1. neoclassical growth model

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.


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.


• Further features:

– The macroeconomic output of an economy is created


– 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 …


… On the one hand, the input factor labour can be

increased quantitatively due to an increase of the


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.


Neoclassical Growth Model

1.1 Baseline Model without Technological Progress

5


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


6. Optimal Saving Rate and Golden Rule Capital Stock


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

∂ ∂

∂ ∂= < = <

∂ ∂.


2. Inada-conditions:


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= ∀ > .


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.


• Example: Cobb-Douglas production function (A = level of

knowledge):


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−

= = =∂

∂= = − = −

∂.

,


K Y KF α α⋅ = =KYK

• Income shares of output:


( )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

= ⋅ = ⋅ =

= ⋅ = ⋅ = −

,

.


• Because of characteristic 3 (linear homogeneity) it is

possible to derive the production function in its intensive


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= ⋅ .


•

•


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= .


• We know:

( )( ) ( ), ,

,F K L F K L

Y F K L K LK L

∂ ∂≡ = ⋅ + ⋅

∂ ∂,


• 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

⋅ = − ⋅∂ ∂

∂∂≡ ≡ = ⋅ −

∂ ∂

= − ⋅ .

,

,








15




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



• 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“.



YδK (break-even

investment)

F(K,L)

s·F(K,L) (actual investment)

Y*

18

KK*K1 K2


δKY


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•








20




• 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


• The capital intensity is defined as .

For changes of the capital intensity we get:

k /k K L=


• • • • • •

• 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= + ⋅ = + ⋅

� ��

••

••

.



• 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


• This is the fundamental growth equation:

( ) ( )k s f k n kδ•

= ⋅ − + ⋅•


• 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.








25




• 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.



y ( )n kδ+

( )f k

y∗

( )s f k⋅

( )f k

27

1k

2k kk

∗


Phase diagram for in the Solow Model:k

k•


k Break-even investmentsmaller than actual investment

Break-even investmenthigher than actual investment

28

kk∗


• The long run equilibrium (steady-state) is the state of

condition, where the per capita growth rates are constant


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⋅ = + ⋅


• The steady state growth rate of k is


*

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 .

, ,


• 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..


• Hence, the growth rate of capital must be a falling function

of the capital stock.


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): ]


( ) /s f k k⋅


0kg <

0kg >( )n δ+

( ) /s f k k⋅

33

k∗

k

( ) /s f k k⋅



• 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

⋅= = =

⋅⇒ =

,

.


• Note that


'( ) ( , ) / ( , ) /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⋅ = .


•


'( )

( )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.








37




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*


s


k• t

t0

t

0t

39

k

0t t


yg


t0t

yg

c If the per capita consumption decreases

40

0t t

or increases, depends on the starting conditions!


y (n+δ)k

f2(k)y *


Increase in the technology parameter A:

f1(k)

s·f1(k)

f2(k)

s·f2(k)y1*

y2

y2*

41

kk1* k2*


y(n2+δ)k


Increase in the population growth:

y (n1+δ)k

f(k)

s·f(k)

(n2+δ)k

y1*

y2*

42

kk1*k2*








43




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.


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= − = − ⋅ .


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= − ⋅ = − + .


• 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= = .


• The necessary condition


* ** *( ( )) ( ) ( )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= + .


• Defining the golden capital stock as


* *( )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= − +


f(k)Slope:(n+δ)In the consumption

level is maximal and the

*

goldky


(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


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!


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


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.



1.2 Baseline Model with Technological Progress

54


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


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).


• As the production function is linear homogenous, it follows

that


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++−⋅=•


• In the long run equilibrium we have .

• The equilibrium capital stock per unit of effective labour *

k

ˆ 0k =�•


• 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


• Transitory dynamics (analytically):


.)ng(k

)k(fs

k

kg Ak

δ++−⋅

==

•

59


ˆ( )s f k⋅

• Transitory dynamics (graphically):


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


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


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



1.3 Implications

63


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



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.



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



• 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 <



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


• 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).






Theory


70



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.


• How fast does converge towards ?

• The changes of are:

*kk

k


• • •

• 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=

∂


• Substituting into (2) gives


* *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++= .


• Substituting this into (4) leads to


* *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 = ⋅ .


• Substituting this into (5) yields


( )•

• 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= − + + .


• How fast is the convergence?

• Example:


• 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.


• How much time does it take to reduce the gap between the

current and the steady state capital stock by one half?


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 = .



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*.



• 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.


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.





3. The Solow Model and the Central Questions of

Growth Theory


81



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.



• 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

=



• 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

.

.



– 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.



• 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.



• 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.



• 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.






Theory


89



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.



• 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:

,



• 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.



• 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.



1.4 Conclusions

94


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


... 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)


Exercises

see Romer, Chapter 1

97

1. neoclassical growth model

Documents