part iib. paper 2

Part IIB. Paper 2 Michaelmas Term 2009

Economic Growth

Lecture 2: Neo-Classical Growth Model

Dr. Tiago Cavalcanti

Readings and Refs

Original Articles:Solow R. (1956) ‘A contribution to the theory of economic growth’ Quarterly Journal of Economics, 70, 65-94.Solow R. (1957) ‘Technical change and the aggregate production function’ Review of Economics and Statistics, 39, 312-320.Swan T. (1956) ‘Economic growth and capital accumulation’ Economic Record, 32, 334-361.

Texts: (*)Jones ch.2; BX chs.1,10; Romer ch.1.

The Neoclassical Growth modelSolow (1956) and Swan (1956)

• simple dynamic general equilibrium model of growth

Output produced using aggregate production function Y = F (K , L ), satisfying:

A1. positive, but diminishing returns

FK >0, FKK<0 and FL>0, FLL<0

A2. constant returns to scale (CRS)

0 allfor ),,(),( LKFLKF

– replication argument

Neoclassical Production Function

Production Function in Intensive Form

• Under CRS, can write production function

)1,(.),(LK

FLYLKFY

• Alternatively, can write in intensive form:

y = f ( k )

- where per capita y = Y/L and k = K/L

Exercise: Given that Y=L f(k), show:

FK = f’(k) and FKK= f’’(k)/L .

Competitive Economy

• representative firms maximise profits and take price as given (perfect competition)

• can show: inputs paid their marginal products:

r = FK and w = FL

– inputs (factor payments) exhaust all output:

wL + rK = Y

– general property of CRS functions (Euler’s THM)

A3: The Production Function F(K,L) satisfies the Inada Conditions

0),(lim and ),(lim 0 LKFLKF KKKK

0),(lim and ),(lim 0 LKFLKF LLLL

Note: As f’(k)=FK have that

0)('lim and )('lim 0 kfkf kk

Production Functions satisfying A1, A2 and A3 often called Neo-Classical Production Functions

Technological Progress

= change in the production function Ft

),( LKFY tt

),(),(.1 LKFBLKF tt Hicks-Neutral T.P.

))(,(),(.2 LAKFLKF tt Labour augmenting (Harrod-Neutral) T.P.

)),((),(.3 LKCFLKF tt Capital augmenting (Solow-Neutral) T.P.

A4: Technical progress is labour augmenting

gtt

tt

eAA

LAKFLKF

0

and

))(,(),(

Note: For Cobb-Douglas case three forms of technical progress equivalent:

ttt

tttt

DAB

LKDLAKLKBLKF

)1(

)1()1()1(

when

)()(),(

Under CRS, can rewrite production function in intensive form in terms of effective labour units

)~

(~ kfy

-note: drop time subscript to for notational ease

- Exercise: Show that

KKK FALkfFkf )~

('' and )~

('

AL

Kk

AL

Yy

~ and ~ where

A5: Labour force grows at a constant rate n

ntt eLL 0

A6: Dynamics of capital stock:

KIKdt

dK

net investment = gross investment - depreciation

– capital depreciates at constant rate

Model Dynamics

• National Income Identity

Y = C + I + G + NX• Assume no government (G = 0) and closed

economy (NX = 0)• Simplifying assumption: households save constant

fraction of income with savings rate 0 s 1 I = S = sY

• Substitute in equation of motion of capital:

… closing the model

KALKsFKsYK ),(

Fundamental Equation of Solow-Swan model

kgnyskdt

kd ~)(~~

~

)(~~

)(~

~d

lnd

d

lnd

d

lnd

d

~lnd

lnlnln~

ln~

:

gnk

ysgn

K

sYng

K

K

k

k

t

L

t

A

t

K

t

k

LAKkAL

Kk

Proof

Steady State

0~~ 0~

cyk

0~

)(~ ** kgnksf

Definition: Variables of interest grow at constant rate (balanced growth path or BGP)

• at steady state:

Solow Diagram

0

x 104

0

~

~

~ = (~ )

( + + )~

~ = (~ )

~ (0) ~ ¤

_~ = (~ ) ¡ ( + + )~

Existence of Steady State

• From previous diagram, existence of a (non-zero) steady state can only be guaranteed for all values of n,g and if

0)~

('lim and )~

('lim ~0

~ kfkf kk

- satisfied from Inada Conditions (A3).

Transitional Dynamics

• If , then savings/investment exceeds “depreciation”, thus

• If , then savings/investment lower than “depreciation”, thus

• By continuity, concavity, and given that f(k) satisfies the INADA conditions, there must exists an unique

*~~kk

.0~

~0

~~

k

kgk

k

*~~kk

.0~

~0

~~

k

kgk

k

*** ~)()

~(

~kgnkfthatsuchk

Transitional Dynamics

0

~

~

~ =_~~ = (~ )

~ ¡ ( + + )

~ ¤ ~ 0

~ 0

Properties of Steady State1. In steady state, per capita variables grow at the rate g, and aggregate variables grow at rate (g + n)

Proof:

StateSteady in

~logloglog

logloglog

and ~

as

~

g

ggdt

kd

dt

Ad

dt

kdg

gndt

kd

dt

Ld

dt

Kdg

L

Kk

AL

Kk

kk

kK

2. Changes in s, n, or will affect the levels of y* and k*, but not the growth rates of these variables.

Prediction: In Steady State, GDP per worker will be higher in countries where the rate of investment is high and where the population growth rate is low - but neither factor should explain differences in the growth rate of GDP per worker.

- Specifically, y* and k* will increase as s increases, and decrease as either n or increase

Golden Rule and Dynamic Inefficiency

• Definition: (Golden Rule) It is the saving rate that maximises consumption in the steady-state.

• Given we can use

to find .

)()~

('0~

)(~

~)

~(~

~)()

~()

~()1(~max

***

*

**

****

gnkfs

kgn

s

k

k

kf

s

c

kgnkfkfsc

GR

s

,~*

GRk ** ~)()

~( GRGR kgnksf

GRs

Golden Rule and Dynamic Inefficiency

0 10

~¤

~¤ = (1¡ ) (~ ¤)

Changes in the savings rate

• Suppose that initially the economy is in the steady state:

• If s increases, then

• Capital stock per efficiency unit of labour grows until it reaches a new steady-state

• Along the transition growth in output per capita is higher than g.

*1

*1

~)()

~( kgnksf

0~~

)()~

( *1

*1 kkgnksf

Linear versus log scales

00

x 104

( )

L inear-Scale

00

(())

Log-Scale

() = (0)

( ()) )

( ()) = _

=

Changes in the savings rate

(

())

Log of capital per capita

Log of output per capita

( ())

( ())

Log of consumption per capita

Next lecture

Testing the neo-classical model:

1. Convergence

2. Growth Regressions

3. Evidence from factor prices