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International Journal of Industrial International Journal of Industrial Organization Organization ELSEVIER 14 (1996) 203-226 Technological change and market structure: An evolutionary approach Fernando Vega-Redondo* '* Facultad de Econ6micas and lnstituto Valenciano de lnvestigaciones Econ6micas, Universidad de Alicante, 03071 Alicante, Spain Received 27 April 1993; accepted 6 October 1994 Abstract This paper studies an inter-temporal market context in which firms innovate, imitate, and compete in quantities and technological choices each period. Potential entrants enter if there are profitable opportunities; incumbent firms exit when they go bankrupt. The key aspect of the model is that technological change evolves along a directed graph. This graph reflects both the direction of technological change and the magnitude of costs involved in switching technologies. In this set-up, my main concern is to explore the implications of different technological structures on entry/exit dynamics and on the evolution of market characteristics (market con- centration, profitability variance, etc.). Keywords: Technological change; Evaluation; Path dependence J EL classification: D43; L l l ; 033 Time is invention, or it is nothing at all (Bergson, 1911, p. 361). 1 * Present address: Facultad de Econ6micas, Universidad de Alicante, 03071 Alicante, Spain. t This work was completed while the author was a Research Fellow at the Institute for Advanced Studies at the Hebrew University of Jerusalem. I thank James Friedman for helpful conversations. I also acknowledge financial support from the CICYT (Spanish Ministry of Education), project nos. PB90-0294 and PS90-0156. Quoted in Prigogine and Stengers (1987, p. 92). 0167-7187/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved SSDI 0167-7187(94)00467-6 204 F. Vega-Redondo / Int. J. Ind. Organ. 14 (1996) 203-226 I. Introduction Following Schumpter, the effect of an industry's market structure on its rate of technological change has long been a central concern of the Theory of Industrial Organization. In this paper we focus on a reciprocal issue which, some important exceptions notwithstanding, 2 has attracted much less attention in recent literature. Namely, the role played by technological change in shaping some of the industry's 'structural' characteristics; in particular, of course, its degree of concentration, but also some of its more dynamic features such as the industry's pattern of exit and entry. We shall also be concerned with providing some insight on what the recent handbook survey of Cohen and Levin (1989) has viewed as an important gap in existing literature, namely the lack of theoretical work addressing the persistence of major inter-firm differences in profitability performance. As reported by Schmalensee (1987), 'orthodox' approaches to explain these differences (like linking them to market concentration or some measure of industry-wide differential efficiency) simply do not work. A reason suggested for this failure is that, as Cubbin and Geroski (1987) conclude in another empirical paper in the same volume, " . . . considerable heterogeneities exist within most industries. 3 That is, most firms' profitability experience differs considerably from those of their closest rivals" (emphasis added). Our approach to studying the above set of issues will be to propose a model of market dynamics in which both strategic and path-dependence considerations are jointly taken into account. To incorporate the latter, the model will exhibit a somewhat 'biological' flavor. In particular, the set of technological options will be endowed with a directed-graph structure, each of the firms that are currently active advancing along possibly different edges of it through the accumulation of short-run strategic choices. The graph structure will formalize two key features of the model. On the one hand, the idea that innovation is a gradual accumulation of know-how along a certain technological direction; on the other hand, that imitation may involve switching costs, possibly substantial. Market exit and entry will be modeled explicitly as follows. A potential entrant enters if it can profitably do so given its technological possibilities. On the other side of the coin, an incumbent firm remains in the market only if it is able to meet a certain viability constraint. (For simplicity, we shall require non-negative profits every period.) As we shall see, this leads, under certain technological scenarios, to a continuous turnover in the set of active z See Phillips (1966), Nelson and Winter (1974), Levin (1981), or Mansfield (1983). 3 The authors study a sample of 217 UK firms during the period 1951-77. F. Vega-Redondo / Int. J. Ind. Organ. 14 (1996) 203-226 205 firms. As Schumpeter would phrase it, technological change in such contexts becomes a force of "creative destruction". There exists a rather small literature which shares with this paper a similar approach and concern. Noted representatives of it are Futia (1980), Nelson and Winter (1982), or Iwai (1984a,b). None of them, however, exhibits the path-dependence and strategic considerations which are key to our present approach. The paper is organized as follows. Section 2 describes the different components of the model. In Section 3 we conduct the analysis and discussion. The paper concludes with a summary in Section 4. The proofs of the results are contained in an appendix. 2. The model We divide the presentation of the model into the following subsections: 2.1 the firms; 2.2 the market; 2.3 technology; 2.4 technological change; 2.5 exit and entrance; and 2.6 strategic game. 2.1. The firms There is a set I C ~ of potential firms in the market. As we shall explain below, only a fraction of them will be generally active at any given point in time. Time is measured discretely. At each t = 0, 1, 2 . . . . , the behavior of any firm i ~ I is characterized by the pair (O,(t), xi(t)) E O x ~+ 4 where O/(t) denotes the 'product variety' produced by firm i at t, xi(t) its output, and O stands for the set of all possible product varieties. Implicitly, therefore, we assume that each firm produces only a single variety at each point in time. If O,(t) = I~, we interpret it to mean that firm i is not active at t. This, in particular, means that its output xi(t ) must equal zero. 2.2. The market In the Chamberlinian monopolistic-competition tradition we shall assume that each firm i confronts an/-specific inverse-demand function /~: O' × R~+--,R., (1) which, for each variety and output profiles _~ E ~)l and x ~ R~+, determines the market-clearing price for the product sold by firm i, f~(_~,_x). The 4 R. will denote the non-negative reals; R÷+ the positive reals. 206 F. Vega-Redondo / Int. J. Ind. Organ. 14 (1996) 203-226 dependence of f ( . ) on the market pattern of product varieties is contained in the following assumption: Assumpt ion 1. There is a real function p: 19--+~+ (p(0)=0) such that Vi E N, the/-specific demand function f~(.) has, for any (_O, x) E 19 × R+, the following representation: f/(_O, X) = Pi(P(~)iE,, X_) for some function Pi: •*+ x U*+--+U+ . The functions (P~)~e, are symmetric under index permutation, homogeneous of degree zero in P(~)ie, and, for each i E I, satisfy P~(.)~ 0 as Y_5,ie I xj---+ ~. Moreover, Ve >0, 3~ >0 such that if p(~) /p (g)<-v for some j # i, then Pi(p(Oi)iel; x) <~ e for all x e ~*+. The above assumption has the following interpretation. The effect of product diversity on market demand is fully summarized by the value of some real function p(-), which may be viewed as measuring the market value (for short, we shall speak of the 'quality') of each variety. For technical reasons, we normalize matters and assume that only relative qualities matter [i.e. each Pi(') is homogeneous of degree zero in quality levels]. We shall further suppose that if the quality of the variety produced by some firm i E I deteriorates sufficiently relative to that of a competitor, so do its 'market conditions', as reflected by its corresponding demand function Pi(.). Such market conditions are assumed bounded in the sense that no firm can sell at some positive price if an arbitrarily large aggregate quantity floods the market. By way of illustration, Assumption 1 will be satisfied by any demand function of the form Pi[{P(~l.))jEl; X] = 6 ~ ~_ ) , jc i / where ~b(-) is some appropriately chosen continuous real function that satisfies ~b(0,-) =0 and ~b(-,E)-->0 as 07--->~, with t~(_O) standing for the average quality offered by the active firms in the market. That is, ~i~, P(~) P(-~)- I{i ~ I : ~e}[ ' where I'1 denotes 'cardinality' and, for simplicity, all active firms are given equal weight. {Alternatively, we could contemplate a weighted average where, say, the weight of each firm reflects its share in total production.) F. Vega-Redondo / Int. J. Ind. Organ. 14 (1996) 203-226 207 2.3. Technology For the sake of focus, diversity across firms will be centered on the market value of the variety it produces, not on its production technology. In this latter respect, we shall conveniently assume that all varieties are produced under the same simple cost structure: a fixed cost M and a constant marginal cost c, both positive. Firms will be diverse because, in general, they will confront different decision problems over available product varieties. Such asymmetries among firms will be the result of the following two phenomena. On the one hand, different innovation experiences. On the other hand, the existence of switching costs in implementing imitation choices. In order to formalize these matters, the set t9 of possible product varieties will be endowed with the structure of a directed graph (a 'digraph'), the direction in it reflecting technological precedence. 5 A simple example of a digraph just involving six different varieties is presented in Fig. 1. When two different varieties, O, O ', are adjacent consecutive vertices of 19 (i.e. there is an 'arrow' in the graph going from 0 to O'), we write O/zO' and say that O directly precedes (technologically) O '. Compositions of/Z give rise to the relation of general (as opposed to immediate or direct) technological precedence, as follows. When O and O' are joined by a /z-chain starting at d and ending at O', we simply say that O technologically precedes 0 ' and write O/30'. (For the sake of formal convenience, we make 0/30, i.e. O is joined to itself by a/z-chain of length zero.) Motivated by our interpretation of/3 as a binary relation expressing technological precedence, we assume that /3 is an order ing- in general, a partial one. Or, in the language of Graph Theory, (19,/z) is an acyclic digraph. We now propose a notion of technological distance on (19,/z). If 0 /3-precedes 0 ' , the technological distance d(d, O ' )E ~ is taken to be the usual one considered in Graph Theory, namely the length of the shortest Fig. 1. 5 The elements of O are usually known as 'vertices', the graph specifying which vertices are connected by 'edges' and in which direction. See Berge (1985) for a classical reference on Graph Theory. 208 F. Vega-Redondo / Int. J. Ind. Organ. 14 (1996) 203-226 /x-chain joining O and O '. When O does not/3-precede O ', we generalize the precedent notion by proposing a concept of technological distance remin- iscent of biological contexts. For all O', O"E O denote ~(O', 0") = {0 ~ O: O/30' ,~ O/30"}, (3) i.e. the set of common predecessors of both O' and 0". We define d(O', 0") = min{d(O, 0"): O e ~(O ', 0" )} , (4) where we adopt the convention that d(O, O)= 0 and, if ~(O', 0" )= ~, d(O', 0") = oo. In order to illustrate the concept of technological distance, let us refer again to the digraph depicted in Fig. 1. Here, for example, we have d(O i, 05) > 0 for all ~ • 05, ranging from the maximum d(O0, 05) = 3 to the minimum d(03, 05) = 1, Note that d(.) is not a distance function in the usual mathematical sense. In particular, it is not symmetric. For example, d(Os, 04) = 1 ¢ d(04, 05) = 2. In general, of course, for all O ~ 0 ' such that 0/30', we have d(O, 0 ' )>0, whereas d(O', O)= O. Consider a firm currently producing the variety O and considering whether to change its production to some other variety 0 '. We shall assume that if this shift is performed the firm will have to incur some switching costs. Switching costs may be given different interpretations, not necessarily exclusive. 6 One possible interpretation is that the firm's plant (or any other type of sunk investment) is geared towards producing variety O and needs to be adapted in order to produce the different variety O'. A related explanation has to do with the existence of learning costs in training the firm's workers to produce a different variety. In any event, switching costs will be linked to the technological distance between the former and final varieties, as reflected by the function : N U {0}--, R+, (5) assumed increasing, with ~(1) = ,~(0) = 0 (i.e. 'gradual' adjustment and no adjustment is costless), 7 and the gradient "~(n + 1) - ~/(n) bounded above zero for all n E N. Thus, for any contemplated shift from O to 0 ' the switching cost is given by y(O, 0 ' ) = g/(d(O, 0 ' ) ) , (6) where d(. ) is as defined above. 6 For empirical evidence and a good discussion on the importance of switching costs in the process of technological change, the reader may refer, for example, to David (1985a,b), who discusses specific cases, or Basalla (1988) who provides a general perspective on these issues. 7 We assume that ~(1) = 0 in order to avoid that the innovation process may reach a standstill (see below). F. Vega-Redondo / Int. J. Ind. Organ. 14 (1996) 203-226 209 2.4. Technological change 2.4.1. Innovation Since our concern in this paper is not to explain the rate of innovation but its induced consequences, I shall assume that each firm's innovation is the result of a stochastic process of 'inventing by doing'. Each period, every firm that remains active in the market obtains an invention draw from the set of varieties that are technological successors of the one it previously produced. Such an invention then becomes currently available for adoption if the firm so decides. By Assumption 1, the 'market value' of a given product variety 0 is fully captured by its associated p(O). Let S(O) --- {0' E O: O/zO'} denote the set of direct successors of #, assumed finite for simplicity. We postulate: Assumption 2. (i) At each t, every firm i ~ I with xi(t - 1) > 0 obtains an innovation draw ~(t) from the set S(~(t - 1)) of technological successors of the quality previously produced. These draws are obtained according to the (discrete) density function Aoi(t_l) with A,~(,_I)(O ) i> to for all 0 E S(O,.(t - 1)) and a common to > O. If xi(t q- l ) = O, then ff/(t) = ~. (ii) VO ~ O, 3~ E S(O) such that p(O) 0 such that VO~O, 30ES(O) such that o(o) = p(e) vo ' e so ) . By part (i) of the above assumption, every firm obtains one 8 innovation draw to be used next period if, and only if, it is currently active. The fact that A~iu_1) is assumed with full support implies, in view of part (ii), that every active firm always has positive probability of obtaining an innovation draw which is not 'successful', i.e. yields a variety of lower quality than the one currently produced. By part (iii), on the other hand, there is always a positive probability of obtaining a worthwhile invention with a quality increase ratio of at least ~. We shall also need that any such increase ratio be bounded above. For simplicity, we assume that ~ itself is the upper bound. 2.4.2. Technological diffusion Technological know-how on the arising product varieties will be assumed to filter gradually through the industry with some lag. To model this, denote by K(t) the set of state-of-the-art product varieties at t in the sense that this set is currently available to every firm in the industry (either if it is currently active or a potential entrant). As time proceeds, this set is enlarged by past 8 The fact that the number of draws is one rather than a finite number is inessential for our purposes. 210 F. Vega-Redondo / Int. J. Ind. Organ. 14 (1996) 203-226 inventions as they all become progressively available with some finite lag s ~ N (an arbitrary 9 parameter of the model). The parameter s may be interpreted as reflecting some unavoidable gradualness in the process of diffusion or some legal limits to it induced by, say, a patent system. Formally, we postulate: K(t)= K( t -1 )U( i ?1~(t -s ) ) , (7) for all t E [~, where K(0)= 0, and we use the convention ~( t - s )= ~t if t < s. [See Section 3 for a description of the initial conditions on ~(0) . ] 2.5. Exit and entrance Let F(t) C I denote the set of potentially active firms in the market at time t. Only those firms in F(t) may participate in the market and produce a positive output. The set F(t) is partitioned into two subsets, N(t) and E(t). The set N(t) includes those incumbent firms that entered the market in the past and still remain in it. On the other hand, E(t) includes those firms that are currently considering entry at t. The set F(t) changes through time as a result of the processes of exit and entrance in the industry. Exit is modeled in a straightforward way. If an incumbent firm i E N(t) is not able to make non-negative profits (net of any switching costs), then we assume that it goes bankrupt and is forced to exit the market. For simplicity, we assume it may never participate in the market again. That is, i ,~ F(~-) for all z > t. Entry, on the other hand, is modeled gradually as follows. At each t E N, the set E(t) is composed of only one firm, that indexed e(t), which is the current potential entrant. Assuming firms are listed in the order of potential entrance, we choose e(t) as the firm with lowest index which has never yet entered. If, given the game described in the next subsection, the firm e(t) decides to enter the market, e(t) E N(t + 1) - i.e. it becomes an incumbent next period - and e(t + 1) = e(t) + 1, as long as e(t) was not the last firm in I. If it instead decides to remain away from the market, we shall assume that it still remains the potential entrant at t + 1. Thus, e(t + 1)= e(t). 2.6. Strategic game At each t, the players involved in the game are those in the set F(t). We shall consider a two-stage game, as follows. 9 Any value for s is consistent with our qualitative results. In fact, what matters for them is the ratio ~/s of maximum quality innovation per 'lag period'. F. Vega-Redondo / Int. J. Ind. Organ. 14 (1996) 203-226 211 In the first stage, each firm i ~F( t ) chooses simultaneously a product variety in Oi(t)=-- K(t) U [~ f f i(~-)] , (8) which is its action space in the first stage of the game. Note that for the entrant e(t), Oe(,)(t ) = K(t), since ffe(,)(~')= t~ for all ~-~< t. Also notice from (7) that 0 E K(t) for all t. If the entrant chooses de(,) = ~l, we interpret it to mean that the potential entrant decides to stay out and wait for the next period. Given their decisions in the first stage of the game, firms are assumed to compete as Cournot oligopolists in the second stage, choosing (simul- taneously) how much output to produce of the selected variety. Given that the firms' inverse-demand functions have been selected in Subsection 2.2 as the 'primitives' of the market environment, such Cournot-type competition is adopted here as the simplest modelling option fo