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4 Urban Systems and Historical Path Dependence W. BRIAN ARTHUR If small events in history had been different, would the pattern of cities we have inherited be different in any significant way? Could different "chance events" in history have created a different fot~ation of urban centers than the one that exists today? To a great degree, cities form around and depend on clusters of industry, so that without doing too much injustice to the question, we can ask whether the patterns of industry location follow paths that depend on history. The German Industry Location School debated this question in the earlier part of this century, but it was never settled conclusively. Von Thunen (1826), the early Weber (1909), Predohl (1925), Christaller (1933), and Losch (1944) all tended to see the spatial ordering of industry as preordained by geographical endowments, shipment possibilities, firms' needs, and the spatial distribution of rents and prices that these induced. In their view, history did not matter: the observed spatial pattern of industry was a unique "solution" to a well-defined spatial economic problem. Early events in the configuration of an industry therefore could not affect the result. But others, the later Weber, Englander (1926), Ritschl (1927), and Palander (1935), tended to see industry location as process-dependent, almost geographically stratified, with new industry laid down layer by layer on inherited, previous locational formations. Again, geographical differences and transport possibilities were important, but here the main driving forces were agglomeration economies the benefits of being close to other firms or to concentrations of industry. In the simplest formulation of this viewpoint (Maruyama, 1963), an industry starts off on a uniform, 85

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86 W. BRIAN ARTHUR featureless plain. Early firms put down by "historical accident" in one or two locations; others are attracted by their presence, and others in turn by their presence. The industry ends up clustered in the early-chosen places. But this spatial ordering is not unique: a different set of early historical events could have steered the locational pattern into quite a different outcome, so that settlement history would be crucial. These two viewpoints determinism versus history-dependence, or "necessity" versus "chance" are echoed in current discussions of how modern industrial clusters have come about. The determinism school, for example, would tend to see the electronics industry in the United States as spread over the country, with a substantial part of it in Santa Clara County in California (Silicon Valley) because that location is close to Pacific sources of supply and because it has better access there than elsewhere to airports, skilled labor, and advances in academic engineering research. Any "small events" that might affect location decisions are overridden by the "necessity" inherent in the equilibration of spatial economic forces; and Silicon Valley is part of an inevitable result. lIis- torical dependence, on the other hand, would see Silicon Valley and similar concentrations as largely the outcome of " chance. " Certain key persons- the Packards, the Varians, the Shockleys of the industry happened to set up near Stanford University in the 1940s and l950s, and the local labor expertise and interfirm markets they helped to create in Santa Clara County made subsequent location there extremely advantageous for the thousand or so firms that followed them. If these early entrepreneurs had had other predilections, Silicon Valley might well have been somewhere else. In this argument, "historical chance" is magnified and preserved in the locational structure that results. Although the historical dependence-agglomeration argument is appeal- ing, it has remained problematical. If history can indeed steer the spatial system down different paths, there are multiple " solutions" to the industry location problem. Which of these comes about is indeterminate. In the 1920s, analysts could not cope with this difficulty, and the historical chance argument did not gain enough rigor to become completely re- spectable. This chapter investigates the importance of "chance" (as represented by small events in history) and "necessity" (as represented by determinate economic forces) in determining the pattern of industry location. It con- trasts three highly stylized locational models in which small events and economic forces are both present and allowed to interact. In each model an industry is allowed to form, firm by firm, and build up into a locational pattern. In each model we will examine whether historical chance can indeed alter the locational pattern that emerges. Insights gained from the

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URBAN SYSTEMS AND HISTORICAL PATH DEPENDENCE 87 three models will be used to derive some general conditions under which long-run locational patterns may be affected by small historical events. THE EVOLUTION OF LOCATIONAL PATTERNS: THREE MODELS Model 1. Pure Necessity: Location Under Independent Preferences Let us begin with a very simple model of the emergence of an industry location pattern. Starting from zero firings, we allow an industry to form firm by firm, with each new firm that enters deciding "at birth" which of N possible regions (or sites) it will locate in. Once located, each firm will stay put. Firms in this industry are not all alike; there are I different types. The net present value or payoff to a firm of type i for locating in region j is nil; each firm choosing selects the location with the highest return for its type. In this model, firms are independent: the presence or absence of other firms does not affect what they can earn in each region. We now inject a small element of "chance" by assuming that the particular historical circumstances that lead to the next fimn's being of a particular type are unknown. We do know, however, that a firm of type I will occur next with probability Pi. The question is: What pattern of industrial settlement will emerge in this model, and can it be affected by a different sequence of historical events in the formation of the industry? It is not difficult to work out the probability that at any time of choice, region j will be chosen. This is simply the probability, qj, that the newest firm is of a type that has its highest payoff in region j, which is given by qj = ~Pk for k ~ K, where K is the set of firm types that prefer j. Repeating this calculation for each of the N regions, we have a set of probabilities of choice q = I, q2, . . ., qN) that are constant no matter what the current pattern of location is. Starting from zero firms in any region, concentrations of the industry in the various regions will fluctuate, considerably at first. But the strong law of large numbers tells us that as the industry grows, the proportions of it in the N regions must settle down to the expectation of an addition being made to each region. That is, regions' shares of the industry must converge to the constant vector q. In this simple model then, even though well-def~ned "chance historical events" are present, a unique, predetermined locational pattern emerges and per- sists. Figure 4-1 shows a simple three-region simulation of this process, with three possible firm types that prefer (clockwise from the top) region 1, region 2, and region 3, respectively, with probabilities of occurrence .5, .25, and .25. After 16 firms have located, the regions' shares of the industry are 0.75, 0.125, and 0.125, respectively not yet close to the

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88 r N ~ 16 o o o o o o o oo \ - - \/ W. BRIANARTHUR N ~ 197 of to 8 & o ~ ~ 0 too to ~= o to Do FIGURE 4-1 A three-region example of the independent preferences location model. long-run predicted pattern. After 197 firms have located, however, the shares are 0.528, 0.221, and 0.251 much closer to the predetermined theoretical long-run shares. In this model, chance events, represented as randomness in the sequence of firm types that enter the industry, are important early on. But they are progressively averaged away to become dominated by the economic forces represented by firs' payoffs in each region. Different sequences of firm types caused by different historical events would, with probability one, steer the system into the same locational pattern. Here, historical chance cannot affect the outcome. Necessity dominates. Model 2. Pure Chance: Location by Spin-off We now assume a quite different mechanism driving the regional for- mation of an industry one in which chance events become all-important. Once again the industry builds up firm by firm, starting with some set of initial firms, one per region, say. This time new firms are added by "spinning off" from parent fimns one at a time. (David Cohen t1984] has shown that such spin-offs have been the dominant "birth mechanism" in the U.S. electronics industry.) We assume that each new firm stays in its parent location and that any existing firm is as likely to spin off a new firm as any other. With this mechanism we have a different source of "chance historical events": the sequence in which firms spin off daughter ~- ilrms. It is easy to see that in this case firms are added incrementally to regions with probabilities exactly equal to the proportions of firms in each region at that time. This random process, in which unit increments are added one at a time to one of N categories with probabilities equal to current proportions in each category, is known in probability theory as a Polya process. We can use this fact to examine the long-term locational patterns

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URBAN SYSTEMS AND HISTORICAL PATH DEPENDENCE 89 that might emerge. From Polya theory we know that once again the industry will settle into a locational pattern (with probability one) that has unchanging proportions of the industry in each region. But although this vector of proportions settles down and becomes constant, surprisingly it settles to a constant vector that is selected randomly from a uniform distribution over all possible shares that sum to 1.0. This means that each time this spin-off locational process is "rerun" under different historical events (in this case a different sequence of firms spinning off), it will in all likelihood settle into a different pattem. We could generate a repre- sentative outcome by placing N- 1 points on the unit interval at random and cutting at these points to obtain N "shares" of the unit interval. Figure 4-2 shows four realizations of this location-by-spin-off mecha- nism starting from the same three original firms in a three-region case. Each of the four "reruns" has settled into a pattern that will change but little in regional shares with the addition of further firms. But each pattern is different from the others. In this model, industry location is highly path-dependent. Although we can predict that the locational pattern of industry will indeed settle down to constant proportions, we cannot predict what proportions it will settle into. Any given outcome any vector of proportions that sum to 1.0 is as likely as any other. "History," in the shape of the early random sequence of spin-offs, becomes the sole de- termining factor of the regional pattern of industry. In this model, "chance" dominates completely. O ~,-c,O ~ DO ~ ~ / 04~ FIGURE 4-2 FOUr realizations of location by spin-off. OCi ~ O Do o o O Cl- ~ \ ~1 ~~ - C ~ O / O O / ,~.o' / ~ o trace O O O 2~ o D - o o o \ 'Ro~ _ _ TIC o \ al ~ Alto Rio of 0 to o

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To W. BRIANARTHUR Model 3. Chance and Necessity: Location Under Agglomeration Economies Firms that are not tied to raw material localities and that do not compete for local customers are often attracted by the presence of other firms in a region. More densely settled regions offer better infrastructure, deeper labor markets (David, 1984), more specialized legal and financial services, better local availability of inventory and parts, and more opportunity to do business face to face. For our third model we go back to model 1 and extend it by supposing that new firms gain additional benefits from local agglomerations of firms. Suppose now that the net present value or payoff to a firm of type i for locating in region j iS ~ + g~yj) ~ where the "geographical benefits, " Hi, are enhanced by additional "agglomeration benefits," g~yj)' from the presence of yj firms already located in that region. We can recalculate the probability that region j iS chosen next, given that ye, . ., IN firms are currently in regions 1 through N. once again as qj = ~Pk for k ~ K, where K is now the set of firm types for which H: + g~yj) > rim + gLym) for all regions m ~ j. Notice that in this case the probability that region j iS chosen is a function of the number of firms in each region at the time of choice. Starting from zero firms in the regions, once again we can allow the industry to grow firm by firm, with the appearance of fi~-types subject to known probabilities as in model 1. Again, the pattern of location of the industry will fluctuate somewhat; but in this model, if by a combination of luck and geographical attractiveness a region gets ahead in numbers of firms, its position is enhanced. We can show (see Arthur, 1986, for proof) that if agglomeration benefits increase without ceiling as firms are added to a region (that is, if the function g is monotonically increasing without upper bound), then eventually (with probability 1.0) one of the regions will gain enough firms to offer sufficient locational advantages to shut the other regions out in all subsequent locational choices.* From then on, each entering firm in the industry will choose this region, and this region's share of the industry will tend to 100 percent with the others' shares tending to O percent. Figure 4-3 shows two realizations of a three-region example with agglomeration economies. The first three panels show the buildup of *In the case where g is bounded, several locations can share the industry in the long run. But again, typically, there are multiple possible outcomes, so that chance events matter here too (see Arthur, 1983, 1984, 1986, 1987).

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URBAN SYSTEMS AND HISTORICAL PATH DEPENDENCE Region 1 GO ~Region 2 && G \ ~o Region 3 \ _ Pane'~ Go Go O 0 0 0 \ 0 o\ / o o K~ O yof O O ~ \ 0 - bit ha Panel 3 Go oo 0 0 o o o o Ale ~aBea ~ o \ / o \/ of Oo \ of o Panel 2 0 B `a. / at Panel 4 FIGURE 4-3 TWO realizations of a locational process with agglomeration econ- omies. firms, with geographical preferences dominating in panel 1 but with region 3 in panel 2 by good fortune in the sequence of arrival of firm- types just gaining enough firms to cause another firm-type to favor it instead of its pure geographical preference. In panel 3, region 3 has come to dominate the entire industry in a Silicon Valley-like cluster. Panel 4 shows the outcome of an alternative run. Here the industry is locked in to region 2. In this model of unbounded agglomeration economies, monopoly of the industry by a single region must occur. But which region achieves this "Silicon Valley" locational monopoly is subject to historical luck ire the sequence of firm-types choosing. Chance, of course, is not the only factor here. Regions that are geographically attractive to many f~rm-types- regions that offer great economic benefits will have a higher probability of being selected early on. And this will make them more likely to become the single region that dominates the industry. To use an analogy borrowed from genetics, chance events act to "select" the pattern that becomes "fixed"; but regions that are economically attractive enjoy "selectional advantage," with correspondingly higher probabilities of gaining domi- nance. In this third model the long-run locational pattern is due both to chance and necessity.

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92 W. BRIAN ARTHUR PATH-DEPENDENCE AND CONVEXITY Each of our three stylized industry location models includes both de- terminate economic forces and some source of chance' events. Yet each behaves differently. Determinate forces, or historical chance, or a mixture of the two are in turn responsible for the long-run pattern of industry settlement that emerges. To explain these results and to provide some precise conditions under which historical chance can be important, it is useful to introduce a general framework that encompasses all three models (as well as many others). In this general framework, suppose there are N regions and that industry locates, one firm at a time, starting from a given number of firms in each region. Different economic forces, different sources of chance events, and different mechanisms of locational choice would be possible within this framework, but we do not need to know these. What we do need to know are the probabilities that region 1, region 2, . . ., region N will be chosen next, as a function of current regional shares of the industry X, X2, . . ., XN. Plotting this function (as in Figure 4-4 for the two-region case), we might expect that where the probability of a region's receiving the next firm exceeds its current proportion of the industry, it would tend to increase in proportion; and where the probability is less than its current proportion, it would tend to decrease in proportion. Moreover, as firms are added, each new addition changes proportions or shares by an ever smaller magnitude. Therefore proportions should settle down, and fluc / // Probability Regi o n 1 receives next fi rm. // o / 1 O Proportion of i ndu~trg i n Region 1 FIGURE 4-4 Proportion-to-probability mapping (arrows indicate expected mo- tions).

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URBAN SYSTEMS AND HISTORICAL PATH DEPENDENCE i/ 1 i ndep. prefix. Probability Region 1 receive, next fi rm. Hopi Doff,// /~//69glom. / ~ econ*. o , _ O Proportion of i ndustrg i n Region 1 FIGURE 4-5 Probability mappings for the three models. 93 tuations in proportions should die away. In the long run then, we might expect that regions' proportions (the industry's location pattern) ought to converge to a point-to a vector of locational shares where proportions equal probabilities, a point toward one that expected motions lead toward (point x in Figure 4-41. That is, this process ought to end up at a stable fixed point of the proportions-to-probabilities function. It takes powerful theoretical machinery to prove this conjecture, but it turns out to hold under unrestrictive technical conditions (see Hill, Lane, and Sudderth, 1980; and Arthur, Ermoliev, and Kaniovski 1983, 1986, 19871.* Further, and significantly for us, where there are multiple stable fixed points, each of these would be a candidate for the long-run locational pattern, with different sequences of chance events steering the process toward one of the multiple candidates. We can now see what happened in our three locational models (Figure 4-5~. The first model, "independent-preferences," has constant proba- bilities of choice and thus a single fixed point. Therefore, it has a unique, predete~ined outcome. The second model, "spin-off," with probabilities equal to proportions, has every point a fixed point, so that "chance" could drive this locational process to any outcome. The third model, "agglomeration economies," has O and 1 as two candidate stable fixed The set of fixed points needs to have a finite number of components. Where the proportions-to-probabilities function itself changes with the number of firms located, as in the agglomeration case, the theorem applies to the limiting function of these changing functions, providing it exists. (See Arthur et al., 1986, 1987.)

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94 Probability Region 1 receives next fi rm. 1 / o / 1 O Proportion of i ndu~try i n Region 1 Probability Region 1 receive, next fi rm. o FIGURE 4-6 Convex and nonconvex potential functions. W. BRIANARTHUR - ,~ 1 // O Proportion of i ndu~trg i n Region 1 points. Thus, the outcome is not fully predetermined, and one of the candidate solutions is "chosen" by the accumulation of chance events. When does history matter in the determination of industry location patterns? We can now answer this question, at least for the broad class of models that fit our general framework. History-that is, the small elements outside our economic model that we must treat as random- becomes the determining factor when there are multiple solutions or mul- tiple fixed points in the proportions-to-probabilities mapping. More in- tuitively, history counts when expected motions of regions' shares do not always lead the locational process toward the same share. It is useful to associate with each probability function a potential func- tion V whose downhill gradient equals the expected motion of regions' shares (see Figure 4-61.* Intuitively, we can think of the process as be- having like a particle attracted by gravity to the lowest points on the potential, subject to random fluctuations that die away. If this potential function is convex (looking upward at it), it has a unique minimum; therefore, the locational process that corresponds to it has a unique de- terminate outcome which expected motions lead toward and which his- torical chance cannot influence. If, on the other hand, this potential function is nonconvex, it must have two or more minima with a corresponding *For dimension N > 2, a potential function may not exist. This would be the case if there were cycles or more exotic attractors than the single-point cycles considered here.

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URBAN SYSTEMS AND HISTORICAL PATH DEPENDENCE 95 split in expected motions and with "historical chance" determining which of these is ultimately selected.* To establish nonconvexity, all we need is the existence of at least one unstable point, a "watershed" share of the industry, above which the region with this share exerts enough attraction to increase its share and below which it tends to lose its share. Yet in a way, this is another definition of the presence of agglomeration economies: if above a certain density of settlement a region tends to attract further density, and if below it it tends to lose density, there must be some agglomeration mechanism present. The underlying system will then be nonconvex, and history will count. CONCLUSIONS Whether small events in history matter in determining the pattern of spatial or regional settlement in the economy reduces, strangely enough, to a question of topology. It reduces to whether the underlying structure of locational forces guiding the locational pattern as it forms is convex or nonconvex. And for this structure to be nonconvex, so that history will matter, some mechanism of agglomeration must be present. Our models were highly stylized. They considered populations of firms, not people; they assumed that firms lived forever and never moved; and they dealt with the formation of only one industry over time, not several. Nevertheless, even if the mechanisms creating urban systems in the past and present are a great deal more complex, it is still likely that a mixture of economic determinism and historical chance and not either alone- has formed the spatial patterns we observe. Certain firms, such as steel manufacturers, need to be near sources of raw materials; for them, spatial economic necessity dominates historical chance. Certain other firms, such as gasoline distributors, need to be separated from their competitors in the same industry; for them, the necessity to spread apart again dominates historical chance. But most firms need to be near other firms if not firms in their own industry, then firms in other industries that act as their suppliers of parts, machinery, and services, or as consumers of their products and services. For this reason, firms are attracted to existing and growing agglomerations. After all, it is this need of firms to be near other firms that causes cities agglomerative clusters to exist at all. Thus, it is highly likely that the system of cities we have inherited is only partly the result of industries' geographical needs, raw material lo ~For some early discussion of nonconvexity's importance for the role of history, see David (1975).

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96 W. BRIAN ARTHUR cations, the presence of natural harbors, and transportation costs. It is also the result of where immigrants with certain skills landed, where early settlers met to market foods, where wagon trains stopped for the night, where banking services happened to be set up, and where politics dictated that canals and railroads be built. We therefore cannot explain the observed pattern of cities by economic determinism alone without reference to chance events, coincidences, and circumstances in the past. And without knowledge of chance events, coincidences, and circumstances yet to come, we cannot predict with accuracy the shape of urban systems in the future. REFERENCES Arthur, W. B. 1983. Competing technologies and lock-in by historical small events: The dynamics of choice under increasing returns. Center for Economic Policy Research Paper 43. Stanford University. Arthur, W. B. 1984. Competing technologies and economic prediction. Options. I.I.A.S.A. Laxenburg, Austria, April: 10-13. Arthur, W. B. 1986. Industry location patterns and the importance of history. Center for Economic Policy Research Paper 84. Stanford University. Arthur, W. B. 1987. Self-reinforcing mechanisms in economics. In The Economy as an Evolving Complex System, P. W. Anderson and K. J. Arrow, eds. Forthcoming. New York: Addison-Wesley. Arthur, W. B., Yu. M. Ermoliev, and Yu. M. Kaniovski. 1983. A generalized urn problem and its applications. Cybernetics 19:61-71. Arthur, W. B., Yu. M. Ermoliev, and Yu. M. Kaniovski. 1986. Strong laws for a class of path-dependent urn processes. In Proceedings of the International Conference on Stochastic Optimization, Kiev 1984, Arkin, Shiryayev, and Wets, eds. New York: Springer, Lecture Notes in Control and Information Sciences 81. Arthur, W. B., Yu. M. Ermoliev, and Yu. M. Kaniovski. 1987. Path-dependent processes and the emergence of macro-structure. European Journal of Operational Research 30: 294-303. Christaller, W. 1933. Central Places in Southern Germany. Englewood Cliffs, N.J.: Pren- tice-Hall. Cohen, D. L. 1984. Locational patterns in the electronics industry: A survey. Stanford University. Mimeo. David, P. A. 1975. Technical choice, innovation and economic growth. New York: Cam- bridge University Press. David, P. A. 1984. High technology centers and the economics of locational tournaments. Stanford University. Mimeo. Englander, O. 1926. Kritisches and Positives zu einer allgemeinen reinen Lehre vom Standort. Zeitschrift fur Volkswirtschaft und Sozialpolitik. Neue Folge 5. Hill, G., D. Lane, and W. Sudderth. 1980. Strong convergence for a class of urn schemes. Annals of Probability 8:214-226. Losch, A. 1944. The Economics of Location. Translated by W. G. Woglom from 2nd revised ed. New Haven, Conn.: Yale University Press, 1954. Maruyama, M. 1963. The second cybernetics: Deviation amplifying mutual causal pro- cesses. American Scientist 51: 164-179. Palander, T. 1935. Beitrage zur Standortstheorie. Stockholm: Almqvist and Wicksell.

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URBAN SYSTEMS AND HISTORICAL PA TH DEPENDENCE 97 Predohl, A. 1925. Das Standortsproblem in der Wirtschaftslehre. Weltwirtschaftliches Archiv 21:294-331. Ritschl, H. 1927. Reine und historische Dynamik des Standortes der Erzeugungszweige. Schmollers Jahrbuch 51:813-870. Thunen, J. H. von. 1826. Der Isolierte Staat in Beziehung auf Landwirtschaft und Na- tionalokonomie. Hamburg. Weber, A. 1909. Theory of the Location of Industries. 1929. Chicago: University of Chicago Press.