Flexible versus dedicated technology adoption in the presence of a public firm.

1. Introduction

In the recent past, many firms all over the world have substituted their traditional production processes with more flexible systems. Some of these flexible technologies allow for greater capacity (process flexibility), which can increase the ability of firms to adapt to fluctuations in demand (Boyer and Moreaux 1997; Boyer, Jacques, and Moreaux 2002). In other cases, the advantage of a flexible manufacturing system (FMS) over dedicated equipment (DE) is that the former allows a firm to supply several products and consequently to participate in different markets (in other words, to become a multiproduct or multimarket firm (1)) without having to invest in separated manufacturing processes. This is called product flexibility and is the main focus of our paper. Apart from benefits, flexible technologies also report higher set-up costs, mainly in the form of development or adjustment costs (Jaikumar 1986).

The study of the adoption of FMS by private firms was first introduced by Roller and Tombak (1990) and Kim, Roller, and Tombak (1992) in the context of oligopolistic competition. Their findings indicate that the adoption of flexible technologies requires a sufficiently low adoption cost, sufficiently high product differentiation, and large enough markets, while consumers benefit from the use of FMS due to the increase in competition. (2) In addition, Roller and Tombak (1993) validate these results with an empirical study. Dixon (1994) evaluates the welfare effects of using FMS when the marginal cost of production is increasing in the number of goods produced and when the markets are unrelated. As a result, adopting FMS might lead to welfare losses due to the inefficiency in production. On the other hand, Eaton and Schmitt (1994) point out that the adoption of FMS may correspond to preemptive strategies, leading to higher levels of concentration, in the context of horizontal product differentiation.

To the best of our knowledge, the issue of technology choice as exemplified by the adoption of FMS versus DE technologies has not been studied in the context of a mixed market where private (profit-maximizing) firms coexist with public (not-for-profit) ones. Such mixed markets are quite prevalent in transition economies, but not exclusively so; telecommunications, health services, and the postal sector in many countries are organized as mixed markets. Although many public firms have been privatized in recent years, it is worth pointing out that the behavior of these recently privatized firms remains subject to public regulation.

Our analysis is motivated by a large number of industries in which multiproduct and single-product firms coexist and the presence of public (or newly privatized but still regulated) firms is common. This is the case for industries such as energy supply, transport, telecommunications, or health care. First, consider the case of telecommunications. Traditionally, the provision of internet access, telephone, and TV services required the use of different technologies and separate production processes for each one of them. At present, however, cable technology can be used by firms to provide these three different services using the same production process, thereby enabling firms to be present in all three markets and to exploit economies of scope. In this sense, cable technology can be considered an example of FMS. (3) Interestingly, the matter raised public concerns when the technology first appeared and was made available to firms. In the UK, regulators have encouraged cable companies to provide telephone services, but have not allowed the former public operator, British Telecom, to enter the television business (Waverman and Sirel 1997). Similarly, Spanish Telefonica was not permitted to compete with cable operators for a certain period of time (Cantos-Sanchez, Monet, and Sempere 2003).

Another example draws from the health care sector. There is evidence of economies of scope (Ozcan, Luk, and Haksever 1992), which can be related to the use of FMS. There are several empirical studies stressing that public hospitals provide a wider range of services than private hospitals (Shortell et al. 1986; Shortell et al. 1987; Schlesinger et al. 1997). Moreover, public hospitals tend to provide more innovative services without competition, whereas private hospitals are more likely to add these services when there is competition (Schlesinger 1998). This body of observations suggests that both the public or private character of firms and the degree of competition among them seem to be key factors influencing the adoption of FMS (thus, the multiproduct/multimarket character of firms).

Our main contribution is to introduce the analysis of the choice of production flexibility in the context of a mixed duopoly. Our model consists of two output competing firms (one of them being public) and two markets. Following Roller and Tombak (1990) and Kim, Roller, and Tombak (1992), we assume that there is a degree of product substitutability across markets. Using a flexible technology allows firms to be present in both markets, whereas using a dedicated technology constrains firms to be present in only one of them. We aim at characterizing the market conditions (i.e., market size and substitutability) and technology cost conditions that would lead in equilibrium to the adoption of FMS as opposed to DE. For comparison purposes, we also undertake this characterization for the case of a private duopoly. We find that a configuration where both firms adopt flexible technologies requires less-demanding technology cost conditions in the mixed duopoly than in the private duopoly. A similar result occurs when both firms use a dedicated technology for very low or very high substitutability.

A natural question to address in this context relates to the potential benefits of privatizing the public firm when a flexible technology becomes available. This issue, which has been ignored so far by the literature on mixed oligopoly, is relevant from the practical and policy-making point of view. This is especially so in the light of recent liberalization trends across the world, in many cases in industries where, as exemplified before, multiproduct firms (may) coexist with single-product firms. In the absence of the issue of flexible technology adoption, the literature on mixed oligopoly has shown that privatizing a public firm would be worthy from the social welfare point of view if the public firm is less efficient than the private firm and the marginal cost of production is linear, if there is freedom of entry, or if, with economies of scale, the number of private firms is large enough (de Fraja and Delbono 1989, 1990; Estrin and de Meza 1995; Anderson, de Palma, and Thisse 1997). However, if firms' outputs are subsidized, the effects of privatization are not so positive, with welfare unaffected if firms move simultaneously (White 1996; Pal and White 1998; Poyago-Theotoky 2001, among others) or even reduced if the public leader becomes a private leader postprivatization (Fjell and Heywood 2004). In our paper, in order to isolate the issue of the strategic adoption of flexible technologies, we will abstain from introducing public subsidies. Interestingly, our results indicate that privatization is socially beneficial only when both firms in the mixed duopoly adopt FMS and products are sufficiently differentiated. As we argue later, this corresponds with market and technology conditions that grant high profitability from investing in FMS.

The plan of the paper is as follows: First, we introduce the model (section 2) and then characterize the different equilibria (section 3). Next, we consider social welfare and the question of privatization (section 4). Finally, we summarize our main findings (section 5).

2. The Model

We introduce the study of the mixed duopoly within the framework of Roller and Tombak (1990) and Kim, Roller, and Tombak (1992). Although we keep the main features of these two contributions, we also allow for decreasing returns to scale. This assumption is widely spread in the literature on mixed oligopoly and is useful in order to avoid the case of natural monopolies, which, considering the scope of our paper, is uninteresting.

Consider a duopoly competing in output and facing the choice between adopting an FMS or a DE. The use of FMS allows participation in two existing markets, A and B. The use of the DE constrains firms to be active only in one of the markets. In the case of the mixed duopoly, one of the two firms, denoted by the subscript 2, is public (not-for-profit) and acts as a social welfare maximizer. (4) Assuming that the public firm is a social welfare maximizer is in line with the majority of the literature on mixed oligopoly. (5)

The system of inverse demand functions is given by

[p.sup.A] = a - [Q.sup.A] [gamma][Q.sup.B]

and

[p.sup.B] = a - [Q.sup.B] [gamma][Q.sup.A]

where [p.sup.A] and [p.sup.B] are the prices for products A and B, respectively, [Q.sup.A] and [Q.sup.B] are the total quantities in market A and market B, respectively, and a > 0 measures market potential. The parameter [gamma], measures the substitutability of products A and B, [lambda] [member of] [0, 1); the higher [gamma], the fiercer the competition between firms across markets.

The profit of each firm is given by

[[pi].sub.i,j] = [P.sup.A][Q.sup.A.sub.i,j] + [P.sup.B] [Q.sup.B.sub.i,j] - [C.sub.i]([Q.sup.A.sub.i,j] + [Q.sup.B.sub.i,j]) - [F.sub.k],

where i denotes the firm (i = 1 or 2) and j denotes the state of the industry according to the technologies used by the two firms. In particular,

j = 1 if both firms are using FMS;

j = 2 if firm 1 is using DE and firm 2 is using FMS;

j = 3 if firm 1 is using FMS and firm 2 is using DE;

j = 4 if both firms are using DE.

[Q.sup.A.sub.i,j] and [Q.sup.B.sub.i,j] are the quantities chosen by firm i in state j for markets A and B, respectively. Without loss of generality, we assume that if only one firm is using DE, this firm competes only in market A while the other firm participates in both markets. If both firms use DE, they compete in different markets (without loss of generality, firm 1 in market A and firm 2 in market B). Thus, the use of FMS increases the degree of competition not only in the market where a firm is operating but also across markets (due to product substitutability).

[F.sub.k] are the fixed costs of firms, which are related to the use of the available manufacturing technologies; k = FMS or DE. The costs of using FMS are assumed to be higher than the costs of using DE. For simplicity, we normalize the costs of the dedicated technology to [F.sub.DE] = 1. The costs of the flexible technology are then [F.sub.FMS] = 1 + s, where s captures the extent of the cost differential between the two manufacturing technologies.

[C.sub.i] are the costs of production, which are assumed to be quadratic and separable in output:

[C.sub.i]([Q.sup.A][Q.sup.B.sub.i,j]) = [([Q.sup.A.sub.i,j]).sup.2] + [([Q.sup.B.sub.i,j]).sup.2]

The quadratic cost assumption is widely used in the literature on mixed oligopoly to avoid trivial solutions; for example, if costs are linear and firms are equally efficient, the public firm would practice marginal cost pricing and become a public monopoly, with the private firm producing nothing. Equally important, the above assumption implies that we do not consider the existence of cost complementarity or substitutability. Instead, we focus on the strategic effect of choosing FMS, leaving aside the issue of production inefficiencies arising from the use of FMS. (6) However, in our model, economies of scope appear due to subadditivity of fixed costs if 0 < s < 1. This can be seen quite easily by comparing the costs of serving the two markets by using FMS and DE. Using a flexible technology to produce the two goods yields the following costs:

[C.sub.i]([Q.sup.A.sub.i,j],[Q.sup.B.sub.i,j]) + [F.sub.FMS] = [([Q.sup.A.sub.i,j]).sup.2] + [([Q.sup.B.sub.i,j]).sup.2] + (1 + s),

whereas using a dedicated technology for each of the two goods, the costs are

[C.sub.i]([Q.sup.A.sub.i,j],[Q.sup.B.sub.i,j]) + [F.sub.DE] + [F.sub.DF] = [([Q.sup.A.sub.i,j]).sup.2] + [([Q.sup.B.sub.i,j]).sup.2] + 2,

It follows that if s < 1, it is less costly for the firm to use FMS to serve the two markets than to set up two separate dedicated plants. On the contrary, if s > 1, there are diseconomies of scope, so firms would never favor the use of FMS. In this paper, we restrict our analysis to 0 < s < 1, since for s [greater than or equal to] 1, the technology adoption issue is trivial, and the whole problem is reduced to a simple game of entries. In other words, by focusing on the case 0 < s < 1, we are implicitly considering only the cases where adopting FMS is the sole meaningful way to diversify. Hence, firms will weigh the costs (s) and the benefits of diversification when deciding on the adoption of FMS. (7)

Total surplus (TS) is the sum of consumers' surplus (CS) and producers' profits. Linear demand functions yield

CS = 1/2 ([([Q.sup.A]).sup.2] + [([Q.sup.B]).sup.2]).

Thus, TS is given by

TS = CS + [2.summation over (i=1)] [[pi].sub.i,j].

We consider two versions of a two-stage game: (i) a private duopoly and (ii) a mixed duopoly. In the first stage, firms choose which technology to adopt, FMS or DE. In the second stage firms set quantities (Cournot competition). In each of the two stages, a private firm will maximize profits, while a public firm will maximize total surplus. Decisions in each stage are taken simultaneously. (8) Given technology choices made in stage one, it is straightforward to solve the output stage. (9) We can then derive the relevant payoff functions that firms use in solving the first stage ([pi.sup.*.sub.i,j] for a private firm, [TS.sup.*.sub.i,j] for a public firm). In other words, we use subgame perfection as our equilibrium concept. In Appendix 1 we give the second-stage solutions for profits and total surplus. (10) We can then represent the technology choice stage using a matrix of payoffs such as the one in Table l, where A, B, C, and D (i.e., the payoffs of firm 2), correspond to [[pi].sub.2,1], [[pi].sub.2,3], [[pi].sub.2,2], and [[pi].sub.2,4], respectively, if firm 2 is a private firm and to [TS.sub.2,1], [TS.sub.2,3], [TS.sub.2,2], and [TS.sub.2,4], respectively, if firm 2 is a public firm. Note that for the private duopoly Table 1 is symmetric because [[pi].sub.1,1] = [[pi].sub.2,1], [[pi].sub.1,4] = [[pi].sub.1,3] = [[pi].sub.2,2], and [[pi].sub.1,2] = [[pi].sub.2,3].

3. Equilibria Characterization

In this section, we examine the conditions that guarantee one of the four possible pure-strategy equilibria, that is, (FMS, FMS), (DE, DE), (FMS, DE), and (DE, FMS), in each of the regimes, private or mixed duopoly. Using Table 1, we find the critical value of the technology costs, s, above which investment in FMS becomes unprofitable and compare this critical value across the two regimes. All proofs to lemmata and propositions in this section are included in Appendix 2.

The (FMS, FMS) Equilibrium

Private Duopoly

From Table 1, it is clear that (FMS, FMS) is an equilibrium when (i) [[pi].sup.*sub.1,1] - [[pi].sup.*.sub.1,2] [greater than or equal to] 0 for firm 1 and (ii) [[pi].sup.*.sub.2,1] - [[pi].sup.*.sub.2,3] [greater than or equal to] 0 for firm 2. Using the model outlined previously, these conditions are equivalent to

[a.sup.2](3 + 2[gamma])/[(4 + 3[gamma]).sup.2] - 3[a.sup.2][(2[[gamma].sup.2] + [gamma] - 6).sup.2]/2 [(24 - 11[[gamma].sup.2).sup.2] - s [greater than or equal to] 0.

Let [[sigma].sub.l] denote the critical level in (the difference in) fixed costs s that makes the above expression a strict equality. If s is lower than this critical value [[sigma].sub.1], then both firms will choose FMS, as it improves their profits. From the above expression this critical value is

[[sigma].sub.1] = [a.sup.2][f.sub.1]([gamma])/ 2[(4 + 3[gamma]).sup.2][(24 - 11[[gamma].sup.2]).sup.2], (1)

where [f.sub.1]([gamma]) = 1728 + 288[gamma] - 2172 [[gamma].sup.2] - 324[[gamma].sup.3] + 867[[gamma].sup.4] + 88[[gamma].sup.5] - 108[[gamma].sup.6] > 0. Note that the critical value is increasing in market size, [partial derivative][[sigma].sub.1]/ [partial derivative][[sigma].sub.a] > 0, while it is decreasing in product substitutability, [partial derivative][[sigma].sub.1]/[partial derivative][gamma] < 0. The larger market for either product makes firms wish to participate in flexible production in order to serve both markets. With a low degree of substitutability (small [gamma]), firms' products are perceived as highly differentiated by consumers so that a firm that opts for a dedicated production process (DE) and thus serves only one market effectively loses out. Hence, a larger market size and greater product differentiation point towards the adoption of FMS by the firms. (11)

Mixed Duopoly

From Table 1, (FMS, FMS) is an equilibrium if (i) [[pi].sup.*].sub.1,1] - [[pi].sup.*.sub.1,2] [greater than or equal to] 0 for firm 1 and (ii) [TS.sup.*.sub.1,1] - [TS.sup.*.sub.2,3] [greater than or equal to] 0 for firm 2. The first condition yields

[a.sup.2] (3 + 2[gamma])/[(5 + 2[gamma]).sup.2] - s [greater than or equal to] 0,

which implies a corresponding critical value for s, denoted

[[sigma].sub.2] = [a.sup.2][f.sub.2]([gamma])/50[(5 + 2[gamma]).sup.2], (2)

where [f.sub.2]([gamma]) = 75 + 40[gamma] - 12[[gamma].sup.2] > 0. The second condition is equivalent to

2[a.sup.2](8 + 5[gamma] + [[gamma].sup.2])/(1 + [gamma]) [(5 + 2[gamma]).sup.2] - 2[a.sup.2] (61 - 58[gamma] - 12[[gamma].sup.2] + 16[[gamma].sup.3])/[(15 - 8[[gamma].sup.2]).sup.2] - s [greater than or equal to] 0,

implying an associated critical value for s,

[[sigma].sub.3] = 2[a.sup.2][f.sub.3] ([gamma])/(1 + [gamma])[(5 + 2[gamma]).sup.2][(8[[gamma].sup.2] - 15).sup.2], (3)

where [f.sub.3]([gamma]) = 275 - 1703[gamma] - 2493[[gamma].sup.2] + 883[[gamma].sup.3] + 72[[gamma].sup.4] - 16[[gamma].sup.5] > 0. It is easy to establish that [partial derivative][[sigma].sub.2]/[partial derivative]a > 0, [partial derivative][[sigma].sub.3]/[partial derivative]a > 0, [partial derivative][[sigma].sub.2]/[partial derivative][gamma] < 0, and [partial derivative][[sigma].sub.3]/[partial derivative][gamma] < 0. A larger market (higher a) supports a larger critical difference in the fixed costs of the two different types of technology, whereas increased product substitutability (higher [gamma]) has the opposite effect. Taking the two conditions together implies that (FMS, FMS) is an equilibrium when s < [[sigma].sub.2] and s < [[sigma].sub.3], but it is not an equilibrium if s > [[sigma].sub.2] or s > [[sigma].sub.3]. We then state the following lemma:

LEMMA 1. In the mixed duopoly, (FMS, FMS) is an equilibrium if s < min{[[sigma.sub.2], [[sigma].sub.3]}. In particular, given market size a, there exists a critical value [[gamma].sup.*] such that for [gamma] < [[gamma].sup.*], (FMS, FMS) is an equilibrium ifs < [[sigma].sub.2], and for [gamma] > [[gamma].sup.*], (FMS, FMS) is an equilibrium ifs < [[sigma].sub.3]. This critical value is [[gamma].sup.*] = 0.2432.

This result implies that under low levels of competition the private firm is less likely to have a multiproduct profile than the public firm ([[sigma].sup.2] < [[sigma].sub.3] for [gamma] < [[gamma].sup.*]). On the other hand, the opposite happens for high degrees of competition ([[sigma].sub.2] > [[sigma].sub.3] for [gamma] > [[gamma].sup.*]). (12) Having analyzed both the private and mixed duopoly cases, we now proceed to a simple comparison of the two regimes. First, we consider the conditions for an (FMS, FMS) equilibrium to occur, that is, we compare the three critical levels of fixed costs, [[sigma].sub.1], [[sigma].sub.2], and [[sigma].sub.3] (see Eqns. 1-3). The following proposition summarizes our results regarding the (FMS, FMS) equilibrium.

[FIGURE 1 OMITTED]

PROPOSITION 1. For given [gamma] [member of] [0, 1) and any a > 0, the critical value for the fixed technology costs s is lower in the mixed duopoly than in the private duopoly, that is, min {[[sigma].sub.2], [[sigma].sub.3]} < [[sigma].sub.1]. Hence, from the necessary conditions for an (FMS, FMS) equilibrium:

(i) if s < min {[[sigma].sub.2], [[sigma].sub.3]}, then (FMS, FMS) is an equilibrium in both the mixed and private duopolies;

(ii) if min {[[sigma].sub.2], [[sigma].sub.3]} < s < [[sigma].sub.1], then (FMS, FMS) is an equilibrium in the private duopoly but not in the mixed duopoly;

(iii) if [[sigma].sub.1] < s, then (FMS, FMS) is not an equilibrium.

Proposition 1 implies that an equilibrium in (FMS, FMS) is more likely to arise in a private duopoly than in a mixed duopoly (i.e., it requires less-demanding conditions of the technology costs and size of the market). Even if this result might seem surprising, the intuition behind it is clear. First, consider the case with relatively high substitutability between products. In such a case, the public firm is less inclined to invest in FMS because it is less profitable and also socially not meaningful: Investing in FMS would imply bearing the higher technology costs in order to produce a new good that is perceived by consumers to be a very close substitute to the one already produced by the private firm. (13) Second, consider the case of relatively low substitutability. Here, the public firm produces more in each market to compensate for the low substitutability between products, thereby making it less profitable for the private firm to invest in technology adoption; in essence, the public firm crowds out the private firm's investment.

Proposition 1 is illustrated in Figure 1. The figure depicts [[sigma].sub.1], [[sigma].sub.2], and [[sigma].sub.3] for given a. (14) The area below [[sigma].sub.1] represents combinations of s and [gamma] that guarantee an (FMS, FMS) equilibrium in the private duopoly, and the area below the minimum of [[sigma].sub.2] and [[sigma].sub.3] represents equivalent combinations for the mixed duopoly. Therefore, the shadowed area represents parameter combinations that make (FMS, FMS) an equilibrium in the private, but not the mixed, duopoly. This indicates that, for a given size of the market and product differentiation, lower values of the technology adoption costs correspond to an (FMS, FMS) equilibrium in the mixed duopoly.

The (DE, DE) Equilibrium

Private Duopoly

In the case of the private duopoly, the conditions for (DE, DE) to be an equilibrium (see Table 1) are (i) [[pi].sup.*.sub.1,4]-[[pi].sup.*.sub.1,3] > 0 and (ii) [[pi].sup.*.sub.2,4]-[[pi].sup.*.sub.2,2] > 0 for firms 1 and 2, respectively, implying

- 3[a.sup.2]/2[(3 + [gamma]).sup.2] + [a.sup.2](300 - 276 [gamma] - 85[[gamma].sup.2] +122[[gamma].sup.3] - 21[[gamma].sup.4])/2[(24- 11[[gamma].sup.2]).sup.2]

Letting [[sigma].sub.4] denote the relevant critical value for s in this case, we obtain from the above expression

[[sigma].sub.4] = [[alpha].sup.2][f.sub.4]([gamma])/2[(3 + [gamma]).sup.2][(11[[gamma].sup.2]-24).sup.2], (4)

where [f.sub.4]([gamma]) = 972 - 684[gamma] - 537[[gamma].sup.2] + 312[[gamma].sup.3] + 95[[gamma].sup.4]- 4 [[gamma].sup.5]- 21[[gamma].sup.6] > 0. Ifs is greater than this critical value, [[sigma].sub.4], then (DE, DE) is an equilibrium. It is obvious that this critical value is increasing in market size, [partial derivative][[sigma].sub.4]/[partial derivative][gamma] > 0, and it can be easily established that it is decreasing in the product differentiation parameter, [partial derivative][[sigma].sub.4]/[partial derivative][gamma] < 0. Consequently, (DE, DE) is an equilibrium for relatively smaller a and higher [gamma]. The intuition behind this is clear, since the opposite to the FMS case holds: The smaller the market for either product, the less willing a firm is to participate in flexible technology adoption in order to serve both markets. With a high degree of substitutability (high [gamma]), firms' products are perceived as close substitutes by consumers so that a firm that opts for FMS is bearing a high fixed cost to produce two goods that are almost the same. Hence, a smaller market size and lower product differentiation point toward the adoption of DE by the firms, given s.

Mixed Duopoly

From Table 1, the conditions ensuring that (DE, DE) is an equilibrium are (i) [[pi].sup.*.sub.1,4] - [[pi].sup.*.sub.1,3] > 0 (for the private firm) and (ii) [TS.sup.*.sub.2,4]-[TS.sup.*.sub.2,2] > 0 (for the public firm). The first condition can be written as

3[a.sup.2][(2-[gamma]).sup.2]/8[([[gamma].sup.2]-3).sup.2] - [a.sup.2] (51 - 48[gamma] - 14[[gamma].sup.2] + 16[[gamma].sup.3])/[(15 - 8[[gamma].sup.2]).sup.2] + s [greater than or equal to] 0,

implying that the associated critical value for s is

[[sigma].sub.5] = a.sup.2][f.sub.5]([gamma])/ 8[([[gamma].sup.2] - 3).sup.2][(8[[gamma].sup.2] - 15).sup.2], (5)

where

[f.sub.5]([gamma]) = 972 - 756[gamma] - 1251[[gamma].sup.2] + 576[[gamma].sup.3] + 1032[[gamma].sup.4] - 384[[gamma].sup.5] - 304[[gamma].sup.6] + 128[[gamma].sup.7] > 0.

From the second condition we obtain

[a.sup.2](-57 + 60[gamma] - 4[[gamma].sup.2])/100(-1 + [[gamma].sup.2]) - [a.sup.2](17 - 14[gamma] - [[gamma].sup.2] + 2[[gamma].sup.3])/4[(-3 + [[gamma].sup.2]).sup.2]

with associated critical value

[[sigma].sub.6] = [alpha].sup.2][f.sub.6]([gamma]/ 50[([[gamma].sup.2] - 3).sup.2](1 - [[gamma].sup.2]), (6)

where [f.sub.6]([gamma]) = 44 - 95[gamma] + 72[[gamma].sup.2] - 20[[gamma].sup.3] + 4[[gamma].sup.4] - 5[[gamma].sup.5] + 2[[gamma].sup.6] > 0. Notice that [partial derivative][[sigma].sub.5]/ [partial derivative]a > 0 and [partial derivative][[sigma].sub.6]/[partial derivative]a > 0, while it is relatively easy to check that [partial derivative][[sigma].sub.5]/[partial derivative][gamma], < 0 and [partial derivative][[sigma].sub.6]/[partial derivative][gamma] [??] 0 as [gamma] [??] 0.6669.

Therefore a (DE, DE) equilibrium occurs when both s > [[sigma].sub.5] and s > [[sigma].sub.6]. The following Lemma establishes that the latter inequality is sufficient for a (DE, DE) equilibrium; that is, the critical value in the mixed duopoly is the one corresponding to the public firm.

LEMMA 2. In the mixed duopoly, (DE, DE) is an equilibrium if s > [[sigma].sub.6] for all [gamma] [member of] [0, 1).

In line with the discussion of the (FMS, FMS) equilibrium, we now proceed in comparing the private and mixed duopolies in terms of the critical values for the difference in fixed costs as well as characterizing the (DE, DE) equilibrium.

LEMMA 3. Comparing the critical values for the private duopoly, [[sigma].sub.4], and the mixed duopoly, [[sigma].sub.6], we have: [[sigma].sub.4] [greater than or equal to] [[sigma].sub.6] for [[gamma].sub.1] [less than or equal to] [gamma] [less than or equal to] and [[sigma].sub.4] < [[sigma].sub.6] for 0 [less than or equal to] [gamma] < [[gamma].sub.1] and [[gamma].sub.2] < [gamma] < 1, where [[gamma].sub.1] = 0.0056 and [[gamma].sub.2] = 0.6755.

We summarize the results obtained in this subsection in the following proposition:

PROPOSITION 2. (a) For given a > 0 and [[gamma].sub.1] [less than or equal to] [gamma] [less than or equal to] [[gamma].sub.2]:

(i) if s > [[sigma].sub.4], then (DE, DE) is an equilibrium in both the mixed and private duopolies;

(ii) if [[sigma].sub.4] > s > [[sigma].sub.6], then (DE, DE) is an equilibrium in the mixed duopoly but not in the private duopoly;

(iii) if [[sigma].sub.6] > s, then (DE, DE) is not an equilibrium.

(b) For given a > 0, 0 < [gamma] < [[gamma].sub.1], and [[gamma].sub.2] < [gamma], < 1:

(i) ifs > [[sigma].sub.6], then (DE, DE) is an equilibrium in both the mixed and the private duopolies;

(ii) if [[sigma].sub.6] > s > [[sigma].sub.4], then (DE, DE) is an equilibrium in the private but not the mixed duopoly;

(iii) if [[sigma].sub.4] > s, then (DE, DE) is not an equilibrium.

Figure 2 illustrates Proposition 2 for given a. The white area above [[sigma].sub.4] and [[sigma].sub.6] represents combinations of the parameters s and [gamma] such that a (DE, DE) equilibrium exists for both versions of duopoly. The dark-shadowed area represents combinations that guarantee a (DE, DE) equilibrium in the private duopoly but not in the mixed one. Finally, the light-shadowed area represents parameter combinations that make (DE, DE) an equilibrium in the mixed duopoly only.

[FIGURE 2 OMITTED]

Figure 2 shows that the necessary conditions for a (DE, DE) equilibrium are more stringent in the case of the mixed duopoly for low and relatively high values of substitutability. For low values of substitutability, that is, when products are perceived as highly differentiated by consumers, there is a strong incentive for the public firm to serve both markets and so increase the degree of competition. Thus, a (DE, DE) equilibrium is less likely in the mixed duopoly. For high values of substitutability, because the degree of competition across markets is already very high, either firm in the private duopoly is willing to adopt DE as a way of dampening down competition, provided that its counterpart behaves in the same way. Meanwhile, in the case of the mixed duopoly, if the private firm uses DE, the public firm has strong incentives to adopt FMS in order to increase the degree of competition. For intermediate values of product substitutability, a (DE, DE) equilibrium is more prevalent in the mixed duopoly.

The (DE, FMS) and (FMS, DE) Equilibria

Private Duopoly

From Table 1, (DE, FMS) is an equilibrium when (i) [[pi].sup.*.sub.1,1]-[[pi].sup.*.sub.1,2] [less than or equal to] 0 for firm 1 and (ii) [[pi].sup.*.sub.2,4] - [[pi].sup.*.sub.2,2] < 0 for firm 2. The two conditions taken together imply that if [[sigma].sub.1] < s < [[sigma].sub.4] (DE, FMS) is a Nash equilibrium in the case of a private duopoly. Given symmetry, it follows that (FMS, DE) is an equilibrium under the same conditions as (DE, FMS). Thus, if [[sigma].sub.1] < s < [[sigma].sub.4], there are two Nash equilibria. We then state the following lemma.

LEMMA 4. In the private duopoly, (DE, FMS) and (FMS, DE) are Nash equilibria if [[sigma].sub.1] < s < [[sigma].sub.4]. In particular, given market size a, there exists a critical value [[gamma].sup.**] such that if [gamma] > [[gamma].sup.**], then [[sigma].sub.4] > [[sigma].sub.1] and, therefore, (DE, FMS) and (DE, FMS) are Nash equilibria. This critical value is [[gamma].sup.**] = 0.6442.

It is interesting to note that only relatively high values of product substitutability guarantee the existence of asymmetric equilibria (in the sense that firms make differing technology choices). (15) Intuitively, when there is high substitutability across markets, there are situations in which technology costs are high enough to make unprofitable the investment in FMS when the opponent is present in the two markets; whereas, they are not high enough to make the investment unprofitable when the counterpart is only present in one of the two markets. In such circumstances, the equilibrium outcome will be asymmetric. (16)

Further, it is relevant to remark that for [gamma] < [[gamma].sup.**], the conditions for an equilibrium in (FMS, FMS), that is s < [[sigma].sub.1] and in (DE, DE), that is s > [[sigma].sub.4], may hold at the same time, since [[sigma].sub.4] > [[sigma].sub.1] for that range of values of [gamma]. Therefore, if [gamma] < 0.6442 and [[sigma].sub.4] < s [[sigma].sub.1], there is multiplicity of equilibria; although, (DE, DE) will be preferred from the point of view of the firms, as it provides higher profits for each of them. (17)

Mixed Duopoly

We begin with the analysis of the (DE, FMS) equilibrium. In this case, from Table 1, the necessary conditions are (i) [[pi].sup.*.sub.1,1]-[[pi].sup.*.sub.1,2] and (ii) [TS.sup.*.sub.2,4] - [TS.sup.*.sub.2,2] < 0, implying that if [[sigma].sub.2] < s < [[sigma].sub.6], (DE, FMS) is a Nash equilibrium in the mixed duopoly.

LEMMA 5. (DE, FMS) is a Nash equilibrium in the mixed duopoly only if [[sigma].sub.2] < s < [[sigma].sub.6]. This is satisfied for values of the substitutability parameter [gamma] [less than or equal to] 0.3133 or [gamma] [greater than or equal to] 0.8172. For [gamma] [member of] (0.3133, 0.8172), (DE, FMS) is not an equilibrium.

Interestingly, for very large values of the substitutability parameter ([gamma] > 0.88196), an equilibrium in (DE, FMS) would result in negative profits for the public firm. This is true for any value of the market size parameter a. The intuition behind this situation can be summarized as follows: In a case like this, the intensity of competition faced by the private firm is very high (due to the high value of [gamma] and the presence of the public firm in both markets). As a consequence, not to aggravate the competition problem and bring the prices further down, the private firm produces "too little" from the social welfare point of view. As a reply and in order to maximize total surplus, the public firm tends to "overproduce" and incurs losses.

Next, we consider the case of the (FMS, DE) equilibrium. So that (FMS, DE) is an equilibrium, it is required that (i) [[pi].sup.*.sub.1,4]-[[pi].sup.*.sub.1,3] < 0 and (ii) [TS.sup.*.sub.2,3] - [TS.sup.*.sub.2,1] > 0, implying that [[sigma].sub.3] < s < [[sigma].sub.5] must hold.

LEMMA 6. (FMS, DE) is a Nash equilibrium in the mixed duopoly if [[sigma].sub.3] < s < [[sigma].sub.5]. In particular, given market size, a, there exists a critical value [[gamma].sup.***] such that for ? > [[gamma].sup.***] (FMS, DE) is an equilibrium. This critical value is [[gamma].sup.***] = 0.3133.

Interestingly, we can show that given a set of market and technology conditions (a, s, and [gamma]), asymmetric equilibria never arise simultaneously in the private and in the mixed duopoly.

PROPOSITION 3.

(i) For given a > 0 and [gamma] > [[gamma].sup.**], if [[sigma].sub.1] < s < [[sigma].sub.4], (FMS, DE) and (DE, FMS) are equilibria in the private duopoly, but not in the mixed duopoly;

(ii) For given a > 0 and [gamma] [not member of] (0.3133, 0.8172), if [[sigma].sub.2] < s < [[sigma].sub.6], then(DE, FMS) is an equilibrium in the mixed duopoly, but not in the private duopoly;

(iii) For given a > 0 and [gamma] > [[gamma].sup.***], if [[sigma].sub.3] < s [[sigma].sub.5], then (FMS, DE) is an equilibrium in the mixed duopoly, but not in the private duopoly.

In other words, the space of market and technology conditions required for an asymmetric equilibrium to arise in the private duopoly does not overlap with any of the two (one for (FMS, DE), the other for (DE, FMS)) spaces of market and technology conditions required in the mixed duopoly.

4. Welfare Analysis: Is Privatization Beneficial?

In this section, we examine social welfare across the two market arrangements. In doing so, we address the question of privatization of the public firm. Obviously, privatization is beneficial only if it leads to an increase in social welfare (total surplus).

Note that firms might make a different technology choice in the different regimes. In other words, the technology choice equilibrium outcomes of the mixed duopoly might differ from those of the private duopoly under the same market and technology conditions, as has already been shown in Propositions 1-3. Therefore, in order to make a valid and meaningful comparison of the effects of privatization, we need to identify for given sets of market and technology conditions the resulting technological configuration in equilibrium in the mixed and private duopoly (as they might differ from each other) and compare the resulting equilibrium levels of total surplus across the two regimes. (18) This is what we have done in our welfare analysis, details of which can be found in Appendix 3.

The following proposition summarizes our findings:

PROPOSITION 4. Privatization is beneficial in that it increases social welfare when the equilibrium outcome in the mixed duopoly is (FMS, FMS) and [gamma] > 0.0223. In the remaining cases, privatization of the public firm is detrimental, as it would reduce social welfare.

PROOF OF PROPOSITION 4. This follows from Lemmata 7-12. See Appendix 3 for details. QED.

The results we have obtained regarding welfare comparisons across the two market arrangements have some potential policy implications for the debate about the privatization of a public firm. Privatizing the public firm, that is, switching from a mixed duopoly to a private one, would only enhance social welfare when the outcome in the mixed duopoly is (FMS, FMS), that is, both firms are adopting flexibility in their production, provided that products are not (almost) independent. The private duopoly equilibrium outcome would also be (FMS, FMS), but would result in higher levels of social welfare. In all other cases, a privatization would result in a reduction in social welfare. In fact, the underlying conditions for the (FMS, FMS) equilibrium to arise in a mixed duopoly imply high potential profitability from using the

technology (low technology costs, relative to the size of the market and/or the degree of substitutability between markets). (19) Therefore, larger markets (large a), lower technology costs, and lower substitutability across markets (except when markets are almost independent) point towards the beneficial effects of the privatization of public firms.

Our main result in this section is easier to interpret if we explore what a social planner would choose in both the private duopoly and in the mixed duopoly case. This is tedious but straightforward to do and requires ranking the total surplus expressions from the appendices for the private and the mixed duopoly cases. (20) Interestingly, in the private duopoly, net of s, for any a and [gamma], the highest level of welfare is provided by (FMS, FMS) and the lowest by (DE, DE). It follows that for low technology costs, the social planner would choose (FMS, FMS), and as the technology costs increase it would move towards the asymmetric configuration and if the costs increase further, towards the (DE, DE) configuration. In the case of the mixed duopoly, the optimal choice for the social planner is less straightforward. In fact, net of s, the preferred outcome would be (FMS, FMS) only for almost independent goods. For the rest of the range of values of T, (DE, FMS) would be preferred instead. This indicates that, unless there is a very low degree of competition across markets, "a lot of" flexibility in the mixed duopoly is "too much" in the view of the social planner. In those cases, a privatization is beneficial. The relative strength of Proposition 4 in terms of its policy implications is derived from the fact that it can be used even without knowing the exact values of a, [gamma], and s. It seems quite plausible to assume that policy makers know accurately the strategic plans of public firms, in this case the FMS investment plan in technology choice and the closeness between the markets/goods. If the public firm does not have any intention of replacing DE with FMS, then privatizing it should not be considered.

5. Concluding Remarks

In this paper we have introduced a mixed duopoly in the context of a differentiated product, quantity-setting duopoly facing the decision of whether to adopt a flexible technology (and become a multiproduct or multimarket firm) or a dedicated technology. We have also studied the equivalent private duopoly so as to compare the outcomes of the two different market arrangements and provide some tentative policy guidelines on the privatization of a public firm. In doing this we have combined two different matters, technology adoption (or product flexibility) and the presence of a private versus a public firm, in a single model. Although we have used a simple model to do this, it nevertheless became quite complex to solve. However, we have been able to derive policy implications as to the desirability of pursuing the privatization of the public firm. Our main findings can be summarized as follows: Flexibility is encouraged by low technology costs, large market sizes, and (generally) high degrees of differentiation. An equilibrium with both firms choosing flexible technologies is more likely to arise in the case of the private duopoly. Further, an equilibrium involving the two firms using dedicated technologies is also more likely to arise in the private duopoly when products are very close substitutes or almost independent. Mixed (asymmetric) equilibria with one firm being flexible and the other dedicated are less likely to be obtained in the private duopoly. In the case of a mixed duopoly, the public firm chooses a dedicated technology when products are very close substitutes, because it is not socially profitable to bear higher technology costs in order to produce almost the same good.

Privatization of the public firm is warranted, that is, beneficial, when the market and technology conditions lead to an equilibrium outcome where both firms use flexible technologies and goods are not (almost) independent. The underlying conditions for this equilibrium to arise imply high potential profitability (low technology costs relative to the size of the market and/or the degree of substitutability between markets). In all remaining cases, privatizing the public firm would result in a reduction of social welfare. Thus, our results provide limited support for privatizing the public firm. However, a word of caution is needed here. The results we obtain are based on a simple duopoly model, with linear demand and quadratic costs. It would be interesting to examine the robustness of the model's predictions in a more general setting of an oligopoly with general demand and cost functions and whether the results are sensitive to the mode of competition (quantity vs. price). It would also be relevant to study the adoption of flexible technologies when firms can endogenously determine the degree of product differentiation. We leave the study of these issues for future research.

Appendix 1: Equilibrium Solutions

Private Duopoly

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Mixed Duopoly

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Appendix 2: Equilibria Characterization. Proofs.

PROOF OF LEMMA 1. Note that [partial derivative][sigma].sub.2]/[partial derivative][gamma] < 0 and [partial derivative][[sigma].sub.3]/[partial derivative][gamma] < 0. Further, from Equations 2 and 3, we obtain [[sigma].sub.2] [absolute value of [sub.[gamma]=0] = [0.06a.sup.2], [[sigma].sup.2]][sub.[gamma][right arrow] 1] = [0.042a[.sup.2], [[sigma].sub.3][absolute value of [sub.[gamma]=0] = [0.0977a.sup.2], [[sigma]][sub.3]][sub.[gamma][right arrow]1 = 0, and [[sigma].sub.3][absolute value of [sub.[gamma]=0] > [[sigma].sub.]][sub.[gamma]=0, > [[sigma].sub.2]][sub.[gamma]=0], while [[sigma].sub.2] [absolute value of [sub.[gamma][right arrow] 1 > [[sigma].sub.3]][sub.[gamma][right arrow]1] = 0. Therefore, [[sigma].sub.2] and [[sigma].sub.3] must cross. Setting Equations 2 and 3 equal we obtain [[gamma].sup.*] = 0.2432, where [[sigma].sub.2] and [[sigma].sub.3] cross. The result then follows immediately. QED.

PROOF OF PROPOSITION 1. Lemma 1 establishes that the relevant critical value for s in the mixed duopoly is min {[[sigma].sub.2], [[sigma].sub.3]}; in particular, for [gamma] < [[gamma].sup.*] the relevant critical value is given by [[sigma].sub.2], and for [gamma] [greater than or equal to] [[gamma].sup.*] it is given by [[sigma].sub.3], [[gamma].sup.*] = 0.2432. Thus, we need to show that [[sigma].sub.2] < [[sigma].sub.1] for [gamma] < [[gamma].sup.*] and [[sigma].sub.3] < [[sigma].sub.1] for [gamma] [greater than or equal to] [[gamma].sup.*]. Note that [partial

derivative][[sigma].sub.1]/[partial derivative][gamma] < 0, [partial derivative][[sigma].sub.2]/partial derivative][gamma] < 0, and [partial derivative][[sigma].sub.3]/[partial derivative][gamma] < 0. Further, from Equations 1 and 2, we obtain [[sigma].sub.1][[absolute value of [sub.[gamma]=0] = 0.0937[a.sup.2] and [[sigma].sub.2]], respectively. [[sigma].sub.1] = [[sigma].sub.2] at [gamma] = 0.4593 > [[gamma].sup.*] and [[sigma].sub.2][absolute value of [sub.[gamma]=0] < [[sigma].sub.1]][sub.[gamma]=0]. Therefore, [[sigma].sub.2] < [[sigma].sub.1] when [gamma] < [[gamma].sup.*]. Similarly, from Equations 1 and 3 we obtain [[sigma].sub.1][[absolute value of [sub.[gamma][right arrow]1] = 0.0221[a.sup.2] and [[sigma].sub.3]][sub.[gamma][right arrow]1] = 0, respectively. [[sigma].sub.1] = [[sigma].sub.3] at [gamma] = 0.0393 < [[gamma].sup.*] and [[sigma].sub.3][absolute value of [sub.[gamma][right arrow]l < [[sigma].sub.1]1][sub.[gamma][right arrow]1. Therefore, [[sigma].sub.3] < [[sigma].sub.1] when [gamma] > [[gamma].sup.*], and we have shown that min{[[sigma].sub.2], [[sigma].sub.3]} < [[sigma].sub.1]. The rest of the proposition follows from the relevant equilibrium conditions. QED.

PROOF OF LEMMA 2. From Equations 5 and 6,

[[sigma].sub.6] - [[sigma].sub.5] [a.sup.2][f.sub.5,6][([gamma])/200([[gamma].sup.2]-3).sup.2]([[gamma].sup.2] 1)[(8[[gamma].sup.2] - 15).sup.2].

This is positive as [f.sub.5,6]([gamma]) < 0, where

[f.sub.5.6]([gamma]) = -15300 + 66600[gamma], - 78135[[gamma].sup.2] - 39900[[gamma].sup.3] + 111331[[gamma].sup.4] 14380[[gamma].sup.5] - 49792[[gamma].sup.6] + 13120[[gamma].sup.7] - 1920[[gamma].sup.9] - 512[[gamma].sup.10], and the denominator is negative as [lim.sub.[gamma][right arrow]1] < 0. QED.

PROOF OF LEMMA 3. Note that [[sigma].sub.4][absolute value of [sub.[gamma]=0] = 0.0937[a.sup.2], [[sigma].sub.6]][sub.[gamma]=0] = 0.0978[a.sup.2], [[sigma].sub.4][absolute value of [sub.[gamma]=1] = 0.0246[a.sup.2], and [lim.sub.[gamma][right arrow]1] [[sigma].sub.6] = [infinity]. Therefore, [[sigma].sub.4][absolute value of [sub.[gamma]=0] < [[sigma].sub.6]][sub.[gamma]=0] and [[sigma].sub.4] | [sub.[gamma]=1] < [lim.sub.[gamma][right arrow] 1 [[sigma].sub.6] = [infinity]. [[sigma].sub.6] reaches its minimum at [gamma] = 0.6689, whereas [[sigma].sub.4] [absolute value of [sub.[gamma]=0.6689] = 0.0393[a.sup.2] and [[sigma].sub.6][sub.[gamma]=0.6689] = 0.0388[a.sup.2], meaning that [[sigma].sub.4][absolute value of [sub.[gamma]=0.6689] > [[sigma].sub.6]][sub.[gamma]=0.6689. Hence, [[sigma].sub.4] and [[sigma].sub.6] must cross twice: Setting [[sigma].sub.4] and [[sigma].sub.6] equal, we find that they cross at [gamma].sub.1] = 0.0056 and at [[gamma].sub.2] = 0.6755. The rest of the lemma follows. QED.

PROOF OF PROPOSITION 2. Follows from Lemma 3 and the necessary conditions for equilibrium. QED.

PROOF OF LEMMA 4. Using Equations 1 and 4 we obtain

[[sigma].sub.1] - [[sigma].sub.4] = [a.sup.2][gamma][f.sub.1,4]([gamma])/2[(3 + [gamma]).sup.2][(4 + 3[gamma]).sup.2][(24 - 11[[gamma].sup.2]).sup.2],

where

[f.sub.l,4]([gamma]) = 576 + 168[gamma] - 1608[[gamma].sup.2] - 488[[gamma].sup.3] + 646[[gamma].sup.4] 20[[gamma].sup.6] + 81[[gamma].sup.7] [??] 0 for [gamma] > [[gamma].sup.**] = 0.6442.

The rest of the lemma follows immediately. QED.

PROOF OF LEMMA 5. Note that [[sigma].sub.6] [absolute value of [sub.[gamma]=0] = 0.1[a.sup.2], [[sigma].sub.2][sub.[gamma]=0] = 0.06[a.sup.2], [[sigma].sub.6] [absolute value of [gamma][right arrow]1] = [infinity], and [[sigma].sub.2][sub.[gamma][right arrow]1] = 0.042[a.sup.2]. Further, [partial derivative][[sigma].sub.2]/ [partial derivative][gamma] < 0 and [partial derivative][[sigma].sub.6]/[partial derivative][gamma] [??] 0 for [gamma] [??] 0.6669. Setting [[sigma].sub.2] and [[sigma].sub.6] equal, we find that they cross at [gamma] = 0.3133 and at [gamma] = 0.8172. It is then obvious that [[sigma].sub.2] < [[sigma].sub.6] when [gamma] [less than or equal to] 0.3133 and when [gamma] [greater than or equal to] > 0.8172, and [[sigma].sub.2] > [[sigma].sub.6] when [gamma] [member of] (0.3133, 0.8172). The rest of the lemma follows from the equilibrium conditions. QED.

PROOF OF LEMMA 6. [partial derivative][[sigma].sub.3]/[partial derivative][gamma] < 0 and [partial derivative][[sigma].sub.5]/[partial derivative][gamma] < 0. Furthermore, [[sigma].sub.3] [absolute value of [sub.[gamma]=0]] = [0.0977a.sup.2], [[sigma].sub.3][absolute value of [sub.[gamma][right arrow]1] = 0, [[sigma].sub.5] [absolute value of [sub.[gamma]=0] = [0.06a.sup.2], and [[sigma] [absolute value of [sub.[gamma][right arrow]1] = [0.008a.sup.2], so that [[sigma].sub.3] [absolute value of [sub.[gamma]=0] = [0.06a.sup.2] while [[sigma].sub.3] [absolute value of [sub.[gamma][right arrow]1] = 0 [[sigma].sub.5] [absolute value of [sub.[gamma][right arrow]1. Therefore, [[sigma].sub.5] and [[sigma].sub.3] cross at a critical value of [gamma], [[gamma].sup.***] = 0.3133. Thus, if [gamma] [less than or equal to] [[gamma].sup.***], [[sigma].sub.5] > [[sigma].sub.3]. The rest of the lemma follows from the equilibrium conditions. QED.

PROOF OF PROPOSITION 3. As shown in Lemma 4, for (DE, FMS) or (FMS, DE) to be equilibria in the private duopoly, [[sigma].sub.1] < s < [[sigma].sub.4] must hold; this can only happen for [gamma] > [[gamma].sup.**] = 0.644205. Recall that (DE, FMS) is an equilibrium in the mixed duopoly if [[sigma].sub.2] < s < [[sigma].sub.6]. We know that [partial derivative][[sigma].sub.2]/[partial derivative][gamma] < 0 and [partial derivative][[sigma].sub.4]/[partial derivative][gamma] < 0 and that [[sigma].sub.2] [absolute value of [sub.[gamma]=0] = [0.06a.sup.2], [[sigma].sub.2] [absolute value of [sub.[gamma][right arrow]1] = [0.042a.sup.2], [[sigma].sub.4] [absolute value of [sub.[gamma]=0] = [0.9375a.sup.2], and [[sigma].sub.4] [absolute value of [sub.[gamma][right arrow]1] = [0.02459a.sup.2]. Therefore, [[sigma].sub.2] [absolute value of [sub.[gamma]=0] < [[sigma].sub.4] [ absolute value of [sub.[gamma]=0] while [[sigma].sub.2] [absolute value of [sub.[gamma][right arrow]1] > [[sigma].sub.4] [absolute value of [sub.[gamma][right arrow]1]. Thus, they must cross. Setting [[sigma].sub.2] and [[sigma].sub.4] equal, we know that [[sigma].sub.2] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [[sigma].sub.4] for [gamma] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 0.450595. Therefore, for [gamma] >[[gamma].sub.**], [[sigma].sub.2] > [[sigma].sub.4], implying that [[sigma].sub.1] < s < [[sigma].sub.4] and [[sigma].sub.3] < s < [[sigma].sub.6] can not hold simultaneously. Furthermore, recall that (FMS, DE) is an equilibrium in the mixed duopoly if [[sigma].sub.3] < s < [[sigma].sub.5]. We know that [partial derivative][[sigma].sub.1]/[partial derivative][gamma] < 0 and [partial derivative][[sigma].sub.5]/[gamma] < 0 and that [[sigma].sub.1] [absolute value of [sub.[gamma]=0] = [0.09375a.sup.2], [[sigma].sub.5] [absolute value of [sub.[gamma]=0] = [0.06a.sup.2], [[sigma].sub.1] [absolute value of [sub.[gamma][right arrow]1] = [0.06a.sup.2], and [[sigma].sub.5] [absolute value of [sub.[gamma][right arrow]1] = [0.009328a.sup.2]. Thus, [[sigma].sub.1] > [[sigma].sub.5] for any [gamma] and therefore [[sigma].sub.1] < s < [[sigma].sub.4] and [[sigma].sub.3] < s < [[sigma].sub.5]. The rest of the proposition follows. QED.

Appendix 3: Welfare Analysis

Given that firms might make a different technology choice in the private as compared to the mixed duopoly, it is necessary to identify the equilibrium outcomes of each of the two types of duopoly under the same market and technology conditions in order to make a valid analysis of the effects of privatization. We use the following procedure: We start by considering one of the four possible equilibria in the mixed duopoly, say (FMS, FMS). We know that this equilibrium requires a particular set of conditions related to the parameters of the model, s, a, and [gamma] (as established in Lemma l). Then we identify which would be the corresponding equilibrium outcome in the private duopoly under the same set of market and technology conditions, which might differ from that of the mixed duopoly under the same set of conditions. Having done this, we compare the equilibrium level of total surplus across the two regimes. We then repeat this procedure for the other three possible equilibria in the mixed duopoly (DE, FMS), (FMS, DE), and (DE, DE). We denote by subscripts M (the mixed duopoly) and by P (the private duopoly), followed by 1, 2, 3, and 4 denoting the (FMS, FMS), (DE, FMS), (FMS, DE), and (DE, DE) equilibria, respectively.

(FMS, FMS) Equilibrium in the Mixed Duopoly

Recall from Lemma 1 that (FMS, FMS) is an equilibrium in the mixed duopoly if s < min{[[sigma].sub.2], [[sigma].sub.3]}. The equivalent condition for the private duopoly is s < [[sigma].sub.1], but from Proposition 1 the critical value for the fixed technology costs s is lower in the mixed duopoly than in the private one, min{[[sigma].sub.2], [[sigma].sub.3]} < [[sigma].sub.1]. So (FMS, FMS) is an equilibrium in both the mixed and private duopolies if s < min{[[sigma].sub.2], [[sigma].sub.3}. A straightforward comparison of the total surplus in the two market regimes reveals that welfare is higher in the private duopoly except when products are nearly independent, as the following lemma demonstrates.

LEMMA 7. [TS.sub.p1] [greater than or equal to] [TS.sub.M1] for [gamma] [greater than or equal to] 0.0223 and [TS.sub.p1] < [TS.sub.M1] for [gamma] < 0.0223.

PROOF OF LEMMA 7.

[TS.sub.p1] - [TS.sub.M1] = [2a.sup.2][fp.sub.1][M.sub.1]] ([gamma])/ [(1 + [gamma]).sup.2] [(5 + 2[gamma]).sup.2] [(4 + 3[gamma]).sup.2],

where [fp.sub.1], [M.sub.1]], ([gamma]) = -3 + 128[gamma] + 277[[gamma].sup.2] + 209[[gamma].sup.3] + 67[[gamma].sup.4] + 8[[gamma].sup.5] [??] 0 for [gamma] [??] 0.0223. Hence, [TS.sub.p1] [greater than or equal to] [TS.sub.M1] if [gamma] 0.0223, and [TS.sub.p1] < [TS.sub.M1] if [gamma] < 0.0223. QED.

As a consequence, we can state that under the conditions that lead to an equilibrium in the mixed duopoly in (FMS, FMS), privatization would lead to an increase in surplus unless the products were almost independent.

(DE. DE) Equilibrium in the Mixed Duopoly

As shown in Lemma 3, the relevant condition for a (DE, DE) equilibrium in the mixed duopoly is s > [[sigma].sub.6], while the equivalent condition in the private duopoly requires s > [[sigma].sub.4]. We then distinguish the following cases. Case A: s > [[sigma].sub.6] and s > [[sigma].sub.4]. (DE, DE) is the outcome in both market arrangements. Case B(i): s > [[sigma].sub.6], s < [[sigma].sub.4], and s [greater than or equal to] [[sigma].sub.1]. (DE, DE) obtains in the mixed duopoly, whereas either (DE, FMS) or (FMS, DE) occurs in the private duopoly. Case B(ii): s > [[sigma].sub.6], s < [[sigma].sub.4], and s < [[sigma].sub.1], where (DE, DE) is the mixed duopoly equilibrium and (FMS, FMS) is the private duopoly equilibrium. We next proceed to examine each of these cases in detail.

Case A. (DE, DE) is the equilibrium in both the mixed and private duopolies so we just need to compare [TSp.sub.4], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This is done in the following lemma.

LEMMA 8. For a > 0 and 7 [gamma] 0 and [gamma] [member of] [0, 1), when s > [[sigma].sub.6] and s > [[sigma].sub.4], [TSp.sub.4] < [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

PROOF OF LEMMA 8.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [fp.sub.4][M.sub.4] ([gamma]) = (-9 + 6[gamma] + [[gamma].sup.2] - [2[gamma].sup.3]) < 0 for any [gamma]. Hence [TSp.sub.4] < [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. QED.

Case B(i). The mixed duopoly is characterized by a (DE, DE) equilibrium, whereas the private duopoly equilibrium is either (DE, FMS) or (FMS, DE). Hence, the relevant welfare comparison is between total surplus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the mixed duopoly and total surplus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the private duopoly recall that the private duopoly equilibria are symmetric. The following, Lemma 9, illustrates.

LEMMA 9. For a > 0 and [gamma] [member of] (0.6442, 0.6755), when s > [[sigma].sub.6], s < [[sigma].sub.6], and s [greater than or equal to] [[sigma].sub.1], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

PROOF OF LEMMA 9. From Lemma 4, [[sigma].sub.1] < [[sigma].sub.4] if and only if [gamma] > [gamma] ** = 0.6442. Further, from Lemma 3, [[sigma].sub.6] - [[sigma].sub.4] < 0 if and only if 0.0056 < [gamma] < 0.6755. Hence, the relevant range for [gamma] is [gamma] [member of] (0.6442, 0.6755). It can be checked that the difference [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is decreasing in s and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as f[P.sub.2][M.sub.4]([gamma]) = - 10368 - 4032[gamma] + 30600[[gamma].sup.2] + 9816[[gamma].sup.3] - 29466[[gamma].sup.4] - 6772[[gamma].sup.5] + 12203[[gamma].sup.6] + 1670[[gamma].sup.7] - 2041[[gamma].sup.8] - 110[[gamma].sup.9] + 72[[gamma].sup.10] > 0 for [gamma] [member of] (0.6442, 0.6755). Note also that in this region of [gamma], [[sigma].sub.1] > [[sigma].sub.6]. Then, given that s [greater than or equal to] [[sigma].sub.1] it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. QED.

Case B(ii). In this case, the mixed duopoly equilibrium is (DE, DE), whereas the private duopoly yields (FMS, FMS). In the following lemma, we compare total surpluses [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

LEMMA 10. For a > 0 and [gamma] [member of] (0.0536, 0.6736), when s > [[sigma].sub.6], s < [[sigma].sub.4], and s < [[sigma].sub.1] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. PROOF OF LEMMA 10. From Equations 1 and 6,

[[sigma].sub.1] - [[sigma].sub.6] = [a.sup.2][f.sub.1,6]([gamma])/ 50[(4 + 3[gamma]).sup.2][(3 - [[gamma].sup.2]).sup.2] [(24 - 11[[gamma].sup.2]).sup.2] [(1 - [[gamma].sup.2]).sup.2]

and sign([[sigma].sub.1] - [[sigma].sub.6]) = sign[f.sub.l,6([gamma]), where [f.sub.l,6([gamma]) = -16704 + 332064[gamma] - 343356[[gamma].sup.2] - 744420[[gamma].sup.3] + 706663[[gamma].sup.4] + 634292[[gamma].sup.5] - 531705[[gamma].sup.6] - 252133[[gamma].sup.7] + 180629[[gamma].sup.8] + 44928[[gamma].sup.9] - 24779[[gamma].sup.10] 2563[[gamma].sup.11] + 522[[gamma].sup.12]. Note that [f.sub.1,6([gamma]) > 0 for [gamma] [member of] (0.0536, 0.6736), which is the relevant range for [gamma]. It can be checked that the difference [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is decreasing in s and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as f[P.sub.1][M.sub.4]([gamma]) = 616 - 856[gamma] + 297[[gamma].sup.2] + 662[[gamma].sup.3] - 122[[gamma].sup.4] - 56[[gamma].sup.5] - 183[[gamma].sup.6] - 38[[gamma].sup.7] + 72[[gamma].sup.7] + 72[[gamma].sup.8] < 0. Then, given that s > [[sigma].sub.6], it follows that, in the relevant region of [gamma], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. QED.

To sum up the results of this section, under the market and technology conditions that lead to an equilibrium with both firms choosing DE in the mixed duopoly, privatization will not be welfare enhancing.

(DE, FMS) Equilibrium in the Mixed Duopoly

Next we turn our attention to the (DE, FMS) equilibrium in the mixed duopoly. From Lemma 5, the relevant condition for a (DE, FMS) equilibrium is [[sigma].sub.2] < s < [[sigma].sub.6] and is satisfied when [gamma] [not member of] (0.3133, 0.8173). In this range of values for [gamma], the corresponding equilibrium in the private duopoly would be either (DE, DE), if s > [[sigma].sub.4] (Case C), or (FMS, FMS) if s < [[sigma].sub.1] (Case D). We start by analyzing the first of these cases.

Case C: [[sigma].sub.2] < s < [[sigma].sub.6] and s > [[sigma].sub.4]. (DE, FMS) is the outcome in the mixed duopoly, and (DE, DE) obtains in the private duopoly. Comparing total surplus in the two market regimes yields the following lemma.

LEMMA 11. For a > 0 and [gamma] [not member of] (0.0056, 0.8173), when [[sigma].sub.2] < s < [[sigma].sub.6] and s > [[sigma].sub.4], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

PROOF OF LEMMA 11. From Lemma 3, [[sigma].sub.4] < [[sigma].sub.6] if and only if [gamma] [not member of] (0.0056, 0.8173), and from Lemma 5, [[sigma].sub.2] < [[sigma].sub.6] if and only if [gamma] [member of] (0.3133, 0.8173), so the relevant range for [gamma] is [gamma] [no member of] (0.0056, 0.8173). It can be checked that the difference [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is increasing in s; further

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as f[P.sub.4][M.sub.2]([gamma]) = - 225 + 600[gamma] - 4667[[gamma].sup.2] + 632[[gamma].sup.3] + 6381[[gamma].sup.4] - 414[[gamma].sup.5] - 2901[[gamma].sup.6] + 132[[gamma].sup.7] + 416[[gamma].sup.8] - 14[[gamma].sup.9] - 4[[gamma].sup.10] <0. Then, given that s < [[sigma].sub.6], it follows that, in the relevant region of [gamma], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. QED.

Case D: [[sigma].sub.2] < s < [[sigma].sub.6] and s < [[sigma].sub.1] (DE, FMS) is the outcome in the mixed duopoly and (FMS, FMS) in the private one. The relevant welfare comparison is between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

LEMMA 12. For a > 0 and [gamma] [not member of] (0.3133, 0.8173), when [[sigma].sub.2] < s < [[sigma].sub.6] and s < [[sigma].sub.1], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

PROOF OF LEMMA 12. From Lemma 5, [[sigma].sub.2] < [[sigma].sub.6] if and only if [gamma] [not member of] 60.3133, 0.8173). Further, from the proof of Proposition 1, [[sigma].sub.2] < [[sigma].sub.1] if and only if [gamma] < 0.4593. Therefore, the relevant range for [gamma] is [gamma] [not member of] (0.3133, 0.8173). It can be checked that the difference [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is decreasing in s. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We thank John Beath, Nikolaos Georgantzis, Rafael Moner-Colonques, Vicente Octs, and two anonymous referees for their helpful comments. All remaining errors are our own. We also thank for financial support the Spanish Ministry (SEJ2005-08764/ECON) and the British Academy (Joint Activities Scheme).

Received September 2006; accepted April 2007.

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Maria Jose Gil-Molto, Department of Economics, University of Leicester, University Road, Leicester, LE1 7RH England, United Kingdom: E-mail [e-mail]; corresponding author.

Joanna Poyago-Theotoky, Department of Economics, Loughborough University, Sir Richard Morris Building, Loughborough, LE11 3TU England, United Kingdom; E-mail [e-mail].

(1) This represents an alternative interpretation of our model.

(2) See also Gupta (1998) for some corrections and reinterpretations of the results in Roller and Tombak (1990).

(3) Dial-up internet access can also be provided using traditional telephone technology. In that sense, traditional telephone technology could also be seen as a flexible technology, since it can be used to service two markets: telephone and internet access services. However, cable technology also allows firms to provide TV services, which cannot be provided by using traditional telephone technology.

(4) This implies that the public firm could potentially incur negative profits if by doing so social welfare were maximized. The potential existence of negative profits does not affect our results, as it would only move upwards/downwards the critical value of the technology costs that firms are facing.

(5) An alternative not pursued here is provided by Matsumura (1998): Partially privatized firms are assumed to combine the maximization of social welfare with the maximization of profits.

(6) For an interesting analysis of this in the context of a private duopoly see Dixon (1994).

(7) We are grateful to a referee for pointing this out to us.

(8) Introducing Stackelberg leadership by the public firm does not affect our results qualitatively. On the other hand, the issue of endogenous choice of timing of the production stage, as in Pal (1998) and Matsumura (2003), falls outside the scope of this paper.

(9) Second-order conditions are satisfied in all cases.

(10) Second-order conditions are satisfied in all cases.

(11) Roller and Tombak (1990, 1993) obtain a similar result for a different specification of the variable production costs.

(12) Note that this result is confirmed empirically by Schlesinger et al. (1997) and Schlesinger (1998) in the context of competition among hospitals in the provision of several services.

(13) In such a case, it would be more efficient to produce a higher quantity of the "old" good instead.

(14) The value of a does not affect the diagrams qualitatively, since a is just a scaling parameter. The same remark applies to Figure 2.

(15) This result is in contrast with Kim, Roller, and Tombak (1992), where asymmetric equilibria in pure strategies do not exist.

(16) Here the two firms are interested in being the one using FMS. Given that [[pi].sup.*.sub.1,2] - [[pi].sup.*.sub.1,3] > 0 and [[pi].sup.*.sub.1,3] - [[pi].sup.*.sub.1,4] > 0 must hold, and by definition [[pi].sup.*.sub.1,4] > [[pi].sup.*.sub.1,2]([for all][gamma] [not equal to] 0), then [[pi].sup.*.sub.l,3] > [[pi].sup.*.sub.1,2]. Given the symmetry of the game, the same applies to firm 2. Therefore, in the case of asymmetric equilibria, the firm using FMS obtains higher profits than the one using DE. Therefore, given the multiplicity of equilibria, firms might end up in the worst scenario possible unless some coordination mechanism is used.

(17) It is relatively straightforward to show that [[pi].sup.*.sub.1,1] < [[pi].sup.*.sub.1,4] for [[sigma].sub.4] < s < [[sigma].sub.1].

(18) For instance, this might imply comparing the total surplus provided by a mixed duopoly choosing (DE, DE) with that provided by a private duopoly choosing (FMS, FMS) if for given a, s, and [gamma], (DE, DE) and (FMS, FMS) are equilibria in the mixed and private duopoly, respectively.

technology (low technology costs, relative to the size of the market and/or the degree of substitutability between markets). (19) Therefore, larger markets (large a), lower technology costs, and lower substitutability across markets (except when markets are almost independent) point towards the beneficial effects of the privatization of public firms.

Our main result in this section is easier to interpret if we explore what a social planner would choose in both the private duopoly and in the mixed duopoly case. This is tedious but straightforward to do and requires ranking the total surplus expressions from the appendices for the private and the mixed duopoly cases. (20) Interestingly, in the private duopoly, net of s, for any a and [gamma], the highest level of welfare is provided by (FMS, FMS) and the lowest by (DE, DE). It follows that for low technology costs, the social planner would choose (FMS, FMS), and as the technology costs increase it would move towards the asymmetric configuration and if the costs increase further, towards the (DE, DE) configuration. In the case of the mixed duopoly, the optimal choice for the social planner is less straightforward. In fact, net of s, the preferred outcome would be (FMS, FMS) only for almost independent goods. For the rest of the range of values of T, (DE, FMS) would be preferred instead. This indicates that, unless there is a very low degree of competition across markets, "a lot of" flexibility in the mixed duopoly is "too much" in the view of the social planner. In those cases, a privatization is beneficial. The relative strength of Proposition 4 in terms of its policy implications is derived from the fact that it can be used even without knowing the exact values of a, [gamma], and s. It seems quite plausible to assume that policy makers know accurately the strategic plans of public firms, in this case the FMS investment plan in technology choice and the closeness between the markets/goods. If the public firm does not have any intention of replacing DE with FMS, then privatizing it should not be considered.

5. Concluding Remarks

In this paper we have introduced a mixed duopoly in the context of a differentiated product, quantity-setting duopoly facing the decision of whether to adopt a flexible technology (and become a multiproduct or multimarket firm) or a dedicated technology. We have also studied the equivalent private duopoly so as to compare the outcomes of the two different market arrangements and provide some tentative policy guidelines on the privatization of a public firm. In doing this we have combined two different matters, technology adoption (or product flexibility) and the presence of a private versus a public firm, in a single model. Although we have used a simple model to do this, it nevertheless became quite complex to solve. However, we have been able to derive policy implications as to the desirability of pursuing the privatization of the public firm. Our main findings can be summarized as follows: Flexibility is encouraged by low technology costs, large market sizes, and (generally) high degrees of differentiation. An equilibrium with both firms choosing flexible technologies is more likely to arise in the case of the private duopoly. Further, an equilibrium involving the two firms using dedicated technologies is also more likely to arise in the private duopoly when products are very close substitutes or almost independent. Mixed (asymmetric) equilibria with one firm being flexible and the other dedicated are less likely to be obtained in the private duopoly. In the case of a mixed duopoly, the public firm chooses a dedicated technology when products are very close substitutes, because it is not socially profitable to bear higher technology costs in order to produce almost the same good.

Privatization of the public firm is warranted, that is, beneficial, when the market and technology conditions lead to an equilibrium outcome where both firms use flexible technologies and goods are not (almost) independent. The underlying conditions for this equilibrium to arise imply high potential profitability (low technology costs relative to the size of the market and/or the degree of substitutability between markets). In all remaining cases, privatizing the public firm would result in a reduction of social welfare. Thus, our results provide limited support for privatizing the public firm. However, a word of caution is needed here. The results we obtain are based on a simple duopoly model, with linear demand and quadratic costs. It would be interesting to examine the robustness of the model's predictions in a more general setting of an oligopoly with general demand and cost functions and whether the results are sensitive to the mode of competition (quantity vs. price). It would also be relevant to study the adoption of flexible technologies when firms can endogenously determine the degree of product differentiation. We leave the study of these issues for future research.

Appendix 1: Equilibrium Solutions

Private Duopoly

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Mixed Duopoly

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Appendix 2: Equilibria Characterization. Proofs.

PROOF OF LEMMA 1. Note that [partial derivative][sigma].sub.2]/[partial derivative][gamma] < 0 and [partial derivative][[sigma].sub.3]/[partial derivative][gamma] < 0. Further, from Equations 2 and 3, we obtain [[sigma].sub.2] [absolute value of [sub.[gamma]=0] = [0.06a.sup.2], [[sigma].sup.2]][sub.[gamma][right arrow] 1] = [0.042a[.sup.2], [[sigma].sub.3][absolute value of [sub.[gamma]=0] = [0.0977a.sup.2], [[sigma]][sub.3]][sub.[gamma][right arrow]1 = 0, and [[sigma].sub.3][absolute value of [sub.[gamma]=0] > [[sigma].sub.]][sub.[gamma]=0, > [[sigma].sub.2]][sub.[gamma]=0], while [[sigma].sub.2] [absolute value of [sub.[gamma][right arrow] 1 > [[sigma].sub.3]][sub.[gamma][right arrow]1] = 0. Therefore, [[sigma].sub.2] and [[sigma].sub.3] must cross. Setting Equations 2 and 3 equal we obtain [[gamma].sup.*] = 0.2432, where [[sigma].sub.2] and [[sigma].sub.3] cross. The result then follows immediately. QED.

PROOF OF PROPOSITION 1. Lemma 1 establishes that the relevant critical value for s in the mixed duopoly is min {[[sigma].sub.2], [[sigma].sub.3]}; in particular, for [gamma] < [[gamma].sup.*] the relevant critical value is given by [[sigma].sub.2], and for [gamma] [greater than or equal to] [[gamma].sup.*] it is given by [[sigma].sub.3], [[gamma].sup.*] = 0.2432. Thus, we need to show that [[sigma].sub.2] < [[sigma].sub.1] for [gamma] < [[gamma].sup.*] and [[sigma].sub.3] < [[sigma].sub.1] for [gamma] [greater than or equal to] [[gamma].sup.*]. Note that [partial derivative][[sigma].sub.1]/[partial derivative][gamma] < 0, [partial derivative][[sigma].sub.2]/partial derivative][gamma] < 0, and [partial derivative][[sigma].sub.3]/[partial derivative][gamma] < 0. Further, from Equations 1 and 2, we obtain [[sigma].sub.1][[absolute value of [sub.[gamma]=0] = 0.0937[a.sup.2] and [[sigma].sub.2]], respectively. [[sigma].sub.1] = [[sigma].sub.2] at [gamma] = 0.4593 > [[gamma].sup.*] and [[sigma].sub.2][absolute value of [sub.[gamma]=0] < [[sigma].sub.1]][sub.[gamma]=0]. Therefore, [[sigma].sub.2] < [[sigma].sub.1] when [gamma] < [[gamma].sup.*]. Similarly, from Equations 1 and 3 we obtain [[sigma].sub.1][[absolute value of [sub.[gamma][right arrow]1] = 0.0221[a.sup.2] and [[sigma].sub.3]][sub.[gamma][right arrow]1] = 0, respectively. [[sigma].sub.1] = [[sigma].sub.3] at [gamma] = 0.0393 < [[gamma].sup.*] and [[sigma].sub.3][absolute value of [sub.[gamma][right arrow]l < [[sigma].sub.1]1][sub.[gamma][right arrow]1. Therefore, [[sigma].sub.3] < [[sigma].sub.1] when [gamma] > [[gamma].sup.*], and we have shown that min{[[sigma].sub.2], [[sigma].sub.3]} < [[sigma].sub.1]. The rest of the proposition follows from the relevant equilibrium conditions. QED.

PROOF OF LEMMA 2. From Equations 5 and 6,

[[sigma].sub.6] - [[sigma].sub.5] [a.sup.2][f.sub.5,6][([gamma])/200([[gamma].sup.2]-3).sup.2]([[gamma].sup.2] 1)[(8[[gamma].sup.2] - 15).sup.2].

This is positive as [f.sub.5,6]([gamma]) < 0, where

[f.sub.5.6]([gamma]) = -15300 + 66600[gamma], - 78135[[gamma].sup.2] - 39900[[gamma].sup.3] + 111331[[gamma].sup.4] 14380[[gamma].sup.5] - 49792[[gamma].sup.6] + 13120[[gamma].sup.7] - 1920[[gamma].sup.9] - 512[[gamma].sup.10], and the denominator is negative as [lim.sub.[gamma][right arrow]1] < 0. QED.

PROOF OF LEMMA 3. Note that [[sigma].sub.4][absolute value of [sub.[gamma]=0] = 0.0937[a.sup.2], [[sigma].sub.6]][sub.[gamma]=0] = 0.0978[a.sup.2], [[sigma].sub.4][absolute value of [sub.[gamma]=1] = 0.0246[a.sup.2], and [lim.sub.[gamma][right arrow]1] [[sigma].sub.6] = [infinity]. Therefore, [[sigma].sub.4][absolute value of [sub.[gamma]=0] < [[sigma].sub.6]][sub.[gamma]=0] and [[sigma].sub.4] | [sub.[gamma]=1] < [lim.sub.[gamma][right arrow] 1 [[sigma].sub.6] = [infinity]. [[sigma].sub.6] reaches its minimum at [gamma] = 0.6689, whereas [[sigma].sub.4] [absolute value of [sub.[gamma]=0.6689] = 0.0393[a.sup.2] and [[sigma].sub.6][sub.[gamma]=0.6689] = 0.0388[a.sup.2], meaning that [[sigma].sub.4][absolute value of [sub.[gamma]=0.6689] > [[sigma].sub.6]][sub.[gamma]=0.6689. Hence, [[sigma].sub.4] and [[sigma].sub.6] must cross twice: Setting [[sigma].sub.4] and [[sigma].sub.6] equal, we find that they cross at [gamma].sub.1] = 0.0056 and at [[gamma].sub.2] = 0.6755. The rest of the lemma follows. QED.

PROOF OF PROPOSITION 2. Follows from Lemma 3 and the necessary conditions for equilibrium. QED.

PROOF OF LEMMA 4. Using Equations 1 and 4 we obtain

[[sigma].sub.1] - [[sigma].sub.4] = [a.sup.2][gamma][f.sub.1,4]([gamma])/2[(3 + [gamma]).sup.2][(4 + 3[gamma]).sup.2][(24 - 11[[gamma].sup.2]).sup.2],

where

[f.sub.l,4]([gamma]) = 576 + 168[gamma] - 1608[[gamma].sup.2] - 488[[gamma].sup.3] + 646[[gamma].sup.4] 20[[gamma].sup.6] + 81[[gamma].sup.7] [??] 0 for [gamma] > [[gamma].sup.**] = 0.6442.

The rest of the lemma follows immediately. QED.

PROOF OF LEMMA 5. Note that [[sigma].sub.6] [absolute value of [sub.[gamma]=0] = 0.1[a.sup.2], [[sigma].sub.2][sub.[gamma]=0] = 0.06[a.sup.2], [[sigma].sub.6] [absolute value of [gamma][right arrow]1] = [infinity], and [[sigma].sub.2][sub.[gamma][right arrow]1] = 0.042[a.sup.2]. Further, [partial derivative][[sigma].sub.2]/ [partial derivative][gamma] < 0 and [partial derivative][[sigma].sub.6]/[partial derivative][gamma] [??] 0 for [gamma] [??] 0.6669. Setting [[sigma].sub.2] and [[sigma].sub.6] equal, we find that they cross at [gamma] = 0.3133 and at [gamma] = 0.8172. It is then obvious that [[sigma].sub.2] < [[sigma].sub.6] when [gamma] [less than or equal to] 0.3133 and when [gamma] [greater than or equal to] > 0.8172, and [[sigma].sub.2] > [[sigma].sub.6] when [gamma] [member of] (0.3133, 0.8172). The rest of the lemma follows from the equilibrium conditions. QED.

PROOF OF LEMMA 6. [partial derivative][[sigma].sub.3]/[partial derivative][gamma] < 0 and [partial derivative][[sigma].sub.5]/[partial derivative][gamma] < 0. Furthermore, [[sigma].sub.3] [absolute value of [sub.[gamma]=0]] = [0.0977a.sup.2], [[sigma].sub.3][absolute value of [sub.[gamma][right arrow]1] = 0, [[sigma].sub.5] [absolute value of [sub.[gamma]=0] = [0.06a.sup.2], and [[sigma] [absolute value of [sub.[gamma][right arrow]1] = [0.008a.sup.2], so that [[sigma].sub.3] [absolute value of [sub.[gamma]=0] = [0.06a.sup.2] while [[sigma].sub.3] [absolute value of [sub.[gamma][right arrow]1] = 0 [[sigma].sub.5] [absolute va lue of [sub.[gamma][right arrow]1. Therefore, [[sigma].sub.5] and [[sigma].sub.3] cross at a critical value of [gamma], [[gamma].sup.***] = 0.3133. Thus, if [gamma] [less than or equal to] [[gamma].sup.***], [[sigma].sub.5] > [[sigma].sub.3]. The rest of the lemma follows from the equilibrium conditions. QED.

PROOF OF PROPOSITION 3. As shown in Lemma 4, for (DE, FMS) or (FMS, DE) to be equilibria in the private duopoly, [[sigma].sub.1] < s < [[sigma].sub.4] must hold; this can only happen for [gamma] > [[gamma].sup.**] = 0.644205. Recall that (DE, FMS) is an equilibrium in the mixed duopoly if [[sigma].sub.2] < s < [[sigma].sub.6]. We know that [partial derivative][[sigma].sub.2]/[partial derivative][gamma] < 0 and [partial derivative][[sigma].sub.4]/[partial derivative][gamma] < 0 and that [[sigma].sub.2] [absolute value of [sub.[gamma]=0] = [0.06a.sup.2], [[sigma].sub.2] [absolute value of [sub.[gamma][right arrow]1] = [0.042a.sup.2], [[sigma].sub.4] [absolute value of [sub.[gamma]=0] = [0.9375a.sup.2], and [[sigma].sub.4] [absolute value of [sub.[gamma][right arrow]1] = [0.02459a.sup.2]. Therefore, [[sigma].sub.2] [absolute value of [sub.[gamma]=0] < [[sigma].sub.4] [ absolute value of [sub.[gamma]=0] while [[sigma].sub.2] [absolute value of [sub.[gamma][right arrow]1] > [[sigma].sub.4] [absolute value of [sub.[gamma][right arrow]1]. Thus, they must cross. Setting [[sigma].sub.2] and [[sigma].sub.4] equal, we know that [[sigma].sub.2] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [[sigma].sub.4] for [gamma] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 0.450595. Therefore, for [gamma] >[[gamma].sub.**], [[sigma].sub.2] > [[sigma].sub.4], implying that [[sigma].sub.1] < s < [[sigma].sub.4] and [[sigma].sub.3] < s < [[sigma].sub.6] can not hold simultaneously. Furthermore, recall that (FMS, DE) is an equilibrium in the mixed duopoly if [[sigma].sub.3] < s < [[sigma].sub.5]. We know that [partial derivative][[sigma].sub.1]/[partial derivative][gamma] < 0 and [partial derivative][[sigma].sub.5]/[gamma] < 0 and that [[sigma].sub.1] [absolute value of [sub.[gamma]=0] = [0.09375a.sup.2], [[sigma].sub.5] [absolute value of [sub.[gamma]=0] = [0.06a.sup.2], [[sigma].sub.1] [absolute value of [sub.[gamma][right arrow]1] = [0.06a.sup.2], and [[sigma].sub.5] [absolute value of [sub.[gamma][right arrow]1] = [0.009328a.sup.2]. Thus, [[sigma].sub.1] > [[sigma].sub.5] for any [gamma] and therefore [[sigma].sub.1] < s < [[sigma].sub.4] and [[sigma].sub.3] < s < [[sigma].sub.5]. The rest of the proposition follows. QED.

Appendix 3: Welfare Analysis

Given that firms might make a different technology choice in the private as compared to the mixed duopoly, it is necessary to identify the equilibrium outcomes of each of the two types of duopoly under the same market and technology conditions in order to make a valid analysis of the effects of privatization. We use the following procedure: We start by considering one of the four possible equilibria in the mixed duopoly, say (FMS, FMS). We know that this equilibrium requires a particular set of conditions related to the parameters of the model, s, a, and [gamma] (as established in Lemma l). Then we identify which would be the corresponding equilibrium outcome in the private duopoly under the same set of market and technology conditions, which might differ from that of the mixed duopoly under the same set of conditions. Having done this, we compare the equilibrium level of total surplus across the two regimes. We then repeat this procedure for the other three possible equilibria in the mixed duopoly (DE, FMS), (FMS, DE), and (DE, DE). We denote by subscripts M (the mixed duopoly) and by P (the private duopoly), followed by 1, 2, 3, and 4 denoting the (FMS, FMS), (DE, FMS), (FMS, DE), and (DE, DE) equilibria, respectively.

(FMS, FMS) Equilibrium in the Mixed Duopoly

Recall from Lemma 1 that (FMS, FMS) is an equilibrium in the mixed duopoly if s < min{[[sigma].sub.2], [[sigma].sub.3]}. The equivalent condition for the private duopoly is s < [[sigma].sub.1], but from Proposition 1 the critical value for the fixed technology costs s is lower in the mixed duopoly than in the private one, min{[[sigma].sub.2], [[sigma].sub.3]} < [[sigma].sub.1]. So (FMS, FMS) is an equilibrium in both the mixed and private duopolies if s < min{[[sigma].sub.2], [[sigma].sub.3}. A straightforward comparison of the total surplus in the two market regimes reveals that welfare is higher in the private duopoly except when products are nearly independent, as the following lemma demonstrates.

LEMMA 7. [TS.sub.p1] [greater than or equal to] [TS.sub.M1] for [gamma] [greater than or equal to] 0.0223 and [TS.sub.p1] < [TS.sub.M1] for [gamma] < 0.0223.

PROOF OF LEMMA 7.

[TS.sub.p1] - [TS.sub.M1] = [2a.sup.2][fp.sub.1][M.sub.1]] ([gamma])/ [(1 + [gamma]).sup.2] [(5 + 2[gamma]).sup.2] [(4 + 3[gamma]).sup.2],

where [fp.sub.1], [M.sub.1]], ([gamma]) = -3 + 128[gamma] + 277[[gamma].sup.2] + 209[[gamma].sup.3] + 67[[gamma].sup.4] + 8[[gamma].sup.5] [??] 0 for [gamma] [??] 0.0223. Hence, [TS.sub.p1] [greater than or equal to] [TS.sub.M1] if [gamma] 0.0223, and [TS.sub.p1] < [TS.sub.M1] if [gamma] < 0.0223. QED.

As a consequence, we can state that under the conditions that lead to an equilibrium in the mixed duopoly in (FMS, FMS), privatization would lead to an increase in surplus unless the products were almost independent.

(DE. DE) Equilibrium in the Mixed Duopoly

As shown in Lemma 3, the relevant condition for a (DE, DE) equilibrium in the mixed duopoly is s > [[sigma].sub.6], while the equivalent condition in the private duopoly requires s > [[sigma].sub.4]. We then distinguish the following cases. Case A: s > [[sigma].sub.6] and s > [[sigma].sub.4]. (DE, DE) is the outcome in both market arrangements. Case B(i): s > [[sigma].sub.6], s < [[sigma].sub.4], and s [greater than or equal to] [[sigma].sub.1]. (DE, DE) obtains in the mixed duopoly, whereas either (DE, FMS) or (FMS, DE) occurs in the private duopoly. Case B(ii): s > [[sigma].sub.6], s < [[sigma].sub.4], and s < [[sigma].sub.1], where (DE, DE) is the mixed duopoly equilibrium and (FMS, FMS) is the private duopoly equilibrium. We next proceed to examine each of these cases in detail.

Case A. (DE, DE) is the equilibrium in both the mixed and private duopolies so we just need to compare [TSp.sub.4], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This is done in the following lemma.

LEMMA 8. For a > 0 and 7 [gamma] 0 and [gamma] [member of] [0, 1), when s > [[sigma].sub.6] and s > [[sigma].sub.4], [TSp.sub.4] < [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

PROOF OF LEMMA 8.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [fp.sub.4][M.sub.4] ([gamma]) = (-9 + 6[gamma] + [[gamma].sup.2] - [2[gamma].sup.3]) < 0 for any [gamma]. Hence [TSp.sub.4] < [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. QED.

Case B(i). The mixed duopoly is characterized by a (DE, DE) equilibrium, whereas the private duopoly equilibrium is either (DE, FMS) or (FMS, DE). Hence, the relevant welfare comparison is between total surplus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in

Table 1. Payoffs Matrix

                                   Firm 2

                          FMS                   DE

Firm 1     FMS     [[pi].sub.1,1], A     [[pi].sub.1,3], B
           DE      [[pi].sub.1,2], C     [[pi].sub.1,4], D