Upload
others
View
8
Download
0
Embed Size (px)
Citation preview
Ifo Institute for Economic Research at the University of Munich
1. Neoclassical Growth Model
Literature: David Romer (2001), Advanced Macroeconomics,
McGraw-Hill.
1
McGraw-Hill.
Ifo Institute for Economic Research at the University of Munich
The Neoclassical Growth Model – Idea (1)
• The neoclassical baseline model was developed by
Solow (1956) and Swan (1956).Solow (1956) and Swan (1956).
• Central features:
– Driving factor 1: capital accumulation.
– Driving factor 2: exogenous technological progress.
– Problematic features: saving rate is constant,
technological progress is exogenous.
2
technological progress is exogenous.
Ifo Institute for Economic Research at the University of Munich
• Further features:
– The macroeconomic output of an economy is created
The Neoclassical Growth Model – Idea (2)
– The macroeconomic output of an economy is created
with a technology of constant returns to scale, by
using the two input factors labour and capital.
Empirically, one can observe a constant ratio of the
input factor capital and the output of an economy.
If one assumes from this that capital and output grow
with the same rate, it follows that – due to the
3
with the same rate, it follows that – due to the
technology of constant returns to scale – also the
input factor labour has to increase with this rate …
Ifo Institute for Economic Research at the University of Munich
… On the one hand, the input factor labour can be
increased quantitatively due to an increase of the
The Neoclassical Growth Model – Idea (3)
increased quantitatively due to an increase of the
population of the economy, on the other hand, it can
be increased due to technological progress, which
is given exogenously. However, population growth is
at first a demographic variable, which is difficult to
influence by economic decisions. Therefore, in the
Solow-Swan-Model, long term economic growth is
4
Solow-Swan-Model, long term economic growth is
exogenous, as it is driven by the growth of the
population and the (exogenous) technological
progress.
Ifo Institute for Economic Research at the University of Munich
Neoclassical Growth Model
1.1 Baseline Model without Technological Progress
5
Ifo Institute for Economic Research at the University of Munich
Baseline Model without Technological Progress
1. Production Function
2. Capital Accumulation
3. Fundamental Growth Equation
4. Steady State and Transitory Dynamics
5. Changes in Exogenous Parameters
6
5. Changes in Exogenous Parameters
6. Optimal Saving Rate and Golden Rule Capital Stock
Ifo Institute for Economic Research at the University of Munich
Production Function (1)
The production function of the economy is given by:
( ) ( ) ( )( ),Y t F K t L t= .
Characteristics (neoclassical characteristics):
1. Y is 2-times differentiable with respect to its arguments and
has diminishing, positive marginal returns:
: 0, : 0K L
Y YF F
K L
∂ ∂= > = >
∂ ∂,
( ) ( ) ( )( ),Y t F K t L t= .
7
This property implies the substitutability of the two input
factors capital and labour.
2 2
2 2: 0, : 0
KK LL
K L
Y YF F
K L
∂ ∂
∂ ∂= < = <
∂ ∂.
Ifo Institute for Economic Research at the University of Munich
2. Inada-conditions:
Production Function (2)
lim 0, lim 0F F= =
3. Constant elasticities of scale:
( ) ( )( ) ( ) ( )( ), , , 0F K t L t F K t L tλ λ λ λ= ∀ >
0 0
lim 0, lim 0
lim , lim
K LK L
K LK L
F F
F F
→∞ →∞
→ →
= =
= ∞ = ∞ .
,
8
Production function is linear homogenous
(homogenous of degree 1).
( ) ( )( ) ( ) ( )( ), , , 0F K t L t F K t L tλ λ λ λ= ∀ > .
Ifo Institute for Economic Research at the University of Munich
Sidestep: Linear Homogenous Production Functions (1)
• A linear homogenous production function
( ),Y F K L=
has the following characteristics (Euler‘s Theorem):
( ),Y F K L=
( ) ( ), ,F K L F K LY K L
K L
∂ ∂≡ ⋅ + ⋅
∂ ∂ .
9
The input factors are compensated with respect to their
marginal returns. Output is completely distributed to the
input factors.
Ifo Institute for Economic Research at the University of Munich
• Example: Cobb-Douglas production function (A = level of
knowledge):
Sidestep: Linear Homogenous Production Functions (2)
knowledge):
• Marginal returns:
( ) 1, , 0 1Y F K L A K L
α α α−= = ⋅ ⋅ < < .
( )( ) 1 1
,, /K
F K LF K L AK L Y K
K
α αα α− −∂
= = =∂
,
10
( )
( )( ) ( ) ( )
, /
,, 1 1 /
K
L
F K L AK L Y KK
F K LF K L AK L Y L
L
α α
α α
α α−
= = =∂
∂= = − = −
∂.
,
Ifo Institute for Economic Research at the University of Munich
K Y KF α α⋅ = =KYK
• Income shares of output:
Sidestep: Linear Homogenous Production Functions (3)
( )1 1
K
L
K Y KF
Y K Y
L Y LF
Y L Y
α α
α α
⋅ = =
⋅ = − = −
αα ==⋅Y
K
K
Y
Y
KFK ,
.
• Elasticities of output with respect to the two input factors:
( )dY K KdY dK
11
( )
( )
,
, 1
K
L
dY K KdY dKF K L
Y K dK Y Y
dY L LdY dLF K L
Y L dL Y Y
α
α
= ⋅ = ⋅ =
= ⋅ = ⋅ = −
,
.
Ifo Institute for Economic Research at the University of Munich
• Because of characteristic 3 (linear homogeneity) it is
possible to derive the production function in its intensive
Production Function (3)
possible to derive the production function in its intensive form:
• If y=Y/L is the productivity of labour (per capita output)
and k=K/L is the capital intensity, it follows:
( , ) ( / ,1) ( / )Y F K L L F K L L f K L= = ⋅ = ⋅ .
( )y f k= .
12
• The macroeconomic output of the economy is then
derived by:
( )y f k= .
Y L y= ⋅ .
Ifo Institute for Economic Research at the University of Munich
•
•
Production Function (4)
0<)k(''f .
0>)k('f .
•
• equal to the marginal return of the input factor capital:
• Example: Cobb-Douglas production function
'( )f k
1, 0< 1Y AK L
α α α−= <
( )( )
( )( )
( ) ( )1 1, , , ,1'( )
L L
K KL L
F K L F K L F K L F K LKf k L
K L K K
∂ ∂ ∂ ∂∂= = = =
∂ ∂ ∂ ∂ ∂
0<)k(''f .
.
13
• In intensive form:
1, 0< 1Y AK L
α α α−= < .
y Akα= .
Ifo Institute for Economic Research at the University of Munich
• We know:
( )( ) ( ), ,
,F K L F K L
Y F K L K LK L
∂ ∂≡ = ⋅ + ⋅
∂ ∂,
Production Function (5)
• From this follows:
( )
( ) ( )( ) ( )
,
,, und
Y F K L K LK L
F K L f kF K L L f k
K k
≡ = ⋅ + ⋅∂ ∂
∂ ∂= ⋅ =
∂ ∂and .
,
( )( )
( ), ,,
F K L F K LL F K L K
∂ ∂⋅ = − ⋅
∂ ∂,
14
( )
( )( ) ( )
( ) ( )
,
, 1, '
'
L F K L KL K
F K LY KMPL F K L f k
L L L L
MPL f k k f k
⋅ = − ⋅∂ ∂
∂∂≡ ≡ = ⋅ −
∂ ∂
= − ⋅ .
,
,
Ifo Institute for Economic Research at the University of Munich
Baseline Model without Technological Progress
1. Production Function
2. Capital Accumulation
3. Fundamental Growth Equation
4. Steady State and Transitory Dynamics
5. Changes in Exogenous Parameters
15
5. Changes in Exogenous Parameters
6. Optimal Saving Rate and Golden Rule Capital Stock
Ifo Institute for Economic Research at the University of Munich
Capital Accumulation (1)
• Economic output can be increased by capital
accumulation.accumulation.
• Closed economy: savings = investment.
• Assumption of a constant saving rate s:
• The capital goods depreciate.
( ), , 0 s 1I s F K L= ⋅ ≤ ≤ .
16
• Assumption: constant depreciation rate .
• Development of the capital stock:
KIK δ−=•
.
δ
Ifo Institute for Economic Research at the University of Munich
Capital Accumulation (2)
• From the equation for the capital accumulation
•
follows that the capital stock remains constant, if the
investments are as high as the level of depreciation.
Therefore, the level of investment δK is also called
“break-even investment“.
KIK δ−=•
.
17
“break-even investment“.
Ifo Institute for Economic Research at the University of Munich
Capital Accumulation (3)
YδK (break-even
investment)
F(K,L)
s·F(K,L) (actual investment)
Y*
18
KK*K1 K2
Ifo Institute for Economic Research at the University of Munich
δKY
Capital Accumulation (4)
K•
s·F(K,L)
F(K,L)
Y*
Actual
investments
19
K
KK*K1
Depreciation,
investment
necessary to
maintain
the existing
capital stock
investments
K2
K•
Ifo Institute for Economic Research at the University of Munich
Baseline Model without Technological Progress
1. Production Function
2. Capital Accumulation
3. Fundamental Growth Equation
4. Steady State and Transitory Dynamics
5. Changes in Exogenous Parameters
20
5. Changes in Exogenous Parameters
6. Optimal Saving Rate and Golden Rule Capital Stock
Ifo Institute for Economic Research at the University of Munich
• Change of the capital stock over time equals the gross
investments minus the depreciations :
Fundamental Growth Equation (1)
I Kδ
�•
investments minus the depreciations :
where s is the constant saving rate.
• Dividing by leads to:L
I Kδ
, where ( ), , 0 s 1I s F K L= ⋅ ≤ ≤
,
K I Kδ= −�•
( ),K s F K L Kδ= ⋅ −�•
,
21
( )K
s f k kL
δ= ⋅ −
�•
.
• Dividing by leads to:L
Ifo Institute for Economic Research at the University of Munich
• The capital intensity is defined as .
For changes of the capital intensity we get:
k /k K L=
Fundamental Growth Equation (2)
• • • • • •
• The population grows with a constant rate n:
( ) 0 00 , ( )
ntL L L t L e L L n
•
= = ⇒ = .
2
K K L K L K K L K Lk k
t L L L L L L L
• • • • • •• ∂ −
= = = − ⋅ = − ⋅ ∂
..
22
• Transforming the above equation yields
( ) 0 00 , ( ) L L L t L e L L n= = ⇒ = .
K Lk k k k n
L L= + ⋅ = + ⋅
� �� �
••
••
.
Ifo Institute for Economic Research at the University of Munich
Fundamental Growth Equation (3)
• From slide 18 we know:
( )K
s f k kδ= ⋅ −
�•
.
• Substituting the equation for from the previous slide in
the equation above leads to
( )s f k kL
δ= ⋅ − .
L
K•
sk)k(sfknk −=+•
.
23
( ) ( )k s f k n kδ= ⋅ − + ⋅�•
.
• Transformation yields
Ifo Institute for Economic Research at the University of Munich
• This is the fundamental growth equation:
( ) ( )k s f k n kδ•
= ⋅ − + ⋅•
Fundamental Growth Equation (4)
• This is a nonlinear differential equation: difficult to solve
( ) ( )k s f k n kδ= ⋅ − + ⋅
Gross in-vestments
Depreciations: - physical depreciations of the capital stock δ- dilution of capital stock due to population growth n
.
24
• This is a nonlinear differential equation: difficult to solve
analytically.
Ifo Institute for Economic Research at the University of Munich
Baseline Model without Technological Progress
1. Production Function
2. Capital Accumulation
3. Fundamental Growth Equation
4. Steady State and Transitory Dynamics
5. Changes in Exogenous Parameters
25
5. Changes in Exogenous Parameters
6. Optimal Saving Rate and Golden Rule Capital Stock
Ifo Institute for Economic Research at the University of Munich
• The fundamental growth equation
What is the Steady State? (1)
( ) ( )k s f k n kδ•
= ⋅ − + ⋅
implies that the economy converges to a state in which the capital intensity remains constant. This is the steady state. It is thus defined as
( ) ( )k s f k n kδ= ⋅ − + ⋅
* *0 ( ) ( )k s f k n kδ
•
= ⇒ ⋅ = + ⋅ .
26
• To see the convergence, look at the dynamics:
1) k<k*: actual investment > break-even investment: k increases.
2) k>k*: actual investment < break-even investment: k decreases.
Ifo Institute for Economic Research at the University of Munich
What is the Steady State? (2)
y ( )n kδ+
( )f k
y∗
( )s f k⋅
( )f k
27
1k
2k kk
∗
Ifo Institute for Economic Research at the University of Munich
Phase diagram for in the Solow Model:k
k•
What is the Steady State? (3)
k Break-even investmentsmaller than actual investment
Break-even investmenthigher than actual investment
28
kk∗
Ifo Institute for Economic Research at the University of Munich
• The long run equilibrium (steady-state) is the state of
condition, where the per capita growth rates are constant
What is the Steady State? (4)
condition, where the per capita growth rates are constant
(balanced growth path).
• The capital intensity k converges towards the equilibrium
capital intensity k*, which does not depend on the starting
point k(0)>0.
• In the steady state the gross investments are just enough
to compensate for the depreciations.
29
*0k =
�•
to compensate for the depreciations.
• The Solow-Swan model implies convergence towards the
steady state .
• From this follows: .* *
( ) ( )s f k n kδ⋅ = + ⋅
Ifo Institute for Economic Research at the University of Munich
• The steady state growth rate of k is
What is the Steady State? (5)
*
0k
g
•
= = .
• As k* is constant in the steady state, also the per capita
variables y*=f(k*) und c*=(1-s)y* are constant. The total
output Y as well as the capital stock K grow with rate n
along the balanced growth path:
* *0
k
kg
k= = .
30
( analogously)
along the balanced growth path:**
yLY ⋅= **kLK ⋅=
nnk
k
L
L
K
Kg
*
*
*
*
*
*
K* =+=+==
•••
0 *Yg .
, ,
Ifo Institute for Economic Research at the University of Munich
• Convergence can also be shown analytically.
• In general, the economy grows along the transitory path,
The Transitory Dynamics (1)
• In general, the economy grows along the transitory path,
hopefully towards the steady state.
( )( )
( ) ( )k
s f k n kk s f kg n
k k k
δδ
⋅ − + ⋅= = = − +
�•
.
31
• Does the economy converge towards the steady state?
This would require the growth rate of capital be larger
(smaller) than zero if the capital stock k is smaller (larger)
than in the steady state..
Ifo Institute for Economic Research at the University of Munich
• Hence, the growth rate of capital must be a falling function
of the capital stock.
The Transitory Dynamics (2)
of the capital stock.
• We check this:
( )( )
'( ) ( ) /0
kg s f kn
k k k
f k k f k Y Ls s
δ∂ ∂ ⋅
= − + ∂ ∂
⋅ − ∂ ∂= = − <
32
2 2
'( ) ( ) /0
f k k f k Y Ls s
k k
⋅ − ∂ ∂= = − < .
( ) '( )MPL f k k f k= − ⋅[Remember from above (slide 14): ]
Ifo Institute for Economic Research at the University of Munich
( ) /s f k k⋅
The Transitory Dynamics (3)
0kg <
0kg >( )n δ+
( ) /s f k k⋅
33
k∗
k
( ) /s f k k⋅
Ifo Institute for Economic Research at the University of Munich
The Transitory Dynamics (4)
• So far, we analyzed the convergence behavior of capital.
What about per capita income?What about per capita income?
• The growth rate of the per capita income on the way to
equilibrium is
'( )
'( ) '( )
dy dy dky f k k
dt dk dt
y f k k f k kg k
= = = ⋅
⋅= = =
� �
� ��
• •
••
•
,
,
34
'( ) '( )
( ) ( )
'( )
( )
y
y k
y f k k f k kg k
y f k f k k
k f kg g
f k
⋅= = =
⋅⇒ =
,
.
Ifo Institute for Economic Research at the University of Munich
• Note that
The Transitory Dynamics (5)
'( ) ( , ) / ( , ) /k f k K F K L K K F K L K⋅ ∂ ∂ ⋅∂ ∂= =
is the capital rate of return, i.e., the share of capital income
in total income.
• For a Cobb-Douglas production function the capital rate of
'( ) ( , ) / ( , ) /
( ) /
k f k K F K L K K F K L K
f k L Y L Y
⋅ ∂ ∂ ⋅∂ ∂= =
35
return is constant and equal to the capital coefficient α.
• Assume, Cobb-Douglas is a good approximation to the
true production function.
( )'( )k f k f k α⋅ = .
Ifo Institute for Economic Research at the University of Munich
•
The Transitory Dynamics (6)
'( )
( )y k k
k f kg g g
f kα
⋅= =
.
This implies that the ratio of the growth rate of the output
and of the growth rate of the capital return is constant.
Empirically, the capital return ratio is relatively stable,
implying that the Cobb-Douglas framework is a good
approximation.
• From this follows that g increases (decreases), if g
( )f k
36
• From this follows that gy increases (decreases), if gk
increases (decreases).
• At equilibrium, the growth rates of the per capita measures
are equal to zero.
Ifo Institute for Economic Research at the University of Munich
Baseline Model without Technological Progress
1. Production Function
2. Capital Accumulation
3. Fundamental Growth Equation
4. Steady State and Transitory Dynamics
5. Changes in Exogenous Parameters
37
5. Changes in Exogenous Parameters
6. Optimal Saving Rate and Golden Rule Capital Stock
Ifo Institute for Economic Research at the University of Munich
Change in Exogenous Parameters (1)
y (n+δ)k
Increase of the saving rate:
*
1c
f(k)
s0·f(k)
s1·f(k)
*
0c
38
kk0* k1*
Ifo Institute for Economic Research at the University of Munich
s
Change in Exogenous Parameters (2)
k• t
t0
t
0t
39
k
0t t
Ifo Institute for Economic Research at the University of Munich
yg
Change in Exogenous Parameters (3)
t0t
yg
c If the per capita consumption decreases
40
0t t
or increases, depends on the starting conditions!
Ifo Institute for Economic Research at the University of Munich
y (n+δ)k
f2(k)y *
Change in Exogenous Parameters (4)
Increase in the technology parameter A:
f1(k)
s·f1(k)
f2(k)
s·f2(k)y1*
y2
y2*
41
kk1* k2*
Ifo Institute for Economic Research at the University of Munich
y(n2+δ)k
Change in Exogenous Parameters (5)
Increase in the population growth:
y (n1+δ)k
f(k)
s·f(k)
(n2+δ)k
y1*
y2*
42
kk1*k2*
Ifo Institute for Economic Research at the University of Munich
Baseline Model without Technological Progress
1. Production Function
2. Capital Accumulation
3. Fundamental Growth Equation
4. Steady State and Transitory Dynamics
5. Changes in Exogenous Parameters
43
5. Changes in Exogenous Parameters
6. Optimal Saving Rate and Golden Rule Capital Stock
Ifo Institute for Economic Research at the University of Munich
What is the ”Optimal Saving Rate“?
• An economy is getting richer the more it saves. But
eventually, we are interested in consumption.eventually, we are interested in consumption.
• Question: Is there an ”optimal saving rate“, which allows
a maximal consumption level over all generations?
• To answer this question, we proceed as follows:
– First, we need the relationship between consumption and the saving rate.
44
rate.
– Then, we maximize consumption with respect to the saving rate.
Ifo Institute for Economic Research at the University of Munich
Relationship between Consumption and Saving (1)
• We already know that there is a positive relationship between the saving rate s and the steady state capital between the saving rate s and the steady state capital intensity k*:
• The equilibrium per capita consumption level is a function of the capital intensity:
*( )k s with
*( ) / 0dk s ds > .
* * * *( ) (1 ) ( ( )) ( ( )) ( ( ))c s s f k s f k s s f k s= − = − ⋅
45
• Hence, steady state consumption is also a function of s.
* * * *( ) (1 ) ( ( )) ( ( )) ( ( ))c s s f k s f k s s f k s= − = − ⋅ .
Ifo Institute for Economic Research at the University of Munich
Relationship between Consumption and Saving (2)
• Substituting the steady state condition
yields
* *( ) ( )s f k n kδ⋅ = + .
* * * * *( ) ( ( )) ( ( )) ( ( )) ( ) ( )c s f k s s f k s f k s n k sδ= − ⋅ = − + .
46
( ) ( ( )) ( ( )) ( ( )) ( ) ( )c s f k s s f k s f k s n k sδ= − ⋅ = − + .
Ifo Institute for Economic Research at the University of Munich
• Now we have a relationship between steady state
consumption and the saving rate.
Optimal Saving Rate and Golden Rule Capital Stock (1)
consumption and the saving rate.
• To find the optimal saving rate, the necessary condition for
a maximum of the per capita consumption level is:
* * *
*0
dc dc dk
ds dk ds= = .
47
*0
ds dk ds= = .
Ifo Institute for Economic Research at the University of Munich
• The necessary condition
Optimal Saving Rate and Golden Rule Capital Stock (2)
* ** *( ( )) ( ) ( )d f k s n k sdc dkδ − +
implies that the optimal (“golden“) saving rate is chosen
!
* ** *
*
* *
0
0
( ( )) ( ) ( )
'( ( )) ( ) / 0
d f k s n k sdc dk
ds dk ds
f k s n dk ds
δ
δ>
=
− + =
= − + ⋅ = 123144424443
48
implies that the optimal (“golden“) saving rate is chosen
such that
*'( ( )) ( )
goldf k s n δ= + .
Ifo Institute for Economic Research at the University of Munich
• Defining the golden capital stock as
Optimal Saving Rate and Golden Rule Capital Stock (3)
* *( )k k s=
we have the optimality condition for the golden capital
stock:
*'( )
goldf k n δ= +
* *( )gold goldk k s=
.
49
• The golden consumption level thus is
.* *
( ) ( )gold gold gold
c f k n kδ= − +
Ifo Institute for Economic Research at the University of Munich
f(k)Slope:(n+δ)In the consumption
level is maximal and the
*
goldky
Optimal Saving Rate and Golden Rule Capital Stock (4)
(n+δ)k
*
goldc
*
1c
sgold·f(k)
s1> sgold
s0< sgold
*
0c
level is maximal and the slope of the production function is n+δ.
50
*
0k
*
1k
*
goldk
0c
k
Ifo Institute for Economic Research at the University of Munich
Case 1: Initial capital stock larger than golden rule capital
stock (initial saving rate too high).
Welfare Analysis of the Golden Rule Capital Stock (1)
c
y
tt
,c y
0c
51
t0t0
c
Pareto improvement: Every generation (also the current
generation) improves if the saving rate decreases. Transition
from an over-capitalized equilibrium to the golden rule equili-
brium is possible with the agreement of all generations!
Ifo Institute for Economic Research at the University of Munich
Case 2: Initial capital stock lower than golden rule capital
stock (initial saving rate too low).
Welfare Analysis of the Golden Rule Capital Stock (2)
c
y,c y
0c
52
Generational conflict: Future generations improve, but
the current generation has to give up some consumption!
t0t
0
Ifo Institute for Economic Research at the University of Munich
Policy Conclusion from Golden Rule
• Policy phrase: “We have to invest more to grow faster!“
Is this sensible?
1) Increase in savings rate (=more investment) leads to 1) Increase in savings rate (=more investment) leads to
higher growth of output only temporarily.
2) In the long run, the growth rate remains unchanged.
But the level of output is higher.
3) If consumption is the ultimate target, more investment
is not necessarily the right thing to do.
53
is not necessarily the right thing to do.
a) Starting point below Golden Rule level: sensible,
but intergenerational conflict.
b) Starting point above Golden Rule level: not
sensible to invest even more.
Ifo Institute for Economic Research at the University of Munich
Neoclassical Growth Model
1.2 Baseline Model with Technological Progress
54
Ifo Institute for Economic Research at the University of Munich
Baseline Model with Technological Progress
1. Fundamental Growth Equation with Labour-1. Fundamental Growth Equation with Labour-
Augmenting Technological Progress
2. The Steady State
55
Ifo Institute for Economic Research at the University of Munich
Fundamental Growth Equation with Labour-Augmenting Technological Progress (1)
• The production function is given by:•
• The technology grows with a constant rate .
• Multiplying the input factor L(t) and the efficiency
measure A(t) yields the effective amount of labour
Ag
( ) ( ( ), ( ) ( )), A( ) 0Y t F K t A t L t t•
= ⋅ > .
56
measure A(t) yields the effective amount of labour
A(t)·L(t).
Ifo Institute for Economic Research at the University of Munich
• As the production function is linear homogenous, it follows
that
Fundamental Growth Equation with Labour-Augmenting Technological Progress (2)
that
where is the level of capital per unit of effective labourk
,
⋅⋅⋅= 1,
)t(L)t(A
)t(KF)t(L)t(A)t(Y
))t(k(f,)t(L)t(A
)t(KF
)t(L)t(A
)t(Y)t(y =
⋅=
⋅= 1
57
and is the output per unit of effective labour.
• Fundamental growth equation with technological progress:
y
.k)ng()k(fsk A δ++−⋅=•
Ifo Institute for Economic Research at the University of Munich
• In the long run equilibrium we have .
• The equilibrium capital stock per unit of effective labour *
k
ˆ 0k =�•
Fundamental Growth Equation with Labour-Augmenting Technological Progress (3)
• The equilibrium capital stock per unit of effective labour
satisfies the following condition:
*k
Gross in-vestments
Depreciations: - physical depreciations of the capital stock δ
.*
A
*k)ng()k(fs δ++=⋅
58
vestments - physical depreciations of the capital stock δ- dilution of capital stock due to population growth n
and the technological progress
Ifo Institute for Economic Research at the University of Munich
• Transitory dynamics (analytically):
Fundamental Growth Equation with Labour-Augmenting Technological Progress (4)
.)ng(k
)k(fs
k
kg Ak
δ++−⋅
==
•
59
Ifo Institute for Economic Research at the University of Munich
ˆ( )s f k⋅
• Transitory dynamics (graphically):
Fundamental Growth Equation with Labour-Augmenting Technological Progress (5)
ˆ( )s f k⋅
(gA+n+δ)
ˆ 0k
g >
ˆ 0k
g <
kg
ˆ( )
ˆ
s f k
k
⋅
60
*kˆ(0)k
ˆ( )
ˆ
s f k
k
⋅
k
ˆ 0k
Ifo Institute for Economic Research at the University of Munich
Baseline Model with Technological Progress
1. Fundamental Growth Equation with Labour-Augmenting 1. Fundamental Growth Equation with Labour-Augmenting
Technological Progress
2. The Steady State
61
Ifo Institute for Economic Research at the University of Munich
Growth in Levels, per Capita and per Unit of Effective Labour
In the Solow model with technological progress, in equilibrium…
… the variables given in units of effective labour … the variables given in units of effective labour
( , and ) do not change.
… the variables given in per capita units (k, y und c) grow
with the rate of the technological progress .
y ck
Ag( ) ( )ˆ
( ) ( ) ( )L L
K t k tk
A t L t A t= = ,
62
… the variables in levels (K, Y und C) grow with the
rate .ng A +
* *
ˆ
ˆ 0
L
L
L L
k Ak
Ak k
g g g
g g g x
= −
= ⇒ = = .
,
Ag
Ifo Institute for Economic Research at the University of Munich
Neoclassical Growth Model
1.3 Implications
63
Ifo Institute for Economic Research at the University of Munich
Convergence and Speed of Convergence
1. Relative and Absolute Convergence1. Relative and Absolute Convergence
2. Speed of Convergence
3. The Solow Model and the Central Questions of Growth
Theory
4. The Feldstein-Horioka Puzzle
64
4. The Feldstein-Horioka Puzzle
Ifo Institute for Economic Research at the University of Munich
Relative Convergence (1)
• Do poor countries grow faster than rich countries to catch
up?
• The Solow-Swan model implies that countries converge to
their steady-state.
• Comparison of two economies with
– different initial capital intensity,
– identical saving rate and population growth.
• In per capita units, the poor country grows faster than the
richpoor )(k)(k 00 <
65
• In per capita units, the poor country grows faster than the
rich country.
• When both countries have reached their equilibrium growth
path, they grow with the same rates.
Relative convergence.
Ifo Institute for Economic Research at the University of Munich
Relative Convergence (2)
ˆ( )
ˆ
s f k
k
⋅richpoor )(k)(k 00 <
ˆ( )
ˆ
s f k
k
⋅
k
poorg
richgδ++ ng A
66
*k
k
krich)(k 0poor)(k 0
Ifo Institute for Economic Research at the University of Munich
Relative Convergence (3)
• The neoclassical growth model implies relative
convergence.convergence.
• For different parameters, from the concept of relative
convergence one can not conclude that countries, which
have a larger distance to their steady state, grow faster than
countries, which are closer to their steady state.
• One only can conclude that countries grow the faster the
larger the distance to their individual steady state is.
67
larger the distance to their individual steady state is.
• Now: Comparison of a poor and a rich country, which also
differ with respect to their saving rates .)ss( richpoor <
Ifo Institute for Economic Research at the University of Munich
Relative Convergence (4)
ˆ( )
ˆ
s f k
k
⋅richpoor )(k)(k 00 <
richpoor ss <and
δ++ ng A
k richpoor ss <and
poorgrichg
)k(fs ⋅k
)k(fsrich ⋅
68
richg
rich)(k 0rich
* )(k 0
k
)k(fspoor ⋅
poor)(k 0 poor*
k
Ifo Institute for Economic Research at the University of Munich
• The concept of relative convergence has to be distinguished
from the concept of absolute convergence.
Relative vs. Absolute Convergence
from the concept of absolute convergence.
• Absolute convergence means that poor economies grow faster
than rich economies, independently of the parameters like the
saving rate or the growth of the population.
• The concept of absolute convergence is not supported by the
Solow-Swan model.
69
• Hence, if a country suffers from a low saving rate and slow
technological progress, per capita income will remain relatively
low (compared to a country with more favorable parameters).
Ifo Institute for Economic Research at the University of Munich
Convergence and Speed of Convergence
1. Relative and Absolute Convergence1. Relative and Absolute Convergence
2. Speed of Convergence
3. The Solow Model and the Central Questions of Growth
Theory
4. The Feldstein-Horioka Puzzle
70
4. The Feldstein-Horioka Puzzle
Ifo Institute for Economic Research at the University of Munich
Speed of Convergence (1)
• How much time does a country need to reach its steady
state?
• If the convergence process is fast, one can concentrate • If the convergence process is fast, one can concentrate
on the behaviour of the economy with respect to the long-
run equilibrium.
• If the convergence process is slow, the transition dynamics are important.
The question can be answerd only numerically.In general,
71
the analysis are performed by using linear approximations at
the long-run equilibrium.
Ifo Institute for Economic Research at the University of Munich
• How fast does converge towards ?
• The changes of are:
*kk
k
Speed of Convergence (2)
• • •
• A Taylor approximation of order 1 of at yields:*ˆ ˆk k=
ˆ ˆ ˆ ˆ ˆ ˆ( ( )) ( ) ( ) ( ) (1)Ak s f k t g n k t k k kδ• • •
= ⋅ − + + ⋅ ⇒ =
)k(k
•
*ˆ ˆ( )ˆ ˆ ˆ( ) (2)
ˆ
k kk k k
k
••
∂≈ ⋅ −
∂
72
• Differentiating (1) with respect to at point yields:k *k
*
*
ˆ ˆ
ˆ ˆ( ) ˆ'( ) ( ) (3)ˆ Ak k
k ks f k g n
kδ
•
=
∂= ⋅ − + +
∂
*ˆ ˆk k
k=
∂
Ifo Institute for Economic Research at the University of Munich
• Substituting into (2) gives
Speed of Convergence (3)
* *ˆ ˆ ˆ ˆ'( ) ( ) ( ) (4)k s f k g n k kδ•
≈ ⋅ − + + ⋅ −
• However, it is difficult to observe the quantities in (4). To
facilitate computations, remember that in the steady state we
have
* *ˆ ˆ ˆ ˆ'( ) ( ) ( ) (4)Ak s f k g n k kδ ≈ ⋅ − + + ⋅ −
* * *ˆ ˆ ˆ0 ( ) ( ) 0k s f k g n kδ•
= ⇒ ⋅ − + + ⋅ =
73
and thus
* * *ˆ ˆ ˆ0 ( ) ( ) 0A
k s f k g n kδ= ⇒ ⋅ − + + ⋅ =
)k(f
k)ng(s
*
*
A δ++= .
Ifo Institute for Economic Research at the University of Munich
• Substituting this into (4) leads to
Speed of Convergence (4)
* *ˆ ˆ'( )f k k• ⋅
• Finally, remember that is the share of
capital income in total income. According to the Cobb-
Douglas function, this quantity equals α and is constant.
Empirically, the capital share is typically around one third.
* **
*
ˆ ˆ'( )ˆ ˆ ˆ( ) ( ) ( ) (5)ˆ( )
A A
f k kk g n g n k k
f kδ δ
• ⋅≈ + + − + + ⋅ −
* * *ˆ ˆ ˆ'( ) / ( )k f k f k⋅
74
Empirically, the capital share is typically around one third.
• Thus, let us assume a constant share:
* * * *ˆ ˆ ˆ ˆ( ) '( ) / ( )K
k k f k f kα = ⋅ .
Ifo Institute for Economic Research at the University of Munich
• Substituting this into (5) yields
Speed of Convergence (5)
( )•
• What does this mean?
• The speed of convergence of is proportional to the
distance from the equilibrium capital stock .
( )( )* *ˆ ˆ ˆ ˆ( ) 1 ( ) (6)K Ak k g n k kα δ•
≈ − + + ⋅ −
k*
k
75
distance from the equilibrium capital stock .
• The parameter that governs the speed is
k
( )( )*ˆ1 ( )K Ak g nλ α δ= − + + .
Ifo Institute for Economic Research at the University of Munich
• How fast is the convergence?
• Example:
Speed of Convergence (6)
• Example:
– population growth of 2% p.a.: n=0.02
– technological progress 1% p.a.: gA=0.01
– depreciation rate 3% p.a.; δ=0.03
– capital income share one third: α=1/3
• This yields λ=0.04. Thus, the gap between the current and
76
• This yields λ=0.04. Thus, the gap between the current and
the steady state capital stock is reduced only 4% per year.
• If the dynamics of capital and income (per capita) are
similar as argued above, the same convergence speed
applies. Hence, convergence takes time.
Ifo Institute for Economic Research at the University of Munich
• How much time does it take to reduce the gap between the
current and the steady state capital stock by one half?
Speed of Convergence (7)
ˆ ˆ• Define the gap as
• From this follows .
• For an initial value d(0) this equation is solved as
)t(d)t(d ⋅−=•
λ
*ˆ ˆ( ) ( )d t k t k= −
( ) (0)t
d t d eλ−
�= .
.
77
• So we want to find period τ, in which
1( ) (0)
2d dτ = .
Ifo Institute for Economic Research at the University of Munich
Speed of Convergence (8)
1 1 1( ) (0) (0) (0)
2 2 2
1
d d d e d eλτ λττ − −= ⇒ = ⇒ =
• For λ=0.04 this yields τ=17.3 years.
1ln
2
1 1 0.69ln
2
λτ
τλ λ
⇒ − =
⇒ = − ≈
78
• Result: Only after 18 years the gap between the current and the steady state capital stock has decreased by more than 50%.
• Again, y will behave similarly. It will move 4% of the remaining distance towards y* each year and need 18 years to get halfway to y*.
Ifo Institute for Economic Research at the University of Munich
Speed of Convergence (9)
• As an example, consider the case that an economy decides to increase the saving rate by 10% (e.g., from 20% to 22%).
• In the long run, this raises per capita output by about 5% relative to • In the long run, this raises per capita output by about 5% relative to
the old path given the parameter values used above (for the calculation, see Romer, p. 22-23).
• After one year, output is 0.04·5% = 0.2% above its previous path.
• After 4 years, output is (1-exp(0.04·4))·5% = 0.15·5% = 0.7% above its previous path.
• After 18 years, output is approximately 0.5·5% = 2.5% above its
previous path
79
previous path
• Not only is the the overall impact of a substantial change in the saving rate modest, it also does not occur quickly.
Ifo Institute for Economic Research at the University of Munich
JapanSouth Africa
Argentina
Germany
USA
Venezuela
Empirical Convergence?
Growth by Level of GDP per Person, 1960 – 2000
South Korea
Malaysia Singapore
Hong Kong
JapanSouth Africa
Nicaragua
Madagascar
Niger
80
China
India
Thailand
-2 0 2 4 6 8
Madagascar
Nigeria
percent
Note: Average annual growth rate of real GDP per person (RGDPCH) by level of GDP per person
(PPP) in 1960, 107 countries. Source: PWT6.1.
Ifo Institute for Economic Research at the University of Munich
Convergence and Speed of Convergence
1. Relative and Absolute Convergence1. Relative and Absolute Convergence
2. Speed of Convergence
3. The Solow Model and the Central Questions of
Growth Theory
4. The Feldstein-Horioka Puzzle
81
4. The Feldstein-Horioka Puzzle
Ifo Institute for Economic Research at the University of Munich
The Solow Model and the Central Questions of Growth Theory (1)
• The Solow model identifies two possible sources of
variation – either over time or across countries – in variation – either over time or across countries – in
output per worker:
– differences in capital per worker (K/L) and
– differences in the effectiveness of labour (A).
• However, only differences in the growth rates of the
effectiveness of labour can lead to permanent
differences in the growth rates of output per worker.
8282
differences in the growth rates of output per worker.
• There arise two problems when trying to account for
large differences in income levels on the basis of
differences in capital.
Ifo Institute for Economic Research at the University of Munich
The Solow Model and the Central Questions of Growth Theory (2)
• Problem 1: The required differences in capital are far
too large.too large.
– Consider, for example a tenfold difference in output per
worker (this is, e.g, Switzerland vs. India)
– Recall that αK is the elasticity of output with respect to the capital stock.
– Thus, accounting for a tenfold difference in output per worker requires a difference of a factor of in capital per worker.
– Cobb-Douglas: ˆˆK
y kα
83
– Cobb-Douglas:
– For αK = 1/3, we need a value of 1000 for the capital rato to obtain an per capita income ratio of 10, which is far too large!
ˆˆ
ˆˆ
K
S S
I I
y k
y k
α
=
Ifo Institute for Economic Research at the University of Munich
The Solow Model and the Central Questions of Growth Theory (3)
• Problem 2: Attributing differences in output to differences
in capital without differences in the effectiveness of labour in capital without differences in the effectiveness of labour
implies immense variation in the rate of return on capital.
– If markets are competitive, the rate of return on capital equals its
marginal product, , minus depreciation, δ.
– Suppose that the production function is Cobb-Douglas, which in intensive form is
)(' kf
αkkf =)( .
84
– With this production function, the elasticity of output with respect to
capital is simply α.
– Furthermore, the marginal product of capital is
kkf =)(
ααα αα )1(1)('
−− == ykkf
.
.
Ifo Institute for Economic Research at the University of Munich
The Solow Model and the Central Questions of Growth Theory (4)
– This implies that the elasticity of the marginal product of capital with respect to output is –(1-α)/α.respect to output is –(1-α)/α.
– If αK = 1/3, a tenfold difference in output per worker arising from differences in capital per worker thus implies a hundredfold difference in the marginal product of capital:
– This implies that the marginal product of capital in India must be
.
( 1)( 1)
2
( 1)
'( )10 0.01
'( )
S S S
I I I
f k y y
f k y y
α αα α
α α
α
α
−−
−
−
= = = =
.
85
– This implies that the marginal product of capital in India must be 100 times as large as that in Switzerland. But international evidence on the return on capital suggests only modest differences.
Ifo Institute for Economic Research at the University of Munich
The Solow Model and the Central Questions of Growth Theory (5)
• As a result, only differences in the effectiveness of
labour have any reasonable hope of accounting for
the vast differences in wealth across time and space.the vast differences in wealth across time and space.
• However, the Solow models‘s treatment of the
effectiveness of labour is highly incomplete.
– Most obviously, the growth of the effectiveness of
labour is exogenous: the model takes as given the
behaviour of the variable that it identifies as the
driving force of growth.
8686
driving force of growth.
– More fundamentally, the model does not identify
what the „effectiveness of labour“ is; it is just a
catchcall for factors other than labour and capital
that affect output.
Ifo Institute for Economic Research at the University of Munich
The Solow Model and the Central Questions of Growth Theory (6)
• There are two ways to proceed:
• The first way is to take a stand concerning what we • The first way is to take a stand concerning what we
mean by the effectiveness of labour and what causes
it to vary.
– One possibility is that the effectiveness of labour
corresponds to abstract knowledge.
– Other possible interpretations of A are education
and skills of the labour force, the strenght of
87
and skills of the labour force, the strenght of
property rights, the quality of infrastructure, cultural
attitudes towards entrepreneurship and work, and
so on.
Ifo Institute for Economic Research at the University of Munich
The Solow Model and the Central Questions of Growth Theory (7)
• The second way is to consider the possibility that
capital is more important than the Solow model capital is more important than the Solow model
implies.
– If capital encompasses more than just physical
capital, or if physical capital has positive
externalities, then the private return on physical
capital is not an accurate guide to capital‘s
importance to production.
88
importance to production.
– In this case, the conclusions we derived may be
missleading, and it may be possible to resuscitate
the view that differences in capital are central to
differences in incomes.
Ifo Institute for Economic Research at the University of Munich
Convergence and Speed of Convergence
1. Relative and Absolute Convergence1. Relative and Absolute Convergence
2. Speed of Convergence
3. The Solow Model and the Central Questions of Growth
Theory
4. The Feldstein-Horioka Puzzle
89
4. The Feldstein-Horioka Puzzle
Ifo Institute for Economic Research at the University of Munich
The Feldstein-Horioka Puzzle (1)
• Consider a world where every country is described by the Solow model and where all countries have the same amount of capital per unit of effective labour.amount of capital per unit of effective labour.
• If the saving rate in one country rises and all aditional saving is invested domestically, the marginal product of capital in the country falls below that in other countries.
• Therefore, the country‘s residents have an incentive to invest abroad.
• Moreover, the investment resulting from the increased saving is spread uniformly over the whole world.
90
saving is spread uniformly over the whole world.
→Thus, in the absence of barriers to capital move-ments, there is no reason to expect countries with
high saving to also have high investment.
Ifo Institute for Economic Research at the University of Munich
The Feldstein-Horioka Puzzle (2)
• Feldstein and Horioka (1980):
Association between saving and investment rates.
Specifically, Feldstein and Horioka run a cross-country
regression for 21 industrialized countries of the average
share of investment in GDP during 1960-1974 on a
constant and the average share of saving in GDP over the
same Period.
SI Result:
91
( ) ( )910
0.074
8870
0.018
03502
.RY
S..
Y
I
ii
=
+=
where the numbers in parentheses are standard errors.
,Result:
,
Ifo Institute for Economic Research at the University of Munich
The Feldstein-Horioka Puzzle (2)
• Thus, rather than there being no relation between saving
and investment, there is an almost one-to-one relation.
• Possible Explanations:• Possible Explanations:
– There exist significant barriers to capital mobility.
– There exist underlying variables that affect both saving
and investment. For example, high tax rates can reduce
both saving and investment.
– The strong association can arise from government
policies that offset forces that otherwise make saving and
92
policies that offset forces that otherwise make saving and
investment differ. The government will chose to adjust its
own saving behaviour or its tax treatment of saving or
investment to bring them rough into balance.
Ifo Institute for Economic Research at the University of Munich
The Feldstein-Horioka Puzzle (3)
• Conclusion:
– The strong relationship between saving and investment – The strong relationship between saving and investment
differs dramatically from the predictions of a natural
baseline model.
– Whether this differences reflects major departures from
the baseline (such as large barriers to capital mobility) or
something less fundamental (such as underlying forces
affecting both saving and investment) is not clear.
93
affecting both saving and investment) is not clear.
Ifo Institute for Economic Research at the University of Munich
Neoclassical Growth Model
1.4 Conclusions
94
Ifo Institute for Economic Research at the University of Munich
Conclusion: What the Solow-Model predicts...
1. Why does growth exist (in terms of output per capita)?• Because there exists technological progress.
2. Why are some countries rich and some countries poor?• Because rich countries invest more (in physical capital),
have a lower population growth and, in particular, have better
„technology“.
3. Why do growth rates differ?
• Because of transitory dynamics towards the steady state.• Because of differences in the speed of technological progress.
95
• Because of differences in the speed of technological progress.
4. Do we have to think about stabilisiation?• No, there exists an inherent tendency that drives the economy
towards its steady state
Ifo Institute for Economic Research at the University of Munich
... and what the problems are from an empirical perspective.
• Time dimension: Not all underdeveloped countries grow fast. Have they already reached their steady state?they already reached their steady state?
• Cross country dimension: To explain the significant international differences in income, the capital stock or the saving rate would have to vary much more from country to country than they actually do.
• The assumption of a exogenous saving rate is not realistic. Perhaps, the puzzles empirically observed can be explained by the introduction
of an endogenous saving rate.
96
of an endogenous saving rate. -> Ramsey model (Section 2)
Ifo Institute for Economic Research at the University of Munich
Exercises
see Romer, Chapter 1
97