Intermediate Macroeconomics，Assignment 3
Due April 15th (Friday), in-class
1. The residents of a certain dormitory have collected the following data: People who
live in the dorm can be classified as either involved in a relationship or uninvolved.
Among involved people, 50 percent experience a breakup of their relationship every
month. Among uninvolved people, 10 percent will enter into a relationship every
month. What is the steady-state fraction of residents who are uninvolved?
Answer:
In this example, the break-up rate is similar as a job separation rate and the newly-engage
rate is similar as a job finding rate. Hence the steady-state fraction of residents who are
not involved is similar as the natural rate of unemployment that equals
50%
50%+10%=83.3%
2. In this chapter we saw that the steady-state rate of unemployment is U/L = s/(s +
f ). Suppose that the unemployment rate does not begin at this level. Show that
unemployment will evolve over time and reach this steady state. (Hint: Express the
change in the number of unemployed as a function of s, f, and U. Then show that if
unemployment is above the natural rate, unemployment falls, and if unemployment
is below the natural rate, unemployment rises.)
Answer: Assuming the total labor force to be constant, then the change of the number
of unemployed workers is
∆Ut+1 = s(L−Ut)−fUt .
Divide both sides with L, we have
∆Ut+1/L = s(1−Ut/L)−fUt/L,
which leads to
∆Ut+1
L=(s + f)[s
s+f −Ut
L].
Therefore ∆Ut+1
L> 0 when Ut
L<s
s+f , and ∆Ut+1
L< 0 when Ut
L>s
s+f. That is to say
when the unemployment rate is above the natural rate, unemployment falls. Otherwise,
unemployment rises.
3. Consider how unemployment would affect the Solow growth model. Suppose that
output is produced according to the production function
Y = Kα[(1 −u)L]1−α,
where K is capital, L is the labor force, and u is the natural rate of unemployment. The
national saving rate is s, the labor force grows at rate n, and capital depreciates at rate d.
1). Express output per worker (y = Y/L) as a function of capital per worker (k = K/L)
and the natural rate of unemployment. Describe the steady state of this economy.
