lecture 7 economic growth. it’s amazing how much we have achieved
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Lecture 7
Economic Growth
It’s amazing how much we have achieved
But huge difference across countries
Country comparisons
GDP
http://www.google.com/publicdata?ds=wb-wdi&met=ny_gdp_mktp_cd&idim=country:USA&dl=en&hl=en&q=gdp
GDP growth rates http://www.google.com/publicdata?ds=wb-wdi&met=ny_gdp_mktp_kd_zg&idim=country:USA&dl=en&hl=en&q=gdp+growth+of+us
Growth and differences
Nigeria is only 1/43 of the US.
We study Why so much growth Why so much difference
Robert Solow: 1924 --
Won Nobel Prize in economics in 1979 for his contribution in the growth theory.
Basic idea
Previously we know output is mainly determined by Capital stock Labor
We focus on capital stock.
Basic idea
Solow considers How capital stock increases How capital stock decreases
Equilibrium is reached when:Increase of the capital stock = decrease of the capital stock
The accumulation of capital stock
Per capita production function
The per worker production function is:
1, LKLKFY
kL
K
L
L
L
K
L
LK
L
Yy
1
11
Per capita production function.
The marginal production of capital, MPK:
MPK is obtained by taking the first derivatives from the aggregate production function, or from the per capita production function.
k
yk
L
KLK
K
YMPK
11
11
Per capita production function
Per capita production function
At per capita level: y = c + i
Per capita consumption is:c = (1 – s) y.
Rearrange terms, we get: i = s*y=s*f(k).
Output/investment graph
Evolvement of capital stock
Capital stock: Increases if investment. Decrease if depreciation.
Each period, Amount of increase: It Amount of depreciation: δK
ttt IKK 11
If labor force does not change
At per capita level
Rearrange this:
ttt ikk 11
11 )( ttttt kksfkik
Equilibrium:
At the equilibrium, we must have:
At the steady state level k*, we have:
*1
1 0
kkk
kksfk
tt
ttt
** kksf
Equilibrium
Discussions:
Why steady state?
If k > k*: From the graph, Depreciation > investment k level would decrease.
If k < k*:From the graph, Depreciation < investment k level would increase.
** kksf
** kksf
Discussions: an increase in saving rate
Saving rate and per capita output
A key prediction of the Solow model is that higher saving would be the cause of higher per capita output.
Investment and per capita output
Investment and per capita output
Discussions:
A higher level of saving would lead to a higher level of per capita output
The most important growth policy is the policy of raising the saving rate.
China experience: GDP growth rates
http://www.google.com/publicdata?ds=wb-wdi&met=ny_gdp_mktp_kd_zg&idim=country:CHN&dl=en&hl=en&q=china+gdp+growth+rates
Example: China’s gross national saving as a percentage of GDP
Public policies that affect saving rates
Public policies that may raise savings rates: Tax benefits for IRA, Roth IRA, 401K, 403B, and
529 raise private saving rate. Reducing budge deficit would raise the public
saving and hence the total saving. Reducing trade deficit would raise the total both
public and private saving. Reducing capital gains tax.
Establishing social security and Medicare would reduce demand for precautionary saving.
28
China’s saving rates
Households
Enterprises
Government
China’s problem: saving is too high
Various measures of reducing savings are apparently not successful. Expand the enrollment of higher
education and raise the tuition for higher education.
However, it creates wrong incentives – some parents now would save for higher education while others pay for higher educations.
New enrollment is six times as much in 2010 than in 1999
New College Enrollments in China
0
1000
2000
3000
4000
5000
6000
7000
1985 1990 1995 2000 2005 2010
Year
En
rollm
en
ts (
in 1
,00
0)
Reducing saving in China
Establishing social safety network Nationwide health insurance
1998 – urban employees 2003 – rural residents 2007 – urban residents (non-
employees)
It helped reducing save rate but not much (increasing consumption by roughly 10%).
Reducing saving rate in China
This is important for US because of the large trade deficit between US and China.
So far, nothing worked.
Compromise between saving and consumption
A higher saving rate higher per capita output in the future but a lower consumption rate.
In the extreme case, a saving rate 100% no current consumption.
The Golden Rule level of capital stock
“Golden rule” the steady level consumption is the highest.
At steady state, we have:
** kksf
The Golden Rule level of capital stock
The steady state level of consumption:
Maximizing c* to get the Golden rule level of consumption:
****** 1 kkfksfkfysc
0' **
*
kfk
c
The Golden Rule level of capital stock
The Golden rule of capital stock is given by:
MPK = δ
The Golden Rule level of capital stock
A numerical example
Production function: Depreciation rate: δ = 0.1 At the optimum:
k* = 25
2/1ky
2/15.0 kMPK
225.0* k
A numerical example
In the steady state:
s = 0.5
** kksf
2/1**
*
kkf
ks
Summarize:
Two unknowns, saving rate s, and optimal level of capital stock k.
Two equations: Golden rule equation:
Steady state equations:
0' Gkf
GG kksf
Population growth
Assume population grows at n, ΔL/L = n.
The evolvement of capital stock remains at:
ΔK = I – δK
The evolvement of per-labor capital stock is more complicated:
Evolvement of per-labor capital stock
knksf
nkkiL
K
L
L
L
KI
L
LK
L
K
L
Kk
2
Population growth
The steady state is determined by: Δk=0
Therefore, at the steady state,
** knksf
Population growth
Population growth
Prediction: a higher population growth rate, a lower level of per capita capital stock and output.
Population growth
Population growth
Discussions
Causality: here it is suggested that a higher population growth rate a lower per capita output.
It is possible that the reverse causality is true:a higher per capita output a lower population growth
Discussions
Reasons for reverse causality: In poor countries, children sometimes
serve as the saving for retirement. A higher income would reduce such demand.
Richer people would enjoy leisure more and hence less likely to have more children.
US data: income and number of children
Technology
To introduce technology growth, we introduce a concept of efficiency labor, E. A higher E means that labor becomes more effective.
Production function now becomes:
1, ELKELKFY
Technology
We now work with per-efficiency laborer capital stock:
Define:
Let the growth rate of E be g:
EL
Yy
EL
Kk ,
E
Eg
Technology
The evolvement of aggregate capital stock remains the same: ΔK = I – δK
The evolvement of the little k:
kgnksf
ngkkys
L
L
E
E
EL
K
EL
KsYEL
LELE
EL
K
EL
KI
EL
LELEK
EL
K
EL
Kk
2
Technology
ELK
EL
K
EL
Kk
1
EL
1
Technology
At the steady state
Δk =0
** kgnksf
The steady state
The Golden rule
c = y – i (output – investment)
= y – sy (output – saving)
(at steady state, sy = (δ + n + g) k)
c = y – (δ + n + g) k = f(k) - (δ + n + g) k
The Golden rule
The first order condition:f’(k) - (δ + n + g) = 0
The Golden Rule level capital stock:
Or: MPK = δ + n + g
gnkf G '
Summary
Symbol steady-state growth rate
capital per effective worker k = K/(E*L) 0 output per effective worker y = Y/(E*L) = f(k) 0 output per worker Y/L = y * E g capital per worker K/L = k * E g total output Y = y * (E * L) g+n total capital K = k * (E * L) g+n
Discussions:
To calculate the Golden Rule saving rate, two equations and two unknowns:
The Golden rule equation:
Steady state equation:
GG kgnksf
gnkf G '
A numerical example
In the US, we have:
k = 2.5y δk = 0.1y depreciation MPK*k = 0.3y income for the owners of
capital stock
What is the Golden-rule saving rate?
A numerical example
The depreciation rate: δ = 0.1y/k=0.1/2.5y = 0.04
MPK = 0.3y/k = 0.3y/2.5y = 0.12 The Golden rule level:
Given n = 0.01, δ = 0.04, and g < 0.07 We have:
gnMPK G
12.0 MPKMPK G
A numerical example
Current MPK is too high, suggesting
We should invest more
Our saving rate is probably too low.
A numerical example:
Now to obtain the Golden rule saving rate:
The Golden-rule:
MPK = δ + n + g
0.3y/k = δ + n + g
k(δ + n + g) = 0.3y
A numerical example
At the steady state:
sy = k(δ + n + g)
the golden rule saving rate is at:
s = k(δ + n + g) / y
Therefore:
3.0Gs
A numerical example
The optimal (the Golden rule) saving rate for US is roughly 30%
Current US saving rate is:
http://www.bea.gov/briefrm/saving.htm
US net national savings rate
US net private savings rate
Current US savingWhy increase?
US government savings
Discussions
Convergence: the Solow model suggests convergence to the steady state where the growth rate would tend to be the same.
Evidence from states within US support this.
Evidence across countries not necessarily true.
Discussions
Solow model can only explain a small portion of the variations across countries.
Consider US and Mexico.
Per capita income: US/Mexico = 4
Numerical example
(1) U.S. and Mexico both have the Cobb-Douglas production function Y= K1/2L1/2.
(2) Suppose technology growth is zero in both countries.
(3) Other information:
US Mexico n 0.01 0.025 δ 0.04 0.04 s 0.22 0.16
per capita income: yus/ymexico= 4
What is the ratio of the two countries according to the Solow model?
A numerical example
At the steady state:sy = (n+δ)k
y = s/(n+δ)
knsk 2/1
nssk /2/1
US-Mexico
79.1
04.0025.0
16.004.001.0
22.0
Mexico
Mexico
US
US
Mexico
US
n
sn
s
y
y
US-Mexico
Therefore, according to the Solow model, the ratio between US and Mexico is 1.79, much smaller than the actual GDP per capita, which is 4.
So the Solow model can only explains a small portion of the ratio.
What is the potential problem?
US-Mexico
A potentially different efficiency E.
A different level of E
Consider the Solow model with technology growth
sy = (n+δ+g)k
US
Mexico
US
Mexico
MexicoMexico
USUS
MexicoMexico
Mexico
USUS
US
Mexico
US
E
E
E
E
LY
LY
LE
YLE
Y
y
y 4
/
/79.1
A different level of E
A Mexico worker is 45% of efficiency of a US worker’s level.
4475.4
79.1
US
Mexico
E
E
Endogenous growth theory
Basic idea: investment, especially investment in R&D, would lead to higher productivity.
Suppose E = B* K/L
AKKBLLKBKLEKY 111 /
Endogenous growth model
Consider the evolvement of capital stock:
ΔK = sY – δK = sAK – δK = (sA – δ)K
Increase of capital stock: sAK Decrease of capital stock: δK
Endogenous growth model
Increase in k = sk
Decrease in k = δk
No steady state equilibrium
If sA > δ No steady state, capital stock will continue to rise forever.
The DOTCOM Bubble in 1990s
The DOTCOM Bubble
The spectacular rise and fall of the NASDAQ (tech-heavy):
In 1995 – NASDAQ at 900 March 10, 2000 – NASDAQ rose to 5,048 Oct 4, 2002 – NASDAQ down to 815 Oct 4, 2010 – NASDAQ 1,975
Individual stock – example: Microstrategy
http://www.google.com/finance?q=mstr
Michael Saylor – lost 6 billion dollars in one single day.http://www.slate.com/id/77774/
Broadcast.com and Facebook.com
Broadcast.com – in 1999, $50 million revenue, 330 employees.
Facebook.com – in 2010, $1 billion revenues, 1500 employees.
Broadcast.com was sold to yahoo.com at the peak of the internet bubble at US$ 5.9 billion. One third of employees are millionaires on paper.
If evaluation based on broadcast.com, Facebook.com would be worth 118 billion. Currently Facebook.com is evaluated at US$ 11 billion.
Example
Whole market becomes crazy during the internet boom.
Market evaluation of internet companies is completely wrong.
Mark Zuckerberg Mark Cuban
Could worth 25 billion (if in 1999)
But only worth 7 billion today
Actually worth 2.5 billion
Would only worth 200 million if sold today
Broadcast.com and Facebook.com
Mark Cuban purchased NBA Dallas Mavericks for $285 million.
He wouldn’t be able to do that if based on the current evaluation.
No steady-state equilibrium Capital stock keeps rising no
steady state equilibrium.
During the DOTCOM bubble in 1990’s, it is widely believed that growth is unlimited.
The key person is Paul Romer, a Stanford economist, one of TIME Magazine’s 25 most influential economists in 1997.
http://www.time.com/time/magazine/article/0,9171,986206-10,00.html
No longer – burst of the DOTCOM Bubble
Summary
Amount of per effective capital stock increase due to investment:
Amount of per effective capital stock decrease due to depreciation, population growth, and technology growth.
Equilibrium condition:
kgnks
ks
kgn
The Solow model
Decrease of k: (δ+g+n)k
Increase of k: sy
Steady state equilibrium:decrease of k = increase of k
k*
Summary
Golden rule: the steady state equilibrium where the consumption is maximized.
Golden rule condition:
gnkMPK 1
Summary
Policy implications: The most important long-run economic
policy is to encourage both public and private savings.
This is particularly important for the United States since our saving rate is too low.