Monetary Theory First Session – Spring Semester 2015
This is a closed book exam. You have 2 hours, 10 questions to answer and a maximum of 70 points to be awarded. Before you start please read carefully the following: • Read the whole problem before starting, as answers to some questions may be found later in the problem. • All questions call for short answers, no need to write a long paragraph (you increase your probability of saying something wrong and therefore have points discarded.) • Any non justified answer will be considered as wrong. So explain, and give economic intuition to any result you obtain. This is an economic course, not a math course. (In particular when an economic interpretation is requested) • The exam is written in such a way you should not be stuck if you cannot solve a question. So if you cannot find the solution to a question, move to the next one. • Make sure you have your name and ID number on your exam sheet. • Any attempt to cheat will grant you 0 immediately.
We consider the case of a closed economy populated by a continuum of household and a continuum of firms. Each continuum has mass one, and we will make the representative agent hypothesis when appropriate. Households have preferences over consumption, ct , and hours worked, ht , described by the utility function Et
∞ X τ =0
h1+ν β τ exp(χt+τ ) log(ct+τ ) − t+τ 1+ν
where β ∈ (0, 1), ν > 0 and χt = ρχ χt−1 + (1 − ρχ )χ + εχ,t 1
!
where εχ,t ; N (0, σχ ) and |ρχ | < 1. Households receive income from the labor they supply on the market, dividends, Πt , they receive from firms and asset income in the form of interest Rt−1 they accrue from bonds, Bt−1 , bought in period t − 1. This income is used to purchase consumption goods at price Pt and bonds for the following period. Question 1: Write the problem of the household and find the first order conditions associated to this problem and give the economic intuition.
(5 points)
Question 2: Show that the system of first order conditions admits the following log–linear representation νb ht = w bt − b ct
(1)
bt − Et π b ct − Et b ct+1 + R bt+1 + Et χt+1 − χt = 0
(2)
where wt is the real wage and πt is the inflation rate.
(5 points)
The economy comprises 2 types of firms. The first one produces a final good that can be used for consumption purposes or to pay for any additional costs in the economy. The final good is obtained by combining intermediate goods according to a CES technology of the form Z yt =
1
yt (i)
θ−1 θ
θ θ−1 di
0
The final good firms are price takers on the intermediate good market. The price of each intermediate good is denoted Pt (i). Question 3: What assumption do you need to place on θ for yt to be a proper final good aggregate? (5 points) Question 4: Show that the optimal demand for intermediate good i is given by yt (i) =
Pt (i) Pt
−θ yt
Show that the aggregate price index is given by Z
1
Pt =
1 1−θ Pt (i)1−θ di
0
(5 points) The second type of firms produces intermediate goods by means of labor input according to a constant returns to scale technology of the form yt (i) = exp(zt )ht (i) 2
Each firm has local monopoly power, but faces a real cost of adjusting its price of Pt (i) Φ yt γ πt−1 Pt−1 (i) where Φ0 (x) > 0, Φ00 (x) > 0, Φ(1) = Φ0 (1) = 0 and Φ00 (1) = ϕ > 0. Note that implicit in this formulation is that inflation is zero in a deterministic steady state (π = 1). You will denote Ψt,t+s the discount factor of the firm between period t and t + s. Question 5: Find the optimal demand for labor, and show it depends linearly on the real marginal cost st .
(5 points)
Question 6: Show that, in a symmetric equilibrium, the optimal price setting behavior of the firm is described by the following relationship πt exp(χt+1 )ct πt+1 0 πt+1 yt+1 πt 0 + βEt Φ =0 1 − θ + θst − γ Φ γ exp(χt )ct+1 πtγ yt πt−1 πt−1 πtγ Give economic intuition for this equation. (Hint: It may be useful to consider extreme cases.)
(5 points)
Question 7: Show that at the general equilibrium of the economy, we can derive Phillips curve of the form β γ π bt−1 + Et [b πt+1 ] 1 + βγ 1 + βγ Give the exact expression for κ. Re-express the Phillips curve in terms of the output gap. In π bt = κb st +
particular you give a precise expression of the natural output.
(5 points)
Question 8: The central bank is assumed to follow a monetary policy of the form bt = µb R πt + ηt where ηt is a monetary policy shock represented by an AR(1) process. Under what conditions on µ is the equilibrium determinate? Why?
(5 points)
bt . In particular you bt , ybt and R Question 9: Discuss the form of the solution decision rules for π will propose a guess for the solution. (Note that I am not asking you to find the solution). (5 points) Question 10: In the next figure, we present 4 potential set of impulse response functions to a monetary policy shock. The monetary policy shock is expansionary, and therefore, Ceteris Paribus, cuts the interest rate. Which one are correct and derived from the model? Give the conditions on the parameters for these IRFs to be meaningful, and justify your answer intuitively. (25 points)
3
Figure 1: Potential IRFs (a) Output
Inflation
Nominal Interest Rate
0.8
0
0
0.6
−0.1
−0.1
0.4
−0.2
−0.2
0.2
−0.3
−0.3
0 0
10
20
−0.4 0
10
20
−0.4 0
10
20
(b) Output
Inflation
1
Nominal Interest Rate
1.5
1.5
1
1
0.5
0.5
0.5 0 −0.5 −1 0
10
20
0 0
10
20
0 0
10
20
(c) Output
Inflation
1
Nominal Interest Rate
0.8
0.6 0.4
0.6 0.5
0.2 0.4 0
0 0.2 −0.5 0
10
20
0 0
−0.2 10
20
−0.4 0
10
20
(d) Output
Inflation
Nominal Interest Rate
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1 0
10
20
−1 0
10
4
20
−1 0
10
20
