Econ 905 Macroeconomics
An Introduction to neo-classical
growth theory.
Outline
• Solow’s Growth model
– Balanced growth path
– Effect of Savings
• Output
• Consumption
Central Questions of Growth Theory.
1. Why are some countries so rich and others so
poor?
2. Why do growth rates vary across countries and
over time?
3. What are the policies that can change growth in
the short and long run?
4. Why do some countries “take off” while others
fall behind?
Think Sub-Saharan Africa, East Asia, OECD...
Kaldor’s (1961) list of stylized
facts
1. Per capita output Y/L grows over time. [Growth
rates need not be constant over time.]
2. Physical capital per worker K/L grows over time.
3. Rate of return to capital r nearly constant.
4. Ratio of physical capital to output K/Y nearly
constant.
5. Shares of labour wL/Y and physical capital rK/Y
in national income nearly constant.
The Solow-Swan growth model
The basic assumptions of the model
1. Focuses on four variables: output (Y),
capital (K), labour (L), and “knowledge”
or the “effectiveness of labour” (A).
Y(t) ═ F((K(t), A(t), L(t))
A and L enter multiplicatively. AL is referred
to as “effective labour”.
2. Constant returns to scale in its two
arguments, capital and effective labour.
• F(cK, cAL) ═ cF(K, AL) for all c ≥ 0
Assumptions (...cont.)
• Production function in intensive form:
Setting c ═ 1/AL in above equation yields
F(cK, cAL) = F(k,1) = f (k), where k = K/AL
likewise: y = Y/AL = f (k)
3. The intensive-form production function, (k), is assumed to
satisfy f
1. f (0) = 0
2.
f '(k) >0
3.
f ''(k)<0
Marginal Products
Note that since: (1/AL)F(K,AL) = f (k)
We have: F(K,AL) = AL f (K/AL)
It follows that:
F(K,AL)/K = AL f '(K/AL)(1/AL)
= f ' (k)
4. Inada Conditions are satisfied.
limk0 f '(k) = 
limk f '(k) = 0
This & what we know about f '(k) & f ''(k)
implies shape of production function
The Production Function
y
f(K/AL)
k
Assumptions (...cont.)
• Factors of production are fully and
efficiently employed.
• Labour and knowledge grow at constant
rates n and g respectively.
• Capital depreciates at a constant rate .
• Capital markets clear so S = I.
Growth of A and L
L and A grow at constant rate over time:
dLt
 nLt
dt
dAt
 gAt
dt
n and g are the growth rates.
Capital Accumulation
• National income accounting
Y = C + I + G + NX.
• Assume for now no government G = 0 and start
with closed economy, NX = 0.
• Capital stock, K(t), depreciates at constant rate
δ>0
Capital stock: dK/dt = I(t) − δK(t).
• In equilibrium: I(t) = S(t) = sY(t).
Therefore, dK/dt = sY(t) − δK(t).
Capital Accumulation
The growth rate of capital is determined by savings
and depreciation.
dK/dt = sYt – Kt
• With both s and  exogenous constants.
Capital stock will grow so long as:
sYt > Kt
Evolution of K/AL depends on n, g and the rate of
depreciation  as well as savings.
Differentiating k
dt
dt
K
A
L

K
(
A
L
)
t t
t
t t
k dt 
2
( At Lt )
K At Lt  K t ( At L  A Lt )

( At Lt ) 2
dt
dt
dt
K dt At Lt K t ( At Ldt  Adt Lt )


2
2
( At Lt )
( At Lt )
Differentiating k
dt
Kt
K
dt
dt
dt
k 

[
A
L

A
Lt ]
t
2
( At Lt )
At Lt
dt
dt
dt
K t At L
K t A Lt
K



2
2
( At Lt )
( At Lt )
At Lt
dt
dt
dt
Kt L
Kt A
K



At Lt Lt At Lt At
At Lt
Differentiating k
dt
K
k dt 
 kt n  kt g
At Lt
sYt  K t

 kt n  kt g
At Lt
Yt
Kt
s

 kt n  kt g
At Lt
At Lt
Differentiating k
k dt  syt  kt  kt n  kt g
 sf kt     n  g kt
Since yt = f (kt)
Recall: break even investment occurs where
savings are just sufficient to stabilize K/AL,
i.e. where:
syt = (n + g + )kt
Break Even Investment
sy
(n+g+)k
k
Solow Growth Model
sy
(n+g+)k
sf(K/AL)
k*
k
Balanced Growth Path
Growth converges to balanced growth path.
On this growth path:
- Kt grows at rate n + g
- So does Yt
- yt stagnates at this point
- Y/L grows at rate g
Equilibrium Convergence
sy
(n+g+)k
sf(K/AL)
X
k
Change in s
sy
X2
(n+g+)k
X1
k
Change in Savings Rate
Increase in s is instantaneous:
s
s1
s0
t0
t
Impact on k
dk
0
k
t0
t
t0
t
Impact on yL
dyL
g
0
yL
t0
t
t0
t
Change in Savings Rate
Produces one-off convergence to new
balanced growth path
This is at higher K/AL
Long term growth rate still depends on A
Formalizing the long term effect
Using the chain rule we have in general:
y *
k *
 f ' k *
s
s
Where the arguments of k are the constants
s,g,n and , and:
f k *
f ' k * 
k
The long term effect on y
Since k is stable on the balanced growth path
we know that:
sf k *  n  g   k *
Which can be differentiated with respect to s
using the product and chain rules.
The long term effect on y
k *
k *
sf ' k *
 f k *  n  g   
s
s
k *
k *
f k *  n  g   
 sf ' k *
s
s
k *
f k *  n  g     sf ' k *
s
The long term effect on y
k *
f k *

s n  g     sf ' k *
This can be substituted into the initial
expression:
y *
k *
 f ' k *
s
s
The long term effect on y
y *
f ' k * f k *

s n  g     sf ' k *
This can be re-expressed in elasticity terms.
The long term effect on y
First, pre-multiply by s/y*:
s y *
s
f ' k * f k *

y * s
f k * n  g     sf ' k *
The long term effect on y
then, noting that sf(k*) = (n+g+)k*, we have:
s y *

y * s
n  g   k * f ' k *

sf k * f ' k * 
f k *n  g    

f k * 

The long term effect on y
or:
s y *

y * s
n  g   k * f ' k *


n  g   k * f ' k * 
f k *n  g    

f k *


which can be simplified by multiplying out
(n+g+) and rearranging.
The long term effect on y
n  g   k * f ' k *
s y *
f k *

y * s
 k * f ' k * 
n  g   1 

f k * 

k * f ' k *
f k *

k * f ' k *
1
f k *
The long term effect on y
y *
Considerin g that f ' k * 
we now have
k *
k * y *
s y *
f k * k *

y * s 1  k * y *
f k * k *
The long term effect on y
k * y *
s y *
y * k *

y *  s 1  k * y *
y * k *
k * y *
and :  k k * 
y * k *
The long term effect on y
We then obtain:
 k k *
s y *

y * s 1   k k *
with k(k*) the elasticity of y* with respect to
k*.
Effect on Consumption
Consumption per unit of effective labour:
c = (1-s)f (k) = f (k) –sf (k)
and
c* = f (k*) –(n+g+)k*
Effect on Consumption
noting that k* = k*(s,n,g,), we have:
c *
k *
k *
 f ' k *
 (n  g   )
s
s
s
c *
k *
  f ' k *  (n  g   )
s
s
Effect on Consumption
If ...
f ' k *  n  g     c  c
*
1
*
0
f ' k *  n  g     c  c
*
1
*
0
Maximum Consumption
F.O.C.:
c *
k *
  f ' k *  (n  g   )
0
s
s
This holds where:
f ' k *  (n  g   )
Golden Rule Level of k*
y
(n+g+)k
f(K/AL)
f'(k*) = (n+g+)
k