Transcript Solow Growth Model

ECO 402
Prof. Erdinç
Fall 2013
Economic Growth
The Solow Model
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
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 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
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 )
Y
K
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
sy
  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: Steady State
ibreakeven  n  g   k
y, i, sy
sk *  n  g   k *

y  k
ss
k*
sy  sk 
k
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
lim k 0 f ' (k )   and lim k  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
*
*
*
k
such
that
sf
(
k
)

(
n



g
)
k
exists an unique


k


k
Properties of Steady State
1. In steady state, per capita variables
grow at the rate g, and aggregate
variables grow at rate (g + n) Proof:
K
as k 
AL
d log K d log A d log L d log k
gK 



 g  n  gk
dt
dt
dt
dt
d log k d log K d log A d log L
gk 



dt
dt
dt
dt
 0 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.
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.
max c*  (1  s) f (k * )  f (k * )  (n  g   )k *
s
c* f (k * ) k *
k *
*


(
n

g


)

0

f
'
(
k
GR )  (n  g   )
*
s
k s
s
*
k
• Given GR ,we can use
to find sGR .
*
*
sf (kGR
)  (n  g   )kGR
Golden Rule and Dynamic
Inefficiency
• If our savings rate is given by sGR then our savings
rate is optimal and
f ' (k * )  (n  g   )
GR
• If
• If
*
f ' (kGR
)  (n  g   ) then we must be under-saving
*
f ' (kGR
)  (n  g   ) then we must be over-saving
• Check why this is the case!
Is Golden Rule attained in the
US? Is it Dynamically Efficient?
• Let us check: Three Facts about the US Economy
a) k  2.5 y
The capital stock is about 2.5 times the GDP
b) k  0.1 y
About 10% of GDP is used to replace depreciating capital
'
MP
.
k
f
(k ).k
'
c) f (k ).k  0.3 y
k

   0.3
OR
y
f (k )
Capital income is 30% of GDP: Note alpha also measures
the elasticity of output with respect to capital!
Is Golden Rule attained in the
US? Is it Dynamically Efficient?
Since
k
k

0.1y
   0.04
2.5 y
MPk .k 0.3 y

 MPk  0.12
k
2.5 y
US real GDP grows on average at 3% per year, i.e. n  g  0.03
Hence, US economy is under-saving because
MPk    (n  g )
Changes in the savings rate
• Suppose that initially the economy is in the
*
*
sf
(
k
)

(
n

g


)
k
steady state:
1
1

*
*
• 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.