Download The Solow Growth Model (part two)

The Solow Growth
Model (Part Two)
The golden rule level of capital,
maximizing consumption per
worker.
Model Background
• As mentioned in part I, the Solow growth model allows us a
dynamic view of how savings affects the economy over
time. We also learned about the steady state level of capital.
• Now, we assume policy makers can set the savings rate to
determine a steady state level of capital that maximizes
consumption per worker. This is known as the golden rule
level of capital (k*gold)
Building the Model:
• We begin by finding the steady state
consumption per worker.
From the national income accounts
identity,
y=c+i
we get
c=y–i
• We want steady state “c” so we
f(k*),δk*
substitute steady state values for both
output (f(k*)) and investment which
equals depreciation in steady state (δk*)
giving us
c*=f(k*) – δk*
δk*
f(k*)
• Because, consumption per worker is the
difference between output and
investment per worker we want to
choose k* so that this distance is
maximized.
• This is the golden rule level of capital
k*gold
• A condition that characterizes the
golden rule level of capital is
MPK = δ
c*gold
k*
k*gold
Below k*gold,
increasing k*
increases c*
Above k*gold,
increasing k*
reduces c*
Building the Model:
• While the economy moves
toward a steady state it is
not necessarily the golden
rule steady state.
• Any increase or decrease
f(k*),δk*
δk*
f(k*)
in savings would shift the
sf(k) curve and would
result in a steady state
with a lower level of
consumption.
sgoldf(k*)
sgoldf(k*)
k*
k*gold
To reach the
golden rule
steady state…
The economy
needs the right
savings rate.
A Numerical Example
• Starting with the Cobb-Douglas production function from
part I,
(1)
y=k1/2
recall that the following condition holds in steady state,
(2)
s/δ = k*/f(k*)
• assume depreciation is 10% and the policy maker
chooses the savings rate and thus the economy’s steady
state. Equation (2) becomes,
s/.1 = k*/√k*
Squaring both sides yields,
k* = 100s2
• With this we can compute steady state capital for any
savings rate.
A Numerical Example
• Using the functions from the previous slide and solving for
a range of savings rates …
• We can see that at s=.5 we get c*=2.5 so at savings rate of
.5 consumption per worker is maximized. Also note that at
that level MPK–δ=0 and k*=25.
s
k*
y*
δk*
c*
MPK
MPK-δ
0
0
0
0
0
∞
∞
.1
1
1
.1
.9
.5
.4
.2
4
2
.4
1.6
.25
.15
.3
9
3
.9
2.1
.167
.067
.4
16
4
1.6
2.4
.125
.025
.5
25
5
2.5
2.5
.1
0
.6
36
6
3.6
2.4
.083
–.017
.7
49
7
4.9
2.1
.071
–.029
.8
64
8
6.4
1.6
.062
–.038
.9
81
9
8.1
.9
.056
–.044
1.0
100
10
10
0
.05
–.05
A Numerical Example
• Another way to identify the golden rule
steady state is to choose the level of capital
stock where MPK – δ = 0
• In this example MPK = 1/(2√k) – .1 = 0
so…
and…
and…
1 = .1(2√k)
5 = √k
25 = k*
A Numerical Example
• But what is the time path toward k*? To get
this use the following algorithm for each
period.
•
•
•
•
•
•
k = 4, and y = k1/2 so, y = 2.
c = (1 – s)y, and s = .5 so c = .5y = 1.0
i = s*y, so i = 1.0
δk = .1*4 = .4
Δk = s*y – δk so Δk = 1.0 – .4 = .6
so k = 4+.6 = 4.6 for the next period.
A Numerical Example
• Repeating the process gives…
period
k
y
c
i
δk
Δk
1
2
.
10
.
∞
4
4.6
.
10.12...
.
25
2
2.144...
.
3.087...
.
5
1.0
1.072...
.
1.543...
.
2.5
1.0
.536…
.
1.543...
.
2.5
.4
.46…
.
.953…
.
2.5
.6
.612…
.
.590…
.
0.0
And we converge to k=25
The Transition to the Golden Rule Steady State
• Suppose an economy starts
with more capital than in the
golden rule steady state.
• This causes an immediate
increase in consumption
and an equal decrease in
investment.
Output, y
• Over time, as the capital
stock falls, output,
consumption, and
investment fall.
Consumption, c
Investment, i
• The new steady state has a
higher level of consumption
than the initial steady state.
t0
At t0, the savings
rate is reduced.
Time
The Transition to the Golden Rule Steady State
• Suppose an economy starts
with less capital than in the
golden rule steady state.
• This causes an immediate
decrease in consumption
and an equal increase in
investment.
Output, y
• Over time, as the capital Consumption, c
stock grows, output,
consumption, and
investment increase.
• The new steady state has a
higher level of consumption
than the initial steady state.
Investment, i
t0
At t0, the savings
rate is increased.
Time
Conclusion
• In this section we used our knowledge that savings
affects the steady state and chose the savings rate to
maximize consumption per worker. This is known as
the golden rule level of capital (k*gold)
• In the next section we augment this model to include
changes in other exogenous variables; population and
technological growth.