by Ajay Agrawal
t
rate of 0.4 percent per year. Regimes last for thirty years. Period 100 is highlighted with a black circle. At this point in time, the capital share is relatively high and growth is relatively low.
the capital share in the future is lower than in period 100 instead of higher
can be reversed.
Summing Up
Automation—an increase in —can be viewed as a “twist” of the capital-
t
and labor- augmenting terms in a neoclassical production function. From
250 Philippe Aghion, Benjamin F. Jones, and Charles I. Jones Uzawa’s famous theorem, since we do not in general have purely labor-augmenting technical change, this setting will not lead to balanced growth.
In this particular application (e.g., with < 0), either the capital share or the
growth rate of GDP will tend to increase over time, and sometimes both.
We showed one special case in which all tasks are ultimately automated that
produced balanced growth in the limit with a constant capital share less than
100 percent. A shortcoming of this case is that it requires automation to be
the only form of technological change. If, instead, the nature of automa-
tion itself improves over time—consider the plow, then the tractor, then the
combine- harvester, then GPS tracking—then the model is best thought of
as featuring both automation and something like improvements in A . In
t
this case, one would generally expect growth not to be balanced. However,
a combination of periods of automation followed by periods of respite,
like that shown in fi gure 9.3 does seem capable of producing dynamics at
least superfi cially similar to what we have seen in the United States in recent
years: a period of a high capital share with relatively slow economic growth.
9.3 Artifi cial Intelligence in the Idea Production Function
In the previous section, we examined the implications of introducing AI in
the production function for goods and services. But what if the tasks of the
innovation process themselves can be automated? How would AI interact
with the production of new ideas? In this section, we introduce AI in the
production technology for new ideas and look at how AI can aff ect growth
through this channel.
A moment of introspection into our own research process reveals many
ways in which automation can matter for the production of ideas. Research
tasks that have benefi ted from automation and technological change include
typing and distributing our papers, obtaining research materials and data
(e.g., from libraries), ordering supplies, analyzing data, solving math prob-
lems, and computing equilibrium outcomes. Beyond economics, other ex-
amples include carrying out experiments, sequencing genomes, exploring
various chemical reactions and materials. In other words, applying the same
task- based model to the idea production function and considering the auto-
mation of research tasks seems relevant.
To keep things simple, suppose the production function for goods and
services just uses labor and ideas:
(19)
Y = A L .
t
t
t
But suppose that various tasks are used to make new ideas according to
1
1/
(20)
A = A
X di
where
< 0.
t
t
it
0
Artifi cial Intelligence and Economic Growth 251
Assuming some fraction of tasks have been automated—using a similar
t
setup to that in section 9.2—the idea production function can be ex-
pressed as
(21)
A = A ( B K ) + ( C S )
(
)1/ A F BK , C S
(
),
t
t
t
t
t
t
t
t
t
t
t
where S is the research labor used to make ideas, and B and C are defi ned t
t
t
as before, namely, B ≡ (1 ) and C ≡ (1
)(1 )/ .
t
t
t
t
Several observations then follow from this setup. First, consider the case
in which is constant at some value but then increases to a higher value
t
(recall that this leads to a one- time decrease in B and increase in C ). The t
t
idea production function can then be written as
BK
(22)
A
= A S F
t , C
t
t
t
St
~ A CS ,
t
t
where the “~” notation means “is asymptotically proportional to.” The
second line follows if K / S is growing over time (i.e., if there is economic t
t
growth) and if the elasticity of substitution in F(·) is less than one, which we have assumed. In that case, the CES function is bounded by its scarcest
argument, in this case researchers. Automation then essentially produces a
level eff ect but leaves the long- run growth rate of the economy unchanged
if < 1. Alternatively, if = 1—the classic endogenous growth case—then
automation raises long- run growth.
Next, consider this same case of a one- time increase in , but suppose the
elasticity of substitution in F(·) equals one, so that F(·) is Cobb- Douglas. In this case, as in the Zeira model, it is easy to show that a one- time increase in
automation will raise the long- run growth rate. Essentially, an accumulable
factor in production (capital) becomes permanently more important, and
this leads to a multiplier eff ect that raises growth.
Third, suppose now that the elasticity of substitution is greater than one.
In this case, the argument given before reverses, and now the CES function
asymptotically looks like the plentiful factor, in this case K . The model will t
then deliver explosive growth under fairly general conditions, with incomes
becoming infi nite in fi nite time.9 But this is true even without any automa-
tion. Essentially, in this case researchers are not a necessary input and so
standard capital accumulation is enough to generate explosive growth. This
is one reason why the case of < 1—that is, an elasticity of substitution
less than one—is the natural case to consider. We focus on this case for the
remainder of this section.
9. A closely related case is examined explicitly in the discussion surrounding equation (27) below.
252 Philippe Aghion, Benjamin F. Jones, and Charles I. Jones 9.3.1 Continuous Automation
We can now consider the special case in which automation is such that the
newly automated tasks constitute a constant fraction, q, of the tasks that
have not yet been automated. Recall that this was the case that delivered a
balanced growth path back in the Balanced Growth section
In This Case, B → 1 and (C / C ) → g = – [(1 – )/ ]· > 0 Asymptotically t
t
t
c
The same logic that gave us equation (22) now implies that
B K
(23)
/> A
= A C S F
t
t ,1
t
t
t
t
C S
t
t
~ A C S ,
t
t
t
where the second line holds as long as BK / CS → ∞, which holds for a large class of parameter values.10
This reduces to the Jones (1995) kind of setup, except that now “eff ective”
research grows faster than the population because of AI. Dividing both sides
of the last expression by A gives
t
A
S
(24)
t = Ct t .
A
A 1
t
t
In order for the left- hand side to be constant, we require that the numerator
and denominator on the right side grow at the same rate, which then implies
+ g
(25)
g = gC
S .
A
1
In Jones (1995), the expression was the same except g = 0. In that case, the
C
growth rate of the economy is proportional to the growth rate of research-
ers (and ultimately, the population). Here, automation adds a second term
and raises the growth rate: we can have exponential growth in research eff ort
in the idea production function not only because of growth in the actual
number of people, but also as a result of the automation of research implied
by AI. Put another way, even with a constant number of researchers, the
number of researchers per task S/ (1 – ) can grow exponentially: the fi xed
t
number of researchers is increasingly concentrated onto an exponentially
declining number of tasks.11
10. Since B → 1, we just require that g > g . This will hold—see below—for example if > 0.
k
c
11. Substituting in for other solutions, the long- run growth rate of the economy is g =
y
{– [(1 – )/]· + n}/(1 – ), where n is the rate of population growth.
Artifi cial Intelligence and Economic Growth 253
9.4 Singularities
To this point, we have considered the eff ects of gradual automation in
the goods and idea production functions and shown how that can poten-
tially raise the growth rate of the economy. However, many observers have
suggested that AI opens the door to something more extreme—a “techno-
logical singularity” where growth rates will explode. John Von Neumann is
often cited as fi rst suggesting a coming singularity in technology (Danaylov
2012). I. J. Good and Vernor Vinge have suggested the possibility of a self-
improving AI that will quickly outpace human thought, leading to an “intel-
ligence explosion” associated with infi nite intelligence in fi nite time (Good
1965; Vinge 1993). Ray Kurzweil in The Singularity is Near also argues for
a coming intelligence explosion through nonbiological intelligence (Kurz-
weil 2005) and, based on these ideas, cofounded Singularity University with
funding from prominent organizations like Google and Genentech.
In this section, we consider singularity scenarios in light of the produc-
tion functions for both goods and ideas. Whereas standard growth theory is
concerned with matching the Kaldor facts, including constant growth rates,
here we consider circumstances in which growth rates may increase rapidly
over time. To do so, and to speak in an organized way to the various ideas
that borrow the phrase “technological singularity,” we can characterize two
types of growth regimes that depart from steady- state growth. In particular,
we can imagine:
• a “Type I” growth explosion, where growth rates increase without
bound but remain fi nite at any point in time; and
• a “Type II” growth explosion, where infi nite output is achieved in fi nite
time.
Both concepts appear in the singularity community. While it is common
for writers to predict the singularity date (often just a few decades away),
writers diff er on whether the proposed date records the transition to the
new growth regime of Type I or an actual singularity occurring of Type II.12
To proceed, we now consider examples of how the advent of AI could
drive growth explosions. The basic fi nding is that complete automation of
tasks by an AI can naturally lead to the growth explosion scenarios above.
However, interestingly, one can even produce a singularity without relying
on complete automation, and one can do it without relying on an intelligence
explosion per se. Further below, we will consider several possible objections
to these examples.
12. Vinge (1993), for example, appears to be predicting a Type II explosion, a case that has been examined mathematically by Solomonoff (1985), Yudkowsky (2013), and others. Kurzweil (2005), by contrast, who argues that the singularity will come around the year 2045, appears to be expecting a Type I event.
254 Philippe Aghion, Benjamin F. Jones, and Charles I. Jones 9.4.1 Examples of Technological Singularities
We provide four examples. The fi rst two examples take our previous mod-
els to the extreme and consider what happens if everything can be auto-
mated—that is, if people can be replaced by AI in all tasks. The third ex-
ample demonstrates a singularity through increased automation but without
relying on complete automation. The fi nal example looks directly at “super-
intelligence” as a route to a singularity.
Example 1: Automation of Goods Production
The Type I case can emerge with full automation in the production for
goods. This is the well- known case of an AK model with ongoing techno-
logical progress. In particular, take the model of section 9.2, but assume that
all tasks are automated as of some date t . The production function is there-0
after Y = A K and growth rates themselves grow exponentially with A .
t
t
t
t
Ongoing productivity growth—for example, through the discovery of new
ideas—would then produce ever- accelerating growth rates over time. Spe-
cifi cally, with a standard capital accumulation specifi cation ( K = sY – K ) t
t
t
and technological progress proceeding at rate g, the growth rate of output
becomes
(26)
g = g + sA
,
Y
t
which grows exponentially with A .
t
Example 2: Automation of Ideas Production
An even stronger version of this acceleration occurs if the automation
applies to the idea production function instead of (or in addition to) the
goods production function. In fact, one can show that there is a mathe-
matical singularity: a Type II event where incomes essentially become infi -
nite in a fi nite amount of time.
To see this, consider the model of section 9.3. Once all tasks can be auto-
mated, that is, once AI replaces all people in the idea production function,
the production of new ideas is given by
(27)
A = K A .
t
t
t
With > 0, this diff erential equation is �
�more than linear.” As we discuss next,
growth rates will explode so fast that incomes become infi nite in fi nite time.
The basic intuition for this result comes from noting that this model is
essentially a two- dimensional version of the diff erential equation A = A 1+
t
t
(e.g., replacing the K with an A in equation [27]). This diff erential equation can be solved using standard methods to give
1/
(28)
A =
1
.
t
A
t
0
Artifi cial Intelligence and Economic Growth 255
And it is easy to see from this solution that A( t) exceeds any fi nite value before date t* = (1 / A). This is a singularity.
0
For the two dimensional system with capital in equation (27), the argu-
ment is slightly more complicated but follows this same logic. The system of
diff erential equations is equation (27) together with the capital accumulation
equation ( K = sY – d K , where Y = A L). Writing these in growth rates gives t
t
t
t
t
A
(29)
t = Kt A ,
A
A
t
t
t
K
A
(30)
t = sL t
.
K
K
t
t
First, we show that ( A / A ) > ( K / K ). To see why, suppose they were equal.
t
t
t
t
Then equation (30) implies that ( K / K ) is constant, but equation (29) would t
t
then imply that ( A / A ) is accelerating, which contradicts our original t
t
assumption that the growth rates were equal. So it must be that
( A / A ) > ( K / K ).13 Notice that from the capital accumulation equation, this t
t
t
t
means that the growth rate of capital is rising over time, and then the idea
growth rate equation means that the growth rate of ideas is rising over time
as well. Both growth rates are rising. The only question is whether they rise
suffi
ciently fast to deliver a singularity.
To see why the answer is yes, set = 0 and sL = 1 to simplify the algebra.
Now multiply the two growth rate equations together to get
A
K
(31)
t
t = A .
A
K
t
t
t
We have shown that ( A / A ) >