W O R K I N G P A P E R
ON C O W R I N G TECH)JOLLK;lXS AND HEIDIUCAL SULL EVEXI'S:
THE
DYNAMICS OF CHOICE UNDER INCREAS[NG RRVRNSW. Brian Arthur Stanford University
September 1983 WP-83-90
i n t e r n a t i o n a l I n s t i t u t e for Applied Systems Analysis
NOT FOR QUOTATION WITHOUT PERMISSION OF
THE
AUTHORON COMlTlTNG TECHNOlDGIES AND HlSTOKlCAL SMALL EVENT3 THE DYNAMICS OF CHOICE UNDER INCREASING RJTlWWS
W. Brian Arthur Stanford University
September 1983 WP-83-90
Working R p e r s are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of t h e Institute or of its National Member Organizations.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria
This paper explores t h e dynamics of microeconomic choice between objects with increasing r e t u r n s . I t finds t h a t s u c h dynamics possess four features: (a) a potential inefficiency of aggregate outcome, even where individual choices a r e perfectly rational; (b) a n inflexiblilty of outcome, in t h a t m a r k e t s h a r e s become locked-in--they c a n n o t always be influ- e n c e d by standard, marginalist policy measures; (c) a non-predictability, in t h a t knowledge of supply and demand conditions does not suffice to predict u l t i m a t e m a r k e t shares; and (d) a non-ergodicity, in t h a t small historical events a r e not always averaged away, b u t c a n determine t h e p a t h of m a r k e t shares. These properties a r e d e m o n s t r a t e d within a sim- ple model where agents choose between technologies competing for
adoption.
Choice u n d e r increasing r e t u r n s appears t o raise serious questions for policy prescription, for t h e interpretation of economic history, and for t h e possibility of constructing models for a c c u r a t e economic predic- tion.
ON COMPETING TECHNOLOGIES AND IUSTOFUCAL SMALL
EVENTS:
THE
DYNAMICS OF CHOICE UNDER INCREASING FEIVRNSW. Brian Arthur
A t t m p t s t o describe t h e dynamics of m a r k e t s with increasing returns' (or decreasing supply costs) have long b e e n f r u s t r a t e d by a n analytical difficulty. Where objects with increasing r e t u r n s c o m p e t e , t h e m a r k e t o u t c o m e is usually i n d e t e r m i n a t e . It is not difficult t o s e e why.
With increasing r e t u r n s p r e s e n t in a given problem, non-convexities appear, so t h a t multiple equilibria a r e called into being. Information on preferences, endowments a n d tranformation possibilities e n a b l e s u s t o locate t h e s e long-run equilibria, but it is often insufficient t o tell us which one will be " s e l e c t e d . From m a n y initial positions of i n t e r e s t , t h e system--like a pencil perfectly balanced on i t s point-is equally
"attracted" by several equilibrium outcomes. We c a n n o t say which way i t will "fall"; we c a n n o t describe uniquely which p a t h i t will follow; h e n c e we c a n n o t p u r s u e conventional And t h e o r y h a s little f u r t h e r t o say.
As a simple example of t h i s type of problem, consider a n island in which c a r s a r e introduced, all a t m o r e or less t h e s a m e time. Drivers 1 By contrast, the s t a f i c s of economies with increasing returns is the subject of a vigorous literature. For general increasing-return studies, see among others, Arrow and Hahn (Ch.7, 1971), Beato (1982), Brown and Heal (1976, 1979), Flaherty (1980), Guesneries (1975), Krug- man (1980), Scarf (1983), Schelling (1978), Spence (1981), and Weitzman (1982). Economies with increasing returns (or decreasing supply cost) commodities possess several interesting properties: for example, Pareto optima are not always equilibria; conversely, equilibria may satisfy standard first-order conditions, yet be inefficient; and not all endowment distribu- tions permit the attainment of optimality through trading.
2 In the Principles (9th Ed.,p.BOs), for example, Marshall tried t o analyze the case where several firms with decreasing long-run cost curves compete for shares of an industry. One firm, he showed, eventually prevails. But which one achieves the monopoly he could not determine : the outcome depended on "whichever firm first gets a good start".
a r e f r e e t o choose between t h e right- and left-hand sides of t h e road a n d have no in-built bias toward e i t h e r . Each side possesses increasing r e t u r n s : a s a higher proportion of drivers chooses one side, t h e very real r e t u r n s t o choosing t h a t side rapidly rise. Casual t h o u g h t tells u s t h a t we would observe a good deal of randomness t o t h e proportions ini- tially driving on e a c h side, b u t t h a t , if one side by c h a n c e g o t sufficiently ahead, o t h e r drivers would "fall in" on this side, so t h a t eventually all c a r s would drive on t h e s a m e side of t h e road. Of course, t h e side t h a t
"winsw--that comes t o "dominate t h e marketw--cannot be deduced in advance. The outcome is i n d e t e r m i n a t e . And perhaps we m u s t conclude t h a t t h e r e is little m o r e t o be said in t h i s case.
But notice, i n this r a t h e r artificial example, four a s p e c t s of t h e out- come in themselves worth investigation. First, in c o n t r a s t t o t h e u s u a l diminishing-returns situation, t h e o u t c o m e need possess n o efficiency properties--the side t h a t "takes t h e m a r k e t " need not, from any long- t e r m collective viewpoint, be t h e b e t t e r of t h e two. Second, driving is now locked-in t o t h e "chosen" side. The outcome is s t r u c t u r a l l y rigid, in t h a t marginal i n d u c e m e n t s t o individual drivers t o change sides would likely prove ineffective a n d policy m u s t find o t h e r m e a n s . Third, even t h o u g h we know drivers' preferences a n d possibilities, e z a n t e t h e out' come would be h a r d t o predict. "Small events" outside t h e model-- perhaps s o m e drivers' reactions, perhaps a dog r u n n i n g into t h e road.
perhaps t h e timing or" positioning of c e r t a i n traffic lights--may be c r u - cial in deciding t h e outcome. And fourth, e z post, e x a c t causality would be h a r d t o assign--it would certainly be a mistake t o ascribe i t t o t h e
"superiority" of t h e outcome.
What of t h e indeterminacy? One way t o bring i t within analytical s c r u t i n y would be t o make explicit t h e "small events", ad.d t h e m t o t h e model, a n d examine in detailed "slow-motion" t h e dynamic process by which t h e y c u m u l a t e into a n aggregate outcome. This would be difficult in o u r imaginary example. But if it were possible in s o m e better-defined case, we should want t o a s k by what process "small events" t e n d t o m a k e a large difference where t h e r e a r e increasing r e t u r n s , b u t n o t usually where t h e r e a r e diminishing r e t u r n s , a n d t o what degree knowledge of supply a n d d e m a n d functions enables u s t o predict m a r k e t shares.
In t h i s p a p e r we a t t e m p t t o explore, by way of a m a t h e m a t i c a l l y simple model, t h e dynamics of choice between objects with increasing r e t u r n s . We find t h a t such dynamics typically possess four features: (a) a potential inefficiency of aggregate outcome, even where individual choices a r e perfectly rational; (b) a potential inflexibility of outcome, in t h a t u l t i m a t e m a r k e t s h a r e s c a n n o t always be influenced by s t a n d a r d , m a r g i n a l i s t policy m e a s u r e s ; (c) a non-predictability, in t h a t complete knowledge of supply a n d d e m a n d functions does not suffice t o predict t h e p a t h of m a r k e t s h a r e s ; a n d (d) a non-ergodic property, in t h a t "small events" a t t h e o u t s e t a r e n o t averaged o u t a n d "forgotten", b u t m a y
"decide" t h e p a t h of m a r k e t s h a r e s . P ~ ~ o v i d e d t h e notion of "historical small events" is carefully defined, "indeterminacy" t u r n s o u t t o be both amenable to analysis a n d interesting in i l s own right. And we find t h a t choice u n d e r increasing r e t u r n s appears t o raise serious questions in t h e i n t e r p r e t a t i o n of economic history, in policy prescription, a n d in t h e possibility of c o n s t r u c t i n g models for a c c u r a t e economic prediction.
To keep t h e discussion c o n c r e t e , we s e t u p a model where a g e n t s choose between technologies competing for adoption. This use of t e c h - nologies r a t h e r t h a n goods a s t h e objects of choice h a s a p a r t i c u l a r advantage. Where m o s t goods show diminishing r e t u r n s (in t h e form of increasing supply costs), very m a n y technologies show increasing r e t u r n s : often t h e m o r e a technology is adopted, t h e m o r e it is improved, a n d t h e g r e a t e r i t s payoff. Assuming e a c h a g e n t ' s m o m e n t of choice t o be s u b j e c t t o small, b u t unknown events, we find t h a t t h e m a r k e t s h a r e of e a c h technology follows a stochastic proccess--in this case a r a n d o m walk. If both technologies show s t a n d a r d diminishing r e t u r n s , t h i s r a n - dom walk has r e f l e c t i n g barriers. The aggregate outcome is efficient, flexible, predictable, a n d ergodic. If both technologies show increasing r e t u r n s , t h e r a n d o m walk has a b s o r b i n g b a r r i e r s . The aggregate out- come is n o t necessarily efficient, n o r flexible, nor predictable, n o r ergodic.
We begin by introducing t h e notion of "competing technologies", t h e n go on t o s e t u p t h e simple model of choice.
1. C O W r n G TECHNOLOGIES
A Preliminaries
Usually t h e r e a r e several ways t o c a r r y t h r o u g h a n y given economic purpose. We shall call t h e s e "ways" (or m e t h o d s ) t e c h n o l o g i e s a n d we will say t h a t m e m b e r s of t h e s e t of technologies t h a t c a n fulfill a p a r t i c u l a r purpose c o m p e t e , if adoption of one technology by a n economic a g e n t t e n d s t o displace or preclude t h e adoption of a n o t h e r . Competition is m e a n t h e r e in t h e unconscious, passive sense: technologies, whether incorporated in physical plant or m a c h i n e r y or existing as p u r e m e t h o d or p u r e information, a r e a s s u m e d in t h i s paper t o be ope y available t o all, a n d n o t subject t o s t r a t e g i c influence o r manipulation.
9
In t h i s paper competition a s s u m e s a s t r o n g e r f o r m t h a n t h e s t a n - dard diffusion c a s e where a new a n d superior technology c o m p e t e s with an old a n d inferior one. Here two or m o r e s u p e r i o r technologies com- pete with e a c h o t h e r t o replace a n outmoded horse-and-buggy technology.
Thus, i n t h e 1890s, t h e s t e a m engine, t h e e l e c t r i c motor, a n d t h e gaso- line engine competed, in t h e passive sense, a s power sources for t h e new automobile. In t h e 1800s a n d on i n t o this c e n t u r y , spinning m u l e s com- peted with ring-frames in cotton manufacturing (Saxonhouse a n d Wright 1983). More r e c e n t l y t h e n u c l e a r technology "competes" with hydroelec- tric. coal, a n d o t h e r technologies, for p a r t c a p t u r e of t h e electricity gen- eration m a r k e t . And gallium arsenide c o m p e t e s with doped silicon in t h e m a n u f a c t u r e of f a s t semiconductors.
In g e n e r a l i t n e e d n o t be t h e case t h a t t h e n u m b e r of technologies competing for a given purpose is few. If we consider t h e a r r a n g e m e n t of the 40 or s o k e s on a typewriter a s a technology, t h e n in principle 40- factorial o r
lo4'
possible keyboards compete with t h e s t a n d a r d QWERTY3 Patentable techniques, proprietary methods, and "trade secrets" do not fulfill this a s sumption. They can be transferred a t a manipulable price, and they are not open t o all.
keyboard. More properly, we should call this typewriter c a s e one of
"competing standards" or "competing conventions": h e r e t h e technolog- ical choices a r e given a n d fixed.
A technology which is not a m e r e s t a n d a r d or convention t e n d s t o be fluid: it m u t a t e s , changing in design and s o m e t i m e s in use, typically existing in several or m a n y "variants". In 1904, for example, t h e s t e a m automobile c a m e in several s c o r e of forms, among which t h e b e t t e r known were t h e Stanley, t h e White, t h e Chelmsford, t h e Gardner- Serpollet a n d t h e Toledo (Fletcher 1904). Choice between competing technologies t h e r e f o r e m a y involve selection from two o r m o r e collec- tions of "variants".
We shall t h i n k of a given technology ( o r v a r i a n t of i t ) a s combining a c e r t a i n vector of economic inputs or factors for a given a m o u n t of desired "output", s o t h a t m o n e t a r y returns-in-use or payoff t o adoption to a p a r t i c u l a r a g e n t a r e simply t h e value of t h e o u t p u t less factor c o s t over a n appropriate t i m e horizon. Almost by definition a new technology is subject to u n c e r t a i n t y , s o t h a t i t s m o n e t a r y r e t u r n will have a proba- bility distribution. Choice between competing technologies is therefore normally a choice between competing lotteries.
As a p a r t i c u l a r technology spreads in use, t h e payoff t o adopting it may change considerably. Much of t h e individual's utility in adopting a s t a n d a r d , for example, depends on t h e degree t o which o t h e r s have adopted i t o r will follow suit. In t h e c a s e of a m u t a b l e technology, increased adoption brings a growing accumulation of experience a n d knowledge; a n d t h i s "learning by using" (Rosenberg 1982) in t u r n becomes i n c o r p o r a t e d i n t o m o r e efficient a n d reliable v a r i a n t s of t h e technology. (Supersonic a i r c r a f t , for example, improved rapidly after a c t u a l designs a c c u m u l a t e d in-the-air experience.) Not all technologies.
of course, enjoy increasing r e t u r n s with adoption. The very popularity of a factor-intensive technology m a y bid its inputs u p in price, so t h a t diminishing r e t u r n s accompany adoption. (Hydroelectric power, for example, becomes costlier with i n c r e a s e d u s e a s suitable dam sites become s c a r c e r a n d hydrodynamically less efficient.) Time itself may of course be a m a j o r f a c t o r in changing r e t u r n s t o adoption; i n o u r model we will a b s t r a c t from t h i s a n d suppose r e t u r n s t o depend only on t h e n u m b e r s who have c h o s e n a technology.
B. A Simple Model with Heterogenous Adopters
Two technologies, A a n d B, compete for adoption by a large n u m b e r of economic a g e n t s who a r e c u r r e n t l y using a n o u t m o d e d technology of t h e one-horse-shay type. (For simplicity we t r e a t t h e pool of agents a s infinite in size.) I t pays a g e n t i t o r e t a i n his obsolete e q u i p m e n t until i t s demise a t t i m e ti; b u t h e c a n n o t afford t o be without working machinery, s o t h a t a t t h i s p o i n t h e adopts e i t h e r technology A or technology B a n d holds i t t h e r e a f t e r . The v a r i a n t e a c h a g e n t chooses is fixed or frozen i n design a t h i s t i m e of choice, s o t h a t his payoff is not affected by f u t u r e changes i n o r f u t u r e adoption of e i t h e r technology. Agents a r e rational, they obey t h e von Neumann-Morgenstern axioms, a n d t h e y a r e perfectly informed a t t h e i r m o m e n t of choice: t h e y h o w of all c u r r e n t l y available v a r i a n t s of both technologies a n d t h e i r payoff distributions. Agents fall
i n t o two types, R and S , with equal n u m b e r s in each, t h e two types independent of t h e times of choice but differing in t h e i r preferences, or in t h e i r degree of risk-aversion, or in t h e i r economic environment.
With t h e s e assumptions e a c h a g e n t can form a well-defined scalar utility for each technology o r variant of it; when his t i m e comes, h e chooses t h e highest utility variant available a n d r e m a i n s a t t a c h e d t o it.
For simplicity we suppose t h e payoff-utility or r e t u r n s t o adopting A or B t o vary linearly with t h e n u m b e r s n~ and ng who have chosen each, as in Table 1.
Table 1. Returns t o ~ d o ~ t i o n ~
We c a n c o n t r a s t t h e dynamics of t h e adoption process u n d e r dimin- ishing, increasing, or c o n s t a n t r e t u r n s regimes by allowing r a n d s t o be simultaneously negative, positive or zero. We a s s u m e aR
>
b R a n das
<
b S SO t h a t R-agents have a n a t u r a l preference for A , a n d S-agentshave a n a t u r a l preference for B. (Later we relax linearity a n d several of t h e o t h e r simplifying assumptions.)
We now have a well-defined, neoclassical model of choice: two types of agents choose between A a n d B. The supply cost (or r e t u r n s ) func- tions a r e h o w n , a s is t h e d e m a n d (each a g e n t d e m a n d s one unit inelast- ically). Of i n t e r e s t a r e t h e properties of t h e m a r k e t outcome--the pro- portion of t h e total technologies adopted t h a t belongs t o type A or type B a s t h e n u m b e r s of adopters increase.
Notice t h a t we have avoided several complexities. Agents a r e economically forced to choose, h e n c e waiting for r e t u r n s t o rise is not a n option; r e t u r n s t o a given adopter do not depend on f u t u r e choices (but t h e y do depend on past choices), hence expectations a r e not a problem;
a n d technologies c a n n o t be priced or manipulated, h e n c e game- theoretical s t r a t e g i c maneuvering does not e n t e r . Each of t h e s e assumptions could be relaxed a t some analyti.ca1 cost. But in t h i s exploratory paper we deliberately keep t h e analysis simple.
To complete t h e model, i t r e m a i n s t o define a s e t of "historical small events". Recall t h a t in t h e earlier side-of-the-road example, our lack of knowledge of c e r t a i n events-drivers' reactions, weather condi- tions, traffic-light timings--caused t h e o u t c o m e Lo be i n d e t e r m i n a t e .
4 More in keeping with the learning-effects literature (Steinmueller 1983) we could modify these returns-versus-adoption functions to be log-linear (or exponential) in form: e.g.
ln(Uo
- u) =
a~ - an^ (for technology A) and ln(lJO- U ) =
b R-
ang (for tech- nology B. The results t h a t follow would however be the same.Technology A Technoloay B R-Ag en t
S-Agent
a R + m d a , ~
+
sndb R
+
rn,b s
+
s n ,Were we to have infinitely detailed knowledge of s u c h e v e n t s a n d condi- tions, t h e outcome--the side of t h e roa t h a t would be selected--would presumably be determinable in a d v a n c e 3 We c a n conclude t h a t o u r lim- i t e d discerning power, or more precisely t h e limited discerning power of a n implicit o b s e r v e r , c a u s e d t h e indeterminacy. We m a y t h e r e f o r e define "historical small events" t o be those events or conditiocs t h a t a r e outside t h e knowledge of t h e observer- beyond t h e resolving power of his
"model" or abstraction of t h e situation.
To r e t u r n t o OUT model, we a s s u m e a n observer who h a s full knowledge of all t h e conditions a n d r e t u r n s functions, e x c e p t t h e s e t of t i m e s of choice
i t i ] .
The observer t h u s "sees" t h e choice o r d e r as a binary sequence of R a n d S types with t h e property t h a t a n R or a n S s t a n d s in t h e n t h position in t h e line with equal likelihood, t h a t is, with probability one half.In t h e analysis t h a t follows, we c o n t r a s t t h e outcome--the m a r k e t s h a r e s gained by each technology after n agents have chosen--under r e g i m e s of c o n s t a n t r e t u r n s , diminishing r e t u r n s , a n d increasing r e t u r n s .
I t is useful t o view t h i s 'choice process as a s e a r c h procedure--the agents, by t h e i r choices, adopting o r "exploring" along a path t h a t con- s i s t s of a m i x t u r e of A or B variants. ln principle, a t stage n , n out of a t o t a l possible Zn A a n d B v a r i a n t s could be "explored". Accordingly, we shall say t h a t t h e o u t c o m e of t h e choice process, viewed a s a s e a r c h pro- c e d u r e , is e f f i c i e n t , if a t e a c h s t a g e n , n o n e of t h e n choices actually adopted ( o r " e x p l o r e d ) have lower r e t u r n s t o e i t h e r a g e n t type t h a n t h e n missing good options n o t adopted. If t h ' s is t r u e , t h e p a t h t h e choice process takes is not
6 .
We have built some "small-event" u n c e r t a i n t y i n t o t h e process--at l e a s t as f a r as t h e observer is concerned--so t h a t we c a n n o t e x p e c t per- f e c t prediction of z,, t h e m a r k e t s h a r e of A a f t e r n choices have been made. But we would hope t h a t historical fluctuations m a t t e r less, o r average away, a s t h e adoption process proceeds. Accordingly, we will say t h a t t h e o u t c o m e of t h e choice process i s p r e d i c t a b l e if t h e observer can, e z a n t e , c o n s t r u c t a forecasting s e q u e n c e
i Z n ]
t h a t comes ever closer t o t h e exact m a r k e t s h a r e , t h a t is, with t h e p r o p e r t y t h a t1 Sn -
z ,1
goes t o zero, with probability o n e , a s n goes t o infinity.We will say t h a t t h e outcome is f l e x i b l e , o r amenable t o marginalist policy intervention, if changing one of t h e technologies' r e t u r n s func- tions by a n a r b i t r a r y small a m o u n t E from a n y s t a g e n onward, c a n affect t h e n u m b e r s of a g e n t s choosing A o r B a t s o m e s t a g e in t h e f u t u r e .
5 This i s not t o deny t h a t "God plays dice"; i t is merely t o take the Laplacian position t h a t , given complete knowledge of t h e world, t h e dice become determinate. Randomness follows t h e n from lack of knowledge, and t h e notion of "pure chance" need not be invoked.
8 Except i n the corlstant returns case this "greedy" choice process of adopting t h e highest- r e t u r n option a t hand does not guarantee maximum aggregate payoff; hence we use this less stringent criterion of efficiency.
Finally, we will say t h a t t h e outcome is ergodic if different sequences of historical small events, in all likelihood, lead t o t h e s a m e m a r k e t outcome--that is, if two "samples" from t h e observer's s e t of pos- sible historical events, with corresponding time paths jz,] a n d jz',], have t h e property t h a t
1
z,-
z',I
goes t o zero, with probability one, as n goes t o infinity. If this is t h e case, small events average o u t a n d become"forgotten" a s t h e m a r k e t expands.
II. MARKET
SHARINGAND
MARKE3 EXCLUSIONk Dynamics and Properties of the Three Regimes
Before looking a t t h e outcome of choices in our R a n d S agent model, i t is instructive t o take a glance a t how t h e dynamics would r u n were all a g e n t s of one type only. Here choice order does n o t m a t t e r ; a g e n t s a r e homogeneous a n d indistinguishable; a n d t h e r e a r e no unk- nown events so t h a t ergodicity is not an issue. We bypass t h e trivial con- s t a n t r e t u r n s case where a g e n t s always choose t h e higher payoff technol- ogy-
Where both technologies show diminishing returns--the s t a n d a r d textbook case--market-sharing in general t a k e s place. As d e m a n d increases, adoption follows t h e composite supply curve obtained from lateral addition of t h e s e p a r a t e r e t u r n s curves for each technology.
Adopt i o n s
n n
A ' B Figure 1.
The outcome is predictable--our observer can d e t e r m i n e in advance m a r k e t s h a r e s a f t e r n choices exactly in this situation--and i t is easy t o show i t i s efficient. I t is also flexible: marginal a d j u s t m e n t of e i t h e r r e t u r n s curve will shift t h e composite supply c u r v e a n d h e n c e m a r k e t
share.
Where both technologies show increasing r e t u r n s , t h e result is more interesting. The first agent chooses t h e more favorable technology, A say. This enhances t h e r e t u r n s to adopting A . The next agent a-fortiori chooses A too. This continues, with A chosen each time, a n d B incapable of "getting started". The end result is t h a t A "corners t h e market" and B is excluded. This outcome is trivially predictable, and efficient if r e t u r n s rise a t the same rate. Notice though t h a t if r e t u r n s increase a t different rates, the adoption process may easily become inefficient, as a cursory inspection of Fig. 1 shows. In this instance, not only a r e unadopted options better, but choices of B's only would have produced higher aggregate r e t u r n s . But this situation cannot, in general, be corrected by marginalist policy; after n choices t h e finite gap between t h e r e t u r n s t o A after n adoptions and t h e r e t u r n s t o B a t t h e starting point would have t o be closed. Flexibility is not present; and choice becomes increasingly
"locked-in" to A .
Now let u s r e t u r n t o t h e case of i n t e r e s t , where t h e unknown choice sequence of two types of agents allows us t o include some notion of his- torical "small-events". Begin with t h e constant-returns situation, and let n A ( n ) and nB(n) be t h e n u m b e r of choices of A and B respectively, when n choices in total have been made. We write t h e difference, n A ( n ) -nB(n), as d , . (Note t h a t through t h e variables
4
and n--the difference and total--we c a n fully describe t h e dynamics of t h e adoption of A versus B: in particular z,, t h e m a r k e t s h a r e of A , is 0.5+
d , / 2 n . ) In this constant r e t u r n s situation R-agents always choose A , and S agents always choose B, regardless of t h e n u m b e r of adopters of either technology. Thus t h e way in which adoption of A a n d B cumulates is determined simply by t h e sequence in which R and S agents "line up" t o make their choice, n A ( n ) increasing by one unit if t h e next agent in line is a n R , nB(n) increasing by one u n i t if t h e next agent in line is an S , with t h e difference in adoption, d,, moving upward by one unit or down- ward one u n i t accordingly.To our observer, t h e choice-order is random, with both agent types equally represented. Hence t o him, t h e "state"
4
appears t o perform a simple coin-toss gambler's random walk with each "move" having equal probability 0.5.In the diminishing-returns situation, these simple dynamics a r e modified. Figure 2 illustrates t h e r e t u r n s functions of each agent type.
Observe t h a t , although a t t h e outset R-agents will choose the higher- r e t u r n s ( t o t h e m ) technology A , adoption bids its r e t u r n s downward, so t h a t future R-agents will switch their preference t o B if the numbers using A become sufficiently greater t h a n t h e numbers using B. That is, R-agents will "switch" t h e i r preferred choice in our model if
R e t u r n s t o R-Agents L
I
R e t u r n s t o S-AgentsFigure 2
Adopt i o n s
Similarly S a g e n t s will switch preference t o A if n u m b e r s adopting B become sufficiently ahead of t h e numbers adopting A , t h a t is, if
dn
=
n A ( n )-
n g ( n )<
AS-
-
(as-
bS)S (2)
Adopt i o n s
We s e e now (in Fig. 3) t h a t t h e r e a r e t h r e e distinct regions in t h e d n , n plane where t h e directions of choice differ. In region I, where adoption of both technologies shows little difference R-types choose A and S-types choose B. But in regions I1 and 111 both a g e n t types choose t h e s a m e technology--the one t h a t is "behind". Thus d, m a y wander a t will in
n n
A ' B "A
'
"B. -
region I b u t c a n n o t e n t e r regions I1 o r 111. The competitive choice pro- cess with diminishing r e t u r n s appears to o u r observer a s a random walk with reflecting barriers.
We obtain slightly different dynamics in t h e increasing r e t u r n s situation. Now R-agents, who s t a r t with n a t u r a l preference for A , will
"switch allegiance" if adoption pushes B f a r enough ahead of A in numbers and i n payoff. Similarly, S-agents, with a n a t u r a l preference for
B ,
will switch t h e i r choices t o A if adoption pushes A far enough a h e a d of B. Regions of choice again appear in t h e 4 , n plane (see Fig. 4). defined by inequalities similar to (1) a n d (2). Once region I1 or 111 is entered, both a g e n t types choose t h e s a m e technology, but in this case t h e differ- e n c e is t h a t they will choose t h e technology t h a t is " a h e a d , with t h e r e s u l t t h a t this technology f u r t h e r i n c r e a s e s its l e a d The choice pro- cess is "locked into" e i t h e r region 11 or region 111 from t h e n on. In the 4 , n plane t h e boundaries of t h e s e regions become b a r r i e r s which"absorb" t h e process. Once e i t h e r is r e a c h e d by random movement of d n , t h e process ceases to involve both technologies-it is "locked-in" to one technology only.
I11 B l e a d s
( D i f f e r e n c e i n
Figure 3.
\R
I I
\ s
We a r e now in a position t o use t h e elementary theory of random walks ( a s in Karlin and Taylor 1975 say) t o derive t h e properties of this choice process under t h e different linear r e t u r n s regimes. For con- venient reference we summarize t h e m in Table 2.
A d o p t i o n )
n
I
T o t a l Adoptions
Table 2. Properties of the Three Regimes
In t h e increasing-linear-returns situation, we know t h a t
&
becomes absorbed with probability one-that is, t h a t the m a r k e t s h a r e of A m u s t eventually become zero o r one, s o t h a t the two technologies cannot coexist indefinitely and one m u s t exclude t h e other. But as in t h e side- of-the-road example, o u r observer cannot e z ante predict which technol- ogy will predominate. He can predict t h a t one technology will take t h e market; if h e knows random walk theory he can predict t h a t i t will be A with probability T (aS - bS)/ ((s (aR-
bR)+
r ( a S-
bS)); b u t he cannotNecessarily Necessarily Predictable Ergodic Efficient Flexible
Constant Returns Diminishing Re t u r n h c r e asing Re t u r n s
No Yes
No Yes
Yes No
Yes Yes No
Yes Yes No
(Difference dR
I1in
dnI _ _ - - - A leads - - - _ _ _ - - -
Adoption)
$5-
- -
'
B l e a d s Figure 4.predict t h e actual market-share outcome with any a c c u r a c y a t all--in spite of his knowledge of supply and demand functions. This s t a t e of affairs is quite different where r e t u r n s a r e c o n s t a n t , o r diminishing. In t h e constant-returns case ( n o barriers), t h e standard deviation of
%
increases with fi, so t h a t
G / n
t e n d s to zero, with probability one, as n increases; and in t h e diminishing r e t u r n s situation (reflecting bar- riers) d , is trapped between Finite constants, so t h a t again % / n t e n d s t o zero a s n increases. In both cases our observer c a n predict t h a t m a r k e t shares will become asymptotically equal with probability one, t h e fifty-fifty m a r k e t split resulting from t h e i n h e r e n t s y m m e t r y of t h e prob- lem in this case.Ergodicity follows easily i n t h e constant and diminishing r e t u r n s cases. Any sequence of historical events--any line-up of t h e agents-- drives t h e m a r k e t t o fifty-fifty in t h e diminishing r e t u r n s case; a n d only truly extraordinary happenstance events (for example, twice as m a n y R- agents as S-agents joining t h e line indefinitely) with associated probabil- ity zero can cause deviation from fifty-fifty in t h e c o n s t a n t r e t u r n s case.
The line-up caused by t h e historical timing of agent choices therefore has no effect on eventual m a r k e t s h a r e s , and t h e process is ergodic-it forgets its small-event history. In t h e increasing r e t u r n s case t h e situa- tion is quite different. A sizeable proportion of t h e choice sequences causes t h e m a r k e t outcome t o "tip" toward A , t h e remaining proportion causes i t to ''tip" toward B. (The extraordinary line-ups--say S followed
by R followed by ,C followed by R a n d so on indefinitely--that cause m a r k e t s h a r i n g , have proportion or s t r i c t l y speaking, m e a s u r e , zero.) Thus, t h e historical s e q u e n c e of t h e choices (which depends on t h e small e v e n t s
I t i ] )
d e c i d e s t h e p a t h of m a r k e t s h a r e s , a n d t h e process is non- ergodic- i t r e m e m b e r s i t s small-event history.Marginalist policy a d j u s t m e n t s t o t h e r e t u r n s trivially have no effect in t h e c o n s t a n t - r e t u r n s situation. In t h e two o t h e r c a s e s t h e y correspond t o a marginal shift of one o r both of t h e barriers. Once t h e i n c r e a s i n g - r e t u r n s process is "locked-in" t o A or B , however, t h e s a m e technology i s c h o s e n with a n ever widening returns-to-adoption differ- e n c e between i t a n d o t h e r , a n d marginal subsidies or taxes can have n o p u r c h a s e o n t h e dynamics of choice in t h e f u t u r e . They m u s t however affect f u t u r e choices i n t h e diminishing-returns situation (in absolute n u m b e r s , if n o t i n m a r k e t s h a r e s ) , because reflecting barriers continue t o influence t h e process (with probability one) a t t i m e s in t h e future.
The efficiency issue is different, having nothing t o do with random- ness.
I t
is simple t o show t h a t choices a r e always efficient in t h e diminishing-returns case. But with r e t u r n s increasing, i t is very easy t o c o n s t r u c t a low-payoff "locked-in" m a r k e t t h a t leaves options unadopted t h a t deliver a h i g h e r payoff. Increasing r e t u r n s do n o t g u a r a n t e e effi- ciency.B. Extensions and Variations
Would t h e s e r e s u l t s have been materially different if we h a d m a d e weaker a s s u m p t i o n s in o u r model? The answer is a qualified no.
To begin with, we c a n easily show t h a t t h e s a m e qualitative r e s u l t s hold for N technologies in competition, a n d for a g e n t types in unequal proportions ( h e r e t h e random walk "drifts"). Where a g e n t n u m b e r s a r e finite, t h e " e x t r a o r d i n a r y paths" occupy a definite proportion of t h e pos- sible paths, s o t h a t absorption or reflection now have probability some- what less t h a n one a n d properties t h a t a s s e r t themselves asymptotically m a y no longer fully hold. Where r e t u r n s t o one technology depend also on t h e n u m b e r s adopting t h e other technology, switching barriers again a p p e a r , causing corresponding behavior in t h e dynamics.
Our l i n e a r - r e t u r n s r e s u l t s e x t e n d t o t h e nonlinear case, providing t h e r e t u r n s functions a r e "parallel"
-
in t h e s e n s e t h a t4
causes"switching" t o o c c u r always a t t h e s a m e n u m e r i c a l difference in adoption between t h e two technologies. (The log-linear learning-curve r e t u r n s of footnote 4 a r e parallel in t h i s s e n s e , for example.) Then t h e associated b a r r i e r s a r e c o n s t a n t , and t h e random-walk r e s u l t s above again obtain.
In t h e m o r e g e n e r a l case, where r e t u r n s have no c o m m o n p a t t e r n besides a similar monotonicity, t h e r e s u l t s a r e weaker. Here t h e cross- over point a t which t h e d i f f e r e n c e in adoption
4
c a u s e s switching m a y vary a s t o t a l choices n increase. The b a r r i e r s m a y widen or narrow wit n , a n d if t h e y widen a t a sufficient r a t e n o switching m a y o c c u r a t all.9
Market-sharing is still g u a r a n t e e d in t h e diminishing-returns case; b u t 7 Switching occurs, with probability one, only i f the non-constant barriers lie within iterated-logarithm-law limits, from some finite stage onward.
market-exclusion n o longer, in t h e increasing-returns case. For exam- ple, if t h e increasing payoffs to learning-by-using become exhausted so t h a t r e t u r n s t o adoption level off a t different levels for e a c h technology, t h e situation gradually becomes akin t o constant r e t u r n s . Both techno- logies can, in t h i s case, s h a r e t h e market.
Three variations t h a t m e r i t f u r t h e r study a r e worth some specula- tive c o m m e n t s . Consider first a variation where all agents differ in preferences, in risk aversion, or in economic c i r c u m s t a n c e s (see David 1969). There is now a distribution of agents over r e t u r n s a t any stage in t h e adoption process. I t appears t h a t this i n c r e a s e d heterogeneity changes l i t t l e t h e mechanism t h a t c a u s e s market-sharing o r m a r k e t exclusion. The h a r d barriers between regions of choice now disappear, t o be replaced by a gradation where m o r e a n d m o r e agents have switched preference a s d, becomes larger. The s a m e r a n d o m walk appears, b u t now with changing s t e p probabilities instead of barriers. Market sharing a n d m a r k e t exclusion can again be shown y d e r appropriate conditions, but m o r e sophisticated machinery is needed
.
In a quite different variation, we might suppose t h a t adoption does n o t necessarily m e a n a technology will be improved, it m e r e l y increases t h e chances t h a t i t will be improved (see Sahal 1981, David 1975, and Nel- son and Winter 1982, for example). There may be a wide class of s u c h
"probabilistic increasing-returns" models, with "small-events" now becoming t h e discovery of improved variants. Exact market-exclusion or market-sharing conditions would depend on t h e n a t u r e of t h e model.
Finally, where conventions or standards compete, r e t u r n s usually become a function of f u t u r e a s well as pas adoptions, so t h a t technologi-
b
cal expectations (Rosenberg 1982) e n t e r . We c a n n o t say m u c h in this case without f u r t h e r information on how expectations form a n d a r e modified a s r e t u r n s change. But i t is likely t h a t in m o s t c a s e s increasing r e t u r n s again c a u s e m a r k e t exclusion: given sufficient increasing r e t u r n s , expectations of what is likely t o prevail, even if founded on very little, can become self-fulfilling, so t h a t t h e fundamental m a r k e t insta- bility is f u r t h e r exacerbated.
8 Such as the path-dependent strong law theorem of Arthur, Ermoliev and Kaniovski (1983).
Market exclusion here appears to depend crucially on the extent to which agents have corre- lated preferences for t h e two technologies.
B For an analysis of t h e expectations case, see the forthcoming dissertation of Hanson
III. DISCUSSION
k S o m e Technological Examples
In our various theoretical models, t h e economy, under cir- cumstances of increasing r e t u r n s , can become "locked-in" t o a future technological path t h a t is neither guaranteed to be efficient nor entirely predictable in advance. The most common real-world case actually con- forms to none of t h e above illustrative models. A technology is initially adopted as most suited t o prevailing conditions; but after some time these conditions change. Because users and ancillary machinery h ve 7 0 become a c c o m o d a t e d l o t h i s technology, however, i t is now locked in
.
Better alternatives cannot make a s t a r t . The 1950's programmi g 1
f.
language FORTRAN; t h e excessively narrow gauge of British railroads , t h e U.S. color television system--all technologies initially adopted for sound engineering reasons--show t h a t initial adoption can carve an inef- ficient groove t h a t t h e f u t u r e finds h a r d to escape.
The QWERTY typewriter keyboard, manifestly inefficient for modern touch-typing, is a case in point. Before 1073, early typewriters displayed a variety of keyboard arrangements, t h e most common being alphabeti- cal order for easy reference. In 1873, however, Christopher Sholes found t h a t t h i s arrangement caused his up-strike key mechanism t o jam. After considerable experimentation and on t h e advice of his brother-in-law, a school teacher and mathematician, Sholes minimized jamming by selecting a keyboard t h a t caused t h e typing bars t o come u p from dif- ferent directions on most words. The first six l e t t e r s were "QWERTY".
Approximately 1000 of t h e s e "type-writers" were mass-produced in t h e Remington sewing-machine factory in New York. In due course, employers bought QWERTY; typists learned QWERTY; and teachers taught QWERTY; so t h a t "...other manufacturers adopted t h e arrrange- m e n t with only slight variations. Those who failed t o do so disappeared
'1 2 without a trace." (Beeching 1974) ,
Examples need n o t b e confined t o engineering standards nor t o trivial technologies. If road and rail "compete" a s alternative possibili- ties for a sizeable portion of freight transported, if each mode exhibits increasing r e t u r n s in t h e form of long-run decreasing costs per ton-mile as its freightage increases, and if e a c h s t a r t s with roughly similar costs, then small events --a timely lobbying effort perhaps, or t h e opening u p of a new industrial region--may favor freightage on one mode, causing its costs t o fall and c u s t o m e r s to switch their patronage towards it. Freight density on this favored mode further increases and its costs fall further.
10 We could treat this case within our framework as one of myopic homogeneous agents operating in an R-environment t h a t a t some time changes to an S-environment; or equivalently, as a finite sequence of R-agents, followed by an indefinite sequence of S-agents.
11 Veblen writes in 1915 of "the silly little bobtail carriages used in British goods traffic;
which were well enough in their time, before American or German traffic was good for much, but which have at best a playful air when brought up against the requirements of today."
(See also Frankel 1955.)
12 Not quite. The Dvornk keyboard, invented in 1932, and reported to be 35% faster than QWERTY, still struggles on. But of forty-five nations using Roman-alphabet languages, only Belgium, Portugal and Turkey today possess standard alternatives t o the QWERTY keyboard, (See the Olympia Lnternational standard keyboards in Beeching 1974.)
Eventually t h e advantaged mode comes t o dominate m u c h of t h e m a r k e t , b u t which m de this is m a y differ in different countries. Under t h e s e suppostionsl', where road is relatively healthy, rail would be chronically under-invested, requiring periodic subsidies t o maintain some degree of efficiency; a n d vice-versa.
This is not t o say, of course, t h a t every case of competing technolo- gies shows tendencies toward m a r k e t exclusion. Most power-generation technologies, for example, a r e factor-intensive a n d show eventual dimin- ishing r e t u r n s . We would expect t h e s e t o s h a r e t h e m a r k e t in a more-or- less predictable a n d efficient way. Similarly, ring a n d m u l e technologies s h a r e d t h e cotton-spinning m a r k e t up t o t h e 1920s. The ring could spin successfully f r o m a narrow r a n g e of cotton grades, whereas t h e mule, although less efficient, could perform over a wider range (Saxenhouse a n d Wright 1983), with t h e r e s u l t t h a t m a n u f a c t u r e r s ' different access t o qualities of raw cotton m a i n t a i n e d a s h a r e d m a r k e t .
If it is t r u e t h a t competing technologies a r e often of t h e increasing- r e t u r n s type, t h e n we would e x p e c t t h e past to contain a "fossil r e c o r d of discarded o r excluded technologies t h a t would have been a s good as, or, given equal development, might have been b e t t e r t h a n , those t h a t eventually predominated. As a p u r e example of this, consider t h a t in t h e past, t h e h a n d s on c e r t a i n clocks ( t h e Uccello clock of 1433 in Florence cathedral, for example) t u r n e d anticlockwise. (See also Cipolla 1967.p.65.) After about 1550 t h i s convention was excluded.
B.
On
Historical ExplanationThe a r g u m e n t of t h i s paper suggests caution in t h e i n t e r p r e t a t i o n of economic history. Often, where we observe t h e predominance of one technology over i t s competitors--say gasoline over s t e a m a s t h e propul- sion device for automobiles--we t e n d t o look for reasons why t h e predom- i n a n t technology was superior, a n d for the m e a n s by which t h i s i n n a t e superiority c a m e t o be t r a n s l a t e d into adoption. But t h i s form of reason- ing is valid only for c o n s t a n t a n d diminishing r e t u r n s technologies.
Where technologies exist potentially i n e v e r m o r e efficient variants, superiority becomes itself a function of adoption or use. Although we should be c a u t i o u s about engineering claims, r e c e n t evidence (Burton 1976; S t r a c k 1970) suggests t h a t had t h e more efficient steam-cycle been properly h a r n e s s e d a n d developed for automotive t r a n s p o r t , i t might well have been preferable t o t h e gasoline technology ( s e e also Fletcher 1904)14. Gasoline in North America s e e m s t o have gained i t s decisive edge between 1896 a n d 1898 when one or two variants of t h e gasoline technology appeared t h a t were temporarily superior t o contemporary s t e a m v a r i a n t s . Larger e n t r e p r e n e u r s like Ransom Olds " s w i t c h e d i n t o
13 In practice, of course, this simple mechanism would be somewhat complicated by govern- ment regulation, cartel agreements, inter-regional differences, and t h e multi-product na- ture of freight. Note however t h a t from 1870 onward, econometric studies "all give strong evidence of increasing returns" t o U.S. railroad freight density (Keeler 1883).
14 The Rankine (steam) cycle is thermodynamically more efficient than the Otto (gasoline) cycle. In a 1970 NASA study Strack concluded t h a t a "steam propulsion system could b e designed t o weigh approximately t h e same as a conventional automobile propulsion system.
The overall fuel cost wuld be n o greater, and perhaps less, than today's average case."
gasoline, and magnified its prevalence in production runs. Gasoline gained a small temporary lead t h a t subsequently proved unassailable (May 1977).
Seen from our analysis, t h e issue in historical interpretation of choice of t e c
!?E
ology is not quite "market determinism" versus "histori- cal accident".
More precisely i t is whether t h e outcome is built in a- priori t o t h e endowments, opportunity s e t s , and preferences existing in t h e economy, with aggregate choice being guided to a n inevitable con- clusion by a n invisible hand of market determination, or whether una- voidable fluctuations--small events outside t h e given economic conditions-can c u m u l a t e to sway the f u t u r e technological s t r u c t u r e of t h e economy. The former'iBses
ergodicity, obtains i n constant and diminishing r e t u r n s regimes.
Small events a r e "averaged away" and forgotten--the dynamics do not "notice" t h e presence or absence of headaches a n d horseshoe-nails, and causality lies with t h e superiority of t h e m a r k e t outcome. But while this is comforting, history here is reduced t o t h e s t a t u s of m e r e carrier--the deliverer of t h e inevitable.The l a t t e r case, non-ergodicity, obtains i n increasing-returns regimes.
Micro-events become magnified by positive feedbacks; their cumulation decides t h e outcome and forms t h e causality. Insignificant cir- c u m s t a n c e s become cemented into t h e technological s t r u c t u r e of the economy; a n d history, in a sense, becomes destiny.
Historical explanation, we can conclude, should be different in t h e different r e t u r n s regimes.
C. Policy and Prediction
The two main regimes of diminishing and increasing r e t u r n s call for different policy action. In t h e diminishing r e t u r n s case of two objects of choice competing for t h e m a r k e t , it is usually best to l e t t h e superior aggregate choice, or t h e superior mix of choices, reveal itself in t h e out- come t h a t eventually dominates. But if this policy is applied in t h e increasing r e t u r n s c a s e t h e r e is no g u a r a n t e e t h a t t h e "fittest" (in t h e long r u n sense) will be t h e one t h a t survives. Further, if government seeks t o maintain a healthy balance between increasing-return choices ( road a n d rail, say) by subsidizing the choice t h a t h a s fallen behind, i t pushes policy onto a razor-edge. Small subsidies to t h e excluded choice will not re-establish it; large subsidies, on t h e o t h e r hand, will swing t h e m a r k e t and drive t h e dominant choice out.
15 Positions on this ancient debate run all the way from Engels, 1894 ( "In default of Na- poleon, another would have filled h s place,
..
.") to Croce, 102 1 ( "The material of history.. .
[is] the fleeting network of a human world which drifts like clouds before the wind and is often totally changed by unimportant events."). Modem historiography ( e.g. Conrad and Meyer 1061) takes a comfortable compromise position t h a t causality is part deterministic, part "random". Interestingly w e find in our variable-returns models no such compromise.
Causality resides either deterministically within the given economic structure, or "random- ly" in the small events and circumstances outside t h e given structure. Strictly speaking,
"random events" are not invoked or defined in this papel--only circumstances t h a t lie out- side the main description of the dynamic structure.
16 For an earlier recognition of ergodicity a s a useful concept in economic history see David (1075, p. 16)
More effective policies i n t h e i n c r e a s i n g - r e t u r n s case would be predicated on t h e n a t u r e of t h e m a r k e t breakdown: in o u r model a g e n t s myopically chose t h e best v a r i a n t a t hand; t h e r e was no inter-agent m a r k e t t o induce t h e m t o explore promising b u t less-developed infant technologies. One possibility t h e n , would be t h e a s s i g n m e n t of l i m i t e d rights t o compensation by l a t e r users. But t h i s is only partially effec- tive. Restricting s u c h "patents" t o tightly defined v a r i a n t s allows easy bypass by l a t e c o m e r s ; widening i t t o whole technologies ( s t e a m propul- sion, for example) r e s t r i c t s explaration by o t h e r s . As a second possibil- ity, t h e c e n t r a l a u t h o r i t y could itself underwrite adoption a n d explora- tion along promising b u t less popular technological paths. But again s u c h policies c a n be problematic. Eventual r e t u r n s t o a technology m a y be h a r d t o a s c e r t a i n --witness t h e controversy over solar e n e r g y subsi- dies, for example. And while t h e r e a r e obvious c o s t s t o being locked-in t o a n inferior technology, t h e r e a r e equally obvious costs t o exploring large n u m b e r s of unknown technological paths. Where g o v e r n m e n t does have a clear favored outcome, t h e b e s t c o u r s e i s t o "tilt" t h e process economi- cally toward t h e favored technology a t t h e o u t s e t , t h i s being especially effective if e v e n t s a r e r u n n i n g close t o locking-in t h e p r e f e r r e d choice.
With increasing r e t u r n s , events a t only certain t i m e s influence t h e out- come s o t h a t timing becomes all-important, whereas with diminishing r e t u r n s timing m a t t e r s little.
Finally, a word or two about economic prediction. We have s e e n , in t h e illustrative model, t h a t where increasing r e t u r n s a r e p r e s e n t , m u c h of t h e l a t e r development of an economy m a y depend upon "small events"
beneath t h e resolution of a n observer's model a n d so m a y be impossible t o predict with a n y degree of certainty. This suggests t h a t t h e r e m a y be t h e o r e t i c a l limits, a s well as practical ones, t o t h e predictability of t h e economic f u t u r e . Suppose we g r a n t econometricians, for a m o m e n t , full knowledge of f u t u r e wars, of t h e timing of e a r t h q u a k e s , of t h e formation of cartels, a n d of t h e technological possibilities over t h e horizon. Sup- pose we g r a n t t h e m c o n s u m a t e skill in finding c o r r e c t e c o n o m e t r i c descriptions of supply a n d d e m a n d functions a n d m a r k e t conditions.
Suppose t h e economy c o n t a i n s processes of choice which operate subject t o increasing r e t u r n s . And suppose t h a t e c o n o m e t r i c models-whether computer-based o r not--are of finite size a n d h e n c e of finite resolution, so t h a t t h e r e a r e real-world micro-events t h a t lie b e n e a t h t h e i r notice.
Then t h e i n h e r e n t potential amplification of t h e s e unnoticeable small event m a y bring into being a corresponding region of u n c e r t a i n out- comeP7. We c a n speculate t h a t a n e c o n o m e t r i c model t h a t predicts a c c u r a t e l y with c e r t a i n t y is, u n d e r t h e s e m o s t favorable c i r c u m s t a n c e s , an impossibility.
17 Similar arguments apply ( Leith 1066; Lorenz 1863) to t h e theoretical possibility oi accu- rate meteorological forecasting. The obsemiitional n e t would have to be finer than t h e ra- dius of the smallest eddy, else these "small events" become amplified by inherent positive feedbacks into large uncertainties.
IV.
CONCLUSIONIn this paper we constructed a simple theoretical model of compet- ing technologies t h a t showed t h e dynamics of choice under increasing r e t u r n s to have several important properties: historical small events can determine t h e future technological path of t h e economy; knowledge of preferences, endowments, and transformation possibilities is not always sufficient t o predict the path t h e economy will follow; a n d t h e economy c a n over time become locked in t o a rigid and not always effi- cient technological s t r u c t u r e .
There remains a large number of open questions, to which t h e con- ceptual framework developed h e r e might apply. Do t h e s e properties obtain in all increasing r e t u r n s dynamics? If not, c a n we characterize t h e situations where they do? What happens when we allow strategic manipu'lation, as would be likely with competing products? What happens if r e t u r n s depend also on future choices? What difference does it make if agents can wait for r e t u r n s t o increase? Under what circumstances would we see similar, increasing-returns properties in t h e dynamics of international trade, of economic development, or of industrial struc- t u r e ? What policies might prove effective in particular cases?
Whether t h e theoretical properties obtained in this paper apply t o some portion of t h e actual economy remains to be empirically proven; if they do standard conventions i n policy prescription, historical interpre- tation and economic prediction may have to be revised.
A C K N O W L E D G ~
The author would like to t h a n k Paul David, Ward Hanson, Richard Nelson, Nathan Rosenberg, Martin Shubik, Gavin Wright, and t h e members of t h e Technological Innovations Project Workshop a t Stanford for useful suggestions, comments, and criticisms.
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