Bent E. Sørensen
ECONOMICS 7344 – MACROECONOMIC THEORY II, Spring 2006
Homework 4. Thursday February 12. Due Wednesday February 19 (NOTE: The mate-
rial in questions 3–5 will be covered in class Monday so allocate time Monday afternoon
or Tuesday for doing these questions.)
1. Compare the formulas (6.50) and (6.51) in the text. Calculate the profits πFIXED
and πADJ for a 10 percent increase in real demand (just start from M/P = 1) for η = 5
and ν = 0.1. Sketch the labor supply curve for this value of ν. Redo the calculations
for η = 2. Interpret why the result change. Then assume that ν = 5 and calculate the
profits for this value (keep η = 2) and interpret. Sketch the labor supply curve for this
value and interpret why it is different from the previous one.
2. Define the lag polynomial a(L) = 1 + .5L and b(L) = 1 − L + L2. Define the
z-transform a(z) corresponding to a(L) and b(z) corresponding to b(L) and find the
roots [meaning the solution(s) to, say, a(z) = 0] in each polynomial. Find the polyno-
mial c(z) = a(z) ∗ b(z). Define the lag-polynomial c(L) using the coefficients from c(z)
and verify that for a given time series xt:
c(L)xt = a(L) [ b(L) xt ] .
3. (24% of midterm 1, Spring 2005) Assume that income follows the AR(1) process
yt = 2 + 0.4yt−1 + et (∗)
where et is white noise with variance 3.
a) Is this time-series process stable?
b) Assume that y0 is a random variable. For what values of the mean E(y0) and the
variance var(y0) will the time series yt; t = 0, 1, 2, ... be stationary?
c) What is E1y3 if y1 = 5 and y0 = 2?
d) Write the infinite Moving Average model that is equivalent to the AR(1) model (*)
[assuming that the process now is defined for any integer value of t]. (Half the points
are from getting the correct mean term.)
4. (4% Core Spring 2004) Assume that income follows the ARMA process
yt = 3 + 2.0 yt−1 + et
where et is white noise.
a) Is this time-series process stable?
b) If y0 = 2, what is E0y1?
5. (12% Final Exam 2004) Assume that income follows the ARMA process
yt = 3 + 0.3yt−1 + et
w