Economic Growth I - Princeton University Press

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Models of Economic Growth A
Outline:
Because this area is complex and mathematical
there are two files of slides for this topic
Lecture A
• Introduction – trends in growth
• Neoclassical growth models
Lecture B
• Endogenous growth models
• The convergence debate
Below are slides for lecture A
See next file for lecture B
Introduction
• Need to define ‘economic growth’ (in book
this is growth in GDP per capita, not GDP
growth).
• Some background on history of economic
growth – including own country data
• Also, worthwhile to stress importance of
small differences in growth rates e.g.
2% growth per year  GDP p.c. increases 7.4 fold in 100 years
0.6%  GDP per capita increase 1.8 times in 100
.... 72 / growth rate = no. of years to double, hence China’s 10% p.a. implies 7.2 years
The very long run
Growth of GDP per capita (average annual percentage changes)
1500-1820
OECD
Non-OECD
World
0.04
1820-1900
1.2
0.4
0.8
1900-2000
2.0
0.6
1.9
Source: Boltho and Toniolo (1999, Table 1) OECD refers to North America, Western
Europe, Japan, Australia and New Zealand.
5000
15000
25000
35000
USA, UK and EIRE
1965
1970
1975
1980
1985
1990
1995
2000
year
GBR
USA
IRL
Growth of GDP p.c: USA=2.2%, GBR=2.0%, Ireland=3.7% (but post-93, 8.5%)
GDP per capita is US$ 1996 constant prices. Source: Penn World Table 6.1
0
1000
2000
3000
4000
China and India
1965
1970
1975
1980
1985
1990
1995
2000
year
CHN
IND
Growth: pre-90 China 3.7%, India 4.4%. 1990-2000: China 7.0%, India 4.4%
Source: Penn World Table 6.1
0
5000
10000
15000
Brazil, S. Korea, Philippines
1965
1970
1975
1980
1985
1990
year
BRA
PHL
KOR
Source: Penn World Table 6.1 (http://pwt.econ.upenn.edu/aboutpwt.html)
1995
2000
Other data
• Above are from Penn World Table 6.1, now
6.3 is available
http://pwt.econ.upenn.edu/
Some further links at:
http://users.ox.ac.uk/~manc0346/links.html
GDP per capita growth not everything
• Focusing on ‘economic growth’ does neglect
health, the environment, education, etc
• UN’s Human Development Index (HDI) gives equal
weight to life expectancy, education and GDP per
capita (http://hdr.undp.org/reports/global/2004/)
• Ultimate interest ‘well-being’ or ‘happiness’.
Layard, R. (2003). "Happiness: Has Social Science
a Clue?" http://cep.lse.ac.uk/events/lectures/layard/RL030303.pdf.
• GDP measures aggregate value added – whether
coal power station or wind farm
• Friedman, Ben (2005) The Moral Consequences of
Economic Growth argues growth is important for
‘stable’ societies
Neoclassical model
• There are many ways to teach this. Book tends
to use equations, but can do a great deal with
intuition and few diagrams.
• This model most often attributed to Robert Solow
(1956) – US Nobel prize winner …. but Trevor
Swan (1956) (a less well known Australian
economist) published (independently) a very
similar paper in the same year – hence refer to
Solow-Swan model
Neoclassical growth model
• Model growth of GDP per worker via capital accumulation
• Key elements:
– Production function (GDP depends on technology,
labour and physical capital)
– Capital accumulation equation (change in net capital
stock equals gross investment [=savings] less
depreciation).
• Questions:
– how does capital accumulation (net investment) affect
growth?
– what is role of savings, depreciation and population
growth?
– what is role of technology?
Solow-Swan equations
Y  Af (K , L)
(production function)
Y  G D P , A  technology,
K  capital, L  labour
dK
 sY   K
(capital accum ulation equation)
dt
s  proportion of G D P saved (0  s  1)
  depreciation rate (as p roportion) (0    1)
Solow-Swan analyse how these two equations interact.
Y and K are endogenous variables; s,  and growth rate of L
and/or A are exogenous (parameters).
Outcome depends on the exact functional form of production
function and parameter values.
Neoclassical production functions
Solow-Swan assume:
a) diminishing returns to capital or labour (the ‘law’ of
diminishing returns), and
b) constant returns to scale (e.g. doubling K and L, doubles Y).
For example, the Cobb-Douglas production function

1
w here 0    1
Y  AK L
y
Y
L


1 
AK L
L

AK

L


K 

 A

A
k

L


Hence, now have y = output (GDP) per worker as
function of capital to labour ratio (k)
GDP per worker and k
output per worker
Assume A and L constant (no technology growth or
labour force growth)
y
y=Af(k)=Ak
concave slope reflects
diminishing marginal
product of capital
dY/dK=dy/dk=Ak-1
k
(capital per worker)
Accumulation equation
If A and L constant, can show*
dk
 sy   k
dt
This is a differential equation. In words, the change in
capital to labour ratio over time = investment (saving) per
worker minus depreciation per worker.
Any positive change in k will increase y and generate
economic growth. Growth will stop if dk/dt=0.
*accu m u latio n eq u atio n is:
dK
 sY   K , d ivid e b y L yield s
dK
dt
A lso n o te th at,
dK
 K 
 d 
/
d
t

/L

dt
L
d
t


dk
dt
sin ce L is a co n stan t.
/ L  sy   k
Graphical analysis of
dk
(Note: s and  constants)
dt
 sy   k
output per worker
y
k
(depreciation)
sy
net investment
(savings = gross
investment)
k*
k
(capital per worker)
Solow-Swan equilibrium
y=Ak
output per worker
y
y*
consumption
per worker
k
sy
k*
k
GDP p.w. converges to y* =A(k*). If A (technology) and L
constant, y* is also constant: no long run growth.
What happens if savings increased?
• raising saving increases k* and y*, but long run
growth still zero (e.g. s1>s0 below)
• call this a “levels effect”
• growth increases in short run (as economy moves to
new steady state), but no permanent ‘growth effect’.
y=Ak
y
y1 *
y0 *
k
s1y
s0y
k0*
k1*
What if labour force grows?
dk
Accumulation eqn now
 sy  (  n ) k
w here n 
dt
/L
(m ath note 2)
dt
y
y
output per worker
Population growth
reduces
equilibrium level
of GDP per
worker (but long
run growth still
zero) if
technology static
dL
(n)k
k
sy
Population growth (n>0)
pivots the ‘depreciation’
line upwards, and reduces
k and y steady state
kn*
k
Analysis in growth rates
Distance between lines is
growth rate of capital per worker
Can illustrate above
with graph of gk and k
net
investment
dk
dk
 sy  (  n ) k 
dt
dt  g  s y  (  n )
k
k
k
Distance
y between lines represents growth
s per
s  average
of capital
in capital
worker (gproduct
k)
k
n
net disinvestment
k*
k
capital per worker
Rise in
savings rate
(s0 to s1)
gk
y
s0
k
s1
y
k
B
C
A
n
k
k*
g Y, g k
gY=(MPK/APK) gk = sk gk
(sk =  in Cobb-Douglas case)
NB: This graph of
how growth rates
change over time
0%
Saving change
y
s1
k
Time
Golden rule
• The ‘golden rule’ is the ‘optimal’ saving rate (sG)
that maximises consumption per head.
• Assume A is constant, but population growth is n.
• Can show that this occurs where the marginal
product of capital equals (  n)
P roof:
dk
 sy  (  n ) k  0 at steady state,
dt
hence sy   (  n ) k  , w here * indicates steady state equilibrium value
T he problem is to:
m ax c  y  sy  y *  (   n ) k 
k
First order condition :
0
dy *
dk *
 (  n )
hence M Pk =
dy *
dk *
 n
Graphically find the maximal
distance between two lines
y
output per worker
y
slope=dy/dk=n+
y**
(n)k
maximal
consumption
per worker
sgold y
k**
k
capital per worker
… over saving
y=Ak
output per worker
y
slope=dy/dk=n+
y**
sovery
maximal
consumption
per worker
(n)k
k**
sgoldy
k*
k
capital per worker
Economies can over save. Higher saving does increase GDP
per worker, but real objective is consumption per worker.
Golden rule for Cobb Douglas case
•
•
•
•
Y=KL1-
or
y = k
Golden rule states: MPk = (k*)-1 =(n + )
Steady state is where: sy* = ( +n)k*
Hence, sy* = [(k*)-1]k*
or s = (k*) / y* = 
Golden rule saving ratio =  for Y=KL1- case
Assuming perfect competition, and factors are paid
marginal products,  is share of GDP paid to capital
(see C&S, p.481). Expect this to be 0.1 to 0.3.
Solow’s surprise*
• Solow’s model states that investment in capital cannot
drive long run growth in GDP per worker
• Need technological change (growth in A) to avoid
diminishing returns to capital
• Easterly (2001) argues that “capital fundamentalism”
view widely held in World Bank/IMF from 60s to 90s,
despite lessons of Solow model
• Policy lesson: don’t advise poor countries to invest
without due regard for technology and incentives
* This is title of Chapter 3 in Easterly (2001), which is worth a quick read for
controversy surrounding growth models and development issues
What if technology (A) grows?
• Consider y=Ak, and sy=sAk, these imply
that output can go on increasing.
• Consider marginal product of capital (MPk)
MPk=dy/dk =Ak1,
if A increases then MPk can keep increasing (no
‘diminishing returns’ to capital)
• implies positive long run growth
…. graphically, the production function
simply shifts up
y
Technology growth:
A2 > A1 > A0
2
y=A1k
1
output per worker
y=A2k
y=A0k
0
k
k
Capital to labour ratio
…. mathematically

Y  K ( AL)
E asier to use
1 
w here 0    1
(T his assum es A augm ents labour (H arrod-neutral technological change)

C an re-w rite
dA
A ssum e
dt
K ( AL)
/ A  gA
1 
 A
1 

1
K L
(for reference this sa m e as A t  A o e
g At
)
T rick to solving is to re-w rite as
y
Y


K ( AL)
AL
AL
1 

 K 

=

(
k
)

A
L


w here y = output per 'effective w orker', an d k  capital per 'effective w orker'
C an show
dk

/ k  s(k )  (n  a   )k
dt
T h is can be solved (plotted) as in sim pler S olow m odel.
Output (capital) per effective worker diagram
output per effective worker
Y/AL
(Y/AL)*
Y/AL
NOTE: ‘dilution’ line now
includes technology growth (a)
(na)k
s(Y/AL)
(K/AL)*
K/AL
capital per effective worker
If Y/AL is a constant, the growth of Y must equal the growth rate
of L plus growth rate of A (i.e. n+a)
And, growth in GDP per worker must equal growth in A.
Summary of Solow-Swan
• Solow-Swan, or neoclassical, growth model,
implies countries converge to steady state GDP
per worker (if no growth in technology)
• if countries have same steady states, poorer
countries grow faster and ‘converge’
– call this classical convergence or ‘convergence to
steady state in Solow model’
• changes in savings ratio causes “level effect”, but
no long run growth effect
• higher labour force growth, ceteris paribus, implies
lower GDP per worker
• Golden rule: economies can over- or under-save
(note: can model savings as endogenous)
Technicalities of Solow-Swan
• Textbooks (Jones 1998, and Carlin and Soskice 2006) give
full treatment, in short:
• Inada conditions needed ( “growth will start, growth will
stop”)
dY
dY
lim
 0,
lim
 ,
K   dK
K  0 dK
• It is possible to have production function where dY/dK
declines to positive constant (so growth declines but never
reaches zero)
• Exact outcome of Solow model does depend on precise
functional forms and parameter values
• BUT, with standard production function (Cobb-Douglas)
Solow model predicts economy moves to steady state
because of diminishing returns to capital (assuming no
growth in technology A)
Endnotes
M ath note 1: y t  y 0 e
gt
can be used to analyse i m pact of grow th over tim e
L et y= G D P p.w ., g= grow th (e.g. 0.02  2% ), t= tim e.
H ence, for g  0.02 and t  100, y t / y 0  e  7.39
2
M ath N ote 2:
S tart w ith
dK
 sY   K , divide by L yields
dt
N ote that
sim plify to
hence
dk
dt
hence
dk
dt
dK
dt
dL  2
K 
 dK
d
/
dt

L

K  /L

 dt
dt
L
dt




dk
 dL
K
/L
/L
dt
 dt
 L
dK
+ nk =
/ L  sy   k
dK
/ L  sy   k
dt
= sy  (  n ) k
or
dK
dt
/ L  nk
(quotient rule)
(since n is labour grow th and K / L  k )
Questions for discussion
1. What is the importance of diminishing marginal
returns in the neoclassical model? How do other
models deal with the possibility of diminishing
returns?
2. Explain the effect of (i) an increase in savings
ratio (ii) a rise in population growth and (iii) an
increase in exogenous technology growth in the
neoclassical model.
3. What is the golden rule? Can you think of any
countries that have broken the golden rule?
References
Boltho, A. and G. Toniolo (1999). "The Assessment: The
Twentieth Century-Achievements, Failures, Lessons." Oxford
Review of Economic Policy 15(4): 1-18.
Easterly, W. (2001). The Elusive Quest for Growth: Economists’
Adventures and Misadventures in the Tropics. Boston, MIT Press.
Swan, T. (1956). "Economic Growth and Capital Accumulation."
Economic Record 32: 344-361.
Jones, C. (1998) Introduction to Economic Growth, (W.W. Norton,
1998 First Edition, 2002 Second Edition).
Carlin, W. and D. Soskice (2006) Macroeconomics:
Imperfections, Institutions and Policies, Oxford University Press.
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