The Solow model Stylised facts of growth The Solow model Steady state and convergence

The Solow model

Stylised facts of growthThe Solow model

Steady state and convergence

The Solow model

Until now, when output was changing, it was due to economic fluctuations in the IS-LM or AS-AD models.

Long run growth, however, determines the capacity of the economy to produce goods and services, and ultimately welfare:

1913 : Argentina’s GDP is 70% larger than Spain’s. 2000 : Spain’s GDP is 50% larger than Argentina’s.

1945 : Ghana’s GDP is 60% larger than Korea ’s. 2000 : South Korea’s GDP is 100% larger than Ghana’s.

1970 : Italy’s GDP is 50% larger than Ireland’s. 2000 : Ireland’s GDP passes Italy’s GDP.

What are the causes of economic growth? How can one maintain growth?

The Solow model

5 Stylised facts

The Solow model

Convergence to the steady state

Growth and convergence

Stylised fact 1 :Sudden acceleration of output

US Industrial production index (Source: NBER)

0

500

1000

1500

2000

2500

1790

1795

1800

1805

1810

1815

1820

1825

1830

1835

1840

1845

1850

1855

1860

1865

1870

1875

1880

1885

1890

1895

1900

1905

1910

1915

Stylised fact 2 :Medium run fluctuations in growth

Real GDP per capita (1950 =100)

0

200

400

600

800

1000

1200

1950

1953

1956

1959

1962

1965

1968

1971

1974

1977

1980

1983

1986

1989

1992

1995

1998

CAN FRA GBR ITA JPN USA

Source: Penn Tables 6.1

Stylised fact 3 : Persistent lags and catch-up

ARG

AUSAUT BEL

BOL

BRA

CAN CHE

COLCRI

DNK

EGY

ESP

ETH

FINFRA GBR

GTMHNDIND

IRLISL

ISR

ITA

JPN

KENLKA

LUX

MAR

MEX

MUS

NGANIC

NLD

NOR

NZL

PAK

PANPER

PHL

PRT

SLV

THA

TTO

TUR

UGA

URY

USA

VENZAF

01

00

00

20

00

03

00

00

40

00

0

GD

P p

er

capit

a

20

00

0 2000 4000 6000 8000 10000

GDP per capita 1950

Stylised fact 3 :Persistent lags (USA=100)

0

10

20

30

40

50

60

70

80

90

100

1960

1962

1964

1966

1968

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

Cameroon Ivory Coast Gabon Rwanda Senegal

Real GDP per capita Source: Penn Tables 6.1

Stylised fact 3 :Catch-up (USA=100)

0

10

20

30

40

50

60

70

80

90

100

1950

1952

1954

1956

1958

1960

1962

1964

1966

1968

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

China India Japan Singapore Thailand

Real GDP per capita (1950 =100) Source: Penn Tables 6.1

Stylised fact 4 :Increased inequality between countries

Source: Bourguignon & Morrison (2003)

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1820 1850 1870 1890 1910 1929 1950 1960 1970 1980 1992

Inequality between countries Inequality within countries

Stylised fact 5 :Biased technical change

The technological evolutions linked to growth seem to favour skilled labour, leading to a loss of jobs in traditional sectors

This is called “skill-biased technical change”. This increases income inequality because it changes the structure of the demand for labour. Keeping labour supply unchanged this leads to either An increase in unemployment A fall in relative wage between skilled/unskilled labour

This phenomenon is neither universal or permanent The post-war boom did not affect unskilled labour

negatively

5 Stylised facts

1. World output has seen an abrupt acceleration over the long run.

2. GDP per capita and productivity can fluctuate significantly in the medium run. These fluctuations are not necessarily synchronised across countries.

3. Some countries have been able to catch up with the living standards of the richest countries, while other countries have stagnated relative to rich countries.

4. Inequalities have increased and shifted from inequalities within countries to inequalities between countries. This has slowed down since the 90’s, mainly because of the take-off of the Chinese and Indian economies.

5. Technical progress is biased as in increases income inequalities, either by reducing the wages of the unskilled labourers, either by increasing unemployment (i.e. By reducing their employability).

The Solow model

5 Stylised facts

The Solow model



The Solow model

The Solow model is based on several simplifying assumptions Joan Robinson ironically referred to the lack of

realism of these assumptions by referring to the “Kingdom of Solovia”

A1 Factors of production are substitutes and not complements.

A2 Savings generate investments, which is consistent with the neoclassical interpretation of the savings/investment

balance.

A3 The interest rate is perfectly flexible and instantaneously adjusts investment and savings.

A4 Wages adjust such that the supply of labour (set exogenously by the growth rate of the population) and the demand for labour adjust perfectly

The Solow model

The macroeconomic production function Production is a function of capital K and L (with

exogenous growth rate n ) It exhibits constant returns to scale

Simplification : By dividing by the amount of labour L, one can express the variables “per capita”:

,Y F K L

,1Y K

FL L

y f kYy

L K

kL

The Solow model

y

Capital per worker

Out

put

per

wor

ker

k

Decreasing marginal returns: each extra unit of capital per worker reduces the marginal productivity of capital

1

Output y = f(k)

The Solow model

y c i

i s y

y f k

y

Capital per worker

Out

put

per

wor

ker

k

Output y = f(k)

Investment i= s × f(k)y

Output per worker

c

Consumption per worker

iInvestment per worker i s f k

Income is either spent or saved :

Additionally, savings are equal to investment :

Therefore:

The Solow model

This tells us that given a production technology and a level of population, the level of output will depend only on the available stock of capital.

This stock is determined by two flows: Investment : the capital stock increases when firms

purchase new equipment . We have just seen how this is determined.

Capital consumptions, which reduce the stock of capital available to workers. This is what we look at next.

Capital stock per worker

Investment

Capital consumptions

The Solow model

Capital consumptions 1: Discounting Capital stock is reduced by depreciation. As the

capital stock grows older, its value is discounted The amount of discounting is given by the

discount rate δ. For example, if the expected life of a piece of

equipment is 20 years, the discount rate is around 5%. This gives δ≃0,05.

With a capital stock k, the size of the discount is equal to δk

The Solow model

Capital consumptions 2: Population growth In the long run, populations are not constant.

This creates a second capital consumption, as one needs to provide capital to the new workers: Lets assume a fixed capital stock K :

If the population grows at a rate n, the expenditure required to keep the the capital stock per worker equal to k is equal to nk

Kk

L

The Solow model

Capital consumptions 3: Technical progress

If new technologies are introduced, workers become more productive.

Less labour is required to produce the same amount of output ⇒ Some workers become available for other uses

Technical progress is therefore equivalent to an increase in the number of workers, in other words to population growth (we shall call this growth g).

The net variation of the capital stock per worker is therefore given by the following equation :

Δk = i – (δ+n+g)k

The Solow model

( )g n k

Capital per worker

Cap

ital c

ons

umpt

ion

k

Capital consumption (δ+n+g)k

Expenditure required to maintain this level of capital per worker

The Solow model

5 Stylised facts

The Solow model




Capital per worker (k)

Investment & consumption flows

Investment

i = s×f(k)

Capital consumptions (δ+n+g)×k

k2

i2

(δ+n+g) ×k2

k1

i1

(δ+n+g) × k1

The capital stock increases as investment is higher than capital consumptions

The capital stock falls as consumptions are higher than investment

(δ+n+g)×k*=i*

k*

Steady-state level of capital per worker


Capital per worker (k)

s1×f(k)

…increases the steady-state capital stock

k2*k1*

New steady-state

s2×f(k)

Capital consumptions (δ+n+g)k

Initial steady-state

An increase in the savings ratio…



ALB

ARG

ARM

ATG

AUS

AUT

AZEBDI

BEL

BEN BFABGD

BGR

BLRBLZ

BOL

BRA

BRB

BWA

CAN CHE

CHL

CHNCIVCMR COG

COL

COM

CPV

CRI

CZE

DNK

DOM DZAECUEGY

ESP

EST

ETH

FIN

FJI

FRA

GAB

GBR

GEO

GER

GHAGIN

GMB GNBGNQ

GRC

GRDGTM GUY

HKG

HND

HRVHUN

IDNIND

IRL

IRN

ISL

ISR

ITA

JAMJOR

JPN

KAZ

KENKGZ

KHM

K KOR

LBN LCA

LKALSO

LTU

LUX

LVA

MAC

MARMDA

MDG

MEX

MKD

MLIMOZ MRT

MUS

MWI

MYS

M

NERNGANIC

NLD

NOR

NPL

NZL

PAK

PANPER

PHLPNG

POL

PRT

PRY ROM

RUS

RWASEN

SLV

SVK

SVN

SWE

SWZ

SYC

SYR

TCDTGO

THA

TJK

TTO

TUNTUR

TZAUGA

UKR

URY

USA

VCT VEN

YEM

ZAF

ZMBZWE

010

000

2000

030

000

4000

0

Inco

me p

er

capit

a in 1

99

9

0 10 20 30 40

Investment as a percentage of output (1960-1999)


k

s×f(k)

(δ+n1+g)×k

Capital per worker

k1*

1. A higher growth rate of the population…

(δ+n2+g)×k

2... Reduces the capital stock per worker…

k2*

3. …And therefore reduces the steady-state capital stock.

The Solow model predicts that countries with high demographic growth rates should have a lower level of per-capita income, ceteris paribus.



ARG

AUSAUTBEL

BOL

BRA

CAN

COL CRI

DNK

EGYSLV

ETH

FINFRA

GTMHND

ISL

IND

IRL

ISR

ITA

JPN

KEN

LUX

MUS

MEX

MAR

NLD

NZL

NICNGA

NOR

PAK

PANPER

PHL

PRT

ZAF

ESP

LKA

CHE

THA

TTO

TUR

UGA

GBR

USA

URY

VEN

010

000

2000

030

000

4000

0

Inco

me p

er

capit

a 2

00

0

0 1 2 3

Demographic growth rate (average annual growth rate)


The concept of steady-state has three central implications : An economy at steady state no longer changes. An economy that isn’t at the steady-state will

tend to move towards it. It therefore defines the long run equilibrium of

the economy. However: the steady state depends on the

savings ratio, therefore there is space for an economic growth policy.


Output, investment, and consumption flows

k

c2

i2


c1

i1

Investment i2= s2 × f(k)


Which of the 2 steady states is socially preferable ?

Production y = f(k)

The savings ratio and the golden rule



Capital stock per worker k

Investment i2= s2 × f(k)c2

i2



c1

i1

Which of the 2 steady states is socially preferable ?

Production y = f(k)




pmk n g k

Production y = f(k)

Investment i*= s* × f(k*)c*

i*


The optimal steady-state maximises consumption

This occurs when the slope of the production function is equal to the slope of the capital consumption function

yn g

k





t

Fall in the savings ratio

Investment (i)

Consumption (c)

Production (y)

t0

Transition to the golden rule steady-stateStarting off with too much Capital


t

Increase in the savings ratio

Investment (i)

Consumption (c)

Production (y)

t0

Transition crisis, which requires political intervention and arbitrage

Transition to the golden rule steady-stateStarting off with too little Capital

The Solow model

5 Stylised facts

The Solow model




Empirical analysis of growth (%Δ real GDP)

Country 1948-1972 1972-1995 1995-2000

Germany 5.7 2.0 1.7*

Canada 2.9 1.8 2.7

United States 2.2 1.5 2.9

France 4.3 1.6 2.2

Italy 4.9 2.3 1.4

Japan 8.2 2.6 1.1

United Kingdom 2.4 1.8 2.5


ARG

AUS

AUT

BDI

BEL

BEN

BFABGD

BOL

BRABRB CAN

CHE

CHL

CHN

CIV

CMR

COGCOL

COM

CPV

CRI

DNK

DOM

DZA

ECU

EGY

ESP

ETH

FINFRA

GAB GBR

GHAGIN

GMBGNB

GNQGRC

GTM

HKG

HND

IDN

IND

IRL

IRN ISLISR ITA

JAM

JOR

JPN

KEN

KOR

LKALSO

LUXMAR

MDG

MEX

MLI

MOZ

MUS

MWI

MYS

NER

NGA

NIC

NLD

NOR

NPL

NZL

PAKPAN

PER

PHL

PRT

PRY

ROM

RWA

SEN

SGP

SLV

SWE

SYC

SYR

TCD

TGO

THA

TTO

TUR

TZA

UGA URY

USA

VENZAF

ZMB

24

68

10

Ave

rag

e a

nn

ua

l gro

wth

ra

te

0 1000 2000 3000 4000GDP per capita (1960)Source: Penn Tables 6.1

Convergence (All countries)


ARG

AUS

AUTBEL

CAN

CHE

DNK

ESP

FIN

FRA

GBR

GRC

IRL

ISL

ISRITA

JPN

LUX

NLD

NOR

NZL

PRT

SWE

USA

56

78

Ave

rag

e a

nn

ua

l gro

wth

ra

te

1000 1500 2000 2500 3000 3500

GDP per capita (1960)Source: Penn Tables 6.1

Convergence (OECD Countries)


BDI BEN

BFABGD

BOL

BRABRB

CHL

CHN

CIV

CMR

COGCOL

COM

CPV

CRI

DOM

DZA

ECU

EGY

ETH

GAB

GHAGIN

GMBGNB

GNQ

GTM

HKG

HND

IDN

INDIRN

JAM

JOR

KEN

KOR

LKALSO

MAR

MDG

MEX

MLI

MOZ

MUS

MWI

MYS

NER

NGA

NIC

NPL

PAKPAN

PER

PHL

PRY

ROM

RWA

SEN

SGP

SLV

SYC

SYR

TCD

TGO

THA

TTO

TUR

TZA

UGA URYVENZAF

ZMB

24

68

10

Ave

rag

e a

nn

ua

l gro

wth

ra

te

0 500 1000 1500

GDP per capita (1960)Source: Penn Tables 6.1

Convergence (Non OECD countries)


Convergence, as predicted by the Solow model, is not a universal phenomenon. Not all countries seem to be converging…

Disparities between groups of countries can be explained by differences in the determinants of the steady state. Rate of investment Growth rate of the population Level of technology

Convergence only occurs between countries that have the same steady state!

Documents

The Solow model Stylised facts of growth The Solow model Steady state and convergence