Tutorial to the Monetary Policy Lecture May 24-28, 2004

1. Tutorial to the Monetary Policy Lecture May 24-28, 2004 Dr. Julian von Landesberger HVB Group Economics julian.vonlandesberger@hvb.de julian.von-landesberger@gmx.de

2. 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.

3. 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 ?

4. 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.

5. 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.

6. 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 pt = b/a Take expectations... pte = b/a Compute the loss... lt = (1/2)(b)2/a

7. 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.

8. 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

9. 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

10. 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.

11. 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.

12. 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.

13. Taylor overlapping wage model Overlapping nominal wage contracts. In period t, set (log) nominal wage wtfor two periods. Average (log) wage Set wages according to expected average nominal wages

14. Taylor overlapping wage model

15. Rotemberg’s quadratic price-adjustment costs model optimal unrestricted (log) price, price of particular firm, pt (log) price level. First-order condition for Optimal unrestricted price:

16. Rotemberg’s quadratic price-adjustment costs model All firms are identical, therefore The Phillips curve can be derived as follows:

17. Calvo’s staggered contracts model optimal unrestricted (log) price, price of particular firm is adjusted in period t with prob q, pt (log) price level. First-order condition for

18. Calvo’s staggered contracts model

19. Calvo’s staggered contracts model Aggregate price level (not all firms equal)

20. Calvo’s staggered contracts model Insert into definition of the price level

21. The economy Say that the economy is described by: utis the unemployment rate,is the natural rate of unemployment, ptis the inflation rate and its expected value etis a supply shock, i.i.d. with mean 0 and variance s2 Agents have rational expectations.

22. Policymaker‘s objective The policymaker’s loss function is given by: is the target unemployment rate which for now we take as being below the natural rate: The target for inflation is normalized to zero, without loss of generality.

23. 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 etwhen setting policy ptat 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 ELtgiven this policy?

24. 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.

25. Question 1 and 2: 2. The discretionary Central Bank solves: (1) with F.O.C yielding the optimal policy rule: (2) This is a simple form of a countercyclical policy. Replace utfrom the Phillips curve into the expression and take expec-tations to obtain: (3)

26. Question 1 and 2: Replace this, together with the Phillips Curve into equation (2), to obtain: (4) which, after rearranging, gives the solution for inflation. Plugging this into the Phillips Curve (together with equation 3) you obtain unemployment: (5) (6)

27. Question 1 and 2: Plug these into the loss function to obtain the expected loss: (7) Take the expectations taking into account that E(et)=0 and E(e2t)=s2to obtain the ex ante expected loss under discretion: (8)

28. Question 3: Assume now the policymaker can commit ex ante to a linear state contingent rule: (3) pt= c + bet 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.

29. Answer to 3: Replace the inflation rule into the ex ante expected loss, and take expectations to obtain: (9) Minimizing this with respect to b and c yields the optimal rule: c =0 and b = l /(1+l). Equilibrium unemployment and inflation are: (10) (11) Clearly, since this policy leads to the same unemployment but lower inflation than the discretionary one, it achieves a superior outcome.

30. 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.

31. Question 4: Just set b =0 in (9) Minimize with respect to c to obtain the optimal policy rule: c = 0. Equilibrium unemployment and inflation are: pt= 0 (12) (13)

32. 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+ l). 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: (14)

33. Question 4: (15) The non-state-contingent 0-inflation rule is therefore preferrable to discretion if: LD>LR (17) (18)

34. 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.

35. 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 l*. Find the optimally appointed Central Banker’s l*(you do not need to find a closed form solution). Show that 0 <l*< l.

36. Question 5: From question 2, we know the appointed Central Bank will follow the policy: (19) (20) Plug this into the loss function, noting crucially that the social loss function still involves land not l*.

37. Question 5: Take expectations to obtain the Government’s ex ante expected social loss function: (21) (22) Minimize this with respect to l* to obtain the F.O.C that implicitly defines the optimally appointed Central Banker: (23)

38. Question 5: To prove the claim in the text, note that: G (0) = - ls2<0 (24) G (l) = lk2> 0 (25) Moreover, differentiate G(.) with respect to its argument to obtain the slope of the function: (26)

39. Question 5: Note that in the interval [0, l] then G’0(.)>0, i.e. the function is monotonically increasing. But, if the function in the interval [0, l] 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 l*such that: 0 < l*< l,as we wanted to show.

40. Question 6: • Assume instead now that the Government cannot appoint a Central Banker with a ldifferent 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?

41. Question 6: The discretionary Central Bank now minimizes the loss function: (27) Follow exactly the same steps as in question 1, to obtain, respectively, the policy rule, the equilibrium inflation and equilibrium unemployment: (28) (29) (30)

42. Question 6: So immediately note that by setting w=lk, we reach the first-best policy defined in question 3. Intuitively, note that the infla-tion bias problem is non-state contingent (it is lk whatever et), 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 (l*/1+l<l/1+l).

43. 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 state-contingent optimal policy. But it decreases the loss from the non-state-contingent, constant, inflation bias, and if adequately set can fully eliminate it.

44. Question 7: Alternatively, say the Government can give the Central Bank an ex-plicit inflation target 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). Again derive the optimal and discuss the relation to the previous question.

45. 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: (31) But, just expand the quadratic to see this is just: (32)

46. Question 7: Yet, the last term ( ) is not under the control of the Central Bank and so can be dropped from the minimisation. Set 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.

47. 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 . 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?

48. Question 8: The Central Bank now minimizes: (33) (34) The FOC is: (35)

49. Question 8: Taking expectations gives: (36) The solution for inflation is therefore: (37) First, see that underestimating the natural rate leads 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 lk.

50. 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.