Problem Set 2
FE411 Spring 2008
Rahman
Partial Answer Key
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 96 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.
k  k (1 / 3)  (n   )k . In order for steady-state to hold, this expression must equal
zero.
output per worker
output
depreciation
yss
investment
kss
capital per worker
b. Solve for the steady-state value of y as a function of γ, n, and δ.
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Problem Set 2
FE411 Spring 2008
Rahman
You should be able to show the steps on how to solve for kss and yss yourselves. The
final solution will be:
k
ss
  


n  
3/ 2
  
y 

n  
1/ 2
ss
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?
ss
Let’s call the steady-state level of income in the developed country y rich ,and the
ss
steady-state level of income in the less-developed country y poor . Taking the ratio of
these levels and plugging in our parameter values gives us
1/ 2
y rich
ss
y poor
ss
1/ 2
  rich 
 0.45 




1/ 2
nrich   
0.01  0.04 
9





  3
1/ 2
1/ 2
1
0.1
  poor 






n

 0.06  0.04 
 poor   
Thus, based on these two differences alone, the prediction is that the developed
country will be three times as wealthy (in per person terms) as the less developed one.
d. What policies might the less-developed country pursue to raise its level of
income?
The goals would be to have the less-developed country save and invest more of its
wealth, or lower its rate of population growth, or some combination of both. Just
HOW this could be accomplished is the REAL question. For example, see Easterly’s
discussion of the role of incentives for both investment and limiting fertility. We will
discuss this more later in the course.
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
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Problem Set 2
FE411 Spring 2008
Rahman
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?
Greater savings allows you to maintain a higher level of k in the long-run, and k is
really the only factor in production. While the benefit of a larger γ is more output per
person in the long run (and possibly more long-run consumption per person,
although this is not necessarily true), the costs are lower consumption per person in
the short term.
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?)
Notice that now our solution for steady-state values are:
k
ss
  


n  
3
y
ss
  


n  
2
Redoing our ratio calculations then gives us:
2
y rich
ss
y poor
ss
2
  rich 
 0.45 




2
nrich   
0.01  0.04 
9




    81
2
2
1
0.1
  poor 






n

 0.06  0.04 
 poor   
Woah, now the prediction is that the developed country is a whopping 81 times as
wealthy as the less developed one! Why is our calculation so sensitive to the
exponent we put on capital per person in the production function (recall that we have
been calling this α)? When α gets larger, k becomes more important in production,
and so the things that make k larger become more important. These include higher
investment rates (since as γ rises K rises and thus k rises) and lower population
growth rates (since as n rises L rises and thus k falls).
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Problem Set 2
FE411 Spring 2008
Rahman
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).
Before beginning the analysis, we define two new variables. The level of capital per
worker, necessary to achieve consumption level c* is denoted k*. If the initial level of
capital, ki, is above k*, savings will be positive, and if ki is below k*, savings will be 0.
The second variable I define is k^ (refer to the second figure). It 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.
Case1:
Depreciation is always greater than Savings: k  f (k ), k
In the figure below, at any initial level of ki, 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 capital
stock are zero.
output per worker
output
depreciation
investment
capital per worker
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Problem Set 2
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Rahman
Case 2:
Depreciation is not always greater than Saving: k  f (k ), for some k.
The figure below shows two possible scenarios. If the initial level of capital, ki, is
equal to or below k*, then savings in the economy will be zero. The level of capital in the
economy falls due to depreciation and we achieve the same result as in the first case. On
the other hand, if ki > k^, then the level of savings exceeds or will exceed the level of
depreciation and the capital stock rises over time. The capital stock will reach a steadystate value as will income. If ki < k^, 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, commensurate with
the initial level of income.
output
output per worker
depreciation
investment
k^
kss
capital per worker
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.
It must be that Country X has a higher rate of investment than Country Y. That
is,  x   y .
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Problem Set 2
FE411 Spring 2008
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.
First, in steady-state, the change in capital stock will be zero:
k  0  f (k ss )  k ss
Rearranging this, we know that investment must equal depreciation:
f (k ss )  k ss
Second, given that any output not saved is consumed, we can write an equation for
consumption as,
C  y  f (k )  Ak   k
In the last part of the previous equation, we replace savings with depreciation and write
output in functional form. In this form, we are able to take the derivative to find the
necessary condition that will guarantee consumption maximization. Taking the
derivative with respect to k and rearranging,
d
d
C
Ak   k  Ak  1   .
dk
dk
Ak  1   .
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Problem Set 2
FE411 Spring 2008
Rahman
That is, the marginal product of capital must equal the rate of depreciation. Combining
the consumption maximization condition ( Ak  1   ) with the steady-state condition
( f (k ss )  k ss ), we get:
saving = y  f (k )  k  Ak  1k  Ak   f (k )  y.
Therefore, it is easy to see the  must equal  by the above string of equalities. In any
steady-state level of consumption per worker, the investment/saving rate must equal the
value  .
5) Convergence vs. Divergence
Read Chapter 3 of the Easterly book. The economist William Baumol argues that there
has been a general tendency toward convergence of national incomes. The economist
Lant Pritchett, on the other hand, argues 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.
In short, it is all about sample selection. As we discussed before, Baumol picked
those economies for which we have data back in 1870 – these were the economic
successes; the fact that these countries are now wealthy allows us to have the data for
these economies in the first place. Pritchett on the other hand as a broad range of
countries (for a shorter time period, naturally), and here it appears that countries
diverge in living standards.
b. The Solow model illustrated in Weil’s Chapter 3 can illustrate 2 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 a variety of results (no equations necessary, just intuition)?
In short, there are differences between underlying features of the economies (different
investment rates, population growth rates, etc.) and differences in INITIAL income
levels. So for example, two countries with similar incomes per person but different
investment rates will experience DIVERGENCE, while two countries with different
incomes per person but similar investment rates will experience CONVERGENCE.
Finally, two countries that have different investment rates and are each at their own
steady-state levels were have a PERMANENT income gap between them.
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