“selected.” The small events of history become important11 Where we observe
the predominance of one technology or one economic outcome over its com-
petitors we should thus be cautious of any exercise that seeks the means by
which the winner’s innate “superiority” came to be translated into adoption.
(3) The usual policy of letting the superior technology reveal itself in the outcome that dominates is appropriate in the constant- and diminishing-returns
cases. But in the increasing-returns case laissez-faire gives no guarantee that the “superior” technology (in the long-run sense) will be the one that survives.
Effective policy in the (unsponsored) increasing-returns case would be predi-
cated on the nature of the market breakdown: in our model early adopters
impose externalities on later ones by rationally choosing technologies to suit 10. Amusingly, Fletcher (1904) writes: “ . . . unless the objectionable features of the petrol carriage can be removed, it is bound to be driven from the road by its less objectionable rival, the steam-driven vehicle of the day.”
11. For earlier recognition of the significance of both non-convexity and
path-dependence for economic history see David (1975).
[ 82 ] Complexity and the Economy
only themselves; missing is an inter-agent market to induce them to explore promising but costly infant technologies that might pay off handsomely to
later adopters.12 The standard remedy of assigning to early developers (pat-
ent) rights of compensation by later users would be effective here only to the degree that early developers can appropriate later payoffs. As an alternative, a central authority could underwrite adoption and exploration along promising
but less popular technological paths. But where eventual returns to a tech-
nology are hard to ascertain—as in the U.S. Strategic Defence Initiative case
for example—the authority then faces a classic multi-arm bandit problem of
choosing which technologies to bet on. An early run of disappointing results
(low “jackpots”) from a potentially superior technology may cause it perfectly rationally to abandon this technology in favour of other possibilities. The fundamental problem of possibly locking-in a regrettable course of development
remains (Cowan, 1987).
IV. CONCLUSION
This paper has attempted to go beyond the usual static analysis of
increasing-returns problems by examining the dynamical process that “selects”
an equilibrium from multiple candidates, by the interaction of economic forces and random “historical events.” It shows how dynamically, increasing returns can cause the economy gradually to lock itself in to an outcome not necessarily superior to alternatives, not easily altered, and not entirely predictable in advance.
Under increasing returns, competition between economic objects—in
this case technologies—takes on an evolutionary character, with a “founder
effect” mechanism akin to that in genetics.13 “History” becomes important. To
the degree that the technological development of the economy depends upon
small events beneath the resolution of an observer’s model, it may become
impossible to predict market shares with any degree of certainty. This sug-
gests that there may be theoretical limits, as well as practical ones, to the predictability of the economic future.
Stanford University
Date of receipt of final typescript: May 1988
12. Competition between sponsored technologies suffers less from this missing market. Sponsoring firms can more easily appropriate later payoffs, so they have an incentive to develop initially costly, but promising technologies. And financial markets for sponsoring investors together with insurance markets for adopters who may make the
“wrong” choice, mitigate losses for the risk-averse. Of course, if a product succeeds and locks-in the market, monopoly-pricing problems may arise. For further remarks on policy see David (1987).
13. For other selection mechanisms affecting technologies see Dosi (1988), Dosi et al. (1988), and Metcalfe (1985).
comPe t ing t ecHnologie s , incr e a sing r e turn s [ 83 ]
APPENDIX A
A. DEFINITIONS OF THE PROPERTIES
Here I define precisely the properties used above. Denote the market share of
A after n choices as x . The allocation process is:
n
(i) predictable if the observer can ex-ante construct a forecasting sequence
{ x∗ with the property that | x − ∗ → 0 with probability one, as n → ∞; n
xn|
,
n }
(ii) flexible if a given marginal adjustment g to the technologies’ returns can alter future choices;
(iii) ergodic if, given two samples from the observer’s set of possible historical events, { t } and { t′ }, with corresponding time-paths { x } and { x′ }, i
i
n
n
then | x′ − x | → 0, with probability one, as n →∞;
n
n
(iv) path-efficient if, whenever an agent chooses the more-adopted
technology α, versions of the lagging technology β would not have delivered more had they been developed and available for adoption. That is,
path-efficiency holds if returns Π remain such that Π α( m) ≥ Max {Π
j
β( j)} for
k ≤ j ≤ m, where there have been m previous choices of the leading technology and k of the lagging one.
B. THE EXPECTATIONS CASE
Consider here the competing standards case where adopters are affected by
future choices as well as past choices. Assume in our earlier model that R-agents receive additional net benefits of Π R Π R
,
, if the process locks-in to their
A
B
choice, A or B respectively; similarly S-agents receive Π S Π S
,
. (Technologies
A
B
improve with adoption as before.) Assume that agents know the state of the
market ( n , n ) when choosing and that they have expectations or beliefs that A
B
adoptions follow a stochastic process Ω. They choose rationally under these
expectations, so that actual adoptions follow the process Γ(Ω). This actual process is a rational expectations equilibrium process when it bears out the expected process, that is, when Γ(Ω) ≡ Ω.
We can distinguish two cases, corresponding to the degree of heterogeneity
of preferences in the market.
Case ( i). Suppose initially that a
R
−
> Π and b
S
− > Π and that R
S
a
R
bR
B
S
A
and S-types have beliefs that the adoption process is a random walk Ω with absorption barriers at Δ′ , Δ′ , with associated probabilities of lock-in to
R
S
[ 84 ] Complexity and the Economy
A, P(n , n ) and lock-in to B, 1 − P(n , n ). Under these beliefs, R-type expected A
B
A
B
payoffs for choosing A or B are, respectively:
a
R
+
+ ( , )Π (4)
R
rnA P nA nB
A
b
R
+
+[1 −
( ,
)]Π . (5)
R
rnB
P nA nB
B
S-type payoffs may be written similarly. In the actual pr
ocess R-types will switch to B when n and n are such that these two expressions become equal.
A
B
Both types choose B from then on. The actual probability of lock-in to A is zero here; so that if the expected process is fulfilled, P is also zero here and we have n and n such that
A
B
a
R
+
=
+
+ Π
R
rnA bR rnB
B
with associated barrier given by
∆ =
−
= −( − − R
n
n
a
b
Π ) r
/ . (6)
R
A
B
R
R
B
Similarly S-types switch to A at boundary position given by
∆ =
−
= ( − − S
n
n
b
a
Π )/ s. (7)
S
A
B
S
S
A
It is easy to confirm that beyond these barriers the actual process is indeed
locked in to A or to B and that within them R-agents prefer A, and S-agents prefer B. Thus if agents believe the adoption process is a random walk with absorbing barriers Δ′ , Δ′ given by (6) and (7), these beliefs will be fulfilled, R
S
and this random walk will be a rational expectations equilibrium.
Case ( ii). Suppose now that a
R
−
< Π and b
S
− < Π . Then (4) and
S
a
R
bR
B
S
A
(5) show that switching will occur immediately if agents hold expectations
that the system will definitely lock-in to A or to B. These expectations become self-fulfilling and the absorbing barriers narrow to zero. Similarly, when
non-improving standards compete, so that r and s are zero, in this case again beliefs that A or B will definitely lock-in become self-fulfilling.
Taking cases (i) and (ii) together, expectations either narrow or collapse the switching boundaries. They exacerbate the fundamental market instability.
C. THE PATH-DEPENDENT STRONG-LAW THEOREM
Consider a dependent-increment stochastic process that starts with an initial
vector of units b , in the K categories, 1 through K. At each event-time a unit 0
comPe t ing t ecHnologie s , incr e a sing r e turn s [ 85 ]
is added to one of the categories 1 through K, with probabilities p = [ p ( x), 1
p ( x), . . . , p ( x)], respectively. (The Borel function p maps the unit simplex of 2
K
proportions SK into the unit simplex of probabilities SK.) The process is iterated to yield the vectors of proportions X , X , X , . . . .
1
2
3
THEOREM. Arthur, Ermoliev, and Kaniovski (1983, 1986)
(i) Suppose
p: SK → SK is continuous, and suppose the function p(x) − x possesses a Lyapunov function (that is, a positive, twice-differentiable func-
tion V with inner product <[ p(x) − x], V > negative). Suppose also that the x
set of fixed points of p, B = { x: p(x) = x} has a finite number of connected components. Then the vector of proportions { X } converges, with prob-n
ability one, to a point z in the set of fixed points B, or to the border of a connected component.
(ii) Suppose p maps the interior of the unit simplex into itself, and that z is a stable point (as defined in the conventional way). Then the process has
limit point z with positive probability.
(iii) Suppose z is a non-vertex unstable point of p. Then the process cannot converge to z with positive probability.
(iv) Suppose probabilities of addition vary with time n, and the sequence { p }
n
converges to a limiting function p faster than 1/ n converges to zero. Then the above statements hold for the limiting function p. That is, if the above conditions are fulfilled, the process converges with probability one to one
of the stable fixed points of the limiting function p.
The theorem is extended to non-continuous functions p and to non-unit
and random increments in Arthur, Ermoliev and Kaniovski (1987 b). For the case K = 2 with p stationary see the elegant analysis of Hill et al. (1980).
[ 86 ] Complexity and the Economy
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