Flexible vs Dedicate Technologies in a mixed duopoly

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1 Flexible vs Dedicate Technologies in a mixed duopoly Maria Jose Gil Molto and Joanna Poyago-Theotoky, Loughborough University March 15, 005 Abstract We compare the choice of exible manufacturing technologies in mixed and private duopolies and relate our results to the empirical literature on competition among hospitals. We nd that an equilibrium involving the two rms using exible technologies is more likely to arise in private duopolies than in mixed ones. The same applies to the equilibrium with both rms being dedicated when products are almost independent or highly substitute. Privatizations are only recommendable for market and technology conditions that allow both the private and the public rm to be exible as long as goods are not (almost) independent. Keywords: Flexible Technology, Privatization, Mixed Oligopoly. J.E.L. Classi cation: L11, L13, L33. 1 Introduction In the recent past, many rms all over the world have substituted their traditional production processes by more exible systems. One of the advantages of a exible manufacturing system (FMS) over a dedicated equipment (DE) is that the former allows a rm to supply several products and consequently to participate in di erent markets without having to incur additional production Corresponding author: Department of Economics, Loughborough University, Loughborough LE11 3TU, United Kingdom. m.j.gil-molto@lboro.ac.uk, tel.: + 44 (0) , fax: + 44 (0) We thank Nikos Georgantzis, Vicente Orts Rios, Rafael Moner-Colonques and John Beath for helpful suggestions that have improved this paper. The usual disclaimer applies. We also thank the British Academy for nancial support under the Joint Activities Scheme. 1

2 costs. 1 A signi cant motivation for our study of the choice between manufacturing technologies, that is, FMS versus DE, comes from the health care sector and speci cally from observing competition among hospitals in the provision of several services. First, there is evidence of economies of scope in health care centers (Oczan et al. (199)), that can be related to the adoption of exible technologies (Cantos-Sánchez et al. (003)). Second, there are several studies stressing the fact that public (non-for-pro t) hospitals provide a wider range of services than private (for-pro t) hospitals (Shortell and Morrison (1986, 1987)), i.e. these hospitals are more likely to have a multiproduct character. Further, Schlesinger et al. (1997) show that public hospitals provide more innovative services without competition, but private hospitals are more likely to add these services when there is competition, that is, they are more likely to introduce new products when competition is intense (see also Schlesinger (1998)). This body of observations suggest that public hospitals are more likely to have a multiproduct pro le than private hospitals and that the degree of competition seems to be a key factor encouraging the multiproduct character of public and private hospitals. From a theoretical point of view, the study of the adoption of FMS by private rms was rst introduced by Röller and Tombak (1990) and Kim, Röller and Tombak (199), in the context of oligopolistic competition. Their ndings indicate that the adoption of exible technologies requires a su ciently low adoption cost, su ciently high product di erentiation and large enough markets. Consumers bene t from the use of FMSs, due to the increase in competition 3. In addition, Röller and Tombak (1993) validate these results by an empirical study. 4 To the best of our knowledge, there has been no attempt to thoroughly analyze the adoption of FMS versus DE in the presence of a public rm competing with (at least) a private rm. That is, the issue of technology choice as exempli ed by the adoption of FMS versus DE technologies has not been studied in the context of a mixed oligopoly. 5 Nevertheless, a mixed oligopoly seems to 1 Boyer and Moreaux (1997, 00) report additional bene ts of using a FMS related to capacity exibility: FMS can increase the capacity of rms to adapt to uctuations in demand. Garcia-Gallego and Georgantzis (1996) use similar modeling to assess multiproduct activity from a competition policy point of view. 3 See also Gupta (1998) for some corrections and reinterpretations of the results in Röller and Tombak (1990). 4 Eaton and Schmitt (1994), in the context of horizontal product di erentiation, point out that the adoption of FMSs may correspond to pre-emptive strategies leading to higher levels of concentration. 5 For an excellent review on mixed oligopoly modelling see De Fraja and Delbono (1990).

3 be an ideal framework to study the health care sector, in which typically public and private rms compete in the same markets. Our aim is to provide an initial analysis into the choice of production exibility within a simple duopolistic market consisting of either a public and a private rm (mixed duopoly) or two private (pro t-maximizing) rms. We derive the market conditions that would lead public and private rms to adopt FMS and in doing so we provide some initial guidelines for economic policy. In order to do that we keep the main features from Röller and Tombak (1990) and Kim, Röller and Tombak (199) but consider decreasing returns to scale. 6 In our setting, the public rm acts as social-welfare maximizer. We nd that an equilibrium outcome involving both rms using FMS is more prevalent in a private duopoly than in a mixed duopoly. However, an outcome with both rms using DE prevails under less demanding conditions in the case of the private duopoly, when products are (almost) independent or close substitutes. We do nd support for our results in the empirical literature on hospital competition. Moreover, from a social welfare point of view, we demonstrate that the private duopoly is better than the mixed duopoly in a variety of situations; however, we cannot recommend privatization as a general rule. The remainder of this paper is structured as follows: rst we introduce the model (section ) and then characterize the di erent equilibria (section 3). Next we analyze the behavior of rms in mixed and private duopolies and consider social welfare (section 4). Finally we summarize our main ndings (section 5). The Model Consider a duopoly competing in output and facing the choice between using a exible manufacturing system (FMS) and a dedicated equipment (DE). The use of a FMS allows participation in two existing markets, A and B. The use of the DE constraints rms to be active only in one of the markets. In the case of a mixed duopoly, one of the two rms, denoted by the subscript, is public (non-for-pro t). In line with the majority of the literature, the objective of the public rm is to maximize social welfare. 7 6 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. 7 Partially privatized rms are assumed to combine the maximization of social welfare with the maximization pro ts (see Matsumura 1998). We do not contemplate the latter case in this paper. 3

4 The system of inverse demand functions is given by: and P A = a Q A Q B (1) P B = a Q B Q A () where P A and P B are the prices for products A and B respectively, Q A and Q B the total quantities in market A and market B respectively and a > 0 measures market potential. The parameter measures the substitutability of products A and B, [0; 1). We consider 1 >, that is, the own price e ect dominates the cross-price e ect (). It is intuitive that the higher the, the ercer the competition between rms across markets. The pro t of each rm is given by: i;j = P A Q A i;j + P B Q B i;j F k C i (Q A i;j + Q B i;j) (3) where i denotes the rm (i = 1 or ) and j denotes the state of the industry according to the technologies used by the two rms. In particular, j = 1 j = j = 3 j = 4 if both rms are using FMS; if rm 1 is using DE and rm is using FMS; if rm 1 is using FMS and rm is using DE; if both rms are using DEs. Q A i;j and QB i;j are the quantities chosen by rm i in state j for markets A and B respectively. Without loss of generality we assume that if only one rm is using DE, this rm competes only in market A while the other rm participates in both markets. If both rms use DEs, they compete in di erent markets (without loss of generality, rm 1 in market A and rm in market B). Thus, the use of FMS increases the degree of competition not only in the market where a rm is operating but also across markets (due to product substitutability). F k are the xed costs of rms, which are related to the use of the available manufacturing technologies; k = F M S or DE. The costs of using a FMS are higher than the costs of using a DE. 8 For simplicity, we normalize the costs of the dedicated technology to equal one, F DE = 1, and the costs of the exible technology to be F F MS = 1 + s, where s captures the extent of the 8 Developments costs are higher for a FMS than a DE; see Jaikumar (1986). 4

5 cost di erential between the two manufacturing technologies. C i are the costs of production, which are assumed to be quadratic and separable in the outputs in di erent markets: C i (Q A i;j + Q B i;j) = (Q A i;j) + (Q B i;j) : (4) Total Surplus (T S) is the sum of consumers surplus (CS) and producers pro ts. Linear demand functions lead to an equilibrium CS given by: CS = 1 ((QA ) + (Q B ) ) (5) where Q A and Q B are total equilibrium quantities in markets A and B respectively. Thus, T S is given by: X T S = CS + i;j : (6) We consider a two-stage game: in the rst stage, rms choose technologies and in the second stage, output. Decisions are taken simultaneously by the two rms in each of the stages. 9 In the appendix 10, we provide the equilibrium solutions for the pro ts and total surplus, for given combination of technology choice in the private and the mixed duopolies. Using these equilibrium pro ts we can then represent the technology choice stage using this simple matrix: i=1 F irm F MS DE F irm 1 F MS 1;1 ; ;1 1;3 ; ;3 DE 1; ; ; 1;4 ; ;4 Table 1 We note here that in the case of a private duopoly, the above matrix is symmetric since 1;1 = ;1, 1;4 = ;4, 1;3 = ; and 1; = ;3. 9 We have analyzed variants of this game by considering sequential moves: (i) with the public rm being leader in the two stages, and (ii) with the public rm being Stackelberg leader only in technology choice. The results we have obtained do not di er in a qualitative way from the ones we report here. For a discussion on endogenous timing in mixed duopolies see Pal (1998). 10 The second-order conditions are ful lled in all cases. 5

6 3 Equilibria Characterization In this section we proceed to establish the conditions under which each of the combination of strategies in technology choice is a Nash equilibrium. We proceed by nding the critical value of the technology costs, s; above which investment in FMS becomes unpro table. First, we consider the case of a private duopoly; both rms are pro t-maximizers. Next, we characterize the possible equilibria when the market is organized as a mixed duopoly; i.e. rm 1 is private (pro t-maximizing) and rm is public (welfare maximizing). Using Table 1, we examine the conditions that guarantee one of four possible pure-strategy equilibria in each of the regimes, private or mixed duopoly: (FMS, FMS) where both rms choose a exible production technology and serve both markets, (DE, DE) where both rms choose a dedicated production process and (FMS, DE), (DE, FMS) where one rm chooses FMS and the other DE. For each such equilibrium we rst solve the second stage market (Cournot) game for both markets and then we use the solutions of this in the payo functions to obtain the various i;js for all states j of the rst stage (technology choice) game. In other words, we use subgame perfection as our equilibrium concept. 3.1 The (FMS,FMS) Equilibrium Private Duopoly. From Table 1, it is clear that (FMS, FMS) is an equilibrium when (i) 1;1 1; 0 for rm 1 and (ii) ;1 ;3 0 for rm. Using the model outlined previously, these conditions are equivalent to the following: a (3 + ) (4 + 3) 3a ( + 6) (4 11 ) s 0: Let 1 denote the critical level in (the di erence in) xed costs; this is where FMS becomes unpro table, i.e. the level of xed costs, s; that makes the above expression a strict equality. If s is lower than this critical value 1 then both rms will choose FMS. From the above expression this critical value is, 1 = a f 1 () (4 + 3) (4 11 ) (7) where f 1 () = > 0. Note that the critical value is increasing in market 1 =@a > 0, while it is decreasing in product 1 =@ < 0. The larger market for either product makes rms wish to participate in exible production in order 6

7 to serve both markets. With a low degree of substitutability (small ) rms products are perceived as highly di erentiated by consumers so that a rm that opts for a dedicated production process (DE) and thus serves one market only e ectively looses out. Hence a larger market size and greater product di erentiation point towards the adoption of FMS by the rms. 11 Mixed Duopoly. From Table 1, (FMS, FMS) is an equilibrium if (i) 1;1 1; 0 for rm 1 and (ii) ;1 ;3 0 for rm. The rst condition yields a (3 + ) 3a (5 + ) 50 s 0 which implies a corresponding critical value for s denoted; = a f () 50(5 + ) (8) where f () = > 0. The second condition is equivalent to a ( ) a ( ) (1 + )(5 + ) (15 8 ) s 0 implying an associated critical value for s, 3 = a f 3 () (1 + )(5 + ) (8 15) (9) where f 3 () = > 0: It is easy to establish =@a > 3 =@a > =@ < 0 3 =@ < 0. larger market (higher a) supports a larger critical di erence in the xed costs of the two di erent types of technology while increased product substitutability (higher ) has the opposite e ect. Taking the two conditions together we can say that (FMS, FMS) is an equilibrium when s < and s < 3 while it is not an equilibrium if s > or s > 3. We then state the following Lemma 1 : Lemma 1 : In a mixed duopoly, (FMS, FMS) is an equilibrium if s < minf ; 3 g: In particular, given market size a; there exists a critical value such that for < (FMS, FMS) is an equilibrium if s < and for > (FMS, FMS) is an equilibrium if s < 3 : This critical value is = 0: This result is also obtained by Röller and Tombak (1990, 1993) for a di erent speci cation of the variable production costs. 1 For the sake of readability of the paper, we con ne all the proofs of our lemmata and propositions to the mathematical appendix. A 7

8 It is interesting to note that this result is con rmed by Schlesinger et al. (1997), in the context of hospital competition for services, where it is reported that under low levels of competition the private rm is less likely to have a multiproduct pro le than the public rm ( < 3 for < ). On the other hand, the opposite happens for high degrees of competition ( > 3 for > ). This is also further re ected in empirical studies, as in Schlesinger et al. (1998), where it is found that the private rm is more likely to introduce new services as the degree of competition increases. 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, i.e. we compare the three critical levels of xed costs, 1 ; and 3 (see expressions (7),(8) and (9)). The following proposition describes this comparison. Proposition 1: For given [0; 1) and any a > 0 the critical value for the xed technology costs s is lower in the mixed duopoly than in the private duopoly, that is minf ; 3 g < 1. Hence from the necessary conditions for an (FMS, FMS) equilibrium, (7),(8) and (9): (i) if s < minf ; 3 g then (FMS, FMS) is an equilibrium in both the mixed and private duopolies; (ii) if minf ; 3 g < s < 1 then (FMS, FMS) is an equilibrium in the private duopoly but not in the mixed duopoly; (iii) if 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 the products. In such a case, it is less likely that a public rm will invest in FMS. Investing in FMS is less pro table for the public rm and also socially not meaningful; it does not make sense to bear the associated with FMS costs in order to produce goods that are perceived by consumers almost as perfect substitutes. Second, consider the case of relatively low substitutes. In such a case, the public rm produces more 8

9 in each market to compensate for the low substitutability between products, therefore making it less pro table for the private rm to invest in technology adoption; in essence the public rm crowds out the private rm s investment. Thus, for relatively low substitutability, it is more likely to observe (FMS, FMS) in the private duopoly than in the mixed one. Proposition 1 is illustrated in gure 1. The gure graphs 1 ; ; 3 for given a (we have chosen a = 1; a is just a scaling parameter). The area below 1 represents combinations of s and that lead to an (FMS, FMS) equilibrium in the private duopoly and the area below the minimum of and 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 in the mixed duopoly. This indicates that, for given size of the market and product di erentiation, lower values of the technology adoption costs correspond to an (FMS, FMS) equilibrium in the mixed duopoly. [Insert gure 1 about here] 3. 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) 1;4 ; > 0, for rms 1 and respectively, implying ;4 1;3 > 0 and (ii) 3a (3 + ) + a ( ) (4 11 ) + s 0: Letting 4 denote the relevant critical value for s in this case, we obtain from the above expression, 4 = a f 4 () (3 + ) (11 4) (10) where f 4 () = > 0. If s is greater than this critical value, 4 ; then (DE, DE) is an equilibrium. It is obvious that this critical value is increasing in market 4 =@a > 0, while it can be easily established that it is decreasing in the product di erentiation 4 =@ < 0. Consequently, (DE, DE) will be an equilibrium for relatively smaller a and higher. The intuition behind this is clear, since the opposite to the FMS case holds: The smaller the market for either product makes rms less willing to participate in exible technology adoption in order 9

10 to serve both markets. With a high degree of substitutability (high ) rms products are perceived as close substitutes by consumers so that a rm that opts for a FMS is bearing a high xed cost to produce two goods that are almost the same. Hence a smaller market size and lower product di erentiation point towards the adoption of DE by the rms, given s. Mixed Duopoly. From Table 1, the conditions ensuring that (DE, DE) is an equilibrium are (i) 1;4 1;3 > 0 (for the private rm) and (ii) ;4 ; > 0 (for the public rm). The rst condition can be written as 3a ( ) 8( 3) a ( ) (15 8 ) + s 0 implying that the associated critical value for s is 5 = a f 5 () 8( 3) (8 15) (11) where f 5 () = > 0. From the second condition we obtain a ( ) 100( 1 + ) with associated critical value a ( ) 4( 3 + ) + s 0 6 = a f 6 () 50( 3) (1 ) (1) where f 6 () = > 0. Notice 5 =@a > 0 6 =@a > 0 while it is relatively easy to check 5 =@ < 0 6 =@ S 0 as S 0:6669. Therefore a (DE, DE) equilibrium occurs when both s > 5 and s > 6. The following Lemma establishes that the latter inequality is su cient for a (DE, DE) equilibrium; that is, the critical value in the mixed duopoly is the one corresponding to the public rm. Lemma : In a mixed duopoly (DE, DE) is an equilibrium if s > 6 = maxf 5 ; 6 g for all [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 di erence in xed costs as well as characterizing the (DE, DE) equilibrium fully. 10

11 Lemma 3 Comparing the critical values for the private duopoly, 4, and the mixed duopoly, 6, we have: 4 6 for 1 and 4 < 6 for 0 < 1 and < < 1; where 1 = 0:0056 and = 0:6755. We summarize the results obtained in this subsection in the following proposition: Proposition : (a) For given a > 0 and 1 : (i) if s > 4 then (DE, DE) is an equilibrium in both the mixed and private duopolies, (ii) if 4 > s > 6 then (DE, DE) is an equilibrium in the mixed duopoly but not in the private duopoly, (iii) if 6 > s then (DE, DE) is not an equilibrium; (b) For given a > 0, 0 < < 1 and < < 1 : (i) if s > 6 then (DE, DE) is an equilibrium in both the mixed and the private duopolies, (ii) if 6 > s > 4 then (DE, DE) is an equilibrium in the private but not the mixed duopoly, (iii) if 4 > s then (DE, DE) is not an equilibrium. Figure (plotted for a = 1), illustrates proposition. The white area above 4 and 6 represents combinations of the parameters s and such that that a (DE, DE) equilibrium exists for both versions of duopoly. The dark grey area represents combinations that guarantee a (DE, DE) equilibrium in the private duopoly but not in the mixed one. Finally, the pale grey area represents parameter combinations that make (DE, DE) an equilibrium in the mixed duopoly only. [Insert Figure about here] Figure illustrates that the necessary conditions for a (DE, DE) equilibrium are more stringent in the case of the mixed duopoly for low and high values of substitutability. The intuition for this is rather clear. For low values of substitutability, i.e. when products are perceived as highly di erentiated by consumers, there is a strong incentive for the public rm to serve both markets and so increase the degree of competition. Thus, a (DE, DE) equilibrium is less easily observed in a mixed duopoly. For high values of substitutability, since the degree of competition across markets is already very high, either rm in a 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 rm uses dedicated equipment, the public rm has strong incentives to become exible in order to increase the degree of competition. For intermediate values of product substitutability a (DE, DE) equilibrium is more prevalent in a mixed duopoly. 11

12 3.3 The (DE, FMS) and (FMS, DE) Equilibria Private Duopoly. From Table 1, (DE, FMS) is an equilibrium when (i) 1;1 1; 0 for rm 1 and (ii) ;4 ;3 > 0 for rm. The rst condition requires that s is higher than the critical value 1 while the second condition implies that s is lower than 4 : Therefore, if 1 < s < 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 1 < s < 4 there are multiple(two) Nash Equilibria. We can state the following lemma: Lemma 4 In a private duopoly, (DE, FMS) and (FMS, DE) are Nash Equilibria if 1 < s < 4 : In particular, given market size a, there exists a critical value such that if > then 4 > 1 and therefore, (DE, FMS) and (DE, FMS) are Nash Equilibria: This critical value is = 0:644: It is interesting to note that only relatively high values of product substitutability guarantee the existence of asymmetric equilibria, in the sense that rms make di ering technology choices. 13 Intuitively, we can argue that these equilibria arise in situations in which when one rm is exible, there are no incentives for the other rm to be exible given the high substitutability between markets, but also when one of the rms is dedicated, there are incentives for the other rm to be exible, since the technology costs are not too high relatively to the size of the market. 14 Mixed Duopoly. We begin with the analysis of the (DE, FMS) equilibrium. In this case, from Table 1, the necessary conditions are (i) 1;1 1; 0 and (ii) ;4 ; < 0: This implies that s must be above and below 6 : Thus, if < s < 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 < s < 6. This is satis ed for values of the substitutability parameter such that 0:3133 or 0:817. For (0:3133; 0:817); (DE, FMS) cannot be an equilibrium. 13 This result is in contrast with Kim, Röller and Tombak (199) where asymmetric equilibria in pure strategies do not exist. 14 In a situation like this, the two rms are interested in being the one using FMS. Given that 1; 1;1 > 0 and 1;3 1;4 > 0 must hold, and by de nition 1;4 > 1; (8 6= 0), then 1;3 > 1;. Given the symmetry of the game, the same applies to rm. Therefore, in the case of asymmetric equilibria the rm using FMS obtains higher pro ts than the one using DE. Therefore, given the multiplicity of equilibria, in their search of maximum pro t rms might end up in the worst scenario possible unless some coordination mechanism is used. 1

13 Next we consider the case of the (FMS, DE) equilibrium. So that (FMS, DE) is an equilibrium it is required that (i) 1;4 1;3 > 0 and (ii) ;3 ;1 > 0, implying that 3 < s < 5 must hold Lemma 6 (FMS, DE) is a Nash equilibrium in the mixed duopoly if 3 < s < 5 : In particular, given market size, a;there exists a critical value such that for > (FMS, DE) is an equilibrium: This critical value is = 0:3133: The intuition behind this result is clear: As substitutability between products increases the public rm tends to choose DE as a response to FMS, as long as the increase in the competition due to the use of a FMS is not o set by the technology costs: the public rm is less willing to bear the technology costs, since they are socially less pro table, when the private rm is already producing the two goods and that there is high substitutability between them. In the following propositon we state that asymmetric equilibria never simultaneously (i.e. under the same market and technology conditions) in the private duopoly and in the mixed duopoly. Proposition 3: (i) For given a > 0 and > if 1 < s < 4 (FMS, DE) and (DE, FMS) are equilibria in the private duopolies but not in the mixed duopoly; (ii) For given a > 0 and = (0:3139; 0:8176) if < s < 6 then (DE, FMS) is an equilibrium in the mixed duopoly but not in the private duopoly; (iii) For given a > 0 and > if 3 < s < 5 then (FMS, DE) is an equilibrium in the mixed duopoly but not in the private duopoly. 4 Analysis of Social Welfare In this section, we examine social welfare across the two market arrangements and in doing so we aim to provide some guidelines with respect to the issue of privatizing the public rm. Our procedure is the following: We start by considering one of the four possible equilibria in the mixed duopoly (FMS, FMS), (DE, FMS), (FMS, DE) and (DE, DE) which correspond to di erent conditions related to the parameters of the model, s; a and. We then establish which would be the corresponding equilibrium outcome in the private duopoly. Having done this, we compare the level of total surplus (social welfare) achieved in the two types of duopoly. We denote by subscripts M the mixed duopoly and by P the private duopoly, followed by 1,, 3 and 4 denoting the (FMS, FMS), (DE, FMS), (FMS, DE) and (DE, DE) equilibria respectively. 13

14 4.1 (FMS, FMS) Equilibrium in the mixed duopoly From Lemma 1 recall that (FMS, FMS) is an equilibrium in the mixed duopoly if s < minf ; 3 g. In addition, the equivalent condition for the private duopoly is s < 1 while from Proposition 1 the critical value for the xed technology costs s is lower in the mixed duopoly than in the private one, minf ; 3 g < 1. So (FMS, FMS) is an equilibrium in both the mixed and private duopoly if s < minf ; 3 g. A straightforward comparison of the total surplus in the two market setups reveals that welfare is higher in the private duopoly except when products are nearly independent, as the following Lemma demonstrates: Lemma 7 T S P 1 T S M1 for 0:03 and T S P 1 < T S M1 for < 0:03: 4. (DE, DE) Equilibrium in the mixed duopoly The relevant condition for a (DE, DE) equilibrium in the mixed duopoly is s > 6 while the equivalent condition in the private duopoly requires s > 4. We can then distinguish the following cases. Case A: s > 6 and s > 4 ; (DE, DE) is the outcome in both market arrangements. Case B(i): s > 6, s < 4 and s 1 ; (DE, DE) obtains in the mixed duopoly while either (DE, FMS) or (FMS, DE) occurs in the private duopoly; Case B(ii): s > 6 ; s < 4 and s < 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. In this case (DE, DE) is the equilibrium in both the mixed and private duopolies so we just need to compare welfare in terms of total surplus, i.e. T S P4 and T S M4. This is done in the following Lemma. Lemma 8 Given s > 6 and s > 4,a > 0 and [0; 1); T S P4 < T S M4. Case B(i). Here the mixed duopoly is characterized by a (DE, DE) equilibrium while the private duopoly equilibrium is either (DE, FMS) or (FMS, DE). Hence the relevant welfare comparison is between total surplus T S M4 in the mixed duopoly and total surplus T S P in the private duopoly - recall that the private duopoly equilibria are symmetric. This comparison constitutes our lemma 9. Lemma 9 Given s > 6, s < 4 and s 1 ; a > 0 and (0:644; 0:6755), T S P < T S M4 : 14

15 Case B(ii). In this case the mixed duopoly equilibrium is (DE, DE) while the private duopoly yields (FMS, FMS). In the following lemma, we compare total surpluses T S M4 and T S P1. Lemma 10 Given s > 6, s < 4 and s < 1, a > 0 and (0:0536; 0:6736), T S P1 < T S M4 : 4.3 (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 < s < 6 and this is satis ed when = (0:3133; 0:8173). In those ranges of values of, the corresponding equilibrium in the private duopoly would be either (DE, DE), if s > 4, or (FMS, FMS) if s < We start by analysing the rst of these cases. Case C : < s < 6 and s > 4 ; (DE, FMS) is the outcome in the mixed duopoly while (DE, DE) obtains in the private duopoly. Comparing total surplus in the two market set-ups yields the following. Lemma 11 Given < s < 6 and s > 4, a > 0 and = (0:0056; 0:8173); T S P4 < T S M : Case D: < s < 6 and s < 1 ; (DE, FMS) is the outcome in the mixed duopoly and (FMS, FMS) in the private one. The relevant welfare comparison is between T S P1 and T S M and is presented in the following Lemma. Lemma 1 Given < s < 6 and s < 1, a > 0 and = (0:3133; 0:8173); T S P1 < T S M : 4.4 (FMS, DE) Equilibrium in the mixed duopoly Finally, let us consider the case of the (FMS, DE) equilibrium in the mixed duopoly. Lemma 6 requires that in this case 3 < s < 5 ; which is guaranteed as long as > = 0:3133. In proposition 3 we have shown that asymmetric 15 Recall that in proposition 3 we have shown that asymmetric equilibria do not arise as equilibria in both types of duopoly for given a, s and. 15

16 equilibria do not arise as equilibria in both types of duopoly for a given set of technology and market conditions. Moreover, it can be easily checked that 4 > 5 and thus, (DE, DE) is never an equilibrium in the private duopoly for values of s such that 3 < s < 5 : On the contrary the conditions for (FMS, FMS) to be an equilibrium in the private duopoly are compatible with 3 < s < 5, since 1 > 5. Hence, whenever the equilibrium in the mixed duopoly is (FMS, DE) the counterpart in the private duopoly is (FMS, FMS). Therefore, the only comparison that proceeds in this case is between T S P1 and T S M3. Lemma 13 Given 3 < s < 5 ; s < 1 and a > 0, T S P1 < T S M3 : The results we have obtained regarding welfare comparisons across the two market arrangements have some potential implications for the debate about the privatization of a public rm. Based on our analysis the following table summarizes. [Table about here] Privatization will enhance social welfare when the outcome in the mixed duopoly is (FMS, FMS), i.e. both rms are choosing exibility in their production, provided that products are not (almost) independent. The private duopoly outcome would also be (FMS, FMS) but would result in higher levels of social welfare. In all other cases, privatization would result in a reduction in social welfare. The following Proposition summarizes. Proposition 4: Privatization increases social welfare when the equilibrium outcome in the mixed duopoly is (FMS, FMS) and > 0:03. In the remaining cases privatization of the public rm would reduce social welfare. The relative strength of the proposition and its policy implications derive from the fact that it can be used even without knowing the exact values of a, and s, requiring only knowledge of the observed behavior of both parties in the mixed duopoly. It seems quite plausible to assume that policy makers know accurately the strategic plans of public rms, in this case the FMS investment plan in technology choice and the similarity of the produced goods. If the public rm has plans to undertake FMS investment, policy makers should consider the possibility of a privatization. If the public rm does not have any intention 16

17 of replacing its DE with FMS, then a privatization should not be considered. However, a word of caution is warranted here. The results we obtain are based in 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 also whether the results are sensitive to the mode of competition, i.e. quantity versus price. We leave this aside for future research. 5 Concluding Remarks In this paper we have introduced a mixed duopoly in the context of a Cournot di erentiated-goods duopoly facing the decision of whether to use a exible or a dedicated technology. We have also analyzed the private duopoly case so as to compare the outcomes of the two di erent market arrangements and give some tentative guidelines on the privatization of a public rm. Our main ndings can be summarized as follows: an equilibrium with the two rms choosing exible technologies is more likely to arise in the private duopoly than in the mixed one. Further, an equilibrium involving the two rms 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 rm being exible and the other dedicated, are less likely to obtain in the private duopoly. In the case of a mixed duopoly, the public rm chooses a dedicated technology when products are very close substitutes, since it is not pro table from the social welfare point of view to bear higher technology costs in order to produce almost the same good. Our results con rm the empirical ndings in the literature on competition between public and private hospitals, which show that public hospitals o er a wider range of services than private hospitals and that private hospitals are more likely to provide new services when there is intense competition. Privatization of the public rm is warranted when the market and technology conditions lead to an equilibrium outcome such that both rms use exible technologies and goods are not (almost) independent. In all remaining of cases, privatizing the public rm would result in a reduction of social welfare. 17

18 References [1] Boyer, M. and M. Moreaux, 1997, Capacity-Commitment versus Flexibility, Journal of Economics and Management Strategy, 6, [] Boyer, M., Jacques, A. and M. Moreaux, 00, Observation, Flexibilité et Structures Technologiques des Industries, Cahiers de la Série Scienti que/scienti c Series, 1, Cirano, University of Montreal. [3] Cantos-Sánchez, P., R. Moner-Colonques and J.J. Sempere-Monerris, 003, Competition Enhancing Measures and Scope Economies: A Welfare Appraisal, Investigaciones Económicas, 7, [4] De Fraja, G. and F. Delbono, 1990, Game-Theoretic Models of Mixed Oligopolies, Journal of Economic Surveys, 4, [5] Eaton, B.C. and N. Schmitt, 1994, Flexible Manufacturing and Market Structure, American Economic Review, 84, [6] García-Gallego and N. Georgantzís, 1996, Multiproduct Activity and Competition Policy: The Tetra-Pack Case, European Journal of Law and Economics, 3, [7] Gupta, S., 1998, A Note on "Strategic Choice of Flexible Production Technologies and Welfare Implications", Journal of Industrial Economics, 46, 403. [8] Jaikumar, R., 1986, Postindustrial Manufacturing, Harvard Business Review, Nov.-Dec., [9] Kim T., Röller, L-H. and M. Tombak, 199, Strategic Choice of Flexible Production Technologies and Welfare Implications: Addendum and Corrigendum, Journal of Industrial Economics, 40, [10] Matsumura, T., 1998, Partial Privatisation in mixed Duopoly, Journal of Public Economics, 70, [11] Ozcan, Y.A., R.D. Luk, C. Haksever, 199, Ownership and Organizational Performance. A Comparison of Technical E ciency Across Hospital Types, Medical Care, 30, [1] Pal, D., 1998, Endogenous Timing in a Mixed Oligopoly, Economics Letters, 61, [13] Röller, L-H. and M. Tombak, 1990, Strategic Choice of Flexible Production Technologies and Welfare Implications, Journal of Industrial Eco- 18

19 nomics, 38, [14] Röller, L-H. and M. Tombak, 1993, Competition and Investment in Flexible Technologies, Management Science, 39, [15] Schlesinger M., 1998, Mismeasuring the Consequences of Ownership: External In uences and Comparative Performance of Public, For-Pro t and Private Nonpro t Organizations, in Walter Powell and Elizabeth Clemens (eds.), Private Action and the Public Good, New Haven: Yale. [16] Schlesinger, M., R. Dorwart, C. Hoover and S. Epstein, 1997, Competition, Ownership and Access to Hospital Services: Evidence from Psychiatric Hospitals, Medical Care, 35, [17] Shortell, S.M., E.M. Morrison, S.L. Hughes, B.S. Friedman and J.L. Vitek, Diversi cation of Health Care Services: The E ects of Ownership, Environment and Strategy, 1987, Advances in Health Economics & Health Services Research, L. Rositer and R. Sche er eds., 3-40 [18] Shortell, S.M., E.M. Morrison, S.L. Hughes, B.S. Friedman, J. Coverdill and L. Berg, 1986, The E ects of Hospital Ownership on Nontraditional Services, Health A airs, 5,

20 6 Mathematical appendix: 6.1 Private duopoly. Equilibrium solutions (FMS, FMS) Firm 1 Firm T S a (3+) a (4+3) (1 + s) (3+) (4+3) (1 + s) a (5+) (4+3) (1 + s) (DE, DE) Firm 1 Firm 3a (3+) 1 3a (3+) 1 T S 4a (3+) (FMS, DE) Firm 1 Firm T S a(300 (3 )(9+(59 1)) 4 11 (1 + s) a (+)(154 (15 (81 4))) (4 11 ) ( + s) (DE, FMS) Firm 1 Firm T S 3a (6 ) (4 11 ) 1 a(300 (3 )(9+(59 1)) 3a (6 ) (4 11 ) (1 + s) a (+)(154 (15 (81 4))) (4 11 ) ( + s) 6. Mixed duopoly: Equilibrium solutions (FMS, FMS) Private Firm Public Firm T S a (3+) 4a (1+3+ ) (5+) (1 + s) (1+)(5+) (1 + s) a (8+(5+)) (1+)(5+) (1 + s) (DE, DE) Private Firm Public Firm T S 3a ( ) a 8(3 ) 1 (9 (5 )) 8(3 ) 1 a (17 (14+(1 ))) 4(3 ) (FMS, DE) Private Firm Public Firm T S a ( ) a (18+(3 (9 4)) (15 8 ) 1 (15 8 ) (1 + s) a (61 (9+(3 4)) (15 8 ) ( + s) (DE, FMS) Private Firm Public Firm 3a 50 1 T S a (41 (143 0(6 )) 00(1 ) (1 + s) a (57 4(15 )) 100(1 ) ( + s) 0

21 6.3 Proofs to Lemmata and Propositions: Proof Lemma 1: Note =@ < 0; 3 =@ < 0: Further, from (8) and (9) we obtain j =0 = 0:06a, j!1 = 0:04a, 3 j =0 = 0:0977a, 3 j!1 = 0 and 3 j =0 > j =0 while j!1 > 3 j!1 = 0. Therefore and 3 must cross. Setting (8) and (9) equal we obtain = 0:43 where and 3 cross. The result then follows immediately (Q.E.D). Proof Proposition 1: Lemma 1 establishes that the relevant critical value for s in the mixed duopoly is minf ; 3 g; in particular, for < the relevant critical value is given by and for it is given by 3, = 0:43: Thus, we need to show that < 1 for < and 3 < 1 for : Note 1 =@ < =@ < 3 =@ < 0: Further, from (7) and (8) we obtain 1 j =0 = 0:0937a and j =0 = 0:06a respectively. 1 = at = 0:4593 > and j =0 < 1 j =0 : Therefore, < 1 when <. Similarly, from (7) and (9) we obtain 1 j!1 = 0:01a, and 3 j!1 = 0 respectively. 1 = 3 at = 0:0393 < and 3 j!1 < 1 j!1 : Therefore, 3 < 1 when and we have shown that minf ; 3 g < 1. The rest of the proposition follows by considering the conditions for (FMS, FMS) equilibrium, i.e. conditions (7), (8) and (9) (Q.E.D). Proof Lemma : From (11) and (1), 6 5 = a f 5;6() 00( 3) ( 1)(8 15) : This is positive as f 5;6 () < 0; where f 5;6 () = ; and the denominator is negative as lim!1 < 0. (Q.E.D). Proof Lemma 3: Note that 4 j =0 = 0:0937a, 6 j =0 = 0:0978a, 4 j =1 = 0:046a and lim!1 6 = 1. Therefore, 4 j =0 < 6 j =0 and 4 j =1 < lim!1 6 = 1. 6 reaches its minimum at = 0:6689 while 4 j =0:6689 = 0:0393a and 6 j =0:6689 = 0:0388a, meaning that 4 j =0:6689 > 6 j =0:6689 : Hence, 4 and 6 must cross twice: setting 4 and 6 equal, we nd that they cross at 1 = 0:0056 and at = 0:6755: The rest of the lemma follows (Q.E.D). Proof Proposition : Follows from Lemma 3 and the necessary conditions for equilibrium, i.e. conditions (7), (8) and (9) (Q.E.D). 1

22 Proof Lemma 4: Using (7) and (10) we obtain 1 4 = a f 1;4() (3+) (4+3) (4 11 ) where f 1;4 () = R 0 for R = 0:644: The rest of the lemma follows immediately. (Q.E.D) Proof Lemma 5: Note that 6 j =0 = 0:1a, j =0 = 0:06a ; 6 j!1 = 1 ; and j!1 = 0:04a : =@ < 0 6 =@ Q 0 for Q 0:6669. Setting and 6 equal, we nd that they cross at = 0:3133 and at = 0:817: It is then obvious that < 6 when 0:3133 and when 0:817 and > 6 when (0:3133; 0:817): The rest of the lemma follows from the equilibrium conditions. (Q.E.D) Proof Lemma 3 =@ < 0 5 =@ < 0: Furthermore, 3 j =0 = 0:0977a, 3 j!1 = 0, 5 j =0 = 0:06a and 5 j!1 = 0:0083a ;so that 3 j =0 > 5 j =0 = 0:06a while 3 j!1 = 0 < 5 j!1 : Therefore, 5 and 3 cross at a critical value of the parameter of substitutability, = 0:3133: Thus, if ; 5 3 and if > ; 5 > 3 : The rest of the lemma follows from equilibrium conditions (9) and (11). (Q.E.D) Proof Proposition 3: As seen in lemma 4, so that (DE, FMS) or (FMS, DE) are equilibria in the private duopoly it is needed that 1 < s < 4 and this can only happen for > = 0:64405: Recall that (DE, FMS) is equilibrium in the mixed duopoly if < s < 6. We know =@ < 0 4 =@ < 0 and that j =0 = 0:06a, j!1 = 0:04a, 4 j =0 = 0:9375a, 4 j!1 = 0:0459a. Therefore j =0 < 4 j =0 while j!1 > 4 j!1 :Thus, they must cross at a certain value of. Setting and 4 equal, we know that Q 4 for Q 0:450595: Therefore for >, > 4, meaning this that 1 < s < 4 and < s < 6 can not hold simultaneously. Furthermore recall that (FMS, DE) is equilibrium in the mixed duopoly if 3 < s < 5. We know 1 =@ < 0 5 =@ < 0 and that 1 j =0 = 0:09375a, 5 j =0 = 0:06a ; 1 j!1 = 0:06a and 5 j!1 = 0:00938a : Thus, 1 > 5 for any and therefore 1 < s < 4 and 3 < s < 5. The rest of the proposition follows. (Q.E.D). Proof Lemma 7: T S P 1 T S M1 = a f P1 M 1 () (1+) (5+) (4+3) where f P1;M 1 () = Q 0 for Q 0:03: Hence T S P 1 T S M1 if 0:03 and T S P 1 < T S M1 if < 0:03. (Q.E.D). Proof Lemma 8: T S P4 T S M4 = a (1 ) f p4 M 4 () 4(3+) (3 ) where f P4S 4 () = ( ) < 0 for any : Hence T S P4 < T S M4 :(Q.E.D).

23 Proof Lemma 9: From Lemma 4, 1 < 4 if and only if > = 0:644. Further, from Lemma 3, 6 4 < 0 if and only if 0:0056 < < 0:6755. Hence the relevant range for is (0:644; 0:6755). It can be checked that the di erence T S P T S M4 is decreasing in s and T S P T S M4 j s=1 = a f P M 4 () 4(4+3) (3 ) (4 11 ) > 0 as f PM 4 () = > 0 for (0:644; 0:6755): Note also that in this region of ; 1 > 6. Then, given that s 1, T S P < T S M4 : Proof Lemma 10: From (7) and (1) 1 6 = a f 1;6() 50(4+3) (3 ) (4 11 ) (1 ) and sign( 1 6 ) = signf 1;6 (), where f 1;6 () = Note that f 1;6 () > 0 for (0:0536; 0:6736), which is the relevant range for. It can be checked that the di erence T S P1 T S P4 is decreasing in s and a T S P1 T S M4 j s=6 = f P1 M 4 () 100(4+3) (3 ) (1 ) < 0 as f P 1M 4 () = < 0: Then, given that s > 6 it follows that, in the relevant region of, T S P1 T S M4 < 0:(Q.E.D) Proof Lemma 11: From Lemma 3, 4 < 6 if and only if = (0:0056; 0:8173) and from Lemma 5, < 6 if and only if = (0:3133; 0:8173) so the relevant range for is = (0:0056; 0:8173). It can be checked that the di erence T S P4 T S M is increasing in s; further a T S P T S M4 j s=6 = f P4 M () 100(3+) (3 ) (1 ) < 0 as f P4M () = < 0. Then, given that s < 6 it follows that, in the relevant region of, T S P < T S M4 :(Q.E.D) Proof Lemma 1: From Lemma 5, < 6 if and only if = (0:3133; 0:8173). Further, from the proof of Proposition 1, < 1 if and only if < 0:4593: Therefore the relevant range for is = (0:3133; 0:8173). It can be checked that the di erence T S P1 T S M is decreasing in s. Then T S P1 T S M j s= = a f P1 M () 100(5+) (4+3) (1 ) < 0 as f P1M () = < 0. Hence, given that s > it follows that, in the relevant range for, T S P1 < T S M. (Q.E.D) Proof Lemma 13: It can be checked that the di erence T S P1 T S M3 is decreasing in s. Then, T S P1 T S M3 j s=3 = f P1 M 3 () a (1+)(5+) (4+3) < 0 as f P1M 3 () = 3

24 > 0: Hence, given that s > 3 it follows that T S P1 < T S M3 :(Q.E.D) Proof Proposition 4: Follows from lemmata (Q.E.D) 4

25 7 Table : Summary of policy guidelines Mixed Private Conditions Duopoly Duopoly s < minf ; 3 g (FMS, FMS) (FMS, FMS) Recommendable Privatization Yes (if > 0:03) (DE, DE) if s > 4 s > 6 (DE, DE) (FMS, FMS) if s < 1 No (FMS, DE) if 1 < s < 4 (DE, FMS) if 1 < s < 4 3 < s < 5 (FMS, DE) (FMS, FMS) No < s < 6 (DE, FMS) (DE, DE) if s > 4 No (FMS, FMS) if s < 1 5

26 6

27 7