THE NEOCLASSICAL SOLOW-SWAN GROWTH MODEL
Sarantis Kalyvitis
Contents:
- Introduction
- The simple Solow-Swan model without technological progress
- The Solow-Swan model with technological progress
- Convergence
Some general remarks about models and assumptions in economics:
•
What is a model? A mathematical description of the economy.
•
Why do we need a model? The world is too complex to describe it in every
detail.
•
What makes a model successful? When it is simple but effective in
describing and predicting how the world works.
•
A model relies on simplifying assumptions. These assumptions drive the
conclusions of the model. When analyzing a model it is crucial to spell out
the assumptions underlying the model.
•
Realism may not outweigh the property of a good assumption.
Introduction
Useful definitions: the growth rate
Growth rate of output =
Yt − Yt −1
Yt −1
In continuous time:
•
dY / dt Y
Growth rate of output =
=
Y
Y
In per capita terms:
•
•
•
•
y (Y / N ) Y N
Growth rate of per capita output = =
= −
y Y/N
Y N
…and some main stylized facts in growth:
Per capita income rises
Growth rates between countries differ
Capital-output ratio is steady
Capital-labour ratio increases
Shares of capital and labour are constant
2
The basic structure of the model
Aggregate production function:
Yt = F ( K t , Lt )
where Yt : output, Kt : capital stock, Lt : labour force
Neoclassical assumptions on the production function:
(1) Positive and diminishing marginal returns of factor inputs:
∂F ( K t , Lt )
MPK t =
> 0,
∂K t
∂MPK t ∂ 2 F ( K t , Lt )
=
<0
∂K t
∂K t2
∂F ( K t , Lt )
> 0,
MPLt =
∂L t
∂MPLt ∂ 2 F ( K t , Lt )
=
<0
2
∂Lt
∂Lt
(2) Constant returns to scale w.r.t capital and labour:
F ( λ K t ,λ Lt ) = λ F ( K t , Lt ) όπου λ>0
(3) Inada conditions:
lim ( MPK t ) = lim ( MPLt ) = ∞
K t →0
Lt →0
lim ( MPK t ) = lim ( MPLt ) = 0
Kt →∞
Lt → ∞
In per capita terms by condition (2):
K 
Yt
= F  t ,1 ⇒ y t = f (k t )
Lt
 Lt 
where yt=Yt/Lt, kt=Kt/Lt, and f(kt)=F(kt,1). (Note: yt can increase only if kt
increases over time). By the neoclassical assumptions:
∂F ( K t , Lt ) ∂ [Lt f (k t )]
=
= f ′(k t ) > 0
∂K t
∂K t
∂MPK t ∂f ′(k t ) 1
=
=
f ′′(k t ) <0 ⇒ f ′′(k t ) <0
Lt
∂K t
∂K t
MPK t =
3
An example: The Cobb-Douglas production function
Y = AK α L1−α
where Α>0 is a technology parameter and 0<α<1.
(1) The Cobb-Douglas production function has positive and diminishing
marginal production w.r.t. capital and labour:
∂Y
L
= αA 
∂K
K
1−α
∂ 2Y
= −α (1 − α ) AK α − 2 L1−α <0
2
∂K
> 0,
α
∂Y
K
= (1 − α ) A  > 0,
∂L
L
∂ 2Y
= −α (1 − α ) AK α L−1−α <0
2
∂L
(2) The Cobb-Douglas production function exhibits constant returns to scale:
A(λ K ) (λL )
α
1−α
= λα λ1−α AK α L1−α = λAK α L1−α = λY
(3) The Cobb-Douglas production function satisfies the Inada conditions:
∂Y
L
lim
= lim αA 
K →0 ∂K
K →0
K
1−α
∂Y
L
lim
= lim αA 
K →∞ ∂K
K →∞
K
α
=∞
∂Y
K
lim
= lim (1 − α ) A  = ∞
L →0 ∂L
L →0
L
=0
∂Y
K
lim
= lim (1 − α ) A  = 0
L →∞ ∂L
L →∞
L
1−α
α
In per capita terms the Cobb-Douglas becomes:
K
Y = A( )α L = Ak a L ⇒ y = Ak a
L
By the neoclassical assumptions:
∂ [Lt f (k t )]
= f ′(k t ) = aAk a −1 > 0
∂K t
∂MPK t ∂f ′(k t ) 1
=
=
f ′′(k t ) <0 ⇒ f ′′(k t ) = (a − 1)aAk a − 2 <0
Lt
∂K t
∂K t
MPK t =
4
Capital accumulation:
•
K = It − δ Kt
•
where It is investment, δ is the depreciation rate and K = dK t dt .
Investment funding via savings:
I t = S t και S t = sYt
where 0<s<1 is the marginal propensity to save. By the last two eqs:
•
K = sYt − δ K t
In per capita terms
•
•
Y
K
K
K
= s t −δ t ⇒
= s f (k t ) − δ k t
Lt
Lt
Lt
Lt
K
d  t
•
L
k≡  t
dt

 L dK t − K dLt
•
•
t
t
•
=
dt
dt ⇒ k = K − k L
t
2
Lt
Lt
Lt
• 


L
⇒ k = s f ( k t ) −  δ + k t
L



•
•
L
where = n is the labour force growth rate. Then, the law of motion for the
L
capital stock in per capita terms becomes:
•
k = s f ( k t ) − ( n + δ )k t
(FUNDAMENTAL DIFFERENTIAL EQUATION FOR CAPITAL STOCK
ACCUMULATION IN THE SIMPLE NEOCLASSICAL SOLOW-SWAN
MODEL)
5
Equilibrium in the Solow-Swan model
(n+δ) kt
s f(kt)
k
k
•
•
[s f(kt)] > [(n+δ)kt ] ⇒ k > 0 and [s f(kt)] < [(n+δ)kt ] ⇒ k < 0
•
Steady state ( k = 0 ): ⇒ s f (k t ) = (n + δ )k t
•
Explanation:
f (k t )
k
=s
− (n + δ )
k
kt
Dynamic analysis in the Solow-Swan model
•
k >0
n+δ
•
k <0
k
s f(kt)/kt
kt
Conclusion by the solution of the fundamental differential equation of capital
accumulation:
THE STEADY-STATE OUTPUT IS CONSTANT (THE GROWTH RATE IS
ZERO) DUE TO DIMINISHING RETURNS!
6
A rise in the savings rate
(n+δ) kt
s´ f(kt)
s f(kt)
k
k´
k
If the savings rate changes from s to s´ the [s f(kt)] curve moves upwards.
⇒ The steady state capita-labour ratio rises from
⇒ Per capita income rises.
k
to
k ´.
Implication: If two countries have the same characteristics (parameters) but one
of them has a higher savings rate, then it will have a higher per capita income
(though both will have a zero growth rate).
Question: How much should a country save to achieve maximum welfare?
Criterion: Maximize per capita consumption (a function of income ofcourse)
(
)
c ( s) = f k ( s) − (n + δ )k ( s)
[
]
dk
dc
= f ′(k ) − (n + δ )
ds
ds
Since
dk
> 0 , we focus on [f´( k )-(n+δ)]. Maximum consumption is attained
ds
when:
dc
= 0 ⇒ f ′(k * ) = n + δ
ds
“Golden Rule” of capital accumulation
7
A contrast: the AK model
Production function:
Yt = AK t
Αt : technological constant, so that MPK is constant and equal to Α (i.e. the
capital-output ratio is constant). In per capita terms:
Yt
K
=A t
Lt
Lt
⇒
y t = Ak t
The fundamental differential equation for capital accumulation becomes:
•
k
= sA − (n + δ )
k
When sA > n + δ ⇒ the growth rates of capital and output are constant and
positive
Dynamic analysis in the AK model
sA
•
k
>0
k
n+δ
k0
kt
8
The basic structure of the model with exogenous technological progress
Aggregate production function:
Yt = F (K t , At Lt )
•
A
with = g > 0
A
Neoclassical assumptions are satisfied:
(1) Positive and diminishing marginal returns of factor inputs:
(2) Constant returns to scale w.r.t capital and labour:
F (λK t , At (λLt )) = λF (K t , At Lt ),
λ >0
(3) Inada conditions
By assumption (2):
 K

Yt
~
= F  t ,1 ⇒ ~
yt = f (k t )
At Lt
 At Lt 
~
where ~
yt = Yt At Lt is output per effective unit of labour, k t = K t At Lt and
~
~
f (k t ) = F (k t ,1) .
Capital accumulation equation:
•
K = sYt − δ K t
In terms of effective units of labour ΑtLt:
•
•
Y
Kt
K
K
~
= s t −δ
⇒
= s~
yt − δ k t
At Lt
At Lt
At Lt
At Lt
9
By definition we have
 Kt

d
•
AL
~
k ≡  t t
dt
•
 •
•
 K At Lt − K t  A Lt + At L 
•
• •
•

 ⇒ k~ = K − k~  A + L 
=
t

2
At Lt
( At Lt )
 A L


•
•
•
L
K
~
~
If
= n , then k =
− ( g + n )k t ⇒
At Lt
L
•
( )
~
~
~
k = s f k t − (n + g + δ )k t
(FUNDAMENTAL DIFFERENTIAL EQUATION FOR CAPITAL STOCK
ACCUMULATION IN THE NEOCLASSICAL SOLOW-SWAN MODEL
WITH EXOGENOUS TECHNOLOGICAL PROGRESS
Equilibrium in the neoclassical model
with exogenous technological progress
(n + g + δ )k~t
( )
~
sf k t
~
kt
~
k
•
Steady state:
~
~
~
k = 0 ⇒ sf (k ) = (n + g + δ )k
•
•
Y
y
y A
~
~
y = f (k ) = t = t = constant ⇒ = = g .
At Lt At
y A
10
Speed of convergence
~
Around the steady-state k , the fundamental equation is given by a Taylor
approximation:
 •

 ~ ~

•
~
~ ~
~  ∂ k (kt )
~ ~

k =
(kt − k ) ⇒ k depends upon the distance (kt − k )
~
 ∂kt


~ ~
kt = k 

•
The first term is given by:
•
~
∂k
~
∂k t
~
= sf ′(k ) − (n + g + δ )
~ ~
kt = k
~
(n + g + δ )k
, we get:
Since s =
~
f (k )
•
~ ~
∂ k (k )
~
∂k t
~ ~
kt = k *
~ ~
(n + g + δ )k f ′(k )
~
=
−
(
n
+
g
+
δ
)
=
[
ε
(
k
) − 1](n + g + δ )
ΥΚ
~
f (k )
~
~
~ k
where ε ΥΚ (k ) = f ′(k ) ~ is the elasticity of the production function w.r.t
f (k )
capital. Replacing in the Taylor approximation:
•
[
](
~
~
~ ~
k = [ε ΥΚ (k ) − 1](n + g + δ ) k t − k
)
Solution:
~
~
~
−[1−ε ΥΚ ( k )]( n + g +δ )t ~
kt − k = e
(k 0 − k )
Conclusion: The rate at which
~
to [1 − ε ΥΚ (k )](n + g + δ ) .
~
kt
approaches equilibrium is constant and equal
11
Relaxing the two input constraint: human capital in the Solow model
Question: Will an increase in the number of production factors eliminate the
impact of diminishing returns?
An obvious candidate: Human capital
(covers knowledge through education and/or experience, skills, ideas and
anything else that can be accumulated by the individual and used in the
production process –and is not covered under ‘physical’ capital)
A digression: Why do individuals accumulate human capital?
Again, an obvious answer:
In order to gain from increased returns through the rise in the wage rate
… and some difficulties associated with human capital
It is practically impossible to measure the human capital stock
We can only proxy it (say, through the number of years of education)
Utilizing its concept in the context of a growth model is necessarily vague
The basic structure of the Solow-Swan model with human capital
Aggregate production function:
Yt = F ( K t , H t , Lt )
where Ht : human capital stock
Additional assumptions on the production function:
(1) positive and diminishing marginal products with respect to human capital:
∂Yt
>0
∂H t
∂ 2Yt
<0
∂H t2
(2) constant returns to scale:
F (λK t , λH t , λLt ) = λF ( K t , H t , Lt ),
(3) Inada conditions: Hlim
→0
t
∂Yt
=∞
∂H t
λ >0
lim
H t →∞
∂Yt
=0
∂H t
12
Mechanics of the model:
By constant returns to scale:
K H 
Yt
= F  t , t ,1 ⇒ y t = f (k t , ht )
Lt
 Lt Lt 
•
•
Capital accumulation: K = ( I K ) t − δ K K t ,
H = (I H ) t − δ H H t
Investment financing: ( I K ) t = s K Yt , ( I H ) t = s H Yt
0 < s K , s H < 1,
sK + sH < 1
Fundamental equations of the Solow-Swan model with human capital:
•
k = s K f (k t , ht ) − (n + δ K )k t
•
h = s H f (k t , ht ) − (n + δ H )ht
Equilibrium:
•
k =0
•
h=0
⇒
s K f (k t , ht ) = (n + δ K )k t will yield k
⇒
s H f ( k t , ht ) = ( n + δ H ) ht will yield h
Result. The Solow-Swan model with human capital does not predict long-run
economic growth.
Intuition: why do diminishing returns to physical and human capital lead to
equilibrium where the essential variables of the model (y, k, h) remain constant?
•
k s K f (k t , ht )
− (n + δ K ) ,
We have: =
k
kt
•
h s H f (k t , ht )
=
− (n + δ H )
h
ht
•
•
•
h k
f ( k t , ht )
k
> (n + δ K ) , that cannot hold unless > > 0 , which in
If > 0 ⇒ s K
kt
h k
k
f ( k t , ht )
turn would lead to a continuous decline in
ht
•
•
h k
⇒ only = = 0 is the feasible equilibrium.
h k
13
CONVERGENCE IN THE NEOCLASSICAL MODEL
Definition of convergence: the path of the variable towards a specific value.
Various aspects of convergence can be examined:
National: convergence across countries
Domestic: convergence across regions
Global: convergence across groups of countries
Convergence in the Solow-Swan model
Cobb-Douglas production function:
Yi = AK ia L1i − a
1
Steady-state capital-labor ratio in this economy:
Steady-state income per capita:
yi =
1
1
−
A α
Production function in per capita terms
⇒ growth rate gy of country i : g y = ag k
i
 sA  1−α
ki = 

n + δ 
a
 s  1−α
n + δ 


y i = Ak ia
i
At the steady state g y = g k = 0 with the capital-labor ratio k increases at a
diminishing rate while approaching the steady-state point k (if k (0) < k ).
Result. Two economies that have the same production function and the same
values of the parameters s, n, δ will exhibit the same the long-run equilibrium
level of income per capita.
This results says that structurally similar economies will converge in the long
run to the same income level.
14
Dynamics of the growth rate of k along the transition path:
g k = sAk − (1− a ) − (n + δ )
Dynamics of the growth rate of y along the transition path:
g y = asAk − (1− a ) − a (n + δ )
⇒
∂g y
∂k
< 0,
i.e. growth rate is inversely proportional to the capital-labor ratio
along the economy’s transition to the steady-state (the lower the initial capitallabor ratio, the higher the growth rate will be).
‘Absolute Convergence’ hypothesis: In the neoclassical model of exogenous
growth, the economy tends to grow faster for a low level of initial capitallabor ratio compared to the case of a higher capital-labor ratio.
Explanation:
If capital exhibits diminishing returns (α<1), an economy with lower capitallabor ratio exhibits a higher marginal product of capital and thus, grows faster
compared to a similar economy with a higher capital-labor ratio ⇒ the
differences across countries will tend to fade out over time, with per capita
income and its growth rate gradually converging until reaching an identical
long-run equilibrium level for both countries, respectively.
Otherwise stated: if all economies have similar characteristics, the ‘poor’
(‘rich’) countries with low (high) initial capital k0, and, thus, low (high) initial
income y0, will display a higher (lower) growth rate (the two variables are
expected to be negatively correlated).
But: can it be always true that two economies with different levels of initial
income will exhibit different (and inversely related) growth rates? (or, is it
possible for a typical African country with low capital and income levels to
grow faster than Switzerland?)
Answer: ‘NO!’
15
Recall: ‘Absolute Convergence’ applies to economies with identical structural
parameters
(If, for example, if the population growth rate n is higher in a country, then it
will exhibit lower steady-state income and growth rate along transition to
equilibrium than the latter.)
⇒ Convergence should only be anticipated across structurally ‘similar’
economies (i.e. with identical parameter values).
‘Conditional Convergence’ hypothesis: In the neoclassical model of
exogenous growth, economies tend to converge faster to their own steady
state the further away they are from it.
An economy is likely to exhibit both a high initial income and fast growth
simply because its steady-state income is simultaneously high and relatively far
from its current income.
Example: large values of the technology constant A result in high steady-state
capital-labor ratio and income.
⇒ A simple comparison with an economy having the same initial capital stock
and income level without looking at technology (parameter A) would lead to
inaccurate conclusions regarding its convergence pattern.
16
Empirical tests on convergence
General formulation: g yi , t , t + T = f [log( yi ,t )]
g y i , t , t + T = log( yi ,t +T / yi ,t ) / T : average growth rate of economy i from t to t+T
log( yi ,t ) : log of income of economy i in period t
Cobb-Douglas production function: linear approximation around the steady-state
yields of the capital growth rate (which determines the income growth rate):
gk =
d log(K )
k
≅ −(1 − a)(n + δ ) log( )
dt
k
y
k
y
Using log( ) = a log( ) , the income growth rate is: g y ≅ −(1 − a)(n + δ ) log( )
y
k
y
y
g
=
−
β
log(
)
General specification: y
y ,
where β=(1-α)(n+δ).
Convergence is determined by β: for β>0, if the initial income is lower (higher)
than the steady-state income, the economy will grow faster (more slowly)
⇒ β-convergence.
Convergence speed: coefficient β shows how fast the economies approach their
long-run equilibrium: higher (lower) β implies faster (lower) convergence.
Solving the last equation yields: log( y t ) = (1 − e − βt ) log( y ) + e − βt log( y 0 )
Income at any time t depends on the constant value of the initial income y0
and of the steady-state income y .
For higher values of β, income yt will be closer to its steady-state level.
yT
)
y0
y
(1 − e − βT )
=
log( )
T
T
y0
log(
At t=T:
The average growth rate after T periods is inversely related to the ratio of the
initial and the steady-state income.
17
‘Absolute’ and ‘Conditional’ β-convergence: In the neoclassical model of
exogenous growth, absolute β-convergence is measured in terms of the initial
income of the economy, whereas ‘conditional’ β-convergence is given by the
deviation of the initial income level from its steady-state level.
Question: If each economy tends to converge to its steady-state income level,
will convergence emerge at the global level, i.e. will the inequality of income
distribution be worldwide diminished?
Intuition: for a small income gap across economies worldwide, individual
economies should tend to converge to equilibrium income per capita.
But: does β-convergence lead necessarily to small income dispersion?
Answer: ‘NO!’: This income gap can stay constant (or increase) despite the
presence of β-convergence.
Example: a poor country may grow much faster than a rich one so that the
income of the former may turn out to exceed the level of the latter and thus the
income gap will increase while there is β-convergence in both countries
(Galton’s Fallacy).
Consider the case of successive time periods (Τ=1) between t-1 and t:
log( yt / yt −1 ) = (1 − e − β ) log( y ) − (1 − e − β ) log( yt −1 ) ⇔ log( yt ) = e − β log( yt −1 ) + c
where c = (1 − e − β ) log y . Income dispersion is then given by the variance:
Var[log( y t )] = e −2 β Var[log( y t −1 )]
2
Setting Var[log( yt )] ≡ σ t :
σ t2 = e −2 β σ t2−1
⇒ in order for income dispersion to be reduced intertemporally we must have
2
2
that β>0 (β-convergence), in which case σ t < σ t −1 (σ-convergence)
Relation between β-convergence and σ-convergence: In the neoclassical
model of exogenous growth β-convergence is a necessary but not sufficient
condition for σ-convergence.
18
Log of real per capital income and growth rate in 119 countries
8
Κατά κεφαλή εισόδηµα (λογάριθµος) 1960
7
6
5
4
3
2
1
0
-2
-1
0
1
2
3
4
5
6
7
8
Ρ υθµ ό ς µ εγέθυνσ ης 19 60 -19 89
Source: Levine και Renelt (1992).
Log of real per capital income and growth rate in OECD countries
6
6
Κατά κεφαλή εισόδηµα (λογάριθµος) 1960
5
5
4
4
3
3
2
2
1
1
2
3
4
5
6
7
8
Ρ υ θ µ ό ς µ εγ έθ υ ν σ η ς 1 9 6 0 -1 9 8 9
Source: Levine και Renelt (1992).
Intertemporal dispersion of log of real per capital income
in all economies and OECD countries
1 ,2 0
1 ,0 0
0 ,8 0
0 ,6 0
0 ,4 0
0 ,2 0
1950
1955
1960
1965
W o rld
1970
1975
1980
1985
1990
OECD
Source: Sala-I-Martin (1996).
19