From: Robin Hanson <hanson@econ.berkeley.edu>

Date: Thu Mar 26 1998 - 13:05:31 PST

Date: Thu Mar 26 1998 - 13:05:31 PST

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

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