Problem Set 3
FE411 Fall 2014
Rahman
To Be Completed: September 22nd
1) The Solow Growth Model
Assume that production is a function of capital and labor, and that the rate of savings (γ),
depreciation (δ), and population growth (n) are all constant, as described in Weil’s
version of the Solow Model in section 4.2. Further, assume that the production per
worker can be described by the function:
y  f (k )  k (1 / 3)
where k is capital per worker.
a. Write the Law of Motion of Capital (this is the equation on page 94 in Weil, but
use the specific functional form for y stated above). 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 effective worker.
b. Solve for the steady-state value of y as a function of γ, n, and δ.
c. A developed country has a saving rate of 45% and a population growth rate of 1%
per year. A less-developed country has a saving rate of 10% and a population growth
rate of 6% per year. In both countries, δ = 0.04. Find the steady-state value of y for
each country. How much more wealthy is the developed country compared with the
less-developed country?
d. What policies might the less-developed country pursue to raise its level of
income?
e. 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?
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Problem Set 3
FE411 Fall 2014
Rahman
f. Finally, redo part c, 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 less-developed one? Why is your answer so
different from your answer in part c? (Explain the intuition – what is it about the new
production function that makes the difference?)
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*; any
income less than this, and 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 Weil text to analyze the steady states of this economy (Note the plural steady
STATES).
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? Illustrate diagrammatically.
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Problem Set 3
FE411 Fall 2014
Rahman
4) The Golden Rule of Capital Accumulation
The production function per worker is, as described in Weil’s 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.
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