Monetary Policy and Rational Asset Price Bubbles Redux

Transcription

1 Monetary Policy and Rational Asset Price Bubbles Redux Jianjun Miao Zhouxiang Shen Pengfei Wang This version: January 208 First Version: November 207 Abstract We show that Galí s (204) counterintuitive results are driven by his choice of a backwardlooking sunspot solution around a stable bubbly steady state. Under adaptive learning, such a steady state and the associated sunspot solutions under optimal monetary policy are not E-stable. When deriving the unique forward-looking minimum stable variable (MSV) solution around a bubbly steady state, we obtain results that are consistent with the conventional views. Such a steady state and the associated MSV solution are E-stable. JEL Classification: E3, E32, E44, E52, G2 Keywords: asset bubble, monetary policy, learning, multiple equilibria, E-stability We thank Jess Benhabib for helpful comments. Department of Economics, Boston University, 270 Bay State Road, Boston, MA Tel.: miaoj@bu.edu. Homepage: Department of Economics, Boston University. Department of Economics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong. Tel: (+852) pfwang@ust.hk

2 Introduction Due to the recent financial crisis during , there is a renewed interest in understanding the role of asset bubbles for business cycles and the associated policy implications. Galí (204) presents an elegant overlapping generations model with nominal rigidities to study the impact of monetary policy on rational asset bubbles. He finds some provocative results that are inconsistent with conventional views. These results are summarized below: A stronger interest rate response to bubble fluctuations (i.e., a leaning against the wind policy ) may raise the volatility of asset prices and of their bubble component. The optimal monetary policy strikes a balance between stabilization of current aggregate demand and the bubble. If the average size of the bubble is sufficiently large, the latter motive will be dominant, making it optimal for the central bank to lower interest rates in the face of a growing bubble. In this paper we revisit Galí s analysis. We argue that his results are driven by his particular choice of the equilibrium solution. In his model there are multiple steady states and multiple equilibria. In particular, there is a continuum of stable bubbly steady states and a continuum of unstable bubbly steady states. He focuses on a backward-looking sunspot solution around a stable bubbly steady state. For this solution the asset bubble is predetermined and does not respond to shocks on impact. Thus an increase in interest rates raises the future bubble size. By contrast we analyze the forward-looking minimal state variable (MSV) solution around an unstable bubbly steady state. For this solution the asset bubble responds to shocks on impact just like any asset prices. An increase in interest rates dampens the asset bubble on impact. We find results that are consistent with conventional views and are different from Galí s results mentioned above. In particular, the optimal policy calls for a leaning-against-the-wind rule. Given that there are multiple steady states and multiple equilibria in Galí s model, which one should we pick? Following the methodology surveyed by Evans and Honkapohja (999, 200), we use learning as a selection device to select a particular steady state and a particular equilibrium. The idea is that agents of the model do not initially have rational expectations and they instead form forecasts by using some adaptive learning rules such as recursive least squares based on the data. The question is whether the agents can learn a particular equilibrium or a particular steady state. Marcet and Sargent (989) and Evans and Honkapohja (999, 200) show that the notion of expectational stability (E-stability) determines local convergence of real time recursive learning algorithms in a wide variety of economic models. See Bullard and Mitra (2002), Adam (2003), Woodford (2003), Duffy and Xiao (2007), Benhabib, Evans, and Honkapohja (202), among others, for the application of learning to select equilibrium in macroeconomic models.

3 We find that the sunspot equilibrium solution adopted by Galí (204) is not E-stable under his optimal monetary policy rule, but the forward-looking MSV solution is E-stable. We also find that the unstable bubbly steady state Pareto dominates the stable bubbly steady state. Moreover the former steady state is E-stable, but the latter is not. Our analysis suggests that the MSV equilibrium around the unstable bubbly steady state studied in this paper is more appealing from the welfare and learning perspective and should be the focus for policy analysis. 2 Solving Galí s Model We first present the equilibrium system of Galí s model without spelling out its detailed setup. A key feature of his model is that once an old bubble bursts, a new bubble can be created as in Martin and Ventura (202) and Miao, Wang, and Xu (205). The equilibrium system consists of eight equations C t +C 2t =, D t +W t =, C t +Q t = W t +U t, [ ] Ct B t +U t = βe t B t+, C 2t+ Q t = B t +U t, [ ] βct 0 = E t ( MW t ), C 2t ( ) Πt ln(+i t ) = lnr+φ π ln +φ Π b ln = β(+i t )E t [ Ct C 2t+ Π t+ ], ( Qt Q ) +lne t Π t+, for nine stochastic processes {C t },{C 2t },{D t },{W t },{Π t },{i t },{Q t },{B t }, and {U t }, where these variables denote respectively the young agent s consumption, the old agent s consumption, dividends, the real wage, the inflation rate, the nominal interest rate, the aggregate bubble, the old bubble, and the new bubble. Moreover, M = ɛ/(ɛ ) denotes the markup. A variable without a time subscript denotes its deterministic steady state value. Define the gross real interest rate as R t = (+i t )E t Π t+. Since there are eight equilibrium conditions for nine variables, the equilibrium system cannot determine the size of the old bubble and the new bubble independently. Following Galí (204), we assume that the value of the new bubble U t follows an exogenous stochastic process. In the deterministic case where U t = U > 0 for all t, Galí (204) shows that the old bubble {B t } satisfies the difference equation B t+ = ( /M)(B t +U) β/m (+β)b t U H(B t,u). () 2

4 He also shows that a necessary and sufficient condition for the existence of a deterministic bubbly steady state is given by M < +β. (2) Furthermore, when this condition is satisfied there exists a continuum of stable bubbly steady states indexed by U, { (Bs (U),U) : B s (U) = H(B s (U),U) for U ( 0,Ū )}, and a continuum of unstable bubbly steady states also indexed by U, { (Bu (U),U) : B u (U) = H(B u (U),U) for U [0,Ū), B u(u) > B s (U) }, where Ū = β +(+β)( W) 2 β(+β)( W) > 0 and W = M. The economy also has a bubbleless steady state in which B = U = 0. In this steady state we can show C = M, C 2 = M M, and R =. β Thus condition (2) is the same as R <, which is the standard condition in the literature (Tirole (985)), i.e., the bubbleless equilibrium is dynamically inefficient. Galí (204) shows that the log-linearized system around a bubbly steady state for any fixed U ( 0,Ū) is given by 0 = c,t +βrc 2,t, (3) c,t = E t c 2,t+ r t, (4) c 2,t = ( Γ)d t +Γb t, (5) q t = Rb t +( R)u t, (6) q t = E t b t+ r t, (7) E t w t = E t d t = 0, (8) r t = φ π π t +φ b q t, (9) where we use a lower case variable x t to denote the log deviation from its steady state value, x t = ln(x t /X). Moreover, we define Γ = ɛb/(ɛb +) and show that the bubbly steady-state real interest rate is given by R = β /M+B /M B = B B +U. Note that there are two bubbly steady states for a fixed U ( 0,Ū). Without risk of confusion, we use the same notation B to represent either one of the steady-state size of the old bubble in the analysis below. 3

5 In Appendix A we show that the above log-linearized equilibrium system can be reduced to a unidimensional system b t = R(φ b +) E tb t+ + φ b ɛb(+β) E t b t (0) φ b + + R R u t + (ɛb φ b)(r ) E t u t. (φ b +)R Our objective is to solve for a rational expectations equilibrium (REE) using (0). Galí (204) assumes that u t is IID. We consider a more general AR() process u t = ρu t +e t, ρ (0,), () where e t is an IID random variable with mean zero and variance σ 2 e. Galí (204) focuses his analysis on a sunspot solution around a stable bubbly steady state. Given (), we can derive the following more general solution. Its proof and the proofs of the remaining propositions in the paper are given in Appendix B. Proposition Fix U ( 0,Ū). For any b 0, there is a linear sunspot solution in a neighborhood of the bubbly stable steady state given by b t = χb t +( R)(+ɛB)ρu t 2 + 2e t + 3e t + 4ξ t + 5ξ t, where ξ t denotes a sunspot shock satisfying E t ξ t = 0, 3 and 5 are arbitrary real numbers, and 2 = 3 +(R )(+φ b), R(φ b +) χ 4 = 5 R(φ b +) χ, χ = R(+ɛB(+β)) (0,). Galí (204) shows that χ = H(B,U)/ B. For a stable bubbly steady state, we must have χ (0, ), which also implies that the backward-looking solution in Proposition is stationary. Galí (204) defines a sunspot variable ξ t = b t E t b t. Substituting this variable into (0) yields a particular solution b t = χb t +(φ b +)( R)u t (φ b ɛb)( R)ρu t 2 (2) which can also be obtained by setting +ξ t +(φ b ɛb(+β))rξ t, 2 = 0, 3 = ( R)(+φ b), 5 = (φ b ɛb(+β))r 4

6 in our general solution given in Proposition. The solution in Galí (204) corresponds to ρ = 0 in (2). For this solution, the initial value b 0 is indeterminate. Galí (204) derives all his results for a fixed b 0. From (2) we can see that monetary policy only affects the anticipated component of the old bubble E t b t through the interest rate coefficient φ b. In the case of ρ = 0, Galí (204) shows that a leaning-against-the-wind policy which corresponds to φ b > 0 generates a larger volatility in the bubble than a policy of benign neglect (φ b = 0). Now we consider the solution in the neighborhood of the unstable bubbly steady state. Proposition 2 Fix U ( 0,Ū). There is a unique forward-looking linear solution in a neighborhood of the unstable bubbly steady state given by where χ = R(+ɛB(+β)) >. ɛb + b t = (R ) χ ρ ρu t + R R [ ] ρ +ɛb +φ b χ ρ + e t, (3) In a neighborhood of the unstable bubbly steady state, we have χ >. The backward-looking solution in (2) is not stationary. We must solve for b t forward to obtain the forward-looking solution in (3) so that b t is stationary. This solution is also called the minimal state variable (MSV) solution in the literature (e.g., Evans and Honkapohja (200)). In the next section we will focus our analysis on this solution. Note that if ρ = 0 as in Galí (204), then b t = e t (R )/R. In this case monetary policy through φ b does not affect bubble dynamics. We thus assume ρ (0,) throughout the paper. 3 Monetary Policy What is the impact of the monetary policy on bubble dynamics? From equations (6) and (3), we can derive the volatility of the aggregate bubble [ ] Var(q t ) = (R ) 2 ɛb + 2 [ ] (R )ρ R χ ρ ρ 2 ( ρ 2 ) σ 2 +ɛb 2 e + σ 2 +φ b χ ρ e. Thus a leaning-against-the-wind policy (i.e., φ b > 0) generates a lower volatility of the aggregate bubblethanapolicyofbenignneglect(φ b = 0), contrarytogalí sresult. Thevolatility isminimized when φ b. Interestingly, when φ b decreases to negative infinity, the bubble volatility also decreases to zero. We use (3) to compute the volatility of the old bubble Var(b t ) = ( ) ɛb + 2 χ ρ ρ (R ) 2 σ 2 e ρ 2 + ( ) R 2 [ R ρ +ɛb +φ b χ ρ + ] 2 σ 2 e. 5

7 It is minimized at φ b = ρ(+ɛb) χ ρ < 0. (4) Figure presents the relation between φ b and the volatilities of the old and aggregate bubbles. We choose the same parameter values as in Galí (204) by setting β =, ɛ = 6, U = These values imply B s = 0., B u = 0.458, and M =.2. While Galí (204) studies equilibria around the stable bubbly steady state B s = 0., we focus on the solution around the unstable bubbly steady state B u = Standard Deviation Standard Deviation Figure : Monetary Policy and Bubble Volatility Note: This figure plots the standard deviations of the aggregate bubble q t and old bubble b t for various coefficients φ b. The vertical line indicates the value of φ b that minimizes the standard deviation of the old bubble. The parameter values are β =, ɛ = 6, U = 0.75, φ π = 2, ρ = 0.8, and σ 2 e = 0.0. We focus on the unstable bubbly steady state with B = To understand Figure, we consider the economy s responses to an exogenous positive bubble shock to u t. By equations (6) and (7), we see that the old bubble satisfies the asset pricing equation b t = R E tb t+ R r t ( R) R u t. (5) Solving forward shows that the old bubble is equal to the (negative) discounted value of future interest rates and new bubbles. Since 0 < R <, new bubbles {u t } act as negative dividends. An increase in u t has a direct effect of lowering b t and an indirect effect through the change in the interest rate r t. In contrast to Galí (204), b t is a jump variable and responds to shocks on impact. Figure 2 presents the impulse response functions for {b t },{q t },{r t }, and {π t } given a % shock to e 0. When monetary policy does not respond to bubbles (φ b = 0), a positive shock to expand the new bubble u 0 at date 0 crowds out the value of old bubbles b 0 and dampens the aggregate bubble q 0. When φ b > 0, the central bank will cut the interest rate and hence the fall of the old and 6

8 aggregate bubbles is mitigated. Thus a leaning-against-the-wind policy lowers the bubble volatility in response to the bubble shock. When φ b < 0, the old and aggregate bubbles may rise on impact in response to a positive bubble shock. When the central bank cuts the interest rate to encourage bubbles, this effect may dominate the direct negative effect of the rise in the new bubble on the old bubble as shown in equation (5). As shown in Figure 2, when φ b decreases from 2 to 5, the old and aggregate bubbles are dampened and the fall of interest rate is also mitigated. If bubbles expanded, the central bank would cut the interest rate more, which in turn would encourage bubbles further. This positive feedback effect might make the bubble explode percent 0-2 percent percent b = -5 b = -2 b = 0 b = 2 b = 5 percent Figure 2: Impulse Responses to A New Bubble Shock Note: This figure plots the impulse response functions for a one percent positive new bubble shock, in percentage deviation from the steady state. The parameter values are β =, ɛ = 6, U = 0.75, φ π = 2, ρ = 0.8, and σ 2 e = 0.0. We focus on the unstable bubbly steady state with B = Since firms adjust price one period in advance before shocks are realized, the inflation rate π t is predetermined. Thus it does not respond to the bubble shock on impact. As shown in Figure 2, 7

9 it may rise or fall in the second period depending on the value of φ b. In Appendix C we show that the inflation rate around the unstable bubbly steady state is given by π t = ρ(r )[ρ(ɛb +)+(+φ b)(βɛbr ρ)] u t. φ π (χ ρ) If φ b = 0, the inflation rate falls in the second period because R < and χ >. The central bank can stabilize inflation by two strategies: First, it can set φ π at an arbitrary large value and set φ b at a finite value. Second, it can set φ π at a finite value and set φ b = ρ(ɛb +)/(ρ βɛbr). In Galí s (204) model, inflation is not a source of welfare losses given synchronized price-setting and an inelastic labor supply. Thus it is not optimal for the central bank to stabilize inflation. To study optimal monetary policy, we follow Galí (204) to take the unconditional mean of an agent s lifetime utility as a welfare criterion. In a neighborhood of a steady state, we can derive the second-order approximation to the mean: E[lnC,t +βlnc 2,t+ ] lnc +βlnc 2 2 (Var(c,t)+Var(c 2,t )). By the resource constraint C,t +C 2,t =, Var(c,t ) is proportional to Var(c 2,t ). Thus the optimal monetary policy that maximizes welfare will minimize the variance of In Appendix C we show that c 2,t = ( Γ)d t +Γb t. d t = χ(r )[φ b(ρ ɛbβr) ɛb(βr+ρ)] βr 2 e t. (+φ b )(χ ρ) Thus minimizing the volatility of dividends calls for setting φ b = ɛb(βr+ρ) ɛbβr ρ. However this policy would raise the volatility of the old bubblebecause it is minimized at a different value given in (4). Thus optimal monetary policy trades off between the volatility of dividends and the volatility of the old bubble. and Note that b t and d t are also correlated. In Appendix C we derive that c 2,t = ɛbρ(r ) u t + (R )ρ(φ b ɛb) χ ρ βr(+φ b )(χ ρ) e t, ( ) ɛbρ(r ) 2 [ ] Var(c 2,t ) = ( ρ 2 ) σ 2 e χ ρ + (R )ρ(φb ɛb) 2 σ 2 e βr(+φ b )(χ ρ). From this equation we can show that the optimal coefficient is given by φ b = ɛb > 0. Thus the leaning-against-the-wind policy is optimal. Moreover the optimal coefficient increases with the size of the bubble. Figure 3 illustrates the relation between φ b and Var(c 2t ). The welfare loss is minimized at φ b =

10 .5 4 Standard Deviation 0.5 Standard Deviation Figure 3: Monetary Policy and Welfare Note: This figure plots the standard deviations of dividend d t and consumption of the old c 2,t for various values of φ b. The vertical lines indicate the values of φ b that minimize the standard deviation of consumption and dividend respectively. The parameter values are β =, ɛ = 6, U = 0.75, φ π = 2, ρ = 0.8, and σ 2 e = 0.0. We focus on the unstable bubbly steady state with B = Learning and Equilibrium Selection There are multiple (deterministic) steady states and multiple REE solutions in Galí (204). We will use learning as a selection device to select a particular steady state and a particular REE solution. To understand the basic idea, we consider an economic model with a solution described as a particular parameter vector (e.g., the parameters of an autoregressive process or a steady state). Under adaptive learning agents do not know but estimate it from data using a statistical procedure such as least squares. This leads to estimates t at time t and the question is whether t as t. Evans and Honkapohja (200) show that, for a wide range of economic examples and learning rules, convergence is governed by the corresponding E-stability condition, i.e., the local asymptotic stability of under the differential equation d dτ = T (), (6) where τ denotes notional or virtual time, T () is the mapping from the perceived law of motion (PLM) to the implied actual law of motion (ALM) T (). In the following analysis we will check the E-stability condition. 4. Learning a Steady State We start by the steady states. It is clear that any bubbly steady states Pareto dominates the bubbleless steady state. The following result shows that the unstable bubbly steady state Pareto 9

11 dominates the stable bubbly steady state for a fixed size of new bubble. Proposition 3 For any fixed U ( 0,Ū), the bubbly unstable steady state Pareto dominates the bubbly stable steady state. Which steady state is E-stable? We consider the deterministic dynamical system in () where we replace B t+ by a forecast B e t+. Suppose that the PLM is B t+ = Φ for an arbitrary Φ. Then the ALM is B t = H (Φ,U) by (). The differential equation is given by dφ dτ = H (Φ,U) Φ. Since 0 < H(B,U) B < at the stable steady state and H(B,U) B immediately obtain the following result. > at the unstable steady state. We Proposition 4 For any fixed U ( 0,Ū), the bubbly unstable steady state is E-stable and the bubbly stable steady state is not E-stable. 4.2 Learning MSV Solution In this subsection we study the MSV solution in (3). Suppose the PLM is given by b t = u t + 2 e t. Substitute this PLM into the right-hand side of (0), we can derive the ALM b t = u t + 2 e t, where (, 2 ) = T(, 2 ) for some mapping T given in Appendix B. Let = (, 2 ). We can analyze the asymptotic stability of the system of differential equations in (6) and derive the following result. Proposition 5 The MSV solution in Proposition 2 is E-stable if and only if φ b >. In the previous section we have shown that the optimal coefficient φ b is positive for the MSV solution. The preceding proposition shows that the MSV solution under optimal monetary policy is E-stable. 4.3 Learning Sunspot Solution Now we check the E-stability of the sunspot solutions in Proposition. Suppose the PLM is b t = 0 b t + u t e t + 3 e t + 4 ξ t + 5 ξ t, (7) 0

12 Substitute this PLM into the right-hand side of (0), we can derive the ALM b t = 0 b t + u t e t + 3 e t + 4 ξ t + 5 ξ t, (8) where( 0,,..., 5 ) = T( 0,..., 5 )forsomemappingt given inappendixb. Let = ( 0,..., 5 ). We can analyze the asymptotic stability of the system of differential equations in (6) and derive the following result. Proposition 6 For φ b > the sunspot solution in Proposition is not E-stable. Galí (204) shows that the optimal response coefficient φ b that minimizes the welfare loss is greater than. Proposition 6 shows that the equilibrium under this optimal policy is not E-stable. 5 Conclusion In this paper we have shown that Galí s (204) counterintuitive results are driven by his choice of a backward-looking sunspot solution around a stable bubbly steady state. His model also features a continuum of bubbly steady states. When deriving the unique forward-looking MSV solution around a bubbly steady state, we obtain results that are consistent with the conventional views. We apply learning as a selection device to select steady state and equilibrium. We find that the unstable bubbly steady state is E-stable and the associated MSV equilibrium is E-stable under optimal monetary policy. But the stable bubbly steady state is not E-stable and the associated sunspot equilibrium is not E-stable under optimal monetary policy. Thus the unstable bubbly steady state and the associated MSV equilibrium should be the focus for policy analysis. In an infinite-horizon framework without recurrent creation of new bubbles, Miao and Wang (208) prove that the economy has two steady states. The local equilibrium around the bubbly steady state is unique and the local equilibrium around the the bubbleless steady state is indeterminate of degree one. Miao, Wang, and Xu (205) and Dong, Miao, and Wang (207) incorporate recurrent bubbles and show that the economy has a continuum of bubbly states as in Galí (204). However, they are unable to prove the stability of these steady states analytically due to the complexity of their multi-dimensional equilibrium systems. In contrast to Galí (204), their numerical results indicate that each bubbly steady state is a saddle point and the local equilibrium around each bubbly steady state is unique. We suspect that the difference in results may be due to the difference in the infinite-horizon and overlapping-generations frameworks. Further theoretical research is needed to understand this issue.

13 Appendix A Deriving Equilibrium Bubble Dynamics Combining (3) to (5) we can obtain r t = ( Γ)E t d t+ +ΓE t b t+ +βr(( Γ)d t +Γb t ) = ΓE t b t+ +βr(( Γ)d t +Γb t ), where we have used E t d t+ = 0 by (8) in the second equality. Combining the equation above with (6) and (7) yields r t = Γ(r t +Rb t +( R)u t )+βr(( Γ)d t +Γb t ). We substitute Γ = ɛb/(ɛb +) into the equation above to obtain r t = ɛbr(+β)b t +ɛb( R)u t +βrd t. (A.) Taking expectations conditional on information at time t we obtain E t r t = ɛbr(+β)e t b t +ɛb( R)E t u t, (A.2) where we have used E t d t = 0. We use equation (A.2) and interest rate rule (9) to derive r t E t r t = φ π (π t E t π t )+φ b (q t E t q t ) = φ b (q t E t q t ) = φ b R(b t E t b t )+φ b ( R)(u t E t u t ), (A.3) where the second equality follows from π t = E t π t due to price stickiness and we use (6) to substitute for q t to derive the third equality. Using (A.2) and (A.3) we derive r t = r t E t r t +E t r t = φ b Rb t +(ɛb(+β) φ b )RE t b t +φ b ( R)u t +(ɛb φ b )( R)E t u t. Now we substitute the equation above into (7) and use (6) to derive E t b t+ = Rb t +( R)u t +φ b Rb t (φ b ɛb(+β))re t b t +φ b ( R)u t (φ b ɛb)( R)E t u t = (φ b +)Rb t (φ b ɛb(+β))re t b t +(φ b +)( R)u t (φ b ɛb)( R)E t u t. We then obtain (0). Q.E.D. 2

14 B Proofs Proof of Proposition : Conjecture that the solution takes the form b t = 0 b t + u t e t + 3 e t + 4 ξ t + 5 ξ t, where 0,, 2, 3, 4, and 5 are coefficients to be determined. Substituting this solution into (0) yields b t = R(φ b +) [ 0b t + u t + 3 e t + 5 ξ t ] + φ b ɛb(+β) ( ) 0 b t + φ b + u t e t + 5 ξ t + R R (ρu t +e t )+ (ɛb φ b)(r ) ρu t. (φ b +)R That is, b t = R(φ b +) [ ( ) 0 0 b t + u t e t + 3 e t + 4 ξ t + 5 ξ t + (ρu t 2 +e t )+ 3 e t + 5 ξ t ] + φ b ɛb(+β) ( ) 0 b t + φ b + u t e t + 5 ξ t + R ( ρ 2 ) (ɛb φ u t 2 +ρe t +e t + b )(R ) ( ρ 2 ) u t 2 +ρe t. R (φ b +)R Using the conjectured form for b t again and matching coefficients, we obtain 0 = R(φ b +) φ b ɛb(+β) φ b + 0, (B.) = R(φ b +) ( 0 +ρ )+ φ b ɛb(+β) φ b + + R R ρ2 + (ɛb φ b)(r ) ρ 2, (φ b +)R (B.2) 2 = R(φ b +) ( )+ R R, (B.3) 3 = R(φ b +) ( )+ φ b ɛb(+β) φ b R R ρ+ (ɛb φ b)(r ) ρ, (φ b +)R (B.4) 4 = R(φ b +) ( ), (B.5) 5 = R(φ b +) φ b ɛb(+β). φ b + (B.6) There are two solutions for 0 : 0 = 0 and 0 = χ = R(+ɛB(+β)). In a neighborhood of the stable bubbly steady state, we have χ (0,). The only stationary solution must corresponds to 0 = χ as Galí (204) points out. We can then solve for the other 3

15 coefficients: = ( R)(+ɛB)ρ, and 3 and 5 are arbitrary numbers. Q.E.D. 2 = 3 +(R )(+φ b ), R(φ b +) χ 4 = 5 R(φ b +) χ, Proof of Proposition 2: We take expectations conditional on information at time t on both sides of (0) to obtain [ E t b t φ ] b ɛb(+β) φ b + This implies that E t b t = = + R(φ b +) E t b t+ [ R R + (ɛb φ ] b)(r ) ρu t. (φ b +)R R[+ɛB(+β)] E ( R)(ɛB +) t b t+ R(+ɛB(+β)) ρu t. By iterating the equation above forward we can derive ( ( R)(ɛB +) E t b t = R(+ɛB(+β)) ( R)(ɛB +) = ρu t, χ ρ ρ/r[+ɛb(+β)] under the condition χ R[+ɛB(+β)] >. Therefore we also have E t b t+ = ) ρu t ( R)(ɛB +) ( R)(ɛB +) ρu t = (ρ 2 u t +ρe t ). χ ρ χ ρ Substituting the preceding expressions for E t b t+ and E t b t into (0), we obtain the rational expectations solution in (3). Q.E.D. Proof of Proposition 3: We use lifetime utility as the welfare criterion. Define the steady state welfare as W f ln(c )+βln(c 2 ), where C and C 2 denote the steady-state consumption of a consumer in his young and old. In a steady state we have C = /M B and C 2 = /M+B. Therefore W f = ln( M B)+βln( M +B). We can compute W f B = ( M C C 2 (+β) +β ) B. 4

16 Denote B /M /(+β). Note that B > 0 under the condition M < +β. This implies that welfare is increasing with B when B < B. As shown in Galí (204) Lemma, for any U (0,Ū) themodel has two bubblysteady states. Moreover the stableone B s is always less than theunstable one B u. Thus to show the welfare is greater at B u than at B s, it suffices to show that B u < B. Since B u is the larger root of equation H(B,U) = B, we have Therefore B u = B u B = +β (+U M )+ (+U +β M )2 4(+β)( M )U 2(+β) ( U +β M )+ (+U +β M )2 4(+β)( M )U 2(+β) Note that U +β M < 0 by (2). To show B u < B, it suffices to show that ( U +β M )2 > (+U +β M )2 4(+β)( M )U... This inequality is equivalent to 4(+β)U > 4U, which holds true since U,β > 0. Q.E.D. Proof of Proposition 4: See the main text. Proof of Proposition 5: Conjecture the solution takes the form b t = u t + 2 e t. This is the perceived law of motion (PLM). Substitute this PLM into the right hand side of (0) to get the actual law of motion (ALM): b t = R(φ b +) (ρu t +e t )+ φ b ɛb(+β) φ b + u t + R R (ρu t +e t )+ (ɛb φ b)(r ) ρu t (φ b +)R = u t + 2 e t. This defines a mapping from the PLM to the ALM, (, 2 ) = T(, 2 ), where [ ρ = R(φ b +) + φ ] b ɛb(+β) φ b + + R ( ) ɛb R ρ φb φ b + +, 2 = R(φ b +) + R R. 5

17 Expectational stability is determined by the following matrix differential equation: d(, 2 ) dτ = T(, 2 ) (, 2 ) ) + R = [( ρ χ R(φ b +) ( R ρ ɛb φb φ b + )u + ] t. R(φ b +) 2 + R R The Jacobian matrix (evaluated at the REE solution) is given by [ ρ χ ] R(φ J = b +) 0. R(φ b +) The eigenvalues of the Jacobian matrix are λ =, λ 2 = ρ χ R(φ b +). E-stability requires that all eigenvalues are less than zero. Since in the unstable steady state χ >, E-stability requires φ b >. Q.E.D. Proof of Proposition 6: Suppose that the PLM for b t is given by (7). Substituting it into (0) yields the ALM b t = 0 b t + u t e t + 3 e t + 4 ξ t + 5 ξ t, (B.7) where 0,..., 5 are given by the expressions on the right-hand sides of equations (B.) through (B.6). Let denote the vector of 0,..., 5 and T () denote the vector of 0,..., 5. We consider the differential equation where T () can be explicitly written as d dτ = T (), 0 = = 2 = 3 = 4 = 5 = R(φ b +) φ b ɛb(+β) φ b + 0, R(φ b +) ( 0 +ρ) + φ b ɛb(+β) φ b + + R(φ b +) ( )+ R R, R(φ b +) ( )+ φ b ɛb(+β) φ b R(φ b +) ( ), R(φ b +) φ b ɛb(+β) φ b + 5. (ɛb +)(R ) (φ b +)R ρ2, (ɛb +)(R ) (φ b +)R ρ, 6

18 The Jacobian matrix of d/dτ evaluated at the sunspot solution in Proposition ( 0 = χ, = ( R)( ɛb), 2 = 2, 3 = 3, 4 = 4, 5 = 5 ) is χ R(φ b +) ( R)( ɛb) ρ R(φ b +) R(φ b +) J = R(φ b +) χ 2 0 R(φ b +) R(φ b +) 0 0 R(φ b +) 3 R(φ b +) R(φ b +) R(φ b +) χ R(φ b +) R(φ b +) χ The eigenvalues are R(φ b +), ρ R(φ b +), χ χ R(φ b +), 0, R(φ b +), and 0. Notice that there are two zero eigenvalues because 3 = 3 and 5 = 5. The other eigenvalues are negative if φ b <. We then complete the proof. Q.E.D. C Deriving MSV Equilibrium From Proposition 2 we have the forward-looking MSV solution for the old bubble: (R )(ɛb +) b t = ρu t + R [ ] ρ(ɛb +) χ ρ R (φ b +)(χ ρ) + e t.. (C.) We use this solution to derive solutions for other variables in the model. By (6) we obtain the solution for q t : q t = Rb t +( R)u t [ = ( R) By (7) we obtain the solution for r t : R(ɛB +) χ ρ r t = E t b t+ q t [ (ɛb +)(ρ R) = (R ) χ ρ By (9) we obtain the solution for π t : ] [ ρu t +(R ) ] + ρ(+ɛb) (φ b +)(χ ρ) ] e t. ρu t + φ bρ(r )(ɛb +) e t. (φ b +)(χ ρ) π t = (R )[(ɛb +)ρ+(φ b +)(ɛbrβ ρ)] ρu t. φ π (χ ρ) Substituting (C.3) and (C.) into (A.) we obtain the solution for d t : By (5) we obtain the solution for c 2,t : d t = χ(r )[φ bρ ɛb(βr(+φ b )+ρ)] βr 2 e t. (+φ b )(χ ρ) c 2,t = ( Γ)d t +Γb t = ɛbρ(r ) u t + ρ(r )(φ b ɛb) χ ρ βr(+φ b )(χ ρ) e t. (C.2) (C.3) (C.4) 7

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