fin 30220: macroeconomic analysis long run growth
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FIN 30220: Macroeconomic Analysis
Long Run Growth
The World Economy
Total GDP (2012): $83T Population (2012):7B GDP per Capita (2012): $12,500 Population Growth (2012): 1.1% GDP Growth (2012): 3.3%
GDP per capita is probably the best measure of a country’s overall well being
Region GDP % of World GDP
GDP Per Capita
Real GDP Growth
United States $15T 18% $48,000 1.3%
European Union $16T 19% $33,000 1.0%
Japan $4.3T 6% $34,200 -.4%
China $7.8T 11% $6,000 9.8%
India $3.2T 5% $2,800 6.6%
Ethiopia $66.3B .09% $800 8.5%
Note. However, that growth rates vary significantly across countries/regions. Do you see a pattern here?
Source: CIA World Factbook
At the current trends, the standard of living in China will surpass that of the US in 25 years! Or, will they?
Per
Cap
ita
Inco
me
That is, can China maintain it’s current growth rate?
Income GDP/Capita GDP Growth
Low $510 6.3%
Middle $2,190 7.0%
High $32,040 3.2%
As a general rule, low income (developing) countries tend to have higher average rates of growth than do high income countries
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”
Some countries, however, don’t fit the normal pattern of development
SudanGDP: $80B (#80)GDP Per Capita: $2,400 (#184)GDP Growth: -11.2% (#219)
QatarGDP: $150B (#59)GDP Per Capita: $179,000 (#1)GDP Growth: 16.3% (#1)
So, what is Sudan doing wrong? (Or, what is Qatar doing right?)
At current trends, Per capita income in Qatar will quadruple to $716,000 over the next decade. Over the same time period, per capita GDP in Sudan will drop by roughly 40%to $670!!!
To understand this, let’s look at the sources of economic growth….where does production come from?
LKAFY ,,Real GDP
“is a function of”
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:
LAKY 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)
3
2
3
1
LAKY
Suppose we have the following Cobb-Douglas production function:
A 1% rise in employment raises GDP by 2/3%
A 1% rise in capital raises GDP by 1/3%
We can rewrite the production function in terms of growth rates to decompose GDP growth into growth of factors:
LKAY %3
2%
3
1%%
Real GDP Growth (observable) Employment
Growth (observable)
Capital Growth (observable)
Productivity Growth (unobservable)
Year Real GDP (Billions of 2000 dollars)
Real Capital Stock (Billions of 2000 dollars)
Employment (thousands)
1939 1,142 1,440 29,923
2006 11,257 12,632 135,155
2007 11,467 12,883 137,180
Lets decompose some recent data first…
85.1100*257,11ln467,11ln% Y
97.1100*632,12ln883,12ln% K
48.1100*155,135ln180,137ln% L
Note that capital is growing faster than employment
20.48.13
297.1
3
185.1% A
Year Real GDP (Billions of 2000 dollars)
Real Capital Stock (Billions of 2000 dollars)
Employment (thousands)
1939 1,142 1,440 29,923
2006 11,257 12,632 135,155
2007 11,467 12,883 137,180
Now, lets look at long term averages
39.3100*
68
142,1ln467,11ln%
Y
22.3100*
68
440,1ln883,12ln%
K
23.2100*
68
923,29ln180,137ln%
L
84.23.23
222.3
3
139.3% A
1939 - 1948 1948 - 1973 1973-1993 1993-2007
Output 5.79 4.10 1.96 2.63Capital 3.34 4.24 2.10 2.94Labor 4.46 2.10 1.86 1.60Productivity 1.71 1.28 0.02 0.59
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!
Contributions to growth from capital, labor, and technology vary across time period
Our model of economic growth begins with a production function
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
3
2
3
1
LAKY
We are concerned with capital based growth. Therefore, growth in productivity and employment will be taken as given
Productivity grows at rate
AgPop
Pop
LF
LF
LL
Population grows at rate
Lg
Employment
Labor Force= Employment Ratio
( Assumed Constant)
Labor Force
Population= Participation rate
( Assumed Constant)
3
2
3
1
LAKY
Our simple model of economic growth begins with a production function with one key property – diminishing marginal product of capital
Y
K
),,( LKAF
As the capital stock increases (given a fixed level of employment), the productivity of capital declines!!
K
YMPK
Change in Capital Stock
Change in Production
An economy can’t grow through capital accumulation alone forever!
3
2
3
1
LAKY
K
Y
K
Y
Everything in this model is in per capita terms
3
2
3
1
LAKY Divide both sides by labor to represent our variables in per capita terms
3
13
1
3
2
3
1
3
2
3
1
AkL
KA
LL
LAK
L
Yy
Capital Per Capita
Productivity
Per capita output
In general, let’s assume lower case letters refer to per capita variables
3
1
Aky
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
k
Capital stock per capita
Output per capita
3
1
Aky
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
k
000,1
000,8
6
%2
0
L
K
A
g
g
L
A
8k
1286 3
1
y
Next, assume that households save a constant fraction of their disposable income
TYS
Savings
Income Less Taxes
Constant between zero and one
Again, convert everything to per capital terms by dividing through by the labor force
L
T
L
Y
L
S tys
KEY POINT:
Savings = Household income that hasn’t been spentInvestment = 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
tys S = I
Firms access these funds by taking out loans, issuing stocks and bonds, etc. and use the funds for investment activities
iL
I
Investment per capita
3
1
Aky sy,
k
i s y t
Continuing with our example:
0
%10
t
8k
1286 3
1
y
.10 12 0 1.2i
Investment represents the purchase of new capital equipment. This will affect the capital stock in the future
IKK )1('
Future capital stock current capital stock
Annual Depreciation Rate
Investment Expenditures
We need to write this out in per capita terms as well…
L
I
L
K
L
K )1(
' Divide through by labor to get things in per capita terms
L
I
L
K
LL
LK )1(
'
'' Multiply and divide the left hand side by future labor supply
L
I
L
K
L
L
L
K
)1('
'
'
Recall that labor grows at a constant rate
)1('
LgL
L
ikgk L )1(1' Lg
ikk
1
)1('
We need to write this out in per capita terms as well…
lg
ikk
1
)1('
Future capital stock per capita
Annual depreciation rateCurrent capital per capita
Investment per capita
In our example…
10%
2%
8
1.2
Lg
k
i s
Given
Calculated
The evolution of capital per capita…
Annual population growth rate
24.802.1
2.18)10.1('
k
ikgk L
~)1(1'
kk '
kgi L ~
Just as a reference, lets figure out how much investment per capita would be required to maintain a constant level of capital per capita
Evolution of per capita capital
Assume constant capital per capita
Solve for investment
In our example…
200,1
000,8
000,1
%2
%10
SI
K
L
gL
Given
Calculated
96.802.10.~ i
Just to make sure, lets check our numbers…
In our example…
96.802.10.~ i
000,8
000,1
%2
%10
K
L
gL
lg
ikk
1
)1('
The evolution of capital per capita…
kk
8
02.1
96.8)10.1('
3
1
Aky iiy~,,
k
tysi
kgi L ~
1286 3
1
y
2.1si
96.~ i
8k
Let’s update our diagram…
Actual investment
“break even” investment
Now we have all the components to calculate next years output per capita and the rate of growth
3
1
''' kAy
0
2%
6
8
1.2
' 8.24
A
L
g
g
A
k
i
k
Given
Calculated 11.1224.86' 3
1
y
%91.100*12ln11.12ln ygOutput per capita growth
24.802.1
2.18)10.1('
k
3
1
Aky iiy~,,
k
tysi
kgi L ~
12y
2.1si
96.~ i
8k
Let’s update our diagram…
24.8'k
11.12y
020,1
405,8
6
%2
0
L
K
A
g
g
L
A
Let’s repeat that process again…
24.8'k 11.1224.86 3
1
y
0
%10
tT
211.1011.1210. s
46.8
02.1
211.124.810.1
1
)1('
Lg
ikk
Capital Savings = Investment
Evolution of Capital
Output
New Output
22.1246.86' 3
1
y
Output Growth
%90.100*11.12ln22.12ln yg
Growth is slowing down…why?
3
1
Aky iiy~,,
k
tysi
kgi L ~
i~
The rate of growth depends on the level of investment relative to the “break even” level of investment.
k
si
y
Level of investment needed to maintain current capital stock
Actual investment based on current savings
3
1
Aky iiy~,,
k
tysi
kgi L ~
Eventually, actual investment will equal “break even” investment and growth ceases (in per capita terms). This is what we call the steady state.
ssk
isi~
ssy
The steady state has three conditions….
1
kgi L
Investment is sufficient to maintain a constant capital/labor ratio
2
Savings per capita is a constant fraction of output per capita
tys 3
1
Aky Output is a function of capital per capita
3
Recall that, in equilibrium, savings equals investment
is
With a little algebra, we can solve for the steady state in our example.
%10
%10
0
6
%2
0
t
A
g
g
L
A kgi L Start with condition 3
kgty L Use condition 2 and the fact that savings equals investment
kgtAk L
3
1
Substitute condition 1
kgAk L 3
1Recall that taxes are zero in our example
2
3
Lg
Ak
Solve for k
Plugging in our parameters gives us steady state values.
%10
%10
0
6
%2
0
t
A
g
g
L
A
18.1102.10.
6*10. 2
32
3
Lg
Ak
Steady state per
capita capital
40.1318.116 3
13
1
AkySteady state per capita output
34.1040.1310. tys Steady state per capita savings/investment
18.11
02.1
34.118.1110.1
1
)1('
Lg
ikk
06.12040.139.1 tyc 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.
3
1
Aky iiy~,,
k
tysi
kgi L ~
18.11ssk
34.1si
40.13ssy
In the steady state (with no productivity growth), all per capita variables have zero growth!
Suppose we started out example economy above it’s eventual steady state…
%10
%10
0
000,1
000,15
6
%2
0
t
L
K
A
g
g
L
A
15k
8.14156 3
13
1
Aky
48.108.1410. tysi
7.14
02.1
48.11510.1
1
)1('
Lg
ikk
7.147.146'' 3
13
1
Aky
%68.100*8.14ln7.14ln% y
An economy above its steady state shrinks (in per capita terms) towards its steady state.
3
1
Aky
iiy~,,
k
tysi
kgi L ~
15k
48.1i
8.14y
An economy above its steady state shrinks (in per capita terms) towards its steady state.
8.1~ i
An economy above its steady state can’t generate enough savings to support its capital stock!
y
kSteady State
Countries below their eventual steady state will grown towards it
Investment needed to maintain current capital/labor ratio
Actual investment (equals savings)
Countries above their eventual steady state will shrink towards it
Investment needed to maintain current capital/labor ratio
Actual investment (equals savings)
Countries at their eventual steady state will stay there
“Absolute convergence” refers to the premise that every country will converge towards a common steady state
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!
Productivity
3
1
Aky
Does the economy have a “speed limit”?
Economic Growth can be broken into three components:
GDP Growth
= Productivity Growth + (2/3)Labor Growth + (1/3)Capital Growth
In the Steady State, Capital Growth = Labor Growth
GDP Growth
= Productivity Growth + Employment 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…
Country A
000,1
000,4
%10
%10
0
6
%15
0
L
K
t
A
g
g
L
A
000,1
000,8
%10
%10
0
6
%2
0
L
K
t
A
g
g
L
A
Country B
We already calculated this!
12y%91.yg
52.946 3
13
1
Aky
952.052.910. tysi
95.3
15.1
952.410.1
1
)1('
Lg
ikk
48.995.36'' 3
13
1
Aky
41.100*52.9ln48.9ln yg
Even though Country B is poorer, it is growing slower than country A (in per capita terms)!
iiy~,,
k
itys
kiA 10.02.~
18.11ssAk
With a higher rate of population growth, country B has a much lower steady state than country A!!!
kiB 10.15.~
71.315.10.
6*10. 2
3
2
3
L
ssB
g
Ak
4Bk 8Ak
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.
y
Steady State
(Haiti)
High Population Growth (Haiti)
Low Population Growth (Argentina)
Haiti
Population Growth: 2.3%
GDP/Capita: $1,600
GDP Growth: -1.5%
Argentina
Population Growth: .96%
GDP/Capita: $13,700
GDP Growth: 8.7%Steady State
(Argentina)
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.
y
Steady State
(Zimbabwe)
High Savings Rate (Hong Kong)
Low Savings Rate (Zimbabwe)
Zimbabwe (until recently)
GDP/Capita: $2,100
GDP Growth: -7%
Investment Rate (%0f GDP): 7%
Hong Kong
GDP/Capita: $37,400
GDP Growth: 6.9%
Investment Rate (% of GDP): 21.2%Steady State
(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.
y
Steady State
(France)
Small Government (US)
Large Government (France)
France
GDP/Capita: $30,000
GDP Growth: 1.6%
Government (%0f GDP): 55%
USA
GDP/Capita: $48,000
GDP Growth: 2.5%
Government (% of GDP): 18%Steady State
(USA)
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
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?
3
1
Aky
icy
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)
kgcy L In the steady state, investment simply maintains the existing steady state
kgAkkgyc LL 3
1 Maybe we should be choosing a steady state with the highest level of consumption!
kgAkc L 3
1
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
LgAkdk
dc
3
2
3
1
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
3
1
Aky iy~,
k
kgi L ~
*k
03
1 3
2
LgAkdk
dc
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
Consumption equals zero – capital maintenance requires all of production
Steady state consumption is maximized!!!
03
1 3
2
LgAk
Let’s go back to our example…
2
3
3*
Lg
Ak
%10
%10
0
6
%2
0
t
A
g
g
L
A
We can solve for the steady state capital that maximizes consumption
6802.10.3
6*
2
3
k
3
1
Aky
iiy~,,
k
tysi 10.
kgi L ~
353max k
Maximum sustainable capital stock – consumption equals zero
68*k11*k
Steady state capital that maximizes consumption
Steady state with a 10% investment rate
%10
%10
0
6
%2
0
t
A
g
g
L
A
Using our example, lets compare consumption levels…
6802.10.3
6*
2
3
k18.1102.10.
6*10. 2
3
k
40.1318.116 3
1
y 5.24686 3
1
y
34.118.1110.02. i 16.86810.02. i
06.1234.140.13 c
Steady State with Savings Rate = 10% “Optimal” Steady State
34.1616.85.24 c
In this example, we could increase consumption by 30% by altering the savings rate!!
%10
%10
0
6
%2
0
t
A
g
g
L
A
By comparing steady states, we can find the savings rate associated with maximum consumption
2
3
31
*
Lg
Ak
2
3
Lg
Ak
Steady State with a given Savings Rate “Optimal” Steady State
To maximize steady state consumption, we need a 33% savings/investment rate!!
3
1*
So, where does the US stand?
Production (2008) Consumption (2008) Investment (2008) Government Purchases (2008)
$14,264B $10,057B $1,994B $2,882B
The savings rate in the US is currently around 4%, but what we really want is the investment rate
%14100*264,14
994,1
%10
%2
%5.11
L
A
g
g
3
1
Aky
At our steady state, GDP growth should be 3-3.5% (Per capita GDP will grow at 1-1.5%)
We need to calibrate the model to the data…
Production (2008) Consumption (2008) Investment (2008) Government Purchases (2008)
$14,264B $10,057B $1,994B $2,882B
%14
%10
%2
%5.11
L
A
g
g2
1
2
33
1
2
3
3
1
LL g
Ag
AAAky
In the steady state…
2
3
Lg
Ak
We know everything but the value for productivity
GDP Per Capita (2008) = $47,000
2
1
2
3
02.10.
14.000,47
A 194,1A
Now, suppose that we could increase the investment rate to 33% as our model prescribes
Production (2008) Consumption (2008) Investment (2008) Government Purchases (2008)
$14,264B $10,057B $1,994B $2,882B
%33
%10
%2
%5.11
L
A
g
g
075,6802.10.
33.194,1
2
1
2
3
y
In the steady state…
We could raise per capita income to close to $70,000
Investment expenditures would be 33%, or $22,464.
610,45$464,22075,68 iygc
Private consumption per capita
government purchases per capita
Currently, government plus private consumption per capita is around $43,000
yi 33.iiy~,,
k
yi 14.
ki 10.02.~
k
Projected Steady State (y=$68,000)
Current Steady State (y = $47,000)
Should we pursue policies to raise the investment rate in the US?
'k
c
time
Is a higher steady state worth the transition?
Growth of per capita consumption under old policy regime = 1.5%
Immediate drop in consumption as economy responds to policy change
Growth of per capita consumption increases during transition period
Growth of per capita consumption returns to 1.5%
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