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Fall Semester ’05-’
Akila Weerapana
Lecture 10: Comparative Static Analysis
I. OVERVIEW
- In the last lecture we derived the Solow model with technology. We drew a modified version
of a Solow diagram in terms of capital per effective worker and then looked at the properties
of the steady state. In the steady state, capital per worker and output per worker grow at
the rate of technology while capital and output grow at the rate of population + the growth
rate of technology.
- Today’s lecture looks at the results of some comparative static exercises using the Solow
model with technology. As with the basic Solow model, we can look at the economic impact
of changes in the rate of population growth, the saving rate, the rate of depreciation or the
level of capital or labor in the economy.
- In addition to these changes, we can also look at changes in the growth rate of technology as
well as the impact of changes in the level of technology.
- As I mentioned earlier, hopefully, adding technology to the model will preserve the sensible
predictions of the basic Solow model and improve some of the less-sensible predictions of the
basic model.
II. SAMPLE COMPARATIVE STATICS EXERCISE
An Increase in the Saving Rate
- Suppose the economy is at the steady state (
k
∗
0
) when the saving rate increases from s to s
′ .
For simplicity, assume that this is the only change in the economy.
- We can use the Solow diagram to illustrate the effects of the change: in the diagram the
saving per worker line shifts upwards from sy to sy
′
. At the old steady state,
k
∗
0
, saving per
worker exceed the level of break-even investment per worker (s
′ y˜
∗
0
(g + n + δ)
k
∗
0
): therefore
k increases.
- This process continues until the new steady state
k
∗
1
is reached. At this new steady state,
the saving per worker line is once more equal to break-even investment per worker (s
′ y˜
∗
1
(g + n + δ)
k
∗
1
) therefore
k does not change.
- This is illustrated in the diagram below.
6
Saving,Investment per effective worker
kt
s
k
α
s
′ ˜ k
α
(g + n + δ)
k
k
∗
0
k
∗
1
6
6
6
be the point at which the saving rate increased and t 1
be the time at which the economy
returned to steady state. We can draw the following graphs to show the path of
k and ˜y
6
ln
k
t
ln
k
∗
0
ln
k
∗
1
t 0 t 1
q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q
6
ln ˜y
t
ln ˜y
∗
0
ln ˜y
∗
1
t 0 t 1
q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q
k
- Before t 0
the economy is at steady state:
k is constant at
k
∗
0
- Between t 0 and t 1 ,
k is increasing over time (see Solow diagram).
- After t 1
the economy is back at steady state:
k is constant at
k
∗
1
- NOTE: The dotted line shows the path of the variable in the absence of an increase in the
saving rate.
k
α and α is constant, the graph for ˜y looks exactly the same as the graph for
k.
- Since nothing happened to technology or labor (they continued to grow exogenously at rate
g and n respectively) the graphs for A and L can be drawn easily.
6
ln K
t
t 0 t 1
g + n
g + n
g + n
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
6
ln Y
t
t 0 t 1
g + n
g + n
g + n
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
An Increase in the Growth Rate of Technology
- Suppose the economy is at the steady state (
k
∗
0
) when the growth rate of technology increases
from g to g
′
. For simplicity, assume that this is the only change in the economy.
- Then the break-even investment per effective worker line becomes steeper. At the old steady
state, saving per effective worker lies below the level of break-even investment per effective
worker: therefore
k decreases.
- Why does this happen? Well, at the old steady state, the break-even requirements are higher:
with better technology, more investment needs to be used to keep capital per effective worker
constant. Essentially, more investment is needed to make use of the faster growing technology.
- This process continues until the new steady state (
k
∗
1
) is reached. At this new steady state,
saving per effective worker is once more equal to break-even investment per effective worker.
The Solow diagram would look as follows:
6
Saving,Investment per effective worker
k t
s
k
α
(g
′
k
(g + n + δ)
k
k
∗
1
k
∗
0
6
6
6
- The dynamics of the endogenous variables
k and ˜y can be determined using the Solow diagram.
k should then be constant at
k
∗
0
until t 0
, at which point
k begins to decline
until t 1 when it reaches its new steady state
k
∗
1
where
k becomes constant again.
k
α ,and α has not changed, the graph for ˜y looks identical to the graph for k.
6
ln
k
t
k
∗
1
k
∗
0
t 0
t 1
q q q q q q q q q q q q q q q q q q q q q q q q q
6
ln ˜y
t
y ˜
∗
1
y ˜
∗
0
t 0
t 1
q q q q q q q q q q q q q q q q q q q q q q q q q
- The evolution of technology is more interesting here. There is an increase in the growth rate
of technology at t 0 causing the slope of the curve to change from g to g
′
g. The growth of
the labor supply is constant at a rate n
6
ln A
t
˙ A
A
= g
˙ A
A
= g
′
@
@I
@
@I
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
6
ln L
t
˙ L
L
@ = n
@I@
- As before, we can determine the time paths of the variables k and y from the time paths of
k, ˜y and A.
- Consider the behavior of k where that
k =
k
A
. Before t 0
k is constant. Therefore since A is
increasing at a rate g then k must also be increasing at a rate g.
and t 1
k is decreasing. Given that A is increasing at a rate at a rate g
′
g then
k must be increasing at a rate < g
′
6
ln K
t
t 0 t 1
g + n
g
′
< g
′
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
6
ln Y
t
t 0 t 1
g + n
g
′
< g
′
q
q
q
q
q
q
q
q
q
q
q
q
q
q
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