Monetary and Fiscal Policy under Imperfect Knowledge

Transcription

1 Monetary and Fiscal Policy under Imperfect Knowledge Monash University Mini-course Part III Bruce Preston July 16, 2012

2 Motivation Macroeconomic models depend on household and Örm expectations Policy design ñ Not enough to observe some history of beliefs ñ Need a theory of the determination of beliefs: rational expectations Rational expectations policy design ñ Heavy reliance on managing expectations through announced commitments ñ What are the consequences of imprecise control of beliefs?

3 Motivation II Reasonable to suppose agents have limited information about policy regime ñ US Financial Crisis witnessed dramatic change in the nature and conduct of stabilization policy ñ Does imperfect knowledge limit the e cacy of policy? For example, Bernanke (2004): ì[...] most private-sector borrowing and investment decisions depend not on the funds rate but on longer-term yields, such as mortgage rates and corporate bond rates, and on the prices of long-lived assets, such as housing and equities. Moreover, the link between these longer-term yields and asset prices and the current setting of the federal funds rate can be quite loose at times.î

4 The Agenda Part 3 of this mini-course proceeds as follows ñ Lecture 1: Tools and conceptual issues ñ Lecture 2: Consequences of learning for the choice of monetary policy rule ñ Lecture 3: Monetary and Öscal interactions

5 Introducing Learning Dynamics Two foundations of rational expectations equilibrium analysis ñ Optimization ñ Mutual consistency of beliefs Strong knowledge assumptions ñ Agents know more than the modeler

6 ScientiÖc Method Principle components ñ Maintained theories ñ Procedures for collecting data ñ Procedures for confronting theory with data ñ Procedures for updating theory given discrepancy between theory and data

7 Bounded Rationality: Learning Dynamics Attempts to place the agent and modeler on equal footing Agents might proceed as: ñ Classical econometrician: know model but unsure about parameters ñ Bayesian econometrician: unsure about model and parameters but understand how they are unsure

8 Learning Dynamics Analyze broad class of model that relaxes the second pillar of REE analysis ñ Retain optimization ñ Permit non-rational expectations Beware of theorists bearing free parameters! ñ Constrain the analysis by requiring the procedure to nest rational expectations

9 Why learning? Permits analysis of a broader class of problems May resolve/elucidate ñ Questions of indeterminacy ñ Questions of robustness ñ Questions of dynamics ñ Questions of regime change/regime uncertainty

10 Today Mathematical foundations ñ Learning renders model dynamics self-referential ñ Requires new set of tools to analyze model properties Conceptual issues in modeling multi-period decision problems

11 A Simple Model Consider a variant of the Cobweb model p t = + E t1 p t + w t1 + t (1) where ; a; > 0; w t and exogenous process; and t a bounded iid disturbance Unique bounded solution p t =a + bwt1 + t where a =(1) 1 and b =(1) 1

12 Statistical Learning Suppose agents forecast p t as a linear function where fa t1 ;b t1 g are currently maintained beliefs ^E t1 p t = a t1 + b t1 w t1 (2) Price dynamics then p t =( + a t1 ) +( + b t1 ) w t1 + t ñ Self-referential system ñ Data generating process is non-stationary

13 Least Squares Learning Beliefs fa t1 ;b t1 g estimates using data fp i ;w i g i=t1 i=0 Ordinary least squares implies 0 a t1 t1 X b t1 i=1 z i1 z 0 i1 where z i =[1 w i ] ñ Fully speciöed system: relations (1) - (3) 1 A1 0 t1 i=1 1 z i1 p i A (3)

14 Question Under what conditions do a t! a and b t! b as t!1?

15 Theorem Consider the model (1) - (3). If <1 then (a t ;b t )! a; b with probability one. If >1 then convergence occurs with probability zero. ñ Property <1 referred to as expectational stability ñ E-Stability principle governed by mapping between agent beliefs and true model coe cients

16 E-Stability principle Recall ^E t1 p t = a t1 + b t1 w t1 (4) and p t =( + a t1 ) +( + b t1 ) w t1 + t Latter expression implies optimal rational forecast E t1 p t =( + a t1 )+( + b t1 ) w t1

17 Mapping DeÖned Agentsí beliefs and optimal forecast deöne the mapping ñ An REE is a Öxed point of this mapping T a b! = + a + b ñ Application of method of undetermined coe cients!

18 E-Stability principle II DeÖne the associated ordinary di erential equation where is notional time d d a b! = T a b! a b! ñ Local stability properties govern E-Stability principle

19 E-Stability principle III In particular, the REE is E-Stable if and only if the ODE is local stable at a; b Hence requires <1 da d db d = + ( 1) a = + ( 1) b

20 Recursive Least Squares Ordinary least squares regression can be written as where t =[a t b t ] 0 t = t1 + t 1 R t z t1 pt 0 t1 z t1 R t = R t1 + t 1 z t1 zt1 0 R t1

21 Recursive Least Squares II Since p t = ( + a t1 )+( + b t1 ) w t1 + t = T t1 0 zt1 + t we have t = t1 + t 1 Rt 1 z t1 z 0 t1 T t1 t1 + t R t = R t1 + t 1 z t1 zt1 0 R t1 ñ Question: does this system converge??

22 Stochastic Approximation Methods Provide results characterizing convergence of such systems ñ Ljung (1977), Marcet and Sargent (1989) Consider stochastic recursive algorithm t = t1 + t Q (t; t1 ;X t ) where t - parameter estimates (a t ;b t ;R t ) X t - state vector (e ects of p t, z t and t ) t - deterministic sequence of gains (t 1 )

23 Stochastic approximation approach associates the ODE where d d = h ( ()) h () lim t!1 EQ(t; ; X t) ñ E denotes the expectation of Q (t; ; X t ) with respect to the invariant distribution of X t for Öxed

24 Stochastic Approximation Results Under suitable assumptions: ñ If is a locally stable equilibrium point of the ODE, then is a possible point of convergence of the SRA ñ If is not a locally stable equilibrium point of the ODE, then is not a possible point of convergence of the SRA. That is t! with probability zero.

25 Technical Assumptions Stochastic approximation results rely on assumptions regarding ñ Regularity conditions on Q ñ Conditions on the rate at which t! 0 ñ Assumptions on the properties of X t Evans and Honkapohja (2001, Chapter 6 & 7)

26 Example Revisited To write our RLS algorithm in SRA form deöne S t1 R t and write the system as t = t1 + t 1 St1 1 z t1 z 0 t1 T t1 t1 + t so that S t = S t1 + t 1 t zt z 0 t S t1 t +1 t = vec ( t ;S t ) X t = [1 w t w t1 t ] 0 t = t 1

27 Example Revisited II Hence Q (t; t1 ;X t ) = St1 1 z t1 z 0 t1 T t1 t1 + t Q S (t; t1 ;X t ) = vec t zt zt 0 S t1 t +1 ñ To do: compute the associated ODE! ñ Fix t and compute expectation over X t

28 Example Revisited III Fixing (; S) implies h (; S) = lim t!1 ES1 z t1 z 0 t1 (T () )+ t h S (; S) = t zt lim t!1 E zt 0 S t +1 Since " Ez t zt 0 = Ez t1 zt = 0 where =E w t wt 0 ; Ezt1 t =0and lim t!1 # M t t+1 =1 h (; S) = S 1 M (T () ) h S (; S) = M S

29 The Associated ODE is then d d ds d = S 1 M (T () ) = M S ñ The latter is globally stable for any initial S: S! M ñ Therefore S 1 M! I Hence d d = T () ñ Intuition?

30 Summary Have motivated the concept of E-Stability ñ Governed by eigenvalues of the associated ODE Remainder ñ Microfoundations and modeling issues ñ Subsequent lectures: Applications in policy design

31 Modeling Learning Dynamics: Some Issues Intertemporal optimization implies multi-period forecasts ñ Fundamental to macroeconomic analysis Transversality conditions central to this theory ñ Partial Equilibrium: Marcet and Sargent (1989) ñ General Equilibrium: Preston (2005) Yet most of the macro learning literature ignores this requirement of optimality

32 Beliefs Under rational expectations: 1. Agents optimize given beliefs 2. The probabilities they assign to future state variables coincide with the predictions of the model This paper retains (1) and replaces (2) with 2 0. Future state variables outside agentís control are forecasted using an econometric model.

33 Knowledge Know own preferences and constraints Do not know true economic model of determination of variables outside their control. ñ Uncertain about the equilibrium mechanisms determining prices and policy variables ñ E.g. even if Y i t = Y j t = Y t in EQ ; agents know Y i t = Y j t Observe aggregate variables and disturbances Forecast variables outside their control using an atheoretical VAR ñ Takes the minimum state variable form

34 Townsend Investment Problem Consider a representative Örm solving max fk T g ^E t 1 X T =t T t p T k T 2 kt k T 1 2 where 0 < < 1 and ; > 0: The price of output, k t, is determined on competitive markets according to p t = K t + u t u t = u t1 + " t where 0 <<1 and K t the aggregate capital stock. Equilibrium requires: k t = K t = Z1 k t (i) di ñ Assumption: Örms do not know market mechanism determining prices 0

35 Townsend Investment Problem II To solve this problem a Örm requires forecasts of future capital prices and state variables ñ Prices are exogenous to their decision problem ñ As are the aggregate capital stock and stochastic disturbance Denoting the set of variable beyond the Örmís control as z t fp t ;K t ;u t g assume forecasting model of the form z t = a t + b t z t1 + t ñ Use data available until period t 1

36 Forecasts are then constructed according to ^E t z T =(I 3 ) 1 I 3 b T t t1 at1 + b T t t1 z t ñ Note that it is assumed that when making decisions over the forecast horizon beliefs are held Öxed ñ Formally we solve an anticipated utility problem in the sense of Kreps and Sargent ñ That agents do not account for subsequent revisions in their beliefs when making decisions in the current period represents the non-rationality of agents decisions

37 Townsend Investment Problem III The Örst-order conditions of the Örmís problem are k t k t1 = ^E t (k t+1 k t )+ p t and lim E t T k T =0 T!1 Solving the Örst condition forward and applying the second gives the optimal investment rule k t = k t1 + ^E t 1 X T =t T t p T ñ This is the optimal rule conditional on arbitrary beliefs: holds under rational expectations

38 Rational Expectations Solution and Learning Straightforward to show capital is determined by where 0 <<1 k t = K t1 + ( ) u t ñ Under learning dynamics, optimal decisions converge to this allocation with probability 1

39 Alternative Approaches The predominant approach to modeling learning in macroeconomics is to commence with Euler equations implied by a rational expectations analysis. In the context of the Townsend problem, commence with as the primitive behavioral assumption k t k t1 = ^E t (k t+1 k t ) + p t Requires a forecasting model k t =~a t + ~ bt K t1 +~c t u t1 + t so that ^E t k t+1 =~a t1 + ~ bt1 K t +~c t1 u t

40 Alternative Approaches II This second approach is fundamentally distinct from the Örst. transversality condition agents make suboptimal decisions By ignoring the Represents a basic confusion of the economics: Örms forecast their own endogenous decisions variable using an arbitrary model ñ Represents a contradiction of what agents know to be true about their decision problem ñ Forecasts of endogenous variable induces by optimal decision rules and primitive assumptions about forecasting variables beyond the control of the decision maker

41 It is not that the Euler equation is irrelevant. optimal decision rule yields Note that quasi di erencing the k t = k t1 + ^E X 1 t T t p T T =t = k t1 + p t + X 1 ^E t ^E t+1 T =t+1 = k t1 + p t + ^E t (k t+1 k t ) which is the Euler equation. T t p T Optimality requires capital to be forecast in a particular way. SpeciÖcally ^E t k t+1 = k t + ^E t 1 X T =t+1 T t p T ñ In general this will not be equal to ^E t k t+1 =~a t + ~ bt K t +~c t u t

42 Model Agents Households Firms Monetary authority Fiscal authority

43 Model Features Money or cashless limit Monopolistic competition/nominal rigidities Incomplete asset markets No capital Non-rational expectations

44 Beliefs Under rational expectations: 1. Agents optimize given beliefs 2. The probabilities they assign to future state variables coincide with the predictions of the model This paper retains (1) and replaces (2) with 2 0. Future state variables outside agentís control are forecasted using an econometric model.

45 Knowledge Know own preferences and constraints Do not know true economic model of determination of variables outside their control. ñ E.g. even if Y i t = Y j t = Y t in EQ ; agents know Y i t = Y j t Observe aggregate variables and disturbances Forecast variables outside their control using an atheoretical VAR ñ Takes the minimum state variable form

46 Household Problem Households seek to maximize subject to ^E t i 1X T =t T t h U C i T ; T v h i T ; T i Z 1 Bt i = (1 + i t1 ) Bt1 i + w th i t + t (j) dj T t P t Ct i 0 = (1 + i t1 ) Bt1 i + PYi t T t P t Ct i ñ Beliefs: a-theoretical VAR of exogenous variables ó nests MSV REE ñ Assume agents forecast period income

47 First-order conditions Log-linear approximation yields ^C i t = ^E i t ^C i t+1 i t ^E i t t+1 g t + ^E i tg t+1 and 1X 1X where ^E i t T =t T t ^C i T = Ai t + ^E i t T =t T t ^Y i T A i t = B i t=p t Y ; g t = t u c =u c ; 1 = u cc Y =u c ñ Assume government debt in zero net supply

48 Deriving the Consumption Decision Rule Solving the Euler equation backwards recursively from date T to date t and taking expectations at that time gives ^E i t ^C i T = ^C i t g t + ^E i t 2 4g T + TX 1 T =t (^{ t ^ t+1 ) ñ Use this to eliminate future expected consumption in the intertemporal budget constraint 3 5

49 Optimal decision rule First-order conditions imply ^C i t = (1 ) A i t + ^E i t 1X T =t T t h (1 ) ^Y i T ^{ T T +1 + gt g T +1 i ñ This is an example of permanent income theory ñ Forecasts about prices into the indeönite future matter! ñ A particularly important source of uncertainty are anticipated future interest rates

50 Comparison to Existing Approaches Are decisions implied by the optimal decision equivalent to those implied by ^C i t = ^E i t ^C i t+1 i t ^E i t t+1 g t + ^E t g t+1 ñ Where forecasts determined by the model z t = a t1 + b t1 z t1 + " t with z t = h ^C i t t i t g t i 0 and zt = h ^Y t t i t g t i 0

51 Comparison to Existing Approaches II Are decisions implied by the optimal decision equivalent to those implied by ^C i t = ^E i t ^C i t+1 i t ^E i t t+1 g t + ^E t g t+1 ñ Where forecasts determined by the model z t = a t1 + b t1 z t1 + " t with z t = h ^C i t t i t g t i 0 and zt = h ^Y t t i t g t i 0 In general - No! Optimal decision rule if expectations treated correctly ñ i.e. the conditional expectation of consumption induced by model primitives and optimality

52 Forecasting Consumption The optimal forecast ^E i t ^C i t+1 = (1 ) ^E i ta i t+1 + ^E t i 1X T =t+1 T t h (1 ) ^Y i T ^{ T T +1 + gt g T +1 i Not the same as ^E i tz t+1 = a t1 + b t1 z t ñ Conditional expectations of endogenous variables induced by policy function and forecasts about variable the are exogenous to the agentís decision problem

53 Commentary Market clearing conditions are part of the REE One-period-expectations approach does not deliver optimal decision rule ñ Confuses endogenous decision variables and exogenous state variables ñ The correct distribution with respect to which expectations are taken is induced from the decision problem and properties of exogenous variables ñ In general intertemporal budget constraint will be violated in this formulation Expected future consumption plans are inconsistent with what agents know to be true about their own future behavior given their own decision problem

54 Aggregate Consumption Dynamics Aggregating over the continuum provides X 1 ^C t = ^E t T t h (1 ) ^Y T ^{ T T +1 + gt g i T +1 T =t where ^C t Z ^C i tdi; ^E t Z ^E i tdi; 0 Z A i tdi ñ For any variable X t, ^E t ^E t+1 X t+1 6= ^E t X t+1

55 Summary Under rational expectations equilibrium probability laws by construction satisfy all relevant constraints ñ In particular: Transversality conditions Under learning dynamics expectations about future endogenous decision variables are taken with respected to a distribution ñ Induced by beliefs about exogenous states ñ Induced by optimal decisions

56 Firms Continuum of Örms maximize ^E j t 1X T =t T t Q t;t h j T (p t(j)) i where j T (p) = Y T P T p1 w i T f 1 Y T P T p =A T y i t = A t f h i t

57 Optimal Price Setting Log-linear approximation implies p j t = ^E j t 1X T =t () T t 2 4 (1 )! + 1 x T + T 1+! 3 5 ñ InÖnite horizon forecasts matter ñ x t = Y t ^Y n t is the output gap

58 Model Summary Aggregate demand and supply: x t = X 1 ^E t T t (1 ) x T +1 i T T +1 r T T =t t = X 1 x t + ^E t () T t x T +1 +(1) T +1 + u t T =t where r t = r t1 + " t and 0 <<1 u t = u t1 + t and 0 <<1

59 Rational Expectations Under this assumption ^E t ^E t+1 X t+1 = ^E t X t+1. Hence x t = E t [(1 ) x t+1 (i t t+1 r t )] + ^E t 1 X T =t+1 T t (1 ) x T +1 i T T +1 r T = E t (1 ) xt +1 (i t t+1 r t ) + E t x t+1 = E t x t+1 (i t E t t+1 r t )

60 Rational expectations analysis gives x t = E t x t+1 (i t E t t+1 r t ) t = kx t + E t t+1 Bullard and Mitra (2002, JME) and Evans and Honkapohja (2003, ReStud) x t = ^E t x t+1 i t ^E t t+1 r t t = kx t + ^E t t+1

61 Does any of this matter?

62 Objectives Implications of Learning for the choice of monetary policy rule The role of communication in policy design

63 Benchmark Model: Optimal Decisions Aggregate demand and supply: X 1 x t = ^E t T t (1 ) x T +1 i T T +1 r T T =t X 1 t = x t + ^E t () T t x T +1 +(1) T +1 + u t T =t where r t = r t1 + " t u t = u t1 + t are known exogenous processes with 0 < ; < 1 and " t, t iid disturbances Implicitly assumes Ricardian Öscal policy ñ Agents need not forecast debt or taxes

64 Alternative Model: ìone-period-ahead Forecastsî Interested in comparing policy emerging under the anticipated utility approach to the more familiar approach given by x t = ^E t x t+1 i t ^E t t+1 r t t = kx t + ^E t t+1 ñ Bullard and Mitra (JME, 2002), Evans and Honkapohja (ReStud, 2003)

65 Issues in Policy Design Early literature concerned with policy robustness and equilibrium selection ñ Given that there are a multiplicity of rules consistent with implementing a given equilibrium, can impose additional desiderata ñ Robustness to small expectational errors Models with learning also permit addressing critiques of policy ñ Friedman (1968) ó Monetary policy procedures based on interest-rate rules prone to Wicksellian cumulative process

66 An Interest-Rate Peg: Optimal Decisions Consider the policy i t = r r t where r t is iid and no u t ñ Indeterminacy of rational expectations equilibrium: Sargent and Wallace (1975) Agent forecasts given by ^E t z T = a t1 for z t = [ t x t i t ] 0 : Combined with structural model gives mapping da d db d = T (a) a = T (b) b ñ E-Stability determined by the eigenvalues of the Jacobian of these ordinary di erential equation

67 Conditions on Eigenvalues Consider the matrix A with dimension (3 3) : From ja Ij =0the characteristic equation is 3 c c 2 c 3 =0 where c 1 =Trace(A), c 2 is the sum of all second-order principal minors of A and c 3 = jaj. The following restrictions on the coe cients c i must be satisöed for all eigenvalues to have negative real parts: c 1 < 0 c 3 c 1 c 2 > 0 c 3 < 0:

68 For a matrix A with dimension (2 2) ; ja Ij =0implies the characteristic equation is 2 c 1 + c 2 =0 where c 1 =Trace(A) and c 2 = jaj. For both eigenvalues to have negative real parts c 1 < 0 and c 2 > 0 must be satisöed

69 Stability Analysis Straightforward to show and T 0 (a) = det(t 0 (a) I) = (1) ñ Expectations driven áuctuations (1 ) (1 ) > 0 ñ What about the Euler equation approach: Evans and Honkapohja (2003)

70 An Interest Rate Peg: One-Period-Ahead Expectations Consider the policy i t = r r t with r t i.i.d. Agent forecasts given by ^E t z t+1 = a t1 for z t =[ t x t ] 0 : Combined with structural model gives mapping da d db d = T (a) a = T (b) b

71 An Interest Rate Peg II The true data generating process is " # " xt 1 = ( + ) t which imply forecasts E t " xt+1 t+1 # = #" ax;t1 a ;t1 # " 1 ( + ) " (r 1) ( r 1) #" ax;t1 a ;t1 # # r t

72 DeÖning the T-map T a x a! = " 1 ( + ) #" ax a # Expectations stability requires the real parts of the eigenvalues of T 0 (a) I to be negative ñ That is: det T 0 (a) I > 0 and trace T 0 (a) I < 0 The determinant is det T 0 (a) I = < 0 ñ The minimum-state-variable solution is not learnable

73 Taylor Rule McCallum (1983): conditioning on endogenous variables resolves indeterminacy Consider the rule i t = t + x x t ñ Requires ( 1) + (1 ) x > 1 for determinacy under REE and stability under learning ñ Bullard and Mitra (2002), Preston (2005)

74 Emerging di erences Policy actions that depend on misspeciöed beliefs can lead to self-fulölling expectations Investigate a range of policies that are consistent with implementing the optimal commitment equilibrium under rational expectations ñ Implementation of such policy involve a dependence of interest-rate policy on expectations ñ This will reveal some sharp di erences across the anticipated utility approach and the one-period-ahead expectations approach

75 Optimal Commitment Policy under Rational Expectations Central bank is assumed to minimize the loss function E t0 X 1 tt 0 t=t 0 2 t + x 2 t where 0 <<1 for some weight >0 subject to (5) t = kx t + E t t+1 + u t t0 = t0 =(1 ) x t1 + 1 u t ñ Optimality from the ìtimeless perspectiveî: Woodford (2003, chap. Giannoni and Woodford (2002, 2012) 6) and

76 Optimal Commitment Solution Standard methods show that the optimal state-contingent paths of f t ;x t ;i t g are given by the following relations for all t t: t = (1 ) x t1 + 1 u t and x t = x t1 1 u t i t = (1 ) x t1 + ( + 1) 1 u t + 1 rn t where 0 <<1 is the modelís only eigenvalue within the unit circle.

77 Optimal Commitment Solution II Useful to represent this solution in terms of a particular exogenous state variable Solving the output gap relation backwards recursively gives x t = 1 1X j=0 j u tj : DeÖning the exogenous state variable t 1X j=0 j u tj ; which satisöes the process t = t1 + u t

78 Allows the optimal solution to be written as t = 1 [u t (1 ) t1 ] x t = 1 [ t1 + u t ] i t = 1 [u t (1 )( t1 + u t )] + 1 rn t where denotes the optimal solution when expressed as a linear function of f t1 ;u t ;r t g : ñ Note the question of what policy implements the optimal commitment equilibrium is yet to be answered ñ We have only described what that equilibrium looks like

79 Candidate Policy Rules Consider the following instrument rule as a means to implement the optimal equilibrium: i t = i t + x (E cb t x t+1 E t x t+1 )+ (E cb t t+1 E t t+1 ) where E cb t = { f t + xe cb t x t+1 + E cb t t+1 denotes the forecasts that the central bank intends to respond to and { l t = i t xe t x t+1 E t t+1 collects exogenous terms and E t x t+1 and E t t+1 ñ Recall f t ;x t ;i t g denote the optimal paths for each endogenous variable in the optimal commitment equilibrium when written as a linear function of the exogenous state variables f t1 ;u t ;r t g.

80 Candidate Policy Rules II Expectations based rules argued to be desirable and given empirical support ñ Batini and Haldane (1999), Levin, Wieland and Williams (2003), Clarida, Gali and Gertler (1998, 2000) Assume E cb t = ^E t where the forecasts are described below The policy parameters ; x determine the response to deviations of the actual path of the ináation rate and output gap expectations from that path consistent with the optimal equilibrium ñ In the desired equilibrium it is clear that Et cb t+1 = E t t+1, Ecb t x t+1 = E t x t+1 and i t = i t

81 Forecasts Agents therefore estimate the linear model z t = a t + b t z t1 + c t u t + d t r t + t (6) where z t =( t ;x t ;i t ; t ) 0, t is the usual error-vector term, fa t ;b t ;c t ;d t g are parameters to be estimated of the form and a t = a ;t a x;t a i;t a ;t ; c t = b t c ;t c x;t c i;t c ;t ; d t = b ;t b x;t b i;t b ;t d ;t d x;t d i;t d ;t 3 7 5

82 Forecasts II Believes revised according to recursive least squares formulation t = t1 + t 1 Rt 1 w t1 (z t1 wt1 0 t1) R t = R t1 + t 1 (w t1 wt1 0 R t1) where t =(a t ;b 4t ;c t ;d t )and w t f1; t1 ;u t ;r t g 0. Given homogeneity of beliefs, average forecasts can then be constructed by solving (9) backward and taking expectations to give ^E t z T = (I 4 b t1 ) 1 (I 4 b T t t1 )a t1 + b T t t1 z t +u t (I 4 b t1 ) 1 T t I 4 b T t t1 ct +r t (I 4 b t1 ) 1 T t I 4 b T t t1 dt for T t, where I 4 is a (4 4) identity matrix:

83 Properties A unique bounded rational expectations equilibrium will obtain i 0 <( 1) + (1 ) x < 2 (1 + ) Under learning a necessary condition for E-Stability is + x > (1 ) : ñ Clearly violates the Taylor principle; consider the limit case of! 1 ñ Di culty: by having central bank behavior depend on ìmisspeciöedî private beliefs makes self-fulölling expectations more likely Note central regulator of how shifting expectations impact current outcomes

84 Properties II What about the one-period-ahead expectations models? Can show that under the same rule expectation stability obtains if and only if ñ That is ó the Taylor principle holds ( 1) + (1 ) x > 1 ñ Fundamentally di erent predictions: arises from the one-period-ahead expectations approach gutting the model of economics. For example, expectations of future interest are irrelevant to dynamics in that approach

85 Raises important questions What is the appropriate use of forecasts in policy design? Can instability be mitigated? ñ Optimal policy/targeting rules imply a particular kind of dependency of interest rate decisions on forecasts ñ The role of communication ñ The role of debt policy Di erent statistical properties ñ Forecasts, dynamics and policy evaluation

86 Optimal Internal Forecasts Consider a sophisticated central bank that understands private-sector expectations formation Substituting the private forecasts into the structural relations implies the output gap and ináation to be given by t = a + b t1 +c u t + d r t +e i t (7) x t = a x + bx t1 +c x u t + dx r t +e x i t (8) where the coe cients a i ; bi ; c i ; di ; e i for i 2f; xg are functions of the coe cients (a t ;b t ;c t ;d t ) These equations give the period t ináation rate and output gap realizations, conditional on the current choice of the interest rate.

87 Can then compute ìoptimalî forecasts as Et cb t+1 = a + b t +c u t + d r t +e Et cb i t+1 Et cb x t+1 = a x + bx t +c x u t + dx r t +e x Et cb i t+1 : Substitution of these relations into yields an equation of the form i t ={ f t + xe cb t x t+1 + E cb t t+1 i t = ( a + x a x )+( b + x bx ) t +( c + x c x )u t +( d + xdx )r t +( e + x e x )Et cb i t+1 ñ Has unique bounded solution if j e + x e x j < 1 ñ Determines an interest-rate decision that is a function of agentsí learning behavior

88 Optimal Internal Forecasts II If the monetary authority constructs optimal internal forecasts then the Taylor principle ( 1) + (1 ) x > 0 is necessary and su cient for stability under learning dynamics. ñ Underscores an important point: understanding the details/mechanics of expectations formation critical ñ Policy needs to depend on expectations in a very speciöc way to ensure stability

89 Implementing Optimal Policy using Target Criteria Recent monetary economics literature has proposed implementing policy using target criteria ñ See, for example, Svensson and Woodford (2005), Svensson (2003), Giannoni and Woodford (2002, 2012) and Woodford (2003, chp. 7) The approach speciöes a criterion that must be checked each time an interest-rate decision is made ñ For example: Bank of Englandís ináation forecast targeting ñ Requires assumption about what model the central bank has when implementing policy ñ The approach gives very speciöc guidance on how forecasts should be incorporated in policy decisions

90 Deriving the Target Criterion Recall the optimal commitment equilibrium implies the dynamics t = (1 ) x t1 + 1 u t x t = x t1 1 u t i t = (1 ) x t1 + ( + 1) u t rn t

91 Deriving the Target Criterion II Giannoni and Woodford (2002) and Woodford (2003, chap 7) show t = (x t x t1 ) fully characterizes the optimal equilibrium from the timeless perspective in the sense that the previous dynamics hold for all t t 0 if and only if the above target criterion holds ñ Take this as the desired target criterion: under rational expectations implies determinacy for all parameter values ñ It is a restriction on endogenous variables ó central bank requires a model to implement ñ History dependence of optimal commitment captured by the lagged output gap

92 Forecasts Agents therefore estimate the linear model z t = a t + b t z t1 + c t u t + d t r t + t (9) where z t =( t ;x t ;i t ) 0, t is the usual error-vector term, fa t ;b t ;c t ;d t g are parameters to be estimated of the form and a t = a ;t a x;t a i;t ; c t = b t c ;t c x;t c i;t 3 0 b ;t 0 0 b x;t 0 0 b i;t ; d t = d ;t d x;t d i;t 3 7 5

93 Forecasts II Believes revised according to recursive least squares formulation t = t1 + t 1 Rt 1 w t1 (z t1 wt1 0 t1) R t = R t1 + t 1 (w t1 wt1 0 R t1) where t =(a t ;b 2t ;c t ;d t )and w t f1;x t1 ;u t ;r t g 0. Given homogeneity of beliefs, average forecasts can then be constructed by solving (9) backward and taking expectations to give ^E t z T = (I 3 b t1 ) 1 (I 3 b T t t1 )a t1 + b T t t1 z t +u t (I 3 b t1 ) 1 T t I 3 b T t t1 ct +r t (I 3 b t1 ) 1 T t I 3 b T t t1 dt for T t, where I 3 is a (3 3) identity matrix:

94 The Central Bankís Model A number of assumptions could be made. The central bank could: ñ Make projections based on minimum-state-variable rational expectations equilibrium ñ Make projections based on the structural equations implied by the learning model ñ Make projections based on the assumption that the Euler equations implied by rational expectations are valid.

95 Projecting under Rational Expectations Suppose the central bank mistakenly assumes agents have rational expectations If it projects the evolution of the economy assuming the minimum-state-variable equilibrium the target criterion will be satisöed under the interest-rate policy i t = (1 ) x t1 + ( + 1) 1 u t + 1 rn t ñ This is not an exogenous instrument rule ó depends on lagged output gap ñ Concerns of Sargent and Wallace (1975) need not be valid ñ Result: >su cient for instability under learning

96 Projecting under the True Model Suppose the central bank understands the true structural model ó that is, the aggregate demand and supply equations, and the precise details of agents belief formation The central bank can then guarantee satisfaction of the target criterion by setting interest rates according to i t = 1 ^x t + 1 X 1 ^E t T t (1 )x T +1 ( i T +1 T +1 )+r T T =t where ^x t = + 2x t1 + 2 ^E t 1X T =t () T t x T +1 +(1 ) T +1 + u T ñ Second relation is the value of the output gap that jointly satisöed the aggregate supply curve and the target criterion

97 This approach to implementing policy guarantees satisfaction of the target criterion regardless of the nature of expectations formation ñ Requires an interest-rate setting that depends upon expectations into the indefinite future ñ Is consistent with implementing the optimal commitment equilibrium if agents have rational expectations Under learning dynamics implies the underlying rational expectations equilibrium is E-Stable for all parameter values

98 Projecting using Euler Equations Suppose the central bank thinks the true model is x t = ^E t x t+1 i t ^E t t+1 r t t = kx t + ^E t t+1 Then the target criterion implies the instrument rule i t = 1 ^E t x t+1 + 2x t ^E t t u t + r n t What are the stability properties?

99 Final Remark on Target Criteria Under rational expectations the target criterion p t = x t implies the same state-contingent responses to disturbances as t = (x t x t1 ) Under learning they imply di erent outcomes ó that is, out of rational expectations equilibrium they imply di erent policy responses when the central bank Önds its target criterion is not met because expectations evolved in a way that was not anticipated ñ It turns out the price-level targeting is more conducive to stability

100 Communication as Stabilization Policy The discussion so far emphasizes that policy must be conditioned on expectations in certain ways An important source of instability is that agents do not know how interest rates are determined ñ This is one of the many equilibrium restrictions that they are attempting to learn ñ Agents beliefs need not be consistent with monetary policy strategy; coherent statement of ìunanchoredî expectations ñ But what if agents understood some details of policy? Does this improve the stability properties of forecast based instrument rules?

101 Communication as Stabilization Policy II Suppose the central bank declares that it interest rate decisions are determined by i t = E t t+1 Agents can use this information to restrict their forecasting model. SpeciÖcally, policy-consistent forecasts of the nominal interest rate can be determined as ^E t i T = ^E t T +1 ñ Such ináation and interest-rate forecasts are consistent with monetary policy strategy ñ They ensure projections satisfy the Taylor principle

102 Communication as Stabilization Policy III Policy consistent forecasts imply aggregate demand is X 1 x t = ^E t T t h (1 ) x T +1 ^E i t T +1 T +1 r T T =t ñ Agents need only forecast ináation and the output gap ñ Can show that E-Stability obtains so long as the Taylor principle holds ñ Communication about future policy intentions can resolve important uncertainty Taken up in detail by Eusepi and Preston (2010, AEJM) Now consider simpler analysis by Orphanides and Williams (2005)

103 Part III: Monetary and Fiscal Interactions The analysis so far has pushed Öscal policy to the back ground The remainder of the lecture will consider the consequences of debt management policy for the implementation of monetary policy ñ Interested in the role of scale and composition of the public debt ñ What kind of constraints do Öscal policy impose on monetary policy?

104 Motivation US Financial Crisis witnessed dramatic change in the nature and conduct of stabilization policy ñ Large-scale asset purchases intended to ináuence long interest rates ñ Under what conditions can this be e ective? Standard models satisfy an irrelevance proposition ñ Ricardian equivalence; more generally state-contingent consumption independent of government liabilities

105 What we do Simple model of output gap and ináation determination One informational friction: ñ Agents have an incomplete knowledge about the economy ñ Implication: departures from Ricardian Equivalence Explore constraints imposed on stabilization policy by choice of Öscal policy ñ SpeciÖcally: Scale and composition of Debt ñ Are there additional consequences of shortening the maturity structure of debt, over and above the presumed stimulus of lower interest rates

106 Literature Monetary and Öscal interactions: ñ Unpleasant monetary arithmetic: Sargent and Wallace (1980) ñ FTPL: Leeper, Sims and Woodford (early 90s) Learning: Marcet and Sargent (1989), Evans and Honkapohja (2001) Extensive literature on stability of expectations in NK models ñ Howitt (1992), Bullard and Mitra (2002) ñ Evans and Honkapohja (2011): survey of recent literature

107 Model Agents Households Firms Monetary authority Fiscal authority

108 Beliefs Under rational expectations: 1. Agents optimize given beliefs 2. The probabilities they assign to future state variables coincide with the predictions of the model This paper retains (1) and replaces (2) with 2 0. Future state variables outside agentís control are forecasted using an econometric model.

109 Knowledge Know own preferences and constraints Do not know true economic model determining variables outside their control. ñ E.g. even if Y i t = Y j t = Y t in EQ ; agents know Y i t = Y j t Observe aggregate variables and disturbances Forecast variables outside their control using an atheoretical VAR ñ Takes the minimum state variable form

110 Asset Structure and the Fiscal Authority Exogenous purchases of G t per period Issue two kinds of debt ñ B s t : One period debt in zero net supply with price P s t = (1 + i t) 1 ñ B m t : An asset in positive supply that has the payo structure T (t+1) for T t +1 Let P m t denote the price of this second asset. Asset has the properties: ñ Price in period t +1of debt issued in period t is P m t+1 ñ Average maturity of the debt is (1 ) 1

111 Monetary and Fiscal Authorities Flow budget constraint P m t B m t = B m t1 (1 + P m t ) P t S t Fiscal policy maintains intertemporal solvency (ëpassiveí) i t = i Bt1 m! i l ; i B m l >;i l Monetary policy controls ináation (ëactiveí) 1+i t 1+{ = t ; > 1 Under rational expectations: standard view of monetary policy

112 Household Problem Saving and work decision. No capital in the model, only gov. bonds. Householdsí preferences! U(C t ;H t ) = (1 ) 1 C t C 1+ tht 1+ 1, >0 where ñ =0! GHH; =1! KPR preferences ñ 1! IES 1; C t ;H t complements

113 Household Problem Household i maximizes (1 ) 1 ^E (i) t 1X T =t! 1 T t C t (i) C 1+ t(i)ht 1+ (i) subject to P s t B s t (i)+p m t B m t (i) (1 + P m t ) B m t1 (i)+bs t1 (i)+ 1 W t Wt H t (i) and No-Ponzi condition +P t t LS t P t C t (i) lim T!1 ^E (i) t Q i t;t h (1 + P m T ) B m T 1 (i)+bs T 1 (i)i =P T 0

114 Key Equation 1: Consumption Decision Combining Euler equations, labor supply, budget constraint to log-linear approximation provides ^C (i) t (1 ) ^H (i) t =s 1 P C (; ) m B m! P Y 1 1 t1 ^ t + ^P t m ^E (i) t 1X T =t T t h T LS^ LS T + T W ^ W T ^{ T ^ i 1 A T +1 {z } Gov. Debt and Taxes +s 1 C (; ) (1 ) ^E (i) t 1X T =t T t I W ^w T + I ^T + 1 ^E (i) t 1X T =t T t ^{ T ^ T +1

115 Key Equation 1: Consumption Decision Combining Euler equations, labor supply, budget constraint to log-linear approx. provides ^C (i) t (1 ) ^H (i) t = s 1 P C (; ) m B m! P Y 1 1 t1 ^ t + ^P t m ^E (i) t 1X T =t T t h T LS^ LS T + T W ^ W T ^{ T ^ i 1 A T +1 {z } Gov. Debt and Taxes +s 1 C (; ) (1 ) ^E (i) t 1X T =t T t I W ^w T + I ^T + 1 ^E (i) t 1X T =t T t ^{ T ^ T +1

116 Key Equation 1: Consumption Decision Combining Euler equations., labor supply, budget constraint to log-linear approx. provides ^C (i) t (1 ) ^H (i) t =s 1 P C (; ) m B m! P Y 1 1 t1 ^ t + ^P t m ^E (i) t 1X T =t T t h T LS^ LS T + T W ^ W T ^{ T ^ i 1 A T +1 {z } Gov. Debt and Taxes +s 1 C (; ) (1 ) ^E (i) t 1X T =t T t I W ^w T + I ^T + 1 ^E (i) t 1X T =t T t ^{ T ^ T +1

117 Key Equation 1: Consumption Decision Combining Euler equations., labor supply, budget constraint to log-linear approx. provides ^C (i) t (1 ) ^H (i) t =s 1 P C (; ) m B m! P Y 1 1 t1 ^ t + ^P t m ^E (i) t 1X T =t T t h T LS^ LS T + T W ^ W T ^{ T ^ i 1 A T +1 {z } Gov. Debt and Taxes +s 1 C (; ) (1 ) ^E (i) t 1X T =t T t I W ^w T + I ^T + 1 ^E (i) t 1X T =t T t ^{ T ^ T +1

118 Key Equation 2: Public Debt Price of government debt X ^P t m 1 = ^E t () T t ^{ T T =t {z } Expectation Hypothesis Evolution of public Debt ^bm t = 1 ^bm t1 ^ t + (1 )^{t 1 1 ^s t + (1 ) ^E t 1 X T =t () T t ^{ T +1

119 Firms Monopolistic competition + nominal rigidities (Rotemberg) Firm j has produces according to Y t (j) =H t (j) =(p t (j) =P t ) Y t p t (j) chosen to maximize expected proöts, where t (j) =[p t (j) W t ](p t (j) =P t ) Y t [p t (j) =p t1 (j) 1] 2

120 Firms Monopolistic competition + nominal rigidities (Rotemberg) In aggregate, the Phillips curve Y ^ t = +! C ^Y t + X 1 + ^E t () T t T =t ^w T +1 +(1 ) ^ T +1

121 Learning and Expectations (use ináation as an example) Rational Expectations (RE) ^E RE t T +1 = 0 + RE b ^bm t1, where T>t Learning ^E t T +1 = L c;t1 + L b;t1^b m t1 + lagged variables, Convergence to RE? E-Stability ñ Use methods of Marcet and Sargent (1989), Evans and Honkapohja (2001)

122 E-Stability and Learning Actual Law of Motion: t = T ( L c;t1 ; L b;t1 ) " 1 ^bm t1 # + ::: Convergence under least-squares learning IFF the associated ODE: d d ( c; b )= T ( L c ; L {z b ) } Actual Law of Motion is locally stable at the REE of interest: T ( RE c ( L c ; L b ) {z } Perceived Law of Motion ; RE b ) = ( RE c ; RE b ). Source of instability: economy as a self-referential system

123 E-Stability Numerical study: model with LUMP SUM taxation Key parameters: 1 =1=4,and P m B m 4 P Y =1:5 Others: ñ Preferences and technology: =0:99; =0:8 (Price stickiness) ñ Fiscal Policy: LS l =1:5 ñ Steady State: C=Y =0:78

124 1.6 E-stability: KPR vs. GHH preferences KPR preferences GHH preferences φ π *(1- βρ) - 1 Figure 1: E-stability properties in monetary policy ñ debt maturity space. Instability regiones lie above the contours.

125 Aggregate Demand: Increase in Ináation Expectations Ricardian Households ^C t (1 ) ^H t = 1 ^E t 1 X T =t T t ^ T ^ T s 1 C (; ) ^E t 1 X T =t T t I T ñ we have substituted for the bond price equation and the monetary policy rule, ^{ t = ^ t

126 Aggregate. Demand: Increase in Ináation Expectations Baseline ^C t (1 ) ^H t = 1 s 1 C (; ) S! Y ^E t 1 X T =t T t ^ T ^ T +1 + s 1 C (; ) S X 1 Y ^E t () T t ( ^ T )+ T =t +s 1 C (; ) S Y ^b m t1 +s1 C (; ) (1 ) ^E t 1 X T =t T t (I T T T ) :

127 Aggregate. Demand: Increase in Ináation Expectations Baseline ^C t (1 ) ^H t = 1 s 1 C (; ) S! Y ^E t 1 X T =t T t ^ T ^ T +1 + s 1 C (; ) S X 1 Y ^E t () T t ( ^ T )+ T =t +s 1 C (; ) S Y ^b m t1 +s1 C (; ) (1 ) ^E t 1 X T =t T t (I T T T ) :

128 Aggregate. Demand: Increase in Ináation Expectations Baseline ^C t (1 ) ^H t = 1 s 1 C (; ) S! Y ^E t 1 X T =t T t ^ T ^ T +1 + s 1 C (; ) S X 1 Y ^E t () T t ( ^ T )+ T =t +s 1 C (; ) S Y ^b m t1 +s1 C (; ) (1 ) ^E t 1 X T =t T t (I T T T ) :

129 Government Debt: Increase in Ináation Expectations Using the bond price equation and the monetary policy rule, ^{ t = ^ t, permits the evolution of real debt to be written as ^bm t = 1 ^bm t1 ^ t +(1 ) ^ t 1 1 ^s t + (1 ) ^E t 1 X T =t () T t ^ T +1 which depends on the maturity structure

130 Government Debt: Increase in Ináation Expectations Using the bond price equation and the monetary policy rule, ^{ t = ^ t, permits the evolution of real debt to be written as ^bm t = 1 ^bm t1 ^ t +(1 ) ^ t 1 1 ^s t + (1 ) ^E t 1 X T =t () T t ^ T +1 which depends on the maturity structure through (with exp. constant ináation) X 1 (1 ) ^E t () T t ^ T +1 = T =t (1 ) 1 ^E t ñ For =0:99 the right hand side peaks ' 0:9: average maturity of 2 years ñ For =1: dynamics independent of its own price.

131 1.8 E-stability: KPR preferences with σ = B/4Y =.7 B/4Y = 1.5 B/4Y = 2.2 φ π *(1- βρ) - 1 Figure 2: E-stability properties in monetary policy ñ debt maturity space. Instability regiones lie above the contours which are indexed by average level of indebtedness..

132 4.5 E-stability: KPR preferences B/4Y =.7 B/4Y = 1.5 B/4Y = 2.2 φ π σ Figure 3: E-stability properties in monetary policy ñ elasticity of substitution space. Instability regiones lie above the contours which are indexed by average level of indebtedness.

133 /s KPR C 1/s GHH C 1/σ- δ/s KPR C 1/σ- δ/s GHH C σ Figure 4: Decomposing wealth and subtitution e ects. The black lines plot s 1 C, which indexes the scale of wealth e ects from holdings of the public debt, for KPR and GHH preferences. The blue lines give the elasiticy of substitution of consumption deemand with respect to the real interest rate. Both are plotted as function of.

134 Forward-looking policy rules Alternative policy rules that responds to expectations ^{ t = ^E t^ t+1 Here we assume that agents know the policy rule

135 E-stability: forward-looking vs. baseline monetary policy rules Baseline rule Forward-looking rule φ π *(1- βρ) - 1 Figure 5: E-stability properties in monetary policy ñ debt maturity space for a standard Taylor rule and an expectations-based Taylor rule. Instability regions lie above the contours.

136 Intuition: Debt Dynamics Using the bond price equation and the monetary policy rule, ^{ t = ^E t^ t+1, permits the evolution of real debt to be written as ^bm t = 1 ^bm t1 ^ t + (1 ) ^E t^ t ^s t +(1 ) ^E t 1 X T =t () T t ^ T +2 ñ Debt depends on expectations regardless of ñ Destabilizing e ects on consumption and debt decline monotonically with

137 Distortionary taxation Consider the model labor taxes ( w t )......and assume now: 1 =1=2

138 1.9 E-stability: distortionary taxes B/4Y =.7 B/4Y = 1.5 B/4Y = 2.2 φ π *(1- βρ) - 1 Figure 6: E-stability properties in monetary policy debt maturity space. Instability regions lie above the contours indexed by average indebtedness.

139 Intuition: Distortionary taxation Same mechanism as before but with a ëtwistí Phillips curve has an extra term ^ t = " + Y C! ^Y t + w (1 w )^ w t # + ^E t 1 X T =t () T t ^w T +1 +(1 ) ^ T +1 Changes in tax rates as cost-push ìshocksî, reinforcing the drift in expectations

140 Conclusion Uncertainty about policy regime can induce drift in expectations High debt levels and short to medium maturity debt induce instability ñ Instability generated through wealth e ects Instability of expectations as ëöscal limití Fundamentally changes the nature of household and Örm responses to shocks ó even if expectations stable in the long-run

141 Motivation Revisited Macroeconomic models depend on household and Örm expectations Policy design ñ Not enough to observe some history of beliefs ñ Need a theory of the determination of beliefs: rational expectations Rational expectations policy design ñ Heavy reliance on managing expectations through announced commitments ñ What are the consequences of imprecise control of beliefs?

142 Motivation II For example, Bernanke (2004): ì[...] most private-sector borrowing and investment decisions depend not on the funds rate but on longer-term yields, such as mortgage rates and corporate bond rates, and on the prices of long-lived assets, such as housing and equities. Moreover, the link between these longer-term yields and asset prices and the current setting of the federal funds rate can be quite loose at times.î

143 The Agenda Simple model of output gap and ináation determination One informational friction: ñ Agents have an incomplete knowledge about the economy Explore constraints imposed on stabilization policy by Önancial market expectations ñ What if the expectations hypothesis of the yield curve does not hold? ñ Is the zero lower bound on nominal interest rates an important constraint?

144 Asset Structure and the Fiscal Authority Exogenous purchases of G t per period Issue two kinds of debt ñ B s t : One period debt in zero net supply with price P s t = (1 + i t) 1 ñ B m t : An asset in positive supply that has the payo structure T (t+1) for T t +1 Let P m t denote the price of this second asset. Asset has the properties: ñ Price in period t +1of debt issued in period t is P m t+1 ñ Average maturity of the debt is (1 ) 1

145 Asset Pricing: Theory I Log-linear approximation of Euler equations for each kind of bond holdings implies ^{ t = ^P t s = ^E t i ^P t m ^P t+1 m ñ Restriction on asset price movements ñ Combined with transversality X ^P t m 1 = ^E t () T t ^{ T T =t ñ Asset price is a function of fundamentals: the expectations hypothesis of the term structure holds; does not imply interest-rate expectations consistent with monetary policy strategy ñ Call: Anchored Financial Market Expectations

146 Optimal Spending Plans I Assume agents understand Öscal policy is Ricardian When the expectations hypothesis of the term structure holds ó aggregate demand relation of the form ^C i t = ^E i t +s 1 C 1X T =t T t i T T +1 (1 ) ^E i t " 1X T t T =t ñ Example of permanent income theory! # ^w T + 1^ T (10) ñ Independent of the average maturity structure of debt; Isomorphic to an economy with one-period debt ó i.e. =0 ñ Independent on long-debt-price expectations

147 Asset Pricing: Theory II Under non-rational expectations and incomplete markets: exist alternative ways to impose no-arbitrage Consider pricing the asset directly in all periods t ^{ t = ^E i t ^P m t ^P m t+1 ñ Given monetary policy and expectations, asset price determined. No-arbitrage consistent forecasts of the short-rate can then be determined by ^E t^{ i T = ^E t i ^P T m ^P T m +1 ñ Because interest rate projections are not directly related to current interest rates ó expectations hypothesis need not hold

148 Demand then determined by ^C i t = ^E i t +s 1 C 1X T =t T t h ^P m T +1 ^P m T (1 ) ^E i t " 1X T t T =t T +1 i! # ^w T + 1^T (11) ñ Nest anchored expectations model when =0;since ^E i t^{ T = ^E i t ^P T ñ In this case forecasting bond prices and forecasting interest rates equivalent As maturity increases this divergence increases

149 Beliefs Formation Agents construct forecasts according to ^E tx i t+t = a X t1 where X = n ; ^w; ^;^{; ^P mo for any T>0. ñ In period t forecasts are predetermined. ñ Beliefs are updated according to the constant gain algorithm a X t =(1 g) a X t1 + gx t where g>0 ñ With i.i.d. shocks and zero debt nests the REE Learning only about the constant

150 Beliefs Formation II Under anchored expectations, based on interest-rate data: ^E i t^{ T = a i t1 Under unanchored expectations, based on bond-price data: ^E t^{ T = ^E t ^P m T ^P m T +1 ñ In general they are not equal ñ Equivalent when =0 = (1 ) a P m t1

151 Calibration Discount factor is =0:99 Labor supply elasticity 1 =2 Nominal rigidities =0:75 Elasticity of demand across di erentiated goods =8

152 Unanchored Expectations and Simple Rules Consider the case of a simple Taylor rule ^{ t = ^ t + y x t Use notion of ìrobust stabilityî ñ Dynamics converge for a given gain coe cient ñ Distinct from E-Stability

153 φ π (policy resp. to inflation) φ x = 0 φ x =.5/4 φ x =.5 Stability with a Taylor Rule ρ (debt duration) Figure 7: Robust stability regions for di erent maturity structures. The three contours correspond to di erent Taylor distinguished by their respond to the current output gap.

154 Intuition Aggregate demand can be written x t = ^{ t + ^E t ^P m t+1 + ^E t 1 X T =t T t h (1 ) ^P m T +1 + T +1i ^A t +s 1 1 " X C (1 ) ^E t T t T =t! ^w T ^T +1# : ñ For inönite-period debt =1: depends only on ^E t ^P m t+1 ñ Requires aggressive adjustment of current nominal interest rates Key mechanism: increasing debt maturity implies arbitrage restrictions in the expectations hypothesis are weaker