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

Problem Set 2
FE411 Spring 2009
Rahman
Some Answers
1) Steady-State Comparisons using Solow
Suppose that there are two countries, X and Y, that differ in both their rates of investment
and their population growth rates. In country X, investment is 20% of GDP and the
population grows at 0% per year. In country Y, investment is 5% of GDP, and the
population grows at 4% per year. In both countries, the rate of depreciation of physical
capital is 5%.
a. Use the Solow model to calculate the ratio of their steady-state levels of income
per capita, assuming that α = ½. How much wealthier is country X relative to country
Y?
 i    j 
,

 / 

ss
n  
n


yj
 i
  j

where i is the developed country and j is the developing country (be able to solve this
from the Δk equation). Plugging in values, the developed country is thus 7.2 times as
wealthy in the long run.
Relative steady-state income in this case is given by:
yi
ss
b. Use the Solow model to calculate the ratio of their steady-state levels of income
per capita, assuming that α = ¾. How much wealthier is country X relative to country
Y in this case? Why is your answer so different from your answer in part a? (beyond
simply saying that α is different – what is it about this assumption that changes our
prediction of income differences?)
3
    j 
 . In this case the developed country is a
Now we have ss   i  / 
 n  
n


yj
 i
  j

whopping 373 times wealthier than the developing country in the long run! Again,
this is a world where capital is hugely important, and so labor is fairly unimportant.
A country that saves more ends up with more capital, and this capital contributes
hugely to production. On the other hand, a country with a higher population growth
rate will end up with more mouths to feed, but very little extra production potential.
Thus, differences in savings rates and population growth rates will create bigger
differences in steady-state incomes the greater is α.
yi
ss
3
Page 1 of 6
Problem Set 2
FE411 Spring 2009
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).
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.
Production
y
Depreciatio
n
Investment
k*
k
Page 2 of 6
Problem Set 2
FE411 Spring 2009
Rahman
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
Depreciation
Investment
k*
kthreshhold
kss
k
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 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.
Page 3 of 6
Problem Set 2
FE411 Spring 2009
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.
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:
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.
Page 4 of 6
Problem Set 2
FE411 Spring 2009
Rahman
5) Solow with Different Population Growth Rates
Consider the Solow model with population growth, as presented in the text. Assume that
population can grow at two different rates n1 and n2, where n1 > n2. The population
growth rate depends on the level of output per capita (and therefore the level of capital
per capita). Specifically, population grows at rate n1 when k < k*, and slows down to rate
n2 when k > k*.
Draw a diagram for this model. Assume that (n1 + δ)k* > γf(k*) and that (n2 + δ)k* <
γf(k*). Explain what the diagram says about the steady state of the model.
y
(n2 + δ)k
(n1 + δ)k
γf(k)
kAss
kthreshhold
kBss
k
Multiple values of population growth will yield multiple steady states under some
conditions. In the diagram, point A and point B are the multiple capital per capita
steady-state values and hence the resulting incomes per capita steady-state values. In
addition, for a country with an initial capital stock below kthreshhold, the dynamics will
move the country to point A. That is, in this specific case of multiple steady-states, initial
capital levels completely determine whether a country achieves a high capital and low
population growth equilibrium or a low capital and high population growth equilibrium.
6) 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.
Page 5 of 6
Problem Set 2
FE411 Spring 2009
Rahman
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 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)?
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.
Page 6 of 6