Download Lecture Note #3 Solow Growth Model

Solow’s Growth Model
Solow’s Economic Growth Model
‘The’ representative Neo-Classical Growth Model:
focusing on savings and investment.
It explains the long-run evolution of
economy quite well with all being held
Constant – Dynamic Model
Features
Focusing on capacity of Savings to meet the
demand for Investment as Capital Requirements
and, beyond that, as Capital Accumulation for
expansion of Production capacity;
Convergence is usual but possibility of Technical
Innovation for Sustained Economic Growth
1. Math
Assumptions of the model
• Population grows at rate n
L’ = (1 + n)L
• Population equals labor force
• No productivity growth
• Capital depreciates at rate 
1) Per-capital Income
• Production function: Y = F(K, L)
• In “per worker” terms: y = f(k)
• Relationship between variables:
Y
y 
L
K
k 
L
From the above we can get:
• Per-person or per-capita income level (y)
depends on each worker’s capital
equipment(k).
• y=f(k) shows DMR.
Can you draw the graph with y and k?
– Growth rate is measured by the slope of the tangent line of
the y or f(k) curve.
– Growth rate decreases as the per-capita capital stock rises.
It is true for all countries- “Convergence”
– Countries that start further away from the steady state grow
faster
2) Actual Supply of Capital
Assume FIXED SAVINGS RATE or APS:
s =S/N/Y/N = savings /income
• Given an income of y
– Actual savings= s · y = s f(k)
EXAMPLE
• Savings rate of 40%
– s = .4 (you save a fraction of your income)
Can you draw the actual savings curve in the
previous graph you have drawn?
3) Required Capital for Just Keep-Up
Minimum Capital Requirement to just keep up for each work
is proportional to population growth rate(n) and capital
depreciation rate()
Minimumk  (  n)k
*if you do not replenish the economy with the minimum
requirement of capital, then the level of capital and thus the level
of production or income fall.
• Investment above and beyond this requirement
will lead to Accumulation of Capital
and Expansion of Production and
an Increase in per-capita Income.
(but still the growth rate continues to fall;
per-capita income rises in decelerating manners)
• Example)
• Y = 100; L = 20; K = 10
y = Y/L = 5
k = K/L = 10/20 = 0.5
n = 3% ;   5%
Then you need 8% of capital every year to keep
constant each worker’s capital equipment.
4) Equilibrium or Not
• The Change in capital per worker is the actual
supply of capital over the minimum required
capital
k  s  y  (  n) k
k  s  f ( k )  (  n) k
We may call this net investment.
• Thus:
– If k > 0: economy accumulates capital per worker
– If k < 0: economy reduces capital per worker
– If k = 0: constant capital per worker: steady state
2. Graphically
f(k)
(+n)k
k < 0
s f(k)
k > 0
k0
k*
k
• Steady-state Per-capita Income or y* = Y/N is
determined where s f(k*) = (+n)k*.
3. Implications of the model
• The economy converges, over time, to its
steady state.
– If the economy starts BELOW the steady state, it
accumulates capital until it reaches the steady
state.
– If the economy starts ABOVE the steady state, it
reduces capital until it reaches the steady state.
• Growth rates
– Capital per worker grows at rate 0
– Output per worker grows at rate 0
– Total capital: K = k · L grows at rate n
– Total output: Y = y · L grows at rate n
4. Comparative statistics
1) Parameters of the model: s, n, 
2) Once-and-for-all increase in K
3) Technical Innovations
(1) Savings rate and growth
(+n)k
s2f(k)
s1f(k)
kss
kss2
k
• Note that an increase in savings rate does
increase the level of income, but not the rate of
growth of income.
*Is there the optimal savings rate?
• Yes, there is the Dynamically Optimal savings
rate:
The savings rate that maximizes
consumption(= Income– Investment or savings
in the steady state.
• It is the Golden Rule Savings Rates
*Golden Rule
The Golden Rule of Savings Rate is such that
MPk = n + .
This is the condition that the economy is dynamically
efficient.
If   0 for simplicity, then
C1 is consumption for Golden Rule where MPk = n +  , which is lager
than C2 or any others.
Note: This graph is for the case where there is no depreciation; add  to n
everywhere.
(2) Population growth rate and growth;
A lower rate of n raises the growth rate of Y.
(+n2)k
(+n1)k
sf(k)
kss2
kss
: Population control raises the growth rate of national income
k
(3) An Injection of Capital
It really does not do anything for the economy and
income in the long-run.
Try the llustration.
(4)Technical Innovations
• How is this different for the y curve from an increase in
savings rate?
*Solow Model discounts the importance of
Capital and carves up technical innovation
in the long run.
• Technological development will be the only
motor of economic growth in the long run.
The bigger questions are:
What is technical innovation?
How does it happen?
How can we promote it?
These are not easy questions at all.
Practice
• Click here for Solow’s Model simulation for
Economic Growth
Note: The production function used in the above illustration is
not the original Solow model, but is an revised version with
the later developed theory of Human Capital. Thus it is a
hybrid of Solow model and Endogenous Growth Model.