Download Neoclassical Growth Model

Part IIB. Paper 2
Michaelmas Term 2009
Economic Growth
Lecture 2: Neo-Classical
Growth Model
Dr. Tiago Cavalcanti
Readings and Refs
Texts:
(*)Jones ch.2; BX chs.1,10; Romer ch.1.
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.
The Neoclassical Growth model
Solow (1956) and Swan (1956)
• simple dynamic general equilibrium model of
growth
Neoclassical Production Function
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)
F (K , L)  F ( K , L), for all   0
– replication argument
Production Function in Intensive Form
• Under CRS, can write production function
K
Y  F ( K , L)  Y  L.F ( ,1)
L
• 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
lim K 0 FK ( K , L)   and lim K  FK ( K , L)  0
lim L0 FL ( K , L)   and lim L FL ( K , L)  0
Note: As f’(k)=FK have that
lim k 0 f ' (k )   and lim k  f ' (k )  0
Production Functions satisfying A1, A2 and A3
often called Neo-Classical Production Functions
Technological Progress
= change in the production function Ft
Yt  Ft ( K , L)
1. Ft ( K , L)  Bt F ( K , L)
Hicks-Neutral T.P.
2. Ft ( K , L)  F ( K , ( At L))
Labour augmenting
(Harrod-Neutral) T.P.
3. Ft ( K , L)  F ((Ct K ), L)
Capital augmenting
(Solow-Neutral) T.P.
A4: Technical progress is labour augmenting
Ft ( K , L)  F ( K , ( At L))
and
At  A0 e gt
Note: For Cobb-Douglas case three forms of
technical progress equivalent:
Ft ( K , L)  Bt K  L(1 )  K  ( At L) (1 )  ( Dt K ) L(1 )
when Bt  At
(1 )
 Dt

Under CRS, can rewrite production function in
intensive form in terms of effective labour units
~
~
y  f (k )
~ K
Y
~
where y 
and k 
AL
AL
-note: drop time subscript to for notational ease
- Exercise: Show that
~
~
f ' (k )  FK and f ' ' (k )  AL FKK
Model Dynamics
A5: Labour force grows at a constant rate n
Lt  L0e
nt
A6: Dynamics of capital stock:
dK 
 K  I  K
dt
net investment = gross investment - depreciation
– capital depreciates at constant rate 
… closing the model
• 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:
K  sY  K  sF ( K , AL)  K
Fundamental Equation of
Solow-Swan model
~
~
dk ~
~
 k  sy  (n  g   )k
dt
~ K
~
Proof : k 
 ln k  ln K  ln A  ln L
AL
~
d ln k d ln K d ln A d ln L




dt
dt
dt
dt
~
k K
sY
s~
y
 ~  g n 
 (n  g   )  ~  (n  g   )
K
k K
k
Steady State
Definition: Variables of interest grow at
constant rate (balanced growth path or BGP)
~
k  0 ~
y  c~  0
• at steady state:
 
~*
~*
sf k  (n  g   )k  0
Solow Diagram
_ = ( ~
~
~


) ¡ (  + + 
)


~ = ( ~
)

~

( +  + ) ~
~ = ( ~


)
0
0
~

(0)
~

~¤

4
x 10
Existence of Steady State
• From
previous diagram, existence of a (nonzero) steady state can only be guaranteed for all
values of n,g and  if
limk~0
~
~
f ' (k )   and limk~ f ' (k )  0
- satisfied from Inada Conditions (A3).
Transitional Dynamics
•
~ ~*
If k  k ,
then savings/investment exceeds
~
“depreciation”, thus k~  0  g  k~  0.
k
~ ~*
• If k  k , then savings/investment lower than
~
“depreciation”, thus k~  0  g  k~  0.
k
• By continuity, concavity, and given that f(k)
satisfies the INADA conditions, there must
~*
~*
~*
exists an unique k such that f (k )  (n    g )k
~
k
~
k
Transitional Dynamics
_
~

=
~

~

)
 (~
¡ (  + + 
)
~

~ =


~ 0

¤
~

0
~


~ 0

Properties of Steady State
1. In steady state, per capita variables
grow at the rate g, and aggregate
variables grow at rate (g + n)
~ K
K
and k 
Proof: as k 
AL
L
d log K d log L d log k
gK 


 n  gk
dt
dt
dt
~
d log k d log A d log k
gk 


 g  g k~
dt
dt
dt
 g in Steady State
2. Changes in s, n, or  will affect the
levels of y* and k*, but not the growth
rates of these variables.
- Specifically, y* and k* will increase as s
increases, and decrease as either n or  increase
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.
Golden Rule and Dynamic
Inefficiency
• Definition: (Golden Rule) It is the saving rate
that maximises consumption in the steady-state.
~*
~*
~*
*
~
max c  (1  s) f (k )  f (k )  (n  g   )k
s
~* ~*
~*
*
~
~*
c
f (k ) k
k
 ~*
 (n  g   )
 0  f ' (kGR )  (n  g   )
s
s
s
k
~*
• Given kGR ,we can use
to find sGR .
~*
~*
sf (kGR
)  (n  g   )kGR
Golden Rule and Dynamic
Inefficiency
¤


~¤ = (1 ¡  ) ( ~
)

~¤
0
0
 

1
Changes in the savings rate
• Suppose that initially the economy is in the
~*
~*
steady state: sf (k1 )  (n  g   )k1
~*
~*
~
• If s increases, then sf (k1 )  (n  g   )k1  k  0
• 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.
Linear versus log scales
4
x 10
Log-Scale
L inear-Scale

( 
( )) )



( ) = 
(0)
=

_
=
()
(())
( 
( ) )


0
0
0

0



Changes in the savings rate
(  ())
Log of capit al per capit a
(  ())
 per capit a
Log of out put
(  ())
ion per capit a
Log of consumpt

Next lecture
Testing the neo-classical model:
1. Convergence
2. Growth Regressions
3. Evidence from factor prices