Growthin Dynamic General Equilibrium

8/3/2019 Growthin Dynamic General Equilibrium

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Growth in dynamic general equilibrium(a) The Solow model in continuous time

Outline

Preamble: some stylised facts

Main features and properties of the Solow model in continuous

17/10/2011 L13501 - Adv. Macroeconomics 1

Basic Facts on Growth and Development

(i) Per capita output grows over time, and its growth rate does not tend to diminish

- Kaldors stylised facts of economic growth

Most developed countries exhibit the following characteristics in the long run:

(ii) Capital per worker grows over time

(iii) The real rate of return to capital is nearly constant

(v) The shares of labour and physical capital in national income are nearly constant

(iv) The ratio of physical capital to output is nearly constant


,

Balanced growth refers to an allocation where output grows at a constant rate

and capital-output ratio, the interest rate, and factor shares remain constant


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1. There is an enormous variation in per capita income across economies. The

poorest countries have per capita incomes that are less than 5 percent of per

capita incomes in the richest countries.

- Romers additional stylised facts of economic growth

2. Rates of economic growth vary substantially across countries.

3. Growth rates are not necessarily constant over time.

4. A countrys relative position in the world distribution of per capita income is

not immutable. Countries can move from being poor to being rich, and vice-

versa.


5. Growt n output an growt n t e vo ume o nternat ona tra e are c ose yrelated

6. Both skilled and unskilled workers tend to migrate from poor to rich countries

or regions

Overview

Main assumptions: - Exogenous labour augmenting technological progress

- Exo enous savin rate

THENEOCLASSICALGROWTH MODEL(Solow(1956)&Swann(1956))

Mainfocus:

Main conclusions:

- Investigate the effects of division of output between

C and I on capital accumulation and growth.

- Differences in capital/labour ratio should account for

differences in real income across countries.

-


Main weakness:

-

growth in output per capita.

- It takes as exogenous the variable that is identified

as the driving force behind sustained growth.


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The model(in continuous time)

Key notion: capital and labour are substitutable

, ,= ec no ogy:

CRS

Saving: ( ) ( ) ( ) ( ) , 0 1S t Y t C t sY t s = < diving by A(t)L(t) we get, .

( ) ( ( ), ( ) ( )) ( )K t sF K t A t L t K t =

( )( ( )) ( )

( ) ( )

K tsf k t k t

A t L t =

ln ( )Note: Taking log and exploiting the property we have:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( )( ) . ,

( ) ( )

( ) ( )( ) ( ) ( )

( ) ( ) ( )

d x t x

dt x

k t K t L t k t K t n g

k t K t L t k t K t

K tk t

A t L t

A t K t k t n g k t

A t A t L t

=

= =

= +

?


T e equilibrium dynamics o t e mo e n cont nuous t me s g ven y,

)()())(()( tkgntksftk ++=


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?Is there asteady state (or b.g.p) with constant positive per capita growth ?

Re-write the fundamental dynamic equation as follows:

( ) ( ( ))( )

( ) ( )

k t f k t s n g

k t k t = + +

( )( )

( )

y ts n g

k t= + +

0)()(

)(=++ gn

tk

tys

Taking log and differentiating w.r.t time: ( ) ( ) 0( ) ( )

y t k t s s

y t k t =

Along the b.g.p we have: .


Satisfied if and only if :( ) ( )

0( ) ( )

k t y t

k t y t = =

( ) ( )/ // /

Y L K L A

Y L K L A

= =

Considering per capita units, rather than units of effective labour, we have:

( ) ( )/ /0

/ /

Y L K LA A

Y L A K L A

=

Along the b.g.p per capita

GDP grows at rate g !

In the Solow model, sustained growth can only occur in the presence of

technological progress.


Why?Because as capital per capita accumulates, the tendency of the

marginal product of capital to decrease is offset by technological

progress.


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Comparative dynamics

( ) ( ( ))( )

( ) ( )

k t f k t s n g

k t k t = + +

Recall: The law of motion ofK/ANin the continuous time Solow model:

- Permanent chan e e. . decrease ins:

( )k t

( )

( )

k t

k t

shifts curve to the left; on impact kfixed;

in subsequent periods kfollows dashed arrows

and converges monotonically to new eqm.

- Change (e.g. increase ) in n ():

identical comparative dynamics


*k( ( ))

( )( )

f k t s n g

k t + +

( ( ))' ( )

( )

f k t s n g

k t + +

**k

Further property of the Solow model

The golden rule of capital accumulation: dynamic inefficiency possible.

At steady state (on the b.g.p): ( *) ( ) *sf k n g k = + +

* (1 ) ( *) ( *) ( ) *c s f k f k n g k = = + +

=> k* is determined bys and the other parameters of the model: * *( , , , )k k s n g =

=> ( )

*( , , , )

* ( , , , )

*

'( )

s n g

k s n g

dc k

f n g ds s

= + +

The golden rule saving rate is such that c* is maximised . Since*( , , , )

0s n g k

s

>

*


* ( )'( )GRk sf n g = + +

In the Solow models is exogenous and it could well be above or below GRs

*

* ( )If is such that '( ) 0k sdc

s f n g ds

< + + < dynamic inefficiency

0 'iff f n g s

= = + +

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Growthin Dynamic General Equilibrium