Problem Set 7 Solutions
Jeffery is responsible for this answer sheet. If you have any questions about the solutions, please
email him at jeffery.zhang@yale.edu.
1. Shorties
a. Assuming that the savings rate, the depreciation rate, and the population growth rate do not
change, then long-run investment and output per worker will be the same. Hence, consumption
per worker will be the same in steady state.
Long-run Effects of Losing Capital Stock
0.6
Break-even Investment per Worker, (δ + n)k
Investment per Worker, sf(k)
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.5
1.0
1.5
2.0
2.5
Capital per Worker, k
3.0
3.5
4.0
b. The effect on consumption per worker depends on where the economy is relative to the
“Golden Rule” steady state, which is the steady state in which consumption is maximized. If the
economy has less than or equal to the Golden Rule capital stock, then a drop in the savings rate
will decrease consumption per worker. However, if the capital stock is greater than the Golden
Rule level, then a drop in the savings rate may increase consumption per worker.
Long-run Effects of Lower Savings Rate
0.6
Break-even Investment per Worker, (δ + n)k
Investment per Worker, sf(k)
0.5
Investment per Worker with Lower s
0.4
0.3
0.2
0.1
0.0
0.0
0.5
1.0
1.5
2.0
2.5
Capital per Worker, k
3.0
3.5
4.0
2. Economic Growth with Alligators
To answer this question, we need to first consider how capital stock per worker changes with
lower population growth.
Long-run Effects of Lower Population Growth
0.6
Investment per Worker, sf(k)
Break-even Investment per Worker, (δ + n)k
0.5
Break-even Investment per Worker with Lower n
0.4
0.3
0.2
0.1
0.0
0.0
0.5
1.0
1.5
2.0
2.5
Capital per Worker, k
3.0
3.5
4.0
From this diagram, we see that capital per worker increases when population growth is lower.
Note that the presence of n is not driven by any behavioral changes but rather by a purely
mechanical channel since grows at rate .
Next, let’s look at national income accounting:
.
Using
and
, and letting * denote steady state, we have
.
Thus, given a higher level of
, we see that consumption per worker increases in the long run.
To analyze the effect on the marginal product of capital, we solve for as a function of s, n, and
. In steady state, we have
, so
. Next, we can rearrange the
expression to get
.
Since, the marginal product of capital is
, we see that
. Thus, a fall in
reduces the marginal product of capital.
3. Spreadsheet Modeling
a. Initial output is
, so initial output per worker is
.
b. See the accompanying Excel spreadsheet.
c. From the calculations in the Excel spreadsheet:
Per Capita Output, Capital, and Consumption in the
Neoclassical Growth Model
3.0
y
k
c
2.5
2.0
1.5
1.0
0.5
0.0
0
5
d. Recall that
also know that
10
15
20
25
30
35
, and that, in steady state,
, so, in steady state, we must have
(
) .
40
45
In our case, we
50
With
,
,
, and
,
(
)
.
(
)
.
e. Using the same logic as before,
Per Capita Output, Capital, and Consumption in the
Neoclassical Growth Model with Increased Savings
4.5
y
k
4.0
c
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0
5
10
15
20
25
30
35
40
45
50
Here is the same graph, but measured as percent change from the baseline, with baseline being
the results from part (3b). (Note: This version of the graph is not necessary.)
Per Capita Output, Capital, and Consumption in the
Neoclassical Growth Model with Increased Savings
60%
y
k
Percent Change from Baseline (3b)
50%
c
40%
30%
20%
10%
0%
-10%
0
5
10
15
20
25
30
35
40
45
50
f. The growth rate in the long-run is zero for any savings rate. This is because there is no
technological change. Note that the savings rate affects the level of output but not the growth of
output in the long run.