Market size, technology choice, and market structure

Transcription

1 Market size, technology choice, and market structure Walter Elberfeld and Georg GÄotz y June 29, 2001 Abstract We introduce technology choice into a model of monopolistic competition and analyze the structural e ects of changes in market size. A larger market leads to the adoption of a large scale technology. If a technology switch occurs, the number of rms decreases, and a rationalizing e ect arises: individual and aggregate output increases; prices fall. This need not bene t consumers since a technology switch is associated with a decrease in product variety. Keywords: technology choice, monopolistic competition, shakeout, variable elasticity of substitution JEL classi cation: L10 Staatswissenschaftliches Seminar, UniversitÄat zu KÄoln, Albertus-Magnus-Platz, D KÄoln, Germany, elberfeld@wiso-r10.wiso.uni-koeln.de y Department of Economics, University of Vienna, BWZ - BrÄunnerstr. 72, A-1210 Vienna, Austria, georg.goetz@univie.ac.at 1

2 1 Introduction Market size and demand growth are key variables in the explanation of market structure, technological change and industry evolution. The size of the market limits the advantages of specialized work and machinery. This relationship was already stated by Adam Smith in his famous theorem that `the division of labor is limited by the extent of the market. Increases in demand often induce the adoption of new and usually more specialized technology, which are typically characterized by higher xed costs and lower variable costs. The introduction of a new technology with higher xed costs necessitates a larger rm size. The implications of the technology switch for market structure are, however, ambiguous. On the one hand, non-switching rival rms may lose market share, and perhaps have to exit. This would reduce the number of rms. On the other hand, a larger market can be expected to accommodate a larger number of rms if free entry prevails. Thus, it is not clear, a priori, whether the introduction of a new technology increases or decreases the number of active rms in the market. Empirically, it has been observed that rm numbers rst rise, then later fall, as an industry evolves. In a study of 46 major developing industries, Gort and Klepper (1982) found that most of these markets have experienced some degree of shakeout in the number of rms at some point in their history. In extreme cases the number of producers fell by 90 % or more over 15 to 20 years despite robust growth in output. A number of authors has attempted to explain the phenomenon of industry shakeouts theoretically. For instance, Klepper and Graddy (1990) rely on chance events and exogenous factors. These e ects in uence the number of potential entrants to the industry, the growth rate of incumbents, as well as the ease of imitation of industry leaders and thereby a ect the ultimate number and the size distribution of rms in the industry. Jovanovic and MacDonald (1994) explain a non-monotonic time path in the number of rms using a competitive model in which innovation opportunities fuel entry and relative failure to innovate prompts exit. According to the model of Carree and Thurik (2000), the non- 2

3 monotonicity in rm numbers found in many young industries is a consequence of the gradual decline in unit costs. The common feature of all these papers is that they use a dynamic framework to explain characteristic features of an industry life cycle. In the present article, we propose a static model to answer the question why industry shakeouts can occur. We set out a model of monopolistic competition, in which technology choice is endogenous. Firms are allowed to choose from a binary set of technologies. They may choose either a technology with high xed costs and low marginal costs (largescale technology) or one with low xed but high marginal costs (small-scale technology). Within this framework, we show that technology choice leads to a non-monotonic relation between market size and the number of rms. If market size reaches a critical value and a technology switch occurs, some rms have to exit so that the total number of rms in the industry decreases. By giving an example, we demonstrate that the fall in the number of rms may be drastic. A technology switch and the resulting market consolidation are accompanied by a rationalizing e ect: individual and aggregate output increases; prices fall. Our approach of explaining industry shakeouts can be seen as complementary to the above cited models. Its distinguishing features are the reliance on the importance of demand growth for the adoption of new technologies as well as its simplicity. Both the theoretical and empirical literature on the relationship between market size and market structure is extensive; see, for example, Schmalensee (1989), and Scherer and Ross (1990). Among the most important work, using a static model, has been done by Shaked and Sutton (1987), and Sutton (1991), (1998). Sutton (1991) has argued that the relationship between market size and concentration varies between industries, depending on the existence of exogenous or endogenous sunk costs. Using an oligopolistic model of vertical product di erentiation, he showed that the traditional growth-entry mechanism may not work in industries in which advertising and/or R&D outlays play an important role. Neumann et al. (2001) have analyzed a static Cournot oligopoly with homogenous prod- 3

4 ucts to relate horizontal concentration to market size and xed costs. Assuming that higher xed costs yield a lower level of marginal costs, a vertical and horizontally growing market is shown to cause horizontal concentration to decline if there is free entry. Both Sutton (1991) and Neumann et al. (2001) use a continuous set of technologies from which rms can choose. This implies that their models do not exhibit the drastic nonmonotonicity in rm numbers, which is an important feature of our model, where the technology set is discrete. Whether the process of technology adoption in an industry is better described by continuous or discrete opportunity sets, is ultimately an empirical question. While the assumption of a continuous set is more appropriate if innovations result from a large number of small changes, discrete sets should be assumed if the improvement in productivity results from major technological breakthroughs. The market model we use is a derivation of the Dixit/Stiglitz (1977) model of monopolistic competition. However, rather than restricting ourselves to the well-known and often used case of a utility function yielding a constant elasticity of substitution (CES), we consider the more general case of variable elasticity of substitution (VES) proposed by Dixit and Stiglitz (1977). Contrary to the CES-speci cation, the VES-case exhibits the comforting feature of a negative relationship between the number of rms and the price they charge. The "large group" assumption (Chamberlin, 1965) underlying the market structure of monopolistic competition ensures virtually no strategic interaction among rms. Yet, individual rms have monopoly power because they produce a special brand of a differentiated product. The former feature allows us to focus on the e±ciency argument, that is the cost minimization argument, for a technology switch. By the latter feature, we capture the market power e ects of a technology switch. In view of the trade-o between productive e±ciency and product variety, our welfare analysis answers the question whether rms make e±cient technological choices in the decentralized solution. The paper is organized as follows. Section 2 sets up the framework, and characterizes the free entry equilibrium. In Section 3, we analyze the e ect of changes in market size. In Section 4, we brie y investigate social welfare. Section 5 concludes. 4

5 2 The framework We consider an industry in which a continuum of potential rms exists. The market structure is assumed to be monopolistically competitive, and entry into the industry is free. A rm can choose between two available technologies, S and L, forproducingits variety. Technology S entails low xed costs, f S, and high (constant) marginal costs, c S. Technology L exhibits high xed costs, f L, but low (constant) marginal costs, c L.Thus, we have f S <f L and c S >c L. Each active rm produces a single, unique variety of a di erentiated product. Therefore, the measure of rms equals the measure of brands available to the consumers. For simplicity, we refer to this measure as the "number of rms". The measure (number) of consumers is A. Each consumer maximizes the quasi-linear utility function of the form U = y + bc e =e 0 <e<1; (1) where y denotes the quantity of a numeraire. The consumption index C is of the Dixit- Stiglitz (1977) type and de ned as C = µz n where x(j) is the quantity consumed of variety j, and 0 v(x (j)) dj 1= 0 < <1; (2) v (x (j)) = ( k + mx(j) for x>0 0 for x =0 (3) with k 0andm>0. This functional form is proposed by Dixit and Stiglitz (1977) as an example of a function which entails variable elasticity of substitution (VES) between any two varieties. The well known CES-speci cation arises as a special case, if k =0and m = 1. As will become clear below, the VES-speci cation has certain advantages over the CES-speci cation. Aside from being more general, it implies the intuitive feature that an increasing number of rms leads to lower prices. The budget constraint reads as y + Z n 0 p (j) x (j) dj = I; (4) 5

6 where I denotes the consumer's income and p(j) is the price of variety j. We assume that income is large, so that the numeraire will always be demanded by the consumer. A consumer's demand for variety i takes the form µz n e x(i) = k=m +(m e b) 1 1 e p(j) e 1 dj p(i) 1 1 : (5) 0 Since b and k in (5) are arbitrary constants, without loss of generality, we can set m =1. Furthermore, we de ne B = b 1=(1 e) and P = µz n 0 p(j) 1 dj ( 1)= ; where P denotes the Dixit-Stiglitz price index. With these de nitions, overall demand for variety i denoted by X(i) can be written as X(i) =Ax(i) =A ³ k + BP 1=(1 ) 1=(1 e) p(i) 1=(1 ) ; (6) which is a standard demand function with a nite cut-o price. The supplier of variety i treats P as a constant, and so perceives a demand elasticity of j (1 ) 1 f(p i )j( k + f(p i )) 1,wheref(p i )=BP 1=(1 ) 1=(1 e) p(i) 1=(1 ). Observe that the price sensitivity of demand increases with k, sothatitislowest,ifk = 0. In this case (CES-case), the elasticity of demand is a constant and equal to (1 ) 1. Finally, note that demand of a rm would be a decreasing function of P and thus a decreasing functions of the rivals' prices, if e>. In this case, the di erentiated goods would be complements rather than substitutes. To exclude this case, we assume e<. 2.1 Prices and pro ts Suppose that an industry structure has emerged, where n S rms produce with technology S (S- rms), and n L with technology L (L- rms). Firms' pro ts depend on the chosen technology. The pro t function of rm i, who chooses technology j, isgivenby (i) =(p(i) c j )X(i) f j ; j 2fS; Lg: (7) Firms maximize pro ts by choosing a price p i. A price equilibrium, where rms with the same technology choose the same price, must solve the following two rst order conditions: 6

7 0 k BP1=(1 ) 1=(1 e) ( p i c i )p (2 )=( 1) i A =0; i = S; L: (8) 1 In (8), the price index P can be computed as 1 P = ³ n S p =(1 ) S + n L p =(1 ) (1 )= L : If k = 0 (CES-case), we obtain the familiar pricing rule p i = c i =, which is independent of the number of rms. On the other hand, if k>0 (VES-case), the equilibrium prices p S and p L are decreasing functions of n S and n L. The fact, that prices decrease with the number of rms is a desirable and plausible feature of the model. It states that entry of new rms leads to more competition, resulting in lower prices. Since a rms's perceived elasticity of demand is increasing in k, prices in the VES-case are always lower than in the CES-case. Substituting p S and p L into (7) yields the equilibrium pro ts S = A (p S c S ) ³ k + Bp S 1 1+ P 1=(1 ) 1=(1 e) f L (9) for a rm using technology S, and L = A (p L c L ) ³ k + Bp L 1 1+ P 1=(1 ) 1=(1 e) f L (10) for a rm using technology L. 2.2 Entry and technology choice For any given industry structure (n S ;n L ), the equilibrium prices are functions of (n S ;n L ). Thus, reduced form pro ts are functions of the numbers of rms, ¼ i = ¼ i (n S ;n L ), i = S; L. Of course, an arbitrary industry structure need not represent an equilibrium state. If a rm using technology S can improve its pro ts by switching to technology L, itwantsto use technology L. Likewise, a rm using technology L maywanttoswitchtotechnology S. Thus, an equilibrium (industry) structure must satisfy the following no-switching condition: ¼ S (n S ;n L )=¼ L (n S ;n L ): (11) 7

8 Moreover, since entry into the market is free, in equilibrium the pro ts in (11) must be zero. Proposition 1 In equilibrium, all rms will (almost) always choose the same technology. 1 Proof see Appendix. According to Proposition 1, only two types of equilibrium structures can arise. One where all rms use technology S, and one where all rms produce with technology L. The outcome where S- rms and L- rms coexist can arise only as a knife-edge case. The non-existence of equilibrium structures with heterogeneous rms is due to our assumption that the market is monopolistically competitive. 2 In such a market, agents are `small', so that a single rm, when it switches from one technology to another, does not a ect the pro ts of its rivals. Therefore, the no-switching condition (11) requires that pro ts of all rms are exactly the same. This implies that equilibrium structures with heterogeneous rms, can arise only on a set with measure zero. 3 When agents are `large', a switch of technology by one rm in general a ects the pro ts of all rms, including those of the switching rm. Thus, an equilibrium is possible in which the pro ts of the two di erent types are di erent. In fact, this would happen if the switch to a more pro table technology reduced the pro ts of the latter type so much that the rm is better o by not switching. 4 To obtain further insight into the determinants of technology choice implied by Proposition 1, it is instructive to consider the special CES-case (k = 0). In this case, the 1 Exceptions can occur only on a parameter set of measure zero. 2 Using a model with monopolistic competition, Grossman and Helpman (1999) obtain a similar knifeedge result with respect to vertical integration. 3 The assumption of `small' agents also leads to robustness of the equilibrium with respect to the modes of entry and the timing structure of the model. It is irrelevant whether rms decide about entry, technology and pricing simultaneously or sequentially. As entry deterrence cannot occur, it does not matter whether rms enter simultaneously or sequentially. With `large' agents, for instance in a Cournot oligopoly, all these di erences lead to di erent results (see, for instance, Brander and Spencer (1983) and Dasgupta and Stiglitz (1980) for issues regarding the timing structure, and McLean and Riordan (1989) for entry deterrence). 4 see Mills and Smith (1996), Elberfeld (2001) and GÄotz (2001). 8

9 no-switching and the zero-pro t conditions can be satis ed if, and only if, := ( c L c S ) 1 f S f L =0: (12) If <0 only L- rms are active, whereas only S- rms are active if >0. 5 This implies that technology choice depends on the cost structure in a way one would expect. The use of a particular technology is more likely the smaller its marginal or xed costs. Another determinant of technology choice is the degree of product di erentiation measured by the parameter. A market in which the nal goods are close substitutes ( close to 1) is likely to be one where rms use the technology, which entails lower marginal costs. The reason is that close substitutability makes pro ts highly sensitive to relative prices, and so the lower prices and the economies of scale associated with technology L become more important. 3 E ects of increasing market size In this section we analyze the structural e ects of changes in market size A. In the CEScase the answer to this question is simple. If k = 0, technology choice is independent of market size. Formally, this is an immediate consequence of the fact that A does not enter the parameter. Since rms use a mark-up pricing rule, which is independent of the number of rms, individual outputs do not change with market size. Otherwise, pro ts could not be kept at zero. Thus, if market size increases, the rise in aggregate output is solely caused by an increase in the number of rms. The VES{case is more complicated and more interesting. If k>0, individual prices are decreasing in the number of rms, implying that individual output increases with market size. A higher output provides an incentive to incur xed costs in order to lower marginal costs. Thus, in order to realize economies of scale, one should expect rms to choose the L{technology, when the market becomes su±ciently large. 5 In the general case, we can not calculate the dividing hyperplane, because the pricing rule can not be explicitly determined. 9

10 Proposition 2 For any market sizes A 1 >A 2 such that di erent technologies are used L is used in A 1 and S in A 2. Proof see Appendix. Proposition 2 con rms our intuition that in larger markets rms adopt technologies, which allow them to better achieve economies of scale. It may, however, be the case that technology S is used independently of market size. To understand this, recall that in the borderline case k = 0 individual output, and therefore technology choice, is independent of market size A (see (12)). If k is greater than zero, output per rm increases with A so that, in general, for market size su±ciently large, the adoption of the large scale technology becomes pro table. We cannot exclude, however, that for positive, but small k the increase in individual output due to increases in market size is too small to induce aswitchintechnology. Next, we explore the e ect of a technology switch on the equilibrium structure of the industry. Proposition 3 Suppose that due to an increase in market size a technology switch occurs. Then, prices and the number of rms decrease, whereas individual and aggregate output increases. Proof see Appendix. Propositions 2 and 3 provide us with the following story of the evolution of an industry. If demand is small, the level of production is too small to support technologies which rely on highly specialized equipment. Firms, therefore, use small scale technologies. However, as the market grows, the number of rms and output per rm increases. With increasing rm size, rms are better able to achieve scale economies. As a result, they may adopt a technology with higher xed costs and lower marginal costs. If a technology switch occurs, some rms will exit. The market wide adoption of the large scale technology entails a rationalizing e ect: individual and aggregate output increase; prices fall. 10

11 All these properties are described as stylized facts for the U.S. automobile tire industry by Jovanovic and MacDonald (1994, p. 324). Furthermore, the shakeout which took place in that industry is explained as the result of a single technological improvement, which is also the driving force in our model. Our approach di ers from the one Jovanovic and MacDonald (1994) in that the passing of a critical market size rather than the random arrival of a new technology causes the shakeout. Our explanation relates to the `demand pull' theory of technical change: Increases in demand induce the introduction of new technology. 6 The results of a simulation, presented in Figures 1-5, show in more detail how industry evolution looks like in our framework. The parameters used are =0:5;c L =1;c S =2;f L =50;f S =20;k =1;b = 100;e=0:25: (13) n A 7 A SB 12 A Figure 1. The equilibrium number of rms (strong line) and the second-best 7 (dashed line) number of rms as a function of market size. 6 For a survey of the 'demand pull' theory of technical change see Thirtle and Ruttan, The second-best criterion requires that rms break even. See Section 4 for a discussion of the welfare properties. 11

12 p A 7 A SB 12 A 2 Figure 2. The equilibrium price (strong line) and the second-best (dashed line) price as a function of market size. x A 7 A SB 12 A Figure 3. The equilibrium output per rm (strong line) and the second-best (dashed line) output per rm as a function of market size. Aggregate Output A Figure 4. Aggregate output of the di erentiated product, i.e. nx, as a function of market size. 12

13 Aggregate Expenditure A 7 A SB 12 A Figure 5. Aggregate expenditure for the di erentiated product, i.e. pnx, as a function of market size. As long as market size is smaller than A =4:5, all rms use the small scale technology S. However, if market size reaches the critical value A, all rms adopt technology L. The technology switch leads to a decline in the number of rms (see Figure 1), from n S =98:7 ton L =56:2, an increase in individual output (see Figure 2), from X S =15:8 to X L =57:8. Price declines from p S =3:3 top L =1:9. As c L =1andc S = 2, (relative) markups and price-cost margins (de ned as (p c)=p) increase. The diagrams relating to variables for the whole industry show that the fall in the number of rms is more than compensated by the rise in individual output. Industry output increases from to (Figure 4). The lower prices induce consumers to spend more on the variants of the di erentiated product, implying an increase in aggregate expenditure from to (Figure 5). With further increases of A, the number of rms rises again, price decreases further, outputs and aggregate expenditure increase further. 8 A notable feature of the model is the extent of the change in the number of rms. In the example, the number of rms decreases by as much as 40 percent. Thus, our model is able to generate severe shakeouts, as they are found in the life cycle of many industries (see, for instance, Klepper and Graddy (1990)). 8 Although our analysis is restricted to the case of two alternative technologies, one can imagine, that the presence of a third technology with smaller marginal costs than c L = 1 and larger xed costs than f L = 50, would generate another switch point A >A, at which the number of rms would jump down again. Thus, the speci c path of rm numbers is likely to be very sensitive to the set of available technologies. In contrast, outputs and prices are always monotonic functions of market size. 13

14 Our model sheds new light on a recent contribution by Neumann et al. (2001). These authors examine the relationship between market size and market structure in a Cournot oligopoly, where rms, by investing in R & D, generate process innovations. They show that larger markets induce both more process innovations and an increase in the number of rms. Neumann et al. argue that the elasticity of the number of rms with respect to market size should be between.5 and 1. Our analysis suggests that their upper bound is sensitive to the speci cation of their model. This can be most easily seen by considering the CES-case. If A increases by a factor t, the number of rms increases by more than t. The increase in the number of rms leads to a decrease of the price index P and thus to increased aggregate expenditure even though individual prices are constant. Thus, as long as the elasticity of the aggregate demand for the di erentiated product is greater than 1, i.e. e>0, the number of rms rises more than proportional. Our model puts also the lower bound into doubt. The non-monotonicity result of Proposition 3 shows that the number of rms might even fall. Neumann et al. estimate an elasticity of the number of rms with respect to market size (denoted by B in their model) of about.5 for a cross-section of 291 German manufacturing industries. Furthermore, 'estimates for individual industries from the balanced sample... indicate substantial di erences in the size of B across markets' (Neumann et al., 2001, p. 835). In view of our results, these di erences across industries are not surprising. In general, technological opportunities di er among industries. Therefore, di erent industries should be expected to move along di erent technological trajectories, implying diverse relationships between market size and market structure. A nal point we want to discuss in this section concerns the relationship between market size and competition. Asplund and Sandin (1999) argue that 'competition intensi es with market size' (p. 70). They use variable pro ts per capita as a measure of competition. In our model increasing market size leads to more competition in the sense of lower prices and lower price-cost margins in the regions where no technology switch occurs. However, comparing points just to the right and to the left of A in our example shows that market 14

15 power increases with market size in the sense that price-cost margins increase. 9 At the same time, prices continue to fall. Interestingly, this prediction ts quite well with some empirical results of Asplund and Sandin. They also obtain the result that prices in a large market (a duopoly, in their case) are lower than in a small market (with a monopoly rm), while pro ts are higher in the large market. They try to explain this 'unreasonable' result by referring to higher xed cost (in the duopoly case) giving lower marginal costs (p. 80). 4 Comparing market outcomes to the second best: an example In this section, we brie y analyze social welfare. We ask whether the technology choice in the market solution coincides with that of a social planner. As a welfare criterion we use the Dixit-Stiglitz (1977) second best solution, where rms must break even. Since free entry leads to zero pro ts of all rms, social welfare coincides with consumers' surplus: CS A(U I) = 1 e ABP e 1 e + Ak(nL p L + n S p S ): (14) e In the derivation of the r.h.s we used the indirect utility function and the budget constraint. The planner's optimization problem consists of maximizing consumers' surplus (see 14) subject to the zero-pro t conditions for the rms. For the CES-case, technology choice in the decentralized solution coincides with the choice of the social planner. The reason is that, in the market solution, individual output is the same as the socially optimal one (see Dixit and Stiglitz, 1977 and Beath and Katsoulacos, 1991). Given that rms are cost-minimizing, technology choices must coincide. Figure 1 shows that the technology choice of the social planner, in general, does not coincide with the market solution. It indicates that the second-best solution puts variety before economies of scale. There are more and smaller rms than in the market solution. The switch to the large scale technology occurs at a greater value of A (A SB =9:60). As 9 In our example the price-cost margin (p c)=p increases from 39% to 46%. 15

16 a result, prices are higher and rm output is smaller in the second-best solution compared to the market solution (see Figures 2 and 3). The fact that the planner's and the decentralised technology choice do not coincide has important consequences for the evaluation of the technology switch in the market solution. The switch causes a reduction in welfare. To see this, note that at A the price index P is the same irrespective of the technology used. It follows from (14) that consumers prefer a situation with many (small) rms charging high prices over a scenario with a small number of rms charging low prices. Thus, we obtain a seemingly surprising result that although a technology switch leads to lower prices and greater aggregate output, welfare decreases. The reason is that the loss in product variety resulting from a decrease in the number of rms dominates the gains from the improvements in productive e±ciency implied by the rationalizing e ect. This result sheds an interesting light on the evolution, for instance, of the retail sector. The superseding of corner shops by a much smaller number of supermarkets can be seen as a technology switch from a small-scale to a large scale technology. Our results show that, at least initially, this kind of structural change need not be an improvement from a social point of view. It takes quite an increase in market size (in our example until A = 9:10!) until the e±ciency gains of the large-scale technology dominate the loss in product variety. A further consequence of a technology switch is that trade liberalization need not be welfare improving for all countries. To see this, note that in our example a consumer's surplus, i.e. utility per head, is the same in a country of size A ~ =4:05, in which only the S-technology is used, and in a country of size A =4:53, in which the L-technology is used. If a country of size A 2 [ A; ~ A ) were to open up free trade with a (small) country of size A A, welfare in the former country would decrease due to the technology switch. In our example, the maximum welfare loss, incurred by a country of size A, which is joined by country of size ", amounts to 4:7% of the utility level before the technology switch. A welfare loss cannot arise if the size of the integrated economy is greater than A =5:00. 16

17 5 Conclusions In this paper we have investigated the relationship between market size and market structure, when technology choice is endogenous. Our framework yields a simple and fairly intuitive explanation for the phenomenon of industry shakeouts. An increase in the market size induces entry, and thus more competition. A smaller mark-up requires a larger output in order to cover the xed costs. With increasing production the switch to a large scale technology may become pro table. We showed that a technology switch always leads to a reduction in the number of active rms and is accompanied by a rationalizing e ect (individual and aggregate output increase). Our paper highlights also the welfare implications of the often severe structural change associated with growth. We showed that consumers need not bene t from a switch in technology. The reason is that the loss in product variety resulting from a decrease in the number of rms may dominate the gains from the improvements in productive e±ciency implied by the rationalizing e ect. Appendix Proof of Proposition 1 In equilibrium, we must have ¼ S (n S ;n L )=¼ L (n S ;n L )=0: (15) where ¼ S and ¼ L take the values given in (9) and (10), respectively. Now, suppose that the pair (¹n L ; ¹n S ) satis es these conditions, and de ne K ¹n L p L 1+ +¹nS p S 1+. Then, each element of the set Z = n (n L ;n S ):n L 2 h i 0; Kp L 1 ;ns = ³ K n L p L 1 ps 1 o (16) satis es (15), and therefore constitutes an equilibrium structure. Since all (n S ;n L ) 2 Z involve the same price index P = K (1 )=, the equilibrium prices p S and p L are constant on Z. To show that the set of parameters for which coexistence occurs has measure zero, consider the xed cost parameter f L associated with a particular equilibrium. Suppose 17

18 there exist other values of f L such that the conditions in (15) are satis ed. Since (n S ; 0) 2 Z, the zero-pro t condition ¼ S (n S ; 0) = 0 must be satis ed. Obviously, neither the equilibrium number n S nor the equilibrium price p S depend on f L. It follows that K = n S p S 1+ does not depend on fl either. The fact that K is independent of f L implies that that equilibrium prices and numbers of rms are independent of f L,forany(n S ;n L ) 2 Z. However, as (0;n L ) 2 Z, this is a contradiction, since the zero-pro t condition ¼ L (0;n L )= 0 is a function of f L. 2 Proof of Proposition 2 We proof the Proposition by contradiction. Suppose, contrary to the Proposition, that at A 1 technology S is used and technology L at A 2. Then, (0;n L ) satis es the no-switching and no-entry conditions ¼ S (0;n L ) <¼ L (0;n L ) = 0 (17) at A 2,wheren L denotes the equilibrium number of L{ rms. By de nition of n L we know that d L (0;n L )=da = 0. We prove that for all A>A 2 technology L is used. For this to hold, it su±ces to show that the total derivative d¼ S da à S L @n ; (18) if evaluated at (0;n L ), is negative. Note that the third term in brackets is zero, since due to the envelope S =@p S = 0. Next, we calculate the various partial derivatives in (18). Using the implicit-function theorem, we obtain from the free-entry condition = (1 e) n L µ 1 ³ e +e 1+Bkp L 1 nl p L 1+ A ( e) Substituting p L into the rst-order condition (8), and using the implicit-function theorem, (19) L = ( e) p L (c L p L L n L ((2 e) c L p L ) (20) The partial derivatives of S with respect to A, p S and n S can be easily inferred from (9). Substituting all these derivatives into (18), we obtain, after some tedious calculations: d¼ S (0;n L ) da = (p S c S )h (1 )((2 e)c L p L ) ; (21) 18

19 where h = B ( e) ³ e e 1 n L p L 1+ (c L p L ) p S 1+ k (1 ) ((2 e)c L p L ) +k (1 e)(p L =p S ) 1 1 ((2 )cl p L ) : Note that p L c L < 0 (see the main text). Thus, the sign of (21) is negative if and only if h<0. This inequality is certainly ful lled if, in h, wereplacetheterm(p L =p S ) 1 1 (< 1) by 1. The resulting inequality can be transformed into ( e) (c L p L ) µ k + BP e p 1+ S < 0 (22) This inequality holds, since >e, p L <c L =, and demand of each consumer (third term) is positive. We have shown that if technology L is used at A 2 it is also used at A 1 >A 2. A contradiction! 2 Proof of Proposition 3 The starting point of the proof is the fact that for A = A we obtain a knife-edge solution. If A = A, the following equations hold: L (0;n L )= S (0;n L ) and S (n S ; 0) = L (n S ; 0): (23) The left equation in (23) states that (0;n L ) constitutes an equilibrium structure, while the right equation says that (n S ; 0) is an equilibrium. From the proof of Proposition 1 we know that the price index in both equilibrium structures must be the same. It follows that n L p =( 1) L = n S p =( 1) S : (24) Using the implicit function theorem, the rst order condition for any rm type i in (8) implies that p i increases with c i. Thus, since c L <c S, it follows that p L <p S. This shows that prices fall, when a technology switch occurs. The inequality p L <p S and (24) imply that n L <n S, i. e. a technology switch leads to a decrease in the number of rms. It remains to be shown that aggregate output increases. n L X L >n S X S is equivalent to ( n L p 1=(1 ) L n S p 1=(1 ) S )P 1=(1 ) 1=(1 e) + k(n S n L ) > 0: (25) 19

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