Download Due Date: Thursday, September 8th (at the beginning of class)

Problem Set 2
FE411 Spring 2007
Rahman
Due Date: Thursday, February 8th (at the beginning of class)
INSTRUCTIONS: Use your own paper to answer the questions. Please turn in your
problem sets with your name clearly marked on the front page and all pages stapled
together. You are encouraged to work together, but you must hand in your own work.
You must show your work for credit and answer in complete sentences when appropriate
(such as when the question asks you to “describe” or “explain”).
1) The Solow Growth Model
Assume that production is a function of capital and labor, and that the rate of savings and
depreciation are both constant, as described in Chapter 3’s version of the Solow Model.
Further, assume that the production per worker can be described by the function:
y  f (k )  k (1 / 3)
where k is capital per effective worker.
a. Using the functional form for y, write the functions for both investment and
depreciation in the Solow Model.
i = γk0.33
d = δk
b. Using the functional form above, write the Law of Motion of Capital (Δk = …).
What condition for the Law of Motion of Capital must hold for the economy to be in
a steady state? Show the steady state condition on a graph with the investment
function and the depreciation of capital per worker.
Δk = γk0.33 – δk
For steady state, this expression must equal zero. [Graph not shown, see book &
notes].
c. Solve for the steady-state value of y as a function of γ and δ.
Ans: y =( γ/δ)0.5
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Problem Set 2
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d. A developed country has a saving rate of 28%. A less-developed country has a
saving rate of 10%. In both countries δ = 0.04. Find the steady-state value of y for
each country. How much richer is the developed country compared to the lessdeveloped one?
Plug into equation above.
Developed country: y = 2.65
Developing country: y = 1.58
2.65/1.58 ≈ 1.67. The developed country is 67% richer than the developing
country.
e. What policies might the less-developed country pursue to raise its level of income?
Induce savings: tax incentives, balancing budget, etc.
f. Explain why the savings rate in an economy is so important for the steady state. If
the less-developed country increases its savings rate, show on the graph from b what
happens to the investment function and the steady state level of capital. What is the
cost today of increasing the savings rate? What are the benefits of doing so?
Savings translates into permanent increases of living standards, through raising
capital stocks per person. On the graph [not shown], you would draw a new
investment line, and illustrate a higher kss and yss.
The cost of raising γ is a lower consumption per person today. The benefits come
in the form of higher output per person. Whether or not this also translates into
higher long-run consumption per person is uncertain.
g. Finally, redo part d, but now assume that production is described by:
y  f ( k )  k ( 2 / 3)
Now, finding the new steady-state values for each country, how much richer is the
developed country compared to the developing one? Why is your answer different
from your answer in part d?
y =( γ/δ)2
Plug into equation above.
Developed country: y = 49
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Problem Set 2
FE411 Spring 2007
Rahman
Developing country: y = 6.25
49/6.25 ≈ 7.84. The developed country now is almost 8 times richer than the
developing country. With α = 2/3, capital has become more important in
production, which means investment becomes more important. Differences in
investment then will magnify differences in steady-state output.
2) The Possibility of Poverty Traps
Suppose that there are no investment flows among countries, so that the fraction of output
invested in a country is the same as the fraction of output saved. Saving in an economy is
determined as follows: There is a subsistence level of consumption per worker, c*,
people will consume all of their income. All income per worker in excess of c* will be
split between consumption and investment, with a fraction γ going to investment and the
rest going to consumption. Use a diagram like Figure 3.4 from the text to analyze the
steady states of this economy.
First, define k* as the level of capital per worker necessary to achieve the
consumption level c*. If the initial level of k is above k*, savings will be positive, and
if k is below k*, savings will be zero. Let’s also define another variable, kthreshhold.
This is the level of capital at which depreciation is equal to savings and distinct from
the steady-state level of capital, if it exists. We are now ready to begin our analysis.
There are two cases.
Case 1: Depreciation is always greater than savings: δk > γf(k), for all k.
In the figure below, at any initial level of k, depreciation is always greater than
savings. The level of capital falls over time, as does the level of income per worker.
Consequently, the economy will continue to stagnate until the level of income and the
capital stock are zero.
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Problem Set 2
FE411 Spring 2007
Rahman
Production
y
Depreciatio
n
Investment
k*
k
Case 2: Depreciation is not always greater than savings: δk ≤ γf(k), for some k.
The figure below shows two possible scenarios. If the initial level of capital k is
equal to or below k*, then savings in the economy will be zero. The level of capital in
the economy remains zero and we achieve the same result as in the first case. On the
other hand, if k > kthreshhold, then the level of savings exceeds the level of depreciation
and the capital stock rises over time. The capital stock will reach a steady-state
value, as will income. Finally, if k* < k < kthreshhold, then the amount of savings does
not exceed the amount of depreciation. The level of capital stock begins to fall and
we are in the first case where both income and capital go to zero. In the end, the
ultimate determinant of where the economy rests is determined by the initial level of
capital. Here history matters – you need to start “rich enough” to stay rich.
y
Depreciatio
n
Investment
k*
kthreshhold
kss
k
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Problem Set 2
FE411 Spring 2007
Rahman
3) Divergence
Country X and Country Y have the same level of output per worker. They also have the
same values for the rate of depreciation, δ, and the measure of productivity, A. In
Country X output per worker is growing, while in Country Y it is falling. What can you
say about the two countries’ rates of investment?
It must be the case that Country X has a higher investment rate than Country Y,
and that both countries are out of their steady-states. In the long run, when both
countries are in their steady states, Country X will be permanently richer than
Country Y.
4) The Golden Rule of Capital Accumulation
The production function per worker is, as described in Chapter 3:
y  f (k )  Ak 
Note that, in our simple closed economy case, any output that is not saved is consumed.
In other words:
c  Ak   Ak 
where c is consumption per worker, and γ is the savings/investment rate.
Find the value of γ that will maximize the steady-state level of consumption per worker.
The key to this problem is realizing that in steady-state, investment equals
depreciation. Thus, we can write:

c ss  Ak ss  k ss
In this form, we are able to take the derivative to find the necessary condition that
will guarantee consumption maximization. Taking the derivative of above with
respect to k and setting equal to zero, we get:
Ak ss
 1

That is, the marginal product of capital must equal the rate of depreciation.
Combining this consumption maximization condition with the steady-state
condition:
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Problem Set 2
FE411 Spring 2007
Rahman
saving  y  f (k )  k  Ak ss
 1

k  Ak ss  f ( y)  y
Thus it is easy to see that γ must equal α by the above string of equalities. In any
steady-state level of consumption per worker, the investment rate must equal the
value α to maximize this long-run consumption.
5) Convergence vs. Divergence
Read Chapter 3 of the Easterly book. The economist William Baumol argued that there
has been a general tendency toward convergence of national incomes. The economist
Lant Pritchett, on the other hand, argued the opposite – that the gap between rich and
poor countries has only grown over the past century.
a. In a sense they are both correct. In two or three sentences, explain how.
See pages 60-65 of Easterly for full discussion.
b. The Solow model illustrated in Weil’s Chapter 3 can illustrate 2 two countries
that are converging, or two countries diverging, or even two countries whose
differences in incomes are permanent. In a sentence or two, how is the Solow
model able to achieve such variety in results (no equations necessary, just
intuition)?
Convergence or divergence can occur, depending on differences on initial
conditions and differences in investment rates. See pages 67-69 of Weil for
further discussion.
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