Eco 504, Part 1, Spring 2006 504_F1s_S06.tex Lars Svensson 3/16/06 Eco 504, Macroeconomic Theory II Final exam, Part 1, Monetary Theory and Policy, with Solutions Answer all questions. You have 120 minutes to complete the exam, and the questions are worth a total of 120 points. Please allocate your time accordingly. Short-answer questions (10 points each). Give short answers to these questions (not longer than half a page). Note any important qualifications that you consider necessary. 1. Let Q t,τ (s τ t ) denote the stochastic nominal discount factor between period t and period τ t for the history s τ t. How are the stochastic real discount factor, q t,τ (s τ t ), andthe Arrow-Debreu price, p t,τ (s τ t ), related to the stochastic nominal discount factor? 2. Explain intuitively why increased inflation variance might reduce the inflation risk premium. 3. Define predetermined variables, forward-looking variables, instruments, and target variables. Does inflation targeting mean that inflation is the only target variable? 4. Under optimization under discretion, does the central bank treat current expectations of future variables as exogenous? 5. Must the central bank be independent for inflation targeting to work in emerging market countries? Answers 1. We have q t,τ (s τ t ) = Q t,τ (s τ t ) P τ(s τ t ) P t (s t ), p t,τ (s τ t ) = Q t,τ (s τ t ) P τ(s τ t ) P t (s t ) π t,τ (s τ t ), where P τ (s τ t ) is the price level in period τ conditional on history s τ t and π t,τ (s τ t ) is the probability of history s τ t conditional on event s t. See Lecture Notes 2 p. 28 29. 2. Under the assumption of joint lognormality of the price level and the real discount factor, we derived the following expression for the inflation risk premium: 1 2 Var[π t+1]+cov[logq t,t+1,π t+1 ], where increased variance of inflation (π t+1 log(p t+1 /P t )) for unchanged covariance with the log real discount factor (q t,t+1 ), reduces the inflation risk premium. An intuitive explanation for the fact that the inflation risk premium is decreasing in the c 2006 Lars E.O. Svensson. This document may be reproduced for educational and research purposes, as long as the copies contain this notice and are retained for personal use or distributed free.
variance of inflationis the following: Consider a nominal bond in period t that pays one unit of money in period t +1. The real return on this bond is proportional to P t /P t+1 e π t+1. Note that e π t+1 is a convex function of π t+1. By Jensen s inequality, a mean-preserving spread of π t+1 will, everything else equal, increase the mean real return on the nominal bond and make it more attractive relative to an indexed bond with a constant real return. 3. See Lecture Notes 6 for these definitions. Inflation targeting includes flexible inflation targeting, where also the output gap or other variables representing the real economy enters. 4. Depends. Under discretionary optimization in the linear-quadratic model we have used in class, in the optimization on period t the central bank assumes that expectations of future forward-looking variables, x t+1,aregivenby x t+1 x t+1 t = GX t+1, where the matrix G is taken as known in period t, andx t+1 is the vector of predetermined variables. The matrix G is the fixpoint (G, V ) of a mapping from matrices (G t+1,v t+1 )to (G t,v t ), as explained in class. Only if GX t+1 consists only of exogenous variables (which is the case in the simple New Keynesian model) does this mean taking expectations as exogenous. If GX t+1 includes endogenous predetermined variables (wich is the case in more general models), the expectations are taken as endogenous. 5. No. Chapter IV in IMF s WEO of September 2005 shows that inflation targeting has worked well in several emerging market countries where the central bank is not very independent. However, an independent central bank makes it more likely that inflation targeting will work well. Problems (70 points). 1. (50 points) Let P t denote the optimal price set in period t by a typical firm in a Calvo model and suppose that it satisfies the first-order condition Ã! Ã! θt+τ X Pt Pt τ=0 ατ q t,t+τ μ P t+τ S t+τ Y t+τ =0. t+τ (a) (5 points) For this first-order condition, what is the implicit assumption about any indexing of prices by firms that are not allowed to set their price optimally in a given period? What does S t+τ denote? (b) (5 points) Consider a steady state with q t,t+τ = δ τ, Pt = P, P t+τ = P, μ t+τ = μ, S t+τ = S, Y t+τ = Ȳ, θ t+τ = θ. Howare S, μ, and θ related? (c) (15 points) Log-linearize the first-order condition around this steady state. (d) (10 points) Suppose that the consumer price index (CPI) in period t satisfies P t = h αpt 1 1 θt +(1 α) P 1 θt t P t+τ i 1 1 θ t. Log-linearize the CPI around the above steady state. (e) (15 points) Combine the log-linearized first-order condition and CPI and derive a Phillips curve in terms of π t log(p t /P t 1 ), ŝ t log(s t / S), andˆμ t log(μ t / μ). 2
2. (20 points) Consider the following model, Solutions π t = δπ t+1 t + κx t + ε t, x t = x t+1 t σ(i t π t+1 t r η t ), where π t is inflation, x t is the output gap, i t is the instrument rate, 0 <δ<1, κ>0, σ>0, r >0, andε t and η t are i.i.d. with mean zero. Suppose that the central bank has the intertemporal loss function, where the period loss is X τ=0 (1 δ)δτ L t+τ, L t = 1 2 [π2 t + λ(x t x ) 2 ], where x > 0. Consider steady states with constant inflation, output gap, and instrument rate. What is the optimal steady-state inflation level? 1. (a) The implicit assumption about the indexation is that prices are held fixed when they are not allowed to be set optimally. Otherwise P t would be replaced by P t Π τ, for instance, as in the derivation in class. S t+τ is the real marginal cost in period t + τ. (b) We have 1= μ S. Furthermore, μ t is the gross markup and θ t the price elasticity of demand, which satisfy μ t = θ t /(θ t 1) (where θ t > 1). Therefore, μ = θ/( θ 1). (c) Since the first parenthesis in the first-order condition is zero in the steady state, we need not loglinearize with respect to q t,t+τ, Y t+τ, or θ t+τ. Therefore, loglinearization simply gives rise to X ³ p τ=0 ατ δ τ t p t+τ ˆμ t+τ Ŝt+τ =0. We can write this as (d) In the steady state we have p t =(1 αδ) X τ=0 ατ δ τ ³ p t+τ +ˆμ t+τ + Ŝt+τ P = hαp 1 θ +(1 α)p 1 θ i 1 1 θ.. (0.1) Since the right side in steady state does not depend on θ, the partial derivative with respect to θ is zero, and we need not loglinearize with respect to θ t. Then loglinearization results in p t = 1 1 θ [α(1 θ)p t 1 +(1 α)(1 θ) p t ]=αp t 1 +(1 α) p t. (0.2) (e) Substituting (0.1) for p t in (0.2), we get X ³ p t αp t 1 = (1 α)(1 αδ) τ=0 ατ δ τ p t+τ +ˆμ t+τ + Ŝ t+τ = (1 α)(1 αδ) ³ p t +ˆμ t + Ŝt X ³ +(1 α)(1 αδ)αδ τ=0 ατ δ τ p t+1+τ +ˆμ t+1+τ + Ŝt+1+τ ³ = (1 α)(1 αδ) p t +ˆμ t + Ŝt + αδ(p t+1 t αp t ) 3
We can rewrite this as ³ α(p t p t 1 ) = (1 α)(1 αδ) ˆμ t + Ŝt + αδ(p t+1 t p t ), (1 α)(1 αδ) ³ p t p t 1 = ˆμ α t + Ŝt + δ(p t+1 t p t ), where π t p t p t 1. π t = δπ t+1 t + (1 α)(1 αδ) α ³Ŝt +ˆμ t, 2. This problem can be interpreted and solved in two different ways, with two different answers. (The first is what I had in mind, but both are arguably correct, given some leeway in interpreting the question): (i) Consider only steady-state equilibria, π t = π, x t = x, for all t. The steady-state Phillips curve is π = δπ + κx, κ π = x αx. (0.3) 1 δ The steady-state Phillips curve has a positive slope, α κ/(1 δ) > 0, given that δ<1 and the implicit assumption in this particular New Keynesian Phillips curve that prices are held constant when they are not set optimally. Consider the period loss in steady state, L = 1 2 [π2 + λ(x x ) 2 ]. Minimize this with respect to x subject to (0.3), L x = πα + λ(x x )=0, απ = λ(x x )= λ( π α x ) Hence, the optimal steady-state inflation and output gap are given by π = λα λ + α 2 x > 0, x = λ λ + α 2 x > 0. (ii) Assume that the central bank optimizes under commitment and consider the corresponding (asymptotic) steady state. Optimization under commitment results in the first-order condition (for periods after the period of commitment, or under the timeless perspective), π t + λ κ (x t x t 1 )=0. Consider the unconditional mean E[π t + λ κ (x t x t 1 )] = E[π t ]=0. Hence, the (asymptotic) optimal steady-state inflation and output gap are given by π =E[π t ]=0, x =E[x t ]=0. 4
The difference between these two cases is that the first considers only actual steady states with constant inflation and output gap and then finds the steady state that minimizes the steady-state loss. The second case does not initially restrict equilibria to be steady states, assumes commitment, and looks at the asymptotic steady state under commitment. 5