Homework 1 Macroeconomic II | Assignments Economics

Prof. X. Lei

Department of Economics and Finance

University of Guelph

Advanced Macroeconomic Theory II (ECON 6040)

Problem Set 1

1. Question 1 Hansen (1985) We have seen in the lecture that a baseline RBC model generates little

labor market movement (even if capital does not fully depreciates). One way to generate more

movement in labor input is by differentiating the extensive margin (work or not) and the intenstive

margin (hours). To see this, consider an economy populated with a continuum of infinitely-lived

representative households whose preference is given by

u(ct, lt) = ln(ct) + bln(1 −lt) (1)

where ct, ltrepresent consumption and labor input respectively. Suppose that ltcan only take the

value of 0 <¯

l < 1 with probability ptor 0 with probability (1 −pt). Intuitively, you can think that

that labor is indivisible: you either don’t have a job, or you have a job and work 8 hours a day!

Hansen argues that instead of having households to choose ctand lt, we can allow them to choose

ctand pt. He also assumes that there is perfect insurance, i.e: whether you are drawn to work or

not, you will always be getting paid the same wage rate. (This ensures that that the competitive

equilibrium is also Pareto optimal.) Firms rent capital and labor for production with a production

technology given by

f(kt, lt) = Atkα

tl1−α

t(2)

and where the aggregate resource constraint is again given by

yt=ct+it(3)

Physical investment depreciates at the rate of δ.

(a) Write down the (ex-ante) expected flow utility of the households at the micro level. What’s

the value of Frisch elasticity at the micro level and the macro level, respectively?

(b) What are the relevant state variables? What are the control variables? Using Lagrangian

method, write down the system of equations that characterize the solution of the model in a

decentralized economy (Hint: this should include the stochastic shock process, all the necessary

first order conditions, aggregate resource constraint, as well as the transversality condition).

(d) Log-linearize the F.O.Cs and the aggregate resource constraint around the steady state values

(You can eliminate wtas what we did in the lecture. That should leave you with four equations

to log-linearize).

(e) Let’s making some fun predictions of the model without fully solving it. What results do

you expect to see in terms of labor input as compared with the baseline RBC model that we

discussed in the lecture? Interpret.

2. Question 2 Now let’s go back to the baseline RBC model with log preferences and fully-depreciated

capital.

Homework 1 Macroeconomic II, Assignments of Economics