ECON 343
Economic Development and Growth
Lecture Notes
Francis T. Lui
Department of Economics
Hong Kong University of Science and Technology
(Spring 2008)
Chapter 1
Introduction
Social scientists often classify countries into “developed” and “less
developed” ones. What exactly do we mean by being more developed? A
wide range of criteria have been used, e.g., income level, educational level
or literacy rate, degree of political freedom or democratization, life
expectancy, density of telephone lines, penetration rate of mobile phones or
computers, etc. Among these, income level seems to be most commonly
used to define the developmental stage of a country. It is something that
every country wants to increase. It also shows significant co-movements
with many variables that are regarded as good indicators of the stage of
development.
Quantitatively, we observe great diversity in both income levels and
income growth rates across countries. (See Table 1.1) For example, the per
capita GNI of Malawi in 2006 (using “purchasing power parity,” or PPP,
measurement) was US$720, which was only 1.63% of that of the United
States.
The huge differences in the average long-term growth rates across
countries are often less recognized. From 1978 to 2006, China’s average
growth rate in per capita real GDP was 8.46%, while that for aggregate real
GDP was 9.69%. This means, per capita real GDP in 2006 was 9.73 times
that of 1978 (13.34 times for aggregate GDP). In another period, 185-1994,
growth rate in real GDP per annum in the United States was 1.3%,
Switzerland 0.5%, Hong Kong 5.3% and South Korea 7.8%. Many of the
African countries, however, experienced negative long-term growth.
1
Table 1.1: Per Capita GNI in current US$ and PPP estimates (2006)
Countries
Per Capita GNI
Per Capita GNI
PPP estimates as
(in current US$)
(in PPP US$)
% of US
United States
44,970
44,260
100.0
Norway
66,530
43,820
99.0
Hong Kong
28,460
38,200
86.3
United Kingdom
40,180
35,580
80.4
France
36,550
33,740
76.3
Japan
38,410
33,150
74.9
Singapore
29,320
31,170
70.4
China
2,010
7,740
17.5
India
820
3,800
8.6
Malawi
170
720
1.6
Source: World Bank, World Development Report 2008.
Differences in long-term growth could have far-reaching
consequences. If a country today has an income level of $1, but will grow at
1% a year for 50 years, then its income will be $1.64 after 50 years. If it
grows at 7% a year, it will be $29.46! What is significant is that differences
in long-term growth rates tend to persist. Many poor countries, which have
experienced negative or very slow growth, seem to have fallen into some
“poverty trap.” On the optimistic side, the kind of high growth rates that we
observe today is a modern phenomenon. Take the example of China again.
Table 1.2 presents the estimates by Angus Maddison (1998) for China’s per
capita real GDP in “international dollar” from 50 AD to 1978 AD using
1990 prices. As we can readily see, there had been clear stagnancy for a long
2
period of time. Rapid growth only occurred in recent years. Similar
phenomenon can be found in Western countries before and after the
Industrial Revolution. This may hint that if a country followed the right
policy, growth rate could substantially improve.
Table 1.2: China’s per capita GDP in international dollars at 1990 prices
50 AD
960
1280
1820
1952
1978
$450
$450
$600
$600
$537
$979
How do we explain the diversity in long-term growth rates and
income levels? This is the main focus of the course. Presumably, if we can
do it, stagnant economies can learn from fast-growing ones on how they can
do better. The implications for human welfare are immense.
To explain growth, we need the concept of “engine of growth.”
Growth means that the economy keeps on producing more and more. This is
like a car that keeps on running. What drives the car or the economy? We
need to identify the engine.
Even if we have the engine, we need to know how the forces work
inside the engine. We need to understand the “mechanics of growth.” In
other words, we want to understand how the forces are transmitted through
the mechanics.
In addition to the mechanical side, we have to pay attention to the
human side. Given that people are rational, why and when do they have the
motive to go fast or slowly?
3
Chapter 2
Neoclassical or Exogenous Growth Models
(2.1) Stylized Facts to be Explained by the Neoclassical Growth Models.
Let Q = output,
L = labor,
K = capital.

Let X denote dX/dt. The growth rate of a variable X can then be expressed

as X /X.1 The following relations are stylized facts that the neoclassical
model seeks to explain.
(1)


Q /Q > L/ L .


Q /Q and L/ L are fairly stable.

(2)
K /K is fairly stable.
(3)
K /K ≈ Q /Q.


In other words, K/Q is fairly stable.

1
Note the following relationships: X /X = (dX/dt)/X = (dX/X)/dt
= (d log X)/dt.
Suppose that multiply the above by 100. Then the result can be read as the
percentage change in X per unit of time.
4
(4) Profit rate is fairly constant in the long run. (This may have to be
modified by more recent experiences, since the profit rates in some countries
may have changed substantially.)
(5) The long-run relative distribution between wages and profits is fairly
stable. (This does not seem to fit Hong Kong’s case too well.)


(1), (2) and (3) together imply that K /K > L/ L .
(2.2) The Solow Model.
The above are derived mainly from data in the US and other highly
developed countries. We should be careful about their applicability to other
economies. I shall first discuss a version of the Solow model (with technical
change), which is meant to provide a consistent explanation of the stylized
facts above.
Let the aggregate production function be
Q(t) = F(K(t), E(t)), where E(t) denotes labor input in “effective
units.” It grows over time because the quality of labor improves.
E(t) = L(t) e
gt
λt gt

= L(0) e e , i.e., E/ E = g + λ.
F is homogeneous of degree one, i.e.,
F(aK, aE) = aF(K, E), for a > 0.
Thus, (1/E) F(K, E) = F(K/E, 1) ≡ f(k), where k ≡ K/E.
5
Define q = Q/E = f(k). Assume that f’(k) > 0, f”(k) < 0, and f’(0) → ∞, f’(∞)
= 0.
q
f(k)
k
Let s be the saving rate, and S be saving, i.e., S(t) = sQ(t).
Assume that investment = saving. Hence,

K = sQ(t).

Therefore, K /K = sQ/K.




Now k /k = K /K – E/ E = (sQ)/K – E/ E = [(sQ/E) / (K/E)] – (g + λ)
= sf(k)/k – (g + λ)
6

Proposition: There exists an equilibrium k* at which k/ k = 0.

At k/ k = 0, f(k) = [(g + λ)k] / s.
At k < k*, f(k) > [(g + λ)k] / s.
This implies that

sf(k)/k – (g + λ) > 0, and k/ k > 0, i.e., k goes up.

Similarly, at k > k*, k/ k < 0 and k goes down. Thus, at k*, K/E is a constant.


K /K = E/ E = g + λ.


This is consistent with K /K > L/ L . (Fact (2)).
q
(g + λ)k
s
f(k)
q*
k*
7
k
Also q* is a constant. So Q/E is a constant, and


Q /Q = E/ E .


Therefore, Q /Q = K /K.


This means that (i) Q /Q > L/ L .
(Fact (1))
(ii) K/Q is a constant.
(Fact (3))
gt
Now Q = Le f(K/E) = E f(K/E).
Profit rate = ∂Q/∂K = E f’(K/E)(1/E) = f’(k).
gt
gt
(Fact (4))
2 gt
gt
Wage rate = ∂Q/∂L = Le f’(K/ Le ) (- K/ L e ) + f(k) e
gt
= e [f(k) – kf’(k)].
At k*, f’(k*) and f(k*) are constants.
Thus, profit rate is a constant. Real wage rate grows at g.
Capital share/Labor share = [K(∂Q/∂K)] / [L(∂ Q/∂L)]
gt
= [K/L]f’(k) / {e [f(k) – kf’(k)]}.
But K/L = K/Ee
-gt
gt
=ke .
Therefore, capital share / labor share = [f’(k) k] / [f(k) – kf’(k)]
= constant at k*.
(Fact (5))
Thus, all the stylized facts are explained by the Solow model. The paths of
Q(t), K(t) and E(t) at equilibrium are called the golden-age paths.
8
(2.3) Phelps’ Golden Rule Path
In the Solow Model, saving rate s is exogenously given. A useful
question to ask is: would it be possible that the government chooses a
socially optimal s such that the long-term consumption possibility for
society is maximized?
Let aggregate consumption in society be C.
C = Q – sQ, where, for convenience, all time subscripts are suppressed here.
C/E = (1 – s)Q/E = (1 – s)f(k).
gt
gt
Per capita consumption = C/L = e (1 – s)f(k) = e [f(k) – sf(k)].
(g + λ)k
s*
q
f’(k*)
f(k)
q*
(g+ λ)k
k*
k
9
By varying s, we get different values of k*. We want to pick the k* (and the
corresponding s*) such that C/L is maximized. Notice that if C/E is
maximized, then C/L is also maximized. From the Solow Model, we know
that at k*, sf(k*) = (g + λ)k*.
Thus, max f(k*) - (g + λ)k*  f’(k*) = g + λ.
Note that the left-hand side of the above expression is C/E.
The k* that satisfies this condition yields an equilibrium. The paths
corresponding to k* are called the “Golden Rule Paths,” i.e., the paths that
maximize the per capita consumption in the set of all golden age or
“balanced growth paths.”
The model by Phelps and others represent an improvement over the
Solow Model. But it disregards the historically given k(0). There is no
guarantee that k(0) is on the golden rule path. For example, if k(0) is
extremely large, we may wonder whether the economy will converge to the
golden rule paths. The Cass-Koopmans Model shows that convergence
would occur.
(2.4) Cass-Koopmans Growth Model
Assume that the economy is closed, i.e., there is no trade and
international capital flow. Population at t is represented by N(t).
N(t) = N(0) e
λt
.
c(t) = stream of real per capita consumption of a single good at t > 0.
10
c(t)
t
Social preferences of the identical rational agents in the economy are given
by the following:


0
e
– ρt
{[c(t)
1-σ
- 1] / [1 – σ]}N(t) dt
(1)
ρ is the discount factor here. The utility function for the representative agent
at t is
u(t) = [c(t)
1-σ
Note that u’ = c
- 1] / [1 – σ].
–σ
> 0,
σ is interpreted as the coefficient of relative risk aversion. Production is
divided into consumption and capital accumulation.
K(t) = stock of capital at t,
11

K (t) = dK/dt = rate of change of K(t).

Production technology is Cobb-Douglas: A

β
1-β
N(t)c(t) + K (t) = A(t)K(t) N(t)
(2)
β is a parameter such that 0 < β < 1. A(t) is a measurement of the level of

technology. Let the exogenously given rate of technical change be μ = A /A,
and without loss of generality assume that μ > 0.
The paths A(t) and N(t) are given exogenously. The problem is to choose a
path c(t) such that (1) is maximized subject to (2) and an initial level of K(0).
The method to solve this problem is to set up the “current-value
Hamiltonian.”
H(K, θ, c, t) = N [c
1-σ
β 1-β
- 1] / (1 – σ) + θ [AK N
- Nc]
(3)
Note that all the time notations have been suppressed here. We shall see how
solving the maximization problem stated above can be reduced to solving
the Hamiltonian (3).
[Mathematical Note on Hamiltonian:
Let us use the optimal control method, which employs the Lagrangian
multiplier.
1-σ
β 1-β
Let I ≡ N [c(t)
- 1] / (1 – σ), and f ≡ AK N
- Nc.
12
Then the Lagrangian for the maximization problem above can be stated as
L=


0
– ρt
e

[I + θ (f - K )] dt.

Noting that K = dK/dt, we can easily get
L=
=


0

e

0
– ρt
e
– ρt
[I + θf ] dt -


0
e
– ρt
θ dK

{[I + θf ] + K  - ρKθ} dt - constant.
The last expression is obtained by applying the method of integration by
parts to the second integral two lines above. [Try this out yourself!] Note
that the Hamiltonian in (3) is H = I + θf. At the maxima, we must have,
0=Δ L
=


0
e
– ρt

{(∂H/∂c) Δc + [(∂H/∂K) – ρθ +  ] ΔK} dt
From this expression, two first-order necessary conditions for maximization
must include
(∂H/∂c) = 0,
and

(∂H/∂K) = ρθ -  .
End of Mathematical note.]
13
According to the formula from the Hamiltonian, the solution to the
Cass-Koopmans Model must include these first-order necessary conditions:
∂H/∂c = N [c

c
–σ
-σ
- θ] = 0.
(4)
= θ.

 = ρθ – θβAK β - 1N 1- β
= θ(t)[ρ – βA(t) K(t)
β-1
N(t)
1- β
]
(5)
In addition to (4) and (5), we also need another necessary condition, the
“Transversality condition.”
lim e
t→∞
– ρt
θ(t) K(t) = 0
(6)
The economic interpretation of this condition is that when time goes
to infinity (or the end-period), the present discounted value of the capital
stock, θ(t)K(t),(which is equivalent to multiplying the shadow price of
capital with the quantity of the capital stock), must go to zero. (Question:
Why must this be the case?)
Given (4), (5) and (6), we want to show (a) the existence of
“balanced paths,” i.e., the particular solution (K(t), θ(t), c(t)) such that the
rates of growth of each of these variables is a constant (the existence result);
and (b) that starting from an arbitrary K(0), the economy will converge to
the balanced path asymptotically (the stability result). This makes the
balanced path empirically meaningful. (Why?)
14
First, recall that A(t) = A(0) e
and
N(t) = N(0) e
μt
λt
.
These are exogenously given paths.
Let κ represent the constant growth rate of c on the balanced path, i.e.,

κ = c/ c . We shall compute κ in terms of μ and λ. From (4),

 /  = - σκ.
[Mathematical Note:
a
a
Let y = x . Then, ln y = ln x = a ln x.

d ln y = a d ln x.
d ln y/dt = a(d ln x)/dt.



y/ y = a( x/ x )
End of Mathematical Note.]

From (5),
A(t)N(t)

 /  = ρ – βA(t)N(t)1-βK(t)β-1
1-β
β-1
K(t)
= (ρ + σκ)/β = constant.

(7)

Thus, A/ A + (1 - β) ( N/ N ) + (β - 1)( K/ K ) = 0

μ + (1 – β)λ = (1 – β) ( K/ K )

K/ K = [(1 - β)λ + μ] / (1 - β) = λ + μ/(1 - β).
15
(8)
Also from (2) to (7),
A(t)N(t)

1-β
β-1
K(t)
= N(t)c(t)/K(t) + K (t)/K(t) =(ρ + σκ)/β

Since (ρ + σκ)/β is a constant and K /K is also a constant on the balanced
path, N(t)c(t)/K(t) must also be a constant. Hence,


c/ c =
N/ N +

K /K.

Thus, K /K = λ + κ
(9)
(8) and (9) implies κ = μ/(1-β).
To summarize, we have,


c/ c = μ/(1-β);

 /  = - σμ/(1-β);
K /K = λ + μ/(1-β)
(10)
By actually computing the above growth rates we have in fact shown the
existence of the balanced paths. Note that the preference parameters ρ and σ
have no effect on the long-run growth rates of consumption c and capital
stock K. In other words, even if our preferences change over time, that
would have no effect on the long-term growth rates.
We can draw some conclusions here. If the economy is on the
balanced path, then the growth rates must be given by (10). Another
important feature of the Cass-Koopmans model is that even if the initial
growth rates are not given by (10), they will still converge towards (10)
asymptotically.
[Sketch of the proof on stability or convergence to the balanced growth
paths:
Along the balanced path, the growth rate of θ(t) is –σκ, while that for K(t) is
λ + κ. Based on these, we can define two new variables:
16
 (t ) ≡ e σκtθ(t)
_
_
K (t ) ≡ e -(λ+κ)tK(t).
_
_
We call  (t ) and K (t ) the normalized variables of θ(t) and K(t), respectively.
Suppose that the economy has already moved to the balanced path, then
_
_
 (t ) and K (t ) must become constants (Why?). If we could show that the
_
_
economy will converge towards  (t ) and K (t ) , given any arbitrary initial
values of θ(t) and K(t), this would be equivalent to showing that the growth
paths converge to the balanced paths. This can be done by the use of phase
diagrams in mathematics.]
One of the results that can be derived from the Cass-Koopmans model
is its saving rate. From (7) and (9), the balanced-path saving rate


1-β β
1-β β-1
s = K / AN K = ( K /K) / AN K = [(λ+κ)β]/(ρ+σκ).
(11)
The parameter σ is a measure of the degree of risk aversion of people. The
higher is σ, the more risk-averse they are. The parameter ρ measures
people’s attitude towards the delay in consumption. When σ or ρ rises,
saving rate declines. Consumption goes up temporarily, but the “level” of
production in the long run will go down. However, we need to remember
that σ and ρ will not affect long-term growth rates. The effects on income
level and growth rate of income can be illustrated by the following diagram.
17
In the neoclassical model, much attention is paid to the difference between
the level and growth effects.
Based on (10) and (11), both the growth rates and saving rate are determined
by parameters in the model. Some of those parameters can be estimated
using real-world data. We can use them to find out whether the κ and s as
calculated from the model are consistent with what have been observed in
reality. If they do not match, the model is not considered to be good enough.
In fact, the quantitative success of the Cass-Koopmans model is limited, at
least when we are using US data.
We have emphasized several times that the objective of studying the
theory of economic growth is to explain why there is such diversity in the
growth rates across countries. Can the Cass-Koopmans model achieve this
end? In other words, can we say that economic growth rates are different
simply because the underlying parameters across countries are different? We
know that λ and 1 – β are not the same in different countries. From (11),
when λ goes up, s will go up, consumption c will go down and output Q will
18
increase. But countries that have experienced rapid growth in population are
not necessarily poorer than those with smaller λ. Another example is that the
values of 1- β (labor share of GDP) in most poor countries tend to be low.


Equation (10) predicts that in economies where 1- β is low, c/ c and K /K
will tend to be higher. But these predictions do not always match reality.
The Cass-Koopmans model belongs to the neoclassical tradition. The
engine of growth is an exogenously given technological change μ. If μ is
zero, there is no growth. But μ is only a parameter. If we cannot explain why
this parameter differs among countries, then using it to explain the diversity
in growth is not really such an impressive approach. We need more powerful
models. The development of endogenous growth models is a response to this
weakness.
The simplest type of endogenous growth model is the so-called AK
model.
19
Chapter 3
Early Endogenous Growth Models
(3.1) AK Models
In the Cass-Koopmans model we need μ > 0 to generate growth. Is it
possible that there is positive growth when μ = 0? The answer is yes, but we
need another model, one that has the property of non-diminishing returns in
capital. (A weaker condition, but one that can still work, is that marginal
product of capital (MPK) is diminishing, but it converges towards a lower
bound.)
Let the utility function be the same as that in the Cass-Koopmans
model:


0
e
– ρt
{[c(t)
1-σ
- 1] / [1 – σ]}N(t) dt.
(1)
The production function now is non-diminishing in K(t):

N(t)c(t) + K (t) =AK(t).

(2)

N/ N = λ, and A/ A = 0.
The problem to solve is to maximize (1) subject to (2) and a given initial
value of K(0). The Hamiltonian for this problem is
H = N [c
1-σ
∂H/∂c = N(c
- 1] / (1 – σ) + θ [AK – Nc].
-σ
- θ) = 0
(3)
(4)

∂H/∂K = θA = ρθ - 
(5)
20
(4) → c
-σ


= θ, and  /  = - σ( c/ c ).

(5) →  /  = ρ – A.


Thus, c/ c = - (  /  )/σ = (A – ρ)/ σ.

From (2), K/ K = A – N(t)c(t)/K(t).
Since the growth rate of K must be a constant on the balanced path, the lefthand side is a constant. Thus, Nc/K is also a constant.



N/ N + c/ c = K/ K

K/ K = λ+ (A – ρ)/σ.

A↑ → c/ c ↑.

ρ↑ or σ↑ → c/ c ↓. (What are the economic
interpretations?)
The above results demonstrate that the growth rates of consumption
and capital are not dependent on the growth rate of technology. The
economy can drive itself forward perpetually, so long as one of the inputs
(and in this case, capital) does not have diminishing returns.
(3.2) Lucas’ Externality Model
We have seen that technology is important for growth. But what is
technology? What is the meaning of saying that the technology of Hong
21
Kong is different from that in the US? Lucas models the externality of
human capital as one form of technology.
Let h(t) be human capital or skill level. A person having h(t) is twice
as productive as a person having only half of h(t). The average level of
human capital in the economy is approximated by
ha = h(t).
We assume here that people are all identical. The production function in the
economy is given by

β
1- β
γ
N(t)c(t) + K (t ) = AK(t) [u(t)h(t)N(t)]
ha (t)
(1)
where u(t) is the proportion of non-leisure time spent directly on working,
and γ is a parameter measuring the external effects of human capital. (How?)
There is also a technology for acquiring human capital:

h(t ) = h(t)δ[1 – u(t)]
(2)
Let social preferences be


0
e
– ρt
ln c(t) N(t) dt.
(3)
Note that this utility function is a special case of the one in the original
Lucas (1988) paper or the one used in the Cass-Koopmans model. It is used
here only for simplicity.
The problem is to maximize (3) s.t. (1), (2) h(0) and K(0). The
current-value Hamiltonian is
β
1- β γ
H = N ln c + θ1 [AK [uhN]
ha – Nc]+ θ2 [ δh(1 – u)].
22
(4)
θ1 can be interpreted as the shadow price of capital and θ2 is the shadow
price of human capital. The decision variables are K(t), h(t), c(t) and u(t).
Here we distinguish between two types of problems:
(1) Optimal path. We choose c(t) and u(t) that maximize (3) s.t. (1) and (2),
the constraint h(t) = ha(t) for all t, and the initial values. (We can just let ha
= h in (4) and then solve the Hamiltonian. This is a social planner’s
problem.)
(2) Equilibrium path. Assume that the private sector consists of many small
households and firms. They do not attempt to change ha(t). Instead they take
ha(t) as given when they are solving the maximization problem. If the
solution path h(t) coincides with the given path ha(t), then the system is in
equilibrium.
The first-order necessary conditions (FONC) for the Optimal

Problem include ∂H/∂c = 0; ∂H/∂u = 0; ∂H/∂K = ρθ1 -  1 and ∂H/∂h =

ρθ2 -  2.
Thus,
1/c - θ1 = 0.
(5)
β -β
1-β γ
(1 – β) θ1AK u (Nh)
h - θ2δh = 0.
(6)
23

β-1
1-β γ
β θ1AK
(uNh)
h = ρθ1 -  1
(7)

β
1-β -β+γ
(1 – β + γ)θ1AK (uN)
h
+ θ2 δ (1-u) = ρθ2 -  2
(8)
For the Equilibrium Problem, (5), (6) and (7) still apply. But (8) will
become

β
1-β -β γ
(1 – β)θ1AK (uN)
h ha + θ2 δ (1-u) = ρθ2 -  2.
At this stage, substitute h into ha. Then,

β
1-β -β+γ
(1 – β)θ1AK (uN)
h
+ θ2 δ (1 - u) = ρθ2 -  2.
(8’)
We shall calculate the balanced paths corresponding to the optimal and

equilibrium problems. Again let κ = c/ c .

(5) →  1 /θ1 = - κ

β-1
1-β 1-β+γ
(7) →  1/θ1 = ρ - βAK (uN)
h
.
Therefore,
β-1
1-β 1-β+γ
βAK (uN)
h
= ρ + κ.
(9)

Let v = h /h.
(2) → v = δ(1 – u)
(10)
Since on the balanced path, v is a constant, u must also be a constant.

(9) → (β - 1) ( K/ K ) + (1 - β)λ + (1 - β + γ)v= 0.

K/ K =λ + [(1 – β + γ)v]/(1 – β)
(11)
24
But from (9) and (1),

(Nc + K )/K = AK
β-1
[uhN]
1- β γ
h = (ρ + κ)/β = constant.
Since the growth rate of K is a constant, Nc/K is also a constant. Therefore,

K /K = κ + λ
(12)
(11) and (12) → κ = [(1 – β + γ)v]/(1 – β)
(13)
We also know that

β-1
1- β γ
s = ( K /K) /{ AK [uhN]
h } = β(κ + λ)/(ρ + κ) = constant.
(14)
From (6),



 1 /θ1 -  2 /θ2 + β ( K /K) + (1 – β)λ + (- β + γ)v = 0.

 2 /θ2 = (β – 1)κ + λ – (β – γ)v = - v + λ
(15)
From (6),
β -β 1-β -β+γ
θ1/ θ2 = δ/ (1 – β)AK u N
h
(16)
Consider the optimal solution first, i.e., use (8) and divide it by θ2 and
substitute (16) into it.

 2 /θ2 = ρ - δu(1 – β + γ)/(1 – β) – δ(1 - u)
= ρ – δ – ( γδu)/(1 – β).
(17)
(15) and (17) → - v + λ = ρ – δ – γδu/(1-β)
25
From (10), - δ + δu + λ = ρ – δ – γδu/(1-β)
Therefore,
u* = [(ρ – λ)(1 – β)] / (1 – β + γ)δ
(18)
v* = δ(1 – u) = δ - [(ρ – λ)(1 – β)] / (1 – β + γ).
(19)
κ* = δ(1 – β + γ)/(1 – β) – (ρ – λ) = δ – ρ + λ + δγ/(1 – β).
(20)
Now consider the equilibrium solution. Divide (8’) by θ2 and substitute
(16) into it.

 2 /θ2 = ρ - δu – δ(1 - u) = ρ – δ.
(21)
Similarly, u = (ρ – λ)/δ
(22)
v=δ–ρ+λ
(23)
κ = v(1 – β + γ)/(1 – β)
= δ – ρ + λ + (δ – ρ + λ)γ/(1 – β)
(24)
For the results to be applicable, we need u* < 1 and u < 1. From (18), for the
optimal path to hold, we need
(ρ – λ)(1 – β)] < (1 – β + γ)δ.
From (22), for the equilibrium path to hold, we need
ρ – λ < δ.
26
Also note that
v* - v = γ(ρ – λ)/(1 – β + γ) > 0 if ρ > λ.
But this is true for u > 0.
Thus, the optimal growth rate of human capital > equilibrium growth rate of
human capital if γ > 0.
This is not a surprising result because the optimal path takes
externalities into account.
27
Chapter 4
The Role of Technological Change
(4.1) Romer’s Endogenous Technological Change Model
Per capita output in the United States went up 10 times in the last 100
years. Per capita capital stock in the United States also went up 10 times in
the last 100 years. If the production function is given by
α 1-α
Y = AK L
and α = 0.3, which is approximately true, then we have,
y = Ak
0.3
,
where k ≡ K/L, and y ≡ Y/L.
From this relationship, if k goes up 10 times,
0.3
y’ = A (10k)
= 1.995y < 10y.
Thus, we cannot just use the increase in per capita capital stock to
explain the increase in per capita output. There must be other important
factors that we need to take into account. These may include human capital,
but according to Romer, it is hard to imagine that the stock of human capital
has gone up by that much. At least, the rate of change in schooling or
experience has little prospects of accounting for the unexplained large
increase in per capita output. (He uses number of years of schooling. This
may not be enough. Better teachers now can teach more effectively. ) Romer
argues that we have to distinguish between embodied and disembodied
technology. His model has three sectors.
28
(a) A research sector uses human capital and existing stock of knowledge to
produce new knowledge.

A = δH2A,
(1)

where H2 represents the human capital stock used for producing A , A is the
stock of existing knowledge, and δ > 0. Knowledge here can be interpreted
as designs for new intermediate inputs in production.
(b) An intermediate goods sector uses capital and the designs from the
research to produce a large number of intermediate inputs. The set of
possible intermediate inputs is indexed by the half line R+, i Є R+, where i
represents an input.


x(i) is an input list. 0
x(i) di is well defined and finite.
x(i)
i
29
(c) A final goods sector uses labor L, human capital H1, and the entire set of
intermediate inputs



α β
Y = H1 L 0
x(i)

0
x(i) di to produce final goods.
1- α - β
di,
(2)
where Y is output of final good, and H1 is human capital stock used for
producing Y.
H1 + H2 = H, which is fixed. L is also fixed.
Output can be saved:

K =Y–C,
(3)
where C is aggregate consumption.
Note that with one design, a firm is free to produce an arbitrary
number of units of an intermediate input. Capital is also required to produce
x units of an intermediate input. Given the design, capital required is ηx.
Because of (1),
1/pA = δA/wH ,
(4)
where pA = price of new designs,
wH = rental rate of one unit of human capital,
1 = price of the capital good.
The reason for (4) is the following. From (1), in one period, ΔA= δH2A. To
produce the new design ΔA, there are two ways. The first is to buy the new
30
design in the market directly. If one unit of capital is spent on buying the
new design, 1/pA units of new design can be brought. The second way is to
rent human capital to produce the new design. If one unit of capital is spent
on renting human capital, then 1/wH units of human capital will be available.
This would produce δA/wH units of new design. For optimality, 1/pA =
δA/wH . Hence,
wH = pA δA
(4’)
The demand for intermediate good i from the final goods sector can be
derived from
Max


0
α β
1- α - β
[H1 L x(i)
– p(i)x(i)] di.
x(i)
α β
-α-β
→ (1 – α – β) H1 L x(i)
– p(i) = 0.
α β
-α–β
→ p(i) = (1 – α – β) H1 L x(i)
(5)
(5) generates a demand curve that the producer of the intermediate input
takes as given. Given one unit of a new design, the producer solves
π = max (px – rηx)
x
α β 1- α - β
= max ((1 – α – β) H1 L x
– rηx)
x
(6)
A producer in the intermediate sector has to decide whether a new
technological design should be purchased. The present value of the stream of
profits from the production of that input is
31


e
t
– r(τ - t)
π(τ)dτ .
To enter, the producer has to pay for the design cost pA. Therefore,


e
t
– r(τ - t)
π(τ)dτ = pA.
(7)
We can apply Leibniz rule to differentiate both sides of (7) with respect to t:
- π(t) + r


e
t
– r(τ - t)
π(τ)dτ = 0
(8)
[Mathematical Note (Leibniz’ Rule):
 (t )
Let g(t) =

(t )
f(x, t) dx.
Then, g’(t) = f(β(t), t) β’(t) – f(α(t), t) α’(t) +
 (t )

(t )
[∂f(x, t)/∂t] dx. ]
From (7) and (8),
π(t) - r pA = 0
(9)
Preferences are given by


e
0
– ρt
{[C(t)
1-σ
- 1] / [1 – σ]}dt.
Here we can allow σ Є [0, ∞). If we solve this model, it is possible to show
that the balanced growth rate




g = C /C = Y /Y = K /K = A /A = (δH – Λρ)/(Λσ+1),
32
(10)
where Λ= α/[(1 – α – β)(α + β)].
If H is too low, a non-negativity constraint will be binding and g = 0. From
(10), it is clear that g depends on H. The level of H matters a lot in this
model. Note that η does not affect g. Subsidization of the accumulation of
capital (which reduces η) will not affect the growth rate.
The model also has implications for international trade and economic
growth. Suppose two identical countries integrate together (because of
trade). In the beginning, there is no effect at all because they share the same
stock of knowledge A and have the same set of intermediate inputs. But
soon afterwards, they can specialize in the production of different
intermediate inputs. Another way to look at it is that the H in (10) has
doubled. Growth rate will therefore increase. Trade has an effect on growth
not because the economy is larger in terms of K and L, but because a larger
amount of human capital can be devoted to research and the production of
new goods.
(4.2) Basu and Weil’s Appropriate Technology Model
In the models we discussed earlier, technology, or rather, blue-print
technology, is freely available to everybody. But even if this were true, in
reality, countries at different levels of development might be using
completely different technologies. A technology that is appropriate for a
country may be totally useless for others. Moreover, after a new technology
has been introduced to a country, it takes time for the latter to be able to
adopt and use it efficiently. When a country is developing very fast, it has to
keep on adopting new technologies. This could pose a problem. It may not
33
have enough time to learn how to use the appropriate technology efficiently
before the latter has become obsolete. We will outline some features of the
model by Basu and Weil and show some of its results.
Let A(k, t) represent the technology at time t for producing output with
a given level of capital-to-labor ratio k. The production function is given by
Y = A(k, t)k
α
(1)
Note that this technology can change over time. Now let A*(k) be the
maximum level of technology at the given value of capital per worker k.
This approach captures the idea that even though technology can keep on
improving, there is an upper bound for the technology that is appropriate for
a given capital-to-labor ratio. Thus, although new technologies for building
spacecraft can be invented, there is some limit to how much we can improve
the technology for using an oxcart.
The growth rate of the technology appropriate for some capital-tolabor ratio j is given by

A( j , t ) = β (A*(j) – A(j, t))
=0
if k – γ < j < k + γ
if otherwise,
(2)
where γ is a positive parameter and k is the current capital-to-labor ratio.
Equation (2) says that for some j within the parametric range of (k – γ, k +
γ), A(j, t) gets closer and closer to the maximum level of technology over
time. However, if j is outside the parametric range, A(j, t) would remain
stagnant. The country cannot adopt any technology that is inappropriate.
Define R as the ratio of technology to its maximum level:
R = A(k, t)/A*(k)
(3)
34
The ratio R in this model affects a country’s growth rate. If a country grows
fast, it has little time to improve the technology at any given capital-to-labor
ratio. R is therefore a negative function of economic growth rate in this
country.
Imagine that a country is the leader in innovating technology, which is
freely available to other following countries that have lower levels of k.
Further suppose that the saving rate in a following country increases. This
means that k and output Y go up faster in this country. As k grows, it has to
adopt new appropriate technologies available from the leader. However, as
argued above, R tends to be lower. The latter has a dragging effect on output.
So part of the growth effect due to faster k is mitigated.
Suppose that the β in (2) increases. The following country has become
more efficient in adopting new technology. Its R will go up. Output growth
rate will also be higher.
Consider now the situation that the following country (Japan) has a
higher saving rate than the leading country (US). From the arguments above,
the following country may overtake the leading country because of the
higher growth rate of the former. Once this happens, the following country
would have to take up the burden of being the leader in innovating new
technology. Japan’s growth rate will become lower, but that of the US will
become faster. This would induce the per capita income in the two countries
to converge to a similar level.
These results demonstrate that following countries that are willing to
save more can enjoy higher output growth rate because they can take
advantage of the technologies that have already been made available by
more advanced countries. However, it would be harder for leading countries
to grow because they have to spend resources in innovating.
35
Chapter 5
Learning-by-Doing Models
(5.1) Lucas’ Learning-by-Doing Model
In many models that use the accumulation of human capital as the
engine of growth, time spent on investing in human capital would be
competing for time that can be spent directly on work. This approach may be
problematic. More working experience could mean faster accumulation of
human capital.
Let there be two consumption goods, but no physical capital.
ci(t) = hi(t)ui(t)N,
u1 + u2 = 1,
i = 1,2
(1)
ui > 0.

h i = hiδiui
(2)
The variable ui can be interpreted as the proportion of time spent on
producing good i. Since there is learning-by-doing, production of human
capital also depends on ui.
Assume that δ1 > δ2. So good 1 is the “fast-learning” good. Let the
utility function be
-ρ
u(c1, c2) = [α1c1
– ρ - 1/ ρ
+ α2c2
]
(3)
36
α1 + α2 = 1, α2 > 0, ρ > - 1, σ ≡ 1/(1+ ρ) is the elasticity of substitution
between c1 and c2. Note that the notations σ and ρ are different from those
in earlier models.
Let (1, q) be the equilibrium price vector of the two goods under
autarky.
q = [∂u/∂c2] / [∂u/∂c1] = (α2/α1) (c2/c1)
- (1+ρ)
= (α2/α1) (c2/c1)
σ -σ
Therefore, c2/c1 = (α2/α1) q
- (1/σ)
(4)
From (4), when q →∞, c2 → 0. There won’t be complete specialization in
good 1 unless the price of good 2 is infinite. When q → 0, c1 → 0. There is
complete specialization in good 2.
Profit maximization implies that q must be determined by relative
factor cost. q = h1/h2. (Price has to be proportional to labor cost and
therefore inversely proportional to labor productivity.)
σ
σ
From (1), c2 /c1 = u2h2 /u1h1 = (α2/α1) (h2/h1)
σ
σ-1
u2/u1 = (1- u1) /u1 = (α2/α1) (h2/h1)
(5)
σ
σ - 1 -1
Thus, u1 = [1 + (α2/α1) (h2/h1)
] .
Since q = h1/h2,
(1/q)(dq/dt) = (1/h1) (dh1/dt) - (1/h2) (dh2/dt) = δ1u1 - δ2u2
= δ1u1 - δ2(1- u1) = (δ1 + δ2) u1 - δ2
σ 1- σ -1
= (δ1 + δ2) [1 + (α2 /α1) q
] - δ2
37
(6)
From (6), (5) and (2), and given h1(0) and h2(0), we can determine the time
paths of h1(t) and h2(t). Let q* be the q on the balanced path.
At q*, (1/q)(dq/dt) = 0.
Consider how q* and u1* can be determined when σ > 1 and when σ
< 1.
Case for σ > 1.
u1
Good 1
1
σ 1-σ -1
[1+(α2/α1) q
]
δ2/(δ1 + δ2)
Good 2
0
q*
q
For σ > 1, the choice of which good to produce depends on q(0). If
q(0) is lower than q*, from (6), (1/q)(dq/dt) < 0. There will be specialization
in good 2. If q(0) is bigger than q*, (1/q)(dq/dt) > 0. There will be
specialization in good 1.
For σ < 1, we have stable allocation. Irrespective of what q(0) is, it
will converge to q*.
38
What will happen when σ = 1? From (5), u1 = α1/(α2+α1).
u1
Case for σ < 1.
1
δ2/(δ1 + δ2)
u*
σ 1-σ -1
[1+(α2/α1) q
]
0
q*
q
If a county specializes in good i, then the growth rate depends on δi.
Initial condition matters here.
Let us introduce international trade. Let the world price be (1, p).
σ -σ
c2 /c1 = (α2 /α1) p
(4’)
For countries whose initial h2/h1 > 1/p, they can produce good 2 at a lower
relative cost. So they specialize in the production of good 2. This induces
greater accumulation of h2. The comparative advantage in producing good 2
improves even further. If p goes up, then more countries will specialize in
good 2. But if more countries produce good 2, p will have to fall. Is it
39
possible that the price movement will lead to a switching of the good
produced?
With no switching, growth rate of the country specializing in good 1
is δ1. Growth rate of the country specializing in good 2 is δ2 + (1/p)(dp/dt).
This is true because p is the price of good 2 measured in units of good 1. It
can also be proved that this is equal to δ2 + (δ1 - δ2)/σ.
Which country will grow faster? The necessary and sufficient
condition for the country producing the high-δ good (good 1) to grow faster
is δ1 > δ2 + (δ1 - δ2)/σ. This is equivalent to σ > 1, given the assumption
that δ1 > δ2.
Now consider a country with q > q* under autarky. This country will
specialize in good 1. Furthermore, assume that p > h1/h2. Under free trade,
this country will specialize in good 2. Lastly, assume that σ > 1. Under these
conditions, autarky will induce the country to only produce the fast-growing
good 1. Once there is free trade, growth rate in this country will fall because
it will produce good 2 instead.
The situation above sometimes is used to justify the policy of importsubstitution. But we should understand the stringent conditions needed for
this result.
(5.2) Stokey’s Learning-by-Doing Model and Introduction of New
Goods
Economic growth is characterized not only by the increase in the
output of goods, but also by the disappearance of old goods and introduction
of new goods. We therefore want to understand the conditions under which
40
this pattern of growth would occur. Only the sketch of the model will be
provided here. The engine of growth in this model is learning-by-doing.
An allocation at time t is represented by the continuous density qt(z).
b
 q ( z)dz represents the quantity of goods of types between a and b at time t.
t
a
q(z)
a
b
z
Preferences of a representative consumer are denoted by

U(qt) =   ( z )u (qt ( z )) dz ,
(1)
0
where u’ > 0, u” < 0, and ρ’(z) > 0. The last condition implies that goods of
higher z are of higher quality in the sense that they are more effective in
making consumers happier.
Cumulative experience for producing good z at t is Qt(z), with z > 0.
Qt+1(z) = Qt(z) + qt(z)
(2)
41
Experience grows over time. If experience lowers the cost of production,
then we have sustained growth.
Assume that there are spillover effects of experience. Let c(z, Qt) be
the labor cost to produce one unit of good z. Labor required for allocation q t

is
 c( z, Q )q ( z )dz
t
(3)
t
0
The assumption that c(.) depends on Qt indicates that there are spillover
effects of experience.
The problem of a representative agent in this economy is to choose
qt(z) so as to maximize (1) subject to

 c( z, Q )q ( z )dz < y,
t
(4)
t
0
and qt(z) > 0, for all z, where y is labor endowment. It can easily be shown
that the first-order conditions are
u’(qt(z)) / λt < c(z, Qt) / ρ(z).
(5)
The = sign holds if qt(z) > 0, and the < sign holds if qt(z) = 0. The righthand side can be interpreted as a cost-to-benefit ratio. If this ratio is small
enough for a particular z, qt(z)will be positive.
To mimic the real world, we want a competitive equilibrium at which
only a finite set of goods is produced, low-quality goods are gradually
dropped out, and new goods are introduced over time. Thus, the competitive
equilibrium should have the properties that in each period t, the set of
42
produced goods is represented by an interval [At, Bt], where both At and Bt
are going up over time.
c(z, Qt) / ρ(z)
u’(0)/ λt
c(z, Qt+1) / ρ(z)
u’(0)/ λt+1
c(z, Qt+2) / ρ(z)
u’(0)/ λt+2
z
The above diagram represents what we need in generating the desired
properties. We know that c(z, Qt) / ρ(z) must be shifting down overtime
because Qt is increasing when experience accumulates. What restrictions on
preferences and technology will give us the desired properties? Consider
restrictions that are insufficient.
Assume that the economy begins with no experience, c(z,0) / ρ(z) is
increasing in z, and lim
c(z, 0) / ρ(z) → ∞. Optimal allocation in the
z 
beginning is [0, Bt].
43
Now assume that learning has no spillovers among goods, i.e.,
experience gained in producing a good does not help the production of any
other good. The interval of z remains the same. No good drops out.
c(z, 0) / ρ(z)
u’(0)/ λt
?
0
B
z
Do new goods come in? This may happen only if λt is decreasing. If
there are no spillover effects, the more are old goods produced, the more
experience will be accumulated and the more it is likely that they will
continued to be produced. Since prices of low-quality goods go down,
people tend to consume less high-quality goods (substitution effect).
However, the lowering of costs would also create an income effect.
Consumers would desire better-quality goods. This effect moves in opposite
direction as the substitution effect. Whether there is introduction of new
goods is uncertain.
44
Again assume that there are no spillovers, but c(z, Qt) /ρ(z) is Ushaped. Even if u’(0) / λt is increasing or decreasing, we do not see
simultaneous rightward shift of both At and Bt.
c(z, Qt) / ρ(z)
u’(0)/ λt
At
Bt
z
Let us introduce spillover effects in learning among goods. Costs in
producing goods that were not produced before will also decline when
experiences accumulate. More goods will be produced. Bt will increase. But
we do not get At increasing. To achieve the latter, we need some additional
restrictions on preferences.
45
Chapter 6
Gains from an Open Economy
We have already discussed some models that are related to an open
economy. In this Chapter, we shall further explore how openness could have
an effect on economic growth.
Openness can integrate different countries into a much bigger regional
or global economy. We know that IRS (increasing returns to scale) can be an
engine of growth. If there is IRS and the economy gets bigger due to
openness, outcome growth will be higher. But is there IRS in the economy?
Romer (1986) presents evidence that the hypothesis of IRS cannot be
rejected.
Let π be the sample estimate for each country of the probability that
for any two consecutive growth rates, the latter one is higher.
UK
France
Denmark
USA
Germany
Sweden
Italy
Australia
Norway
Japan
Canada
Date of 1st
Number of
Observation Observations
1700
20
1700
18
1818
16
1800
15
1850
13
1861
12
1861
12
1861
12
1865
12
1870
11
1870
11
46
π
p-value
0.63
0.69
0.70
0.68
0.67
0.58
0.76
0.64
0.81
0.67
0.64
0.06
0.01
0.02
0.03
0.06
0.25
0.01
0.11
0.002
0.07
0.12
p = 0.06 if π = 0.63
when there are 20
observations
0.5
π
The above results indicate that there is a high likelihood in most
developed countries that growth rates become higher and higher over time.
This hypothesis is also supported by a study by Reynolds, who found that
the median growth rate of 41 less developed countries is lower than the
median of the developed countries.
The data in the above table include the period of the Industrial
Revolution, during which growth accelerated. Thus, it could be argued that
the result should not be generalized. It should be noted that are numerous
studies (e.g., Barro (1991)) showing that once other relevant variables are
controlled for, growth rate generally depends negatively on initial GDP
level. The latter, sometimes regarded as support for the convergence
hypothesis, may be plagued by the fallacy on regression towards the mean,
probably first discovered by Sir Francis Galton in the late 19th century.
Evidence from more careful studies seems to support the idea of
47
“convergence clubs”, rather than that every country’s income level
converges to the same level.
(6.1) Lucas’ Paper on Making a Miracle
Compare South Korea with the Philippines. In 1960, there were 28
million people in the Philippines and 25 million in South Korea. Per capita
GNP was about US$650 (in 1975 prices) in both countries. Degrees of
urbanization were roughly the same. All boys of primary school age were in
college. 5% of Koreans in their early 20s were in college, but in the
Philippines the ratio was 13%. Degrees of industrialization were roughly the
same. However, the growth rate of the Philippines was 1.8% from 1960 to
1988 and that for Korea was 6.2%. Why was there such a big difference in
growth rates? How could countries with similar endowments and facing the
same world prices follow very different growth paths?
If we look at the miracle economies, they seem to share something in
common. They are large-scale exporters of manufactured goods of
increasing sophistication. They have become highly urbanized, and have
high saving rate. (Singapore’s gross domestic saving rate in 1994 was 51%,
that for Hong Kong was 32%, and that for South Korea was 39% in 1994.)
Gross domestic investment rates were similarly high (Singapore 32%, Hong
Kong 31% and South Korea 38% in 1994). The miracle economies also have
pro-business governments.
Let us first consider some factors which do not seem to be sufficient
to explain miraculous growth. Consider the following model:
α
1-α
y(t) = Ak(t) [u h(t)]
(1)
48

k

h
= sy(t)
(2)
= δ(1 – u) h(t)
(3)

This is just a variant of the neoclassical model. From (3),
h / h = δ(1 – u).
The larger is (1-u), the more time is spent on schooling, and the higher
should be the growth rate. But these Asian countries do not seem to have
more schooling than many of the slower growing countries. Differences in
schooling do not look like a sufficient factor.
Can we say that these countries are in the process of catching up and
therefore they grow faster? The process of catching up can be modeled as
follows:
Let H(t) = ∑ ui(t) hi (t) = world supply of effective labor,
Z(t) = H(t) /∑ ui = world average human capital level.

h
1-θ θ
= δ(1 – u) h(t) Z(t)
(4)
Simple calculations show that if ui = u for all i, then Zi (≡ Z/ hi ) will
converge to one. In other words, those with lower levels of human capital
will grow faster and those with higher levels will grow more slowly. We
have in fact assumed spillover effects across countries here. However, this
model does not seem to be sufficient either. The Philippines has not been
growing fast, while some of the richer Asian countries have no sign that
growth has stopped.
How about higher saving rate?

g ≡ y / y =(dy/y)/dt = (dy/dk)(dk)(1/y)/dt = (dy/dk)s/dt,
where s is the saving rate.
49
Compare the Philippines with South Korea. Their difference in
investment rates in 1991 = 0.29 – 0.18 = 0.11. Assume that dy/dk = 0.1,
which is on the high side. Then the difference in the growth rates of these 2
countries, Δg, should be 0.1(0.11) = 0.011, or 1.1%. This does not explain
the difference of 6.2% - 1.8% = 4.4% difference in growth rates.
Can learning-by-doing models provide the proper explanation? Some
empirical evidence show that the learning curve is declining over time, i. e.,
it shows initially high learning rates but slower rates later after more
experiences have accumulated. Even though in the beginning, productivity
may improve rapidly, this is not sustainable.
Another plausible direction is to involve the introduction of new
goods. We note that shifting of workers from old goods with low learning
rates to new goods with high learning rates would involve an initial drop in
productivity because people are more familiar with the production of old
goods. Movement towards the new goods depends on two parameters, the
quality gradient (better goods means better world prices) and the rate (λ) of
new product introduction. We can also imagine that there are forward
spillover effects via the Stokey model. The rate λ should be an increasing
function of the rate of spillover and the learning rate, but negatively related
to the decay rate of experience spillover, and increases when employment is
more heavily concentrated on goods that are closer to the economy’s
production frontier.
In this formulation, we have continual shifting of goods produced.
Comparative advantage is not permanent. A successful theory of economic
miracle should offer the possibility of rapid growth episodes, but should not
imply their occurrence as a simple consequence of backwardness. Suppose
Korea shifts its workforce to the production of goods not formerly produced
50
before and continues to do so, while the Philippines simply produces its
traditional goods. Then according to the learning-by-doing spillovers theory,
Korean production will grow more rapidly. But in 1960, Korean and
Philippine incomes were about the same and the mix of the goods demanded
by their consumers was also about the same. So for this scenario to be
possible, Korea had to open up a large difference between the mix of the
goods produced and the mix consumed. A large volume of trade is essential
to a learning-based growth episode. Import substitution or autarky is not
good for growth despite initial success. Lack of trade would hinder growth.
(6.2) Acemoglu’s Paper on Skill Premia
Trade may induce economic growth, but it does not mean that the
gains are equally distributed among different types of people. Some people
may gain more, and some may gain less or even lose. Simple theory of
international trade would tell us that highly skilled people in advanced
countries gain from trade because they are scarcer in the world economy
than in the home country. Low-skill workers in the advanced countries, on
the other hand, may have to face global competition from workers in less
developed countries. Any possible gains from trade would be at least partly
mitigated for them. Thus, in the advanced economies, trade could increase
income inequality. In less developed countries, trade would raise the demand
for labor-intensive goods and consequently labor as well. Wage rate for lowskill labor would go up. This should lower income inequality in the lessdeveloped countries.
In reality, the rise in income inequality in the developed countries,
measured by skill premium (e.g., defined as wages of college graduates
51
divided by wages of high school graduates) appears to be much faster than
what can be explained by trade. The reduction in income inequality in the
less developed countries also seems to be smaller than that can be predicted
by standard trade theory. We shall discuss part of the model by Acemoglu,
who believes that the standard trade model has not taken into account the
effect due to the skill-biased technical change induced by trade.
Let consumers in all countries have the following identical
preferences:
In this setup, C(τ) is the aggregate consumption at time τ of a labor-intensive
and a skill-intensive good, and r is the discount rate. In country j, aggregate
consumption is
where C l j is total consumption of the labor-intensive good and C hj is total
consumption of the skill-intensive good. ε is the elasticity of substitution
between the labor-intensive and skill-intensive goods. Prices of the two
j
goods are denoted by pl and p hj , respectively. In a competitive equilibrium,
marginal rate of substitution between the two consumption goods must be
equal to the relative price between them.
Relative price is indexed by j because it differs among countries when there
is no international trade.
52
Assume that labor-intensive good is produced using unskilled
workers, and skill-intensive good is produced using skilled workers only.
Let production of these two goods in country j be
where Ahj is the productivity of skilled workers, and Al j is defined similarly.
If we do not introduce international trade, consumption of a good is equal to
production of that good. Thus,
We can normalize the price of the aggregate good to 1.
Skill workers and unskilled workers are paid according to their
marginal products. Skill premium in country j is
This equation shows that the demand curve is downward sloping, i.e.,
relative wage rate of skill labor depends negatively on the relative quantity
of skill workers ( H j / L j ). Most empirical studies show that ε > 1. Given this
assumption, the skill premium depends positively on the skill bias Ahj / Al j . In
the closed economy characterized by this equation, improvements in
productivity of the skilled workers (or skill-complementary technology) will
raise the skill premium  j .
Let us introduce free trade. All countries will face the same prices.
This implies that the consumption ratio C hj / C l j must also be the same across
all countries. In particular, the world equilibrium relative price of skillintensive goods is given by
53
This is true because skilled workers and unskilled workers are pooled
together in the global economy when there is free trade.
With the new world equilibrium price of the skill-intensive good, skill
premium in country j is given by
For simplicity, assume that the skill bias Ahj / Al j is the same for every
country, i.e., if a country is better in skill-complementary technology, it will
also be proportionately better in labor-intensive technology. Also let country
U be the advanced country and all the others are less developed. We have,
where p U is the price of the skill-intensive good in the advanced country
W
W
U
U
when there is no trade, H , L , H , and L are effective world supply of
skills (when there is free trade), effective world supply of labor (when there
is free trade), skill supply in the advanced country (when there is no trade)
and labor supply in the advanced country (when there is no trade),
W W
respectively. We should note that H /L
inequality above.
The post-trade skill premium is
54
U U
< H /L , which leads to the
The world post-trade skill premium is higher than that of the advanced
country U under autarky because trade increases the total demand for the
skill-intensive good. Trade increases inequality in the advanced country
here. However, skill premium in the US has gone up by 20% from 1980 to
1995. But if we substitute in real-world values into the following
,
we can only account for 2 or 3 percentage points of the increase in skill
premium. Thus, trade alone is not sufficient to explain the faster growth is
skill premium.
The bigger than expected rise is skill premium in the advanced
country can be explained by the following. With free trade, the number of
skill workers is larger. The advanced country, which is assumed to be the
sole producer of skill-complementary technology, now has a larger market.
The new opportunity makes it possible for the advanced country to make
more profit when it produces the skill-complementary technology. It will
produce more of such technology. In other words, there is an induced
technological change that favors the ratio Ahj / Al j .
In the less developed countries, they also use the advanced country’s
technologies. Since the latter has become more skill-biased under free trade,
the skill workers in the less developed countries are also becoming more
productive. Skill workers are better off. This fact mitigates part of the
reduction in inequality in less developed countries under free trade.
This model also has the implication that trade can enhance growth
because it provides stronger incentives to produce skill-complementary
technologies.
55
(6.3) McGrattan and Prescott’s Model on Openness, Technology
Capital and Development
It is generally agreed among economists and there is strong empirical
evidence to support that the gains from openness are large. Openness
includes international trade. However, various estimates also show that the
direct gains from trade are more limited than the gains from openness. Thus,
there must be some other avenues for gains due to openness. This paper
provides a theoretical structure showing that openness can lead to
duplications in the use of technology capital in more locations, which will
lead to growth in aggregate production.
Some observations provide the motivation that open countries gain,
but closed countries lose. In 1957, 6 countries, Belgium, France, Italy,
Luxembourg, Netherlands and West Germany signed the Treaty of Rome to
form the European Union, or the so called EU-6. After the economic
integration, they become more open to the investment from other members.
56
The figure above plots EU-6 labor productivity as a percentage of US
productivity against time. EU-6 labor productivity grew faster than US labor
productivity after the integration, even though the ratio between the two was
stagnant before then. Other events of the European Union also support the
same story. For example, in 1973, UK, Ireland and Denmark also joined the
EU. Before that time, their productivity relative to the US had been
declining, but after their joining, it began to rise.
South America in the 20th century was not open. We can see from the
following diagram that South America has been losing ground to the US in
terms of labor productivity.
South American Labor Productivity as % of the US (1900-2000)
57
Asian countries became more open and got more integrated with the
advanced economies in the same period. We can see from the following
figure that their productivity improvement had been faster than the US.
Asian Labor Productivity as % of the US (1900 – 2000)
Let zi represent the aggregate of capital services (ki) and labor services
(li) in country i.
One unit of certain know-how that we call technology capital and zi units of
this aggregate input at one domestic location produces
yi is country i’s final output. A given unit of aggregate input zi can be used at
one and only one location in country i. However, the unit of technology
58
capital can also be used to set up an operation at a location in a foreign
country j. This can produce
The parameter σi can be interpreted as a measure of openness in country i. If
it is 1, country i is totally open. If it is 0, country i is completely closed.
In addition to the openness parameter, there are other parameters. Ai
is a country-specific parameter that reflects differences in legal and
regulatory arrangements. li = Aihi , where hi is hours. Ni is the number of
locations in country i.
If a country i firm operates in a foreign country, then the firm is a
multinational. Let Mi be the stock of technology capital of country i. It is the
sum of the technology capital stocks of all firms in country i. The aggregate
production is the maximum output that can be produced given the quantity of
factor inputs. It is
In this setup, some multinational firms bring in their technology stock
to country i to open operations there. z1 and z2 are the composite input
factors used by each domestic firm and each foreign firm, respectively. Zi is
the total stock of composite inputs in country i. The solution to this is that
country i’s production function is
59
where
fraction of foreign technology capital
permitted. We can interpret ωi as the fraction of foreign technology capital
that can be used in country i. Note that this production function has constant
returns to scale with respect to the inputs M, K, and L.
To complete this model, we have to specify the utility maximization
problem of the household and the laws of motion for the accumulation of the
technology stock and capital stock. We omit these here. Nevertheless, from
the production function above, we can already infer some results. If there is
greater openness, the amount of technology capital that can be used in
country i will go up. The country’s output will increase. Moreover, a larger
population means more customers, which implies that more locations N can
be used. Size matters here.
The paper also uses the method of calibration to estimate the
quantitative effects of various changes. This involves substituting realistic
values of the parameters into the model before making computations. After
calibrating the model, a lot of interesting results can be obtained. Some
examples are:
There is advantage of size when the economy is closed. Suppose
several countries come together and form a union like that of EU-6. Let N
goes up 10 times. Then, according to the calculation, steady state output will
go up by 23.4%, which is highly significant.
60
There is advantage of openness. A large country changing from being
closed to being open will enjoy some gains, which may not be very large.
However, a small country can enjoy larger gains. It has been estimated that
if Canada changes from a completely closed economy to a perfectly open
economy, its productivity will go up by 24.4% in the steady state.
Unilateral opening up while the other countries remain closed will
benefit the country that opens. A small country joining a large union can
catch up rather rapidly.
61
Chapter 7
Human Capital & Fertility Choice
(7.1) Introduction
Human capital, as agreed by most economists and supported by
evidence, is a major determinant of income. Economic growth can be
interpreted as a phenomenon whereby people of the younger generations
possess higher levels of human capital than their parents. But what are the
motives to invest in human capital?
In previous models we have discussed, people are infinitely lived.
They have the incentive to invest in human capital for themselves because
they can always capture the returns later on. In the real world, people do not
live forever. After they have died, their human capital also disappears.
However, if they have invested in the human capital of the next generation,
then economic growth may continue.
There are several issues here. First, how can human capital of the
current generation be transferred to the next? Second, is there any linkage
between the human capital stocks in different generations? Third, what are
the incentives to educate future generations?
There appear to be two main approaches to deal with the above issues.
First, we can assume that parents of the current generation can play the role
of the heads of their respective dynastic families. The objective function of
the head of a representative dynastic family is to maximize his utility
function, which depends not only on his consumption, but also on the utility
functions of the subsequent generations. However, the influences of the
62
utility of distant future generations would be smaller than those closer to the
current one. These models are known as “altruistic models.”
The second approach is the so-called “overlapping-generations
models.” Such models presuppose that at any point of time, there are people
belonging to at least two overlapping generations who co-exist. We can call
them the old and the young. In the next period, the current young will
become old. But then a new generation of young people will emerge.
(7.2) The Ehrlich-Lui Model on Longevity and Growth
The Ehrlich-Lui (1991) model is based on an overlapping-generations
approach. Its main structures are as follows. At any point of time, there are
three overlapping generations of people. The first are children. They do not
work and depend on their parents completely. Parents invest in their
children’s human capital. The second are working adults. They own human
capital which is used to produce consumption goods. Their human capital is
derived from their learning when young. They also educate their children
and take care of their old parents. The third are old people who are taken
care of by their children who are working adults.
The Ehrlich-Lui model deals with several issues. First, it assumes that
there are two types of motives for parents to invest in their children. The first
is old-age security. The more have parents invested in their children, the
more support they can receive in the future. The second motive is that their
children can take care of them or be their companions, so that the utility of
the parents can be increased.
The second issue is parental choices, which have two dimensions.
Young parents can choose the quantity of children and the quality of
63
children. Investment in human capital can raise the quality. Both choices use
up resources, but they can increase parental utility. More children means that
there is a better chance for the parents to be supported when old. If the
children are better educated, they will earn more income, which will also
provide greater returns for the parents.
The third issue is that there is a famous phenomenon called
“demographic transition,” which means that after the death rate has declined,
fertility rate will rise for some time, but after some years, it will fall down
continuously until it has reached a rather low level. At the same time,
economic growth will increase. This is a phenomenon that we need a model
to explain.
Fourth, life expectancy and economic growth appear to have a
positive relationship. Empirical evidence supports this hypothesis of positive
relationship. We also want to generate a theory to account for this.
Fifth, some countries enjoy continuous economic growth for very long
time, while others remain stagnant. We will us the concept of “multiple
equilibrium” to explain the reasons.
We will build a simple version of the model by ignoring first the
question of companionship. Here it is assumed that the only motive to invest
in children is old-age security. We can introduce a more complete analysis
later.
Let the production function of human capital be
(1)
Ht+1 = A(Ht + H*)ht
where Ht = human capital of a representative working adult at time t,
Ht+1 = human capital of a representative working adult in the next
generation at time t+1,
64
H* = raw labor (which implies that even if Ht = 0, Ht+1 can still be
bigger than zero),
ht = the proportion of time that a representative parent at time t invests
in the human capital of each child,
A = technology parameter in the production of human capital.
Consumption of a young adult at time t and the consumption when he is old
at time t+1 are given by
(2)
c1(t) = (Ht + H*)(1 – vnt –htnt) – π2wtHt
(3)
c2(t+1) = π1ntwt+1Ht+1
Each young adult has 1 unit of time. If he uses the entire unit to
produce the consumption good, output is Ht + H*. Even when Ht = 0, raw
labor H* can make output bigger than 0. nt is the number of children borne
by a young parent. The proportion of time spent on raising a child is v. Thus,
vnt is the proportion of time spent on the nt children. In addition, educating
the children requires time. Educating nt of them requires htnt
wtHt.is the amount of consumption good provided by a representative
young adult to support his parent at time t. It can be seen that the larger is Ht,
the larger is the old-age support. wt is the rate committed by the young adult
to support his parent. π2 is the probability that a young adult can survive to
old age. The larger is π2, the longer is the life expectancy of people. The
reason why π2 is included in the term π2wtHt is that a young adult does not
have to pay for the old-age support of his parent if the latter has not
65
survived. Hence, π2wtHt can be interpreted as the expected support for the
parent.
When an adult has turned old, each of his children will provide
support equal to wt+1Ht+1. Even though an adult has given birth to nt
children, some of them cannot survive to adulthood. π1 is the probability that
a child can survive to adulthood and have the chance to work.
The utility function of a representative young adult at t is given by
(4)
ut = [(c1(t)
1-σ
– 1) / (1 – σ)] + δ π2 [(c2(t+1)
1-σ
– 1) / (1 – σ)] ,
where δ represents the discount rate for future consumption. Since the
chance for an adult to survive to old age is π2, we have to multiply old age
utility by δ π2.
The optimization problem of a young adult is to maximize (4) with
respect to nt and ht, subject to the constraints of (1), (2), (3), nt > 0, Ht+1 >
0, and the known values of wt, wt+1, and Ht. The first-order necessary
conditions are
σ
(5)
(c2 / c1) > δA π1π2wt+1 [ht / (v + ht)] ≡ δRn
(6)
(c2 / c1) > δA π1π2wt+1 ≡ δRh.
σ
Rn and Rh can be interpreted as the rates of return of investment in the
quantity of children and in the human capital of children, respectively. If nt
is not a corner solution, then strict equality sign would hold for (5).
Similarly, strict equality would hold for (6) if ht is not a corner solution.
66
From (5) and (6), it can be seen that Rh > Rn, i.e., the rate of return of
investing in human capital is always bigger than the rate of return of
investing in the quantity of children in this model. Thus, parents are not
interested in investing in quantity. However, if nt = 0, parents do not have
the vehicle to invest in human capital. Therefore, we need nt > 0. Again
from (5) and (6), we know that Rn is smaller. We must have a strict > sign
for (5). We can therefore ignore (5). Rearranging (6), we have
(7)
Ht+1 = at+1Ht + bt+1
where at+1 = [AJt+1 (1 – vn - π2wt)] / [ n(A + Jt+1)]
bt+1 = [AJt+1 (1 – vn)H*] / [ n(A + Jt+1)]
Jt+1 ≡ (δA π1
1-σ
1-σ 1/ σ
)
.
π2wt+1
This optimization problem also involves a second stage: how is the
support rate for the parent, wt+1, determined? We can make a simplifying
(but not necessary) assumption here to illustrate. Before the parent invests in
the human capital of the children, the parent will negotiate with the latter a
wt+1 that is mutually beneficial. In case we assume that children do not have
negotiating power, we can interpret the situation as one that the parent
chooses an optimal deal that his children are willing to honor when they
have grown up. This deal should maximize the benefit of the children and at
the same time consistent with the optimal solution for the parent as
represented by (7). Expected utility of a representative child is given by
(8)
u(t+1) = [(c1(t+1)
1-σ
– 1) / (1 – σ)] + δ π2 [(c2(t+2)
67
1-σ
– 1) / (1 – σ)].
We can derive the optimal wt+1 from the following.
(9)
dut+1/dwt+1 = (∂ut+1/∂Ht+1) (∂Ht+1/∂wt+1) + ∂ut+1/∂wt+1 = 0.
We have already made use of the envelope theorem in deriving
equation (9). [We should note that wt+1 affects Ht+1, and Ht+1 affects not
only ut+1, but also Ht+2. The latter can in turn possibly affect ut+1. Thus, we
have to pay attention to the term (∂ut+1/∂Ht+2)(∂Ht+2/∂Ht+1)(∂Ht+1/∂wt+1).
However, since we know that (∂ut+1/∂Ht+2) = 0 (why?), the expression
above must be zero as well. We can therefore ignore it.] The solution to (9)
is
(10) (1 – vn)(1 – σ)A = π2wt+1(A + σJt+1)
Because n is taken as a given constant here (which can be set as 1/ π1),
equation (10) has only one variable, wt+1. We can call the solution w.
Equation (7) can therefore be written as
(7’)
Ht+1 = aHt + b,
where both a and b are constants.
We can ask this question. What is the support rate w that maximizes
the a? We note that a can be interpreted as the long-run growth rate of
human capital. The first-order condition to answer this question is to find the
solution to da/dw = 0. It turns out that the solution of w for this equation is
exactly the same as the w determined by (10). The support rate w optimally
chosen by the parent and the children is also the w that maximizes long-term
growth a.
68
(A)
Ht+1
I
x
slope = a
b
45°
0
Hx
Ht
(B)
Ht+1
II
slope = a
b
45°
0
Ht
69
In Figure A, if H(0) > Hx, then Ht will go down. If H(0) < Hx, then Ht
will go up. Thus, the economy at x is a stable equilibrium, where Ht does not
change and the economy remains stagnant.
Suppose that for some reason, such as an increase in π1 or π2, both a
and b will go up. Line I will move up to become line II in Figure B. After
this has happened, Ht will continue to increase irrespective of what H(0) is.
The economy will grow perpetually. The higher is π1 or π2, the higher is
economic growth. However, the increase in π1 or π2 may also result in a
situation depicted by Figure C. Economic growth rate remains at zero, but
the levels of human capital and output both increase.
(C)
Ht+1
y
I
x
slope = a
b
45°
0
Hx
Ht
By deriving the expressions of da/dπ1 and da/dπ2 , we can easily
show that (da/dπ1)(π1/a) > (da/dπ2)(π2/a) > 0. The effect of improving
70
young people’s survival rate on economic growth is bigger than improving
that for old people. This is a refutable hypothesis that can be tested. In the
empirical part of the Ehrlich-Lui paper, the evidence supports this
hypothesis.
The simple model outlined above has its limitations. We have
assumed that the only motive to raise children is old-age security. This
assumption leads to the result that n is a corner solution. In other words, we
can no longer analyze the changes of n during the course of economic
development. To remedy for this shortcoming, we can introduce the motive
of companionship that parents want from their children. Companionship can
be regarded as a consumption that increases the utility of a retired parent.
Let companionship of an old parent at time t+1 be represented by
β
(11) c3(t+1) = B(π1nt) (Ht+1)
(12) ut* = [(c1(t)
1-σ
α
– 1) / (1 – σ)]
+ δ π2 {[(c2(t+1)
1-σ
– 1) / (1 – σ)] + [(c3(t+1)
1-σ
– 1) / (1 – σ)]}
The first-order necessary conditions are:
σ
1-σ
σ
1-σ
(13) (c2 / c1) > δA π1π2w [1 + β(c3 / c2)
(14) (c2 / c1) > δA π1π2w [1 + α(c3 / c2)
*
] [ht /(v + ht)]≡ δRn
*
] ≡ δRh
*
From (13) and (14), we can see that so long as β > α, Rn is not
*
necessarily smaller than Rh . We can prove that
*
*
Rh > Rn iff (β - α) (c3 / c2)
1-σ
[ (ht/v) - α/(β - α)] < 1.
71
*
If Ht is sufficiently large and α < 1, then Rh will eventually be larger than
*
*
*
Rn . Before reaching this stage, Rh = Rn and n is not a corner solution. We
can use a method similar to that used in the simple version of the model and
obtain
Ht+1 = at+1Ht + bt+1
where at+1 = [AJ (1 – vnt - π2w)] / [ nt(A + J)]
bt+1 = [AJ (1 – vnt)H*] / [nt(A + J)]
1-σ
J ≡ (δA π1
1-σ 1/ σ
π2w )
Suppose that in the beginning, the economy is at a stagnant
equilibrium. Let π1 or π2 go up, and the magnitude of the increase is large
enough. The economy will move into a growth equilibrium. nt will rise in
the beginning, but later on, it will continue to fall, until it has reached a
corner solution. We can set the corner solution at 1/ π1. Once the corner
solution is reached, there is no further decline in nt. The pattern just
described is exactly the demographic transition. Thus, the model can
generate the demographic transition.
The discussion above can be represented by Figures D and E below.
Note that physical capital accumulation has not been taken into account.
However, in the Ehrlich-Lui model, the basic result described just now remains
robust, even if we incorporate the possibility of savings in the economy.
72
(D)
Ht+1
when π1 or π2 goes up
45°
Ht
(E)
nt
when π1 or π2 goes up
1/ π1
0
t
73
Chapter 8
Allocation of Talent
We have gone through some models indicating that human capital is
an important engine of economic growth. In practice, human capital may
consists of more than one kind, some may be socially productive, and some
unproductive. Examples of the first kind include knowledge of science or
management. Knowledge of how to take away the wealth from others (such
as the skills of thieves) is also human capital that is privately productive, but
socially unproductive.
If people allocate their resources to invest in unproductive human
capital, economic growth will not benefit from it. If the institutions in a
society are such that people are encouraged to allocate their talent and
resources to invest in unproductive human capital, economic growth will be
hampered.
The actual implications of bad institutions could be more complicated.
We shall discuss a model that is intended for the study of corruption. This
will help us understand the interactions between institutions, human capital
and economic growth.
(8.1) Ehrlich-Lui Model on Corruption
Corruption can be found in every economy all over the world.
Corruption can be defined in various ways, but it should include at least two
elements. (1) Some people cannot manage the resources they own. They
have to appoint some agents to manage the resources on their behalf. (2)
74
When the agents perform their assigned duties, they have tried to benefit
themselves or other people they favor by violating some rules or laws.
In a society where resource allocation is implemented by a perfectly
competitive market, corruption cannot exist. If a corrupt agent asks for extra
benefits, customers can just acquire the good from another competitor.
To be “qualified” to become a corrupt agent, the person must have
acquired some management power. This is not always easy and may be
costly. Power, like capital, can be inherited. But usually people have to make
investment in order to acquire it. This bears resemblance to “rent-seeking”
activities, which are privately profit-seeking but socially unproductive
activities. They can facilitate wealth re-distribution, but cannot raise
productivity. Murphy-Shleifer-Vishny (1991) and Magee-Brock-Young
(1989) have independently found that the higher is the proportion of lawyers
in an economy, the lower is the economic growth rate of that economy,
when other factors are held constant. On the other hand, when the proportion
of engineers is higher, economic growth will also go up.
In the basic version of the Ehrlich-Lui model on bureaucratic
corruption, all the agents are assumed to be the same. Each can invest either
in some productive human capital or in “political capital” which can help
him get a larger share of the output.
(1)
Ht+1 = A(Ht + H*)ht
(2)
Qt+1 = B(Qt +λH*)qt
where Qt = political capital at time t,
qt = proportion of time spent on acquiring political capital.
75
The reason why H* is incorporated into (2) is to make sure that
political capital can be produced even when Qt = 0. However, the
importance of H* is limited. It does not hurt to assume that λ = 0. Production
Yt and consumption Ct may diverge.
(3)
Yt = (Ht + H*)(1 – ht - qt)
(4)
Ct = [1 + θ ln(Qt / Qt*)] Yt
The parameter θ can represent the relative importance of the
government in the economy. Qt* denotes the average level of political
capital in society. The expression ln(Qt / Qt*) can be interpreted as the
bribery function. If a person has Qt > Qt*, then ln(Qt / Qt*) > 0. His Ct > Yt.
Otherwise, Ct < Yt, i.e., part of his output will be taken away. (What would
happen if a person had zero level of political capital?)
The utility function of a representative agent is

(5)
U=
  [(c
t 1
1
t
t 1
 1) /(1   )] .
The problem for each agent is to choose ht and qt to maximize (5)
subject to (1), (2), (3), (4) and the initial values of Ht and Qt. Qt* is taken as
exogenously given. However, at equilibrium, since the agents are
homogeneous, Qt = Qt*. The first-order necessary conditions are
σ
(6)
(ct+1 / ct ) > βA(1 - qt+1) ≡ β Rh ,
(7)
(ct+1 / ct ) > β(θMt + Nt) ≡ β Rq ,
σ
where ct+1 / ct = (H* + Ht+1 )(1- ht+1 - qt+1)/( H* + Ht )(1- ht - qt)
76
Mt ≡ (H* + Ht+1)(1- ht+1 - qt+1)/(H* + Ht ) qt
Nt ≡ [(H* + Ht+1) qt+1Qt+1] / [(H* + Ht ) qt (H* + Qt+1)].
Note that when qt is close to 0, Mt and, therefore Rq as well, will both
converge to infinity. Investment in political capital will yield very high
returns. Thus, qt will go up.
(6) and (7) imply that there are several equilibria.
(i)
Stable low-level stagnant equilibrium or “poverty trap.” This happens
when there is no human capital investment, but there is political capital
investment. ht = 0, qt = qt+1 = qs, Hs = H*. It can be shown that
qt = βθ / (1 – β + βθ) < 1.
cs = Ys = H*(1 - qs).
dqs /dθ > 0, and dcs /dθ> 0.
(ii)
Unstable stagnant development equilibrium.
ht = ht+1 = hd = 1 – [(βA - 1)/βA][(1 – β + βθ)/βθ] < 1.
qt = qt+1 = qd = 1 – (1/βA) < 1 (assuming that βA > 1).
Hd = (H*Ahd )/(1-Ahd).
(iii)
Persistent growth equilibrium.
When t → ∞, ht → hg and qt → qg. Moreover,
(H*+ Ht+1) /( H*+ Ht) → Ahg = 1+g.
It can be shown that dhg /dθ < 0 (assuming that σ < 1), dqg /dθ > 0 (unless σ
= 0), and d(1- hg - qg) /dθ < 0. We can also prove that
77
qs > qd > qg and hg > hd > hs.
The above model has not fully captured the characteristics of corrupt
bureaucrats. Now let us extend the model by assuming that there are workers
(W) and bureaucrats (B). In the beginning, we have already had
Qb(0) > Qw(0). Let α ≡ Nw / Nb.
Consumption of a representative worker and that of a representative
bureaucrat are given by the following:
_
(8)
Cw(t) = [1 – γ (1 - Q(t ) )] Yw(t)
(9)
Cb(t) ={1 + ln[Qb(t)/ Qb*(t)]} [ Yb(t) + αγ(1 - Q(t ) )Yw(t)]
(Workers)
_
(Bureaucrats)
_
Q(t ) ≡ Qw(t)/ Qb(t).
At equilibrium, Qb = Qb*. The parameter γ can be interpreted as
representing the size of the government.
The utility functions of the workers and bureaucrats are the same as
that represented in (5). They both solve their maximization problems in a
way similar to that in the homogeneous model discussed above. It comes as
no surprise that there are many equilibrium points. The following can be
shown.
1/σ (1-σ)/σ
hw = β
A
1+g = Ahw = (βA)
< 1,
1/σ
> 1.
Now assume that the bureaucrats not only have political capital, but
also a tightly controlled organization in which there is a powerful, but
rational, leadership. The leader can make corruption an organized activity.
78
As such, corruption is different from what is described in the model above,
where individual bureaucrats solve their own maximization problems and
compete for larger shares of outputs. Since they can get rid of the selfdepleting internal competition, (9) becomes
(9’)
_
Cb(t) = Yb(t) + αγ(1 - Q(t ) )Yw(t)
There are only three admissible equilibria:
(i)
Unique poverty trap.
hw = hb = 0,
qw = qb = q > 0.
_
Q = Qw /Qb = constant.
(ii)
Unique stagnant development equilibrium.
hw > 0, and hb > 0,
qw = qb = q > 0.
(iii)
Unique growth equilibrium (Equilibrium with specialization).
Qw = 0,
_
0 < qb* < 1, (i.e., Q = 0)
1/σ (1-σ)/σ
hb = 0, 0 < hw = β
A
< 1.
The growth rate of workers’ output is the same as the growth rate at
the competitive equilibrium. They are both 1+g = Ahw = (βA)
79
1/σ
.
There is another important result here. Since there are three types of
equilibrium, which one will the monopolistic bureaucratic organization
choose? Assume that the parameters in the model are sufficient to support
the growth equilibrium, the leader will make sure that it will be chosen. He
will adjust qb in such a way that it is only slightly bigger than qw. This will
_
cause Q = Qw /Qb to converge to zero. The workers have no incentive to
produce Qw because doing it will be useless. In this way, the workers will
concentrate on producing the consumption good and not waste any resources
on producing Qw.
The above discussion also indicates that in an economy where
corrupt bureaucrats have organized themselves into a monopoly, its growth
rate of output is the same as that under a regime with competitive corruption.
We can also see that in the growth equilibrium, γ does not affect the growth
rate, but it does reduce the level of human capital.
80
Chapter 9
Institutions and Growth
It has often been argued that institutional arrangements can have
fundamental effects on economic growth. For example, private property
rights/public ownership, market mechanism/central planning,
monopoly/competition, or pay-as-you-go retirement schemes/private saving,
can all be factors that affect human/physical accumulation and economic
growth. Because of time constraint, we shall discuss here only the
relationship between the government and economic growth.
(9.1) Barro’s Model on Government Spending and Economic Growth
The question of how government spending affects economic growth
has always been a central issue in economics. Simple Keynesian theory
would tell us that bigger government spending could stimulate the economy
through the so-called multiplier effect. Even if the spending needs to be
financed by higher taxation, the multiplier effect theory, within the context
of the Keynesian approach, does not have to be fundamentally modified.
This theory actually implies that the government is smarter than the people.
The latter would save part of their money, an action which would reduce the
aggregate demand in the economy.
If the above theory were correct, cross-country data would be able to
show that an economy’s growth rate would be positively related to the ratio
of its government spending relative to its GDP. Barro’s paper challenges this
view both from a theoretical and an empirical point of view.
81
Before introducing government spending, let us first introduce a
simple model of economic growth. Let the utility function be
(1)


e
0
– ρt
{[c(t)
1-σ
- 1] / [1 – σ]}dt.
The constraint is
(2)

k = f(k) – c
The notations are similar to those used before.
The Hamiltonian is
1-σ
H = [c(t)
- 1] / [1 – σ] + λ [f(k) – c]
We can easily prove that
(3)

γ ≡ c /c = (f ’ – ρ)/σ.
For example, when f = Ak, γ = (A – ρ). Note that c and k here are based on
individual units. Let us now introduce government spending. Assume that
government spending can provide public service, so that individual output
can be increased.
(4)
y = φ(k, g)
= kΦ(g/k)
Φ’ > 0; Φ’’ < 0.
The function φ above is homogeneous of degree one. y represents per capita
output, and g is government spending per capita. Thus,
y/k = Φ(g/k).
1-α α
If φ(k, g)= k
(5)
g , then
α
y/k = (g/k) , 0 < α < 1.
g is financed by taxes collected in the same period.
82
g = T = τy.
(6)
τ = g/y = g/( k
1-α α
g ) = (g/k)
1-α
The representative agent in the model maximizes (1) subject to (2)
and (6). The Hamiltonian is
1-σ
1-α α
H = [c(t)
- 1] / [1 – σ] + λ [(1 - τ)k g – c].
∂H/∂c = c
–σ
–λ=0

 = ρλ – λ (1 - τ) ∂y/∂k.
(7)


γ ≡ c /c = - (  /λ) /σ
= [(1 - τ) ∂y/∂k – ρ] /σ
α
= [(1 - τ)(1-α)(g/k) – ρ] /σ
α
α/(1-α)
From (6), we know that (g/k) = τ
(8)
. Thus,
α
γ = [(1 - τ)(1-α)(g/k) – ρ] /σ
α/(1-α)
= [(1 - τ)(1-α) τ
– ρ] /σ
The optimal τ can be determined from the following first-order condition:
(9)
∂γ/∂τ = ∂γ/∂(g/y) = (1-α)[(α/(1-α)) (1-τ) τ
-1
 (α/(1-α)) (1-τ) τ
(α/(1-α)) - 1
α/(1-α)
–τ
] /σ = 0
= 1.
 h(1-α)/α = (1-τ)/τ
 τ = α.
We can obtain similar result even if we don’t assume that the production
function is Cobb-Douglas. The optimal growth rate γ will occur at g/y = τ =
elasticity of y with respect to g, which, in this case, is α.
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γ
α
g/y
There is actually no particular reason why we can assume that the
government would choose a (g/y) that maximizes γ. Probably what the
government would do is to choose a (g/y) such that it maximizes the utility
function


e
0
– ρt
{[c(t)
1-σ
- 1] / [1 – σ]}dt. In the model above, we
can prove that whether we want to maximize the utility function or γ, the
result for g/y is the same, i.e., it equals α.
From (8), if g/y or τ has been optimally chosen (i.e., they are both
equal to α), then
2
(10) γ = [(1-α) α
α/(1-α)
– ρ] /σ.
When α goes up, what will happen to γ? Using simple calculus, we
can show from (10) that ∂γ/∂α < 0. That means, when government spending
relative to GDP goes up, the best economic growth rate will decline.
84
γ
when α increases
α1
α2
g/y
Different countries have different α and g/y. We can use econometric
techniques to determine whether there is any negative relationship between γ
and g/y. The evidence does show that this is true.
In fact, g can be classified into consumption-related type (gc) and
investment-related type (gi). The evidence indicates that the negative effect
of gc /y on γ tends to be bigger. The relationship between gi /y is less
significant.
85