It is a well known fact that not all countries develop at the same time. The industrial revolution began over 200 years ago in England and has been spreading over the world ever since. In their paper Barriers to Riches, Parente and Prescott notice that countries that enter the industrial stage later on grow faster than what the early starters did. I present a simple model with learning externalities that generates this kind of behavior. I follow Lucas (1998) and solve the optimization problem of the representative agent under the assumption that the external effect is given by the world leader's human capital.

Alejandro M. Rodríguez

English

Research Papers in Economics

Learning Externalities and Economic Growth
Alejandro M. Rodriguez∗
August 2004
∗I would like to thank Profesor Robert E. Lucas for motivating me to write this paper and for
his helpful comments. Profesor Alvarez also provided some useful insights. Please send comments
to amr@cema.edu.ar. All errors are mine.ABSTRACT
It is a well known fact that not all countries develop at the same time. The
industrial revolution began over 200 years ago in England and has been spreading
over the world ever since. In their paper Barriers to Riches, Parente and Prescott
notice that countries that enter the industrial stage later on grow faster than what the
early starters did. I present a simple model with learning externalities that generates
this kind of behavior. I follow Lucas (1998) and solve the optimization problem of
the representative agent under the assumption that the external eﬀect is given by the
world leader’s human capital.1820 1840 1860 1880 1900 1920 1940 1960 1980
0
10
20
30
40
50
60
70
Year income reached 2,000 (1990 U.S. $)
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Fig. 1. Growth patterns from Parente and Prescott
1. Introduction
In Barriers to Riches, Parente and Prescott observe that ”Countries reaching a
given level of income at a later date typically double that level in a shorter time ” To
support this conclusion they present Figure 1
Figure 1 shows the year in which a country reached the per capita income level of
2,000 of 1990 U.S. dollars and the number of years it took that country to double its
per capita income to 4,000 of 1990 U.S. dollars. Clearly the late starters grew much
faster than the early starters did. This is more than convergence in the traditional
sense. This graph is not only saying that poor countries grow faster than rich countries
do. There is some kind of advantage that the late starters or followers have that early
1starters or leaders do not.
In this paper I present a simple model that generates the behavior shown in Figure
1. The model is a growth model with human capital accumulation and an externality
in the human capital production function. This externality depends on the level of
human capital of the most advance country. The model is similar to the model in
Lucas(1993) with some diﬀerences. On the technology side model has no physical
capital, and the external eﬀect comes from the highest human capital level in the
world. In Lucas’ model is the average human capital level of the world that produces
an external eﬀect. I also introduce a utility maximization agent that optimally chooses
the fraction of time that he allocates to invest in human capital. My model is similar
to Parente and Prescott (1994) and Tamura (1996) in the sense that they also analyze
the behavior of a maximizing agent that faces an external eﬀect on the human capital
production technology. In Section 2 I present the model and solve it analytically, in
Section 3 I present a numerical example. Section 4 concludes the paper.
2. The Model
All countries are identical in preferences and technology. They only diﬀer in their
initial levels of human capital. Each country is in autarchy. There are no international
markets for human capital or consumption goods. The only interaction among them
comes from a learning externality. Each country is represented by a inﬁnitely lived
dynasty with preferences
∞ Z
0
e
−ρtU (c(t))dt, (1)
I assume that U (c) takes the form
U (c)=
c1−σ
1 − σ
(2)
Each country is endowed with one unit of time per period and they can use it
2to produce the consumption good c(t) or to accumulate human capital h(t).T h e
fraction of time spent in the production of the consumption good is u(t) There is an
externality in the human capital sector. The level of human capital from the worlds
most advance economy, H (t), enters into the human capital accumulation function
of all countries. Speciﬁcally, I assume
c(t)=u(t)h(t) (3)
and
˙ h(t)=δh
1−θ (t)H
θ (t)[1− u(t)]. (4)
All other countries have no eﬀect on the human capital of the leader, but I also
assume that all individuals in the leading economy are also aﬀected by this external
eﬀect, and take H (t) a sg i v e na sw e l l .O n ew a yt oi n t e r p r e tt h i si st h a tt h el e a d i n g
country is composed of many identical households and that each household is too
small to aﬀect the aggregate human capital by its own actions.
The human capital of the leader is assumed to grow at a constant rate α. Later, this
guess will be veriﬁed and α will be viewed as an endogenous variable to be determined
b yt h em o d e l .T h i si m p l i e st h a t
˙ H (t)=αH (t),
or alternatively
H (t)=H (0)e
αt.
With this speciﬁcation of preferences and technology I have enough information
to set up and solve the model. Deﬁne
x(t) ≡
h(t)
H (t)
.
so that (1) and (4)c a nb ew r i t t e na s
H1−σ (0)
1 − σ
∞ Z
0
e
−[ρ−α(1−σ)]t [u(t)x(t)]
1−σ dt (5)
3and
˙ x(t)=δx
1−θ (t)[1− u(t)] − αx(t). (6)
The problem faced by each country is to choose the function u(t) to maximize (5)
subject to (6) given x(0)
From now on I assume that ρ − α(1 − σ) > 0, so that the agent’s utility is ﬁnite
under all feasible plans, and δ>ρso that the steady state growth rate is positive.
Deﬁn et h ec u r r e n tv a l u eH a m i l t o n i a nM as
M ≡
(ux)
1−σ
1 − σ
+ ψ
£
δx
1−θ (1 − u) − αx
¤
The maximum of M can be found by setting
∂M
∂u
=0
The rates of change of the co-state variable ψ, and the state variable x are given
by
˙ ψ =[ ρ − α(1 − σ)]ψ −
∂M
∂x
,
and
∂M
∂ψ
= ˙ x.
The transversality condition for this problem is
lim
t→∞e
−[ρ−α(1−σ)]tψ(t)x(t)=0 .
From these equations I obtain
u
−σ = δψx
σ−θ, (7)
˙ ψ
ψ
= ρ + ασ − (1 − θ + θu)δx
−θ, (8)
and
˙ x
x
= δx
−θ (1 − u) − α. (9)
4These equations describe both the behavior of the leading country and the followers.
For the leader h(t)=H (t) for all t so x(t)=1and ˙ x/x =0 .T h e n
u
−σ = δψ, (10)
.
ψ
ψ
= ρ + ασ − (1 − θ + θu)δ, (11)
and
δ(1 − u)=α (12)
Since α is constant, this implies that, for the leader, u(t) is also constant, then
from (10) ψ = u−σ
δ is also constant, so
.
ψ/ψ =0 . Using equation (12) together with
equation (11) set to 0 Ic a ns o l v ef o rα and u. The complete solution to the leader’s
problem is
α =
δ − ρ
σ + θ
,
u
∗ (t)=
ρ + δ(1 − σ − θ)
(σ + θ)δ
,
c
∗ (t)=h
∗ (0)
µ
ρ + δ(1 − σ − θ)
(σ + θ)δ
¶
e(
δ−ρ
σ+θ)t,
and
h
∗ (t)=H (t)=h
∗ (0)e(
δ−ρ
σ+θ)t.
The follower’s behavior is also given by (7) −(9).D i ﬀerentiate (7) with respect to
time, and use that result with (9) in (8) to eliminate the shadow price ψ.S o(7) and
(9) can be written just as functions of u and x
˙ u
u
= −
ρ + αθ − (1 + σu− σ)δx−θ
σ
(13)
˙ x
x
= δx
−θ (1 − u) − α (14)
To ﬁnd a steady state for the follower set ˙ u/u = ˙ x/x =0 .T h e n˙ x =0implies that
u =1−
α
δ
x
θ,
5and
.
u =0implies that
u =
(ρ + αθ)
δσ
x
θ +
σ − 1
σ
.
These two curves are plotted in Figure 2. A steady state is the intersection of this
two curves. The intersection of this two curves when α =( δ − ρ)/(1 − θ + σ) is
x
SS =1 ,
and
u
SS =
ρ + δ(1 − σ − θ)
(σ + θ)δ
This are the same steady state values are the same than the ones from the leader.
Figure 2 plots the phase diagram for the system of diﬀerential equations represented
by (13) and (14).
To analyze the stability of the steady state take a log-linear approximation of (13)
and (14) around the steady state
⎡
⎣
dlogu
dt
dlogx
dt
⎤
⎦ = J ×
⎡
⎣ log
¡
u
uSS
¢
log(x)
⎤
⎦
T h ec h a r a c t e r i s t i cm a t r i xJ is deﬁned as
J =
⎡
⎣ δuSS −(1+σuSS−σ)
σ δθ
−δuSS −δθ
¡
1 − uSS¢
⎤
⎦
with
det(J)=−
δ
2uSSθ
σ
< 0
This implies that there is saddle path stability, i.e. the eigenvalues of J have
opposite sign. So, locally, there is only one path or stable arm that converges to the
steady state solution. I can solve the follower’s problem with initial condition x(0)
by choosing the value of u(0) such that the follower moves along the saddle path.
Choosing an initial condition for u(0) above the saddle path will make u(t) grow
60 1
0
1
x
u
dx/dt=0 
du/dt=0 
saddle path 
Fig. 2. Phase Diagram
7until it eventually becomes constant at a level of one. When u(t)=1then ψ(t)x(t)
grows at the rate ρ−α(1 − σ) and the transversality condition is violated. If instead
we choose an initial condition below the saddle path then x(t) will grow too fast and
in ﬁnite time the country will become the world leader x(t)=1 , and since u(t) <u ss
at that given date, this implies that u(t) must give a discrete jump, violating the ﬁrst
order condition for u.
This model has the property that for all x(0) > 0 both u(t) and x(t) converge to
the leader’s values, if all the parameters of the model are the same. This implies that
there is convergence in growth rates and output. Output convergence is deﬁned as
t h ec o n v e r g e n c eo ft h eo u t p u tr a t i ot oo n ea st i m eg o e st oi n ﬁnity.
3. A Numerical Example
In this section I test if the model can generate the kind of behavior depicted in
Figure 1. I assume that σ =1and ρ =0 .02. I simulate the model for θ =0 .3, θ =0 .1
and θ =0(this corresponds to the case with no learning externality), δ is chosen
such that the long run growth rate α is the same in the three cases. I set α to 0.016
to match the growth rate of the US in the 1820-1990 period1.W i t h t h e n u m e r i c a l
solutions I can calculate how long it takes a country, that reached an income level of
$2000 t years after the leader did, to double its income. The results are compared
to the actual data in Figure 3. To match the model to the data I need to make an
assumption about which country is the leader, so I assume that the US is the world
leader.
T h em o d e lc a p t u r e st h ei d e at h a tl a t es t a r t e r sg r o wf a s t e r ,b u ti tf a i l st oﬁts o m e
aspects data. If I try to ﬁt the data generated by the model to the linear trend in
the Parente and Prescott data then the model would predict a growth rate for the
leader (and long run growth rate for all the countries) of 1%, which is inconsistent
1Data from Maddison (1995)
80
10
20
30
40
50
60
70
80
05 0 1 0 0 1 5 0
Number of years to reach income of 2000 after the leader
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θ=0.3
θ=0.1
θ=0
Linear (Parente & Prescott Data)
Fig. 3. Results of the simulations
9with the growth rate we observe for the US. If I calibrate the model to the have the
observe growth rate of the US then the model predicts that no other country could
have taken longer than 43 years to grow from an income level of 2000 to one of 4000.
Clearly the data shows otherwise. Countries that reached and income level of 2000
on the second half of the nineteenth century experienced growth rates much lower
than those predicted by the model.
4. Conclusions
The model presented in this paper has the property that it can explain the behavior
of a country like the US that has been growing steadily for almost 200 years, and the
more recent miracles of economic growth, like South Korea and Taiwan, that had
yearly growth rates of over 6% for over a decade. Still it fails to explain some other
aspects of the data. These failures lead to new questions and future research.
The key factor of the model are learning externalities. Countries can take advantage
of the human capital of the leader to increase their own. What determines this
externalities is an important question to answer, since the answer will help many
developing economies to make better policies. Lucas (1993) discusses what are the
possible sources of externalities in human capital accumulation. He comes to the
conclusion that models in which the leaders human capital is a public good that is
available to all countries have (like my model) the unpleasant characteristic that all
countries will tend to grow. There is no room in this types of models for countries like
Philippine and Korea, that were very similar in 1960, but experienced very diﬀerent
growth paths there after.
Learning externalities of the type proposed in this paper are not so diﬃcult to
imagine in the real world, but still they are not the only possibility.
10REFERENCES
[1] Barro, Robert J. and Sala-i-Martin Xavier. Economic Growth. MIT Press (1999)
[2] Chiang, Alpha C. Elements of Dynamic Optimization Waveland Press (2000)
[3] Lucas, Robert E. Jr. 1988 “On the Mechanics of Economic Development?” Journal
of Monetary Economics, 22: 3-42.
[4] Lucas, Robert E. Jr. 1993. “Making a Miracle.” Econometrica, 61: 251-272.
[5] Maddison, Angus. Dynamic Forces in Capitalist Development, A Long Run Compar-
ative View. Oxford University Press (1991)
[6] Maddison Angus. Monitoring the World Economy: 1820-1992. Development Centre
Studies (1995)
[7] Parente, Stephen L. and Prescott Edward C. Barriers to Riches. MIT Press (2000)
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Learning Externalities and Economic Growth

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