Transcript Principles of Economic Growth

Economic Growth
Core hypothesis: Economic growth doesn’t just
happen; rather, it is endogenous, and depends on
the choices society makes about political and
economic organization, policies, and history.
In this module, we look at
•History of thought on growth
•Stylized facts of growth
•Early models of growth, from Malthus to Solow
•Current models of endogenous growth
History:
The first revolution:
Adam Smith (1723-1790)
 Theory of wealth
creation, public policy,
and economic growth
size of the market
division of labour
efficiency
Saving and
investment are
by-products and
precursors of
domestic and
foreign trade
The first revolution:
Adam Smith

Saving and investment stimulate growth
direct
effects through accumulation
of capital
indirect
effects through labour productivity
further
indirect effects through interaction
with exchange and trade, through foreign
investment
domestic
market can take the place of
foreign markets
The first revolution:
Adam Smith
Smith’s
reference to ‘private misconduct’ and the
‘publick extravagance of government’

Problem of public corruption and what economists
now call “regulatory capture”
Distinction

between quantity and quality
Quality enhances the productivity of workers and
other technological inputs to production, and permits
further technical innovation to occur
Mutual
advantages of trade and growth, links to
geography

First recognition of the concept of comparative
advantage
The first revolution:
Adam Smith
Benefits from division of labour
If specialization increases efficiency
and wealth and, thereby, economic
growth, then ...
... just about anything that increases efficiency
by the same amount, other things being equal,
should be expected to have the same effect on
growth.
The first revolution:
Adam Smith
 Implications
for growth
If foreign trade enlarges the
market and thus facilitates further
division of labour à la Smith, thereby
increasing wealth and growth, then ...
... all other equivalent means of
increasing the efficiency or quality
of labour, capital, and land should be
expected to affect economic growth in the
same way.
The first revolution:
Adam Smith
 Smith
on education, efficiency, and growth
between the quantity and quality of labour 
education, by increasing labour productivity, also increases
efficiency and growth
 Smith feared the economic, political, and social consequences
of inferior education among the masses
 He favoured public support for education
 Distinction
 First
recognition of the external economic benefit to society of
mandatory universal education
The first revolution:
Adam Smith - Summing up
 Economic
growth = increase in the quantity and
quality of the three main factors of production:
labor, capital, and land
 Growth
 Two
accounting is based on this classification
shortcomings:
 Fixed
quantity of land – diminishing returns
 Increase
in the labour force does not really
count as a source of economic growth
Adam Smith’s followers
Thomas Malthus
Question of population growth and its effect
on economic growth
David Ricardo
Impact of the distribution of wealth and of
foreign trade
Malthus: A Formal Model
Ld=labor demand
Labor
(Pop.)
Ls=labor supply
Real Wage
CBR
CBR, CDR
NRI=0
CDR
Real Wage
w*
Effects of Charity
A Malthusian Perspective
Labor
(Pop.)
The effective
wage falls until
CBR=CDR, leaving
the level of living as
was prior to charity.
Ls2
3
Ls
Ld
Real Wage
CBR
CBR,
CDR
2
Worker receives
w*-c from labor
and c in charity.
Growth shifts
Ls curve up thus
reducing the
effective wage.
CBR>CDR
=> Growth
CDR
Real Wage
(w*-c) w* (w*+c)
1
Malthus: The Plague
Ld
Labor
(Pop.)
Ls
Ls2
2
Real Wage
CBR, CDR
CBR
1
CDR2
CDR
w*
w2*
Real Wage
Technological Advances
Ld
Labor
(Pop.)
Ld2
1
4
Ls2
Ls
Real Wage
CBR
CBR, CDR
3 population grows
NRI=0
CDR
Real Wage
5: wage returns to w*
w* w2
2
Adam Smith’s followers
John Stuart
Mill
 rejected
Malthus’s prediction that
population would outgrow productive
capacity
 more
and better education would
restrain population growth
 distribution
a different matter than
production but can be changed
through policy
Adam Smith’s followers
Karl Marx

Economic mechanisms driving production
and distribution are closely related
 Anticipates
Henry Ford’s comment on the
importance of income as a determinant of
aggregate demand
 General equilibrium effects are important

The limits to growth observed by Malthus
are inescapable  ‘technological
unemployment’
Adam Smith’s followers
Alfred
Marshall
organization
as a fourth factor of production
made
explicit the connection between
education and growth
distribution
of income and wealth matters
for efficiency and growth
‘Knowledge is our most powerful engine of
production ... Organization aids knowledge’
Adam Smith’s followers
Joseph Schumpeter
technology
through invention, innovation,
and entrepreneurship
rent-seekers
motivated by monopoly profits
perfectly
competitive markets may not be very
conducive to economic growth
no
rent to capture under perfect competition
Adam Smith’s followers
John Maynard
Keynes
Accumulation of capital
‘Science and technical inventions’
‘I draw the conclusion that, assuming no
important wars and no important increase
in population, the economic problem may
be solved, or be at least within sight of
solution, within a hundred years.’
Modern Models of Growth
Stylized Facts of Growth
Per capita growth rate
Stylized Facts of Growth
Return to Capital
Stylized Facts of Growth
Why is the rate of depreciation
increasing?
Stylized Facts of Growth
Capital-Output Ratio
Stylized Facts of Growth
Investment rates
Stylized Facts of Growth
Consumption and income
Time-series data 1929-82, in 1982 $$
Enter mathematics: Harrod and
Domar
 Paul

Samuelson’s
Foundations of Economic Analysis (1948)
 laid
the basis for mathematical economics, including the
modelling of dynamic interactions among macroeconomic
variables
Enter mathematics: Harrod and
Domar
Net investment equals the increase in the capital stock
… net of depreciation due to physical or
economic wear and tear
High level of investment entails an
increasing level of the capital stock
High levels of saving and investment are good for growth
even if they are stationary, that is, not increasing
By continuously augmenting the capital stock ...
… even stationary levels of saving and investment
relative to output drive output higher and higher,
thus generating economic growth
•Flows of investment add to the stock of capital
Enter mathematics: Harrod and
Domar
Efficiency is crucial for growth
High level of efficiency stimulates growth by ...
… amplifying the effects of a given level of saving and
investment on the rate of growth of output
All that is required is a steady
accumulation of capital through
saving and investment
A given level of efficiency, including the state
of technology will, then translate the capital
accumulation into economic growth
Enter mathematics: Harrod
and Domar
So, Samuelson’s work neatly formalized,
simplified, and summarized the essence of
almost 200 years’ theorizing about economic
growth
 Harrod
and Domar expressed the dynamic
relationship between saving, efficiency, and
growth in a simple equation
The Harrod-Domar model 
The Harrod-Domar model
 Economic
growth depends on three
factors:
A. the saving rate
B. the capital/output ratio
C. the depreciation rate
The Harrod-Domar model:
Mathematics
Notation:
Y
denotes national income
K denotes capital stock
S denotes saving
Y denotes national income
The Harrod-Domar model:
Mathematics
Assumptions:
Saving
is proportional to income: S=sY
Capital-output ratio is constant: K=vY
Investment (newly produced capital goods) must be
allocated between increasing the stock of capital and
replacing depreciated capital: I=K+K
At equilibrium S=I (desired saving =desired
investment)
The Harrod-Domar model:
Mathematics
Harrod-Domar
equation
From
the capital-output ratio assumption, we can
write K=v Y.
Substituting into the expression for investment, we
have I=v Y+vY
Using the equilibrium condition, we then have
sY= v Y+vY or  Y/Y=s/v-
 Example: s=0.2, v=3, =0.04 yields a growth rate of
roughly 3%.
The Harrod-Domar model

Shortcomings:

Neither theory nor empirical evidence seemed to provide
much support for the capital/output ratio as an exogenous
behavioural parameter in the model
a more elaborate formulation of the link between capital and output
was called for


The model did not leave much room for the other crucial
factor of production, labor
population or labor-force growth is absent from the formula, which
explains output growth solely by saving and efficiency

The second revolution:
The neoclassical model
Since population growth is basically a demographic phenomenon
and, hence, exogenous from an economic point of view, it must
follow that economic growth is also exogenous
According to Solow, saving behaviour was no
longer relevant for long-run growth, nor was
efficiency in a broad sense, except insofar as it
mattered for technology
Economic growth was considered immune
to economic policy, good or bad
Even so, saving and efficiency play an
important role for growth over long periods,
that is, the medium term
The second revolution:
The neoclassical model
Solow showed how the capital/output ratio, rather than being
exogenously fixed as in the Harrod-Domar model,
… is better viewed as an endogenous
variable, which moves over time and
ultimately reaches long-run equilibrium
Once attained, the long-run equilibrium is
consistent with not only a constant
capital/output ratio
… but also with a constant rate of growth
of output per capita, a constant rate of
interest, and a constant distribution of
national income between labour and
capital, all of which seemed to apply to
the real world
The Neoclassical Model
Mathematics
 Output
is produced via a production function which uses
capital and labor as inputs
Y =LK
a
1 a
where the parameter a is between 0 and 1.
 Taking logs on both sides and differentiating yields
Y
K
 g = an  1  a 
Y
K
 Here, g is the rate of growth of output in percentage terms, n is the
exogenously given rate of growth of the labor force (or equivalently, of

population), and KK is the rate of growth of the capital stock.
The Neoclassical Model
Mathematics
 From
empirical work by Kuznets, it is plausible to
assume that the long-run capital-output ratio is constant,
which implies that K
K
 Plugging
=g
into the growth equation, then, we have
K
g = an  1  a  = an  (1  a) g
K
 ag = an
g=n
The Neoclassical Model
Mathematics
Thus, in the Solow model, the long-run rate of growth is
determined entirely by the exogenously given rate of
population growth.
It also follows that in the long-run, there can be no growth in
per capita output
Since we obviously have seen significant increases in
standard of living since the onset of industrialization in
the early 1800’s, the model must be modified if it is to
explain this.
The Neoclassical Model
Mathematics
 We
can explain the observed growth in per capita output by
assuming that technological change makes the labor input more
productive over time, due to factors such as better technology or
better education of the workforce. With this assumption, the
production function becomes
 
a
Y = B e L K 1 a
qt
B
represents some initial state of technology
 e is the base of the natural logarithm
 Labor productivity grows at the rate q
qt
 We refer to e L as the efficiency unit equivalent of the labor input
The Neoclassical Model
Mathematics
 Log
differentiating the production function now gives
K
g = an  q   (1  a)
K
 As
before, taking the long-run capital-output ratio as constant
yields
g =nq
 So,
we now have that growth is exogenous, being driven by
productivity improvements, but per capital growth is now positive
and equal to q.
The Neoclassical Model
Mathematics

Comparing the growth equation for the Solow model with that of
the Harrod-Domar model, we see that we must now have
s
g = n  q = 
v
all the parameters n,q,s,v, and  are exogenously given, then
we would generally not expect the equality above to hold.
 If
 Mathematically, the
Harrod-Domar model is now over-identified.
 Solow resolved this over-identification by assuming that the capital-output
ratio, rather than being exogenously specified, was a function of the other
parameters of the model:
s
s
v=
=
g  n  q 
The Neoclassical Model
Mathematics
 How
do we know the capital-output ratio is the right parameter to
make endogenous?
 Consider the original definition of investment: I = K  K
 This can
 Since
be re-written as
I K  K

= 
 
Y Y  K

saving must equal investment in the long-run, I/Y=s, and we may
then solve the equation above for the capital-output ratio as
K
s
=
Y K  
K
 Hence, changes in any of the right-hand parameters will affect the value of
the capital-output ratio.
The Neoclassical Model
Mathematics
 We can also use this result to solve for the rate of growth of capital in
terms of other parameters of the model:
K
Y
= s 
K
K
 Substituting for the rate of growth of capital in
yields
the Solow growth equation
K
 Y

g = an  q   (1  a) = an  q   (1  a) s   
K
 K

 This equation tells
us that if we increase saving, then the economy will
grow as long as the capital-output ratio remains constant.
 We turn next to the question of whether this ratio will in fact remain
constant.
The Neoclassical Model
Mathematics

Dynamics of the capital-output ratio
 Define the following
 The percentage
ratios of capital and output per efficiency unit at time t:
K
k = qt
Le
Y
y = qt
Le
rate of change of the first ratio is
k K
Y
=  n  q = s   n  q
k K
K
where we use the relationship between the rate of change of capital to the
capital-output ratio to arrive at the right-hand side of the equation.
The Neoclassical Model
Mathematics
 We can
also write the production function in terms of the two ratios as
a
a 1 1 a
Y
B e qt L K 1a
1 a
qt


y = qt =
=
B
e
L
K
=
B
k
e L
e qt L
Y
y
a
 Now, since
= = Bk 
K k
 
 
substituting into the expression for the rate of growth of capital, we get a key
equation:
k
a
= sB k     n  q
k
The Neoclassical Model
Mathematics

The Solow differential equation
k
a
= sB k     n  q
k
is small, so that k  is large, the the rate of change of k will be positive,
a
so the capital stock will increase. On the other hand, if k is large, k  will
be small, so that the rate of change will be negative.
 This means that if we start at a low level of capital, the economy will
accumulate capital, while if we somehow started with a large amount of
capital, we will decumulate it.
 Hence, independently of where the economy starts, it will evolve toward a
steady-state at which
 If k
a
k = 0
The Neoclassical Model
Mathematics
 Steady-state
 Set
the time derivative of k to zero and solve for k
 sB 
ˆ
k=

n  q   
1
a
 Using
the definition of k, we can find the steady-state values of capital and
output:
1
 sB  a qt
ˆ
K =
Le

n  q   
 sB 
ˆ
Y = B

n

q




1 a
a
Le  Le 
qt 1 a
qt a
 sB 
= B

n

q




1 a
a
Le qt
The Neoclassical Model
Mathematics
 Note that the steady-state capital stock and flow of output are actually growing,
but in a balanced way, at the same rate, so that the capital output ratio remains
constant at
K
s
=
Y n  q 

Income distribution in the Solow model
 Standard
results from producer theory tell us that at the competitive equilibrium,
inputs are paid their marginal products. For the simple model with only capital and
labor inputs, these are given by
MPK = 1  a ALa K  a = 1  a 
MPL = aALa 1K 1 a = a
Y
=w
L
Y
= r 
K
The Neoclassical Model
Mathematics
 Hence, for the Cobb-Douglas specification of technology, each factor of production
is paid a constant share (a for labor, 1-a for capital) of output. This is consistent
with data for modern industrial economies, where labor receives 2/3 of total output,
while capital receives 1/3.
 This also gives us a way to calibrate the model, since it says we should set
a=2/3.
 Since the capital-output ratio is constant, it also follows that along a balanced
growth path, interest rates will remain constant.
 For labor, the real wage will grow at the rate g-n=q, since labor productivity is
growing over time.
 Calibration spreadsheet
The third revolution:
Endogenous growth
 The
neoclassical growth model seemed unable to answer some
burning questions about economic growth
 Is
technological change exogenous from an economic point of view?
 Do
economists really have nothing to say about economic growth in the long
run?
 If
output per capita grows at a rate that depends solely on - in
fact, is equal to - the rate of technological progress, then why is
it that the growth performance of different countries differs so
radically over long periods?
 What
does the neoclassical model tell us about relative growth
performance anyway?
The third revolution:
Endogenous growth
Key idea
 Technology
is not exogenous
 Technology
depends on economic factors
 Technological improvement depends on
 Innovation
– “Learning by doing”
 Education
 Basic
research
 Technical
 Basic
innovation is external to firms’ decisions
research generates new technologies available to all
 Education and on-the-job learning spill over from one firm to
the next
Endogenous Growth:
Mathematics
Human
capital investment
 Focus again
on Cobb-Douglas production: Y = AL K
a
1 a
 We assume that some fraction h of the workforce is engaged in
innovation – basic research, fine-tuning technical processes within the
firm, independent invention, or other educational pursuits. The
remaining fraction (1-h) provides labor input for firms.
 The effect of human capital accumulation on production is via A, which
we now assume is an increasing function of the average amount of
human capital accumulation hL
 For specificity, we assume the production function is given by
Y = hL  La K 1 a
a
Endogenous Growth:
Mathematics
Increasing
returns property
 The inclusion
of human capital accumulation effects on productivity
implies that the production function now exhibits increasing returns to
scale. To see this, suppose we increase the labor and capital inputs
by some factor l. This will increase the average labor supply by the
same factor. Hence, the effect on output will be
lhL  lL lK 
a
a
 Because
1 a
= l1 a hL  La K 1 a = l1 aY  lY
a
the human capital effect is external, the increasing returns
will not affect individual firms’ profit maximizations.
Endogenous Growth:
Mathematics

Growth with human capital investment
 Taking
logs and differentiating the production function gives us
Y
L
L
K
= ah  a  1  a 
Y
L
L
K
K
= 1  h an  1  a 
K
 On
a balanced growth path where output and capital grow at the same rate g,
we will have
g = 1  hn

Hence, as in the case of exogenous technical progress, we will have
positive growth per capita, but due in this framework to the productivity
enhancing effects of economic activities associated with human capital
accumulation.
Endogenous Growth:
Mathematics

Income distribution
 As
in the Solow model, factor shares are given by
w=a
Y
L
r   = 1  a 

Y
K
Also as in the Solow model, wages grow over time since output grows
more rapidly than population. Since capital and output grow at the
same rate, interest rates do not grow.
Endogenous Growth:
Mathematics

Steady-state
 We
can also replicate the dynamic analysis from the Solow model.
Define Lˆ = hL L
and
K
Y
k=
y = = k 1 a
Lˆ
Lˆ
 Then

k K Lˆ K
=  =  1  h n
k K Lˆ K
Endogenous Growth:
Mathematics
 Since
K
Y
= s 
K
k
while
Y y
= = k a
K k
k
we have = sk  a    1  h n
k
 Steady-state
capital stock is then


s
ˆ
k=



(
1

h
)
n


1
a
Saving Behavior
Last missing ingredient to
a fully specified
economic model
 Handle
by positing preferences over consumption over time
 A key
parameter is consumer’s degree of patience or impatience,
measured by how little or much they discount future utility
 Consumers
face budget constraints which allow them to trade
off consumption today for consumption tomorrow
 Saving
generates returns in excess of the actual amount saved when
interest rates are positive
Saving Behavior:
Mathematics




Two formulations of consumer model: overlapping generations and dynastic models
In both models, we assume time is split up into discrete periods (planning time).
In overlapping generations model, consumers live finite lives. We will make the
simplifying assumption that the number of periods of life is 2.
Overlapping generations optimization problem is then
subject to
max ln c1   b ln c2 
c 1 , c 2 
c1 
c2
=Y *
1 r
Here, 0<b<1 is the consumer’s discount factor, which measures the degree of her impatience
 r is the market interest rate (which, recall, will be determined by the production side of the
economy), and c2/(1+r) is the present value of second-period consumption determined by the
market interest rate

Saving Behavior:
Mathematics

The easiest way to solve this problem is to substitute for second-period consumption from the
budget constraint into the utility function, and then take a first-order condition with respect to firstperiod consumption:
max ln c1   b ln 1  r Y * c1 
c1

The required first-order condition is
dU 1
1
= b
=0
Y * c1 
dc1 c1
 cˆ1 =
Y*
1 b
and
cˆ2 = 1  r 
b
1 b
Y*
Saving Behavior:
Mathematics

In terms of this model, the per capita rate of growth in consumption is given by
c2
 1 = b 1  r   1
c1

From the growth equations, the per capita growth rate for consumption from the technology side of
the economy is give by g-n. Hence, we will have g  n = b 1  r   1 and we can determine the
equilibrium interest rate as
r=


g  n  1  1
b
So, interaction of growth induced by technology together with degree of patience determines
equilibrium interest rates.
Cases:

Exogenous population growth only: g=n
r=

Growth with technical progress: g-n=q>0
r=
1
b
1
1 q
b
1
Saving Behavior:
Mathematics
 Dynastic
Model
 For
this model, we assume consumers act to optimize the discounted utility
stream of a long-lived family, and hence solve

max  b t ln ct 
c
t =0
subject to

ct
= Y * = PV of income

t
t =0 1  r 
 Analysis
of this model is much harder than for the overlapping generations
model, so we will simplify by looking not at the market version of the model,
but at a social planning version
Saving Behavior:
Mathematics
 Social
planner’s problem:

max  b t ln ct 
c
t =0
subject to
ct = Yt  Kt 1
Yt = La Kt1 a
 Substituting
from constraints into the objective function, the problem
simplifies to


max  b t ln La K t1 a  K t 1
K
t =0

Saving Behavior:
Mathematics

Analyzing the social planner’s problem for this model is not simply a matter of
taking first-order conditions and solving. To see why, we normalize the labor
supply to 1, since it doesn’t change over time. Then, taking first-order
conditions gives us the so-called Euler equations:
b 1  a Kt a
1
Kt1 a  Kt 1
=
Kt11a  Kt
for this module, you show that by letting =1ab and
making a suitable change of variable, the second-order difference equation
generated by the first-order conditions can be converted into the first-order
difference equation

 In the exercises
z t 1 =
1    zt
Saving Behavior:
Mathematics
Kt
.
1 a
K t 1
Graphing the difference equation gives us the following picture
where zt =
1.4
1.2
1
z(t+1)

0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
Z(t)
1
1.2
1.4
Saving Behavior:
Mathematics
Properties of the difference equation
1.4
steady-states, one at z=ab
and the second at z=1
If we start below the first steadystate, or anywhere between it and
the steady-state at 1 and iterate
the difference equation, we will
converge to the first steady-state.
If we start anywhere to the right
of the steady-state at 1, we will
diverge.
Two
1.2
1
z(t+1)

0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Z(t)
Only the lower steady-state is
optimal. At the upper steady-state,
with z=1, we would have K = K 1a
which says that all production is being
devoted to the accumulation of
capital, with none for consumption.
Saving Behavior:
Mathematics

It is possible to show that the planner’s problem can be represented in the simpler form


V Kt  = max ln Kta  Kt 1   bV Kt 1 
K t 1

provided we know the so-called value function V(K). While the math is beyond the
scope of what we can do here, it can be shown that this function will exist under suitable
assumptions about discounting.
This formulation of the optimal capital accumulation program for the economy is known
as the dynamic programming formulation. In the exercises for this module, you showed
that for this model, the value function is
1
V (k ) =
1 b


ab
a
ln 1  ab   1  ab ln ab   1  ab ln k 

