Lecture Notes on Optimal Monetary Policy | EC 861, Papers of Economics

Download Lecture Notes on Optimal Monetary Policy | EC 861 and more Papers Economics in PDF only on Docsity! Chapter 6 Optimal Monetary Policy Models of monetary economics typically involve as much as four distortions. 1. The Þrst distortion derives from the agents’ desire to hold money, given the transaction services that money provides. Since the private cost of holding money is R whereas the social cost of producing it is 0 efficiency would require that the two are set equal by having R = 0. Since the real rate is RR = R−π, this requires inßation to equal minus the real interest rate, that is a steady decline in the price level. 2. The second distortion derives from imperfect competition in the goods market. When α = 1 (no capital), from Y = A η ρ+η−1 X 1 ρ+η−1 < Y ∗ = A η ρ+η−1 we can see that output will be inefficiently low whenever X > 1. To correct this distortion, inßation should be permanently above zero (from the Phillips curve), and this creates in itself a trade-off with the objective in (1) 3. (related to nominal rigidity): Þrms’ inability to adjust prices at any point in time creates a dynamic markup distortion. Markups will ßuctuate over time around their constant frictionless level. 4. (related to nominal rigidity): even in absence of average inßation (π = 0), the lack of synchronization in price adjustments will imply the coexistence of different prices for goods that enter symmetrically agents’ utility functions and which have a one-to-one marginal rate of transformation. This is a static markup distortion. Modern models normally deal with (3) and (4). Both distortions can be corrected at once by a zero inßation policy. It is in this context, and starting from the utility function of the representative individual producer, that Rotemberg and Woodford (1997, NBER Macro Annual) show that the period utility loss resulting from deviations from the X = 1 allocation can by approximated by a quadratic equation of the form: Lt = ucC 2 µ σ2π + λ ε ρ+ η − 1 1 + ε (η − 1)σ 2 x ¶ where λ = θ(1−θ)(1−βθ) is the slope of the Phillips curve 1 1The values in the Þrst version of the paper for the relative weights were slightly different. The reason is explained in footnote 23 of the Woodford book, page 400. 57 58 CHAPTER 6. OPTIMAL MONETARY POLICY 6.1 Deriving the welfare function Woodford derives a quadratic loss function that represents a second-order Taylor series approximation to the level of expected utility of the representative household in the rational expectations equilibrium associated with a given policy. I provide a sketch of the derivation. Appendix 11.6 in Walsh provides a good derivation, as well as Woodford’s book. Remember that the period utility function for the representative individual is, ignoring the real balances term and remembering that in equilibrium Y = C, Y 1−ρt 1− ρ − 1 η (Lt) η in a steady state without distortions, when α = 1: Y = AL (production function) wY −ρ = Lη−1 (labor supply) A = w (labor demand) Y 1−ρ = Lη (equilibrium) Next take a Taylor approximation 6.1.1 LHS u (Y ) = Y 1−ρ 1− ρ + Y −ρ (Yt − Y )− ρY −ρ−1 2 (Yt − Y )2 deÞne now eY = Yt − Y YbYt = log (Yt/Y ) = log(Y/Y ) f(x) + 1 f 0(x) Yt − Y Y x−x − 1 2 f 00(x)/2 µ Yt − Y Y ¶2 so that eY = Yt − Y Y = bY + 1 2 bY 2 and (drop out constant terms): u (Y ) = Y 1−ρ µbYt + 1− ρ 2 bY 2t ¶ (a) 6.1.2 RHS Remember that L is the integral of labor supplied by all households in the economy Z y (z) dz, and each household on the segment produces good Y (z) = y for notational simplicity. Taylor series of v (l) v (l) = lη ³blt + η 2 bl2t´ integrate wrt z across all households, use Y 1−ρ = lηZ v (l) dz = Y 1−ρ ³ Ebyt + η 2 Eby2t ´ = Y 1−ρ ³Ebyt + η2 ³(Ebyt)2 + V AR (byt)´´ Use: Yt = ·Z 1 0 Yt (z) ε−1 ε dz ¸ ε ε−1 bYt = E byt + ε− 1 ε 1 2 V AR (byt) 6.3. OPTIMAL POLICY UNDER DISCRETION 61 6.3 Optimal policy under discretion Under discretion, the central bank expects itself to reoptimize at each successive date, and is unable to commit itself to future paths for inßation and the output gap. That makes the problem rather easy to solve. Easy way to solve the problem (since we choose x π and R) is Þrst to solve under AS constraint only and then to work out optimal R implied by the aggregate demand curve. max π,x −1 2 ¡ π2t + αx 2 t ¢ s.t. πt = λxt + βEtπt+1 + ut where ft ≡ βEtπt+1 + ut is taken as given. The problem then becomes: max π −1 2 à π2t + α µ πt − ft λ ¶2! yielding (since 1λ (πt − ft) = xt) πt + α λ2 (πt − ft) = 0⇒ xbt = − λ α πbt where the superscript b indicates that this is the solution under discretion. To solve the problem now combine this equilibrium condition with the AS curve πt = λxt + βEtπt+1 + ut and impose that expectations are rational. You will get: πt = αβ α+ λ2 Etπt+1 + α α+ λ2 ut ≡ cπt+1 + dut This equation can be solved forward to obtain (under ut+1 = ρut): πbt = c (cπt+2 + dut+1) + dut = = c (cπt+2 + dρut) + dut = = d ¡ ut + cρut + c 2ρ2ut + ... ¢ = d 1− cρut = πbt = α α+λ2 1− αβρ α+λ2 ut = α α (1− βρ) + λ2ut xbt = − λ α (1− βρ) + λ2ut as πt = wut, Eπt+1 = wρut = ρπt. Go back to IS, solved for Rt: φRt = Etxt+1 + φEtπt+1 − xt + gt φRt = −Et λ α πt+1 + φEtπt+1 + λ α πt + gt φRt = µ −λ α + φ+ λ αρ ¶ Etπt+1 + gt Rt = µ 1 + λ (1− ρ) αρφ ¶ Etπt+1 + 1 φ gt Remark 8 Optimal policy responds more than one for one to changes in expected inßation. 62 CHAPTER 6. OPTIMAL MONETARY POLICY 6.3.1 Monetary policy trade-offs under discretion We have found that: −α λ xt = πt = α α (1− βρ) + λ2ut σ2π = µ α α (1− βρ) + λ2 ¶2 σ2u σ2x = µ λ α (1− βρ) + λ2 ¶2 σ2u σ2x = λ2 α2 σ2π Remark 9 α deÞnes the policymaker preferences. For given value of λ, the last equation describes an inverse relationship (Taylor curve) between the two policy objectives. 6.4 Commitment Literature often divided into two strands. The 1980s literature assumes that the output gap goal is to push output permanently above its natural rate. In the Nash equilibrium, this generates inßation with little output gains. The modern literature considers other issues, i.e. optimal rules. 6.4.1 The classic Inßationary Bias Problem Assume demand and supply are given by: xt −Etxt+1 + φ [Rt −Etπt+1]− gt = 0 πt − λxt − βEtπt+1 − ut = 0 Problem is: maxW = −1 2 Et " ∞X t=0 βtLt # where: Lt = π 2 t + α (xt − k)2 k > 0 reßects the presence of distortions so that socially efficient output exceeds natural level. In this case a discretionary central bank faces the following problem: max π −1 2 à π2t + α µ πt − ft λ − k ¶2! yielding: πt + α λ µ πt − ft λ − k ¶ = 0 πt + α λ (xt − k) = 0 ⇒ xt = −λ α πt + k To solve the problem now combine this equilibrium condition with the AS curve πt = λxt + βEtπt+1 + ut = λα α+ λ2 k + βπt+1 + ut 6.4. COMMITMENT 63 and impose that expectations are rational. You will get: πkt = π b t + αλ α (1− β) + λ2 k hence inßation equals inßation under the baseline case plus a term related to k. Solving for xt and remem- bering that xbt = − λα(1−βρ)+λ2ut: xkt = − λ α µ πbt + αλ α (1− β) + λ2 k ¶ + k = −λ α πbt − λ2 α (1− β) + λ2 k + k = = xbt + α (1− β) α (1− β) + λ2 k Remark 10 Under discretionary policy, Inßation is higher and output is slightly above natural level. How- ever the gain disappears as β → 1 6.4.2 The gains from commitment (even) when k = 0 : the optimum within simple rules Here we return to our earlier model but we take into account the possibility that central bank actions might affect private agent expectations. However the solution to this simple problem is not as simple as it might look like.... Consider a rule for the target x of the following form: xct = −ωut this corresponds to the rule under discretion xbt whenever the central bank chooses ω = λ α(1−βρ)+λ2 . Under such a rule inßation is: πct = ut + kx c t + βEtπt+1 = ut (1− λω) + βEtπt+1 = 1− λω 1− βρ ut or differently: πct = 1 1− βρ (ut + λx c t) comparing this with the solution to the discretionary case: πbt = λx b t + βEtπt+1 + ut when the central bank is unable to manipulate expectations, the scale of trade-off is different. Reducing xt by 1% reduces πbt by λ%, rather than λ 1−βρ% as in the commitment case. What is the optimal value of the feedback parameter? Since both πc and xc are multiples of ut, one can 66 CHAPTER 6. OPTIMAL MONETARY POLICY Set up the Lagrangian: L = π2t + µxxt 2 + µrr 2 t + βπ2t+1 + βµxxt+1 2 + βµrr 2 t+1 + −ξ1t (πt − λxt − βπt+1 − ut)− ξ2t (xt − xt+1 + φRt − φπt+1 − gt) −β ¡ξ1t+1 (πt+1 − λxt+1 − βπt+2 − ut+1)− βξ2t+1 (xt+1 − xt+2 + φRt+1 − φπt+2 − gt+1)¢ The central bank minimizes over the whole time period, choosing πt+1 and xt+1 and rt+1. Taking the Þrst-order conditions and scrolling them one period backward yields: πt − φβ−1ξ1t−1 + ξ2t − ξ2t−1 = 0 µxxt + ξ1t − β−1ξ1t−1 − kξ2t = 0 µrrt + φξ1t = 0 together with: πt − kxt − βπt+1 = 0 xt −Etxt+1 + φrt − φπt+1 − gt = 0 This dynamic system of 5 equations in 5 unknowns can be solved for R as a function of existing endogenous variables only. Giannoni and Woodford call the deriving rule a “robustly optimal instrument rule". The idea is that you can play with the Þrst three equations and solve for Rt as a function of the existing endogenous variables only. Rt = µ 1 + kφ β ¶ Rt−1 + β−1∆Rt−1 + kφ µr πt + φµx µr ∆xt A bunch of comments: • the optimal interest rate rule is a function only of the variables in the loss function • the rule requires the interest rate to be positively related to ßuctuations in current inßation, in changes of the output gap, and in lagged interest rates • the rule is super-inertial, in the sense that it requires that the interest rate to vary by more than one for one to past ßuctuations of the interest rate. You will notice that, as µr approaches zero, we are back into the general solution under commitment result. In fact kπt = µx (xt−1 − xt) Pt = −µx k xt The outcome under timeless precommitment is shown in the Figure below (for λr arbitrarily low, Þle giannoni.m). Despite the fact that the cost shock has no persistence, the output gap displays positive serial correlation. By keeping output below potential for several periods into the future after the negative shock, the central bank is able to lower expectations of future inßation. A fall in Etπt+1 at the time of the shock improves the trade-off between inßation and output gap stabilization faced by the central bank. This is unlike the case we obtain under discretion, where we Þnd that πbt = α α (1− βρ) + λ2ut xbt = − λ α (1− βρ) + λ2ut 6.5. WHAT ABOUT THE TAYLOR RULES? 67 hence in that case the variables inherit the persistence properties of the cost-push shock, and there is no inertia in the variables following a shock. 0 2 4 6 -0.01 -0.005 0 0.005 0.01 D EM AN D S H O C K x 0 2 4 6 -0.2 0 0.2 0.4 0.6 0.8 1 r 0 2 4 6 -0.01 -0.005 0 0.005 0.01 π 0 2 4 6 0 0.2 0.4 0.6 0.8 1 u e 0 2 4 6 -0.5 -0.4 -0.3 -0.2 -0.1 0 IN FL AT IO N S H O C K years 0 2 4 6 -0.1 -0.08 -0.06 -0.04 -0.02 0 years 0 2 4 6 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 years 0 2 4 6 0 0.2 0.4 0.6 0.8 1 years OPTIMAL RESPONSES TO PURELY TRANSITORY SHOCKS, (TIMELESS) PRECOMMITMENT Responses to purely transitory shocks, timeless precommitment. 6.5 What about the Taylor rules? Suppose we compare: Rt = µ 1 + kφ β ¶ Rt−1 + β−1∆Rt−1 + kφ µr πt + φµx µr ∆xt (1) Rt = φxPt + φxxt (2) Rt = φxπt + φxxt (3) which policy will yield higher welfare? To begin with, we know that 1 dominates 2 and 3 since it is, by construction, the optimal policy. What about 2 (Wicksellian rule, after Wicksell, 1907) versus 3 (Taylor)? One would believe that 3 is better than 2, since under 2 the policymaker responds to an inßationary shock by bringing about deßation in future periods, hence lowering welfare. However, as shown for instance in Giannoni (2000), “Commitment to an optimal Wicksellian policy allows the policymaker to achieve a response of endogenous variables that is closer to the optimal plan than is the case with the optimal Taylor rule. One particularity of the equilibrium resulting from a Wicksellian policy is that the price level is stationary. This feature turns out to affect the response of endogenous variables in particular when shocks are very persistent [...] [T]he mere expectation of future deßation [..] under the optimal plan and the optimal non-inertial plan already depresses inßation when the shock hits the economy, and is expected to keep inßation below steady-state for several periods. In contrast, under optimal Wicksellian policy, both inßation and the price level rise strongly on impact, but they are expected to return progressively to their initial steady-state" 6.6 A digression on the supply shock The source of the output-inßation variance trade-off for central bank is the “supply” shock. Were there only demand type shocks, there would be no trade-off. This is summarized in Erceg, Henderson and Levin (2000), 68 CHAPTER 6. OPTIMAL MONETARY POLICY Proposition 2. Proposition 2: With staggered price contracts and completely ßexible wages, monetary policy can com- pletely stabilize price inßation and the output gap, thereby attaining the Pareto-optimal social welfare level. However, a price inßation / output gap variance trade-off arises endogenously in the model above with staggered wage and price setting. When both prices and wages are staggered, it is impossible for monetary policy to attain the Pareto optimum except in the special cases where either wages or prices are completely ßexible. Nominal wage inßation and price inßation would remain constant only if the aggregate real wage rate were continuously at its Pareto-optimal level. Such an outcome is impossible because the Pareto-optimal real wage moves in response to various shocks, whereas the actual real wage could never change in the absence of nominal wage or price adjustment. Given that the Pareto optimum is infeasible, the monetary policymaker faces trade-offs in stabilizing wage inßation, price inßation, and the output gap. (B) With staggered wage contracts and completely ßexible prices, monetary policy can completely stabilize wage inßation and the output gap, thereby attaining the Pareto-optimal social welfare level.