The Solow model
Stylised facts of growth
The 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
Stylised fact 1 :
Sudden acceleration of output
2500
2000
1500
1000
500
0
US Industrial production index
(Source: NBER)
Stylised fact 2 :
Medium run fluctuations in growth
1200
CAN
FRA
GBR
ITA
JPN
USA
1000
800
600
400
200
19
50
19
53
19
56
19
59
19
62
19
65
19
68
19
71
19
74
19
77
19
80
19
83
19
86
19
89
19
92
19
95
19
98
0
Real GDP per capita (1950 =100)
Source: Penn Tables 6.1
Stylised fact 3 :
Persistent lags and catch-up
30000
USA
NOR
IRL
20000
JPN
AUT
ITA
ESP
PRT
FIN
ISL
BEL
FRA
DNK CAN
AUS
NLD
CHE
GBR
NZL
ISR
10000
MUS
TTO
ZAF
THA BRA
TUR
PAN
CRI
COL
PER
SLV
EGY
GTM
MAR
PHL
LKA
BOL
IND
PAK
HND
NIC
KEN
UGA
ETH
NGA
0
2000
ARG
URY
MEX
0
GDP per capita 2000
40000
LUX
4000
VEN
6000
GDP per capita 1950
8000
10000
Stylised fact 3 :
Persistent lags (USA=100)
100
90
Cameroon
Ivory Coast
Gabon
Rwanda
80
70
60
50
40
30
20
10
0
Real GDP per capita
Source: Penn Tables 6.1
Senegal
Stylised fact 3 :
Catch-up (USA=100)
100
90
China
India
Japan
Singapore
Thailand
80
70
60
50
40
30
20
10
19
50
19
52
19
54
19
56
19
58
19
60
19
62
19
64
19
66
19
68
19
70
19
72
19
74
19
76
19
78
19
80
19
82
19
84
19
86
19
88
19
90
19
92
19
94
19
96
19
98
20
00
0
Real GDP per capita (1950 =100)
Source: Penn Tables 6.1
Stylised fact 4 :
Increased inequality between countries
1,0
0,9
Inequality between countries
Inequality within countries
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0,0
1820
1850
1870
1890
1910
Source: Bourguignon & Morrison (2003)
1929
1950
1960
1970
1980
1992
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
Convergence to the steady state
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 so 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
Y  F  K , L

Simplification : By dividing by the amount of labour
L, one can express the variables “per capita”:
Y
K 
 F  ,1
L
L 
Y
y
L
y  f k 
K
k
L
The Solow model
Output per worker
y
Output y = f(k)
Decreasing marginal
returns: each extra unit of
capital per worker reduces
the marginal productivity of
capital
1
Capital per worker
k
The Solow model
y
Income is either spent or
saved :
Output y = f(k)
y  ci
Output per worker
Output per
worker
Consumption
per worker
Additionally, savings are
equal to investment :
c
Investment i= s × f(k)
y
i  s  y

 y  f  k 
Therefore:
i
Capital per worker
i  s  f k 
Investment per
worker
k
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.
Investment
Capital stock
per worker
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 :
K
k
L

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
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 consumption
Capital consumption
(δ+n+g)k
Expenditure required
to maintain this level of
capital per worker
Capital per worker
k
The Solow model
5 Stylised facts
The Solow model
Convergence to the steady state
Convergence to the steady state
Investment &
consumption
flows
Capital consumptions
(δ+n+g)×k
(δ+n+g) ×k2
i2
(δ+n+g)×k*=i*
Investment
i = s×f(k)
i1
(δ+n+g) × k1
k1
The capital stock increases as
investment is higher than
capital consumptions
k*
Steady-state level of
capital per worker
k2
Capital per worker (k)
The capital stock falls as
consumptions are higher than
investment
Convergence to the steady state
Investment &
consumption
flows
Capital consumptions
(δ+n+g)k
An increase in the
savings ratio…
s2×f(k)
s1×f(k)
…increases the
steady-state capital
stock
k1*
Initial steadystate
k2*
New steadystate
Capital per worker (k)
LUX
30000
USA
20000
IRL
MAC
GBR
ATG
BRB
10000
MUS
ESP
PRT SVN
CZE
K
NOR
CAN
DNK AUS CHE
HKG
ISL JPN
NLD
SWE
AUTFIN
BEL
GER
FRAITA
NZL
ISR
KOR
GRC
SVK
ARG
SYC
URY
HUN
CHL
MYS
EST
POL
GAB
HRVMEX
BWA
BLR
ZAF
LVA
RUS
LTU
BRA
KAZ
VCT
TUN
TUR
BLZ
VEN
THA
PAN
GRD
GEO
IRN
LBNCRI LCA
FJI
BGR
COL
SWZ
MKD
DOM
PRY
DZA
UKR M
PER
SLVGTM
ROM
EGY
SYR
JOR
MAR
JAM
IDN
GUY
PHL
CHNCPV
ECU
ALB
KGZLKABOL
GIN
PNG
ARM
AZE
INDPAK
MDA
GNQ
CMR
CIVTJK
HNDKEN LSO
COG
COM
SEN
NIC
NPL
KHM
MRT
GHA
GMB
BEN
UGA
MOZ
MLI
TGO
BFA BGD
TCD
MDG
RWA
YEM
NGA
NER
MWI
ZMB
ETH
BDI
GNB
TZA
TTO
0
Income per capita in 1999
40000
Convergence to the steady state
0
10
20
Investment as a percentage of output (1960-1999)
30
ZWE
40
Convergence to the steady state
Investment &
consumption
flows
2... Reduces the
capital stock per
worker…
(δ+n2+g)
×k
(δ+n1+g)
×k
s×f(k)
k2*
3. …And therefore reduces
the steady-state capital stock.
k1*
k
Capital per
worker
1. A higher growth
rate of the
population…
The Solow
model predicts
that countries
with high
demographic
growth rates
should have a
lower level of
per-capita
income, ceteris
paribus.
Convergence to the steady state
30000
USA
20000
DNK
IRL NOR CHE
JPN
NLD
BEL FIN
AUT
FRA
GBR
ITA
ISL
CAN
AUS
NZL
ESP
ISR
PRT
10000
MUS
TTOARG
URY
0
Income per capita 2000
40000
LUX
0
1
MEX
BRA
THAZAF
TUR
VEN
PAN
CRI
COL
PER
SLV
EGY
GTM
MAR
PHL
LKA
IND BOL
HND
PAK NIC KEN
ETHNGA UGA
2
Demographic growth rate (average annual growth rate)
3
Convergence to the steady state

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
tends 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.
Convergence to the steady state
Output, investment, and
consumption flows
Capital consumption
(δ+n+g)k
The savings ratio and
the golden rule
Production y = f(k)
Investment i2= s2 × f(k)
c2
Investment i1= s1 × f(k)
c1
i2
i1
Capital stock per worker
k
Which of the 2 steady
states is socially
preferable ?
Convergence to the steady state
Output, investment, and
consumption flows
Capital consumption
(δ+n+g)k
The savings ratio and
the golden rule
Production y = f(k)
c1
Investment i1= s1 × f(k)
Investment i2= s2 × f(k)
c2
i1
i2
Capital stock per worker
k
Which of the 2 steady
states is socially
preferable ?
Convergence to the steady state
Output, investment, and
consumption flows
Capital consumption
(δ+n+g)k
The savings ratio and
the golden rule
Production y = f(k)
The optimal steadystate maximises
consumption
Investment i*= s* × f(k*)
c*
This occurs when the
slope of the production
function is equal to the
slope of the capital
consumption function
y
 n g
k
i*
pmk    n  g
Capital stock per worker
k
Convergence to the steady state
Transition to the golden rule steady-state
Starting off with too much Capital
Production (y)
Consumption (c)
Investment (i)
t0
Fall in the savings ratio
t
Convergence to the steady state
Transition to the golden rule steady-state
Starting off with too little Capital
Production (y)
Consumption (c)
Transition crisis,
which requires
political intervention
and arbitrage
Investment (i)
t0
Increase in the savings ratio
t