lecture 3. - tu berlinblanchard, chapter 10 – 13 prof. dr. frank heinemann avwl ii seite 2 growth...

AVWL II Prof. Dr. Frank Heinemann Seite 1

Lecture 3.Lecture 3.Growth and

Technological Progress

Blanchard, Chapter 10 – 13


Growth and Technological Progress

3.1 Stylized facts

3.2 Production function

3.3 The Solow model

3.4 Population growth and technological progress in the Solow model

3.5 The role of the technological progress in growth processes

3.6 Determinants of technological progress

3.7 Allocation effects of technological progress


Literature

- Blanchard: Macroeconomics

- Mankiw: Macroeconomics, Chap. 4 – 5

General textbooks on growth theory:

- Barro / Sala-i-Martin: Economic Growth

- Jones: Introduction to Economic Growth

- Romer: Advanced Macroeconomics


3.1 Stylized Facts


3.1 Stylized Facts

Annual growth rate GNP per capitareal GNP per capita (%) (1996 dollars)

Ratio: realGNP per capita

1950-1973 1974-2000 1950 2000 2000 / 1950

France 4,2 1,6 5.489 21.282 3,9

Germany 4,8 1,7 4.642 21.910 4,7

Japan 7,8 2,4 1.940 22.039 11,4

United Kingdom 2,5 1,9 7.321 21.647 3,0

United States 2,2 1,7 11.903 30.637 2,6

Average 4,3 1,8 6.259 23.503 3,7


Growth Forecast EU15 et al.

Quelle: Eurostat

0

1

2

3

4

5

6

7

8

EU

(15

Län

der)

Bel

gien

Dän

emar

k

Deu

tsch

land

Irla

nd

Grie

chen

land

Spa

nien

Fra

nkre

ich

Ital

ien

Luxe

mbu

rg

Nie

derla

nde

Öst

erre

ich

Por

tuga

l

Fin

nlan

d

Sch

wed

en

Ver

eini

gtes

Kön

igre

ich

Isla

nd

Nor

weg

en

Sch

wei

z

Ver

eini

gte

Sta

aten

Japa

nPrognose 2008 Prognose 2009


Growth Forecast of Accession Countries et al.

Source: Eurostat

0

1

2

3

4

5

6

7

8

EU

(27

Länd

er)

Bul

garie

n

Tsc

hech

isch

eR

epub

lik

Est

land

Zyp

ern

Lett

land

Lita

uen

Ung

arn

Mal

ta

Pol

en

Rum

änie

n

Slo

wen

ien

Slo

wak

ei

Kro

atie

n

Maz

edon

ien

Tür

kei

Prognose 2008 Prognose 2009


3.1 Stylized Facts

Convergence of production per capita, OECD countriesConvergence of production per capita, OECD countries


3.1 Stylized Facts

There is no reliable rule of convergence, Countries from all regionsThere is no reliable rule of convergence, Countries from all regions


3.1 Stylized Facts

Data according to country groupsData according to country groups


3.1 Stylized Facts

What growth rates are normal?What growth rates are normal?

• From the end of the Roman Empire till 1500 there was no growth in production per capita in Europe

• 1500-1820 – low growth (0.1% till 1700, afterwards 0.2%)

• 1820-1950 – moderate growth (USA 1.5%)

• The high growth rates of the 50s and 60s are atypical


3.1 Stylized Facts

Reasons for high national income / growthReasons for high national income / growth

Infrastructure

Political und legal stability

Access to the international market

Human capital

Efficient use of scarce resources

Capital accumulation

Technological progress


3.2 Production Function

Output: Y = Aggregate Production, Aggregate Income

Factors: K = CapitalN = Labor

Aggregate production functionAggregate production function

Y = F (K, N)

Growth rate: ΔY / Y



F(K,N) increasing in K and in N

Marginal product of a factor decreases with increasing factor input

Aggregate production functionAggregate production function

Properties of production fun. Y = F (K, N)

Diminishing returns: d2F/dK2 < 0 d2F/dN2 < 0

Positive partial derivative: dF/dK > 0 dF/dN > 0

Why do marginal product decline with increasing factor input?



Inputs from production factors are decided by supply and demand.

Price: Interest rate = Price of the factor capital (r)

Wage rate= Price of the factor labor (w)



Firms carry out investments when they (i) make a contribution to firms’ profit and (ii) are financially feasible.

Assumptions: perfect capital market, constant prices

Firms have access to unlimited credit at interest rate r.

firms implement all projects, whose returns are larger than capital cost r. ⇒



125/100 – 1 = 25%125100C

330/300 – 1 = 10%330300D

290/250 – 1 = 16%290250B

210/200 – 1 = 5%210200A

Returns(~ marginal product of capital)

Paymentin t=2

Required capital in t=1

Project

Example



Projects ordered according to returns

C – B – D – A

Output in t=2

Investmentsin t=1100 350 650 850

125

415

745

955

Approximation by continuous and concave production function

C B D A


A project is profitable when its returns are above the market interest rate.

The most profitable project should be undertaken first. => Marginal product of capital falls with increasing capital input

Returns

C B D A

100 350 650 850

25%

5%

10%

16%

r = 8%

Investmentsin t=1I =



3.2.2. Production Factor Capital

The saving rate (savings as a fraction of national income) 1950-2000

U.S.A. 18,6%Germany 24,6%Japan 33,7%

What do you think…

Would a higher saving rate in Germany lead to lasting higher growth?Would a higher saving rate in Germany lead to lasting higher growth?

Sources of growthSources of growth


3.3 The Slow Model

How does a constant saving rate affect capital

accumulation and growth?

Is there an optimal saving rate?

What effects does technological progress have on

capital accumulation?

Literature: Blanchard/Illing, Chapter 10.3 – 12.Heijdra/van der Ploeg, Ch. 14.Mankiw, Chap. 4 – 5.

Barro / Sala-i-Martin, Chapter 1

Sources of growthSources of growth


3.3 The Solow Model

( , )Y F K N=Production functionProduction function

Diminishing returns: d2F/dK2 < 0 d2F/dN2 < 0

( , ) ( , ) 0F K N F K Nλ λ λ λ= ∀ >

Aggregate production function

An increase in inputs of all production factors by x% raises production likewise by x%.

Assumption 1: constant returns to scale

Positive returns: dF/dK > 0 dF/dN > 0


3.3 The Solow Model

N/1=λ

( , ) ( , ) 0F K N F K Nλ λ λ λ= ∀ >

Consequence: Per capita output Y/N depends only on the ratio of capital to labor K/N:

Assumption 1: constant returns to scale

Choose

N

YNKF

NNKFNKF === ),(

1)1,/(),( λλ

F(k,1) = y=Y/NLet k=K/N.


3.3 The Solow Model

Let y=Y/N Output per work unit,

k=K/N capital intensity

Then: y = F(k,1) = f(k)

Per capita output as a function of capital intensity

Notice: here per capita mean per work unit

Positive, but declining returns of capital:

=> f ’ = dF/dK > 0 , f ’’ = d2F/dK2 < 0

Next we assume constant working population.


3.3 The Solow Model

Output and capital per employeeOutput and capital per employee

y = f(k)

k

y

Capital accumulation


The Solow Model

The long-term relationship between production and capitalThe long-term relationship between production and capital

- The capital stock decides how much to be produced

- The production level determines how much to be saved and to be invested

The Solow model describes these interactions and dependencies.

Assumption 2: The saving rate is constant

Saving rate s = Savings / National income


3.3 The Solow Model

Capital, Production and Savings/Investments

),( NKFY =

),( NKFY tt =

ttt YsSI ==

ondepreciatiIK tt −=Δ

ttt KKK Δ+=+1

),( 11 NKFY tt ++ = tt YYGrowth −= +1

Savings/Invest-ment

Production / Income

Capital stock

Change of capital stock


The Solow Model

GDP Yt = F(Kt,N)

Savings s Yt

Consumption Ct = (1 – s) Yt

Depreciation δ Kt

Saving rate s and depreciation rate δ are constant and between 0 and 1.

Assumption 3: Closed economy with balanced government budget ⇒ Gross investment = savings

Change of capital stock over time:

Kt+1 – Kt = s Yt – δ Kt


The Solow Model

GDP Yt / N = F(Kt / N ,1)

Savings s Yt / N

Consumption Ct / N = (1 – s) Yt / N

Depreciation δ Kt / N

Change of capital stock over time:

Kt+1 – Kt s Yt – δ Kt

N N N N

Per capita values:

=


The Solow model

Per capita values: GDP yt = f (kt)

Gross investment = saving s yt

consumption ct = (1 – s) yt

Depreciation δ kt

Change of capital intensity over time:

kt+1 – kt = s f(kt) – δ kt

Let yt = Yt / N, kt = Kt / N, ct = Ct / N


The Solow Model

yt = f ( kt )

k

y

s yt

Consumption per employee

Savings per employee

capital intensity at period t

kt


The Solow Model

yt = f ( kt )

k

y

savings s yt = gross investment

Depreciation δ kt

Increasing capital intensity

decreasing capital intensity

steady state k*

In steady state k* : gross investment = Depreciation => Net investment. = 0


The Solow Model

Calculation of steady state k*: Change of capital intensity over time:

kt+1 – kt = s f(kt) – δ kt = 0

⇔ s f ( k* ) = δ k*

Solving this equation for k* gives the steady state (= long-term growth equation).

Production level in steady state y* = f(k*)

consumption in steady state c* = (1-s) y*


The Solow Model

Total differentiation of the equation

gives:( *) '( *) * *f k ds s f k dk dkδ+ =

* ( *)0,

'( *)

dk f k

ds sf kδ⇔ = >

−as in steady state δ > s f ‘

( *) *s f k kδ=

Comparative statistics:

How does the steady state react to the saving rate?

> 0


The Solow Model

yt = f ( kt )

k

y

savings s yt

depreciation δ kt

steady state k*

Slope: s f‘ (kt)

Slope δ

In steady state δ > s f‘ (kt)


The Solow Model

k

y

s0 f(kt)

δ k

Increasing the savings rate from s0 to s1 increases the steady state and leads to temporary growth

s1 f(kt)

y0*

y1*


The Solow Model

Following an increase in the savings rate:Increase of production level over time

t

y

In period t0, the saving rate increases from s0 to s1.

y0*

y1*


The Solow Model

““What impact does the savings rate have on the growth rate of What impact does the savings rate have on the growth rate of production?production?””

The previous analysis provides us three answers to these questions:

1. A high savings rate causes a stronger growth of production for a certain period until the new steady state is reached.

2. The savings rate does not affect the long-term growth rate of production per employee. This growth rate is zero.

3. The saving rate determines the size of the long-term production level per employee. Ceteris paribus, countries with a higher saving rate, therefore, have a higher production level.



Examples of the Solow Model

The production function: let F(K,N) = 15 K2/3 N1/3,

Saving rate s = 20%, depreciation rate δ = 10%, labor N = 10 Mio.

a) Determine the intensity form of production function.

b) How high is the capital intensity in steady state?

c) How long will it take to reach steady state?

d) How high is the per-capita-consumption in steady state?

Answer in the lecture


The Solow Model

y = f ( k )

k

y

s yt

c*(s) = per capita consumption in steady state at saving rate s

k*(s) capital intensity in steady state at saving rate s


The Solow Model

y = f ( k )

k*(s)

y

Per capita consumption in steady states at different saving rate (s1 – s4)

s1 f ( k )

s2 f ( k )

s3 f ( k )

s4 f ( k )


The Solow Model

c* (s)

Per capita consumption c

Maximum per capita consumption in a steady state

Saving rate sOptimale saving rate s* 10


The Solow Model

The steady state of the Golden Rule gives a higher per-capita-consumption than every other steady state. [Edmund Phelps, Nobel price 2006]

Lit: Phelps (1961) The Golden Rule of Accumulation: A Fable for Growthman, AER 51, 638-643

The capital intensity in steady state of Golden Rule results from

max ( )k f k kδ−

Optimality condition '( )f k δ=Solving the equation for k gives ** 1' ( )k f δ−=


The Solow Model

How do we figure out the steady state of the Golden Rule? With the optimal savings rate s*, the economy converges towards the steady state of the Golden Rule.

We can calculate s* from k**. In a steady state

( )sf k kδ=* ** **/ ( )s k f kδ=

Thus


The Solow Model

yt = f ( kt )

k

y

s* yt

δ kt

Per capita consumption in steady state of

the Golden Rule

Golden Rule steady state k**

δ

s* = optimal savings rate


The Solow Model

When the savings rate is under s*, a higher future consumption can only be reached by saving more. => Trade-off between current and future consumption

Time preference, distribution between generations is not considered in the Solow Model


The Solow Model

With a saving rate above s*, a drop in capital stocks leads to a higher per capita consumption in (lower) steady state.

A decline in savings goes hand-in-hand with an increase in future consumption.

Current and future consumption can be increased.

Dynamic inefficiency!

The saving rate is too high!


3.4 The Solow Model: Population Growth and Technological Progress

GDP Yt = F(Kt, AtNt)

Labor efficiency At

Savings s Yt

Consumption Ct = (1 – s) Yt

Depreciation δ Kt

Change of capital stocks over time:

Kt+1 – Kt = s Yt – δ Kt

Population growth Nt+1 = (1+n) Nt

Population growth rate n

Technological progress At+1 = (1+g) At

Rate of technological progress g


The Solow Model: Population Growth and Technological Progress

constant return to scale => GDP per unit of labor efficiency


kt+1 – kt = ?

Capital intensity kt = Kt / (AtNt)

Gross investment = savings s yt

Consumption Ct/(AtNt) = ct = (1 – s) yt

Depreciation δ kt

yt = Yt / (AtNt) = F (Kt / (AtNt), 1) = f (kt)


The Solow Model with Population growth and technological Progress

ttt k

ng

ksy−

++−+

=)1)(1(

)1( δ

)1()1(

)(

ng

kgnngsy tt

+++++−

=δ

ttt

ttt k

NA

Kkk −=−

++

++

11

11


Steady state k*: s f(k*) = (δ+g+n) k*

ttt

ttt kNnAg

KsYK−

++−+

=)1()1(

δ

)1)(1(

)1)(1(

)1)(1(

)1(

ng

kng

ng

ksy ttt

++++

−++

−+=

δ

gn is insignificant with small percentage values.

)1()1(

)(

ng

kngsy tt

++++−

≈δ



yt = f ( kt )

k

y

savings s yt

(δ+g+n) kt

Increasing capital intensity

Decreasing capital intensity

steady state k*



GDP per capita

GDP per unit of labor efficiency yt = Yt / (At Nt)

Yt / Nt = At yt

At = (1+g)t A0

At f(k*) = (1+g)t A0 f(k*)

GDP per capita in steady state

Growth rate of GDP per capita in steady state

= Rate of technological progress g

Growth rate of labor efficiency g


The Solow Model: Population Growth and Technological ProgressPer capita magnitudes in steady state by technological progress

t

Per capita magnitudes of capital stock, output and consumption grow with the rate of technological progress in the long term

Kt / Nt = At k*

Y / N Yt / Nt = At f ( k* )

savings s Yt / Nt

consumption (1–s) Yt / Nt



The capital intensity in steady state depends on s, n and g * * *( , , ) : ( ) ( )k s n g sf k n g kδ= + +

Total differentiation gives

* * * *'( ) ( )sf k dk n g dk k dnδ= + + +* *

*0

'( )

k k

n sf k n gδ∂

⇒ = <∂ − − −

* *

*

( )0

'( )

k f k

s n g sf kδ∂

= >∂ + + −

equivalently



Golden Rule

**'( )f k n gδ= + +

Optimality condition

f ‘ = marginal product of capital

*

* *max ( ) ( )k

f k n g kδ− + +

k**:

** 1( ') ( )k f n gδ−= + +



**** 1

''( ) 0''

dkf dk dn

dn f⇒ ⋅ = ⇔ = <

The capital intensity should increase with a decline in population growth.

**'( )f k n gδ= + +From the optimality condition

follows


The Solow Model: Decline in Population Growth

k

y

(δ+n1+g) kt

With a decline in growth of population, less investments are necessary to maintain capital intensity. Therefore a constantsaving rate leads to a higher capital intensity.

s yt

sy0*sy1*

(δ+n0+g) kt



On the one hand, capital intensity should increase when n goes down.

On the other hand, the decline in population growth automatically leads to an increase in capital intensity with a constant savings rate.

How does the savings rate react to a decline in population growth?



Total differential of the equation

* **''( )

k kf ds dn dn

s n

⎡ ⎤∂ ∂⋅ + =⎢ ⎥∂ ∂⎣ ⎦

Comparative statistics:

* *'( ( , , ))f k s n g n gδ= + +

( )''( )

'( )

f ds k dnf dn

n g sfδ⋅ −

⇔ ⋅ =+ + − ⋅

gives

''( ) ( ) ''( ) ( '( ))f f ds f k dn n g sf dnδ⇔ ⋅ ⋅ − ⋅ = + + − ⋅

Inserting the formula from Slide 54 gives



'( ) ''( )

''( ) ( )

ds n g sf f k

dn f f

δ + + − ⋅ + ⋅⇔ =

⋅ ⋅The denominator is negative. The numerator can be either positive or negative!

A clear answer to the question of whether the savings rate rises or falls with a decline in n can be only reached with more information of production function.

When the saving rate cannot adjust, can k increase beyond the Golden Rule?

► Over investment ! ► Japan ?



Consumption per unit of labor efficiency with a decline of n and a constant savings rate.

t

c

At period t0 the growth rate of the working population drops from n0 to n1.

c0*

c1* c1* = (1-s) f(k1*)

c0* = (1-s) f(k0*)



t

C / N

C0*/N = (1-s) At f(k0*)

C1*/N = (1-s) At f(k1*)

Consumption per capita with a decline of n and a constant savings rate

At period t0 the growth rate of the working population drops from n0 to n1.


The Solow Model: Example

Example: f(k) = kα 0 < α < 1

Steady state: ( ) ( )sf k n g kδ= + +

( )sk n g kα δ⇔ = + +

1k n gαα δ−⇔ = + +

'( )f k n gδ= + +

1

1sk

n g

α

δ

−⎛ ⎞⇔ = ⎜ ⎟+ +⎝ ⎠

Golden Rule: 1

1

kn g

ααδ

−⎛ ⎞⇔ = ⎜ ⎟+ +⎝ ⎠


The Solow Model: Example

In steady state the Golden Rule holds:

s α=

1 1

1 1sk

n g n g

α ααδ δ

− −⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟+ + + +⎝ ⎠ ⎝ ⎠

Hence it follows:

The production function f(k) = kα

describes a limiting case where the optimal savings rate is independent of n.


The Solow Model

The Solow model describes the optimal saving in steady state.

Adjustment process takes time though. The Solow model does not describe the optimal adjustment track.

The ‘optimal savings rate’ maximizes the per capita consumption in steady state. The steady state will never be completely reached.

Time preference: future consumption should be discounted. Consumption during the adjustment phase must be considered.

These critiques are considered by the Ramsey model.

Recession studies: business cycles, growth and employment

lecture 3. - tu berlinblanchard, chapter 10 – 13 prof. dr. frank heinemann avwl ii seite 2 growth...

Documents