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
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