Problem Set 7
Professor Nordhaus and staff
Economics 122a
Due: Wednesday November 17, 2010
Problem Set 7 (Ungraded)
1. A growth problem
Do Mankiw, problem 2, p. 245 (under Problems and Applications), parts (a) through (d).
To solve the problem, we use a Cobb-Douglas production function Y = Kα(L x E)1-α so y = kα with α =
0.3. Since the growth rate of output is 3%, we know that n + h = 0.03. We also know δ = 0.04. Since
K / Y = 2.5, we can show that k / y = [K / (L x E)] / [Y / (L x E)] = K / Y, so k / y = 2.5. (Capital-output
ratio is the same in terms of levels as it is in terms of per effective worker).
a. In the steady state sy = (δ + n + h)k. So s = (δ + n + h)(k / y). Plugging in the values above, we get
that s = (0.04 + 0.03)(2.5) = 0.175. So the initial saving rate is 17.5%.
b. Y = Kα(L x E)1-α. So MPK = α[(L x E)1-α / K1-α] = α(Y / K) = α / (K / Y) = 0.3 / 2.5 = 0.12.
c. To satisfy the golden rule, we must have the condition that MPK = δ + n + h. So to reach the golden
rule, MPK = (0.04 + 0.03) = 0.07. Since the MPK that satisfies the golden rule level of capital is 7%,
whereas it is 12% in part b, this means the level of k must increase.
d. From part b, we know that MPK = α / (K / Y), so we can show that 0.07 = 0.3 / (K / Y), so it must hold
that (K / Y) = 4.286, showing that the level of capital must increase relative to the level of output.
For part (e), substitute the following:
(e) Explain in words why a capital-per worker ratio above the golden rule level is inefficient.
e. The golden rule level of capital per worker represents the level of k that maximizes consumption per
worker in the long run. If we have k above k*gold, then we will need to save a greater portion of output
to maintain k, resulting in less consumption per worker, and a less efficient result – the marginal product
of capital will be less than the amount we would need to spend to replace capital to maintain our capitalper worker ratio.
2. More growth, with and without a fence
According to the neoclassical growth model, how would each of the following affect consumption per
worker in the long run (that is, in the steady state)? You should answer this problem using the
graphical approach to the neoclassical growth model.
a. A research breakthrough at university hospitals allows workers to live longer, fuller lives.
This is a permanent increase in L, as each new worker now contributes more labor-hours. So k = K/(E x
L) decreases initially, but rebounds over time, returning to k* as sf(k) exceeds (δ + n + h)k until
equilibrium is reached once again. As such, consumption per worker in the long run returns to (1-s)y*.
Consumption “per worker-hour” reaches its original steady state level as k increases back to k*. If you
were thinking about a very literal interpretation of consumption “per worker,” you would see an increase
(since the increase in L and Y is driven by an increase in hours and not the number of workers) but the
main point is that per unit of effective worker input consumption returns to c*.
b. The Deficit Commission is successful in persuading Congress to lower the full-employment
budget deficit as a percent of GDP with no impact on technological change.
This is an increase in saving, s, which increases sf(k). The effect on consumption per worker in the long
run is ambiguous, depending on whether the increase in savings moves k* closer to the golden rule level
or farther away.
Graph 1 shows an increase in consumption per worker as a result of the increase in saving.
Graph 2 shows a decrease in consumption per worker as a result of the increase in saving.
For the (c), discuss the impact on consumption per worker in both the short-run and the long-run.
Graphically show the transition to the steady state after the policy change.
c. Economists find that the Deficit Commission’s recommendation to cut federal spending reduces
the rate of labor-augmenting technological change.
The reduction in labor-augmenting technological change reduces h, so (δ + n + h)k falls. The economy
is still initially at the original k, so there is more saving and investment than loss of k, so k increases at a
decreasing rate until it reaches the new k*.
In the short run, consumption per worker continues to increase, but at a slower rate, as shown in the
figure below. Consumption per effective worker rises rapidly, but at a decreasing rate. In the long run,
consumption per worker rises at the new slower rate h’, and consumption per effective worker reaches a
new steady state equilibrium at c* (2).
3. An exercise with simulations
a. Reproduce Table 7-2 (Mankiw p. 199) from years 1 to 100 using a spreadsheet. The first
row of your spreadsheet should the year and variable names (year, k, y, s, I, δk, and Δk).
The second row should just be values that you type in for year 1. But beginning with the
second row, you should use formulas so that you can manipulate the initial values in row
2 and see what happens over time. Take a screenshot of the first 20 rows and attach
it to your problem set. (To create a formula linking cells, type “=” followed by the
formula. For example, the following shows a screenshot of the first couple of entries in the
spreadsheet.
b. Assume the initial value of k is 9, so the economy is in steady state. Then, beginning in
year 4, simulate the following situations. Use the chart wizard to graph per capita
consumption in red and per capita GDP in blue from year 1 to year 30. You will have
three different graphs (one for each of i through iii):
i. An earthquake destroys 5 units of capital per worker.
3.5
3
2.5
2
y
c
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Years
ii. The quality of capital improves so the depreciation rate decreases to 0.06.
4.5
4
3.5
3
2.5
y
c
2
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Years
iii. The marginal propensity to consume decreases to 0.5.
5
4.5
4
3.5
3
y
2.5
c
2
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Years