poly: Modeling Economic Singularities

I've been feeling uncomfortable that I've been speculating about
growth issues without having a formal model in mind. I've done
that now, and I have to admit I may have overstated my case.
I'll put the following at: http://hanson.berkeley.edu/fastgrow.html

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There are lots more complex models of economic growth out there,
but the following simple model still embodies many results
relevant for singularity speculations (and many other topics).

First some notation, including standard values where applicable.
n = growth rate of world population, ~1%/yr.
g = growth rate of world per-capita consumption, ~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)).
r = real interest rate, in terms of capital amounts.
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.

Demand for Projects: r = p + a * g

Supply of Projects: A = A(s)

Capital Rental: r = i * A

Accounting: g + n = s * A

DEMAND: The demand for projects depends on how fast consumption is
growing relative to the typical rate at which people discount the
future. If people were risk-neutral, the interest rate would equal
the typical discount rate. But for risk-adverse people who plan to
consume more tomorrow than today, stuff today is worth more to them.

SUPPLY: The supply function A(s) describes the investment projects
that 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 much more capital
a unit of capital can produce, corrected for the fact that an investor
doesn't get all those returns for herself. For example, a project
may 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:
1. We've set the marginal return equal to the average return, so are
   assuming no long-term property rights in projects. A projects
   happens at the first time its return becomes competitive.
2. The return to any one project should rise with time as technology
   improves. The A(s) function, however, describes the distribution
   of returns to not-yet-started projects. Since the best projects
   will be done first, the A(s) function may rise or fall with time.
3. We are using depreciation-corrected parameters A and s in the above
   equations. To have a depreciation d appear explicitly, change to
   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.
4. Our typical utility parameter values are predicted from
   evolutionary selection. When trading resources for a parent now vs.
   for a child a generation from now, note a child shares only half a
   parent's genes. Also 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.

We can use our standard values for some parameters to solve for the
other parameters values. The demand equation implies an interest
rate of r = ~5%/yr, which is roughly within reason. The other
equations then imply s/i = ~80%. So if internality i = ~50%,
then savings s = ~40%, and uncorrected savings s0 = ~60%. This
fits with the standard values that ~20% of non-depreciation-corrected
savings is invested in non-human capital, which gets ~1/3 of income.
(If we assume p = ~2%, doubling in 35 years, we get r = ~4%,
s/i = ~75%, so i = ~50% implies s = ~38%, and s0 = ~61%.)

Assuming that utility parameters p,a don't change a lot, what does
this model say about the possibility of an economic "singularity,"
that is, very large growth rates g >> p in per-capita consumption?
Such growth rates, if they persisted that 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 also can't get very large unless A does.

Using our equations to eliminate r and g, we get a final equation

               A(s)*(i - a * s) = p - a * n

For moderate population growth, not n >> p, the only way to allow very
large total returns A >> p is for s = ~i/a. 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 log risk-aversion an economic singularity requires that
1. Investment projects on net benefit, rather than hurt, non-investors.
2. Savings are carefully balanced to equal the internalization fraction.
3. The returns expected of the worst invested-in project is very large,
   even when most world income is going to such projects, and when
   each project was started as soon as it seemed marginally attractive.

Note also that very fast population growth, n >> p, implies s > i/a,
an even larger savings rate, as well as very high returns A >> p.

The bottom line is that this model does allow for an economic
singularity, if the A(s) function is cooperative. Thus, assuming
this model is reasonable, skepticism about such singularities should
focus on modeling A(s) more explicitly.

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Robin Hanson
hanson@econ.berkeley.edu http://hanson.berkeley.edu/
RWJF Health Policy Scholar, Sch. of Public Health 510-643-1884
140 Warren Hall, UC Berkeley, CA 94720-7360 FAX: 510-643-8614
Received on Thu Mar 26 21:12:54 1998