— MONETARY POLICY — Homework #1: Rational Expectations
The number of * indicates the level of difficulty of the problem.
Exercise 1: Computing conditional expectations *
Consider the following stochastic pro-
cesses xt = ρxt−1 + εt
(1)
xt = (ρ1 + ρ2 )xt−1 − ρ1 ρ2 xt−2 + εt
(2)
xt = εt + θ1 εt−1 + θ2 εt−2
(3)
xt = ρxt−1 + εt + θ1 εt−1 + θ2 εt−2
(4)
where |ρ| < 1, |ρi | < 1, i=1,2, and εt ; iid(0, σ 2 ). 1. Assume that in period t the agent has access to the following information set: Ω = {xt−i , εt−i ; i = 1, . . . , ∞}. For each of the processes (1)–(4), compute E(xt+j |Ω) for j = 0, . . . , ∞. 2. Assume now that in period t the agent has access to the following information set: Ω = {xt−i , εt−i ; i = 2, . . . , ∞}. For each of the processes (1)–(4), compute E(xt+j |Ω) for j = 0, . . . , ∞.
Exercise 2: Nested Processes * Consider the stochastic process yt = φyt−1 + xt + εt xt = ρxt−1 + ηt where |φ| < 1, |ρ| < 1 and εt ; iid(0, σ2 ) and ηt ; iid(0, ση2 ). Denote: Et (zt+j ) ≡ E(zt+j |(xt−i , yt−i , εt−i , ηt−i ; i = 0, . . . , ∞)). Compute Et (yt+j ), for j = 1, . . . , ∞
Exercise 3: Bounded Memory ***
Consider the stochastic process yt = φyt−1 + εt , |φ| < 1
and εt ; iid(0, σ 2 ). Let us assume that the agent has bounded memory in the sense that his information set is given by Ω = {yt−i , εt−i , i = 0, . . . , k} 1
1. Compute the Wold decomposition of the process. 2. Using the Wold decomposition, compute E(yt+1 |Ω). 3. Show that E(yt+1 |Ω) can be expressed as φ(yt − φk+1 yt−(k+1) )
Exercise 4: Signal Extraction ** Consider the case of a firm that wants to predict the demand, d, it will be addressed, but only observes a signal s that is related to d as s=d+η
(5)
where E(d) = δ, E(η) = 0, E(dη) = 0, E(d2 ) = σd + δ 2 < ∞, E(η 2 ) = ση < ∞. Note that this assumption amounts to state that the firm has a access to noisy information in the sense it just observes a signal s that differs from d by a measurement error, η. 1. Given that the firm has information given by Ω = {1, s}, formulate the best prediction for d as E(d|Ω) (Hint: This is just a linear regression)
Exercise 5: Linear R.E. Model * Consider the following linear rational expectation model yt = aEt yt+1 + bxt where |a| < 1, |b| < ∞ and xt is a stationary stochastic process. 1. Show that a bounded solution of the problem takes the form yt = b
∞ X
ai Et xt+i
i=0
2. Solve the model for each of the following stochastic process xt = ρxt−1 + εt
(6)
xt = εt + θ1 εt−1 + θ2 εt−2
(7)
xt = ρxt−1 + εt + θεt−1
(8)
where the properties of each process are the same as in problem 1.
Exercise 6: Linear R.E. Model ** Consider the following model, yt = aEt−1 yt+1 + bxt xt = ρxt−1 + εt 2
where |a| < 1, |b| < ∞, |ρ| < 1 and εt ; iid(0, σ 2 ). Solve the model.
Exercise 7: The Permanent Income Model ***
The aim of this exercise is to familiarize
yourself with the permanent income model and its basic mechanisms. We consider an infinitely– lived household who decides on her consumption by maximizing her intertemporal expected utility subject to her budget constraint. The problem of an agent is then given by "∞ # X 1 2 s β − (c − ct+s ) max Et 2 {ct+s ,at+s+1 }∞ s=0 s=0
subject to at+s+1 = (1 + r)at+s + yt+s − ct+s ∀s 6 0 ∞ X a0 given and E0 β s a2t+s < ∞ s=0
where yt denotes the income of the agent. yt is assumed to be a stochastic process which will be defined later. The information set of the agent consists of all the past history of shocks and asset accumulation. The utility is only increasing for ct 6 c (but it is strictly concave for any value of consumption). We will therefore have to make the assumption that c is large enough to insure this assumption is not violated. In the sequel we will assume that β = (1 + r)−1 1. Show that at the optimum of the agent, consumption follows a martingale — i.e. ct = Et ct+1
2. Show that the solution for consumption takes the form #! "X s ∞ 1 1 ct = r at + Et yt+s 1+r 1+r s=0
Hint: Use the budget constraint and the martingale property. 3. Assume that the income process takes the form yt = ρyt−1 + (1 − ρ)y + εt where |ρ| 6 1 and εt ; N (0, σy2 ). Show that in this case, consumption simplifies to
ct = c + rat + where c ≡
1−ρ 1+r−ρ y.
3
r yt 1+r−ρ
4. Deduce from the previous question that the evolution of assets in the economy is given by at+1 = at +
1−ρ yt − c 1+r−ρ
5. Assume that yt is purely iid (ρ = 0). What is the dynamics of consumption and assets after a positive income shock? Explain. 6. Assume that yt is a pure random walk (ρ = 1).1 What is the dynamics of consumption and assets after a positive income shock? Explain.
1
In this case, we have to assume that y0 is large enough and σ2 is small enough such that we do not end up with a negative income.
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