cant events may by chance give one of them an initial advantage in adoptions.
This technology may then improve more than the others, so it may appeal
to a wider proportion of potential adopters. It may therefore become further
adopted and further improved. Thus a technology that by chance gains an
early lead in adoption may eventually “corner the market” of potential adopt-
ers, with the other technologies becoming locked out. Of course, under dif-
ferent “insignificant events”—unexpected successes in the performance of
prototypes, whims of early developers, political circumstances—a different
technology might achieve sufficient adoption and improvement to come to
dominate. Competitions between technologies may have multiple potential
outcomes.
It is well known that allocation problems with increasing returns tend to
exhibit multiple equilibria, and so it is not surprising that multiple outcomes should appear here. Static analysis can typically locate these multiple equilibria, but usually it cannot tell us which one will be “selected.” A dynamic approach might be able to say more. By allowing the possibility of “random
events” occurring during adoption, it might examine how these influence
“selection” of the outcome—how some sets of random “historical events”
might cumulate to drive the process towards one market-share outcome,
others to drive it towards another. It might also reveal how the two famil-
iar increasing-returns properties of non-predictability and potential inefficiency come about: how increasing returns act to magnify chance events as adoptions take place, so that ex ante knowledge of adopters’ preferences and the technologies’ possibilities may not suffice to predict the “market outcome;”
2. Rosenberg (1982) calls this “Learning by Using” (see also Atkinson and Stiglitz, 1969). Jet aircraft designs like the Boeing 727, for example, undergo constant modification and they improve significantly in structural soundness, wing design, payload capacity, and engine efficiency as they accumulate actual airline adoption and use.
[ 70 ] Complexity and the Economy
and how increasing returns might drive the adoption process into developing a technology that has inferior long-run potential. A dynamic approach might
also point up two new properties: inflexibility in that once an outcome (a dominant technology) begins to emerge it becomes progressively more “locked in”;
and non-ergodicity (or path-dependence) in that historical “small events” are not averaged away and “forgotten” by the dynamics—they may decide the
outcome.
This paper contrasts the dynamics of technologies’ “market shares”
under conditions of increasing, diminishing, and constant returns. It pays
special attention to how returns affect predictability, efficiency, flexibility, and ergodicity; and to the circumstances under which the economy might
become locked-in by “historical events” to the monopoly of an inferior
technology.
I. A SIMPLE MODEL
Nuclear power can be generated by light-water, or gas-cooled, or heavy-water,
or sodium-cooled reactors. Solar energy can be generated by crystalline-silicon or amorphous-silicon technologies. I abstract from cases like this and assume
in an initial, simple model that two new technologies, A and B, “compete”
for adoption by a large number of economic agents. The technologies are
not sponsored 3 or strategically manipulated by any firm; they are open to all.
Agents are simple consumers of the technologies who act directly or indirectly as developers of them.
Agent i comes into the market at time t ; at this time he chooses the latest i
version of either technology A or technology B; and he uses this version there-after.4 Agents are of two types, R and S, with equal numbers in each, the two types independent of the times of choice but differing in their preferences,
perhaps because of the use to which they will put their choice. The version
of A or B each agent chooses is fixed or frozen in design at his time of choice, so that his payoff is affected only by past adoptions of his chosen technology. (Later I examine the expectations case where payoffs are also affected by future adoptions.)
Not all technologies enjoy increasing returns with adoption. Sometimes
factor inputs are bid upward in price so that diminishing returns accompany
3. Following terminology introduced in Arthur (1983), sponsored technologies are proprietary and capable of being priced and strategically manipulated; unsponsored technologies are generic and not open to manipulation or pricing.
4. Where technologies are improving, it may pay adopters under certain condi-
tions to wait; so that no adoptions take place (Balcer and Lippman, 1984; Mamer and McCardle, 1987). We can avoid this problem by assuming adopters need to replace an obsolete technology that breaks down at times { t }.
i
comPe t ing t ecHnologie s , incr e a sing r e turn s [ 71 ]
Table 1. RETURNS TO CHOOSING A OR B GIVEN
PREVIOUS ADOPTIONS
Technology A
Technology B
R-agent
a + rn
b + rn
R
A
R
B
S-agent
a + sn
b + sn
S
A
S
B
adoption. Hydro-electric power, for example, becomes more costly as dam
sites become scarcer and less suitable. And some technologies are unaffected
by adoption—their returns are constant. I include these cases by assuming
that the returns to choosing A or B realised by any agent (the net present value of the version of the technology available to him) depend upon the number of previous adopters, n and n , at the time of his choice (as in Table 1)5
A
B
with increasing, diminishing, or constant returns to adoption given by r and s simultaneously positive, negative, or zero. I also assume a > b and a < b R
R
S
S
so that R-agents have a natural preference for A, and S-agents have a natural preference for B.
To complete this model, I want to define carefully what I mean by “chance”
or “historical events.” Were we to have infinitely detailed prior knowledge
of events and circumstances that might affect technology choices—political
interests, the prior experience of developers, timing of contracts, decisions
at key meetings—the outcome or adoption market-share gained by each
technology would presumably be determinable in advance. We can conclude
that our limited discerning power, or more precisely the limited discerning
power of an implicit observer, may cause indeterminacy of outcome. I therefore define “historical small events” to be those events or conditions that are outside the ex-ante knowledge of the observer—beyond the resolving power of his “model” or abstraction of the situation.
To return to our model, let us assume an observer who has full knowledge of all the conditions and returns functions, except the set of events that determines the times of entry and choice { t } of the agents. The observer thus “sees”
i
the choice order as a binary sequence of R and S types with the property that an R or an S comes n th in the adoption line with equal likelihood, that is, with probability one half.
We now have a simple neoclassical allocation model where two types of
agents choose between A and B, each agent choosing his preferred alternative when his time comes. The supply (or returns) functions are known, as is the
demand (each agent demands one unit inelastically). Only one small element
5. More realistically, where the technologies have uncertain monetary returns we can assume von Neumann-Morgenstern agents, with Table 1 interpreted as the resulting determinate expected-utility payoffs.
[ 72 ] Complexity and the Economy
is left open, and that is the set of historical events that determine the sequence in which the agents make their choice. Of interest is the adoption-share outcome in the different cases of constant, diminishing, and increasing returns,
and whether the fluctuations in the order of choices these small events intro-
duce make a difference to adoption shares.
We will need some properties. I will say that the process is: predictable if the small degree of uncertainty built in “averages away” so that the observer
has enough information to pre-determine market shares accurately in the
long-run; flexible (not locked-in) if a subsidy or tax adjustment to one of the technologies’ returns can always influence future market choices; ergodic (not path-dependent) if different sequences of historical events lead to the same
market outcome with probability one. In this allocation problem choices
define a “path” or sequence of A— and B-technology versions that become adopted or “developed,” with early adopters possibly steering the process onto a development path that is right for them, but one that may be regretted by
later adopters. Accordingly, and in line with other sequential-choice problems, I will adopt a “no-regret” criterion and say that the process is path-efficient if at all times equal development (equal adoption) of the technology that is behind
in adoption would not have paid off better.6 (These informal definitions are
made precise in the Appendix.)
Allocation in the Three Regimes
Before examining the outcome of choices in our R and S agent model, it is instructive to look at how the dynamics would run in a trivial example with
increasing-returns where agents are of one type only (Table 2). Here choice
order does not matter; agents are all the same; and unknown events can make
no difference so that ergodicity is not an issue. The first agent chooses the more favourable technology, A say. This enhances the returns to adopting A. The next agent a-fortiori chooses A too. This continues, with A chosen each time, and B incapable of “getting started.” The end result is that A “corners the market” and B is excluded. This outcome is trivially predictable, and path-efficient if returns rise at the same rate. Notice though that if returns increase at different rates, the adoption process may easily become path-inefficient, as Table 2
shows.
6. An alternative efficiency criterion might be total or aggregate payoff (after n choices). But in this problem we have two agent types with different preferences operating under the “greedy algorithm” of each agent taking the best choice at hand for himself; it is easy to show that under any returns regime maximisation of total payoffs is never guaranteed.
comPe t ing t ecHnologie s , incr e a sing r e turn s [ 73 ]
Table 2. AN EXAMPLE: ADOPTION PAYOFFS FOR HOMOGENEOUS AGENTS
Number of previous 0
10
20
30
40
50
60
70
80
90
100
adoptions
Technology A
10
11
12
13
14
15
16
17
18
19
20
Technology B
4
7
10
13
16
19
22
25
28
31
34
In this case after thirty choices in the adoption process, all of which are
A, equivalent adoption of B would have delivered higher returns. But if the process has gone far enough, a given subsidy-adjustment g to B can no longer close the gap between the returns to A and the returns to B at the starting point. Flexibility is not present here; the market becomes increasingly
“locked-in” to an inferior choice.
Now let us return to the case of interest, where the unknown choice-
sequence of two types of agents allows us to include some notion of histori-
cal “small events.” Begin with the constant-returns case, and let n ( n) and A
n ( n) be the number of choices of A and B respectively, when n choices in B
total have been made. We can describe the process by x , the market share n
of A at stage n, when n choices in total have been made. We will write the difference in adoption, n ( n) ‒ n ( n) as d . The market share of A is then A
B
n
expressible as
x = 0⋅5 +
/ 2 . (1)
n
dn
n
Note that through the variables d and n—the difference and total—we can n
fully describe the dynamics of adoption of A versus B. In this constant-returns situation R-agents always choose A and S-agents always choose B, regardless of the number of adopters of either technology. Thus the way in which adoption of A and B cumulates is determined simply by the sequence in which Rand S-agents “line up” to make their choice, n ( n) increasing by one unit if the A
next agent in line is an R, with n ( n) increasing by one unit if the next agent B
in line is an S, and with the difference in adoption, d , moving upward by one n
unit or downward one unit accordingly. To our observer, the choice-order is
random, with agent types equally likely. Hence to him, the state d appears to n
perform a simple coin-toss gambler’s random walk with each “move” having
equal probability 0·5.
In the increasing-returns case, these simple dynamics are modified. New
R-agents, who have a natural preference for A, will switch allegiance if by chance adoption pushes B far enough ahead of A in numbers and in payoff.
That is, new R-agents will “switch” if
( b − )
R
a
d
R
= ( ) − ( ) < ∆ =
.
(2)
n
nA n nB n
R
r
[ 74 ] Complexity and the Economy
A leads
Difference in
adoptions
of A and B
Both adopter types choose A
R-types choose A. S-types choose B
0
Total adoptions
Both adopter types choose B.
Lock-in
to B
B leads
Figure 1:
Increasing returns adoption: a random walk with absorbing barriers.
Similarly new S-agents will switch preference to A if numbers adopting A become sufficiently ahead of the numbers adopting B, that is, if
( b − )
S
a
d
S
= ( ) − ( ) > ∆ =
.
(3)
n
nA n nB n
S
s
Regions of choice now appear in the d , n plane (see Figure 1), with bound-n
aries between them given by (2) and (3). Once one of the outer regions is
entered, both agent types choose the same technology, with the result that
this technology further increases its lead. Thus in the d , n plane (2) and n
(3) describe barriers that “absorb” the process. Once either is reached by random movement of d , the process ceases to involve
both technologies—it is n
“locked-in” to one technology only. Under increasing returns then, the adop-
tion process becomes a random walk with absorbing barriers. I leave it to the
reader to show that the allocation process with diminishing returns appears
to our observer as a random walk with reflecting barriers given by expressions similar to (2) and (3).
Properties of the Three Regimes
We can now use the elementary theory of random walks to derive the prop-
erties of this choice process under the different linear returns regimes. For
convenient reference the results are summarized in Table 3.
To prove these properties, we need first to examine long-term adoption
shares. Under constant returns, the market is shared. In this case the random
walk ranges free, but we know from random walk theory that the standard
deviation of d increases with √ n. It follows that the d /2 n term in equation n
n
(1) disappears and that x tends to 0·5 (with probability one), so that the mar-n
ket is split 50-50. In the diminishing returns case, again the adoption market comPe t ing t ecHnologie s , incr e a sing r e turn s [ 75 ]
Table 3. PROPERTIES OF THE THREE REGIMES
Predictable
Flexible
Ergodic
Necessarily
path-efficient
Constant returns
Yes
No
Yes
Yes
Diminishing returns
Yes
Yes
Yes
Yes
Increasing returns
No
No
No
No
is shared. The difference-in-adoption, d , is trapped between finite constants; n
hence d /2 n tends to zero as n goes to infinity, and x must approach 0·5. (Here n
n
the 50-50 market split results from the returns falling at the same rate.) In the increasing-returns-absorbing-barrier case, by contrast, the adoption share of
A must eventually become zero or one. This is because in an absorbing random walk d eventually crosses a barrier with probability one. Therefore the two n
technologies cannot coexist indefinitely: one must exclude the other.
Predictability is therefore guaranteed where the returns are constant, or
diminishing: in both cases a forecast that the market will settle to 50-50 will be correct, with probability one. In the increasing returns case, however, for accuracy the observer must predict A’s eventual share either as 0 or 100%.
Complexity and the Economy Page 15