ECO 402
Fall 2013
Prof. Erdinç
Economic Growth
The Solow Model
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
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 firm maximises profits and take price as given (perfect competition)
• Inputs paid by 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
Y and y 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
kgnsykdt
dk)(
)()(
d
lnd
d
lnd
d
lnd
d
lnd
lnlnlnln :
gnk
sygn
K
sYng
K
K
k
k
t
L
t
A
t
K
t
k
LAKkAL
Kk
Proof
Steady State
0y 0
ck
0)( ** kgnksf
Definition: Variables of interest grow at constant rate (balanced growth path or BGP)
• at steady state:
ky syiy ,,
k
sksy
kgni evenbreak
** kgnsk
*k
Solow Diagram: Steady State
ss
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
00
k
kgk k
*kk
*** )()( kgnksfthatsuchk
00
k
kgk k
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 0
loglogloglog
loglogloglog
as
dt
Ld
dt
Ad
dt
Kd
dt
kdg
gngdt
kd
dt
Ld
dt
Ad
dt
Kdg
AL
Kk
k
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
Policies to Promote Growth
1. Are we saving enough? Too much or too little?
2. What policies may change the savings rate?
3. How should we allocate savings between physical and human capital?
4. What policies could generate faster technological progress?
Golden Rule• Definition: (Golden Rule) It is the saving rate
that maximises consumption in the steady-state.• We can use the rule to evaluate if we are saving
too much, too little or about right.
• 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
)()(' * gnkf GR
• If our savings rate is given by then our savings rate is optimal and
• If then we must be under-saving• If then we must be over-saving
• Check why this is the case!
GRs
)()(' * gnkf GR
)()(' * gnkf GR
Is Golden Rule attained in the US? Is it Dynamically Efficient?
• Let us check: Three Facts about the US Economya) The capital stock is about 2.5 times the GDPb) About 10% of GDP is used to replace depreciating capitalc)
OR
Capital income is 30% of GDP: Note alpha also measures the elasticity of output with respect to capital!
yk 5.2
yk 1.0
ykkf 3.0).(' 3.0)(
).(. '
kf
kkf
y
kMPk
Is Golden Rule attained in the US? Is it Dynamically Efficient?
Since
US real GDP grows on average at 3% per year, i.e.Hence, US economy is under-saving because
04.05.2
1.0
y
y
k
k
12.05.2
3.0. k
k MPy
y
k
kMP
03.0 gn
)( gnMPk
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