Lecture 7 Economic Growth. It’s amazing how much we have achieved.

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.