Comparative Statics
Comparative Statics
We look at the following cases:
1) Temporary changes in income: today and tomorrow
2) Permanent changes in income
3) Changes in interest rate
Comparative Statics: Change in Income
0
Increase in todays income (recall
Schooling: Spending
Country
Mongolia
Switzerland (U.S)
Zambia
Ecuador
Source: United Nations
How about private spending?
How about spending per student?
Effect of an Increase in Education
when u is smaller, more people work on producing H and less on
prod
2008-2009
Human Capital and the Recession of
One Consequence of the Recession of 2008-2009
Large decline in employment and the labor force
What did these newly unemployed and out of the labor force
people do?
Some went back to school increase their human
How to find.
Steady state level of capital stock for given saving rate s when there
is population growth n and technology grows at rate g?
sf (k ) = (d + n + g)k
Equilibrium law of motion for k
Growth in capital per effective worker:
= (K ) (N ) (H )
K
NH
Determination of the Equilibrium Real Wage
A Simple Model: The Equilibrium
Market clearing imposes: u H s = u H d = u H
So:
C = zu H
(note w = z) and
H 0 = b(1 u)H
0
H = (H ) = b(1 u) 1
1
H
Computing growth rates of consumption:
(C ) = (zu H ) = (H ) = |c
A Simple Model: The Consumer
Suppose :
1) There is no leisure:
Workers split their time between work
and human capital accumulation.
Let u denote time at work. (1 u) time at school
In principle, u is endogenous, but we will take it as fixed
2) Human capit
What is novel about producitivity?
Why do we get growth in per capita terms with technological
progress?
Answer: No diminishing returns in per capita terms to growth in
productivity
How to see this? Write output per worker:
N
Y
=F
K
K
,HN
and in the long
What is Labor-Augmenting productivity Growth?
Solow does not explain why we have growth:
just assumes g or n
Does not tell us what to do to improve long run growth
Obvious Questions:
How do we improve long-run growth? (esp. if it has slowed down
recently)
Some Lessons from Solow
What drives growth in Solow?
With only population growth in the long run:
Model Results:
1) K is constant
2) K grows at rate n
Implications:
1) No growth in per capita terms
2) Long-run convergence: all countries should have same i
Using the Endogenous Growth Model
Some numbers:
Growth rate of about 3%
Employment to Population ratio: .634 in Dec. 2006, .597 in May
2009
Growth rate of model economy is b(1 u) 1 = .03
Let u = .634 then b = 2.814
Using the Endogenous Growth Model
1 time
Growth Lessons
Malthus:
No growth in per-capita income (even with productivity growth)
Growth in population and aggregate income driven by sustained
increases in productivity
Solow:
With only population growth, no growth in per-capita income, but
growth i
Example: Jacks father
Life-time utility and income:
log(C1) + 0.9 log(C2 )
Y1 = 40000
and Y2 = 10500
Interest rate: r = 5%
Optimality conditions
C1 = 50000 = 26316
1.9
Solution:
C2
= 0.9 50000 = 24868
1.9
Example: Compare the two
Jack and his dad have
Two Period Model: Optimality
Problem of the household:
0
max u(c) + u(c )
c,c0
(1 + r)c + c0 = we(1 + r)
Subject to
From first order conditions:
u0 (c) = (1 + r)u0 (c0 )
Optimality:
0
u (c)
u0cfw_z 0 )
| (c
= (1 + r)
0
c,c
Two Period Model: Optimality
To
Assumptions on utility functions
As before we assume that u(c) is increasing
Household, in any period, prefers more consumption to less
Another way of saying this is that marginal utility, u0 (c),
is positive
We assume diminishing marginal utility
As we i
Interest rate and budget line
Higher interest rate implies that cash flow in the future is
discounted at a higher rate, thus less valuable today.
The interest rate is the relative price between consumption now and
the future
One unit of consumption now wo
A Simple Model: Summary
1) Economy grows indefinitely because of human capital
accumulation
2) Rate of growth determined by intensity and efficiency of human
capital accumulation
A Simple Model: Policy
What are good policies?
1) Increase schooling
2) Incr
Two Period Model: Consumer
Consumers live two periods
New trade-off: consumption today vs consumption tomorrow
(add this to the consumption leisure trade-off studied in part I)
New instrument: bonds s
1) All bonds are identical
2) All bonds are safe
3) No
The Golden Rule with population growth and
technological prgress
To find Golden Rule capital stock, c in terms of k :
c = y i
= f ) i
(k
= f (k ) (d + n + g)k
c is maximized when
M PK = d + n + g
How to find. . .
Golden Rule Stock of Capital when there is
Why?
Think about the programmer example
If they can do more tasks, faster, better, etc
You need more capital
Faster computers More
memory
More storage, etc
To be able to take advantage of their progress.
Technological progress in the S
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Effective worker?
Suppose there are some data in a text file.
We want to enter them in a spreadsheet.
Two workers:
One prints the data and enters them manually. Takes him 10 hours to do
that.
One writes a simple code to import the data. Takes her 1 hour t
Population growth
Assume that the population (and labor force) grow at rate n
Nt+1
N = 1+ n
t
n is exogenous
Now, a steady state equilibrium satisfies
K
0
K
N0
=
constant, or
K
N
Break-even investment
(d + n)kt = break-even investment, the amount of
inves
Depreciation
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The equation of motion for k
0
k k = sf (k) dk
The Solow models central equation
Determ
Starting with too much capital
If k *
gold
*
y
then increasing c
requires a fall in s.
In the transition to
the Golden Rule, c
consumption is
i
higher at all points
in time.
t
0
!me
t
!me
Starting with too little capital
If k *
gold *
then increasing c
re
Long run properties
Long growth rate does not depend on saving rate!
Nor does it depend on initial capital stock
What is long run growth rate?
Long run level of capital (and output) does depend of saving rate
But does not depend on initial capital stock
O
Prediction:
Higher s implies higher k
and since y = f (k), higher k implies higher y
Thus, the Solow model predicts that countries with higher rates of
saving and investment will have higher levels of capital and income per
worker in the long run.
Interna
Solve for the steady state
Need to solve the following equation
sk 0.5 = dk
0.3 k 0.5 = 0.1 k
k
0.5
=3
k=9
Then we can find consumption and output
y = k 0.5 = 3
c = (1 0.3) y = 2.1
Solow Growth Model
Plain Vanilla Model: Optimal Savings
An increase in the
The
steady
state
Invest
ment
and
depreci
ation
!k
sf(k)
*
k
Capital per
worker, k
Moving toward the steady state
Investm
ent and
deprecia
tion
Investmen
t and
depreciati
on
kt+1kt =sf(kt)kt
k
kt+1kt =sf
sf(k)
k
investment
k
d
e
p
r
e
c
i
a
t
i
o
n
*
k1 k
How to find . . . .
Steady state level of capital stock for given saving rate s ?
Answer: solve this equation
sf (k ) = dk
A numerical example
Production function
Y =K
0.5
N
0.5
Find per worker production function
0.5
Y
N
=
0.5
K N
N
Therefore,
K 0.5
= (N