#### Transcript Tutorial Slides

Tutorial to the Monetary Policy Lecture May 24-28, 2004 Dr. Julian von Landesberger HVB Group Economics [email protected] [email protected] Julian von Landesberger 13.04.2015 1 Monetary policy problems Design: The policy design problem is to characterize how the interest rate should adjust to the current state of the economy. Instrument: The instrument problem of monetary policy arises because of the need to specify how the central bank will conduct its open market operations. Intermediate target: The intermediate target problem is the choice of a variable, usually a readily observable financial quantity (or price) that the central bank will treat, for purposes of some interim-run time horizon, as if it were the target of monetary policy. Julian von Landesberger 13.04.2015 2 The structure of the monetary policy problem An important complication of the policy design problem is that the private sector behavior depends on the current and expected course of monetary policy. Therefore credibility is crucial for monetary policy. A key aspect is that wage and price setting today may depend upon beliefs about where prices are headed in the future, which in turn depends on the future course of monetary policy. Are there gains from enhancing credibility either by formal commitment to a policy rule or by introducing some kind of institutional arrangement ? Julian von Landesberger 13.04.2015 3 Discretion In a discretionary regime the central bank can “print” more money and create more inflation than people expect. Why would it do this? •Unanticipated monetary expansions lead to increases in real economic activity. •The natural rate may be viewed as excessive. This can occur through distortions from income taxation, unemployment compensation, which reduce the privately-chosen level of labor and production. •The policy maker can value inflationary finance as a method of raising revenues. Julian von Landesberger 13.04.2015 4 Discretion - Setup The policymaker trades off benefits and costs in each period. The loss function is given by: lt= (a/2) pt2 - bt(pt-pte) The policymaker controls a monetary instrument, which enables him to select the rate of inflation pt in each period. At this point he does not know bt. Similarly people form their expectations pte of the policymakers choice without knowing the parameter. The decision has to be taken every period until infinity. Julian von Landesberger 13.04.2015 5 Discretion - Setup The policymaker treats the current inflationary expectations pte and all future expectations as given when choosing current inflation! pt is chosen to minimize the expected costs for the current period Elt while treating all future costs as fixed. apt - bt= 0 Take expectations... pt = b/a pte = b/a lt = (1/2)(b)2/a Compute the loss... Julian von Landesberger 13.04.2015 6 Commitment People understand the policymaker’s incentives, therefore the surprises - and the benefits - can not arise systematically in equilibrium. Enforced commitment on monetary policy behavior, as embodied in monetary or price rules eliminate the potential for ex post surprises. A commitment to fight inflation in the future can improve the current output/inflation trade-off that a central bank faces. Julian von Landesberger 13.04.2015 7 Commitment - Setup Suppose the policymaker can commit himself in advance to a rule determining inflation. The policymaker conditions the inflation rate on variables that are known also to the private agents. In fact, the policymaker chooses pt and pte together subject to the condition that pt = pte. The inflation surprise term in the loss function is therefore zero by construction. Given the cost term (a/2) pt2 the best inflation rate for the central bank to target is zero. pt* = 0 lRt = 0 Julian von Landesberger 13.04.2015 8 The incentive to cheat If people expect pt = 0, then the policymaker has an incentive to cheat in order to secure some benefits from the inflation surprise. It reflects the distortions that make inflation shocks have a benefit for the policymaker. What does the policymaker gain from cheating: pt = b/a lCt = -(1/2)b2/a The temptation to cheat is E(lR-lC) = (1/2)b2/a Julian von Landesberger 13.04.2015 9 Alternative mechanisms to enhance credibility The costs under the commitment are lower than those under discretion. Without commitment, pt> 0 without benefits resulting. However, no major central bank makes any type of binding commitment over the future course of its monetary policy. What solutions are found in the literature? First-best equilibrium - remove the distortions. Second-best equilibrium - commit to an optimal rule. Third-best equilibrium - delegate monetary policy to a conservative central banker Fourth-best equilibrium - discretionary policy. Julian von Landesberger 13.04.2015 10 Expectations augmented Phillips curve If price setting today depends on beliefs about the future economic conditions, a monetary authority that is able to signal a clear commitment to controlling inflation can improve the short-run output/inflation trade-off. Clarida/Gali/Gertler (1999) argue that this improvement arises even the central bank does not have an incentive to push output above potential. A central bank that commits to a rule is able to credibly signal that it will sustain over time an aggressive response to a supply shock. Julian von Landesberger 13.04.2015 11 Expectations augmented Phillips curve The extra kick in the case of commitment to a policy rule is due to the impact of the rule on the expectations of the future course of the output gap. Since inflation depends on the future evolution of excess demand, commitment to the rule leads to a bigger fall in inflation per unit of output reduction today relative to discretion. Julian von Landesberger 13.04.2015 12 Taylor overlapping wage model Overlapping nominal wage contracts. In period t, set (log) nominal wage wt for two periods. Average (log) wage wt 1 pt wt wt wt 1 2 Set wages according to expected average nominal wages 1 wt wt 1 wt 1 t xt 2 1 pt wt wt 1 2 1 1 1 pt wt 1 wt 1 t xt wt 2 wt t 1 xt 1 2 2 2 Julian von Landesberger 13.04.2015 13 Taylor overlapping wage model 1 pt wt 1 wt 2 wt 1 t wt wt wt t 1 xt xt 1 4 2 1 pt 2 pt 1 2 pt 1 t 2 p pt t 1 xt xt 1 4 2 pt 1 pt 1 pt 1 t p pt t 1 xt xt 1 2 2 pt pt 1 pt 1 t pt xt xt 1 t p t p t 1 t xt xt 1 t t pt pt t 1 t Julian von Landesberger 13.04.2015 14 Rotemberg’s quadratic price-adjustment costs model pt optimal unrestricted (log) price, ~p price of particular firm, t pt (log) price level. 2 1 ~ 2 * ~ ~ min E p p c p p t t t t t 1 ~pt 0 2 0 First-order condition for ~p t ~ pt pt* c ~ pt ~ pt 1 c ~ pt 1 t ~ pt 0 1 ~ ~* ~ ~ p t p t 1 t pt pt c Optimal unrestricted price: pt* pt xt t Julian von Landesberger 13.04.2015 15 Rotemberg’s quadratic price-adjustment costs model ~ ~ All firms are identical, therefore pt pt and p t p t The Phillips curve can be derived as follows: t c c p t p t 1 t xt Julian von Landesberger 13.04.2015 16 Calvo’s staggered contracts model pt optimal unrestricted (log) price, ~p price of particular firm is adjusted in period t with prob q, t pt (log) price level. 2 1 * min (1 q) Et ~ pt pt ~ pt 2 0 First-order condition for ~pt (1 q) ~pt pt* t 0 0 (1 q) ~pt (1 q) pt* t 0 Julian von Landesberger 13.04.2015 17 Calvo’s staggered contracts model (1 q) 1 1 (1 q) ~ pt 1 1 q 1 q p *t t 0 * ~ pt 1 1 q pt 1 1 q 1 q 1 q p *t 1 t 0 ~ p 1 1 q p * 1 q ~ p t Julian von Landesberger 13.04.2015 t 1 t t 18 Calvo’s staggered contracts model pt* pt yt t ~ pt 1 1 q pt yt t 1 q ~ pt 1 t Aggregate price level (not all firms equal) pt q~ pt 1 q pt 1 q~ pt p t qpt 1 q~ pt 1 t p t 1 t qpt Julian von Landesberger 13.04.2015 19 Calvo’s staggered contracts model Insert into definition of the price level p t qpt 1 q1 1 q pt yt t 1 q p t 1 t qpt (1 q )p t q1 1 q yt t 1 q p t 1 t p t p t 1 t Julian von Landesberger 13.04.2015 1 (1 q) yt t q 1 q 20 The economy Say that the economy is described by: ut utN p t p te t ut is the unemployment rate, utN is the natural rate of unemployment, pt is the inflation rate and p te its expected value t is a supply shock, i.i.d. with mean 0 and variance s2 Agents have rational expectations. Julian von Landesberger 13.04.2015 21 Policymaker‘s objective The policymaker’s loss function is given by: Lt p ut u 2 t T 2 t 2 utT is the target unemployment rate which for now we take as bein below the natural rate:utT utN k The target for inflation is normalized to zero, without loss of genera Julian von Landesberger 13.04.2015 22 Question 1 and 2: 1. Given the material covered in the first part of the course,briefly motivate equation 1. Give reasons for why you may argue that k>0. 2. Assume the policymaker observes t when setting policy pt at each period, but rational agents don’t. What is the optimal discretionary policy rule? What are the equilibrium levels of unemployment and inflation? What is the value of the ex ante expected loss ELt given this policy? Julian von Landesberger 13.04.2015 23 Question 1 and 2: Equation (1) is a form of the expectations augmented Phillips Curve, of Friedman and Phelps. It can be justified from micro foundations with rational expectations, via a Lucas islands story. Reasons for a positive wedge between the target social optimum and natural rates of unemployment: - Distortions in the labor market (minimum wage, taxes, subsidies, etc) that push the equilibrium unemployment rate up. - Taxes in the economy, that generally reduce the level of output and employment. - Imperfect competition (e.g. monopoly) so the private production and employment levels are too low. Julian von Landesberger 13.04.2015 24 Question 1 and 2: 2. The discretionary Central Bank solves: 2 T 2 p t ut ut min pt 2 with F.O.C yielding the optimal policy rule: (1) p t ut utT (2) This is a simple form of a countercyclical policy. Replace ut from the Phillips curve into the expression and take expectations to obtain: p te k (3) Julian von Landesberger 13.04.2015 25 Question 1 and 2: Replace this, together with the Phillips Curve into equation (2), to obtain: p t p t k t k (4) which, after rearranging, gives the solution for inflation. Plugging this into the Phillips Curve (together with equation 3) you obtain unemployment: p t k t 1 1 N ut ut t 1 Julian von Landesberger 13.04.2015 26 (5) (6) Question 1 and 2: Plug these into the loss function to obtain the expected loss: 2 2 1 1 E k t t k 2 1 1 (7) Take the expectations taking into account that E(t)=0 and E(2t)=s2 to obtain the ex ante expected loss under discretion: (8) Julian von Landesberger 13.04.2015 1 2 L 1 k s 1 2 D 2 27 Question 3: Assume now the policymaker can commit ex ante to a linear state contingent rule: (3) pt = c + bt In ex ante designing the optimal policy to minimize expected loss ELt, what are the optimal parameters in this rule. Show this policy achieves a superior outcome (in terms of expected loss) to the discretionary one, and explain intuitively why. Julian von Landesberger 13.04.2015 28 Answer to 3: Replace the inflation rule into the ex ante expected loss, and take expectations to obtain: 1 2 2 2 2 2 2 2 c b 1 b s k 2 (9) Minimizing this with respect to b and c yields the optimal rule: c =0 and b = /(1+). Equilibrium unemployment and inflation are: pt t 1 (10) 1 N ut ut t 1 (11) Clearly, since this policy leads to the same unemployment but lower inflation than the discretionary one, it achieves a superior outcome. Julian von Landesberger 13.04.2015 29 Question 4: Say the Central Bank has limited commitment. It can only commit to a non-contingent rule of the form: (4) pt = c Solve for the optimal rule and compare its performance with that of discretion. Julian von Landesberger 13.04.2015 30 Question 4: Just set b =0 in (9) 1 2 c k 2 2 Minimize with respect to c to obtain the optimal policy rule: c = 0. Equilibrium unemployment and inflation are: pt = 0 ut utN t (13) Julian von Landesberger 13.04.2015 31 (12) Question 4: Note immediately that this policy leads to lower inflation than dis-cretion, but unemployment now fluctuates more in response to supply shocks than before (1 >1/1+ ). We expect to find therefore a trade-off between lower inflation and higher variance of unemployment. Plugging the equilibrium into the loss function, and taking expectations, you obtain the loss under a rule: L R Julian von Landesberger 13.04.2015 2 E t k 2 32 (14) Question 4: LR s 2 k2 (15) 2 The non-state-contingent 0-inflation rule is therefore preferrable to discretion if: LD >LR 1 k 2 s s 2 k 2 (17) 1 k 2 s 2 (18) 1 Julian von Landesberger 13.04.2015 33 Question 4: This will hold if: - the wedge between the natural rate and the target rate of unemploy-ment is large (k large) leading to a high inflation bias. - Supply shocks are not very variable. The first factor makes discretion very costly in terms of an increase in inflation, and the second makes the gains from being able to conduct countercyclical policy small, since supply shock don’t lead to a very large variability of unemployment. Discretion therefore becomes undesirable compared with a 0-inflation rule. Julian von Landesberger 13.04.2015 34 Question 5: Now assume that the Central Bank has no commitment ability and so solves every period the problem in question 2 (this will also be true for all the questions until the end of the problem set). Still, the Government has an ability to commit, and it can appoint a Central Banker from a pool of possible candidates. The candidates differ in the weight they give to unemployment vs. inflation variability *. Find the optimally appointed Central Banker’s *(you do not need to find a closed form solution). Show that 0 <*< . Julian von Landesberger 13.04.2015 35 Question 5: From question 2, we know the appointed Central Bank will follow the policy: * p t *k * t 1 ut utN 1 * t 1 (19) (20) Plug this into the loss function, noting crucially that the social loss function still involves and not *. Julian von Landesberger 13.04.2015 36 Question 5: Take expectations to obtain the Government’s ex ante expected social loss function: 2 2 * 1 * 1 E L E k k (21) t t * * 2 1 1 * 2 2 1 * 2 2 (22) E L k s 2 * 2 1 Minimize this with respect to * to obtain the F.O.C that implicitly defines the optimally appointed Central Banker: * 2 G * *k 2 s * 3 1 Julian von Landesberger 13.04.2015 37 (23) Question 5: To prove the claim in the text, note that: G (0) = - s2 <0 (24) G () = k2 > 0 (25) Moreover, differentiate G(.) with respect to its argument to obtain the slope of the function: G. k 2 Julian von Landesberger 13.04.2015 38 1 2* 3 1 * 4 s2 (26) Question 5: Note that in the interval [0, ] then G’0(.)>0, i.e. the function is monotonically increasing. But, if the function in the interval [0, ] is continuous, starts at a negative value, finishes at a positive value, and is monotonically increasing, by an application of Bolzano’s theorem, it must have a unique zero, in the interior of the interval. Thus there is a unique optimal *such that: 0 < *< ,as we wanted to show. Julian von Landesberger 13.04.2015 39 Question 6: Assume instead now that the Government cannot appoint a Central Banker with a different than the social level, but it can offer the Bank a contract. Specifically, it can impose a cost on the Bank for higher inflation (by e.g. negatively indexing the wage of the Banker to inflation, as is the case currently in New Zealand). The modified Central Bank’s Loss function is Lt +wpt. •What is the optimal w ? •Can society achieve the optimal outcome in question 3 now? • Why? Julian von Landesberger 13.04.2015 40 Question 6: The discretionary Central Bank now minimizes the loss function: 2 T 2 p t ut ut Lt wp t 2 (27) Follow exactly the same steps as in question 1, to obtain, respectively, the policy rule, the equilibrium inflation and p t ut utT w equilibrium unemployment: p t k 1 t w 1 N ut ut t 1 Julian von Landesberger 13.04.2015 41 (28) (29) (30) Question 6: So immediately note that by setting w =k, we reach the firstbest policy defined in question 3. Intuitively, note that the inflation bias problem is non-state contingent (it is k whatever t ), but the gains from discretion come from the ability to have state contingent policy. The Barro-Gordon proposal in question 4 for a fixed rule, removes the bias but also state contingency from policy. The Rogoff proposal for appointing a conservative Central Bank, by distorting the relative values of inflation and unemployment variability, reduces the inflation bias but also leads to too little discretionary policy (*/1+ </1+ ). Julian von Landesberger 13.04.2015 42 Question 6: The Walsh proposal for a Central Bank contract, goes to the heart of the problem: the penalty in inflation is linear in the Central Banks’ loss function. Therefore it imposes no extra cost of variable inflation (it is not squared), and so does not change the countercyclical statecontingent optimal policy. But it decreases the loss from the non-state-contingent, constant, inflation bias, and if adequately set can fully eliminate it. Julian von Landesberger 13.04.2015 43 Question 7: Alternatively, say the Government can give the Central Bank an ex-plicit inflation target p around which the variance of inflation must be minimised, together with the variance of unemployment from the target rate. (This is the currently the case in many countries and notably the United Kingdom). p Again derive the optimal previous question. Julian von Landesberger 13.04.2015 and discuss the relation to the 44 Question 7: This has been defended by Svensson (AER) 1997, in the context of a model only slightly different from this. The new loss function the Central Bank minimizes is: Lt 2 1 p t p 2 ut utT 2 (31) But, just expand the quadratic to see this is just: 1 2 T 2 2 Lt p t ut ut 2p p t p 2 Julian von Landesberger 13.04.2015 45 (32) Question 7: Yet, the last term ( p 2 ) is not under the control of the Central Bank and so can be dropped from the minimisation. Set p k w and you are just back in Walsh’s case! So you can again get to the first-best. Therefore, by giving the Central Bank an explicit inflation target that is conservative (below the 0 social optimum inflation rate implicit in the loss function for this question), the Government can gain ensure we obtain the first best. Julian von Landesberger 13.04.2015 46 Question 8: Finally, say that both the Central Bank and private agents do not observe the natural rate of unemployment and the supply shock at t. (Do you know what any of these is, right now?) They only observe the actual value of the unemployment rate. Moreover, the Central Bank targets some optimally formed expected value of the natural rate, so that now . utT EutN a) Derive the discretionary optimal policy rule and the equilibrium level of inflation. How do expectational errors in the forecast of the natural rate affect inflation? Julian von Landesberger 13.04.2015 47 Question 8: The Central Bank now minimizes: 1 2 T 2 Lt p ut ut 2 1 2 e N N 2 Lt p p t p t t ut Eut 2 (33) (34) The FOC is: p t p t p te t utN EutN Julian von Landesberger 13.04.2015 48 (35) Question 8: Taking expectations gives: p te 0 (36) The solution for inflation is therefore: (37) pt 1 u N t EutN t First, see that underestimating the natural urate leads t Eut to higher inflation.Yet, note that this is not an inflation bias as before. In the long-run, because the Central Bank’s expectations are rational, inflation should average to 0, whereas in the discretionary solution in question 2 it averages to k. N Julian von Landesberger 13.04.2015 49 N Question 8: The model predicts high inflation in the 1970s but low inflation in the the 1990s, which fits the data. The Barro Gordon model is still driving the dynamics of inflation, but the “inflation bias” is now time-varying, allowing the model to not only explain the great inflation of the 1970s but also the low inflation of the late 1990s. Julian von Landesberger 13.04.2015 50 Question 8: b) At a given period can this model or the model in question 2 be distinguished from the behavior of inflation? What about in the long-run? c) It has been argued that the 1970s were a period where the natural rate unexpectedly increased and the Central Bank took a while to catch on, making a succession of forecast errors. What does the model predict would happen to inflation? Similarly, during the late 1990s, estimates seem to show the natural rate has fallen but Alan Greenspan repeatedly claimed he believed the economy was over-heated, suggesting he did not believe in such a fall and did not update his natural rate target. What does the model predict then? How do these predictions fit the broad trends in inflation over these periods? Julian von Landesberger 13.04.2015 51 The Canonical Monetary Policy Problem with Serially Correlated Shocks Based on Clarida, Gertler, and Gali (1999). Consider an economy with both supply and demand shocks in which the presence of some form of price rigidity implies the existence of a New Keynesian Phillips Curve. Assume that the policymaker is trying to solve the following problem: 1 2 2 max Et xt p t (1.1) 2 0 s.t. πt =λxt +βEt πt+1 +ut xt = -j[it -Et πt+1]+Etxt+1 +gt where xt is the output gap, πt is the inflation rate, β (0 ,1) the discount factor, it the nominal interest rate, ut a supply shock and gt a demand shock. Julian von Landesberger 13.04.2015 52 The Canonical Monetary Policy Problem - the shocks Where ut ut 1 uˆt , 0,1 and uˆ t ~i.i.d.(0,σu2). Similarly, g t g t 1 gˆ t , 0,1 and gˆ t~i.i.d.(0, σg2). Finally demand and supply shocks are uncorrelated. Julian von Landesberger 13.04.2015 53 The canonical monetary policy problem without commitment Assuming no possibility of commitment, problem (1.1) is equivalent to an infinite sequence of problems defined by: 1 max xt2 p t2 2 πt =λxt +ft (1.2) ft is a given constant from the point of view of the central bank. Why does the absence of commitment imply that problem (1.1) can be written as an infinite sequence of one-period problems like (1.2)? Julian von Landesberger 13.04.2015 54 The Canonical Monetary Policy Problem with Serially Correlated Shocks Without access to a commitment technology, the central bank is free to reoptimize every period taking as given previously formed expectations (discretionary policy). When the expectations in program (1.1) are taken as given, the problem boils down to solve (1.2) for every period. Substituting the Phillips Curve into the one-period loss function, problem (1.2) reduces to: 1 2 max xt xt f t 2 2 Julian von Landesberger 13.04.2015 55 The first order condition The F.O.C. for this problem is: -[αxt +λ (λxt + ft )] = 0 or: λ (λxt + ft )= -αxt Since πt = λxt + ft, the above condition implies that: xt p t (1.5) Substituting (1.5) into the original Phillips curve, we obtain: 2 p t p t Etp t 1 ut Julian von Landesberger 13.04.2015 56 (1.6) Solving by forward substitution The Phillips curve equation is a stochastic first-order difference equation in πt.There are several methods to solve this kind of equations. A simple one is forward substitution: substituting in for πt+1 using (1.6) evaluated at t +1 and then take the expectations, which depend on πt+2, and then repeat the same procedure. Eventually you need to impose some terminal condition to get rid of the last term after an arbitrarily large number of substitutions. An alternative to this method is to use lag (and forward) operators. Julian von Landesberger 13.04.2015 57 Solving by forward operators Define the forward operator as L-kxt =Etxt+k. Using this definition, (1.6) can be written as: 2 1 1 L p t ut 1 L1 p u 2 t 2 t Julian von Landesberger 13.04.2015 58 The canonical monetary policy problem without commitment Show that the optimal policy without commitment implies: xt ut (1.3) pt 2 ut 1 (1.4) 1 2 What is the relationship between these equations and the expressions derived in class? Julian von Landesberger 13.04.2015 59 Solving by forward operators II Since the forward operator is linear, this expression implies that: 1 pt u 2 t 1 L1 2 (1.7) (0,1) With α>0 and , 0 <αβ/(α +λ2) <1. This condition is equivalent to the terminal condition that we need to impose on the problem when we apply forward to solve the substitution 1 1 equation and implies that L 2 1 L1 0 2 (1.8) Julian von Landesberger 13.04.2015 60 Solving by forward operators III Substituting (1.8) into (1.7) yields: 1 pt L ut 2 2 0 pt Et ut 2 2 0 Sinceut ut 1 uˆt know that: 0,1 , where 1 uˆ t , and ut ut j uˆt j j 0 Julian von Landesberger 13.04.2015 61 ~i.i.d.(0,σ2u), we Unwinding the shock Et ut ut Hence and, therefore, pt ut 2 0 2 Since |ρ |<1 and pt Julian von Landesberger 13.04.2015 ut 2 1 2 62 ut 2 1 (1.9) Solution for the output gap Substituting (1.9) into (1.5) yields: ut xt 2 1 (1.10) Note that the presence of supply shocks implies that inflation and output gap move in opposite directions. The expressions derived in class are particular cases of (1.9) and (1.10) when ρ =0, i.e., when there is no persistence in supply shocks. Julian von Landesberger 13.04.2015 63 The second-moment trade-off Show that, given preferences about the inflation-output variability (that is, the parameter α), there is a second-moment efficient frontier characterized by σ (xt)/σ(πt)=λ/α ,where σ(z) denotes the standard deviation of z. •Plot this equation on the (σ(xt)/σ (πt),α)-space. •Why do demand shocks not affect the relative variability of inflation and output and supply shocks do? •What is the optimal variance of inflation when α =0? •What is the optimal variance of xt when α =0? Julian von Landesberger 13.04.2015 64 The second-moment trade-off Given (1.9) and (1.10): and s ut s xt 2 1 s ut s p t 2 1 These two expressions imply that: σ (xt)/σ(πt) = λ/α Julian von Landesberger 13.04.2015 65 The second-moment trade-off sx/sp A B R Julian von Landesberger 13.04.2015 66 The second-moment trade-off When there is a demand shock, the monetary authority adjusts the nominal interest rate to keep xt unchanged (through the IS/Aggregate Demand Curve) and, without any supply shock, inflation does not change (because nothing changes in the Phillips Curve). In contrast, when there is a supply shock the optimal policy for the central bank implies that inflation and output gap are moving in opposite directions. Julian von Landesberger 13.04.2015 67 The second-moment trade-off Equation (1.9) implies that when α = 0 the optimal variance of πt is 0. This means that when the central bank does not care about output, the best policy is total inflation stability. Similarly, making α =0 in equation (1.10), the optimal variance of xt is σ2(ut)/λ2, which is the variability induced on output to achieve total inflation stability using the Phillips Curve. The above figure makes clear that the cost of appointing a conserva-tive central banker (one who has a lower α than the median Julian von Landesberger voter) is higher output volatility. 13.04.2015 68 Inflation targeting Show that the optimal policy, described by (1.3) and (1.4), incorpo-rates inflation targeting in the sense that it implies gradual conver-gence of inflation to its target, i.e., show that, given (1.3), (1.4), and the stochastic process for ut, Et p t 2 ut 1 •What is the rate of convergence of inflation to its target when the supply shock is pure white noise? •What is the rate of convergence of inflation to its target when the central bank does not care about output variability? Julian von Landesberger 13.04.2015 69 Inflation targeting Equation (1.9) implies that Etp t Et ut 2 1 Et ut ut We showed in part (a) that . Hence, Et p t 2 ut 1 Since |ρ |<1, inflation is expected to return to its target level gradually at exponential rate ρ. Julian von Landesberger 13.04.2015 70 Inflation targeting When the shock is pure noise ρ =0, i.e., without persistence, convergence is instantaneous, in the sense that the central bank expects to hit its target in any future period. When the central bank does not care about output variability (when α =0), convergence is instantaneous as well. Et p t 2 ut 1 Julian von Landesberger 13.04.2015 71 The optimal interest rate policy Show that the interest rate policy consistent with (1.3) and (1.4) is given by it Where 1 j gt Etp t 1 1 1 1 j Why is the coefficient on Et πt+1 greater than one? Julian von Landesberger 13.04.2015 72 The optimal interest rate policy Using (1.10), we can write 1 ut xt Et xt 1 2 1 1 xt Et xt 1 ut 2 1 Equation (1.9) implies (1.11) Julian von Landesberger 13.04.2015 1 xt Et xt 1 Etp t 1 73 The optimal interest rate policy The IS/Aggregate Demand Curve can be written as: xt Et xt 1 j it Etp t 1 g t Substituting (1.11) into the above expression yields 1 Etp t 1 j it Etp t 1 gt it Julian von Landesberger 13.04.2015 1 1 Etp t 1 j j gt 74