Problem Set 5
Mr. Nordhaus and staff
Economics 122a
Problem Set 5 (Solutions)
1. Okun’s Law.
a. Define Okun’s Law in a sentence.
Okun’s Law is the empirical regularity that changes in GDP are inversely related to changes in
the unemployment rate.
b. What is your forecast for the unemployment rate on election day 2012?
From the BLS, the current unemployment rate is 9.6%. From the WSJ, the consensus forecast
for 2011 % real GDP growth is 2.8%. Assuming 2012 % real GDP growth will be 2.8% and that
potential GDP will grow 2.5% in both 2011 and 2010, we can use two versions of Okun’s law to
forecast unemployment:
1) Version in Mankiw: % change in real GDP = 3% - 2(change in
unemployment rate). This gives us u(t+1)=u(t) + 1.5 - (.5)(% change in
real GDP). So, we get u(2011) = 9.6 + 1.5 – (.5)(2.8) = 9. 7, and u(2012) = 9.7
+ 1.5 – (.5)(2.8) = 9.8
2) Version in Lecture: change in unemployment rate = (.5)(% change in
potential output - % change in real GDP). So, we get u(2011) = 9.6 +
.5(2.5-2.8) = 9. 45, and u(2012) = 9. 45 + .5(2.5-2.8) = 9. 3.
So, our forecast for election day 2012 is either 9.8% or 9.3%
2. The Phillips curve. Suppose that an economy has the Phillips curve
π(t) = π(t−1) − 0.5[u(t) − un], un = .06
a. What is the natural rate of unemployment (NAIRU)?
The NAIRU is the level of unemployment consistent with stable, long-term inflation. Here, it
is 6%.
b. Graph the short-run and long-run relationships between inflation and
unemployment.
LR

SR
NAIRU
U
c. Assume that inflation is 1 percentage point above the target in 1980. Using the
Phillips curve, give one trajectory for the unemployment rate that will reduce
inflation to the target.
Suppose π(1980) = π(target) + .01. If we set u(1981) = .08 and .06 = u(1982) = u(1983) = …, then
π(1981) = π(target) + .01 - .5(.08-.06) = π(target) and π(target) = π(1982) = π(1983) = …
d. Many people are worried that high unemployment raises the natural rate
because of the erosion of human capital, worker skills, and the like (this being the
hysteresis model). In this model, the natural rate might be:
un = 0.5[u(t-1) + u(t-2)]
Answer part (c) using the Phillips curve with the hysteresis natural rate.
Suppose unemployment is .06 until it falls to .04 in 1980. This means that π(1980) = π(target) –
(.5)(.04 - (.5)(.06 + .06))) = π(target) + .01.
If we set u(1981) = .07, u(1982) = .055, and each subsequent u(t) = .5(u(t-1)+u(t-2)), then
π(1981) = π(target) + .01 – (.5)(.07 – (.5)(.04+.06)) = π(target)
π(1982) = π(target) – (.5)(.055 - (.5)(.07+.04)) = π(target)
π(1983) = π(target)
π(1984) = π(target)
…
The path for unemployment is thus .07, .055, .0625, .0588, .0606, .0597, .0602, .0599, …,
oscillating above and below .06 but converging to .06.
3. The Taylor Rule and the Fed.
a. The objectives of the Fed are explained in a book on monetary policy, Chapter 2 at
http://www.federalreserve.gov/pf/pdf/pf_2.pdf. Explain why the Fed is said to
have a dual mandate.
The Fed has a mandate to both maintain stable prices and promote full employment. This is a
dual mandate in that inflation and unemployment have an inverse relationship.
b. Read the section on the Taylor rule (Mankiw, pp. 415ff.). Explain how the Taylor rule
fits into the mandate that you described in (a).
The Taylor rule says that the Fed generally has increased (and argues that the Fed should
increase) the Fed funds rate (according to the Taylor rule coefficients) in response to both
inflation in excess of the inflation target and output in excess of potential. In light of Okun’s
Law, the Taylor rule dictates expansionary (contractionary) monetary policy in response to low
(high) inflation or high (low) unemployment. This is in line with the mandate.
4. The residents of Yale’s new Boola College decide to study the matching model
popularized by 2010 Nobel Prize recipients Diamond, Mortensen, and Pissarides. They
design a study to estimate the demographics of people who live in Boola College.
a. They first determined whether people are “involved” or “uninvolved” in a
relationship. Among involved people, they estimate that 10 percent experience a
breakup of their relationship every month. Among uninvolved people, 5 percent
enter into a relationship every month. What is the steady-state “uninvolvement
rate,” defined as the fraction of people who are uninvolved?
Suppose there are N people and that u is the steady-state uninvolvement rate. Then, we must
have Nu = (.95)Nu + (.10)N(1-u), or u = .95u + .1(1-u). This gives us u = .1/.15 = 2/3.
b. Assume that people start studying harder because the economy worsens, and the
breakup rate increases to 20 percent per month. What is the new uninvolvement
rate?
Now, we get u = .95u + .2(1-u), which gives us u = .2/.25 = 4/5.