Is a singularity just around the corner?
What it takes to get explosive economic growth
Journal of Evolution and Technology June
1998. Vol. 2 -
 PDF
Version
- (Received 10 April, 1998)
-
- Robin Hanson
School of Public Health
- University of California, Berkeley
Warren Hall, CA 94720-7360
-
ABSTRACT
Economic growth is determined by the supply and demand
of investment capital; technology determines the demand for
capital, while human nature determines the supply. The supply
curve has two distinct parts, giving the world economy two
distinct modes. In the familiar slow growth mode, rates of
return are limited by human discount rates. In the fast
growth mode, investment is limited by the world's wealth.
Historical trends suggest that we may transition to the fast
mode in roughly another century and a half.
Can some new technology switch us to the fast mode more
quickly than this? Perhaps, but such a technology must
greatly raise the rate of return for the world's expected
worst investment project. It must thus be very broadly
applicable, improving almost all forms of capital and
investment. Furthermore, investment externalities must
remain within certain limits.
Introduction
Many technological enthusiasts, impressed by the potential of
various envisioned technologies, have speculated that technology
may soon become so productive as to induce a
"singularity", a period of extremely rapid growth. (A
collection of speculations can be found here and
here.)
How rapid? Over the last few centuries, population has
doubled roughly every 70 years, per-capita consumption has
doubled roughly every 35 years,and scientific progress has
doubled roughly every 15 years. In contrast, computing
power has doubled roughly every two years for the last
half-century. Many have speculated that perhaps economic growth
rates will soon match or even greatly exceed current
computing-power growth rates.
What would it take, exactly, for a technology to make the
economy grow this fast? This paper offers a simple economic
analysis intended to illuminate this question. We will find
that while very rapid growth is possible in principle, this
requires enabling technologies to meet some strong conditions. It
is hard to see how any single new technology could do this,
though historical trends suggest that the accumulation of all new
technologies over the century and a half might.
The supply and demand of investment capital
Growth in consumption over any extended period requires growth
in what economists call "capital", which means any
thing which helps make "products" people want.
These terms are considered abstractly, so a music concert is a
"product", and the skills of a musician are "human
capital".
Capital can be used not only to make products, but also to
make more capital, and the rate at which this happens is called a
"rate of return". The fraction of capital devoted
to making more capital is called the "savings rate",
and the price people pay to rent capital for this purpose is
called a "market rate of return." This is a "real
rate of return" if capital is calibrated so that one more
unit of capital always produces the same "amount" (or
"value" or "utility") of products people
like.
What determines how much people save and the
interest rate they get for it? Supply and demand, naturally.
Technology supplies a pool of investment projects which are
physically possible, and investors demand these projects more or
less depending on how productive they are. Or, looking at it in
terms of investment capital, investment projects demand
capital to be carried out, and investors supply capital to
such projects. This view is illustrated in the following figure.
As is usual in supply and demand graphs, the x-axis is a
quantity axis, here amount of capital, and the y-axis is a price
axis, here a rate of return. The origin where the axes meet
has no capital saved and a zero rate of return.
Supply curves slope up, so that higher returns are required to
induce investors to save larger fractions of their income. Demand
lines slope down, so that the more projects are undertaken, the
worse their return. Projects can be thought of as lined up in
order along the savings axis, with the best projects at the left
and the worst projects at the right. At the point where supply
and demand meet, the marginal investor is just barely willing to
offer capital to get the market return offered by the marginal
project.
While the demand lines shown are fictitious, the shape of the
supply curve shown is true to a reasonable understanding of
what investment supply looks like. This is based both on
observing current human preferences, and on an understanding of
what sort of preferences should have been selected for during our
evolution. Moreover, the dates on the figure show roughly where
the economy has been along this curve. (See the technical
appendix for details.)
The supply curve eventually turns up sharply, shooting off to
infinity at a particular bound. At least it does this if
the population growth rate stays below twice its current value,
and if investments on net benefit rather than harm non-investors
(again, see the appendix). The supply curve is relatively flat,
however, over a large range, at a rate of return determined
mainly by the population growth rate and the degree to which our
genes have taught us to discount risk and time. Since capital
supply should be relatively insensitive to technology, we can
hold this curve fixed in our minds as we consider how technology
might change the demand for capital, i.e., the supply of
investment projects.
When poor technology creates a low demand for investment
capital, as with Demand 1 in the figure, the resulting market
rate of return is not very sensitive to demand. In this case
improving technology mainly just raises the savings rate. Demand
2 in the figure describes an economy nearing the middle of a
transition between the two distinct economic modes, which may
describe our world economy in roughly a century.
When technology is good enough to create a high demand for
investment capital, as with Demand 3, savings becomes insensitive
to demand, and the market rate of return can become sensitive to
technology. In the high demand mode, growth rates can in
principle be very high. Given demand curves with the right
shape, growth rates might increase much faster with time in the
high growth mode than they did in the low growth mode. Such
rapidly increasing growth rates seems most like the imagined
singularity scenarios. With other demand shapes, however, fast
growth rates might increase more slowly than slow growth rates
did.
Burning Up Excess Return in Investment Races
The reader may have noticed that the figure above suggests
that while in the low demand mode the market rate of return stays
roughly constant, the average rate of return of the
projects actually tried might depend dramatically on the demand.
After all, the top of the demand curve could in principle be very
far above the low supply curve. So why couldn't growth rates also
be very large in the low demand mode?
The problem with this argument is that this figure has
neglected to include the time dimension, and a lack of long-term
property rights in most investment projects means that returns
above the market rate are burned up an a race to be first.
There are, in fact, very few long-term property rights
regarding the right to undertake investment projects. Think of
developing a new kind of car, colonizing the moon, developing
specialized CAD software, or making a movie with a certain kind
of gimmick. Each such project requires various forms of capital,
such as machines or skilled labor. While in the short term only
one investor may have the rights to tackle a given project, in
the long term many competing investors could have positioned
themselves to have this short-term opportunity.
For example, Microsoft's dominant position in PC operating
systems now gives it the right to many very attractive
investments. But there was once an open race to become the
dominant operating system, and competitors then tried harder
because of the prospect of later high returns. And when deciding
whether to enter this earlier race and how hard to try, investors
mainly wondered if they could get a competitive rate of return.
Similarly, while one group now has the right to make the next
Batman movie sequel, there have long been open contests to create
popular movie series, and popular comic strips.
Consider a typical as-yet-untried investment project, becoming
more and more attractive with time as technology improves and the
world market grows larger. If there wasn't much point in
attempting such a project very long after other teams tried, then
a race to be first should make sure the project is attempted near
when investors first expect such attempts to produce a
competitive rate of return. This should happen even if the
project returns would be much greater if everyone waited longer.
The extra value the project would have if everyone waited is
burned up in the race to do it first.
Thus most of the return above the market return in our supply
and demand figure above should be burned up, leaving the average
return at about the market return. Thus in the low demand mode
the height of the demand curve is relatively unimportant. If
anticipated, a technology which makes a moderate number of
investment projects much more productive may have no effect on
growth rates or on rates of return.
The width of the demand curve, however, does matter in low
demand mode. If technology creates many new attractive investment
projects, growth rates can rise due to a rise in the savings
rate. This happens by changing the rate of return offered on the
marginal investment project, the project actually tried which is
expected to offer the worst return. To improve the marginal
project, a technology needs to have a very broad range of
productive applications.
Investment Bottlenecks
A big problem with having a very broad range of applications,
however, is the possibility of bottleneck resources. Most
production, for consumption or investment, requires a wide
variety of forms of capital. These include land, raw materials,
energy, various sorts specialized labor and specialized machines,
information sources, places to dispose of waste, access to
channels of distribution and advertising, legal adjudication, and
regulatory approval. A technology that dramatically improves the
productivity of one of these typically gives only a small
improvement to the final output. This is the concept of
diminishing returns in production. If the rate of improvement in
any one input consistently lags behind the rest, that input then
becomes a bottleneck limiting the growth rate for the entire
process.
For example, computer hardware may be getting cheap very fast,
but one cannot now as quickly reduce the costs of training
professionals who understand both computers and application areas
well. Thus such training becomes a bottleneck limiting the
contribution of computers to the larger economy. Similarly,
even having the cost of producing electricity drop to near zero
would only have a minor effect on the economy, at least in the
short term. Most of the price consumers pay for electricity
is to transport it, not to produce it, and new ways of organizing
the transportation and production of electricity would have to
evolve to take more advantage of the new situation.
The notion of bottlenecks applies to timescales in
particular. For example, I think the rapid growth in
computing speed and communication bandwidth, together with loose
notions that most of the economy must be at root computation and
communication, have suggested to many that most all economic
timescales, including economic doubling times, must soon rise to
meet these faster rates. But this ignores the prospect of
other physical processes becoming time bottlenecks, and the
diminishing returns to raw computation. Note also that in the low
growth mode, even changing the total investment time scale by a
large factor, which implies changing the capital demand function
by a large factor, need not change the market return or growth
rate by very much.
It seems hard to escape the conclusion that it just takes a
lot of time for the world economy to absorb even very broadly
applicable technologies like the computer, especially if the
criteria of interest is raising the rate of return on the
marginal investment project worldwide. Thus it seems unlikely
that a single new technology could quickly knock the economy into
a high demand mode with very high growth rates. (For similar
conclusions, see here and here.)
Even if no single technology can create fast growth, the
cumulative effect of all new technologies over many decades,
might, still slowly push the economy up the supply curve into the
high demand mode. This requires that such technologies keep
creating enough productive investment opportunities to keep up
with a very rapidly growing economy, that investments on average
benefit non-investors, and that we don't double population growth
rates. This doesn't seem easy, but it may not be impossible
either.
Conclusion
If we assume that the risk and time preferences of future
investors will not vary much from those which evolution has given
us, then the supply curve for investment capital should say
nearly fixed while technology changes the demand curve. And since
a lack of long-term property rights in investment projects
creates a race to be first, the two parts of the supply curve
create two distinct modes for the economy: a low growth rate mode
and a high growth rate mode. Historical trends on
growth rates suggest that we are slowly moving up the capital
supply curve toward the high growth mode, and may reach it in
roughly a century and a half.
Many have suggested that some special new technology will
induce rapid growth rates much more quickly than this. This is
possible in principle, but we have identified a number of
conditions which such a technology must meet. Investments must on
average benefit, rather than harm, non-investors. And most
difficult, the special technology must have such broad and
attractive enough applications across the world economy that it
dramatically raises the returns on the worst investment anyone
undertakes. To do this, such a technology must dramatically
improve the productivity of almost all forms of capital, not just
a few. And this must be true even though, in a race to be first,
each investment project is started as soon as it seemed
marginally attractive.
These conditions do not appear to have been met by any of the
very diverse range of technologies that have yet appeared. It is
possible that such conditions may be met by some future
technology, but a persuasive argument for this case should
explain, in standard economic terminology, why we should expect
this technology to meet these conditions.
References
Barro, R., Sala-I-Martin, X., Economic Growth, McGraw
Hill, 1995.
Blume, L., Easley, D., "Evolution and Market
Behavior", Journal of Economic Theory, 58:9-40, 1992.
Hansson, I., Stuart, C., "Malthusian Selection of
Preferences," American Economic Review,
80(3):529-544, June 1990.
Maddison, A., Dynamic Forces in Capitalist Development,
Oxford Univ. Press, 1991.
Price, D., Little science, big science... and beyond,
Columbia Univ. Press, 1986.
See also Econ
Growth Web site.
Technical Appendix
There are many more complex models of economic growth out
there, but the following simple model is complex enough to embody
many results relevant to singularity speculations.
This model is one of the simplest models of endogenous
growth, allowing for real long-term growth within the model
and implicitly including all forms of capital improvement,
including research and development. The model's main
simplification is that it does not explicitly distinguish between
various forms of capital and consumption, aggregating them
instead to a single generic form.
First some notation, including standard values where
applicable.
- n = growth rate of world population, now ~1.4%/yr.
- g = growth rate of world per-capita consumption,
now ~2%/yr.
- p = typical discount rate, ~3%/yr (= factor
of 2 in 23 yrs).
- a = typical risk-aversion, ~1 (a =
-c*u''/u' for utility u(c) of consumption rate
c).
- r = real market rate of return (not the risk-less
interest rate).
- s = savings, fraction of world capital making more
capital.
- A = growth rate of invested capital (corrected for
depreciation).
- i = internality, fraction of investment return the
investor gets.
We have five primary equations using these parameters.
Supply of Capital: r = p + a * g
Demand for Capital: A = A(s)
Capital Rental: r = i * A
Accounting: g + n = s * A
SUPPLY: The supply of capital for investment depends on how
fast consumption is growing relative to the typical rate at which
people discount the future. If people were risk-neutral, the
market return would equal the typical discount rate. But for
risk-adverse people who plan to consume more tomorrow than today,
stuff today is worth more.
DEMAND: The demand function A(s) describes the
investment projects that current technology makes possible. It
says what the expected total (private and external) return on the
worst project would be if a fraction s of total income
were invested.
RENTAL: The rental price of capital depends on how fast a unit
of capital can produce more capital, corrected for the fact that
an investor may not internalize all the costs or benefits of a
project. A project may, for example, create improvements in
technology that others can copy.
ACCOUNTING: The total growth rate in consumption and capital
depends on how productive capital is, and on what fraction of
income is saved.
Some comments:
- Typical utility parameter values are predicted from
evolutionary selection. When trading resources for a
parent now vs. for a child a generation from now, it
matters that a child shares only half a parent's genes.
And unit risk-aversion, which is log utility u(c) =
log(c), is selected for, at least regarding shocks to
all copies of a gene, such as the total market risk in
the CAPM.
- These equations allow for different people to own
different amounts of capital, and for different
investment projects to have different returns and
internality. This is because the supply equation is a
very general implication of maximizing expected
discounted utility, and the internality parameter i
can be considered an average over the projects tried.
- We have neglected the probably-weak dependence of
internality i on the savings rate s. Having
i<1 says that investment projects on net
benefit non-investors, while i>1 says that
non-investors are on net harmed.
- We've set the market return equal to the average return,
and so are assuming no long-term property rights in
projects. A projects happens at the first time its return
becomes competitive. Relaxing this assumption is
equivalent to reducing the internality i, which
then becomes more dependent on savings s.
- A(s) summarizes all technical change which creates
economic growth. It does not, however, describe changes
in the growth rate. neralize While the return to any one
not-yet-started project typically rises with time as
technology improves, the best projects will be done
first, so the A(s) function may rise or fall with
time. A(s) also falls if the number of attractive
investments did not grow as fast as the economy did.
- The growth rate g in the accounting equation is
really the growth rate of capital, while the rate g
in the demand equation is the growth rate of consumption.
When savings s is constant, such as with demand A(s)
unchanging with time, these are the same
thing. We are thus assuming that the growth rate of
savings s is much less than the growth rate of
consumption g.
- We are using depreciation-corrected parameters A
and s in the above equations. To have a
depreciation d appear explicitly, change to raw
terms A0, s0, where A0 = A + d and s0 =
(sA+d)/A0. The accounting and rental equations become
n+g = s0*A0 - d and r = i*(A0-d). For
non-human capital, depreciation is typically ~5%/yr.
The first four columns of the following table shows historical
estimates for the annual per-capita and population growth rates g,n
for four differnt dates from 1750 to 1995. (Warning:
the per-capita growth estimate for 1750 is very crude.)
Using the above model and our standard values for preference
parameters a,p, we then infer the next two columns, an
annual rate of return r and a savings fraction s/i
for each date. The return estimates seem roughly
within reason.
| Year
|
# nations in data
|
Per-capita growth rate
|
Popul. growth rate
|
Rate of Return
|
Savings fraction
|
Total return
|
Savings
|
Raw total return
|
Raw savings
|
| |
|
g |
n |
r |
s/i
|
A |
s |
A0
|
s0
|
| 1995
|
100+
|
2.0%
|
1.4%
|
5.0%
|
68%
|
10%
|
34%
|
15%
|
56%
|
| 1905
|
29
|
1.3%
|
0.9%
|
4.3%
|
50%
|
9%
|
25%
|
14%
|
53%
|
| 1825
|
7
|
0.8%
|
0.5%
|
3.8%
|
34%
|
8%
|
17%
|
13%
|
50%
|
| 1750
|
2
|
0.0%
|
0.4%
|
3.0%
|
13%
|
6%
|
7%
|
11%
|
49%
|
Somewhat arbitrarily assuming internality i=50%, and
using the standard value of depreciation d=5%, the rest of
the table gives estimates for total return and savings A,s, and
their raw values A0,s0. A standard estimate of 56%
for current raw total savings roughly fits with the standard
values that ~20% of raw savings is invested in non-human
capital, which gets ~1/3 of income. Note that the savings
growth rate seems to be growing at ~0.3%/yr, which
is much less than the per-capita growth rate, as we had assumed.
What fast growth requires
Assuming that preference parameters a,p don't change
much, what does this model say about the possibility of an
economic "singularity," that is, very large growth
rates g in per-capita consumption? Such growth
rates, if they persisted long, would imply vast changes in
per-capita consumption in a single human generation.
The demand equation says that per-capita consumption growth g
can't get very large unless the interest rate r does, and
the rental equation says that the interest rate r can't
get very large unless the total return A does. The
accounting equation also says that total consumption growth n+g
can't get very large unless total return A does.
Using our equations to eliminate r and g, we get
a market equation, with capital demand on the left and capital
supply on the right,
p - a * n
A(s) = ----------- .
i - a * s
For p < a n and a s < i, this gives the
functional form shown in the figure in the body of the paper.
Thus the relevant savings fraction is actually s/i,
savings relative to internality, rather than savings relative to
total income.
The limits to fast growth appear more directly in
p - n
g = s * ----------- .
i - a * s
Thus the only way to allow very fast growth g>> p
is for s = ~i/a without n = ~p. Thus since s
< 1, we require i < a.
Thus an economic singularity, g>> p, requires:
i < a, s = ~i/a, and A(i/a) >> p .
That is, for unit (i.e., log) risk-aversion, an economic
singularity requires that
- Investment projects on net benefit, rather than hurt,
non-investors.
- Savings must be carefully balanced near the internality
parameter.
- The return expected for the worst invested-in project
becomes very large.
The bottom line is that this model does allow for an economic
singularity under certain circumstances. The historical increase
in the savings fraction has been roughly constant with time for
the last two centuries, suggesting a near 100% savings fraction
near the year 2150. Our ignorance about the internality
parameter is cause for concern. Thus it is possible, though
not obvious, that a continuation of historical trends will result
in an economic singularity of g>> p in roughly a
century and a half.
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