Complexity and the Economy
Page 16
But either choice will be wrong with probability one-half. Predictability is lost.
Notice though that the observer can predict that one technology will take the market; theoretically he can also predict that it will be A with probability s(a − b )/[ s(a − b )+ r(b − a )]; but he cannot predict the actual market-share R
R
R
R
S
S
outcome with any accuracy—in spite of his knowledge of supply and demand
conditions.
Flexibility in the constant-returns case is at best partial. Policy adjust-
ments to the returns can affect choices at all times, but only if they are large enough to bridge the gap in preferences between technologies. In the two
other regimes adjustments correspond to a shift of one or both of the barri-
ers. In the diminishing-returns case, an adjustment g can always affect future choices (in absolute numbers, if not in market shares), because reflecting barriers continue to influence the process (with probability one) at times in the future. Therefore diminishing returns are flexible. Under increasing returns,
however, once the process is absorbed into A and B, the subsidy or tax adjustment necessary to shift the barriers enough to influence choices (a precise
index of the degree to which the system is “locked-in”) increases without
bound. Flexibility does not hold.
Ergodicity can be shown easily in the constant and diminishing returns
cases. With constant returns only extraordinary line-ups (for example, twice as many R-agents as S-agents appearing indefinitely) with associated probability zero can cause deviation from 50-50. With diminishing returns, any sequence
of historical events—any line-up of the agents—must still cause the process
[ 76 ] Complexity and the Economy
to remain between the reflecting barriers and drive the market to 50-50. Both cases forget their small-event history. In the increasing returns case the situation is quite different. Some proportion of agent sequences causes the mar-
ket outcome to “tip” towards A, the remaining proportion causes it to “tip”
towards B. (Extraordinary line-ups—say S followed by R followed by S followed by R and so on indefinitely—that could cause market sharing, have probability or measure zero.) Thus, the small events that determine { t } decide i
the path of market shares; the process is non-ergodic or path-dependent—it
is determined by its small-event history.
Path-efficiency is easy to prove in the constant—and diminishing-returns
cases. Under constant-returns, previous adoptions do not affect pay-off. Each
agent-type chooses its preferred technology and there is no gain foregone by
the failure of the lagging technology to receive further development (further
adoption). Under diminishing returns, if an agent chooses the technology
that is ahead, he must prefer it to the available version of the lagging one.
But further adoption of the lagging technology by definition lowers its payoff.
Therefore there is no possibility of choices leading the adoption process down an inferior development path. Under increasing returns, by contrast, development of an inferior option can result. Suppose the market locks in to tech-
nology A. R-agents do not lose; but S-agents would each gain ( b − a ) if their S
S
favoured technology B had been equally developed and available for choice.
There is regret, at least for one agent type. Inefficiency can be exacerbated if the technologies improve at different rates. An early run of agent-types who
prefer an initially attractive but slow-to-improve technology can lock the market in to this inferior option; equal development of the excluded technology in the long run would pay off better to both types.
Extensions, and the Rational Expectations Case
It is not difficult to extend this basic model in various directions. The same qualitative results hold for M technologies in competition, and for agent types in unequal proportions (here the random walk “drifts”). And if the technologies arrive in the market at different times, once again the dynamics go
through as before, with the process now starting with initial n or n not at A
B
zero. Thus in practice an early-start technology may already be locked in, so
that a new potentially-superior arrival cannot gain a footing.
Where agent numbers are finite, and not expanding indefinitely, absorp-
tion or reflection and the properties that depend on them still assert them-
selves providing agent numbers are large relative to the numerical width of
the gap between switching barriers.
For technologies sponsored by firms, would the possibility of strategic action alter the outcomes just described? A complete answer is not yet known.
comPe t ing t ecHnologie s , incr e a sing r e turn s [ 77 ]
Hanson (1985) shows in a model based on the one above that again market exclusion goes through: firms engage in penetration pricing, taking losses
early on in exchange for potential monopoly profits later, and all but one firm exit with probability one. Under strong discounting, however, firms may be
more interested in immediate sales than in shutting rivals out, and market
sharing can reappear.7
Perhaps the most interesting extension is the expectations case where
agents’ returns are affected by the choices of future agents. This happens for example with standards, where it is matters greatly whether later users fall in with one’s own choice. Katz and Shapiro (1985, 1986) have shown, in a
two-period case with strategic interaction, that agents’ expectations about
these future choices act to destabilise the market. We can extend their find-
ings to our stochastic-dynamic model. Assume agents form expectations in
the shape of beliefs about the type of stochastic process they find themselves in. When the actual stochastic process that results from these beliefs is identical with the believed stochastic process, we have a rational-expectations fulfilled-equilibrium process. In the Appendix, I show that under increasing
returns, rational expectations also yield an absorbing random walk, but one
where expectations of lock-in hasten lock-in, narrowing the absorption barri-
ers and worsening the fundamental market instability.
II. A GENERAL FRAMEWORK
It would be useful to have an analytical framework that could accommodate
sequential-choice problems with more general assumptions and returns
mechanisms than the basic model above. In particular it would be useful to
know under what circumstances a competing-technologies adoption market
must end up dominated by a single technology.
In designing a general framework it seems important to preserve two prop-
erties: ( i) That choices between alternative technologies may be affected by the numbers of each adopted at the time of choice; ( ii) That small events “outside the model” may influence adoptions, so that randomness must be allowed for.
Thus adoption market shares may determine not the next technology chosen
directly but rather the probability of each technology’s being chosen.
Consider then a dynamical system where one of K technologies is adopted each time an adoption choice is made, with probabilities p (x), p (x), . . . , p (x), 1
2
K
respectively. This vector of probabilities p is a function of the vector x, the adoption-shares of technologies 1 to K, out of the total number n of adoptions 7. For similar findings see the literature on the dynamics of commodity competition under increasing returns (e.g. Spence, 1981; Fudenberg and Tirole, 1983).
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the Economy
1
A
p (x)
chooses
2
p (x)
1
Probability next agent
x 2
00
1
Proportion of A in market total
Figure 2:
Two illustrative adoption functions.
so far. The initial vector of proportions is given as x . I will call p(x) the adop-0
tion function.
We may now ask what happens to the long run proportions or adoption
shares in such a dynamical system. Consider the two different adoption func-
tions in Figure 2, where K = 2. Now, where the probability of adoption of A is higher than its market share, in the adoption process A tends to increase in proportion; and where it is lower, A tends to decrease. If the proportions or adoption-shares settle down as total adoptions increase, we would conjecture
that they settle down at a fixed point of the adoption function.
In 1983 Arthur, Ermoliev, and Kaniovski proved that under certain techni-
cal conditions (see the Appendix) this conjecture is true.8 A stochastic process of this type converges with probability one to one of the fixed points of the
mapping from proportions (adoption shares) to the probability of adoption.
Not all fixed points are eligible. Only “attracting” or stable fixed points (ones that expected motions of the process lead towards) can emerge as the long run
outcomes. And where the adoption function varies with time n, but tends to a limiting function p, the process converges to an attracting fixed point of p.
Thus in Figure 2 the possible long-run shares are 0 and 1 for the function
p and x for the function p ). Of course, where there are multiple fixed points, 1
2
2
which one is chosen depends on the path taken by the process: it depends on the cumulation of random events that occur as the process unfolds.
8. See Arthur, Ermoliev, and Kaniovski (1987a) for a readable account of this work.
comPe t ing t ecHnologie s , incr e a sing r e turn s [ 79 ]
We now have a general framework that immediately yields two useful theorems on path-dependence and single-technology dominance.
THEOREM I. An adoption process is non-ergodic and non-predictable if and only if its adoption function p possesses multiple stable fixed points.
THEOREM II. An adoption process converges with probability one to the dominance of a single technology if and only if its adoption function p possesses stable fixed points only where x is a unit vector.
These theorems follow as simple corollaries of the basic theorem above. Thus
where two technologies compete, the adoption process will be path-dependent
(multiple fixed points must exist) as long as there exists at least one unstable
“watershed” point in adoption shares, above which adoption of the technol-
ogy with this share becomes self-reinforcing in that it tends to increase its
share, below which it is self-negating in that it tends to lose its share. It is therefore not sufficient that a technology gain advantage with adoption; the
advantage must (at some market share) be self-reinforcing (see Arthur, 1988).
Non-Linear Increasing Returns with a Continuum of Adopter Types
Consider, as an example, a more general version of the basic model above,
with a continuum of adopter types rather than just two, choosing between K
technologies, with possibly non-linear improvements in payoffs. Assume that
if n previous adopters have chosen technology j previously, the next agent’s j
payoff to adopting j is Π ( n ) = a + r(n ) where a represents the agent’s “natural j
j
j
j
j
preference” for technology j and the monotonically increasing function r represents the technological improvement that comes with previous adoptions.
Each adopter has a vector of natural preferences a = ( a , a , . . . , a ) for the 1
2
K
K alternatives, and we can think of the continuum of agents as a distribution of points a (with bounded support) on the positive orthant. We assume an adopter is drawn at random from this probability distribution each time
a choice occurs. Dominance of a single technology j corresponds to positive probability of the distribution of payoffs Π being driven by adoptions to a
point where Π exceeds Π for all i ≠ j.
j
i
The Arthur-Ermoliev-Kaniovski theorem above allows us to derive:
THEOREM III. If the improvement function r increases at least at rate ε as n increases, j
the adoption process converges to the dominance of a single technology, with probability one.
Proof. In this case, the adoption function varies with total adoptions n. (We do not need to derive it explicitly, however.) It is not difficult to establish that as n
[ 80 ] Complexity and the Economy
becomes large: (i) At any point in the neighbourhood of any unit vector of adoption shares, unbounded increasing returns cause the corresponding technology
to dominate all choices; therefore the unit-vector shares are stable fixed points.
(ii) The equal-share point is also a fixed point, but unstable. (iii) No other point is a fixed point. Therefore, by the general theorem, since the limiting adoption function has stable fixed points only at unit vectors the process converges to one of these with probability one. Long-run dominance by a single technology
is assured. ■
Dominance by a single technology is no longer inevitable, however, if
the improvement function r is bounded, as when learning effects become exhausted. This is because certain sequences of adopter types could bid
improvements for two or more technologies upward more or less in concert.
These technologies could then reach the upper bound of r together, so that none of these would dominate and the market would remain shared from
then on. Under other adopter sequences, by contrast, one of the technolo-
gies may reach the upper bound sufficiently fast to shut the others out. Thus, in the bounded case, some event histories dynamically lead to a shared market; other event histories lead to dominance. Increasing returns, if they are
bounded, are in general not sufficient to guarantee eventual monopoly by a single technology.
III. REMARKS
(1) To what degree might the actual economy be locked-in to inferior technol-
ogy paths? As yet we do not know. Certainly it is easy to find cases where an
early-established technology becomes dominant, so that later, superior alter-
natives cannot gain a footing.9 Two important studies of historical events
leading to lock-ins have now been carried out: on the QWERTY typewriter key-
board (David, 1985); and on alternating current (David and Bunn, 1987). (In
both cases increasing returns arise mainly from coordination externalities.)
Promising empirical cases that may reflect lock-in through learning are the
nuclear-reactor technology competition of the 1950s and 1960s and the US
steam-versus-petrol car competition in the 1890s. The US nuclear industry is
practically 100% dominated by light-water reactors. These reactors were origi-
nally adapted from a highly compact unit designed to propel the first nuclear
9. Examples might be the narrow gauge of British railways (Kindleberger, 1983); the US colour television system; the 1950s programming language FORTRAN; and of course the QWERTY keyboard (Arthur, 1984; David, 1985; Hartwick, 1985). In these particular cases the source of increasing returns is network externalities, however, rather than learning effects. Breaking out of locked-in technolog
ical standards has been investigated by Farrell and Saloner (1985, 1986).
comPe t ing t ecHnologie s , incr e a sing r e turn s [ 81 ]
submarine, the U.S.S. Nautilus, launched in 1954. A series of circumstances—
among them the Navy’s role in early construction contracts, political expedi-
ency, the Euratom programme, and the behaviour of key personages—acted
to favour light water. Learning and construction experience gained early
on appear to have locked the industry in to dominance of light water and
shut other reactor types out (Bupp and Darian, 1978; Cowan, 1987). Yet
much of the engineering literature contends that, given equal development,
the gas-cooled reactor would have been superior (see Agnew, 1981). In the
petrol-versus-steam car case, two different developer types with predilections toward steam or petrol depending on their previous mechanical experience,
entered the industry at varying times and built upon on the best available
versions of each technology. Initially petrol was held to be the less promis-
ing option: it was explosive, noisy, hard to obtain in the right grade, and it required complicated new parts.10 But in the United States a series of trivial circumstances (McLaughlin, 1954; Arthur, 1984) pushed several key developers into petrol just before the turn of the century and by 1920 had acted to
shut steam out. Whether steam might have been superior given equal devel-
opment is still in dispute among engineers (see Burton, 1976; Strack, 1970).
(2) The argument of this paper suggests that the interpretation of economic
history should be different in different returns regimes. Under constant and
diminishing returns, the evolution of the market reflects only a-priori endow-
ments, preferences, and transformation possibilities; small events cannot
sway the outcome. But while this is comforting, it reduces history to the sta-
tus of mere carrier—the deliverer of the inevitable. Under increasing returns, by contrast many outcomes are possible. Insignificant circumstances become
magnified by positive feedbacks to “tip” the system into the actual outcome