topics in the global economyct preface i have prepared this notes for the course of topics in the...

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? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Growth Accounting and Empirical Evidence 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


4 Malthusian Models 54

4.1 The Malthusian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1.1 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

4


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 . . . . . . . . . . . . . . . 82

7 The Role of Governments: Size and Evasion 86

7.1 Notes on Multiple Equilibrium in the Size of Tax Evasions . . . . . . . 86

II International Crisis 93

III 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 interac-

tions 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

3

CHAPTER 1 � MANUSCRIPT

economists. We begin our lectures with the study of the famous Solow-Swan Neoclas-

sical 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) accu-

mulation, 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 pre-

dictions 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 Eco-

nomic Policy of many governments. Therefore, Solow�s model has had a role on the

shape of things today.

4


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 = AtF (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

5


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 = sAtF (Kt; Lt) (1.3)

Ct = (1� s)AtF (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.

6


Exogenous Growth Rates

At+1 = (1 + xt)At (1.4)

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

F (Kt; Lt) := K�t L

(1��)t (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

7


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

LtThen deviding both sides of 3.4 by

Lt and replacing in 1.3 yields:

kt+1 = (1� �) kt + sAtK�t L

(1��)t =Lt

= (1� �) kt + sAtk�t

so if we look at the gross growth of machines per worker we have:

kt+1 � kt = sAtk�t � �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:

8


[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 = sAtk�� k�

Clearing out this equation yields:

k� =

�sAt�

� 11��

(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 accumu-

lation 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 := YtLtas output

per capita, or GDP per capita. From equation 3.3, when dividing both sides by Lt we

obtain:

9


yt = Atk�t (1.8)

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

�sAt�

� �1��

= A1

1��t

�s�

� �1��

B 1.4.2 TheModel�s Predictions (Steady State Comparative Sta-

tics)

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

10


ct :=CtYtwhich we can easily back out from production per capita:

c� = (1� s)A1

1��t

�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 opti-

mizing over the savings rate. We take the derivative of ct with respect to the savings

rate s and set this to 0:

@ct=@s = �A1

1��t

�s�

� �1��

+ (1� s)A1

1��t

�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

11


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

Lt+1= (1� �) Kt

Lt+1+ sAK�

t

L1��t

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� �) kt1

(1 + n)+ sAk�t

1

(1 + n)

and by multiplying both sides by the growth scale of population:

(1 + n)kt+1 = (1� �) kt + sAk�t

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

12


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:

k� =

�sAt� + n

� 11��

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.

y� = A1

1��t

�s

� + n

� �1��

and in the same way consumption per capita:

c� = (1� s)A1

1��t

�s

� + n

� �1��

This result found in theoretical models lead to several policy recommenda-

tions. World Banks policies towards birth control is one example. During the sixtees,

a standard policy recommendation was to apply birth control programs. China estab-

lished the one child policy under a similar philosophy of the model. As we shall discuss

13


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 "irra-

tionality" 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 suggest-

ing 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

14


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:

15


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+1Lt+1: We obtain:

Kt+1

At+1Lt+1= (1� �) Kt

At+1Lt+1+ sAtK

�t

L1��t

At+1Lt+1

and using the same trick we had before of multipliying and deviding by A�t Lt where

needed we obtain:

Kt+1

A1+at+1Lt+1= (1� �) Kt

A1+at Lt

A1+at Lt

A1+at+1Lt+1+ sAtK

�t

L1��t

A1+at+1Lt+1

we de�ne an auxiliary variable kt := Kt

A1+�t Ltand we obtain:

kt+1 = (1� �) kt1

(1 + x)1+a (1 + n)+s

A�tk�t

1

(1 + x)1+� (1 + n)

which yields:

(1 + x) (1 + n) kt+1 = (1� �) kt + sA�at k�t

Take an initial technology level Ao then this becomes:

16


(1 + x) (1 + n) kt+1 = (1� �) kt + sA�a0 (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) kt+1 ~= (1� �) kt

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

17


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.

18


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

� 11��

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.

19


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 KAo KBo

Comercial Balance rBt -rBt

Current Account -rBt +rBt

Wages FAl FBl

Capital Rent FAk FBk

Demographics LA LB

Assume: KAo <KBo and all other variables constant and indentical among

countries at the initial period..Then, by non-arbitrage conditions we have:

20


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:

KAt = K

EAt �Bt

where Bt is the net amount of capital transfers and KEAt 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:

KBt = K

EBt +Bt

and from these two conditions we will obtain the following:

Bt =KEBt �KEA

t

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 Modigliani-

Miller Theorem applies. It does not matter how are we �nancing Bt be it through

21


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:

Y At = F�KEA; LA

�and it is distributed according to the following function:

Y At = wL+ rKA + rBt

so in net terms of value:

Y At = CA + IA +XA �MA

where we set MA = 0 and XA = 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 invest-

ment is constant function of disposable income, and depends on the marginal propensity

to save s:

IA = s�Y At �XA

�22


The evolution of capital we be an analog to the evolution of capital in the

closed economy version of the model:

KAt+1 = (1� �)KA

t + s�wL+ rKA

�KBt+1 = (1� �)KB

t + s�wL+ rKB

�and

r = FK�KEA; L

�= FK

�KEB; L

�and wages equal to:

w = FL�KEA; L

�= FL

�KEB; 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 Coun-

tries 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


Yt = AtK�t L

1��t (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+1 �Kt = it = sAtK�t L

1��t � �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 = �AtK��1t L1��t + (1� �)K�

t L��t + _AtK

�t L

1��t

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


still holds. If we devide this equation by Yt; by using the de�nition of output we will

obtain:

_Y

Yt= �

_K

K+ (1� �)

_L

L+_AtA

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

_YYt;_KKand _L

Lsome approximation of _At

A; The main question now is to determine what

is � because if we have this, we use some arithmetic manupulations to determine _AtA;

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:

�YrKt

= rt ! � = rtYtKt

and an analog can be otained from the labor market�s analog:

(1� �) YrLt= wt ! � = 1� wtLt

Yt

Both, rt and wt are expressed in real terms.For reason regarding the quality of estimates,

26


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 wtLtYt

which is the fraction of wage revenues of total

output.

In his 1957 paper, Solow came up with this decomposition. The term _AtAis

the so called Solow residual, and is obtained as:

_AtA=_Y

Yt� �_K

K+ (1� �)

_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


,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


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


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 Neo-

classical production function of diminishing returns to scale in capital per worker tech-

nology:

31


yt = Atk�t (3.1)

and take the production Taking derivatives we obtain the rate of return of capital of a

given economy:

rt = �Atk��1t

so we can clear out 3.1 we obtain:

rt = �A1=�t y

��1�

t (3.2)

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

rptrust

=

�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


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 theModel to Labor Augmenting Tech-

nology

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 HtLt 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


capital Kt; and labor HtLt:We describe this by the equation:

Yt = AtF (Kt; HtLt) (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; HtLt) := K�t (HtLt)

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


previous lectures. Note that all that has changed from this setup and the Neovlassical

setup is that we�ve replaced the term Lt for HtLt now. Therefore, if we devide the RHS

of 3.3 by the e¤ective units of labor HtLt we would obtain the following:

yt = Atk�t

where kt is Kt=HtLt: 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:

rt = �A1=�t y

��1�

t

Because human capital is greater in the US than in developing countries, the ratio we

had before turns out to be smaller

35


rptrust

=

�yP

yUS

��1:5

=

�yP

yUS

��1:5�HUSt

HPt

��1:5Plugging 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!

rptrust~=

�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 = sAtF (Kt; HtLt) + (1� �)Kt

36


Assuming we have �gures on capital stocks, labor and output, assuming sim-

ilar 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+1Ht

; 1 + n =Lt+1Lt

so we devide again by the e¤ective units of labor to obtain:

kt+1 = sAtk�t

1

(1 + g) (1 + n)+ (1� �) kt

1

(1 + g) (1 + n)

so we have:

kt+1 (1 + g) (1 + n) = sAtk�t + (1� �) kt

which yields the following :

kt+1 � kt = sAtk�t +� (� + g + n+ ng) kt

As in our previous manipulations, we aim at �nding the steady state value of this

equation. We obtain the following:

37


k� =

�sAt

(� + g + n+ ng)

� 11��

If we replace this value into the original production function we would obtain

the following form for output as a function of the primitives:

y = A1

1��t

�s

(� + g + n+ ng)

� �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

�YtLt

�� log (Ht) =

1

1� � log (At)

+�

1� � log (s)��

1� � log (� + g + n+ ng)

Now notice that Ht can be written as Ht = (1 + g)tH0; we obtain:

38


log

�YtLt

�=

1

1� � log (At) + z + t log(1 + g)

+�

1� � log (s)��

1� � log (� + g + n+ ng) + "

To get exactly Mankiw, Romer and Weil�s table you can set z + 11�� 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

�YtLt

�= a+ gt+

�

1� � log�ItYt

��

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 � be the same in the mining industry than

in the manufacturing industry and so on?.

39


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


function be now:

F (Kt; HtLt; Zt) := K�t (HtLt)

1�� Z�t

where Z�t is the externality produced by human capital. The model is identical

to the one we had before with the singularity that:

Zt+1 = shYt +

�1� �h

�Zt

so in addition to this equation, we have the standard �sical capital accumulation equa-

tion:

Kt+1 = sYt + (1� �)Kt

From this equations, steady states should satisfy the following:

Z� =sh

�hY

and again:

K� =s

�Y

so the model�s basic claim is that in steady state both forms of capital should have a

proportionality factor.

41


In addition, we focus on the steady state level of e¤ective human and physical

capital to obtain:

Zt+1Ht+1Lt+1

= shK�t (HtLt)

1�� Z�tHt+1Lt+1

+�1� �h

� ZtHt+1Lt+1

which yield:

zt+1 = shk�t z

� 1

(1 + n) (1 + g)+�1� �h

�zt

1

(1 + n) (1 + g)

and rearranging, in steady state we obtain:

z =shk�t z

��h + n+ g + ng

�and by an analogous argument we can obtain:

k =skk�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 =

shk�t�

�h + n+ g + ng�! 1

1��

we can replace these equation in the later and obtain:

42


k =skk

�1��t�

�k + n+ g + ng� sh�

�h + n+ g + ng�! �

1��

and clearing k we obtain:

k1��1�� =

sk��k + n+ g + ng

� sh��h + n+ g + ng

�! �1��

and if we assume �k = �h :

k =

�sk� 1��1��

�sh� �1��

(� + n+ g + ng)1

1��

and with respect to h; we can the analog by a symmetry argument:

~h =

�sh� 1��1��

�sk� �1��

(� + n+ g + ng)1

1��

we may again substitute these results into the output equationa and obtain the follow-

ing:

y = A

24 �sk� 1��1��

�sh� �1��

(� + n+ g + ng)1

1��

35� 24 �sh� 1��1��

�sk� �1��

(� + n+ g + ng)1

1��

35�

= A

24 �sk� �1��

�sh� �1��

(� + n+ g + ng)�+�

1��

35Under the same assumptions of the previous section, and taking logarithms to both

sides we obtain:

43


log

�YtLt

�= a+ gt+

�

1� �� log�ItYt

�+

�

1� �� log�sh�� + �

1� �� log (� + g + n) + "

Notice that with this setup, investments in physical capita, increase, throuqh an inter-

action 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

1� � log�sk�+

�

1� �� 1

1� � log (� + n+ g + ng)

so replacing this we obtain:

log

�YtLt

�= a+ gt+

�

1� � log�ItYt

�+

� �

1� � 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


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 Impli-

cations on Conditional Convergence:

A 3.4 Why then is there no more investment in Hu-

man 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 inde-

pendent 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 vari-

ables 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 = K�i (BihiLi)

1��

where Bi again represents labor augmentation due to technology di¤erences. The model

45


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:

lnwit = �0 + �A ln(ageit) + �1Y yrexit + �

2Y (yrexit)

2 + �Edyredi + 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 =YiLi= K�

i (Bihi)1��L��i

46


HJ take a di¤erent tack from Solow and MRW. They rewrite yi as:

yi =

�Ki

Yi

��=(1��)hiBi

This works, since substituting from above we get:

yi =

�Ki

K�i (BihiLi)

1��

��=(1��)hiBi

=�K1��i

��=(1��)(hiBi)

��(hiBi)L��i

= K�i (hB)

1��L��i

HJ take the formulation:

yi =

�Ki

Yi

��=(1��)hiBi

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


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 infraestruc-

ture soy they used several proxies for this objective. They use a Government anti-

diversion policies Index (by Political Risk Services - assesses risk to international in-

vestors) 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 mar-

ket 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


where S denotes the social infrastructure and X a collection of other vari-

ables 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 exoge-

nous 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, con-

tiguity, language).

Hall and Jones aim at recovering the the intensity of "Western Europe colo-

nization XVI - XIX centuries" because the underlying argument is that western culture

provided the spread of ideas of Adam Smith, the importance of property rights, sys-

tem of checks and balances in government and these ideas meant a superior, though

49


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:

logY=L = �+ � ~S + ~":

where ~S is the estimate of S via the regression above.

The empirical consists in estimating:

log

�Y

L

�= log(A) +

�

1� � log�K

Y

�+ log

H

L

= �S + "

50


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


[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


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 chil-

dren 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 wel-

54


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 = bBy�Bt Lt

Where Bt is the number of births in the population and yt; is per capita income. The

function: bBy�Bt 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 = bDy��Dt 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


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 + bBy

�Bt � bDy��Dt

�Lt

Steady state calculations imply the following equations:

B� = D�

bB (y�)�B = bD (y

�)��D

y� = (bD=bB)1=(�B+�D)

L� = T (A=y�)1=�

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


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


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)�k�t

and the accumualtion of capital is:

kt+1 = syt + kt (1� �)

Birth rates and death Rates are still the same:

Lt+1 =�1 + bBy

�Bt � bDy��Dt

�Lt

The steady state now is summarized by the following equations:

y� = A(T=L�)�k�� (4.1)

58


k� =s

�y� (4.2)

and

y� = (bD=bB)1=(�B+�D) (4.3)

So replacing ?? in ?? we obtain:

y� = A(T=L�)�

1��

�s�

� �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


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


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 Dy-

namics 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


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)ntVt+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 H0 and a stock Ht. Total

62


human capital is obviously:

H0 +Ht

Income is obtained through a linear production function:

lt(H0 +Ht)

and the evolution of Human capital depends on the forllowing following for-

mula:

Ht+1 = Aht(H0 +Ht)

�

so there are diminishing returns to education.

Consumption is shared among children according to the following formula:

ct + ntf = lt(H0 +Ht)

where f is spending per child.

Finally, altruism is de�ned as:

a(nt) := Cn�"t

B 4.4.1 Solution to the Model

The model is solved by optimizing utility, with the appropiate choice of nt and ht

63


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


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


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


� 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 Colum-

bus), knotweed, maygrass, little barley.

68


� 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.

69


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


� 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


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:

Yt = A1L1

NX1=1

X�i1

A technology in this context is interpreted as the number N of intermediate products.

The �rst order condition for production implies that:

�A1L(1��)1 X

(1��)i1 = 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:

��AL

(1��)1 X

�(1��)i

�Xi1 �Xi1 = �

Mi

73


the FOC here yields the following equation:

��2AL

(1��)1 X

�(1��)i

�= 1

which yields:

Xi =��2Ai

� 1(1��) Li

which is the condition for equilibrium input Xi: Then, we can replicate this result for

all the i inputs and obtain:

Yt = A1

(1��)1 �

2�1��N1Lt

and by this we obtain a per capita income by dividing output by popoluation:

yt = A1

(1��)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


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:

�Mi =

�1� ��

�A

11��1 �

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


following condition holds:

E��Mi (QK)

��QK

QK� rt �

E��Mi (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�

�N2N1�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 �E��Mi �N2

�� 1

V�

�N2N1�N2

�and

76


E��Mi (K)

��K

K�E��Mi �N2

�� 1

V�

�N2N1�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��Mi (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


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 � > 1 :

U =

"nXi=1

c(��1)=�i

#�=(��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

p

�n1=(��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.

78


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:

LN

LS=cNnN

cSnS=

�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 + �n

Nt

79


where � is the rate of di¤usion so that:

nNt+1 = nNt � �nNt + int

= (1� �)nNt + int

De�ne vt = nNt =nt so that:

vt+1 =nNt+1nt+1

=nNt � �nNt + int(1 + i)nt

=(1� �)vt + i(1 + i)

Steady state v doesn�t change so that:

�� =i

i+ �:

Hence in steady state:

nN

nS=i

�

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


y =NYi=1

qi

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

�(q) � maxqfqN � �(q)Ng

the necessary FOC for this problem is

qN�1 � �0(q) = 0

and the SOC:

(N � 1)q � �00(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


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

N= �(q)

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 Np(N). This is a property of preferences and it is

assume ad hoc. Then the �rm�s problem is now:

maxn;qfNp(N)qN � �(q)Ng

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

@x= ax ln a

82


The �rst FOC is for q in the �rm�s problem is:

Np(N)qN�1 = �0(q)

if we are also to have zero pro�ts, we must now have for all q [0, 1]:

�(q) = p(N)qN

Then the �rm�s FOC for N is just:

p(N)qN +Np0(N)qN + qN ln q � �(q) = 0

we can replace in this function the value for �(q) to obtain:

p(N)qN +Np0(N)qN +Np(N)qN ln q � p(N)qN = 0

Cancelling qN :

� 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. Nev-

ertheless we will assume that p(N) guarantees this. Therefore, suppose that:

83


p(N) = N ; < 1

so that

p0(N)

p(N)=�N �1

N=�

N

and a direct condition is obtained:

N =�

� ln q

which is increasing in q, so that better workers work on more complex prod-

ucts. 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 prod-

ucts 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]

84


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.

maxc;n

c1��

1� � �Bn1+1=

1 + 1=

s.t. pc = n

86


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.

maxc;�(1� �)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 equilib-

rium. Since all �rms are equal, either all of them evade, or none of them evade. So we

87


have two possible equilibrium: one with full informality (associated with � = 0), and

one with full formality (associated with � = 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

maxnpAn� � n

FOC in n:

pA�n��1 = 1 (7.2)

It de�nes a labor demand when �e = 0.

Imposing market clearance, and solving equations (1) and (2), one �nds n�0

and p�0. Rational expectations require that �e = � = 0, so we need

n�0 � �n

Case 2: high enforcement (� = 1) - low evasion (� = �)

Let�s assume �e = 1. Thus, it should be clear that � = � . Therefore, �rms

maxn(1� �)pAn� � n

FOC in n:

(1� �)pA�n��1 = 1 (7.3)

It de�nes a labor demand when �e = 1.

88


Imposing market clearance, and solving equations (1) and (3), one �nds n�1

and p�1. Rational expectations require that �e = � = 1, so we need

n�1 > �n

Possibility of multiple equilibrium:

Notice that as long n�0 < n�1, any exogenous �n 2 [n�0; n�1) sustains multiple

equilibrium. The purpose of this note is to �nd conditions on the parameters of the

model that sustain multiple equilibrium.

Calculating n�0; n�1:

Plugging (3) in (1),

(1=(1� �)A�n��1)1�� = (1=B)n��1=

Solving for n, one gets

n�1 = X(1� �)(1��)=[(1��)(1��)+�+1= ]

where X = (1=B)(A�)1��.

Similarly,

n�0 = X

Thus, notice the possibility of multiple equilibrium arise when n�0 < n�1, i.e,

(1� �)(1��)=[(1��)(1��)+�+1= ] > 1

89


Since 1� � < 1, a su¢ cient condition for this is that

(1� �)=[(1� �)(1� �) + � + 1= ] < 0 (7.4)

Let�s re-write the denominator of (4): 1� �+ ��+1= . Moreover, notice that

as long � > 1, since 0 < � < 1, and > 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


Thus, this model generates multiple equilibrium: one with high enforce-

ment/low evasion/high employment and the other with low enforcement/high eva-

sion/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


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


� 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 mar-

ket may imply that massive ownership can be infact a good policy do to a Marshallian

argument.

98


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.

99


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 <

�P

�

� 1

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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

xfor U [0; x]

. The density conditional expectation is obtained by the following integral:

101


E [Z�jP; �] =xZ0

Z�1

xdz =

1

x

xZ0

Z�dz

=1

x

Z�+1

�+ 1jx0 =

x�

�+ 1

and replacing the value of x we obtain:

(P )�

(�+ 1) ��

Therefore the condition to sell will be:

(P )�

(�+ 1) ��

� PK

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

= �1

�K (�+ 1) �

�

� 1��

= �2�� (�� ) (K (�+ 1))

1��

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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; � = 12and 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=

Z�Z0

Z�dz +

1ZZ�

�Z dz

and the amount lost is:

Social Loss=

1ZZ�

[Z� � �Z ] dz

by computing this simple integrals we can compute the following:

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Social Loss as % of Production=

1RZ�[Z� � �Z ] dz

Z�R0

Z�dz +1RZ��Z dz

A 8.3 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


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

V¯ 1be the worst outcome for the sophisticated farmer

Because the game is identical in the next period. Two conditions will be satis�ed:

�V1 � V � �V¯ 1

V¯ 1� V � � �V1

This conditions explain the following. Because, �V1 cannot exceed V ��V¯ 1; 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


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:

V¯ 1+ � �V1 � V

Because of this, we can replace the RHS of this inequality into the �rst in-

equality and it will still hold. We obtain:

�V1 � V¯ 1 + ��V1 � �V¯ 1

which in turn implies:

V¯ 1(1� �) � �V1 (1� �)

but since by construction we have that V¯ 1� �V1 it better be the case that the two

amounts are equal: V¯ 1= �V1: Using this equality, in both inequalities yields:

�V1 �V

1 + �

and

�V1 �V

1 + �

So our only solution is:

106


�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 sophis-

ticated 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 � E�V

1 + �

�

B 8.3.2 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


C N=2

Assume that he already bought the �rst slot. Then the sophisitcated farmer will nego-

tiate and the results will be as before:

V

1 + �;�V

1 + �

This is in the last stage. In the previous stage, the �gure is again repeated, and we

obtain:

V �

(1 + �);�V �

(1 + �)

Note that V � is di¤erent from V: What is the relation of this values? They will satisfy

the following relationship:

V �

(1 + �)= V

so the �nal outcome will be:

V �

(1 + �)2;�V �

(1 + �)2;�V �

(1 + �)

Notice that this fractions add up to 1.

V �

(1 + �)2+

�V �

(1 + �)2+

�V �

(1 + �)=1 + � + � (1� �)

(1� �)2= 1

108


C N=3

Following the previous steps we obtain that the shares will be:

V �

(1 + �)3;�V �

(1 + �)3;�V �

(1 + �)2;�V �

(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


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


the project is veri�able and succesful:

�ARL = 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:

�R� B

(�A � �B)

�= RL � 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

(�A � �B)

�� I � 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


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

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