by Ajay Agrawal
t
t
used in each automated task and L/ (1 – ) units of labor are used on each
t
nonautomated task. The production function can then be written as
1/
(11)
K
Y = A
t
+ 1
(
) L
.
t
t
t
t
1
t
t
5. A technical condition is required, of course, so that tasks that have been automated are actually produced with capital instead of labor. We assume this condition holds.
Artifi cial Intelligence and Economic Growth 243
Collecting the automation terms simplifi es this to
(12)
Y = A
1
K + (1
)1 L
(
)1/ .
t
t
t
t
t
This setup therefore reduces to a particular version of the neoclassical
growth model, and the allocation of resources can be decentralized in a stan-
dard competitive equilibrium. In this equilibrium, the share of automated
goods in GDP equals the share of capital in factor payments:
Y K
K
1
(13)
t
t =
A
t
.
Kt
K Y
t
t
Y
t
t
t
Similarly, the share of nonautomated goods in GDP equals the labor share
of factor payments:
Y L
L
(14)
t
t = 1 A
t
Lt
.
L Y
t
t
Y
t
t
t
Therefore the ratio of automated to nonautomated output—or the ratio of
the capital share to the labor share—equals
1
K
(15)
Kt =
t
t
.
1
L
Lt
t
t
We specifi ed from the beginning that we are interested in the case in which
the elasticity of substitution between goods is less than one, so that < 0.
From equation (15), there are two basic forces that move the capital share (or,
equivalently, the share of the economy that is automated). First, an increase
in the fraction of goods that are automated, , will increase the share of
t
automated goods in GDP and increase the capital share (holding K / L constant). This is intuitive and repeats the logic of the Zeira model. Second, as
K / L rises, the capital share and the value of the automated sector as a share of GDP will decline. Essentially, with an elasticity of substitution less than
one, the price eff ects dominate. The price of automated goods declines rela-
tive to the price of nonautomated goods because of capital accumulation.
Because demand is relatively inelastic, the expenditure share of these goods
declines as well. Automation and Baumol’s cost disease are then intimately
linked. Perhaps the automation of agriculture and manufacturing leads
these sectors to grow rapidly and causes their shares in GDP to decline.6
The bottom line is that there is a race between these two forces. As more
sectors are automated, increases, and this tends to increase the share of
t
automated goods and capital. But because these automated goods experi-
ence faster growth, their price declines, and the low elasticity of substitution
means that their shares of GDP also decline.
Following Acemoglu and Restrepo (2016), we could endogenize auto-
mation by specifying a technology in which research eff ort leads goods to
6. Manuelli and Seshadri (2014) off er a systematic account of the how the tractor gradually replaced the horse in American agriculture between 1910 and 1960.
244 Philippe Aghion, Benjamin F. Jones, and Charles I. Jones be automated. But it is relatively clear that depending on exactly how one
specifi es this technology, / (1 – ) can rise faster or slower than ( K / L )
t
t
t
t
declines. That is, the result would depend on detailed assumptions related to
automation, and currently we do not have adequate knowledge on how to
make these assumptions. This is an important direction for future research.
For now, however, we treat automation as exogenous and consider what
happens when changes in diff erent ways.
t
Balanced Growth (Asymptotically)
To understand some of these possibilities, notice that the production
function in equation (12) is just a special case of a neoclassical production
function:
(16)
Y = A F B K , C L
(
)where B (1 )/ and C (1 )(1 )/ .
t
t
t
t
t
t
t
t
t
t
With < 0, notice that ↑ ⇒ ↓ B and ↑ C . That is, automation is equiva-
t
t
t
lent to a combination of labor- augmenting technical change and capital-
depleting technical change. This is surprising. One might have thought of
automation as somehow capital augmenting. Instead, it is very diff erent: it
is labor augmenting and simultaneously dilutes the stock of capital. Notice
that these conclusions would be reversed if the elasticity of substitution
were greater than one; importantly, they rely on < 0.
The intuition for this surprising result can be seen by noting that automa-
tion has two basic eff ects. These can be seen most easily by looking back at
equation (11). First, capital can be applied to a larger number of tasks, which
is a basic capital- augmenting force. However, this also means that a fi xed
amount of capital is spread more thinly, a capital- depleting eff ect. When the
tasks are substitutes ( > 0), the augmenting eff ect dominates and automa-
tion is capital augmenting. However, when tasks are complements ( < 0),
the depletion eff ect dominates and automation is capital depleting. Notice
that for labor, the opposite forces are at work: automation concentrates a
given quantity of labor onto a smaller number of tasks and hence is labor
augmenting when < 0.7
This opens up one possibility that we will explore next: what happens if
the evolution of is such that C grows at a constant exponential rate? This
t
t
can occur if 1 – falls at a constant exponential rate toward zero, meaning
t
that → 1 in the limit and the economy gets ever closer to full automation
t
(but never quite reaches that point). The logic of the neoclassical growth
model suggests that this could produce a balanced growth path with con-
stant factor shares, at least in the limit. (This requires A to be constant.)
t
In particular, we want to consider an exogenous time path for the fraction
&nb
sp; 7. In order for automation to increase output, we require a technical condition: ( K / ) <
[ L / (1 – )]. For < 0, this requires K/ > L/1 – . That is, the amount of capital that we
allocate to each task must exceed the amount of labor we allocate to each task. Automation raises output by allowing us to use our plentiful capital on more of the tasks performed by relatively scarce labor.
Artifi cial Intelligence and Economic Growth 245
of tasks that are automated, , such that → 1 but in a way that C grows
t
t
t
at a constant exponential rate. This turns out to be straightfoward. Let ≡
t
1 – , so that C = (1 )/ . Because the exponent is negative ( < 0), if falls
t
t
t
at a constant exponential rate, C will grow at a constant exponential rate.
t
This occurs if , = (1 – ), implying that g
t
t
= – . Intuitively, a constant
fraction, , of the tasks that have not yet been automated become automated
each period.
Figure 9.1 shows that this example can produce steady exponential
growth. We begin in year 0 with none of the goods being automated, and
then have a constant fraction of the remainder being automated each year.
There is obviously enormous structural change underlying—and generat-
ing—the stable exponential growth of GDP in this case. The capital share of
factor payments begins at zero and then rises gradually over time, eventually
asymptoting to a value around one- third. Even though an ever- vanishing
fraction of the economy has not yet been automated, so labor has less and
less to do. The fact that automated goods are produced with cheap capital
combined with an elasticity of substitution less than one means that the
automated share of GDP remains at one- third and labor still earns around
two- thirds of GDP asymptotically. This is a consequence of the Baumol
force: the labor tasks are the “weak links” that are essential and yet expen-
sive, and this keeps the labor share elevated.8
Along such a path, however, sectors like agriculture and manufacturing
exhibit a structural transformation. For example, let sectors on the interval
[0,1/3] denote agriculture and the automated portion of manufacturing as of
some year, such as 1990. These sectors experience a declining share of GDP
over time, as their prices fall rapidly. The automated share of the economy
will be constant only because new goods are becoming automated.
The analysis so far requires A to be constant, so that the only form of
t
technical change is automation. This seems too extreme: surely technical
progress is not only about substituting machines for labor, but also about
creating better machines. This can be incorporated in the following way.
Suppose A is capital- augmenting rather than Hicks- neutral, so that the prot
duction function in equation (16) becomes Y = F( A B K , C L ). In this case, t
t
t
t
t
t
one could get a balanced growth path (BGP) if A rises at precisely the rate
t
that B declines, so that technological change is essentially purely labor-
t
augmenting on net: better computers would decrease the capital share at
precisely the rate that automation raises it, leading to balanced growth. At
fi rst, this seems like a knife- edge result that would be unlikely in practice.
However, the logic of this example is somewhat related to the model in
Grossman et al. (2017); that paper presents an environment in which it is
optimal to have something similar to this occur. So perhaps this alternative
8. The neoclassical outcome here requires that not be too large (e.g., relative to the exogenous investment rate). If is suffi
ciently high, the capital share can asymptote to one and the
model becomes “AK.” We are grateful to Pascual Restrepo for working this out.
246 Philippe Aghion, Benjamin F. Jones, and Charles I. Jones
A
B
Fig. 9.1 Automation and asymptotic balanced growth. A, the growth rate of GDP
over time; B, automation and the capital share
Note: This simulation assumes ρ < 0 and that a constant fraction of the tasks that have not yet been automated become automated each year. Therefore C ≡ (1 – β)(1– ρ)/ ρ grows at a constant t
exponential rate (2 percent per year in this example), leading to an asymptotic balanced growth path (BGP). The share of tasks that are automated approaches 100 percent in the limit.
Interestingly, the capital share of factor payments (and the share of automated goods in GDP) remains bounded, in this case at a value around one- third. With a constant investment rate of s , the limiting value of the capital share is ( s / g + δ)ρ .
Y
approach could be given good microfoundations. We leave this possibility
to future research.
Constant Factor Shares
Another interesting case worth considering is under what conditions can
this model produce factor shares that are constant over time? Taking logs
Artifi cial Intelligence and Economic Growth 247
and derivatives of equation (15), the capital share will be constant if and
only if
(17)
g
= 1
(
)
g ,
t
t
1
kt
where g is the growth rate of k ≡ K/ L. This is very much a knife- edge kt
condition. It requires the growth rate of to slow over time at just the right
t
rate as more and more goods get automated.
Figure 9.2 shows an example with this feature, in an otherwise neoclassi-
cal model with exogenous growth in A at 2 percent per year. That is, unlike
t
the previous section, we allow other forms of technological change to make
tractors and computers better over time, in addition to allowing automa-
tion. In this simulation, automation proceeds at just the right rate so as to
keep the capital share constant for the fi rst 150 years. After that time, we
simply assume that is constant and automation stops, so as to show what
t
happens in that case as well.
The perhaps surprising result in this example is that the constant factor
shares occur while the growth rate of GDP rises at an increasing rate. From
the earlier simulation in fi gure 9.1, one might have inferred that a constant
capital share would be associated with declining growth. However, this is
not the case and instead growth rates increase. The key to the explanation is
to note that with some algebra, we can show that the constant factor share
case requires
(18)
g
= g + g .
Yt
A
t
Kt
First, consider the case with g = 0. We know that a true balanced growth
A
path requires g = g . This can occur in only two ways if g = 0: either = 1
Y
K
A
t
or g = g = 0 if < 1. The fi rst case is the one that we explored in the pre-
Y
K
t
; vious example back in fi gure 9.1. The second case shows that if g = 0, then
A
constant factor shares will be associated with zero exponential growth.
Now we can see the reconciliation between fi gures 9.1 and 9.2. In the
absence of g > 0, the growth rate of the economy would fall to zero. Intro-A
ducing g > 0 with constant factor shares does increases the growth rate.
A
To see why growth has to accelerate, equation (18) is again useful. If growth
were balanced, then g = g . But then the rise in would tend to raise g
Y
K
t
Y
and g . This is why growth accelerates.
K
Regime Switching
A fi nal simulation shown in fi gure 9.3 combines aspects of the two pre-
vious simulations to produce results closer in spirit to our observed data,
albeit in a highly stylized way. We assume that automation alternates between
two regimes. The fi rst is like fi gure 9.1, in which a constant fraction of the
remaining tasks are automated each year, tending to raise the capital share
and produce high growth. In the second, is constant and no new automa-
t
tion occurs. In both regimes, A grows at a constant rate of 0.4 percent per
t
248 Philippe Aghion, Benjamin F. Jones, and Charles I. Jones
A
B
Fig. 9.2 Automation with a constant capital share. A, the growth rate of GDP over time; B, automation and the capital share
Note: This simulation assumes ρ < 0 and sets β so that the capital share is constant between t
year 0 and year 150. After year 150, we assume β stays at its constant value; A is assumed to t
t
grow at a constant rate of 2 percent per year throughout.
year, so that even when the fraction of tasks being automated is stagnant,
the nature of automation is improving, which tends to depress the capital
share. 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.
By playing with parameter values, including the growth rate of A and
t
, it is possible to get a wide range of outcomes. For example, the fact that
t
Artifi cial Intelligence and Economic Growth 249
A
B
Fig. 9.3 Intermittent automation to match data? A, the growth rate of GDP over time; B, automation and the capital share
Note: This simulation combines aspects of the two previous simulations to produce results closer in spirit to our observed data. We assume that automation alternates between two regimes. In the fi rst, a constant fraction of the remaining tasks are automated each year. In the second, β is constant and no new automation occurs. In both regimes, A grows at a constant t