Transcript Document

Solow’s Growth Model
The mainline Classical Theory of
Economic Growth
Solow’s Economic Growth Model
‘The’ representative Neo-Classical Growth Model:
focusing on savings and investment, and eventually
on technology.
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: fixed T)
• 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
– Supply of K is from 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 Equilibrium: 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.
What does this mean?
• The Golden Rule Point can be reached with
a lower savings rate than the current
equilibirum.
• The economy has intrinsic incentive to
over-save and under-consume.
(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
How is this relevant to Japan?
1) In the 1950-1970s:
Worked positive.
2) In the 1970sWorked negative.
(3) An Injection of Capital
It really does not do anything for the economy and
income in the long-run; It does not lead to
sustainable economic growth 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 carves up technical innovation
in the long run for
sustainable economic growth; Technical
Innovation fights against Convergence.
MP = f (K/L, T)
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.
Shortcoming
• When it comes to Capital, the Solow Model
emphasizes the importance of an endogenous
factor as opposed to an exogenous factor;
People’s savings rate matters, but Injected
Capital from outside does not lead to
sustained economic growth.
• However, when it comes to Technology, he
just presents it as an important factor like an
exogenous factor or an accident,
and does not explain how it endogenously
evolves from inside of the growing economy.