Transcript Lecture 6 - Economics

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
Y  F K , L  K L
 1

The per worker production function is:
 1
Y K L
y 
L
L

1

K L
K

   1     k
L L
L
Per capita production function.

The marginal production of capital, MPK:
Y
K
 1 1
MPK 
 K L    
K
L

 1
 k
 1
y

k
MPK is obtained by taking the first
derivatives from the aggregate production
function, or from the per capita production
function.
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.
Kt  1   Kt 1  I t

Each period,


Amount of increase: It
Amount of depreciation: δK
If labor force does not change

At per capita level
kt  1   kt 1  it

Rearrange this:
kt  it  kt 1  sf (kt )  kt 1
Equilibrium:

At the equilibrium, we must have:
kt  sf kt   kt 1  0
kt  kt 1  k *

At the steady state level k*, we have:
 
sf k  k
*
*
Equilibrium
Discussions:



Why steady state?
If k > k*:
*
*
From the graph, sf k  k 
Depreciation > investment 
k level would decrease.
 
If k < k*:
From the graph, sf k *  k * 
Depreciation < investment 
k level would increase.
 
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=wbwdi&met=ny_gdp_mktp_kd_zg&idim=country:CHN&dl=en&hl=en&q=china+gdp+grow
th+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.
China’s saving rates
Government
Enterprises
Households
28
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
Enrollments (in 1,000)
7000
6000
5000
4000
3000
2000
1000
0
1985
1990
1995
2000
Year
2005
2010
Reducing saving in China


Establishing social safety network
Nationwide health insurance



1998 – urban employees
2003 – rural residents
2007 – urban residents (nonemployees)
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:
 
sf k  k
*
*
The Golden Rule level of capital stock

The steady state level of
consumption:
 
   
c  1  s y  f k  sf k  f k  k
*

*
*
*
*
*
Maximizing c* to get the Golden
rule level of consumption:
 
c *
*

f
'
k
  0
*
k
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: y  k 1/ 2
Depreciation rate: δ = 0.1
At the optimum:
MPK  0.5k 1/ 2  
 k*  0.25 2
 k* = 25
A numerical example



   k
In the steady state: sf k
s


f k 
k *
*
s = 0.5
 k
* 1/ 2
*
*
Summarize:


Two unknowns, saving rate s, and optimal
level of capital stock k.
Two equations:
 Golden rule equation:

 
f ' kG    0
 
G
G
sf
k


k
Steady state equations:
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
L
 K  K
k     
K 2
L
L
L
I  K L K



L
L L
 i   k  nk
 sf k     n k
Population growth


The steady state is determined by:
Δk=0
Therefore, at the steady state,
 
sf k *    nk *
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:
Y  F K , EL  K  EL
1
Technology

We now work with per-efficiency laborer
capital stock:
K
Y
k
, y
EL
EL

Define:

Let the growth rate of E be g:
E
g
E
Technology


The evolvement of aggregate capital stock remains
the same: ΔK = I – δK
The evolvement of the little k:
E  L  E  L
 K  K
k  
K

EL2
 EL  EL
I  K K E  L  E  L



EL
EL
EL
sY  K K  E L 





EL
EL  E
L 
 s  y  k  k g  n 
 sf k     n  g k
Technology
 K  K
 1 
k  
 K  


 EL  EL
 EL 
 1 


 EL 
Technology
At the steady state

Δk =0

sf k *    n  g k *
 
The steady state
The Golden rule
 c
=y–i
= y – sy
(output – investment)
(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:
    n  g
f' k
G
Or: MPK = δ + n + g
Summary
Symbol
capital per effective worker
output per effective worker
output per worker
capital per worker
total output
total capital
k = K/(E*L)
y = Y/(E*L) = f(k)
Y/L = y * E
K/L = k * E
Y = y * (E * L)
K = k * (E * L)
steady-state
growth rate
0
0
g
g
g+n
g+n
Discussions:


To calculate the Golden Rule saving rate,
two equations and two unknowns:
The Golden rule equation:
 
f ' kG    n  g

Steady state equation:
 
sf k G    n  g k G
A numerical example

In the US, we have:




k = 2.5y
δk = 0.1y
MPK*k = 0.3y
depreciation
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: MPK G    n  g


Given n = 0.01, δ = 0.04, and g < 0.07
We have:
MPK G  MPK  0.12
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:
sG  0.3
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 saving
Why 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
0.01
0.025
n
0.04
0.04
δ
0.22
0.16
s
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
1/ 2
sk
 n   k


sk
1/ 2
 s /n   
 y = s/(n+δ)
US-Mexico
yUS
y Mexico
sUS
0.22
nUS  

 0.01  0.04  1.79
s Mexico
0.16
0.025  0.04
n Mexico  
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
yUS
y Mexico
YUS
EUS LUS
YUS / LUS
E Mexico
E Mexico

 1.79 

 4
YMexico
YMexico / LMexico EUS
EUS
E MexicoLMexico
A different level of E

A Mexico worker is 45% of
efficiency of a US worker’s level.
E Mexico 1.79

 .4475
EUS
4
Endogenous growth theory

Basic idea: investment, especially
investment in R&D, would lead to
higher productivity.
Suppose E = B* K/L
Y  K  E  L
1
 K  B  K / L  L
1
 B1 K  AK
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 (techheavy):




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.
Could worth 25 billion (if in 1999)
Mark Zuckerberg
But
only
worth
7
billion
today
Actually worth 2.5 billion
Mark Cuban
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/magazin
e/article/0,9171,98620610,00.html
No longer – burst of the DOTCOM
Bubble
Summary

Amount of per effective capital stock increase
due to investment:
sk 

Amount of per effective capital stock decrease
due to depreciation, population growth, and
technology growth.
  n  g k

Equilibrium condition:
sk    n  g k

The Solow model
Steady state equilibrium:
decrease of k = increase of k
Decrease of k:
(δ+g+n)k
Increase of k:
sy
k*
Summary


Golden rule: the steady state equilibrium
where the consumption is maximized.
Golden rule condition:
MPK  k  1    n  g
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.