FIN 30220: Macroeconomic
Analysis
Long Run Growth
The World Economy





Total GDP (2013): $87T
Population (2013):7.1B
GDP per Capita (2013): $13,100
Population Growth (2013): 1.0%
GDP Growth (2013): 2.9%
GDP per capita is probably the best measure of a country’s overall well
being
Note. However, that growth rates vary significantly across countries/regions. Do
you see a pattern here?
Region
GDP
% of
World
GDP
GDP Per
Capita
Real GDP
Growth
United States
$17T
20%
$53,000
1.6%
European Union
$16T
18%
$35,000
0.1%
Japan
$4.7T
5%
$36,300
2.0%
China
$13T
15%
$9,800
7.7%
Ghana
$90B
.1%
$3,500
7.9%
Ethiopia
$118.2B
.13%
$1,300
7.0%
Source: CIA World Factbook (2013 Estimates)
Per Capita Income
At the current trends, the standard of living in China will surpass that of the US in
25 years! Or, will they?
That is, can China maintain it’s current growth rate?
As a general rule, low income (developing) countries tend to have
higher average rates of growth than do high income countries
Income
GDP/Capita
GDP Growth
Low
< $1,045
6.3%
Middle
$1,045 - $12,746
4.8%
High
>$12,746
3.2%
The implication here is that eventually, poorer countries should
eventually “catch up” to wealthier countries in terms of per capita income
– a concept known as “convergence”
Source: World Bank (2013 estimates)
Some countries, however, don’t fit the normal pattern of development
Sudan
GDP: $107B (#73)
GDP Per Capita: $5,100 (#159)
GDP Growth: -2.3% (#213)
Macau
GDP: $51.6B (#98)
GDP Per Capita: $88,700 (#3)
GDP Growth: 11.9% (#5)
At current trends, Per capita income in Macau will triple to
$273,000 over the next decade. Over the same time period, per
capita GDP in Sudan will drop by roughly 25%to $4,000!!!
So, what is Sudan doing wrong? (Or, what is Macau doing right?)
To understand this, let’s look at the sources of economic growth….where
does production come from?
“is a function of”
Real GDP
Y  F  A, K , L
Productivity Capital
Stock
Labor
Real GDP = Constant Dollar (Inflation adjusted) value of all goods and
services produced in the United States
Capital Stock = Constant dollar value of private, non-residential fixed assets
Labor = Private Sector Employment
Productivity = Production unaccounted for by capital or labor
A convenient functional form for growth accounting is the Cobb-Douglas
production function. It takes the form:


Y  AK L
where
   1
With the Cobb-Douglas production function, the parameters have
clear interpretations:


Capital’s share of income (what %
of total income in the US accrues to
owners of capital)
Labor’s share of income (what % of
total income in the US accrues to
owners of labor)
Elasticity of output with respect to
capital (% increase in output
resulting from a 1% increase in
capital)
Elasticity of output with respect to
labor (% increase in output resulting
from a 1% increase in labor)
Suppose we have the following Cobb-Douglas production function:
A 1% rise in capital
raises GDP by 1/3%
A 1% rise in
employment raises
GDP by 2/3%
1
3
2
3
Y  AK L
We can rewrite the production function in terms of growth rates to decompose
GDP growth into growth of factors:
1
2
%Y  %A  %K   %L 
3
3
Real GDP Growth
(observable)
Productivity Growth
(unobservable)
Capital Growth
(observable)
Employment
Growth
(observable)
Year
Real GDP (Billions of
2009 dollars)
Real Capital Stock
(Billions of 2005 dollars)
Employment (thousands)
2010
14,939
40,615
130,745
2011
15,190
40,926
132,828
Lets decompose some recent data first…
%Y  ln 15,190   ln 14,939   *100  1.67
%K  ln  40,926   ln  40, 615   *100  .76
%L  ln 132,828   ln 130, 745   *100  1.58
%A  1.67 
1
2
.76   1.58   .36
3
3
*Source: Penn World Tables
Year
Real GDP (Billions of
2009 dollars)
Real Capital Stock
(Billions of 2005 dollars)
Employment (thousands)
1950
2,273
6,328
46,855
2011
15,190
40,926
132,828
Now, lets look at long term averages
ln 15,190   ln  2, 273
%Y 
*100  3.11
61
ln  40,926   ln  6,328 
%K 
*100  3.06
61
ln 132,828  ln  46,855
%L 
*100  1.70
61
%A  3.11 
1
2
 3.06   1.70   .98
3
3
Contributions to growth from capital, labor, and technology vary across time
period
1939 1948
1948 1973
19731990
19902007
20072013
Output
5.79
4.00
3.10
3.60
1.1
Capital
3.34
3.70
4.20
4.10
1.4
Labor
4.46
1.00
1.90
1.60
-0.1
Productivity
1.71
2.1
0.5
1.2
0.7
A few things to notice:
Real GDP growth is declining over time.
Capital has been growing faster than labor
The contribution of productivity is diminishing!
Annual Growth
Generally speaking, productivity growth has been declining since WWII
Our model of economic growth begins with a production function
1
3
2
3
Y  AK L
Real GDP
Productivity Capital
Stock
Labor
Given our production function, economic growth can result
from
• Growth in labor
• Growth in the capital stock
• Growth in productivity
We are concerned with capital based growth. Therefore, growth in productivity
and employment will be taken as given
1
3
2
3
Y  AK L
Population
grows at rate
gL
Productivity
grows at rate
 L  LF 
 Pop
L

 LF  Pop 
gA
Employment
Labor Force
= Employment Ratio
( Assumed Constant)
Labor Force
Population
= Participation rate
( Assumed Constant)
1
3
Our simple model of economic growth
begins with a production function with
one key property – diminishing marginal
product of capital
Change in Production
Y  AK L
Y
F ( A, K , L )
Y
Y
MPK 
K
K
Change in Capital Stock
As the capital stock
increases (given a fixed
level of employment), the
productivity of capital
declines!!
2
3
Y
K
An economy can’t grow through capital accumulation alone forever!
K
Everything in this model is in per capita terms
1
3
2
3
Y  AK L
Divide both sides by labor to represent our variables in per capita terms
1
3
2
3
1
3
Y
AK L
K
3
y


A

Ak


1 2
L
 L
3 3
L L
Per capita
output
1
Capital Per Capita
Productivity
In general, let’s assume lower case letters refer to per capita variables
Again, the key property of production is that capital exhibits diminishing marginal
productivity – that is as capital rises relative to labor , its contribution to production
of per capita output shrinks
y
Output per capita
y  Ak
1
3
Capital stock per
capita
k
Lets use an example. The current level of capital per capita will determine
the current standard of living (output per capita = income per capita)
y
y  Ak
1
3
gA  0
1
y  683  12
g L  2%
A6
K  8,000
L  1,000
k
k 8
Next, assume that households save a constant fraction of their disposable
income
Income Less Taxes
S   Y  T 
Savings
Constant between zero and one
Again, convert everything to per capital terms by dividing through by the
labor force
S
Y T 
  
L
L L
s   y  t 
KEY POINT:
Savings = Household income that hasn’t been spent
Investment = Corporate purchases of capital goods (plant, equipment, etc)
The role of the financial sector is to make funds saved by households
available for firms to borrow for investment activities
Households save
their income by
opening savings
accounts, buying
stocks and bonds,
etc
Firms access these
funds by taking out
loans, issuing stocks
and bonds, etc. and
use the funds for
investment activities
s   y  t 
I
i
L
S=I
Investment
per capita
Continuing with our example:
  10%
t 0
y, s
y  Ak
1
3
1
y  683  12
i  s   y t
i  .10 12  0   1.2
k 8
k
Investment represents the purchase of new capital equipment. This
will affect the capital stock in the future
Annual Depreciation
Rate
K '  (1   ) K  I
Future capital stock
Investment
Expenditures
current capital stock
We need to write this out in per capita terms as
well…
We need to write this out in per capita terms as well…
K'
K I
 (1   ) 
L
L L
K ' L'
K I
 (1   ) 
L' L
L L
Divide through by labor to get things in per capita
terms
Multiply and divide the left hand side by future
labor supply
Recall that labor grows at a constant rate
K '  L' 
K I
   (1   )  
L'  L 
L L
L'
 (1  g L )
L
k 1  g L   (1   )k  i
'
(1   )k  i
k 
1 gL
'
The evolution of capital per capita…
Annual depreciation rate
Current capital per
capita
(1   )k  i
k'
1  gl
Future capital stock per
capita
Investment per
capita
Annual population
growth rate
In our example…
  10%
g L  2%
Given
k 8
i  s  1.2
Calculated
(1  .10)8  1.2
k'
 8.24
1  .02
Just as a reference, lets figure out how much investment per capita would be
required to maintain a constant level of capital per capita
~
k 1  g L   (1   )k  i
'
k k
'
Evolution of per capita capital
Assume constant capital per capita
~
i    g L k
Solve for investment
In our example…
  10%
g L  2%
L  1,000
Given
K  8,000
I  S  1,200
Calculated
~
i  .10  .028  .96
Just to make sure, lets check our numbers…
In our example…
  10%
g L  2%
L  1,000
K  8,000
~
i  .10  .028  .96
The evolution of capital per capita…
(1   )k  i
k'
1  gl
(1  .10)8  .96
k'
8k
1  .02
Let’s update our diagram…
“break even” investment
~
i  g L   k
~
y , i, i
y  Ak
1
3
i  s   y  t 
1
y  683  12
Actual investment
i  s  1.2
~
i  .96
k 8
k
Now we have all the components to calculate next years output per capita and the
rate of growth
(1  .10)8  1.2
k'
 8.24
1  .02
gA  0
g L  2%
A6
k 8
i  1.2
k '  8.24
Output per
capita growth
Given
Calculated
y '  A' k '
1
3
y'  68.24  12.11
1
3
g y  ln 12.11  ln 12 *100  .91%
Let’s update our diagram…
~
i  g L   k
~
y , i, i
y  Ak
1
3
i  s   y  t 
y  12.11
y  12
i  s  1.2
~
i  .96
k 8
k '  8.24
k
Let’s repeat that process again…
  10%
T t 0
gA  0
g L  2%
A6
K  8,405
L  1,020
Capital
k '  8.24
Savings = Investment
Output
1
3
y  68.24  12.11
Evolution of Capital
(1   )k  i 1  .108.24  1.211
k' 

 8.46
1 gL
1.02
s  .1012.11  0  1.211
New Output
1
3
y'  68.46  12.22
Output Growth
g y  ln 12.22   ln 12.11*100  .90%
Growth is slowing down…why?
The rate of growth depends on the level of investment relative to the “break even”
level of investment.
Actual investment
based on current
savings
~
y , i, i
~
i  g L   k
y  Ak
1
3
i  s   y  t 
y
is
~
i
Level of investment
needed to maintain
current capital stock
k
k
Eventually, actual investment will equal “break even” investment and growth ceases
(in per capita terms). This is what we call the steady state.
~
i  g L   k
~
y , i, i
y  Ak
yss
1
3
i  s   y  t 
~
isi
k ss
k
The steady state has three conditions….
1
y  Ak
1
3
2
s   y  t 
Savings per capita is a
constant fraction of
output per capita
Output is a function
of capital per capita
3
i    g L k
Investment is sufficient
to maintain a constant
capital/labor ratio
Recall that, in
equilibrium, savings
equals investment
si
With a little algebra, we can solve for the steady state in our example.
gA  0
g L  2%
A6
t0
  10%
  10%
i    g L k
Start with condition 3
  y  t     g L k
 13 
  Ak  t     g L k


Ak
1
3
   g L k
 A 

k  
   gL 
3
2
Use condition 2 and the fact that
savings equals investment
Substitute condition 1
Recall that taxes are zero in our
example
Solve for k
Plugging in our parameters gives us steady state values.
gA  0
g L  2%
A6
t0
  10%
  10%
3
2
3
2
 A 
 .10 * 6 
  
k  
  11.18
 .10  .02 
   gL 
1
3
y  Ak  611.18  13.40
1
3
Steady state per
capita capital
Steady state per
capita output
s    y  t   .1013.40  0  1.34
Steady state per capita
savings/investment
c  1    y  t   .913.40  0  12.06
k'
(1   )k  i 1  .1011.18  1.34

 11.18
1 gL
1  .02
Steady state per capita
consumption
Constant per capita
capital!!!
Eventually, actual investment will equal “break even” investment and growth ceases
(in per capita terms). This is what we call the steady state.
~
i  g L   k
~
y , i, i
y  Ak
yss  13.40
1
3
i  s   y  t 
i  s  1.34
In the steady state (with no
productivity growth), all per capita
variables have zero growth!
k ss  11.18
k
Suppose we started out example economy above it’s eventual steady
state…
1
3
y  Ak  615  14.8
gA  0
g L  2%
A6
K  15,000
L  1,000
t 0
  10%
  10%
k  15
1
3
i  s    y  t   .1014.8  0  1.48
k'
(1   )k  i 1  .10 15  1.48

 14.7
1 gL
1  .02
1
3
y'  Ak '  614.7   14.7
1
3
%y  ln 14.7  ln 14.8*100  .68%
An economy above its steady state shrinks (in per capita terms) towards its steady state.
An economy above its steady state shrinks (in per capita terms) towards its steady state.
~
i  g L   k
~
y , i, i
y  Ak
1
3
y  14.8
~
i  1.8
i  s   y  t 
i  1.48
An economy above its steady
state can’t generate enough
savings to support its capital
stock!
k  15
k
“Absolute convergence” refers to the premise that
every country will converge towards a common steady
state
y
Investment needed
to maintain current
capital/labor ratio
Actual investment
(equals savings)
Actual investment
(equals savings)
Investment needed
to maintain current
capital/labor ratio
Countries below their
eventual steady state
will grown towards it
Steady
State
k
Countries above their
eventual steady state will
shrink towards it
Countries at their eventual
steady state will stay there
Most countries follow the “usual” pattern of development
1
Developing countries have very little capital, but A LOT of labor. Hence,
the price of labor is low, the return to capital is very high
2
High returns to capital attract a lot of investment. As the capital stock
grows relative to the labor force, output, consumption, and real wages
grow while interest rates (returns to capital fall)
3
Eventually, a country “matures” (i.e. reaches its steady state level of
capital). At this point, growth can no longer be achieved by investment
in capital. Growth must be “knowledge based” – improving
productivity!
y  Ak
Productivity
1
3
Does the economy
have a “speed
limit”?
Economic Growth can be broken into three components:
GDP
= Productivity Growth + (2/3)Labor Growth + (1/3)Capital Growth
Growth
In the Steady State, Capital Growth = Labor Growth
GDP
Growth
= Productivity Growth + Labor Growth
( GDP Per Capita Growth = Productivity Growth)
Developing countries are well below their steady state and, hence should grow
faster than developed countries who are at or near their steady states – a
concept known as absolute convergence
Examples of Absolute Convergence (Developing Countries)
China (GDP per capita = $6,300, GDP Growth = 9.3%)
Armenia (GDP per capita = $5,300, GDP Growth = 13.9%)
Chad (GDP per capita = $1,800, GDP Growth = 18.0%)
Angola (GDP per capita = $3,200, GDP Growth = 19.1%)
Examples of Absolute Convergence (Mature Countries)
Canada (GDP per capita = $32,900, GDP Growth = 2.9%)
United Kingdom (GDP per capita = $30,900, GDP Growth = 1.7%)
Japan (GDP per capita = $30,700, GDP Growth = 2.4%)
Australia (GDP per capita = $32,000, GDP Growth = 2.6%)
Some countries, however, don’t fit the traditional pattern.
Developing Countries with Low Growth
Madagascar(GDP per capita = $900, GDP Growth = - 2.0%)
Iraq (GDP per capita = $3,400, GDP Growth = - 3.0%)
North Korea (GDP per capita = $1,800, GDP Growth = 1.0%)
Haiti (GDP per capita = $1,200, GDP Growth = -5.1%)
Developed Countries with high Growth
Hong Kong (GDP per capita = $37,400, GDP Growth = 6.9%)
Iceland (GDP per capita = $34,900, GDP Growth = 6.5%)
Singapore (GDP per capita = $29,900, GDP Growth = 5.7%)
Qatar (GDP Per Capita = $179,000, GDP Growth = 16.3%)
Consider two countries…
We already
calculated
this!
Country A
Country B
gA  0
gA  0
g L  2%
g L  15%
A6
t0
  10%
  10%
K  8,000
L  1,000
A6
t0
  10%
  10%
K  4,000
L  1,000
1
3
y  Ak  64  9.52
1
3
i  s    y  t   .109.52  0  .952
k'
(1   )k  i 1  .10 4  .952

 3.95
1 gL
1  .15
1
3
y'  Ak '  63.95  9.48
1
3
g y  ln 9.48  ln 9.52 *100  .41
y  12
g y  .91%
Even though Country B is poorer, it is growing
slower than country A (in per capita terms)!
With a higher rate of population growth, country B has a much lower steady state than country A!!!
~
y , i, i
~
iB  .15  .10k
~
iA  .02  .10k
s   y  t   i
k
B
ss
 A 


   gL 
3
2
k 4
B
3
 .10*6  2

  3.71
 .10  .15 
k 8
A
k
A
ss
 11.18
k
Conditional convergence suggests that every country converges to its own unique
steady state. Countries that are close to their unique steady state will grow slowly
while those far away will grow rapidly.
Haiti
Population Growth: 2.3%
High Population Growth
(Haiti)
y
Low Population Growth
(Argentina)
GDP/Capita: $1,600
GDP Growth: -1.5%
Argentina
Population Growth: .96%
GDP/Capita: $13,700
GDP Growth: 8.7%
Steady
State
Steady State
(Argentina)
(Haiti)
Haiti is currently ABOVE its
steady state (GDP per capita
is falling due to a high
population growth rate
Argentina, with its low
population growth is well
below its steady state
growing rapidly towards it
Conditional convergence suggests that every country converges to its own unique
steady state. Countries that are close to their unique steady state will grow slowly
while those far away will grow rapidly.
Zimbabwe (until recently)
GDP/Capita: $2,100
y
GDP Growth: -7%
High Savings Rate
(Hong Kong)
Investment Rate (%0f GDP): 7%
Hong Kong
Low Savings Rate
(Zimbabwe)
GDP/Capita: $37,400
GDP Growth: 6.9%
Investment Rate (% of GDP): 21.2%
Steady State
Steady State
(Zimbabwe)
(Hong Kong)
Zimbabwe is currently
ABOVE its steady state (GDP
per capita is falling due to
low investment rate
Hong Kong, with its high
investment rate is well below
its steady state growing
rapidly towards it
Conditional convergence suggests that every country converges to its own unique
steady state. Countries that are close to their unique steady state will grow slowly
while those far away will grow rapidly.
France
GDP/Capita: $30,000
y
GDP Growth: 1.6%
Small Government (US)
Government (%0f GDP): 55%
Large Government
(France)
USA
GDP/Capita: $48,000
GDP Growth: 2.5%
Government (% of GDP): 18%
Steady State Steady State
(France)
France has a lower steady state due
to its larger public sector. Even
though its per capita income is lower
than the US, its growth is slower
(USA)
The smaller government of
the US increases the steady
state and, hence, economic
growth
Suggestions for growth….
High income countries with low growth are at or near their steady state.
Policies that increase capital investment will not be useful due to the
diminishing marginal product of capital.
Consider investments in technology and human capital to increase
your steady state.
Consider limiting the size of your government to shift resources to
more productive uses (efficiency vs. equity)
Low income countries with low growth either have a low steady state or are
having trouble reaching their steady state
Consider policies to lower your population growth.
Try to increase your pool of savings (open up to international capital
markets)
Policies aimed at capital formation (property rights, tax credits, etc).
Question: Is maximizing growth a policy we should be striving for?
y  Ak
1
3
y  ci
Our model begins with a relationship
between the capital stock and production
These goods and services that we produce
can either be consumed or used for
investment purposes (note: taxes are zero)
y  c    g L k
1
3
In the steady state, investment simply
maintains the existing steady state
c  y    g L k  Ak    g L k
Maybe we should be choosing a steady
state with the highest level of
consumption!
Steady state consumption is a function of steady state capital. If we want to maximize steady
state consumption, we need to look at how consumption changes when the capital stock changes
1
3
c  Ak    g L k
2

dc 1
 Ak 3    g L 
dk 3
Change in
consumption
per unit change
in steady state
capital
Change in
production per
unit change in
steady state
capital
Change in
capital
maintenance
costs per unit
change in
steady state
capital
2

dc 1
 Ak 3    g L   0
dk 3
~
y, i
~
i  g L   k
y  Ak
1
3
Consumption equals zero – capital
maintenance requires all of
production
Steady state
consumption is
maximized!!!
k
k*
In this region, an increase in capital
increases production by more than
the increase in maintenance costs
– consumption increases
In this region, an increase in capital
increases production by less than
the increase in maintenance costs
– consumption decreases
Let’s go back to our example…
2

1
Ak 3    g L   0
3


A

k *  
 3  g L  
3
2
We can solve for the steady state
capital that maximizes consumption
gA  0
g L  2%
A6
t0
  10%
  10%
3
2


6
  68
k *  
 3.10  .02  
~
i  g L   k
~
y , i, i
y  Ak
1
3
i  s  .10 y  t 
k*  11
Steady state with a
10% investment rate
k*  68
Steady state capital
that maximizes
consumption
kmax  353
k
Maximum sustainable
capital stock –
consumption equals
zero
Using our example, lets compare consumption levels…
gA  0
Steady State with Savings Rate = 10%
g L  2%
A6
t0
  10%
  10%
3
2
 .10 * 6 
k 
  11.18
 .10  .02 
“Optimal” Steady State
3
2


6
  68
k *  
 3.10  .02  
y  611.18  13.40
y  668  24.5
i  .02  .1011.18  1.34
i  .02  .1068  8.16
1
3
c  13.40 1.34  12.06
1
3
c  24.5  8.16  16.34
In this example, we could increase
consumption by 30% by altering
the savings rate!!
By comparing steady states, we can find the savings rate associated with
maximum consumption
gA  0
g L  2%
A6
t0
  10%
  10%
“Optimal” Steady State
Steady State with a given Savings Rate
 A 

k  
   gL 
 
 1 A
k*   3 
  g 
L


3
2
* 
1
3
To maximize steady state
consumption, we need a 33%
savings/investment rate!!
3
2