— MONETARY THEORY — Homework #3: Imperfect Information
The number of * indicates the level of difficulty of the problem.
Exercise 1: Solving Models with Lagged Expectations * The aim of this exercise is to show you different ways of solving linear RE models featuring different expectations. We consider the case of an economy in which the equilibrium aggregate price, pt , is a function of the price as expected in period t − 1, Et−1 pt , and a shock xt pt = γEt−1 pt + ϕxt
1. Assume first that xt is a stationary MA(1) process of the form xt = εt + θεt−1 Use an undetermined coefficient method to find the solution of the model. 2. Assume now that xt is a stationary AR(1) process of the form xt = ρxt−1 + εt Assume that the solution of the model is of the form pt = α0 xt + α1 xt−1 Use the undetermined coefficient method to find the solution of the model in this case. 3. Show that assuming a solution of the form pt = β0 εt + β1 xt−1 yield the same solution. (the two solutions are perfectly equivalent) Why? 4. The aim of this question is to approach the problem from another perspective. We will adopt a solution method that relies on expectation errors. In this question, we will assume that xt follows the AR(1) defined in Question 2. 1
a) Show that Et−1 pt =
ϕ 1−γ Et−1 xt
b) Define pbt = pt − Et−1 pt . Show that pbt = ϕt . c) Deduce from the last two questions that the solution of the model corresponds to the one you obtained in Question 3.
Exercise 2: A model with decreasing returns *** We consider an economy populated with an infinite number of identical agents, whose preferences are represented by the following utility function "∞ Z X Et β τ log(Ct+τ ) − 0
τ =0
1
h1+ν i,t+τ 1+ν
!# di
The household is composed of a continuum of individuals, i ∈ (0, 1), who each supply labor, hi,t , on the labor market and get the nominal wage, Wi,t , in return and receives any nominal profits, Πt , realized by firms in the period. The household also hold nominal bonds carried from the previous period that yield a certain gross return Rt−1 .1 These revenus are then used to purchase a consumption bundle, Ct , at price Pt , to get bonds as means to transfer wealth to the next period. The consumption bundle takes the form Z Ct = 0
1
θ−1 θ
Ci,t
θ θ−1 di
On the production side, each firm i produces a specific good by means of labor, according to a decreasing returns to scale technology of the form Yi,t = At hαi,t where α ∈ (0, 1). Since each firm is the sole producer of each good i, it has local monopoly power and is therefore a price setter. At is an exogenous stochastic process that will be specified later on. Finally monetary authorities are assumed to simply set the level of nominal output Mt = Pt Yt where Mt is an exogenous stochastic process whose properties will be given later. In equilibrium, all of the output is used for consumption. 1. Write the optimization program of the household 2. Find the first order conditions of the preceding program 1
Remember that the gross return is 1 + net interest rate.
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3. Assume that firms all have access to full information. Find the optimal behavior of the firms and compute the general equilibrium of this economy. 4. Show that a log–linear version of the general equilibrium is given by νb hi,t + ybt = w bi,t − pbt ybi,t = b at + αb hi,t ybi,t = −θ(b pi,t − pbt ) + ybt α + (1 − α)θ 1−α b at w bi,t − pbt = (b pi,t − pbt ) − ybt + α α α Z 1 pbt = pbi,t di 0
m b t = pbt + ybt bt = Et (b R pt+1 + ybt+1 ) − pbt − ybt
5. Find the solution process for ybt and pbt . 6. What is the impact of a productivity shock on output and prices? What is the effect of an increase in money on output and prices. 7. We now assume that not all firms have access to full information, such that each firm rely on a specific information set. We denote the expectation operator of a specific firm i by Ei,t . A fraction λ ∈ (0, 1) has access to past information only (e.g. Ei,t = Et−1 ), the complementary fraction has access to full information (e.g. Ei,t = Et ) . a) Show that in this case, the log–linear version of the price setting behavior of a firm takes the form pbi,t = Ei,t pbt +
1−α b at α (w bi,t − pbt ) + ybt − α + (1 − α)θ α + (1 − α)θ α + (1 − α)θ
b) Show that the log–linear version of the equilibrium aggregate price index, pbt =
R1 0
takes the form pbt = λ[(1 − γ)Et−1 pbt + γEt−1 (m bt −b at )] + (1 − λ)[(1 − γ)b pt + γ(m bt −b at )] where γ is a constant you will determine.
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pbi,t di,
c) Assume that b at = b at−1 + εa,t m bt = m b t−1 + εm,t
Show that the dynamics of prices is given by pbt = m b t−1 − b at−1 + ξ(εm,t − εa,t ) where ξ is a constant you will determine. d) Deduce that inflation, π bt = pbt − pbt−1 can be expressed as π bt = ξ(εm,t − εa,t ) + (1 − ξ)(εm,t−1 − εa,t−1 )
8. What is the impact of a productivity shock on output, prices and inflation? What is the effect of an increase in money on output, prices and inflation? 9. How persistent are the effects? Why?
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