Models of Economic Growth A

Outline:Because this area is complex and mathematical

there are two files of slides for this topicLecture A• Introduction – trends in growth• Neoclassical growth modelsLecture B• Endogenous growth models• The convergence debateBelow are slides for lecture ASee next file for lecture B

Introduction• Need to define ‘economic growth’ (in book

this is growth in GDP per capita, not GDP growth).

• Some background on history of economic growth – including own country data

• Also, worthwhile to stress importance of small differences in growth rates e.g.

2% growth per year GDP p.c. increases 7.4 fold in 100 years

0.6% GDP per capita increase 1.8 times in 100

.... 72 / growth rate = no. of years to double, hence China’s 10% p.a. implies 7.2 years

The very long runGrowth of GDP per capita (average annual percentage changes)

1500-1820 1820-1900 1900-2000

OECD 1.2 2.0

Non-OECD 0.4 0.6

World 0.04 0.8 1.9

Source: Boltho and Toniolo (1999, Table 1) OECD refers to North America, Western Europe, Japan, Australia and New Zealand.

USA, UK and EIRE5

00

01

50

00

250

00

350

00

Rea

l GD

P p

er

capita

(P

WT

6.1

, ch

ain

)

1965 1970 1975 1980 1985 1990 1995 2000year

GBR IRLUSA

Growth of GDP p.c: USA=2.2%, GBR=2.0%, Ireland=3.7% (but post-93, 8.5%)

GDP per capita is US$ 1996 constant prices. Source: Penn World Table 6.1

China and India

01

00

02

00

03

00

04

00

0R

ea

l GD

P p

er

capita

(P

WT

6.1

, ch

ain

)

1965 1970 1975 1980 1985 1990 1995 2000year

CHN IND

Growth: pre-90 China 3.7%, India 4.4%. 1990-2000: China 7.0%, India 4.4%

Source: Penn World Table 6.1

Brazil, S. Korea, Philippines0

500

01

00

00

150

00

Rea

l GD

P p

er

capita

(P

WT

6.1

, ch

ain

)

1965 1970 1975 1980 1985 1990 1995 2000year

BRA KORPHL

Source: Penn World Table 6.1 (http://pwt.econ.upenn.edu/aboutpwt.html)

Other data

• Above are from Penn World Table 6.1, now 6.3 is availablehttp://pwt.econ.upenn.edu/

Some further links at:

http://users.ox.ac.uk/~manc0346/links.html

GDP per capita growth not everything• Focusing on ‘economic growth’ does neglect health, the

environment, education, etc• UN’s Human Development Index (HDI) gives equal weight to

life expectancy, education and GDP per capita (http://hdr.undp.org/reports/global/2004/)

• Ultimate interest ‘well-being’ or ‘happiness’. Layard, R. (2003). "Happiness: Has Social Science a Clue?" http://cep.lse.ac.uk/events/lectures/layard/RL030303.pdf.

• GDP measures aggregate value added – whether coal power station or wind farm

• Friedman, Ben (2005) The Moral Consequences of Economic Growth argues growth is important for ‘stable’ societies

Neoclassical model

• There are many ways to teach this. Book tends to use equations, but can do a great deal with intuition and few diagrams.

• This model most often attributed to Robert Solow (1956) – US Nobel prize winner …. but Trevor Swan (1956) (a less well known Australian economist) published (independently) a very similar paper in the same year – hence refer to Solow-Swan model

Neoclassical growth model• Model growth of GDP per worker via capital accumulation• Key elements:

– Production function (GDP depends on technology, labour and physical capital)

– Capital accumulation equation (change in net capital stock equals gross investment [=savings] less depreciation).

• Questions:– how does capital accumulation (net investment) affect

growth?– what is role of savings, depreciation and population

growth?– what is role of technology?

Solow-Swan equations( , ) (production function)

GDP, technology,

capital, labour

(capital accumulation equation)

proportion of GDP saved (0 1)

depreciation rate (as p

Y Af K L

Y A

K L

dKsY K

dts s

roportion) (0 1)

Solow-Swan analyse how these two equations interact.

Y and K are endogenous variables; s, and growth rate of L and/or A are exogenous (parameters).

Outcome depends on the exact functional form of production function and parameter values.

Neoclassical production functionsSolow-Swan assume:

a) diminishing returns to capital or labour (the ‘law’ of diminishing returns), and

b) constant returns to scale (e.g. doubling K and L, doubles Y).

For example, the Cobb-Douglas production function

1

1

where 0 1Y AK L

Y AK L AK Ky A Ak

L L L L

Hence, now have y = output (GDP) per worker as function of capital to labour ratio (k)

GDP per worker and kAssume A and L constant (no technology growth or

labour force growth)

y=Af(k)=Ak

k(capital per worker)

y

outp

ut p

er w

orke

r

concave slope reflects diminishing marginal

product of capitaldY/dK=dy/dk=Ak-1

Accumulation equation

If A and L constant, can show* dk

sy kdt

*accumulation equation is: , divide by L yields /

Also note that, / / since is a constant.

dK dKsY K L sy k

dt dtdk K dK

d dt L Ldt L dt

This is a differential equation. In words, the change in

capital to labour ratio over time = investment (saving) per

worker minus depreciation per worker.

Any positive change in k will increase y and generate

economic growth. Growth will stop if dk/dt=0.

Graphical analysis of

k (capital per worker)

y

outp

ut p

er w

orke

r

sy

k

k*

net investment(savings = gross

investment)

(depreciation)

dk

sy kdt

(Note: s and constants)

Solow-Swan equilibrium

k

you

tput

per

wor

ker

sy

k

k*

y*

y=Ak

consumptionper worker

GDP p.w. converges to y* =A(k*). If A (technology) and L constant, y* is also constant: no long run growth.

What happens if savings increased?• raising saving increases k* and y*, but long run

growth still zero (e.g. s1>s0 below)• call this a “levels effect”• growth increases in short run (as economy moves to

new steady state), but no permanent ‘growth effect’.y

s0y

k

k0*

y1*y=Ak

s1y

k1*

y0*

What if labour force grows?Accumulation eqn now ( ) where / (math note 2)

dk dLsy n k n L

dt dt

k

y

outp

ut p

er w

orke

r

sy(n)k

kn*

k

y

Population growth (n>0) pivots the ‘depreciation’ line upwards, and reduces k and y steady state

Population growth reduces equilibrium level of GDP per worker (but long run growth still zero) if technology static

Analysis in growth ratesCan illustrate above with graph of gk and k ( ) ( )k

dkdk ydtsy n k g s ndt k k

kcapital per worker

Dis

tanc

e be

twee

n lin

es is

gr

owth

rat

e of

cap

ital p

er w

orke

r

n

k*

net investment

net disinvestment

average product of capitalys sk

Distance between lines represents growth in capital per worker (gk)

Rise in savings rate

(s0 to s1)

NB: This graph of how growth rates change over time

gY, gk

TimeSaving change

0 %

1

ysk

gY=(MPK/APK) gk = sk gk

(sk = in Cobb-Douglas case)

gk

n

k*

0

ysk 1

ysk

A

B

C

k

Golden rule• The ‘golden rule’ is the ‘optimal’ saving rate (sG)

that maximises consumption per head.

• Assume A is constant, but population growth is n.

• Can show that this occurs where the marginal product of capital equals (n)

Proof: ( ) 0 at steady state,

hence ( ) , where * indicates steady state equilibrium value

The problem is to: max * ( )

*First order condition : 0 ( ) hence

*

k

dksy n k

dtsy n k

c y sy y n k

dyn

dk

*

= *k

dyMP n

dk

Graphically find the maximal distance between two lines

k

y

outp

ut p

er w

orke

r

(n)k

k**

y**maximal

consumptionper worker

slope=dy/dk=n+

capital per worker

y

sgold y

… over saving

Economies can over save. Higher saving does increase GDP per worker, but real objective is consumption per worker.

k

you

tput

per

wor

ker

sgoldy(n)k

k**

y**

y=Ak

maximalconsumptionper worker

sovery

slope=dy/dk=n+

k*capital per worker

Golden rule for Cobb Douglas case

• Y=KL1- or y = k

• Golden rule states: MPk = (k*)-1 =(n + )

• Steady state is where: sy* = ( +n)k* • Hence, sy* = [(k*)-1]k*

or s = (k*) / y* =

Golden rule saving ratio = for Y=KL1-case

Assuming perfect competition, and factors are paid marginal products, is share of GDP paid to capital (see C&S, p.481). Expect this to be 0.1 to 0.3.

Solow’s surprise*• Solow’s model states that investment in capital cannot

drive long run growth in GDP per worker

• Need technological change (growth in A) to avoid diminishing returns to capital

• Easterly (2001) argues that “capital fundamentalism” view widely held in World Bank/IMF from 60s to 90s, despite lessons of Solow model

• Policy lesson: don’t advise poor countries to invest without due regard for technology and incentives

* This is title of Chapter 3 in Easterly (2001), which is worth a quick read for controversy surrounding growth models and development issues

What if technology (A) grows?• Consider y=Ak, and sy=sAk, these imply that

output can go on increasing.

• Consider marginal product of capital (MPk)

MPk=dy/dk =Ak,

if A increases then MPk can keep increasing (no

‘diminishing returns’ to capital)

• implies positive long run growth

…. graphically, the production function simply shifts up

k

y

outp

ut p

er w

orke

r

k

y=A0k

y=A1k

y=A2k

Technology growth:A2 > A1 > A0

Capital to labour ratio

0

1

2

…. mathematically1

1 1 1

Easier to use ( ) where 0 1

(This assumes A augments labour (Harrod-neutral technological change)

Can re-write ( )

dAAssume / (for reference this sa

dt A

Y K AL

K AL A K L

A g

t o

1

me as A A )

Trick to solving is to re-write as

( )= ( )

where =output per 'effective worker', and capital per 'effective worker'

Can show / ( ) ( )

Th

Ag te

Y K AL Ky k

AL AL AL

y k

dkk s k n a k

dt

is can be solved (plotted) as in simpler Solow model.

Output (capital) per effective worker diagram

If Y/AL is a constant, the growth of Y must equal the growth rate of L plus growth rate of A (i.e. n+a)

And, growth in GDP per worker must equal growth in A.

K/AL

outp

ut p

er e

ffec

tive

wor

ker

s(Y/AL)

(na)k

(K/AL)*

(Y/AL)*

Y/ALY/AL

capital per effective worker

NOTE: ‘dilution’ line now includes technology growth (a)

Summary of Solow-Swan• Solow-Swan, or neoclassical, growth model, implies

countries converge to steady state GDP per worker (if no growth in technology)

• if countries have same steady states, poorer countries grow faster and ‘converge’ – call this classical convergence or ‘convergence to steady

state in Solow model’

• changes in savings ratio causes “level effect”, but no long run growth effect

• higher labour force growth, ceteris paribus, implies lower GDP per worker

• Golden rule: economies can over- or under-save (note: can model savings as endogenous)

Technicalities of Solow-Swan• Textbooks (Jones 1998, and Carlin and Soskice 2006) give

full treatment, in short:• Inada conditions needed ( “growth will start, growth will

stop”)

• It is possible to have production function where dY/dK declines to positive constant (so growth declines but never reaches zero)

• Exact outcome of Solow model does depend on precise functional forms and parameter values

• BUT, with standard production function (Cobb-Douglas) Solow model predicts economy moves to steady state because of diminishing returns to capital (assuming no growth in technology A)

0lim 0, lim ,K K

dY dY

dK dK

Endnotes

2

Math Note 2:

Start with , divide by yields /

Note that / / (quotient rule)

simplify to / / or / (since is

dK dKsY K L L sy k

dt dtdk K dK dL

d dt L K Ldt L dt dt

dK dL K dKL L L nk n

dt dt L dt

labour growth and / )

hence + = /

hence = ( )

K L k

dk dKnk L sy k

dt dtdk

sy n kdt

0

20

Math note 1: can be used to analyse impact of growth over time

Let y=GDP p.w., g=growth (e.g. 0.02 2%), t=time.

Hence, for 0.02 and 100, / 7.39

gtt

t

y y e

g t y y e

Questions for discussion

1. What is the importance of diminishing marginal returns in the neoclassical model? How do other models deal with the possibility of diminishing returns?

2. Explain the effect of (i) an increase in savings ratio (ii) a rise in population growth and (iii) an increase in exogenous technology growth in the neoclassical model.

3. What is the golden rule? Can you think of any countries that have broken the golden rule?

ReferencesBoltho, A. and G. Toniolo (1999). "The Assessment: The Twentieth Century-Achievements, Failures, Lessons." Oxford Review of Economic Policy 15(4): 1-18.

Easterly, W. (2001). The Elusive Quest for Growth: Economists’ Adventures and Misadventures in the Tropics. Boston, MIT Press.

Swan, T. (1956). "Economic Growth and Capital Accumulation." Economic Record 32: 344-361.

Jones, C. (1998) Introduction to Economic Growth, (W.W. Norton, 1998 First Edition, 2002 Second Edition).

Carlin, W. and D. Soskice (2006) Macroeconomics: Imperfections, Institutions and Policies, Oxford University Press.