independent Market

Notation (J)                   *I

[P.sup.1j.sub.x],              [[alpha]-[gamma]]/2
[P.sup.2j.sub.x]

[P.sup.*J]                     [3[alpha]+2c+[gamma]]/4

Total profits                  3[([alpha]-[gamma]-2c).sup.2]
                               /16[beta]

[SIGMA]i [[pi].sup.*J.sub.i]

Consumer surplus               [([alpha]-[gamma]-c).sup.2]/
                               /32[beta]

[CS.sup.*1]

Social welfare                 7[([alpha]-[gamma]-2c).sup.2]
                               /32[beta]

[W.sup.*J]

Deadweight loss                9[([alpha]-[gamma]-2c).sup.2]
                               /32[beta]

                                     Integrated market

                               Quantity Competition

Notation (J)                   *Q

[P.sup.1j.sub.x],              [5[alpha]-5[gamma]+c]/11
[P.sup.2j.sub.x]

[P.sup.*J]                     (7[alpha]+4[gamma]+8c/11

Total profits                  56[([alpha]-[gamma]-2c).sup.2]
                               /242[beta]

[SIGMA]i [[pi].sup.*J.sub.i]

Consumer surplus               16[([alpha]-[gamma]-2c).sup.2]
                               /242[beta]

[CS.sup.*1]

Social welfare                 72[([alpha]-[gamma]-2c).sup.2]
                               /242[beta]

[W.sup.*J]

Deadweight loss                49[([alpha]-[gamma]-2c).sup.2]
                               /242[beta]

                                     Integrated market

                               Cournot's model

Notation (J)                   *C

[P.sup.1j.sub.x],              [[alpha]-[gamma]+c]/3
[P.sup.2j.sub.x]

[P.sup.*J]                     (2[alpha]+[gamma]+2c)/3

Total profits                  4[([alpha]-[gamma]-2c).sup.2]
                               /18[beta]

[SIGMA]i [[pi].sup.*J.sub.i]

Consumer surplus               [([alpha]-[gamma]-2c).sup.2]
                               /18[beta]

[CS.sup.*1]

Social welfare                 5[([alpha]-[gamma]-2c).sup.2]
                               /18[beta]

[W.sup.*J]

Deadweight loss                4[([alpha]-[gamma]-2c).sup.2]
                               /18[beta]

This table illustrates that [P.sup.*Q] < [P.sup.*C] < [P.sup.*I],
[[pi].sub.i.sup.*Q] > [[pi].sub.i.sup.*C], [CS.sup.*Q] >
[CS.sup.*C] > [CS.sup.*I] and [W.sup.*Q] > [W.sup.*C] >
[W.sup.*I].
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Date:Sep 1, 2012
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