Topics in the Global Economy
Saki Bigio
June 2008
CT
Contents
I
Growth and Development
1
1 Neoclassical Growth
3
1.1 The Model of Solow and Swan- Why 50 Years after? . . . . . . . . . . .
3
1.2 The Neoclassical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3 Solow Model as a Special Case of Neoclassical Growth
. . . . . . . . .
6
1.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.4.1
What is the Steady State of the Model? . . . . . . . . . . . . .
9
1.4.2
The Model’s Predictions (Steady State Comparative Statics) . .
10
1.5 Growth of Population . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.6 Technological Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.7 What can one say about the assumptions? . . . . . . . . . . . . . . . .
17
1.8 Poverty Traps - Assuming subsistence levels . . . . . . . . . . . . . . .
19
1.9 An Open Economy Version of the Model . . . . . . . . . . . . . . . . .
20
2
CONTENTS — MANUSCRIPT
1.10 What happened to the Industrialized Countries during in the Last 300
Years? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Growth Accounting and Empirical Evidence
23
24
2.1 The Decomposition of Growth . . . . . . . . . . . . . . . . . . . . . . .
24
2.2 Some Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.2.1
U.S. vs. Europe . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.2.2
Is Asia’s a Miracle? . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.2.3
The lost decade in Latin America . . . . . . . . . . . . . . . . .
29
2.2.4
Can China Keep Growing? . . . . . . . . . . . . . . . . . . . . .
29
2.3 Convergence Implications . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.4 Some Examples from Mind the Gap . . . . . . . . . . . . . . . . . . . .
29
2.5 Capital Flows: Calibrating Peru and the U.S. . . . . . . . . . . . . . .
30
3 Human and Social Capital
31
3.1 Human Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.2 Extending the Model to Labor Augmenting Technology . . . . . . . . .
33
3.2.1
Cross Country Variation (Mankiw, Romer and Weil) . . . . . .
36
3.3 Externalities to Human Capital (Low Private Provision) . . . . . . . .
40
3.4 Why then is there no more investment in Human Capital? . . . . . . .
45
3.5 Some thoughts, what are Social Institutions? Is education always Good?
52
3
CONTENTS — MANUSCRIPT
4 Malthusian Models
54
4.1 The Malthusian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1
54
Malthusian Dynamics with Capital . . . . . . . . . . . . . . . .
58
4.2 The Industrial Revolutions . . . . . . . . . . . . . . . . . . . . . . . . .
59
4.3 From Malthus to Solow . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.4 Determinants of the Fertility Rate . . . . . . . . . . . . . . . . . . . . .
62
4.4.1
Solution to the Model . . . . . . . . . . . . . . . . . . . . . . .
63
4.4.2
Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.4.3
Outcome I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.4.4
Outcome II . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
5 Determinants of Initial Conditions
66
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
5.2 So bullets from the book . . . . . . . . . . . . . . . . . . . . . . . . . .
67
6 Technology Di¤usion
72
6.1 Technology Adoption . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
6.2 Technology Adoption and Multiple Inputs . . . . . . . . . . . . . . . .
73
6.2.1
Innovation Process in the Model . . . . . . . . . . . . . . . . . .
75
6.2.2
Immitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
6.3 Innovations in the Quality of Goods . . . . . . . . . . . . . . . . . . . .
77
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CONTENTS — MANUSCRIPT
6.3.1
Static Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . .
79
6.3.2
Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
6.4 Limits to Technology Adoption . . . . . . . . . . . . . . . . . . . . . .
80
6.4.1
Product complexity: Endogenizing N . . . . . . . . . . . . . . .
7 The Role of Governments: Size and Evasion
7.1 Notes on Multiple Equilibrium in the Size of Tax Evasions . . . . . . .
II
International Crisis
III
82
86
86
93
Microeconomic Issues
95
8 Issues in Agriculture
97
8.1 Agriculture Reforms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
8.2 Agriculture Reform Reversals . . . . . . . . . . . . . . . . . . . . . . .
99
8.2.1
Adverse Selection . . . . . . . . . . . . . . . . . . . . . . . . . .
99
8.3 Repeated Bargaining . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.3.1
Rubinstein’s Solution n=1 . . . . . . . . . . . . . . . . . . . . . 104
8.3.2
Rubinstein’s Solution for arbitrary n . . . . . . . . . . . . . . . 107
8.4 Credit Rationing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.4.1
The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5
CT
Preface
I have prepared this notes for the course of Topics in the Global Economy. The notes
are designed for an advanced undergraduate level. Topics include Growth, International
Economics and Microeconomic Issues of Globalization. Some of the Lectures include
extended versions of Jonathan Eaton’s Lecture Notes for the Class during regular
semester so the work is not my own. Eduardo Zilberman contributed with the notes
on Tax Evasions.
Students should note that many of the lectures that are presented in class do
not have a corresponding lecture.
6
Part I
Growth and Development
1
CN
Chapter 1
CT
Neoclassical Growth
"If God had meant there to be more than two factors of production, He would have made
it easier for us to draw three-dimensional diagrams."
Robert Solow
A
1.1
The Model of Solow and Swan- Why 50 Years
after?
In this course we are interested in studying how the di¤erent parties that conform
what we call the Global Economy interact. We start by asking how do the interactions amongst these parties a¤ect the growth of nations. Therefore, we are interested
in studying the sources of economic growth and rely on the classical answers given by
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CHAPTER 1 — MANUSCRIPT
economists. We begin our lectures with the study of the famous Solow-Swan Neoclassical Model of Economic Growth. There are many reasons to study this 50 year old
model in a class of Topics in the Global Economy. Many things have changed since the
war against Korea and the era of the Sputnik. How "global" was the economy of the
50’s as compared to the economy today? Yet to begin with, Solow’s model is still a
useful benchmark model to study growth within and between countries and along time.
It is a benchmark in the sense that it seem’s to work pretty well to explain growth at
least in industrialized countries. As we shall see along the course, it does not apply
well in contexts in which it’s assumptions don’t apply. Solow himself pointed out this
fact, but because it does …t several economies well, it is has been useful as a guidance
for other economists to detect what is it that might not work in these other economies
rather di¤erent economies. For example, Lucas’s model of Human Capital (citet) accumulation, which does a better job in explaining these di¤erences, is an modern version
of the model.
In addition, Solow’s model enables us to decompose the sources of economic
growth which is a useful analytical tool. These tool allows us to do important predictions on long-run growth. Finally, Solow’s model had an substantial impact on the
policies of the World Bank and other lending agencies (citet). It in‡uenced the Economic Policy of many governments. Therefore, Solow’s model has had a role on the
shape of things today.
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CHAPTER 1 — MANUSCRIPT
A
1.2
The Neoclassical Setup
The model is characterized by 3 equations:
Aggregate Production (Flow Equation)
Output Yt ; is produced through some technological process F and two factors,
capital Kt ; and labor Lt :We describe this by the equation:
Yt = At F (Kt ; Lt )
(1.1)
where At is a parameter that scales production and we call technology.
Capital Accumulation (Stock Equation)
Capital evolves according to a stock equation. The stock equation simply
summerizes the fact that capital tomorrow is today’s capital minus a fraction that
depreciates ( ) and today’s investment It
Kt+1 = Kt
Kt + It
(1.2)
Capital tomorrow will be used to production tomorrow.
Aggregate Demand (De…nition)
Finally, production will be distributed among whatever we consume Ct and
whatever we invest.
Yt = Ct + It
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CHAPTER 1 — MANUSCRIPT
A
1.3
Solow Model as a Special Case of Neoclassical
Growth
In Solow’s original 1956 paper, he discusses several variants of the model we presented
above. Virtually all modern Neoclassical growth models have an "endogenous" savings
rate or savings decisions on the consumer side. That is, the consumer’s decision to
save is also modelled not as an additional equation to the system above but posed as a
problem. Nevertheless, we pay attention to a …xed savings rate assumption.
We de…ne the savings rate s as a fraction of output devoted to investment.
This yields:
Savings Rate
It = sAt F (Kt ; Lt )
Ct = (1
(1.3)
s) At F (Kt ; Lt )
The solution to the model consists on …nding a path for capital and consumption given
primitives such as an initial capital K0 ; and some process for the technology paramters
and the growth rate of population. It is standard to assume constant growth rates in
the growth rate of technological progress and the accumulation of labor force.So we
have some additional equations.
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CHAPTER 1 — MANUSCRIPT
Exogenous Growth Rates
(1.4)
At+1 = (1 + xt ) At
Lt+1 = (1 + nt ) Lt
To solve for the model, we need to impose some structure on the production
function. The referential production function is the so called Cobb-Douglas which takes
the name after a U.S. Senator, (Douglas), that assigned a Mathematician, (Cobb) the
task of …nding a good approximation to the "production function" of U.S. …rms. The
function has the form:
Cobb-Douglas Production
(1
F (Kt ; Lt ) := Kt Lt
)
(1.5)
Does this production function seem reasonable? This production function has
some nice or desirable properties that seem reasonable from an intuitive perspective.
First, is satis…es constant returns to scale, or what Barro and Sala-i-Martin call the
replicability. This is intuitive because it means that a factory that doubles it capital
and labor inputs will double production. In addition, it presents diminishing returns
to scale in both inputs holding …xed the other. This property is also intuitive because
it says that, for the same amount of machines, a incrementing the number of workers
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CHAPTER 1 — MANUSCRIPT
will increase output, but this e¤ect is decreasing in the amount of the increment.
Are the assumptions layed so far convincing? Later on we wil deviate from
some of this assumptions by changing the assumptions and noticing how the conclusion
di¤er. But for now, we concentrate on the standard model.
A
1.4
Dynamics
We now look at the dynamics of Output per worker, or GDP per capita. We …rst set
xt = nt = 0: Let’s call capital per worker kt :=
Kt
Then
Lt
deviding both sides of 3.4 by
Lt and replacing in 1.3 yields:
kt+1 = (1
= (1
(1
) kt + sAt Kt Lt
)
=Lt
) kt + sAt kt
so if we look at the gross growth of machines per worker we have:
kt+1
kt = sAt kt
kt
(1.6)
Note that if we have an initial value of k0 we can fully characterize the evolution of
capital.
The following diagram plots this functions. We can …nd steady state output
and capital were the Gross investment function meets net depreciation:
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CHAPTER 1 — MANUSCRIPT
[Solow Graph here]
[Evolution of Capital Through Time Here]
B
1.4.1
What is the Steady State of the Model?
A steady state for the model is a point in which capital per capita is not growing. That
is, the point at which the right hand side of 1.6 is 0:
0 = sAt k
k
Clearing out this equation yields:
k =
sAt
1
1
(1.7)
so k represents the point at which capital per capita does not grow anymore. An
interesting propoperty of the Solow-Swan model is that starting from any point below
kt < k ; the model predicts that capital per capita will eventually attain k as time
approaches to in…nity. The reason is that for any value above k ; the capital accumulation equation 1.6 will predict a decline in capital. The converse is also true. So the
theory predicts that economies with kt less than k will grow while the others decline.
What is then the output per capita of steady state? De…ne yt :=
Yt
Lt
as output
per capita, or GDP per capita. From equation 3.3, when dividing both sides by Lt we
obtain:
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CHAPTER 1 — MANUSCRIPT
(1.8)
y t = At k t
Given this result, we can conclude that capital per capita determines output or gdp
per capita. What would is the steady-state value of GDP per capita? We just replace
1.7 into 1.8 and we obtain:
y = At
B
1.4.2
sAt
1
1
= At1
s
1
The Model’s Predictions (Steady State Comparative Statics)
As we mentioned, the main prediction of the model is that the economy will not grow
in the Long Run above a the steady state as long as the assumptions remain constant.
As vanilla as it is, the model predicts that Output per capita is an increasing function
of the savings rate. Reknown Economists such as Je¤rey Sachs support the idea that
higher savings rate may support growth. On the other hand, the depreciation rate
also plays a role. Depreciation of the same technology may be associated with a harsh
enviorment such as humidity or a war.
But we really don’t care for output per capita, unless, as we read from Paul
Krugman’s the "Truth of Asia’s Miracle", we are a former Soviet dictator interested
in production power rather than welfare. We care more about consumption per capita
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CHAPTER 1 — MANUSCRIPT
ct :=
Ct
Yt
which we can easily back out from production per capita:
1
s) At1
c = (1
s
1
and therefore it is not clear whether savings increase steady state consumption in the
future.
The "maximal" steady state value of consumption can be obtained by optimizing over the savings rate. We take the derivative of ct with respect to the savings
rate s and set this to 0:
1
@ct =@s =
At1
s
1
1
s) At1
+ (1
1
1
(s) 1
1
=0
rearranging this equation yields:
s=
(1
1
s)
which can further be simpli…ed to:
s=
so the "best" savings rate is not 1, but far from it, it should be :
is also the factor
share of capital, that is, the share of output that can be attributed to capital. This
result has lead economists such as Alwyn Young to conclude that many East Asian
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CHAPTER 1 — MANUSCRIPT
economies were growing arti…cially fast through incredible investment rates. The result
also shows that it is not reasonable to try to achieve growth through incrementing the
savings rate above levels.
A
1.5
Growth of Population
So far we have left growth in the population out of the picture. Let’s look at the
situation in which nt is greater than 0. We proceed the same way we do for the no
growth case. Deviding the LHS of the capital accumulation equation 3.4 by Lt+1 yields:
Kt+1
= (1
Lt+1
)
Kt
L1
+ sAKt t
Lt+1
Lt+1
We can rearrange this, by using a simple trick: deviding and multiplying by Lt : We can
obtain the following equation if we remind our de…nitions:
kt+1 = (1
) kt
1
1
+ sAkt
(1 + n)
(1 + n)
and by multiplying both sides by the growth scale of population:
(1 + n)kt+1 = (1
) kt + sAkt
this equation is essentially an analog to 1.6 but the left hand side is multiplied by
a factor. This equation says that the evolution of capital per capita has to be at a
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CHAPTER 1 — MANUSCRIPT
slower rate than without population growth because simply, there is the same amount
of machines for a bigger number of workers.
The steady state will be computed in the same way, and we did before. Note
that if we proceed in the same way as in the case without population growth, the
equations yield the same result except for a term that has to account for population
growth. Capital per capita in steady state is:
sAt
+n
k =
1
1
Notice that n behaves as depreciation factor. Steady state capital per capita decreases
in population and in the same way, output per capita is proven to have a negative
impact.
s
+n
1
y = At1
1
and in the same way consumption per capita:
1
c = (1
s) At1
s
+n
1
This result found in theoretical models lead to several policy recommendations. World Banks policies towards birth control is one example. During the sixtees,
a standard policy recommendation was to apply birth control programs. China established the one child policy under a similar philosophy of the model. As we shall discuss
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CHAPTER 1 — MANUSCRIPT
in the course, China today faces a sever demographic problem since the number of
elderly is growing as a share of total population. In Peru, for example, the Government
of Fujimori engaged itself in birth control plans that were enforced and secret. Doctors
in rural areas especially were told to practice cirgical procedures on women after giving
birth on secret. As we can see, Solow’s is a good representation of population issues.
We will later address other issues when we discuss Malthusian models.
In Botswana for example, though being perhaps an emblema of good economic
policies during the 80’s, today we …nd a situation that by the terrble aids tragedy that
that country is su¤ering, capital per capita is growing fastly and so is output per worker
in spite the fact that nominal output is falling.
When presenting conclusions we have to be very careful. Antropologists such
as Marvin Harris have for decades argued that high birth rates respond not to "irrationality" or lack of birtch control methods. He argues that there are various extreme
examples of societies that applied di¤erent methods, including many that would be
considered a crime in western societies, without an exposure to modern tools such as
Condoms, birth contro pills or medical procedures. NYU’s Bill Easterly, as former
member of the World Bank has also critized several of this programs. His claim is that
people have a rationale to have many children. Even though, Solow’s model is suggesting that consumption levels in per capita terms are decreasing in the rate of growth of
population, it is not by itself useful in addressing the question of why are they what
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CHAPTER 1 — MANUSCRIPT
they are..
A
1.6
Technological Growth
The conclusion of the model we presented in the previous sections predicts that economies
will grow at a decreasing rate. The the poor economies will …nally reach the growth
rates of the rich. The The economic blocks that compose the OECD countries, namely,
the U.S., Canada, Europe and Japan have grown steadily at almost constant rates over
about 150 years now. Average rates in all of these economies, as the model predicts
did decline bet never to zero. We are missing a little piece.
Technology improves. The industrial revolution was brought by a processes
of technological innovations. The use of the steam engine lowered transportation costs
substantially and opened a whole gamma of technological developments that increased
output per worker. We can say similira things about combustion engines, electricity,
telephones, the internet and maybe soon nuclear developments. Nevertheless, many
doubters of the capitalist system claimed that technological progress meant layo¤s and
an increase in productivity. This was the moto of the famous movie by Charlie Chaplin
What does the model predict if the economies productivity of both factors grows?
This case should be treated di¤erently. Let’s think about the case in which
x > 0:Could there be a steady stae for the economy? Let’s suppose there is a Steady
State. That implies that it satis…es equation 1.6 and satis…es the equality kt = kt+1 :
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CHAPTER 1 — MANUSCRIPT
Suppose it does. What happens at period t + 2? It cannot satisfy the equation again
because At grew to become At+1 := (1 + x) At so it can’t satify the equation again. To
address the questions we would like to answer, we use what economists call the big K
little k trick. We will redi…ne the problem again as we did befor by deviding both sides
of 3.4 by the "e¤ective" labor force or At+1 Lt+1 : We obtain:
Kt
L1t
)
+ sAt Kt
At+1 Lt+1
At+1 Lt+1
Kt+1
= (1
At+1 Lt+1
and using the same trick we had before of multipliying and deviding by At Lt where
needed we obtain:
Kt+1
= (1
1+a
At+1 Lt+1
)
A1+a
Lt
L1t
t
+
sA
K
t t
A1+a
Lt A1+a
A1+a
t
t+1 Lt+1
t+1 Lt+1
Kt
we de…ne an auxiliary variable k^t :=
k^t+1 = (1
) k^t
Kt
A1+
Lt
t
and we obtain:
1
1+a
(1 + x)
(1 + n)
+
s ^
1
kt
1+
At (1 + x)
(1 + n)
which yields:
(1 + x) (1 + n) k^t+1 = (1
) k^t + sAt a k^t
Take an initial technology level Ao then this becomes:
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CHAPTER 1 — MANUSCRIPT
(1 + x) (1 + n) k^t+1 = (1
) k^t + sA0 a (1 + x)
t
k^t
and to solve this di¤erence equation we need to expose ourselves into an undisereble
amount of math. Nevertheless, we can see what happens as t ! 1. As time passes
by, for t very big, say 200, for any respectable value of x we can obtain that the second
term will not be important: and we …nd that something approximate to the following
will hold:
(1 + x) (1 + n) k^t+1 =
~ (1
) k^t
which yields something
A
1.7
What can one say about the assumptions?
Solow never attempted to write a theory about every single growth experience. He just
argued that the assumptions he made were possible explaniations of the convergence
phenomenon we …nd in states in the US or counties in Japan. It in fact …ts well in these
experiences though, for example it does a bad job in explaining growth experiences in
di¤erent regions of Italy. Several modi…cations may be done to the assumptions. As I
said in the introduction to this chapter, the model may be a useful tool as to give as a
hint on what is going on. Basically all the equations may be alteres. For example we
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CHAPTER 1 — MANUSCRIPT
may think that it is reasonable to modify the 3.4 equation by introducing adjusment
costs, which would imply that investment takes some time to build. We wouldn’t buy
much from this attempt because time is not part of the conclusions we presented. After
all, India or Peru have had many years to go by.
One interesting change is to assume that the savings rate is not constant. A
reasonable modi…cation is to assume that the poorer you are, the least capacity to save
one hase. We end this lecture by doing so and we observe how important conclusions
we may …nd by altering this assumption slightly.
We can also try to introduce a goverment sector that charges taxes and spends
money in productive or unproductive goods. Corruption can be considered in this
context. In addition we can introduce natural resources and land to study the e¤ects of
these elements. We can look at migration and capital transfers when we open the model
and have two countries (or more). We can also change our de…nition of capital and
adopt a broader de…nition that includes human capital. These issues will be covered in
future lectures. Before doing so, the next lecture we will introduce growth accounting
into the picture. By doing so, we will also build a framework to tell us, what of the
assumptions may be going on. We wrap up this lecture with the study of poverty traps.
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CHAPTER 1 — MANUSCRIPT
A
1.8
Poverty Traps - Assuming subsistence levels
We assume now that s is an increasing function. The basic theoretical support for such
an assumption is that savings are require some subsistence level, in the spirit of classical
keynesian econoics. Assume that the following functional form for savings:
s (y) =
8
>
>
<
0 if y < c
>
>
: s (1
exp [ (y
c)]) otherwise
Then the steady state of the preious section will be a¤ected. Notice that our modi…ed
stady state capital is given by two equations now:
k =
[1
exp [ Ak ]] At
+n
1
1
By using a graphical device we can obtain some surprising conclusions:
[POVERTY TRAP GRAPH GOES HERE]
Let’s look at the fact that there are multiple equilibria. The conclusions of
the neoclassical version of Solow’s model break down! In particular, we …nd multiple
equilibria. One of the equilibrium implies that theres is low income per capita and the
other implies there is high income per capita.
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CHAPTER 1 — MANUSCRIPT
A
1.9
An Open Economy Version of the Model
We can extend solows model to an open economy were capital is tradable and both
countries face di¤erent technologies. We want to ask questions such as what are the
steady of these economies and what are the paths they follow towards this steady state.
The key feature of the model is a non-arbitrage condition, and we want to ask what
are the di¤erent dynamics achieved according to the technology levels.
This model will serve as a basis for the model’s of balance of payments and
technology di¤usion. The following table summarizes the assumptions made here:
Country A Country B
Technology
AA
AB
Savings Rate
sA
sB
Initial Capital
KA
o
KB
o
Comercial Balance
rBt
-rBt
Current Account
-rBt
+rBt
Wages
FA
l
FB
l
Capital Rent
FA
k
FB
k
Demographics
LA
LB
B
Assume: KA
o <Ko and all other variables constant and indentical among
countries at the initial period..Then, by non-arbitrage conditions we have:
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CHAPTER 1 — MANUSCRIPT
rA = rB
Because the model requires that this condition be satis…ed, we have that used
capital must be identical in both countries so the following 2 equations must be satis…ed:
KtA = KtEA
Bt
where Bt is the net amount of capital transfers and KtEA is the e¤ective
amount of capital being used in country A for production in country A. Implicitly we
have assumed that the poor countryu, country A, borrows an amount Bt . Country Bt
will satisfy a similar condition:
KtB = KtEB + Bt
and from these two conditions we will obtain the following:
Bt =
KtEB
KtEA
2
Why 12 ? What is the intuition behind thes balance of capital?
Not that an interesting feature of this model is that a version of the ModiglianiMiller Theorem applies. It does not matter how are we …nancing Bt be it through
21
CHAPTER 1 — MANUSCRIPT
foreign direct investment or directly through external debt of …rms, the balance of
payments will be the same, so one implication of this model is the fact that the source
of production does not really matter for the well being of domestic consumers. Their
resources will be identical in either case unless, of course, there’s autarky.
Total Output in country A will be given by:
YtA = F K EA ; LA
and it is distributed according to the following function:
YtA = wL + rK A + rBt
so in net terms of value:
YtA = C A + I A + X A
MA
where we set M A = 0 and X A = rBt ; so the net ‡ows to to borrowed capital are
rBt ,
so this is the valu of the current account.
Obviously the Balance of Payments are net to 0. Then we have than investment is constant function of disposable income, and depends on the marginal propensity
to save s:
I A = s YtA
22
XA
CHAPTER 1 — MANUSCRIPT
The evolution of capital we be an analog to the evolution of capital in the
closed economy version of the model:
A
Kt+1
= (1
) KtA + s wL + rK A
B
Kt+1
= (1
) KtB + s wL + rK B
and
r = FK K EA ; L = FK K EB ; L
and wages equal to:
w = FL K EA ; L = FL K EB ; L
The steady state will be identical to the close economy version of Solow’s Model, but
the paths will present one of the economies lending the other economy through it’s
convergence path.
A
1.10
What happened to the Industrialized Countries during in the Last 300 Years?
MindTheGap Slide Show
23
CN
Chapter 2
CT
Growth Accounting and Empirical
Evidence
“Everything reminds Milton Friedman of the money supply. Well, everything reminds
me of sex, but I keep it out of the paper.”
Robert Solow
A
2.1
The Decomposition of Growth
In the previous lecture, we brie‡y mentioned that Solow’s model could also be used
to determine the sources of growth. We can assumet that, given what we know from
the solution to the model, all the variables are a smooth function of time. Recall the
production function:
24
CHAPTER 2 — MANUSCRIPT
Yt = At Kt L1t
(2.1)
were now we have included t-subscripts. If we had in our hands, a time series Yt ; Kt
and Lt we could certainly determine what the change in productivity has been provided
that we have the value of . In addition, we could obtain s; by the following formula:
Kt = it = sAt Kt L1t
Kt+1
Kt
and taking a time series average. O¤ course, we usually lack reliable data on Kt
because to have so, we need an initial stock of capital at some point. Regardless
of this downside, good approximations of today’s capital may be obtained from data
on investment and using the fact that depreciation would do the job of making any
initial capital level negligible in determining today’s capital. In addition, we can use a
"competitive markets" argument to obtain a time series for the growth At which will is
something we care about, since, as we learned in our previous discussion, it is the main
engine of long-term economic growth.
Taking derivatives with respect to time of equation (2.1):
Y_ = At Kt
1
L1t
+ (1
) Kt L t
+ A_ t Kt L1t
where I have used the standard notation of Y_ to refere to the derivative: @Yt =@t:Because,
the equation hold’s at all times, while the derivatives may depend on time, the euqation
25
CHAPTER 2 — MANUSCRIPT
still holds. If we devide this equation by Yt ; by using the de…nition of output we will
obtain:
Y_
=
Yt
K_
+ (1
K
)
L_
A_ t
+
L
A
Note that variables expressed as x=x
_
represent the per cent growth of that variable.
Because, technology improvements are not known, we want to compute from data on
_
Y_
; K
Yt K
is
and
L_
L
some approximation of
A_ t
;
A
The main question now is to determine what
because if we have this, we use some arithmetic manupulations to determine
A_ t
;
A
our focus of interest.
Recal standard results in Micro Theory for competitive markets. Assuming
that in the overall, the economy behaves competitively, we have that the marginal
return on capital should equal the interest rates and the marginal return on labor
should equal wages. From the marginal rate of return on capital we have the following
equation:
Yr
= rt !
Kt
= rt
Yt
Kt
and an analog can be otained from the labor market’s analog:
(1
)
Yr
= wt !
Lt
=1
wt Lt
Yt
Both, rt and wt are expressed in real terms.For reason regarding the quality of estimates,
26
CHAPTER 2 — MANUSCRIPT
the term
is obtained from the second equation (but we can always use the …rst
equation as a reference on how robust our estimate is). The way we obtain this is from
the national account’s and more precisely from taxation data. Through information
on the NSA, we obtain the term
wt Lt
Yt
which is the fraction of wage revenues of total
output.
In his 1957 paper, Solow came up with this decomposition. The term
A_ t
A
is
the so called Solow residual, and is obtained as:
A_ t
Y_
=
A
Yt
K_
+ (1
K
L_
)
L
!
or rather simply in words:
Solow Residual = % Output Growth -
% Capital Growth- (1
) % Labor Growth
When analyzing this decomposition for the U.S. economy, Solow found a surprizing
result: 70% of the increase in output in the U.S. output could be attributed to an
increase in productivity. This result is remarkable because it suggested that the U.S.
economy was in a very healthy shape. Interestingly, 1957 was a time in which the Soviet
Union seemed to rapidly catch up with the U.S. The atomic bomb was already in their
hands and they were about to Launch the Sputnik. Many economists and politicians in
the U.S. were con…dent that this was a real trend. Nikita Krushev, the famous soviet
leader was even more con…dent. He would remark "We will soon crush you". Turning
27
CHAPTER 2 — MANUSCRIPT
,to the facts Paul Krugman claims that most of the Soviet increase in output was do
to a forced industrialization. Big capital increases were do to increments in the savings
rate and mass forced exodous from rural areas to industrial clusters. The di¤erence
between the U.S.’s substantial growth and the U.S.S.R.’s impressive growth were the
sources. Without a substantinal increase in productivity, the soviets would inevitably
face the law of diminishing returns to scale, and apparently so they did.
A
2.2
Some Discussions
What can we say about the Solow residual today. The next subsections present diverse
evidence from di¤erent regions and di¤erent areas around the world. Some discussion
is worthwhile.
B
2.2.1
U.S. vs. Europe
[Table Here]
B
2.2.2
Is Asia’s a Miracle?
[Table Here]
28
CHAPTER 2 — MANUSCRIPT
B
2.2.3
The lost decade in Latin America
[Table Here]
B
2.2.4
Can China Keep Growing?
[Alwyn Young - Table Here]
A
2.3
Convergence Implications
The main implication of Solow’s model is the convergence among regions. Here are
some other examples.
A
2.4
Some Examples from Mind the Gap
We look now at a time series for the OECD countries and the rest of the economies.
The motivation is to show that the dynamics that the Solow model predicts are pretty
good for the OECD economies. Countries with higher GDP levels tend to grow at
slower rates that countries with smaller GDP levels.
[Mind the Gap Graphs Here]
Nevertheless the patterns is very di¤erent if we take other regions in the world.
We won’t …nd any pattern’s but wigglings around, with periods of economic growth,
and sudden downturns.
29
CHAPTER 2 — MANUSCRIPT
A
2.5
Capital Flows: Calibrating Peru and the U.S.
So far we have claimed that the Solow model is a good explanation for growth and
convergence in OECD countries. One important question is, if returns on capital are
greater in poor countries, why aren’t there substantial FDI ‡ows? Can taxes be an
explanation? Of course, one direct answer to the question is political instability and
expropiation risks. Venezuela is good recent example. Nevertheless, let’s look at the
di¤erence in the steady states GDP per capita in Peru and in the US. Let’s see what
happens to the rate of return in an investment in Peru, if technology is taken from the
US. Maybe there is a missing piece.
[Table here]
30
CN
Chapter 3
CT
Human and Social Capital
"Once one starts to think about them (economic growth questions) it is hard to think
of anything else." Robert E. Lucas Jr.
A
3.1
Human Capital
This part is based partially on Robert Lucas’s 1990 AER paper titled "Why doesn’t
Capital ‡ow from rich to poor countries?"
Recall from the …rst lecture the fomula for per capita output under the Neoclassical production function of diminishing returns to scale in capital per worker technology:
31
CHAPTER 3 — MANUSCRIPT
(3.1)
y t = At k t
and take the production Taking derivatives we obtain the rate of return of capital of a
given economy:
r t = At k t
1
so we can clear out 3.1 we obtain:
1
1=
(3.2)
r t = At y t
Assuming that technology is the same, and that
is close to 0.4, a 5 fold di¤erence
in capital per worker in the U.S. and Peru imply a would imply that a di¤erence in
reteturns of
rtp
=
rtus
1
5
1:5
=12
~
This, implies that the rate of return to capital per worker should be 12 times higher in
Peru than in the US. Why then do americans invest in the U.S. rather than in Peru.
The neoclassical model implies that there should be a huge ‡ow of capital from rich
to poor countries. Evidently, an answer to the question is that technology is di¤erent
in both countries and moreover, political instability may work as a detrimental factor
32
CHAPTER 3 — MANUSCRIPT
for economic growth, but 12-fold …gures can support any source of political risk. Put a
50% expropiation risk and still, expected returns are 6 times higher.
Regarding technology, why doesn’t an american inverstor come with american
technology. What about public goods such as roads and means of transportation that
htese countries lack? Anyways, 12 fold …gures still seem to be strikingly strong to
support di¤erences in other inputs.
The main focus of Lucas’s paper is that tachnology is inbeeded to human
capital, and that it has positive externalities. We explore this e¤ects in the following
subsection.
A
3.2
Extending the Model to Labor Augmenting Technology
In our …rst lectur, the model Neoclassical model was characterized by 3 equations.
We can have an extension to that model that accounts for a broader version of labor,
that takes into account human capital. We can call that factor, Ht ; and Ht Lt is termed
e¤ective units of labor. Ht is also called the Harrod-Neutral Productivity. The following
set of equations are modi…ed to obtain the following:
Aggregate Production
Output Yt ; is produced through some technological process F and two factors,
33
CHAPTER 3 — MANUSCRIPT
capital Kt ; and labor Ht Lt :We describe this by the equation:
Yt = At F (Kt ; Ht Lt )
(3.3)
where At is a parameter that scales production and we call technology.
We again specify a Cobb-Douglass functional form in terms of the e¤ective
units of e¤ective labor:
F (Kt ; Ht Lt ) := Kt (Ht Lt )1
Capital Accumulation (Stock Equation)
Again, as in the Neoclassical Growth model, capital evolves according to a
stock equation. The stock equation simply summarizes the fact that capital tomorrow
is today’s capital minus a fraction that depreciates ( ) and today’s investment It
Kt+1 = Kt
Kt + It
(3.4)
Capital tomorrow will be used to production tomorrow.
Aggregate Demand (De…nition)
Finally, production will be distributed among whatever we consume Ct and
whatever we invest.
Yt = Ct + It
We can use a similar procedure as the one we followed a couple of times before in the
34
CHAPTER 3 — MANUSCRIPT
previous lectures. Note that all that has changed from this setup and the Neovlassical
setup is that we’ve replaced the term Lt for Ht Lt now. Therefore, if we devide the RHS
of 3.3 by the e¤ective units of labor Ht Lt we would obtain the following:
y^t = At k^t
where k^t is Kt =Ht Lt : One of the …ndings of Lucas’s paper was that once we take this
broader version of human inputs, accounting for a natural version that includes human
capital Ht , the di¤erences in the rates of return are not that important. Lucas used
old calibrations by Anne Krueger from the 60’s. These studies suggested that Human
Capital could be as much as …ve times greater in the US than in India, about the same
with respect to Canada and other developed countries and up that Israel’s would be
around 10% away from the US. What would this …gures mean for the rates of return?
The formula for the rates of return should be again given by equation 3:2:
1=
1
rt = At y^t
Because human capital is greater in the US than in developing countries, the ratio we
had before turns out to be smaller
35
CHAPTER 3 — MANUSCRIPT
rtp
=
rtus
y^P
y^U S
1:5
yP
yU S
1:5
=
HtU S
HtP
1:5
Plugging in some numbers for Peru and the US again, assuming that output in the US
5 times grater and , and human capital is also 5 times greater we obatin that the rates
of return to e¤ective units of labor are about the same!
rtp
=
~
rtus
1
5
1:5
5
1
1:5
=1
When comparing this to factor to India’s, the rates of return where di¤erent by a factor
of 5. So for the Indian case, Lucas’s claim was that the model was not enough. Note
that we have not said anything about the accumulation of human capital. We close the
model and present some evidence in the next section.
B
3.2.1
Cross Country Variation (Mankiw, Romer and Weil)
We can go a little bit further. This section in turn, borrows from the work of Mankiw,
Romer and Weil from the QJE 1992. Recall from Lecture one, the di¤erence equation
that explained growth Assuming the same functional form for, capital accumulation:
Kt+1 = sAt F (Kt ; Ht Lt ) + (1
36
) Kt
CHAPTER 3 — MANUSCRIPT
Assuming we have …gures on capital stocks, labor and output, assuming similar production functions yield we can obtain di¤erent values for Ht :
[Hall and Jones]
Mankiw, Romer and Weil assumed an a constant growth in human capital, so
we have:
1+g =
Ht+1
Lt+1
;1 + n =
Ht
Lt
so we devide again by the e¤ective units of labor to obtain:
k^t+1 = sAt k^t
1
+ (1
(1 + g) (1 + n)
) k^t
1
(1 + g) (1 + n)
so we have:
k^t+1 (1 + g) (1 + n) = sAt k^t + (1
) k^t
which yields the following :
k^t+1
k^t = sAt k^t +
( + g + n + ng) k^t
As in our previous manipulations, we aim at …nding the steady state value of this
equation. We obtain the following:
37
CHAPTER 3 — MANUSCRIPT
sAt
k^ =
( + g + n + ng)
1
1
If we replace this value into the original production function we would obtain
the following form for output as a function of the primitives:
s
( + g + n + ng)
1
y^ = At1
1
Mankiw, Romer and Weil tested the theory by running a regression for this
model. They did assume to things, they used an approximate version of this regression
by assuming g*n is negligeble (in fact they follow a continuous time model in which the
e¤ect is exactly 0) and second, they assumed At is constant. In addition, the assumed
that logH0 = z + ", that is, they assumed that initial human capital conditions were a
country speci…c shock " and a constant z. By taking logarithms to both sides of this
equation they found:
log
Yt
Lt
log (Ht ) =
1
1
+
1
log (At )
log (s)
1
log ( + g + n + ng)
Now notice that Ht can be written as Ht = (1 + g)t H0 ; we obtain:
38
CHAPTER 3 — MANUSCRIPT
log
Yt
Lt
=
1
log (At ) + z + t log(1 + g)
1
+
log (s)
1
1
log ( + g + n + ng) + "
To get exactly Mankiw, Romer and Weil’s table you can set z +
1
1
log (At ) = a; and
noticing that ng is negligible, that log (1 + g) = g for g small we …nally end up with
their equation and that s = I=Y :
log
Yt
Lt
= a + gt +
1
It
Yt
log
1
log ( + g + n) + "
[Mankiw Romer and Weil Table 1] Being careful with table, what they call At
is our Ht
We can brie‡y discuss their results: the good result is that coe¢ cients have
the predicted sign, the bad is that the magnitudes are far from what we would expect
of
being close to 1=2: It is important to discuss this assumption though. Why should
it be the same in all countries. Why should
in the manufacturing industry and so on?.
39
be the same in the mining industry than
CHAPTER 3 — MANUSCRIPT
A
3.3
Externalities to Human Capital (Low Private
Provision)
Why are there exteernalities to education? Think about it. What is the return on
output to an engineer in a country where there aren’t many. It may be high, because
there aren’t many engineer’s and he is highly needed, this is the decreasing rates of
return e¤ect on education as in the previous model. On the other hand, there is an
externality e¤ect that is caused by specialization. In a country with many engineers,
these interact among themselves. They will talk to each other and quickly learn from
each other. They will ask each other if needed. They may specialize in particular duties
or rotate in the labor. They can spend time within the …rm training other engineers
etc.
Therefore, it is natural to assume that the total output of will depend on
the total stock of human capital, in a deminishing way, but in the meantime, human
capital has also an individual e¤ect. Lucas, used this argument to close the gap further
between the model with no externalities to Human Capital and di¤erences in the rate
of return to capital He found that adding this externality e¤ect we could completely
close this gap. We would have to follow the same steps as before.
Here we follow a slightly modi…ed version of the model to obtain some testable
conclusions that were analyzed again by Mankiw Romer and Weil. Let the production
40
CHAPTER 3 — MANUSCRIPT
function be now:
F (Kt ; Ht Lt ; Zt ) := Kt (Ht Lt )1
Zt
where Zt is the externality produced by human capital. The model is identical
to the one we had before with the singularity that:
Zt+1 = sh Yt + 1
h
Zt
so in addition to this equation, we have the standard …sical capital accumulation equation:
Kt+1 = sYt + (1
) Kt
From this equations, steady states should satisfy the following:
Z =
sh
h
Y
and again:
s
K = Y
so the model’s basic claim is that in steady state both forms of capital should have a
proportionality factor.
41
CHAPTER 3 — MANUSCRIPT
In addition, we focus on the steady state level of e¤ective human and physical
capital to obtain:
Zt
+ 1
Ht+1 Lt+1
Zt+1
= sh Kt (Ht Lt )1
Ht+1 Lt+1
h
Zt
Ht+1 Lt+1
which yield:
z^t+1 = sh k^t z^
1
+ 1
(1 + n) (1 + g)
h
z^t
1
(1 + n) (1 + g)
and rearranging, in steady state we obtain:
z^ =
sh k^t z^
h
+ n + g + ng
and by an analogous argument we can obtain:
k^ =
sk k^t z^
k
+ n + g + ng
we can use this equations to clear out both steady states as they represent two equations
in two unknowns. Because, we can clear out the …rst of former of these equations we
obtain the following:
z^ =
sh k^t
h
+ n + g + ng
!11
we can replace these equation in the later and obtain:
42
CHAPTER 3 — MANUSCRIPT
sk k^t1
k^ =
k
h
h
+ n + g + ng
s
+ n + g + ng
!1
and clearing k^ we obtain:
k^
and if we assume
1
=
1
k
=
h
sh
h
+ n + g + ng
sk
k
+ n + g + ng
!1
:
sk
k^ =
1
sh
1
1
( + n + g + ng) 1
1
and with respect to h; we can the analog by a symmetry argument:
sh
~=
h
1
sk
1
1
( + n + g + ng) 1
1
we may again substitute these results into the output equationa and obtain the following:
2
y^ = A 4
2
= A4
s
k
1
1
s
h
( + n + g + ng) 1
s
k
1
s
h
3 2
1
5 4
1
1
( + n + g + ng) 1
+
3
5
s
h
1
1
s
k
1
( + n + g + ng) 1
1
3
5
Under the same assumptions of the previous section, and taking logarithms to both
sides we obtain:
43
CHAPTER 3 — MANUSCRIPT
log
Yt
Lt
= a + gt +
log
1
log sh
1
It
Yt
+
+
log ( + g + n) + "
1
Notice that with this setup, investments in physical capita, increase, throuqh an interaction e¤ect, the human capital and therefor the model predicts, higher growth levels
through this mechanism. Mankiw, Romer and Weil (1992) clear out from the steady
state level of sh , the value of and are able to express this las equation in a version closer
to the one we had in the previous section:
Recall from equation, (), we can obtain the following:
1
log sh =
1
~
log h
1
1
log sk +
1
1
1
1
log ( + n + g + ng)
so replacing this we obtain:
log
Yt
Lt
= a + gt +
1
1
log
It
Yt
+
log ( + n + g + ng) +
1
~ +"
log h
This equation is identical to the equation obtained before, in the previous section
were we did not include the positive externality implied by human capital. The main
contribution on the paper is that that omiting this term will bias the estimates in table
44
CHAPTER 3 — MANUSCRIPT
1. As Robert Lucas had already argued, directly, looking at the rates of return on
human capital. Lucas’s …nding was that this inclusion would close the gap among the
rates of return in underdeveloped and developed economies.
The main conclusion is found in table II and table III which shows the Implications on Conditional Convergence:
A
3.4
Why then is there no more investment in Human Capital?
Mankiw Romer and Weil’s paper may su¤er from a potential endogeneity problem.
They have assumed that the savings rate or investments in Human Capital are independent of output. Hall and Jones take a di¤erent approach. They do include human
capital in their model but, they look at potential explanatory variables that explain
high investment in physical and human capital. Their basic claim is that their are variables related to a "social infraestructure" that explain the di¤erences in investments.
This "social infraestructure" is related to the rule of law, property rights and the time
spent looking for rent seeking activities etc.
Hall and Jones (HJ) start with the same basic production function as MRW:
Yi = Ki (Bi hi Li )1
where Bi again represents labor augmentation due to technology di¤erences. The model
45
CHAPTER 3 — MANUSCRIPT
education slightly di¤erently from MRW, however. Here h is raw labor adjusted for
average years education:
hi = e
(Ei )
where Li is raw labor and Ei average years of education. The function (Ei ) converts
more education into higher productivity. They use results from the returns to education
(Mincerian regressions) to specify the function . The late Jacob Mincer started a …eld
relating earnings to age, education, experience, and other factors:
ln wit =
0
+
A
ln(ageit ) +
1
Y yrexit
+
2
Y
(yrexit )2 +
Ed yredi
+ controls
which can be used to relate education to earnings. The implicit assumption in what Hall
and Jones do is that the e¤ect of education on earnings re‡ects the e¤ect of education
on productivity.
Using the Barro-Lee data rates of return have been computed were the results
of estimations yield: 13.4% per yr.for the …rst1-4 years , 10.1%/yr.for the years 5-8,
and 6.8%/yr >8. That is, an educational level of more that 8 years, explains returns of
about 30%, which implies high marginal returns to human capital, but, this does not
include the social returns studied in the previous section.
Following the same strategies as before we obtain
yi =
Yi
= Ki (Bi hi )1
Li
46
Li
CHAPTER 3 — MANUSCRIPT
HJ take a di¤erent tack from Solow and MRW. They rewrite yi as:
yi =
=(1
Ki
Yi
)
hi Bi
This works, since substituting from above we get:
yi =
=
=(1
Ki
Ki (Bi hi Li )1
Ki1
=(1
= Ki (hB)1
)
)
hi Bi
(hi Bi )
(hi Bi )Li
Li
HJ take the formulation:
yi =
Ki
Yi
=(1
)
hi Bi
as a way of decomposing cross-country di¤erences in yi (which they measure using
output per worker) into (i): di¤erences in (Ki =Yi )
=(1
)
; using
= 1=3; and cross
country data on the capital-output ratio, (ii), hi (using Mincerian results and the
Barro-Lee measure of average years of schooling), and (iii) calculate Bi as a residual.
How much of the di¤erence in income per capita across countries (yi ) can be
attributed to each term?
(Table 1 here)
Hall and Jones question why does the labor augmenting technology Bi vary
so much across countries? So as we said before, Social infrastructure, can encourage:
47
CHAPTER 3 — MANUSCRIPT
accumulation of skills and productive activities or predatory behavior (e.g. rent-seeking,
theft). Social Infraestructure may destroys incentives to produce (tax on output) and
resources are wasted to avoid diversion.
Their empirical strategy consisted in measuring the e¤ects of social infraestructure soy they used several proxies for this objective. They use a Government antidiversion policies Index (by Political Risk Services - assesses risk to international investors) which is an index that measures: law and order, bureaucratic quality,corruption,
risk of expropriation, government repudiation of contracts. In addistion they use the
Openness to international trade (constructed by Sachs & Warner, 1995) because they
claim that openess deceives rent seeking activities. As countries face more competition,
they will spend more time producing and investing rather than loosing resources to
sustain the Statu Quo. This index is constructed with the following variables: nontari¤
barriers cover less than 40% of goods; average tari¤ rates are less than 40%;black market premium less than 20% in the 1970s and 1980s; country not classi…ed as socialist
(by Kornai, 1992);; government does not monopolize major exports.
So HJ hypothesize the following model:
log Y =L =
S=
+ S+"
+ log Y =L + X +
48
CHAPTER 3 — MANUSCRIPT
where S denotes the social infrastructure and X a collection of other variables that in turn explain the social infraestructure. The reason why we need thes two
equations is to avoid sources of endogeneity so this is a 2 stage regression, were the
variables in X, should not be a¤ected by per capita income .The main problem with
estimating the …rst equation on its own is that social infrastructure is endogenous: it
may be caused by output per worker as well. Then, the parameter we are interested in
cannot be estimated through the standard method. This is why we must …nd exogenous variables that are correlated with social infrastructure. These variables are called
instruments.The point that Hall and Jones want to make is that Social infrastructure
is the primary determinant of income, but conversely factors other than income (X)
determine infrastructure. The idea is to avoid chicken and egg arguments such as what
determines what.
Hall and Jones use the following variables: geography (measured as distance
from the equator), languages (English or other European French, German, Portuguese,
Spanish) and predicted trade share based on gravity model (distance, population, contiguity, language).
Hall and Jones aim at recovering the the intensity of "Western Europe colonization XVI - XIX centuries" because the underlying argument is that western culture
provided the spread of ideas of Adam Smith, the importance of property rights, system of checks and balances in government and these ideas meant a superior, though
49
CHAPTER 3 — MANUSCRIPT
they never used this words, underlying infraestructure that promoted growth; Western
Europeans were more likely to settle in sparsely populated regions in XV century were
they did not confront large amounts of enemies: US, Canada, Australia, New Zealand
and Argentina, areas with broadly similar climate.
The important thing is that instruments should be independent of todays
outcome. But do these instruments directly in‡uence income? They claim the answer
is no because European sought areas that were rich in natural resources poor countries
today. We can criticize because, once you control for human capital we do …nd an e¤ect
of natural resources which is positive.
Their …nal regression is:
+ S~ + ~":
logY =L =
where S~ is the estimate of S via the regression above.
The empirical consists in estimating:
log
Y
L
= log(A) +
1
= S+"
50
log
K
Y
+ log
H
L
CHAPTER 3 — MANUSCRIPT
The estimated coe¢ cient
= 5:14 into three parts, observing the impact of
S in each component in the following form:
component =
+ S~ + ~"
(Hall and Jones: Table 4)
Then, they divide the estimated coe¢ cient
= 5:14 into three parts, observ-
ing the impact of S in each component. Regress:
component =
+ S~ + ~"
Acemouglu, Robinson and Johnston take a similar approach than Hall and
Jones but criticize this paper on the grounds of having the incorrect instruments. Their
criticism is build on two facts. First, elements such as distance from the equator and
ethnolingustical fragmentation is not a good proxy of institutions Moreover, both of
this facts may a¤ect output per capita directly. Acemouglu, Robinson and Johnston use
a di¤erent proxy settler mortality because they argue that colonial institutions varied
strongly among regions, some providing strong institutions, such as the pilgrims in the
U.S., and New Zealand, and some very week like the Belgians in Congo. These are
their results, which, di¤er in structure from Hall and Jones but are close in the results.
[Acemoglu Roginson and Johnston Figure 2]
51
CHAPTER 3 — MANUSCRIPT
[Acemoglu Roginson and Johnston Figure 3]
[Acemoglu Roginson and Johnston Table 5]
A
3.5
Some thoughts, what are Social Institutions? Is
education always Good?
Economists at NYU are not so sympathetic to the former results. I myself dare to
criticize these studies.have a di¤erent concern about this models. First of all, these seem
to neglect strong di¤erences within countries. Yes, the British colonized South Africa
bringing institutions whereas King Leopold of Belgium destroyed any infraestructure
in Congo applying brutal laws against aborigens. Nevertheless, within South Africa,
the decendents of the British leave with European standards wereas the original black
population have similar standards of leaving than in Congo or other african countries
that didn’t have the same institutions. The same can be said about Latin America. A
second criticism is that several former colonies already had institutions. The Chineese
promoted trade until the 17th century. Their was a strong mercantile class in China that
arrived to east Africa. The same can be said about african merchants So history has
changed and a lot. In addition, though Acemouglu, Robinson and Johnston manage
to do robustness checks for the e¤ect of the Malaria itself on outpuit per worker, it
does not seem reasonable that it doesn’t have a strong impact once one controls for
52
CHAPTER 3 — MANUSCRIPT
institutions. What about aids then? Most specialists claim that this maladies have had
a strong e¤ects on productivity. Mothers spending time with their children. Goverments
spending enormous amounts of resources in containing epidemics etc.
The main question of development is how to explain a 20 fold di¤erence in
output per person today, when 200 years ago it was only 2 times. The estimations
weve seen are silent about whether the disparities in income are deterministic or not
(or fatalism for the matter). The neglect strong mobility. They don’t answer questions
such as can we change these variables? Not much is said about the evolution of Human
Capital or Social Infraestructure in both sets of countries?
Finally and most importantly, in my view, some fundamental piece is missing.
People in underdeveloped countries, don’t like corruption, they wan’t to see their children educated and property rights respected. Social infraestructure is not generated,
not because people ignore their good properties but because for some reason we need
to study further, we have not identi…ed the mechanisms that prevent them to ‡ourish.
The question for a peruvian economist is not, what policies determine growth but what
prevents good policies to ‡ourish?
53
CN
Chapter 4
CT
Malthusian Models
"I do not know that any writer has supposed that on this earth man will ultimately be
able to live without food."
Thomas Malthus
"Population, when unchecked, increases in a geometrical ratio. Subsistence
increases only in an arithmetical ratio."
Thomas Malthus
A
4.1
The Malthusian Mechanics
Malthuses model is good for understanding population from a historical, rather than
a modern perspective. As opposed to Solow’s model, Malthus’s model proposes that
population growth rates are endogenous and more importantly, they increase with wel54
CHAPTER 4 — MANUSCRIPT
fare. Technological advances are therefore translated into more people and e¤ects acts
as pulling down growth rate per capita because populations are bigger:
The model has the following characteristics:
Birth Rates:
Bt = bB yt B Lt
Where Bt is the number of births in the population and yt ; is per capita income. The
function: bB yt B expresses the idea that the number of births in population will depend
on per capita income. This it says that the rates of birth depend on per capita levels,
regardless of the total amount of population. A similar equation is found for the number
of deaths:
Death Rates:
Dt = bD yt
D
Lt
Instead of capital, the model depends on the total endowment of land which is
not created or does not depreciate. This is again an argument di¢ cult to because land
has expanded. The Incan empire for example expanded its arable land to the mountains
by the construction of andens, which are huge steps carved in the mountains that make
agriculture feasible, and allow the cultivation of diversi…ed products to the di¤erences
in latitude. The dutch expanded their mass of land by building ditches. Land obtained
from the amazon and it erodes. Regardless of these assumptions, Malthuses model
55
CHAPTER 4 — MANUSCRIPT
conclusion don’t change at all. The production function assumed by Malthus should
have looked like:
Output:
yt = A(T =Lt )
Finally, accounting allows us to obtain:
Demographics:
Lt+1 = Lt + Bt
=
Dt
1 + bB yt B
bD y t
D
Lt
Steady state calculations imply the following equations:
B
bB (y )
B
= D
= bD (y )
D
y
= (bD =bB )1=(
L
= T (A=y )1=
B+ D)
The Dynamics of the model are summerized in the following sketch:
[Picture - Malthusian Mechanics]
So there are several conclusions obatain from the model: First, income per
capita depends only on parameters of demography and not on the land endowment or
56
CHAPTER 4 — MANUSCRIPT
technology. So this is very important. Malthuses model predicts that i f land is essential
for the production of food, then changes in the technology levels will only have a medium
run e¤ect on total output per worker. Eventually, the increas in population will be such
that, given the constraint of a …xed amount of land, the increase in population will be
such that output per worker will stabilize again and won’t grow any further in per
capita terms.
Second, an increase in the death rate (Black Death) or decrease in the birth
rate North Western Europe 1400 raises income per capita. Finally, increases in land
(e.g., settlement of North America) or improvements in technology (potato) raised
population but not income per capita.
Malthus had evidence that living standards were very similar around the world
up to 1800, real wages had been the same for a long time, diets were the same for a
long time (meat vs. starches) and there was no evindence in increases in the average
height of population as we do …nd after the industrial revolutions.
Nevertheless, there is evidence that North Western Europe started to pull
ahead after 1400. Why? Perhaps the Black Death played a substantial role in increasing
death rates.There is evidence that shows Lower fertility rates and with the advent of
the Absolute states, there is evidence on Less violence. More saving probably had an
in‡uence too in living standards.
Note that if we could expand the amount of land through investment we would
57
CHAPTER 4 — MANUSCRIPT
be back in Solow’s model so the key feature of the model is the diminishing returns to
scale of labor given a …xed amount of land. Note also, that this model behaves as an
alternative version of the poverty trap model that made savings an endogenous function
of output per capita.
B
4.1.1
Malthusian Dynamics with Capital
What happens to the model when we alter the dynamics and include capital as in
Solow’s model?
The production function becomes:
yt = A(T =Lt ) kt
and the accumualtion of capital is:
kt+1 = syt + kt (1
)
Birth rates and death Rates are still the same:
Lt+1 = 1 + bB yt B
bD yt
D
Lt
The steady state now is summarized by the following equations:
y = A(T =L ) k
58
(4.1)
CHAPTER 4 — MANUSCRIPT
s
k = y
(4.2)
and
y = (bD =bB )1=(
B+ D)
(4.3)
So replacing ?? in ?? we obtain:
s
y = A(T =L ) 1
1
so clearing out this equation we obtain again the steady state labor supply::
L = T (A=y )
1
s
so as we can see, the use of capital is not important in determining the outcomes of the
model.
A
4.2
The Industrial Revolutions
Certainly the Industrial Revolution represents a counterfactual theory regarding output
per person. Industrial revolutions represent a threshold episode in which mankind
abandoned what seemed to be prodominant malthusian dynamics. Is we shall see
later on, Jared Diamond sketched a powerful argument on why more tachnologically
59
CHAPTER 4 — MANUSCRIPT
advanced societies had to be found in Europe or Asia do to a Geographical advantage.
A more complicated question is why it happened in North West Europe and not China
or Japan? A suggest reading David Landes’s the Wealth and Poverty of Nations for a
historical discussion. Nevertheless, we can list some factors such as lower population
growth, tole of institutions and rewards to invention as boosting factors that lead to
solow dynamics.
As we did with the Solow model, we are interested in using Malthuses model,
it’s failures to explain why the western world, and today developing countries have
started to experience a substantial divergence. What explained the end of malthusian
dynamics and its income determinism.
A
4.3
From Malthus to Solow
Several theories have been developed in recent years. Many of them explain a tension
between number of children and investment in human capital. I will discuss several of
them.
Matteo Cervellati and Uwe Sunde (AER 2005) explain a novel mechanism.
They claim that as technology for increasing longevity improved, people had more
incentives to invest in human capital reducing the fertility rates which in turn fostered
greater increases in capital per worker and again higher returns to human capital fuelling
the process again. The argument can be tracked back to Kremer and Chen who argued
60
CHAPTER 4 — MANUSCRIPT
that the direct relation between income per capita and population growth rates are
valid for unskilled workers but not for skilled workers.
Jeremy Greenwood, Ananth Seshadri, Guillaume Vandenbroucke (AER 2005)
support the view that increases in technology provoked substantial increases in output
per worker that eventually lead to an increase in the opportunity cost of having children.
Neverthelless, the baby boom was explained by a sui generis episode in which household
technology increased and allowed for more children.
Oded Galor provides a striking theory. His basic claim is that there was a
self selection processes in technology that eventually allowed for the boost in human
capital and progress towards a Neoclassical dynamics, irregardless of the Malthusian
dynamics.
Finally, Prescott and Parente (2005) build a model in which Malthusian Dynamics and Solow dynamics coexist in a model with 2 sectors. They show that the
Malthusian sector, i.e. with a …xed land factor will always operate whereas the Solow
model appears only if certain conditions for the rates of return occur. They cheat a
little bit by assuming that population growth is a function of what model mechanics
are operating more strongly.
61
CHAPTER 4 — MANUSCRIPT
A
4.4
Determinants of the Fertility Rate
Becker, Murphy, and Tamura (1990) study the detiminants of fertility. Their model
features Parent’s decision about: Number of children and Education of children and
their basic goal is to study how these decisions relates to economic growth.
The model as constructed over several simpli…ying assumptions: agents live
for only two periods of life: Childhood and Adulthood. And only adults make decisions
about: dividing time between work, time with children, and educating children and how
many children to have. In that sense, the model endogeneizes fertility rates as opposed
to the malthusian assumption that people behave, more or less like animals.
Preferences are expressed recursively. The parents utility, Vt is determined
by:
Vt = u(ct ) + a(nt )nt Vt+1
with u an increasing and concave function of the parents utility and and a is
an functional form for altruism which is assumed to be decreasing or constant.
Time Endowmen T is spent in to work hours, lt ; per child spent time v
which is assumed …xed, and education per child ht
T = lt + nt (v + ht )
Human Capital, Income, and Education
Human capital is composed by a genetic endowment H 0 and a stock Ht . Total
62
CHAPTER 4 — MANUSCRIPT
human capital is obviously:
H 0 + Ht
Income is obtained through a linear production function:
lt (H 0 + Ht )
and the evolution of Human capital depends on the forllowing following formula:
Ht+1 = Aht (H 0 + Ht )
so there are diminishing returns to education.
Consumption is shared among children according to the following formula:
ct + nt f = lt (H 0 + Ht )
where f is spending per child.
Finally, altruism is de…ned as:
a(nt ) := Cnt "
B
4.4.1
Solution to the Model
The model is solved by optimizing utility, with the appropiate choice of nt and ht
63
CHAPTER 4 — MANUSCRIPT
B
4.4.2
Implications
Spending time educating children is more worthwhile for parent who is herself educated.
Thus a parent who is educated might …nd it more worthwhile to educate a
child, and choose to spend more time doing so.
Taking time to educate a child is more costly the more children a parent has.
Thus a parent who decides to spend more time educating each child may
choose to have fewer children.
B
4.4.3
Outcome I
If Ht starts out low enough a parent may choose ht = 0:
Such a parent may choose a large nt :
Then by the next generation Ht+1 = 0 from then and forever after.
The economy stagnates
B
4.4.4
Outcome II
If Ht starts out high enough parents may keep adding to the human capital of their
children.
The stock of Ht grows over time.
Parent will choose a smaller nt :
The analysis points to the complementarity between choosing a small number
64
CHAPTER 4 — MANUSCRIPT
of children and educating them more intensively.
65
CN
Chapter 5
CT
Determinants of Initial Conditions
We study the injustices of history for the same reason that we study genocide, and
for the same reason that psychologists study the minds of murderers and rapists... to
understand how those evil things came about.
Jared Diamond
A
5.1
Overview
So far in the course we have discussed Convergence, the role of Human Capital and the
Potential for multiple equilibria under neoclassical dynamics and Malthusian Dynamics.
In this lecture, we will discuss Jared Diamond’s main contribution.
[Place Diagram Here]
66
CHAPTER 5 — MANUSCRIPT
A
5.2
So bullets from the book
The Spread of Humans
Australia 40,000 BC
Americas 12,000-10,000 BC
Greenland 2,000 BC
Polynesia 33,000 BC - 500 AD
Isolation and technological development
Comparative Densities
The Moriori and Maoris on Chatham Island
– Chatham 5/sq. mile
– New Zealand 28 sq. mile
– Tonga, Amita 1100/sq. mile
The bene…ts of a temperate sojourn
Hunting/Gathering vs. Farming
Edible biomass: .1 vs. 90 percent
67
CHAPTER 5 — MANUSCRIPT
A¤ects population density by a factor of 10-100
Domesticated animals: food, power, weapons, vermin control, companionship
Survival of the …ttest vs. survival of the most useful.
Diamond’s Central Hypothesis
Why was there so much more innovation in some areas than others?
Why did innovations di¤use in some directions and not in others?
Why did the Eurasian land mass have a much denser population than the rest of
the world in 1492?
The Geography of Agricultural Innovation: The Fertile Crescent
Geological and Climatic Diversity
Available Seeds and Grasses
Eight founder crops: wheat (2), barley, lentils, peas, chickpeas, bitter vetch, ‡ax,
all by 800 BC.
The Geography of Agricultural Innovation: Northeast North America
Local crops: squash, sun‡ower, sumpweed (which stank; abandoned before Columbus), knotweed, maygrass, little barley.
68
CHAPTER 5 — MANUSCRIPT
pre-Columbian innovations from Mexico
– Corn (200 AD, improvement 900 AD)
– Beans (1100 AD)
– Led to much greater population density but still much less than Eurasia
The Animals
The Dog (10,000 BC) made it to the Americas
The 5 big herbivores
– Sheep
– Goat
– Cow/ox
– Pig.
– Horse
Also camels, llama alpaca, donkey, reindeer, water bu¤alo, yak, Bali cattle,
mithan
Extinction: some potentially domesticatable animals were killed o¤ by hunter/gatherers
before they could be domesticated.
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CHAPTER 5 — MANUSCRIPT
What does it take to make it as a domesticated species?
Don’t grow too slow (no elephants, gorillas)
Breed in captivity (no cheetahs)
Have a good personality (no zebras)
Stay calm
Follow the crowd (the cat exception)
The Di¤usion of Agricultural Innovations
Eurasia and the East-West Axis
The bene…ts of a temperate climate
The isolation of Australia and New Guinea
The Germ Factor and Immunity
Crowd diseases and density
Contact with animals
Writing
Two basic sources: Sumerian and Mesoamerican
70
CHAPTER 5 — MANUSCRIPT
Again, the importance of di¤usion
“blueprint copying”vs. “idea di¤usion.”
The role of social strati…cation.
What makes for an innovative society?
Cheap labor?
Property rights and patents
Institutions and attitudes toward risk
Tolerance of new ideas
East Asia vs. NE Europe: what was the di¤erence?
Issues
Government
Property Rights
Hierarchy and the Division of Labor Religion
71
CN
Chapter 6
CT
Technology Di¤usion
“Once a new technology rolls over you, if you’re not part of the steamroller, you’re part
of the road.”
Stewart Brand
A
6.1
Technology Adoption
Along the course we have studided how in most of the models, technology is a key
engine for growth. How is technology adopted, generated and di¤used along countries?
In this lecture we will study several versions of technology improvements to study under
what circumstances the growth in technology is the same, and the conclusions of the
neoclassical growth literature remain unaltered.
72
CHAPTER 6 — MANUSCRIPT
A
6.2
Technology Adoption and Multiple Inputs
We study a version of the endogenous growth model with multiple inputs along the
lines of work of Barro and Sala-i-Martin and Aghion and Howitt. Production is de…ned
by:
Y t = A1 L 1
N
X
Xi1
1=1
A technology in this context is interpreted as the number N of intermediate products.
The …rst order condition for production implies that:
(1
A1 L1
)
(1
Xi1
)
= Pi
where we assume that Pi is the price of the intermediate good X1i : This condition gives
us the demand equation for this input. We will assume that that technology …rms
invent input technologies, the use of the input grants a patent right for the production
of the intermediate good. For simplicity, we can assume that the marginal cost of
the input is 1: Becuase patents are grant monopoly rights, pro…ts stemming from an
innovation give as the innovator …rms problem once the innovation was produced. The
monopolist will maximize:
(1
AL1
)
Xi
(1
)
73
Xi1
Xi1 =
M
i
CHAPTER 6 — MANUSCRIPT
the FOC here yields the following equation:
AL1
)
Xi =
2
2
(1
Xi
(1
)
=1
which yields:
1
Ai
(1
)
Li
which is the condition for equilibrium input Xi : Then, we can replicate this result for
all the i inputs and obtain:
1
Yt = A1(1
2
1
)
N1 L t
and by this we obtain a per capita income by dividing output by popoluation:
1
yt = A1(1
)
2
1
N1
Note that output per capita is a linear function of function of N1 : The result here
sustains that holding every other variable as constant, per capita income will grow at
the same pace of N1 : Nevertheless conditions innate to the country may have an e¤ect
on output too! A1 captures this feature.
74
CHAPTER 6 — MANUSCRIPT
B
6.2.1
Innovation Process in the Model
What are pro…ts for the i
th intermediate good? We can go back to our …nding of the
optimal monopoly supply of the intermediate good and substitute back in the pro…ts
condition and we will obtain monopoly pro…ts:
M
i
=
1
1
A11
2
(1
)
L1
We can assume that the economies can borrow abroad, namely that the interest rates
are not a¤ected by the countries decision. Investment in a particular innovation requires
a …xed amount of capital K: Some thoughts are worth metioning: …rst, it is important
to determine how costly it is to innovate in a new line of production: Two e¤ects that
work in opposite, …rst it is likely that ideas face diminisihing returns. As di¤erent
processes are discovered it is harder to …nd new and di¤erent innovations. On the
other hand, as ideas are being created they help in …nding new ideas, they inspire new
projects or provide technologies that are later on used for other technologies that won’t
work if the previous steps are not done.
Investments can be either succesful or not, and this basically will responds to
a random process, Bernoulli trials if the probabilities of discoveries are independent.
Thus, investment in technology will have a risky component and this leads us to ask
ourselves questions of risk aversion (or knightian uncertainty even!). We will abstract
from an equilibrium concept in the process of technology and we will assume that the
75
CHAPTER 6 — MANUSCRIPT
following condition holds:
E
M
i
(QK)
QK
QK
E
rt
M
i
(K)
K
K
;Q > 1
This condition expresses a limit for the quantity of investment per period in technology.
This assumption is reasonable, in every period of time there is a limited number of
things that one can do. So the model is constructed in such a way that only one unit
of investment is pro…table in a given period.
B
6.2.2
Immitation
We assume for analogus reasons that there is a function which we call V
N2
N1 N2
that
measures the cost of immitating a technology and it depends crucially on the di¤erences
in the number of products in di¤erent countries N1 and N2 : Here we assume that N1
is the level of technology in the second country and N2 the level of technology the
imitating country’s …rms want to adopt. To motivate this assumption we think og the
same issues in the discussion of pure innovation. A condition for immitation rather
than innovation is to have:
rt
M
i
E
V
N2
N2
N1 N2
and
76
1
CHAPTER 6 — MANUSCRIPT
E
M
i
(K)
K
M
i
E
K
V
N2
1
N2
N1 N2
and since pro…ts are linear the …rst condition insures pro…tability.
It may be the case that for a particular gap (N1
N2 ) ; it may not be prof-
itable to immitate and therefore the country will only have the option to innovate.
Nevertheless, it can also be the case that the country does not innovate at all. because
the condition is not satis…ed do to particular conditions:
rt
E
M
i
(K)
K
K
[INCLUDE GRAPHIC HERE]
We can use a particular choice of functional forms and use a program to …nd
when …rms immitate and …rms innovate. In any case, as long as V is not to "evil"
we there will always be a follow up from the imitators. If conditions are suitable, the
model predicts catching-up and falling behind of leaders.
A
6.3
Innovations in the Quality of Goods
This section follows a version of Krugman (JPE 1979) which is a precursor of the
multicountry endogenous growth models.
There are two countries: N; S with corresponding labor forces: LN ; LS : The is
77
CHAPTER 6 — MANUSCRIPT
a total n of goods: n = nN + nS : Technologies in worker requirements are summerized
by the following matrix:
nN nS
N
1
1
S
1
1
so the basic idea is that N is able to produce any good available and S only those among
the nS set.
Preferences CES with elasticity of substitution
U=
"
n
X
(
ci
1)=
i=1
#
=(
>1:
1)
which expresses love of variety. under some constructions price p; will have
the following structure:
ci =
w
pn
for all i so that
U=
w 1=(
n
p
1)
rises with n:
Additional assumptions of the static model are used: Market Structure is of
Perfect Competition. Since it takes one worker to make one good, any good made in
N costs the wage wN there and any good made in S costs the wage wS there.
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CHAPTER 6 — MANUSCRIPT
B
6.3.1
Static Equilibrium
Relative demands for typical goods:
cN
=
cS
wN
wS
where cN is consumption of a typical N good and cS is the consumption of a typical S
good.
Focus on complete specialization, requiring wN =wS > 1:
Relative labor demands:
cN nN
LN
=
=
LS
cS nS
wN
wS
nN
nS
so that:
wN
=
wS
LN =nN
LS =nS
1=
which we require to exceed 1, meaning that there are relatively more N goods than
workers
B
6.3.2
Dynamics
Innovation is again exogenous and satis…es the following equation:
nt+1 = (1 + i)nt
where i is the rate of innovation. Di¤usion:
nSt+1 = nSt + nN
t
79
CHAPTER 6 — MANUSCRIPT
where
is the rate of di¤usion so that:
N
nN
t+1 = nt
nN
t + int
)nN
t + int
= (1
De…ne vt = nN
t =nt so that:
vt+1 =
=
=
nN
t+1
nt+1
nN
t
nN
t + int
(1 + i)nt
(1
)vt + i
(1 + i)
Steady state v doesn’t change so that:
=
i
:
i+
Hence in steady state:
nN
i
=
S
n
which has to exceed LN =LS if N is to stay ahead.
A
6.4
Limits to Technology Adoption
This section borrow’s from Kremer AER 93 O-ring Production Function theory and
adapted from Boyan Jovanovic’s class. We do the case of no capital and exogenous team
size N. A team consists on a group of workers of di¤erent size and quality. Output is
80
CHAPTER 6 — MANUSCRIPT
y=
N
Y
qi
i=1
where qi is the quality of worker i. Let (q) be wage of workers with quality
q. The …rm’s problem consists on maximizing pro…ts.
Assume that …rms must have all its workers of the same quality, we will go
back to this assumption later on. So its problem is just what quality of team to choose
maxfq N
(q)
(q)N g
q
the necessary FOC for this problem is
qN
0
1
(q) = 0
and the SOC:
(N
00
1)q
(q)
0
Finally, since all the workers must be employed, …rms must be indi¤erent
between what quality of team to choose and so we impose that they should be zero
(q) = 0
A wage function (q) will satisfy the above conditions .In particular, the pro…t
81
CHAPTER 6 — MANUSCRIPT
maximization condiotion will satisfy the following:
qN
1
=
0
(q)
and imposing the condition that workers of no quality have no contribution
(0) = 0. The solution is
qN
= (q)
N
B
6.4.1
Product complexity: Endogenizing N
Here we suppose complex products yield more utility. Let p (N) be the willingness to
pay for a product produced by an N-sized team. Assume that p(N ) is per task, so that
total willingness to pay is N p(N ). This is a property of preferences and it is
assume ad hoc. Then the …rm’s problem is now:
maxfN p(N )q N
n;q
(q)N g
This is not entirely correct but we can N as a continuous variable, and simply
use standard methods in calculus to approximate a solution. Recall the following result
from calculus:
@ax
= ax ln a
@x
82
CHAPTER 6 — MANUSCRIPT
The …rst FOC is for q in the …rm’s problem is:
N p(N )q N
1
=
0
(q)
if we are also to have zero pro…ts, we must now have for all q
[0, 1]:
(q) = p(N )q N
Then the …rm’s FOC for N is just:
p(N )q N + N p0 (N )q N + q N ln q
(q) = 0
we can replace in this function the value for (q) to obtain:
p(N )q N + N p0 (N )q N + N p(N )q N ln q
p(N )q N = 0
Cancelling q N :
ln q =
p0 (N )
p(N )
Therefore N depends on q increasingly. Note that we have not checked the
second order conditions, which require to study cross derivatives of the problem. Nevertheless we will assume that p(N ) guarantees this. Therefore, suppose that:
83
CHAPTER 6 — MANUSCRIPT
p(N ) = N ;
<1
so that
p0 (N )
=
p(N )
N
N
1
=
N
and a direct condition is obtained:
N=
ln q
which is increasing in q, so that better workers work on more complex products. An interesting feature of the model is that the complexity of the process will
depend on the quality of workers. Things a¤ecting the quality of workers will depend
on other issues such as education, health and other institutions. Highly complex products will require highly skilled workers so the choicy of technologies will depend on the
availability of workers. Note that here we have assumed that good workers are paired
with good workers. Indeed this will be the case because of the structure of production
we designed. Some graphs will help us undertand what is going on.
[Distribution and choice of technology]
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CHAPTER 6 — MANUSCRIPT
The main conclusion is that technologies may not be adopted, even if they are
available, if the human resources or quaility of workers are not a good enough match
for a particular choice.
85
CN
Chapter 7
CT
The Role of Governments: Size and
Evasion
A
7.1
Notes on Multiple Equilibrium in the Size of
Tax Evasions
These notes are based on a simple model constructed by Eduardo Zilberman.
Representative consumer problem:
Normalize wages w = 1.
max
c;n
c1
1
B
n1+1=
1 + 1=
s.t. pc = n
86
CHAPTER 7 — MANUSCRIPT
FOC in n (substitute budget constrain and take derivatives w.r.t. n):
p1
= (1=B)n
1=
(7.1)
This equation relates the optimal amount of labor supply to prices (notice that this is
not the labor supply, once it is not a function of w but p).
Firms’problem:
Continuum of …rms in [0; 1]. Given taxes , prices p, and wages w = 1, …rms
choose an percentage ^ of the quantity produced to pay as taxes, and an amount n to
employ.
max(1
c;^
^)pf (n)
n
(n)v((
^)pf (n));
where f (n) is the production function, (n) is the probability of being caught weakly
increasing in n, v(:) is a penalty function with the property that v(x) > x. Notice that
(
^)pf (n) is the amount evaded.
We choose the following parametric forms: f (n) = An , and (n) = 1n>n ,
where 1 is an indicator function (this is a strong assumption). As it is going to be clear
below, the parametric form chosen for (:) implies that the penalty function can be
anything as long as v(x) > x. Thus, although unrealistic, (n) = 1n>n simpli…es a lot
the math.
Let’s solve the …rms’problem. First, we …x n, aiming to …nd multiple equilibrium. Since all …rms are equal, either all of them evade, or none of them evade. So we
87
CHAPTER 7 — MANUSCRIPT
have two possible equilibrium: one with full informality (associated with
one with full formality (associated with
= 0), and
= 1).
Case 1: low enforcement ( = 0) - high evasion (^ = 0).
Let’s assume
e
= 0, where the superscript e denotes expected. Thus, it
should be clear that ^ = 0. Therefore, …rms
max pAn
n
n
FOC in n:
pA n
e
It de…nes a labor demand when
1
(7.2)
=1
= 0.
Imposing market clearance, and solving equations (1) and (2), one …nds n0
and p0 . Rational expectations require that
n0
e
=
= 0, so we need
n
Case 2: high enforcement ( = 1) - low evasion (^ = )
Let’s assume
e
= 1. Thus, it should be clear that ^ = . Therefore, …rms
max(1
)pAn
n
n
FOC in n:
(1
It de…nes a labor demand when
e
)pA n
= 1.
88
1
=1
(7.3)
CHAPTER 7 — MANUSCRIPT
Imposing market clearance, and solving equations (1) and (3), one …nds n1
e
and p1 . Rational expectations require that
=
= 1, so we need
n1 > n
Possibility of multiple equilibrium:
Notice that as long n0 < n1 , any exogenous n 2 [n0 ; n1 ) sustains multiple
equilibrium. The purpose of this note is to …nd conditions on the parameters of the
model that sustain multiple equilibrium.
Calculating n0 ; n1 :
Plugging (3) in (1),
(1=(1
)A n
1 1
)
= (1=B)n
1=
Solving for n, one gets
n1 = X(1
where X = (1=B)(A )1
)(1
)=[(1
)(1
)+ +1= ]
.
Similarly,
n0 = X
Thus, notice the possibility of multiple equilibrium arise when n0 < n1 , i.e,
(1
)(1
)=[(1
)(1
89
)+ +1= ]
>1
CHAPTER 7 — MANUSCRIPT
Since 1
< 1, a su¢ cient condition for this is that
(1
)=[(1
)(1
)+
Let’s re-write the denominator of (4): 1
as long
> 1, since 0 <
< 1, and
(7.4)
+ 1= ] < 0
+
+ 1= . Moreover, notice that
> 0 by de…nition, (4) is automatically satis…ed.
Thus, all we need to generate multiple equilibrium is that
> 1, i.e., the
income e¤ect must dominate the substitution e¤ect.
Intuition: high enforcement ! low evasion ! higher n ! high enforcement
The …rst arrow is from …rm’s optimal decision of how much to evade.
The second arrow comes from the fact that under low evasion, …rms pay
more taxes. Given market clearance in the good markets, the equilibrium price p will
be higher (see …gure 1). Once price is higher, the relative wage (1=p ) will be lower.
Assuming
> 1, the income e¤ect dominates, inducing a negative relationship between
relative wages and labor supply from the consumer’s optimal condition (1). Thus the
higher price p is associated with a higher equilibrium n (see …gure 2).
[width=250pt]FigMultEquil1.pdf
[width=250pt]FigMultEquil2.pdf
Finally, the third arrow implies that for an appropriate choice of n, indeed
n > n.
90
CHAPTER 7 — MANUSCRIPT
Thus, this model generates multiple equilibrium: one with high enforcement/low evasion/high employment and the other with low enforcement/high evasion/low employment.
Possible extensions:
(I) Add heterogeneity among …rms in order to get more plausible equilibrium
instead of only "corners’", i.e, either everybody is formal or every is informal.
(II) Endogeneize n through a political economic model in order to check if we
can sustain multiple equilibrium.
(III) Write a dynamic version of this model in order to adapt this multiple
equilibrium model into a model of multiple steady states, in which initial conditions
are relevant. Here, I can borrow the ideas from Hopenhayn (1992) and Hopenhayn and
Rogerson (1993) and use numerical methods to solve it.
91
CHAPTER 7 — MANUSCRIPT
92
Part II
International Crisis
93
Part III
Microeconomic Issues
95
CN
Chapter 8
CT
Issues in Agriculture
A
8.1
Agriculture Reforms
The Government of Zimbabwe has recently undertaken big reforms in agriculture. That
government is expropiating land from "sophisticated" white farmers and redistributing
land to a bigger number of black "unsophisticated" farmers. The reform is aimed at
a better distribution of the resources but most likely at an important e¢ ciency loss
this policy. Earlier historical reforms were carried out in South Korea, Cuba, Japan,
Taiwan and Mexico.
The most important problems of land is that total productivity may depend
on size do to several factors:
Mechanization is only pro…table at a large scale.
97
CHAPTER 8 — MANUSCRIPT
The rotation of Land is more expensive.
Free riding problems may lead to inne…cient use of pesticides that afect other
crops.
Decentralization of Information may become an important problem.
Risk aversion may have implications on experimentation in new more e¢ cient
crops.
When countries such as Peru undertook similar policies in the late sixties,
the Military Junta divised this problem after a …rst block of privatizations, and to
solve at they created cooperatives which are more or less behaved as Land Pool’s. The
big problem of shared ownership is again free riding, decision taking and other. This
section is not about whether redistribution is good or bad. It is not either about the
scale e¤ects discussed above which seem straight forward arguments. It is more about
how redistributing land, and dividing it into small land slots may complicate it’s resale.
Nevertheless several authors have claimed that imperfections in the labor market may imply that massive ownership can be infact a good policy do to a Marshallian
argument.
98
CHAPTER 8 — MANUSCRIPT
A
8.2
Agriculture Reform Reversals
This section explores the possibility of land reversals. We will discuss three problems
that were caused by the land reforms. We will …rst note that the sale of land can be
problematic by to potential explanations. An adverse selection problem as in Akerlo¤’s
"market for lemmons" and by costly repeated bargaining with multiple minor land
owners. The third section will explain how smaller farms will be less likely to obtain
credit, form projects of similar pro…tability.
B
8.2.1
Adverse Selection
Here we addres the question of whether land reform can be reverted through market
mechanisms such as free purchases. Suppose that for 30 years after a reform, land
is now allowed to be reselled to sophisticated investors. Time has passed by and no
records of the productivy of a land slot are available and neither are any records on
how the land was treated. Land, as you know can be erotioned by a misuse of water,
pesticides and negative rotation.
We will also assume that land is held by "unsophisticated" land owners and
may be potentially bought by "sophisticated" land owners.
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CHAPTER 8 — MANUSCRIPT
C
Output and Utility
We abstract from production functions and labor problems and simple focus in the
value of land as function of quality. We use Z to refer to the quality of land.
We assume that Z is distributed uniform in the unit interval:
Z~U [0; 1]
and that current present value of land is a function of quality. We assume that the
current present value for the unsophisticated farmers are:
Z
and that we for the sophisticated farmers it is equal to:
Z
and we assume that
C
< :
Land Sales
An unsophisticated farmer will sell land if and only if the price of this sale P; satis…es
the following condition:
Z <P !Z<
100
P
1
CHAPTER 8 — MANUSCRIPT
Note that not all the values of Z 2 [0; 1] will satisfy this condition depending
on
P
1
: Therefore, it may be the case, that only land that is not e¢ cient is reselled.
When will the the sophisticated farmers buy? We assumed that the quality of land
cannot be detected by the sophisticted land owner. The condition is needed to satisfy
a pro…table purchase is:
E [Z jZ is sold by unsophisticated]
PK
where E refers to the expectation and P is again the price and K > 1 is a constant
that measures the opportunity cost that amount of money. We are implicitly assuming
that the opportunity cost of the buyer is greater than that of the seller because of the
sophistication argument. Let’s compute the expectation to …nd out what prices could
guarantee a transaction and what "land qualitites" are sold. For simplcity let’s de…ne
the auxiliary variable x =
P
1
Note that this expecation is a conditional so if the
density function was:
Z for U [0; 1]
it is now:
1
for U [0; x]
x
. The density conditional expectation is obtained by the following integral:
101
CHAPTER 8 — MANUSCRIPT
Zx
E [Z jP; ] =
1
1
Z dz =
x
x
Z dz
0
0
1 Z +1 x
j =
x +1 0
=
Zx
x
+1
and replacing the value of x we obtain:
(P )
( + 1)
Therefore the condition to sell will be:
(P )
PK
( + 1)
which in equality gives:
P
= K ( + 1)
This is the equilibrium price that guarantees transactions in the Land Market. We now
turn to look at which land is sold. By the condition for sells:
Land will be sold if:
Z <
=
=
1
P
1
2
(
1
1
K ( + 1)
)
(K ( + 1))
102
1
CHAPTER 8 — MANUSCRIPT
This gives a result for the land that is sold. We can focus on a special case to observe
what happens only depending on
and : Set K = 1;
=
1
2
and the condition gives
us:
Z<
1
2
2
(
)
( + 1)
1
Here we could plot the value of Land that will be selled. More importantly, we can plot
the Welfare loss. De…ne Z as the one that satis…es the above condition with equality.
C
Welfare implications
Total loss will be the computed as how much could be produced if the fraction of land
that will not be in the hands of the "sophisticated" farmers were indeed in the hands
over total output:
Total Output=
ZZ
Z dz +
Z1
[Z
0
Z1
Z dz
Z
and the amount lost is:
Social Loss=
Z ] dz
Z
by computing this simple integrals we can compute the following:
103
CHAPTER 8 — MANUSCRIPT
Social Loss as % of
R1
Production= ZZ
R
0
A
8.3
[Z
Z dz +
Z ] dz
R1
Z dz
Z
Repeated Bargaining
A second reason that complicates the reversal is the fact that a sophisticated farmer that
has the capacity to produce at a large scale has to negotiate with many micro-farmers.
We explore this problem through the solution of the Bargaining problem discovered by
Ariel Rubinstein.
B
8.3.1
Rubinstein’s Solution n=1
The game has the following structure. A sophisticated farmer has to pay a …xed cost
to study where to invest. Suppose the …xed cost is F. When the sophisticated farmer
comes to a single unsophisticated farmer, he has to negotiate with him about a price.
O¤ course, the price cannot exceed the di¤erence in present value of the land, and
cannot be an amount lower. The point here is that there are two individuals that will
bargain about a price. The …nal outcome will be that the price is a function of the pie
total pie the that is to be shared. Let V denote the di¤erence in the total amount that
is to be bargained over. The point is that V is the di¤erence in the net present value
of the land to both farmers.
104
CHAPTER 8 — MANUSCRIPT
The bargaining institution has discounting. That is, the farmers will meet to
bargain, and they if they don’t reach an aggreement, they will discount the value by
: This re‡ects a prference for early resolutions. The protocol is that the sophisticated
farmer will knock on the unsophisticated farmers door to bargain and bargain, if no
agreement is achieved, then the unsophisticated farmer will go to the sophisticated and
so on. We now de…ne two additional objects:
V1 be the best outcome for the sophisticated farmer
and let
V1 be the worst outcome for the sophisticated farmer
¯
Because the game is identical in the next period. Two conditions will be satis…ed:
V1
V
V1
¯
V1
¯
V
V1
This conditions explain the following. Because, V1 cannot exceed V
V1 ; because this
¯
is the worst outcome for the unsophisticated farmer in the second period if there is no
agreement: the worst outcome he will accept cannot be less that the worst outcome
next period of the sophisticated farmer. The reason is that next period, he will knock
105
CHAPTER 8 — MANUSCRIPT
on the sophisticated farme’s door. The second condion will be analogous to the …rst
one. We therefore look at the following. Rearranging the second condition yields:
V1 + V1
¯
V
Because of this, we can replace the RHS of this inequality into the …rst inequality and it will still hold. We obtain:
V1
V1 + V1
¯
V1
¯
which in turn implies:
V1 (1
¯
)
V1 (1
but since by construction we have that V1
¯
)
V1 it better be the case that the two
amounts are equal: V1 = V1 : Using this equality, in both inequalities yields:
¯
V1
V
1+
V1
V
1+
and
So our only solution is:
106
CHAPTER 8 — MANUSCRIPT
V1 =
V
1+
and share of the pie given to the unsophisticated is obviously:
V1 =
V
1+
Not that, regardless of the fact that the di¤erence in wealth is generated by the sophisticated farmer, the pro…ts are shared by both, and interestingly, for
close to 12 ; these
are shared almost evenly.
Now, the …xed cost of the project will be taken if, a priori, the research satis…es
the following condition:
K
B
8.3.2
E
V
1+
Rubinstein’s Solution for arbitrary n
We now look at the same problem but assuming that land is divided into two slots.
Both of which, together sold are valued V, to the sophisticated land owner.We assume
that he has to negotiate with each unsophisticated farmer at di¤erent stages in which
time is discounted
107
CHAPTER 8 — MANUSCRIPT
C
N=2
Assume that he already bought the …rst slot. Then the sophisitcated farmer will negotiate and the results will be as before:
V
V
;
1+ 1+
This is in the last stage. In the previous stage, the …gure is again repeated, and we
obtain:
V
V
;
(1 + ) (1 + )
Note that V is di¤erent from V: What is the relation of this values? They will satisfy
the following relationship:
V
=V
(1 + )
so the …nal outcome will be:
V
V
V
2;
2;
(1 + ) (1 + ) (1 + )
Notice that this fractions add up to 1.
V
V
V
1 + + (1
=
2 +
2 +
(1 + )
(1 + )
(1 + )
(1
)2
108
)
=1
CHAPTER 8 — MANUSCRIPT
C
N=3
Following the previous steps we obtain that the shares will be:
V
V
V
V
3;
3;
2;
(1 + ) (1 + ) (1 + ) (1 + )
C
The N case solution
The Return for the sophisticated farmer will be:
V
(1 + )N
so the condition:
K<E
V
(1 + )N
is less likely to be satis…ed the bigger the number of farmers he has to negotiate with.
C
Some little pies to show how the Cake is Shared
[Insert Pies Here]
A
8.4
Credit Rationing
An interesting point here is that of credit rationing. We will see what determines the
size of loans. As we shall see, the size of a particular loan will be a function of total
109
CHAPTER 8 — MANUSCRIPT
assets that may be used as collaterals for the loans. The point to be made here is that
lending may be rationed to the agriculture sector when the scale is not big enoguh do
to a problem of Moral Hazard. This version of the model is the simplest version of
Townsends 1979 JET paper and can be found in Jean Tirole’s book.
B
8.4.1
The Model
Assume that a farmer has a collateral of value A, say in terms of machines or future
secure production. He has a project which is risky, and requires an amount of loans
equivalent to I. The gross return of the project can be of value R; with probability
A
if he undertakes a "secure" or "responsible" investment. The return will the same
value R; with probability
B
<
A
if he undertakes a more "risky" or "irresponsible"
investment but will secure a value B for his pockets.
We assume that lending is pro…table only if the correct actions are taken:
AR
I>0
but not in the other case including the borrower’s pro…t:
BR
I +B >0
In this setup, no lending contract will be granted.
Competition in the credit market implies that banks ask for an amount RL if
110
CHAPTER 8 — MANUSCRIPT
the project is veri…able and succesful:
A RL
=I
A
The loan has to satisfy a certain compatibility constraint which explains that there
should not be any incentives to undertake the wrong policy. This condition requires
the following inequality to hold:
A
(R
RL
A) >
B
(R
RL
A) + B
So the highest amount of return will satisfy:
B
R
(
= RL
B)
A
A
The right hand side is the return the loaner will have if the investment is pro…table.
Therefore, the individual rationality constraint, or a (0 pro…t condition states that):
A
R
B
(
I
B)
A
A
Therefore loans will be granted if and only if:
A
I
A
R
B
(
A
B)
is satis…ed. Note that this condition means that there could be loans that are still
111
CHAPTER 8 — MANUSCRIPT
pro…table, but are not granted. The idea is that the size of the collateral can be a
problem for certain loans. The model predicts that small farms will be more likley
to be credit rationed. In this setup, there is no rationale for government intervention
in terms of granting loans to the farm sector. In any case, the government will loose
resources. Other than penalizing inmoral behavior or risk taking, the model does not
justify credit by the government to farmers.
112