Asset Price Bubbles and Monetary Policy in a New Keynesian Model with Overlapping Generations

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1 Asset Price Bubbles and Monetary Policy in a New Keynesian Model with Overlapping Generations Jordi Galí December 2016 Abstract I develop an extension of the basic New Keynesian model with overlapping generations, finite lives and retirement. In contrast with the standard model, the proposed framework allows for the existence of rational expectations equilibria featuring asset price bubbles. I examine the conditions under which bubbly equilibria may emerge and the implications for the design of monetary policy. JEL Classification No.: E44, E52 Keywords: volatility. monetary policy rules, stabilization policies, asset price Centre de Recerca en Economia Internacional (CREI), Universitat Pompeu Fabra, and Barcelona GSE. jgali@crei.cat. I am thankful for comments to Davide Debortoli, Alberto Martín, Jaume Ventura, and seminar participants at CREI, NBER Summer Institute, Mannheim, Rome MFB Conference, and Seoul National University. I am grateful to Christian Hoynck, Cristina Manea, and Matthieu Soupré for excellent research assistance. I acknowledge the European Research Council for financial support under the European Union s Seventh Framework Programme (FP7/ , ERC Grant agreement n o ) and the CERCA Programme/Generalitat de Catalunya for funding.

2 Speculative bubbles in asset prices are viewed by many economists and policymakers as an important source of macroeconomic instability, with the bursting of some large bubble often pointed to as a key factor behind many financial crises. A monetary policy that focuses narrowly on inflation and output stability but which neglects the emergence and rapid growth of asset bubbles is often perceived as a potential risk to medium-term macroeconomic and financial stability. 1 Interestingly, the recurrent reference to bubbles in the policy debate contrasts with their conspicuous absence in modern monetary models. A likely explanation for this seeming anomaly lies in the fact that standard versions of the New Keynesian model, the workhorse framework used in monetary policy analysis, leave no room for the existence of bubbles in equilibrium, and hence for any meaningful model-based discussion of their possible interaction with monetary policy. 2 In the present paper I develop a modified version of the New Keynesian model featuring overlapping generations of finite-lived consumers with retirement. 3 The assumption of an infinite sequence of generations makes it possible for the transversality condition of any individual consumer to be satisfied in equilibrium, even in the presence of a bubble that grows at the rate of interest. 4 On the other hand, the assumption of retirement (or, more generally, of an eventual transition to inactivity) can generate an equilibrium rate of interest 1 See, e.g., Borio and Low (2002) for an early statement of that view. Taylor (2014) points to excessively low interest rates in the 2000s as a factor behind the housing boom that preceded the financial crisis of The reason is well known: the equilibrium requirement that the bubble grows at the rate of interest violates the transversality condition of the infinite-lived representative consumer assumed in the New Keynesian model (as well as most macro models). See, e.g., Santos and Woodford (1997). 3 Other authors have extended the New Keynesian model to incorporate overlapping generations of finite-lived agents into the New Keynesian framework, though none of them has allowed for the existence of bubbles. Piergallini (2006) develops a related model with money in the utility function to analyze the implications of the real balance effect on the stability properties of interest rate rules. Nisticò (2012) discusses the desirability of a systematic monetary policy reponse to stock price developments in a similar model, but in the absence of bubbles. Del Negro, Giannoni and Patterson (2015) propose a related framework as a possible solution to the "forward guidance puzzle." None of the previous authors allow for retirement in their frameworks. That feature plays a central role in the emergence of asset price bubbles in the model proposed here. 4 And even though that transversality condition does not hold for the economy as a whole. 1

3 below the economy s trend growth rate, which is a condition necessary for the size of the bubble to remain bounded relative to the size of the economy. Finally, and in contrast with most models with bubbles found in the literature, the assumption of sticky prices a key feature of the New Keynesian model makes monetary policy non-neutral, allowing it to influence the size of the bubble; on the other hand, price stickiness makes it possible for aggregate bubble fluctuations to influence aggregate demand and, hence, output and employment. An appealing feature of the framework developed here is that it nests the standard New Keynesian model as a limiting case, when the probability of death and that of retirement approach zero. After deriving the equations describing the model s equilibrium, I characterize the balanced growth paths consistent with that equilibrium and discuss the conditions under which a non-vanishing bubble may exist along those paths. If the probability of retirement is suffi ciently low (relative to the consumer s discount rate), there exists a unique balanced growth path, and it is a bubbleless one (as in the standard model). On the other hand, if the probability of retirement is suffi ciently high (but plausibly so), a multiplicity of bubbly balanced growth paths is shown to exist, in addition to a bubbleless one (which always exists). Once I characterize the existence and potential multiplicity of balanced growth paths bubbly and bubbleless I turn to the analysis of the stability properties of those paths and the role of monetary policy in determining those properties. Several findings of interest emerge from that analysis. First, even in the absence of bubbles, the possibility of an interest rate below the growth rate, when combined with a suffi ciently low responsiveness of inflation to the output gap, implies that a kind of "reinforced Taylor principle" is needed in order to guarantee a locally unique equilibrium. A second finding of interest relates to the possibility of expectations-driven fluctuations in a neighborhood of a balanced growth path. I show that under some conditions (in particular on the coeffi cients of the interest rate rule) fluctu- 2

4 ations in economic activity and inflation may arise driven by fluctuations in the size of the bubble, even in the absence of any shocks to fundamentals. In some cases, such fluctuations are more likely to arise under a rule that has the central bank raise interest rates systematically in response to an increase in the size of the bubble (i.e. a "leaning against the bubble" policy). In other cases, bubbledriven sunspot fluctuations arise independently of the central bank s response to bubbles. The analysis of simulations of stochastic bubbly equilibria suggest that, contrary to conventional wisdom, a monetary policy that "leans against the bubble" is likely to generate more volatility in output and inflation than a policy that focuses exclusively on stabilizing inflation. The rest of the paper is organized as follows. The next section summarizes the related literature. Section 2 describes the basic framework underlying the analysis in the rest of the paper. Section 3 characterizes the economy s balanced growth paths, bubbleless and bubbly. Section 4 analyzes the equilibrium dynamics in a neighborhood of a balanced growth path, and the role of monetary policy in preventing indeterminacy of equilibria. Section 5 provides an example of aggregate fluctuations driven by a (stochastic) bubble, and of the possible consequences of "leaning against the bubble" policies. Section 6 summarizes and concludes. 1 Related Literature Much of the analysis of rational bubbles in general equilibrium found in the literature has been based on real models. An early reference in that category is Tirole (1985), using a conventional overlapping generations (OLG) framework with capital accumulation. A more recent one is Martín and Ventura (2012), who modify the Tirole model by introducing financial frictions that are alleviated by the existence of a bubble. There is also an extensive literature on bubbles using monetary models with fully flexible prices. In most of those models, including the seminal paper by 3

5 Samuelson (1958), money itself is the bubbly asset. Asriyan et al. (2016) provide a more recent example, introducing the notion of a nominal bubble. While monetary policy is not always neutral in those models, the mechanism through which its effects are transmitted is very different from that emphasized in standard models with nominal rigidities. A number of papers have modified the standard New Keynesian model by introducing overlapping generations à la Blanchard-Yaari, though none of them has considered the possible existence of bubbles. Piergallini (2006) develops a related model with money in the utility function to analyze the implications of the real balance effect on the stability properties of interest rate rules. Nisticò (2012) discusses the desirability of a systematic monetary policy response to stock price developments in a similar model, but in the absence of bubbles. Del Negro, Giannoni and Patterson (2015) propose a related framework as a possible solution to the "forward guidance puzzle." None of the previous authors allow for retirement in their frameworks. That feature plays a central role in the emergence of asset price bubbles in the model proposed here. Bernanke and Gertler (1999, 2001) analyze the possible gains from "leaning against the wind" monetary policies in a New Keynesian model in which stock prices contain an ad-hoc deviation from their fundamental value. The properties of that deviation differ from those of a rational bubble, which cannot exist in their model, which assumes an infinite-lived representative consumer. In Galí (2014) I carried out a similar analysis of monetary policy rules in a sticky price model in which rational bubbles may exist in equilibrium due to the assumption of an infinite sequence of overlapping generations. While closest in spirit to the present paper, the framework used in that paper differed significantly from the New Keynesian framework in many dimensions. In particular, employment and output were constant in equilibrium, independently of fluctuations in the aggregate bubble, which had only a redistributive effect. On the other hand, the assumption of two-period lived individuals, while convenient, cannot be easily reconciled with the frequency of observed asset boom-bust episodes (not to say with the observed duration of individual prices). By way of 4

6 contrast, the model developed here displays endogenous fluctuations in output and employment in response to fluctuations in asset price bubbles, and it is consistent with a calibration of the model to a quarterly frequency (as is convention in the business cycle literature). Finally, an additional advantage of the framework developed below is that it nests the standard New Keynesian model as a limiting case. 2 A New Keynesian Model with Overlapping Generations Next I describe the basic framework underlying the analysis in the rest of the paper. 2.1 Consumers I assume an economy with overlapping generations of the "perpetual youth" type, as in Yaari (1965) and Blanchard (1984). The size of the population is constant and normalized to one. Each individual has a constant probability γ of surviving into the following period, independently of his age and economic status ("active" or "retired"). A cohort of size 1 γ is born (in an economic sense) and becomes active each period. Thus, the size in period t s of a cohort born in period s is given by (1 γ)γ t s. At any point in time, active and retired individuals coexist in the economy. Active individuals supply labor and manage their own firms, which they set up when they join the economy. I assume that each active individual faces a constant probability 1 υ of permanently losing his job and quitting his entrepreneurial activities. That probability is independent of his age. For convenience, below I refer to that transition as "retirement," though it should be clear that it can be given a broader interpretation. 5 5 Gertler (1999) introduces retirement in a similar fashion in a model of social security. More recently, Carvalho et al (2016) have used a version of the Gertler model to analyze the sources of low frequency changes in the equilibrium real rate. Both papers develop real models, in contrast to the present one, and do not consider the possibility of bubbles. 5

7 The previous assumptions imply that the size of the active population (and, hence, the measure of firms) at any point in time is given by α (1 γ)/(1 υγ) (0, 1]. A representative consumer from cohort s, standing in period 0, chooses a consumption plan to maximize expected lifetime utility E 0 t=0 (βγ) t log C t s subject to the sequence of period budget constraints 1 P t α 0 P t (i)c t s (i)di + E t {Λ t,t+1 Z t+1 s } = A t s + W t N t s (1) for t = 0, 1, 2,... β 1/(1 + ρ) (0, 1) is the discount factor. C t s ( ɛ α 1 α ɛ 0 C t s(i) 1 1 ɛ 1 ɛ di) is a consumption index, with C t s (i) being the quantity purchased of good i [0, α], at a price P t (i). P t ( α 1 α 0 P t(i) 1 ɛ di ) 1 1 ɛ is the price index. Complete markets for state-contingent securities are assumed, with Z t+1 s denoting the stochastic payoff (expressed in units of the consumption index) generated by a portfolio of securities purchased in period t, with value given by E t {Λ t,t+1 Z t+1 s }, where Λ t,t+1 is the stochastic discount factor for one-periodahead (real) payoffs. Only individuals who are alive in period t can trade in securities markets. Variable A t s denotes financial wealth at the start of the period. For individuals other than those joining the economy in the current period, A t s = Z t s /γ, where the term 1/γ captures the additional return on wealth resulting from an annuity contract. As in Blanchard (1984), that contract has the holder receive each period from a (perfectly competitive) insurance firm an annuity payment proportional to his financial wealth, in exchange for transferring the latter to the insurance firm upon death. 6 6 Thus, individuals who hold negative assets will pay an annuity fee to the insurance company. The latter absorbs the debt in case of death. The insurance arrangement can also be replicated through securities markets. In that case the individual will purchase a portfolio that generates a random payoff A t+1 s if he remains alive, 0 otherwise. The value of that payoff will be given by E t{λ t,t+1 γa t+1 s } which is equivalent to the formulation in the main text, given that A t s = Z t s /γ. 6

8 Variable W t denotes the (real) wage per hour, and N t s is the number of individual work hours. Both the wage and work hours are taken as given by each worker and assumed to be common to all active individuals, i.e. N t s = N t /α, where N t is aggregate employment. 7 Note that N t s = 0 for retired individuals. Finally, I assume a solvency constraint of the form lim T γ T E t {Λ t,t+t A t+t s } 0 for all t, where Λ t,t+t is determined recursively by Λ t,t+t = Λ t,t+t 1 Λ t+t 1,t+T. 8 The problem above yields a set of optimal demand functions C t s (i) = 1 ( ) ɛ Pt (i) C t s (2) α for all i [0, α], which in turn imply α 0 P t s(i)c t s (i)di = P t C t s. Thus we can rewrite the period budget constraint as: P t C t s + γe t {Λ t,t+1 A t+1 s } = A t s + W t N t s (3) The consumer s optimal plan must satisfy the optimality condition 9 and the transversality condition Λ t,t+1 = β C t s C t+1 s (4) { } lim T γt E t Λt,t+T A t+t s = 0 (5) with (4) holding for all possible states of nature (conditional on the individual remaining alive in t + 1). 2.2 Firms Each individual is endowed with the know-how to produce a differentiated good, and sets up a firm with that purpose when he joins the economy. Each firm 7 By not including hours of work in the utility function I effectively eliminate any wealth effects that would generate systematic counterfactual differences in the quantity of labor supplied by active individuals across age groups. Alternatively one may assume preferences that rule out those wealth effects, but at the cost of rendering the analysis below much less tractable. 8 Note that (Λγ) 1 is the "effective" interest rate paid by a borrower in the steady state. The solvency constraint thus has the usual interpretatin of a no-ponzi game condition. 9 Note that that in the optimality condition the probability of remaining alive γ and the extra return 1/γ resulting from the annuity contract cancel each other. Complete markets guarantee the same consumption growth rate between two different periods for all consumers alive in the two periods. 7

9 remains operative until its founder retires or dies, whatever comes first. 10 All firms have an identical technology, represented by the linear production function Y t (i) = Γ t N t (i) (6) where Y t (i) and N t (i) denote output and employment for firm i [0, α], respectively, and Γ 1 + g 1 denotes the (gross) rate of productivity growth. Individuals cannot work at their own firms, and must hire instead labor services provided by others. 11 firm Aggregation of (2) across consumers yields the demand schedule facing each C t (i) = 1 α ( ) ɛ Pt (i) C t where C t (1 γ) t s= γt s C t s is aggregate consumption. Each firm takes as given the aggregate price level P t and aggregate consumption C t. As in Calvo (1983), each incumbent firm may reset its price only with probability 1 θ in any given period, independently of the time elapsed since the last price adjustment. All newly created firms are allowed to set prices. Accordingly, a fraction 1 υγθ of all firms reset their prices in any given period, while a fraction υγθ keep their prices unchanged. P t The above environment implies that the aggregate price dynamics are described by the equation P 1 ɛ t = υγθp 1 ɛ t 1 + (1 υγθ)(p t ) 1 ɛ where P t is the price set in period t by firms reoptimizing their price. 12 A log-linear approximation of the previous difference equation around the zero inflation steady state yields (letting lower case letters denote the logs of the 10 The assumption of finite-lived firms (or more generally, for firms whose dividends shrink relative to the size of the economy) is needed in order for bubbles to exist in equilibrium. By equating the probability of a firm s survival to that of its owner remaining alive and active I effectively equate the rate at which dividends and labor income are discounted, which simplifies considerably the analysis below. 11 I assume that each firm newly set up in any given period inherits (through random assignment) the index of an exiting firm. 12 Note that the price is common to all those firms, since they face an identical problem. 8

10 original variables): p t = υγθp t 1 + (1 υγθ)p t (7) i.e. the current price level is a weighted average of last period s price level and the newly set price, all in logs, with the weights given by the fraction of firms that do not and do adjust prices, respectively. A firm adjusting the price in period t will choose the price Pt that maximizes { ( )} P max (υγθ) k E t Λ t,t+k Y t t+k t W t+k Pt P t+k k=0 subject to the sequence of demand constraints ( P t Y t+k t = 1 α P t+k ) ɛ C t+k (8) for k = 0, 1, 2,...where Y t+k t denotes output in period t + k for a firm that last reset its price in period t and W t W t /Γ t is the productivity-adjusted real wage (i.e. the real marginal cost). 13 The optimality condition associated with the problem above takes the form { ( )} P (υγθ) k E t Λ t,t+k Y t t+k t MW t+k = 0 (9) P t+k k=0 where M ɛ ɛ 1 is the optimal markup under flexible prices. A first-order Taylor expansion of (9) around the zero inflation steady state yields, after some manipulation: p t = µ + (1 ΛΓυγθ) (ΛΓυγθ) k E t {ψ t+k } (10) k=0 where ψ t log P t W t is the (log) marginal cost, µ log M and Λ 1 1+r is the steady state stochastic discount factor. balanced growth path, w t = w = µ. 14 Note that along a zero inflation Letting µ t p t ψ t denote the average (log) price markup, and combining (7) and (10) yields the inflation equation: π t = ΛΓE t {π t+1 } λ(µ t µ) (11) 13 The firm s demand schedule can be derived by aggregating (2) across cohorts. 14 Throughout assumptions are made that guarantee that ΛΓυγθ [0, 1), so that the infinite sum in (10) converges in a neighborhood of the zero inflation balanced growth path. 9

11 where π t p t p t 1 denotes inflation and λ (1 υγθ)(1 ΛΓυγθ) υγθ Asset Markets In addition to a complete set of state-contingent securities, I assume the existence of markets for a number of specific assets, whose prices and returns must satisfy some equilibrium conditions. In particular, the yield i t on a one-period nominally riskless bond purchased in period t must satisfy 16 1 = (1 + i t )E t { P t Λ t,t+1 P t+1 } (12) thus implying that the relation Λ 1/(1+r) between the discount factor and the real return on the riskless nominal bond (r) will hold along a perfect foresight balanced growth path. Stocks in individual firms trade at a price Q F t (i), for i [0, α], which must satisfy the asset pricing equation: ( ) where D t (i) Y t (i) Pt(i) P t W t Q F t (i) = D t (i) + υγe t { Λt,t+1 Q F t+1(i) } (13) denotes firm i s dividends, and υγ is the probability that firm i survives into next period. Solving (13) forward under the assumption that lim k (υγ) k E t { Λt,t+k Q F t+k (i)} = 0, and aggregating across firms: Q F t = α 0 Q F t (i)di (υγ) k E t {Λ t,t+k D t+k } (14) k=0 where D t α 0 D t(i)di denotes aggregate dividends. Much of the analysis below focuses on intrinsically worthless assets i.e. assets generating no dividend, pecuniary or not. 17 Let Q B t (j) denote the price of one 15 Note that entry and exit of firms and the consequent introduction of new goods each period has the effect of increasing the size of coeffi cient λ, thus raising the "effective" flexibility of prices, for any given value of θ. That effect, however, is likely to be quantitatively small for any reasonable values of γ and υ. 16 Note also that in the asset pricing equations the probability of remaining alive γ and the extra return 1/γ resulting from the annuity contract cancel each other. 17 In Jean Tirole s words, pure bubbly assets are "best thought of as pieces of paper." 10

12 such asset. In equilibrium that price must satisfy the condition Q B t (j) = E t {Λ t,t+1 Q B t+1(j)} (15) as well as the non-negativity constraint Q B t (j) 0 (given free disposal), for all t. Let Q B t denote the aggregate value of bubbly assets in period t. In equilibrium, that variable evolves over time according to the following two equations: Q B t = U t + Q B t t 1 (16) Q B t = E t {Λ t,t+1 Q B t+1 t } (17) where U t is the value of a bubble created by cohort t at birth, 18 and Q B t t k j B t k Q B t (j)dj is the aggregate value in period t of the bubbly assets that were already present in the economy at t k. Note that in the previous environment, the initial financial wealth of a cohort born in period t is given by: A t t = Q F t t + U t/(1 γ) where Q F t t denotes the period t value of a firm that can reset its price in that period (and hence, the value of any newly created firm). A similar environment with bubble creation was first introduced and analyzed in Martín and Ventura (2012) in the context of an overlapping generations model with financial frictions. 19 In the absence of bubble creation by each incoming cohort, the supply of bubbly assets remains constant over time. Accordingly, Q B t = E t {Λ t,t+1 Q B t+1} A t t = Q F t t since Q B t+1 t = QB t+1 and U t = 0 in that case. 18 Think of pieces of paper of a cohort-specific color or stamped with the birth year of their creators. 19 The bubble introduced by each individual can be interpreted as being attached to the stock of his firm and hence to burst whenever the firm stops operating (i.e. with probability 1 υγ). 11

13 2.4 Labor Market The details of wage setting are not central to the main point of the paper. For convenience, I assume an ad-hoc wage schedule linking the productivity-adjusted real wage to average work hours of the form ( ) ϕ Nt W t = (18) α where N t α 0 N t(i)di denotes aggregate work hours. Wage schedule (18), together with the assumption of a constant gross markup M under flexible prices, implies a natural level of output given by Y n t = Γ t αm 1 ϕ Γ t Y. Log-linearizing (18), and combining the resulting expressions with (11) we obtain the following version of the New Keynesian Phillips curve π t = ΛΓE t {π t+1 } + κŷ t (19) where κ λϕ, and ŷ t log(y t /Y n t ) is the output gap. 2.5 Monetary Policy Unless otherwise noted, the central bank is assumed to follow a simple interest rate rule of the form where î t log 1+it 1+r, qb t QB t Γ t Y î t = φ π π t + φ q q B t (20) is the size of the aggregate bubble normalized by trend output, with q B t q B t q B denoting the deviation from its value along a balanced growth path. Note that the previous rule is consistent with a zero inflation target. In much of what follows, I assume the central bank takes as given r and q B, which are assumed to be mutually consistent, as determined by the analysis below. 12

14 2.6 Market Clearing Goods market clearing requires Y t (i) = (1 γ) t s= γt s C t s (i) for all i ( ɛ [0, α]. Letting Y t α 1 α ɛ 0 Y t(i) 1 1 ɛ 1 ɛ di) denote aggregate output, we have: Y t = (1 γ) = C t Note also that in equilibrium N t = α 0 t s= = p t Y t N t (i)di γ t s C t s Y t where Y t Y t /Γ t is aggregate output normalized by productivity and p t α 0 (P t(i)/p t ) ɛ di 1 is an index of relative price distortions, which equals 1 α one up to a first-order approximation. Assuming that all securities other than stocks and the bubbly assets are in zero net supply, asset market clearing requires t (1 γ) γ t s A t s = Q F t s= + Q B t Next I characterize the economy s perfect foresight, zero inflation balanced growth paths. 3 Balanced Growth Paths In a perfect foresight balanced growth (henceforth, BGP) the discount factor is constant and given by Λ t,t+1 = r Λ as implied by (12), and where r is the real interest rate along a BGP. Note also that i = r in the BGP, given the zero inflation assumption. 13

15 Note also that W = 1/M in the zero inflation BGP. Combined with (18) the former condition implies: 1 M = ( ) ϕ Y α Accordingly, output along the BGP, is given by Γ t αm 1 ϕ Γ t Y, which coincides with the natural level of output (and, hence, a constant markup M). Next I describe how aggregate consumption is determined. Details of the derivation can be found in Appendix. Let C j and A i j denote, respectively, consumption and financial wealth along a BGP for an individual aged j, normalized by productivity. Superindex i {a, r} denotes his status as active or retired. The intertemporal budget constraint for a consumer born in period s and who remains active at time t, derived by solving (1) forward, can be evaluated at a BGP as: ( ) (Λγ) k Γ k C t+k s = A a 1 WN t s + 1 ΛΓυγ α k=0 where N and W denote aggregate hours and the wage (the latter normalized by productivity) along the BGP. Note that condition ΛΓυγ < 1, necessary for the consumer s intertemporal budget constraint to be well defined, imposes an exogenous lower bound on the real interest rate, namely, r Γυγ 1 r. Using the fact that C t+k s = [β(1 + r)/γ] k C t s as implied by (4) evaluated at the BGP the following consumption function can be obtained for an active individual aged j: [ ( )] C j = (1 βγ) A a 1 WN j + 1 ΛΓυγ α (21) Thus, consumption for an active individual is proportional to the sum of his financial wealth, A a j, and his current and future labor income (properly discounted), WN/α(1 ΛΓυγ). The corresponding consumption function for a retired individual is given by: C j = (1 βγ) A r j (22) 14

16 Note that in deriving (21) I have assumed that ΛΓυγ < 1, so that the relevant expected present value of future income is bounded. That assumption is maintained throughout the analysis below. It guarantees that the transversality condition is satisfied (as long as individual financial wealth is bounded). Aggregating over all individuals, imposing the asset market clearing condition A = Q F + Q B (with these three variables now normalized by productivity) and using the fact that Q F = D/(1 ΛΓυγ) and Y = WN + D along a BGP, we obtain: [ ] C = (1 βγ) Q B ΛΓυγ Y which can be interpreted as an aggregate consumption function along the BGP. 20 Next I turn to the analysis of "bubbleless" and "bubbly" BGPs. 3.1 Bubbleless Balanced Growth Paths (23) Consider first a "bubbleless" BGP, with Q B = 0. Imposing that condition in (23), together with goods market clearing, C = Y, implies: ΛΓυ = β or, equivalently, r = (1 + ρ)(1 + g)υ 1 Note that the steady state real interest rate is increasing in both υ and g The reason is that an increase in either of those variables raises desired consumption by increasing the expected present discounted value of future income for currently active individuals. In order for the goods market to clear, an increase in the interest rate is called for. 20 In particular, for members of a newly born cohort [ ( )] δuy C 0 = (1 βγ) 1 γ + QF α + 1 WN 1 ΛΓυγ α [ ( )] δu = (1 βγ) 1 γ Y (24) 1 ΛΓυγ α Consumption (normalized by productivity) for an individual aged h is then given by C h = [β(1 + r)] k C 0 15

17 When υ = 1, the real interest rate is given (approximately) by the discount rate plus the growth rate, i.e. r = (1 + ρ)(1 + g) 1 ρ + g, as in the standard model. Note that an increase in the expected lifetime, as indexed by γ, does not have an independent effect on the real interest rate. The reason is that, when ΛΓυ = β, such a change increases in the same proportion the present value of consumption and that of income, making an interest rate adjustment unnecessary. 21 Note for future reference that in the bubbleless BGP considered here the real interest rate r is lower than the growth rate g (i.e. ΛΓ > 1) if and only if υ < β. 3.2 Bubbly Balanced Growth Paths Next I consider the possibility of a BGP with an aggregate bubble growing at the same rate as output, thus implying a constant (productivity-adjusted) bubble size Q B > 0. Letting qt B Q B t /(Γ t Y), qt t k B QB t t k /(Γt Y) and u t U t /(Γ t Y), note first that (17) can be rewritten as: qt B = E t {Λ t,t+1 Γqt+1 t B } = E t {Λ t,t+1 Γ(qt+1 B u t+1 )} (25) Along a bubbly BGP we must have q B t = q B > 0, and u t = u 0 for all t. It follows from (23), together with goods market clearing, C = Y, and (25) that: q B γ(β ΛΓυ) = (1 βγ)(1 ΛΓυγ) ( u = 1 1 ) q B ΛΓ The existence of a BGP with a positive bubble, q B > 0, requires 21 The independence of the steady state real interest rate from γ is a consequence of the log utility specification assumed here. That property is not critical from the viewpoint of the present paper, since there are other factors (the probability of retirement, in particular), that can drive real interest rate to values consistent with the presence of bubbles. 16

18 ΛΓ < β υ On the other hand the non-negativity constraint on newly created bubbles u 0 requires: ΛΓ 1 Accordingly, a necessary and suffi cient condition for the existence of such bubbly BGP is, again, υ < β. If that condition is satisfied, there exists continuum of bubbly BGPs {q B, u} indexed by r ((1 + ρ)(1 + g)υ 1, g]. Furthermore, it can be easily checked that q B is increasing in r, with lim r g q B = γ(β υ) (1 βγ)(1 υγ) qb. Note that the condition for the existence of bubbly BGPs is equivalent to the real interest rate being less than the growth rate in the bubbleless BGP. Note also that q B / υ < 0, i.e. the upper bound on the size of the bubble is decreasing in υ over the range υ [0, β], and converges to zero as υ β. One particular such bubbly BGP has a constant supply of bubbly assets, i.e. u t = 0 all t. Note that in that case ΛΓ = 1 or, equivalently, r = g with the implied bubble size given by q B = q B. Along that BGP any existing bubble will be growing at the same rate as the economy. By contrast, along a bubbly BGP with bubble creation, r < g implies that the size of the aggregate pre-existing bubble will be shrinking over time relative to the size of the economy, with newly created bubbles filling up the gap so that the size of the aggregate bubble relative to the size of the economy remains unchanged. Summing up, one can distinguish two regions of the parameter space relevant for the possible existence of bubbly BGPs: 17

19 (i) β υ 1. In that case, the BGP is unique and bubbleless and associated with a real interest rate given by r = (1 + ρ)(1 + g)υ 1 > g. (ii) 0 < υ < β. Multiple BGPs coexist. One of them is bubbleless, with r = (1 + ρ)(1 + g)υ 1 < g. In addition, there exists a continuum of bubbly BGPs, indexed by a real interest rate r ((1 + ρ)(1 + g)υ 1, g], and associated with a bubble size (relative to output) q B (0, q B ], given by (??). Figures 1 summarizes graphically the two regions with their associated BGPs. Figure 2 illustrates the dependence of bubble size (as a share of output) on υ and r along a bubbly BGP. 3.3 Some Numbers A natural question one may raise at this point is whether the key necessary condition for the existence of bubbly equilibria, namely υ < β, is likely to be satisfied in practice. If one takes the model at face value, υ should be interpreted as the probability of (permanently) dropping out of the labor force and becoming inactive in any given quarter. If we take a narrow perspective and we interpret that transition as corresponding to retirement, information on the average age at retirement can be used to calibrate υ. In what follows we take the age at which individuals enter the economy ("are born" in the model) to be 20. In the U.S., the average age of retirement is 63. That corresponds to an expected active lifetime as of age 20 of 43 4 = 172 quarters (conditional on eff ectively retiring, i.e. not dying earlier). In the model the expected active 1 lifetime (conditional on actual retirement) is given by 1 υγ. On the other hand, life expectancy at age 20 is 60 4 = 240 quarters, which is consistent with a value γ = = It follows that υ = ( ) = , which should be viewed as an upper bound, since some individuals stop working permanently before retirement. Thus, the existence of bubbles calls for β > Unfortunately, the latter condition cannot be verified directly due to the unobservability of that parameter. 22 Instead, one may examine directly the 22 Note that β = implies a discount factor of applied to utility 10 years from today, which seems entirely plausible. 18

20 relation between the average real interest rate, r, and the average growth rate of output, g, two observable variables. As discussed above, the existence of bubbly BGPs requires that r g. Using data on 3-month Treasury bills, GDP deflator and (per capita) GDP, the average values for those variables in the U.S. over the period 1960Q1-2015Q4 are r = 1.4% 4 = 0.35% and g = 1.6% 4 = 0.4% (or, equivalently, Λ = and Γ = ), values which satisfy the inequality condition necessary for the existence of bubbles. Note also that the above calibration implies ΛΓυγ 0.995, thus satisfying the condition for a well defined intertemporal budget constraint. Some of the quantitative analyses below require a calibrated value for β, the discount factor. I set β = , which is consistent with q B = 4, i.e. with an upper bound for the size of the steady state bubble equal to annual steady state output. This is, admittedly, an arbitrary choice, but it allows me to rule out steady state bubbles with an implausible large size. 3.4 Bubbly Equilibria and Transversality Conditions Equilibria with bubbles on assets in positive net supply can be ruled out in an economy with an infinite lived representative consumer (see, e.g. Santos and Woodford (1997)). The reason is that any the bubble component of any asset { must satisfy lim T E t Λt,t+T Q B t+t (j)} = Q B t (j). But any positive net supply of that asset must be necessarily held by the representative consumer, implying { lim T E t {Λ t,t+t A t+t } lim T E t Λt,t+T Q B t+t (j)}. But the consumer s transversality condition requires that lim T E t {Λ t,t+t A t+t } = 0. Given that Q B t (j) 0, it follows that Q B t (j) = 0 for all t. Note that the previous reasoning cannot be applied to the overlapping generations economy with an infinite sequence of agent-types, as the one developed above. The reason is that in that case it is no longer true that the positive net supply of any bubbly asset must be held (asymptotically) by any individual agent, since it can always be passed on to future cohorts (and it will in equilibrium). 19

21 4 Equilibrium Dynamics In the present section I analyze the equilibrium dynamics in a neighborhood of a BGP. In particular, I am interested in exploring the conditions under which equilibrium fluctuations driven by variations in the size of the bubble unrelated to fundamentals may emerge, and the role that monetary policy may play in preventing such fluctuations. As in the standard analysis of the New Keynesian model with a representative agent, 23 I restrict myself to equilibria that remain in a neighborhood of a BGP, with the local equilibrium dynamics being approximated using the log-linearized equilibrium conditions. I leave the analysis of the global equilibrium dynamics (including the possibility of switches between BGPs and other non-linearities) to future research. Secondly, in laying out the model I ignore the existence of fundamental shocks, in order to focus on equilibria driven by fluctuations in the size of the bubble. I start by deriving the log-linearized equilibrium conditions around a BGP. In contrast with the New Keynesian model with a representative agent, the consumer s Euler equation and the goods market clearing condition are no longer suffi cient to derive an equilibrium relation determining aggregate output as a function of interest rates (i.e. the dynamic IS equation). 24 In particular, the derivation of such a relation requires solving for an aggregate consumption function, by aggregating the individual consumption functions obtained from combining the individual Euler equation with the corresponding intertemporal budget constraint. Since no exact representation exists of the individual consumption function, I proceed by deriving a log-linearized approximation of that function around a perfect foresight, zero inflation balanced growth path. Then I aggregate the approximate individual consumption functions to obtain 23 See, e.g., Woodford (2003) or Galí (2015). 24 In particular, one can use the definition of aggregate consumption and the individual Euler equations to derive: E t{ĉ t+1 } = βγ Λ ( ) ( ĉ t + î t E t{π t+1 } + 1 βγ ) E t{ĉ t+1 t+1 } Λ where E t{ĉ t+1 t+1 } cannot be immediately as a function of ĉ t. 20

22 an (approximate) aggregate consumption function. See the Appendix for detailed derivations. Letting ĉ t log(c t /Γ t C) denote log deviation of aggregate consumption from its value along a BGP, one can write the aggregate consumption function as follows: where q B t q B t q B and x t k=0 ĉ t = (1 βγ)( q B t + x t ) (26) (ΛΓυγ) k E t {ŷ t+k } ΛΓυγ 1 ΛΓυγ (ΛΓυγ) k E t { r t+k } is the non-bubbly component of aggregate wealth, expressed in log deviations from its value along a BGP, and with ŷ t log(y t /Γ t Y) denoting the output gap, and r t = î t E t {π t+1 } the real interest rate. Note that x t can also be rewritten in recursive form as: x t = ΛΓυγE t { x t+1 } + ŷ t k=0 ΛΓυγ 1 ΛΓυγ r t (27) Thus, consumption moves in proportion to fluctuations in total wealth, bubbly and non-bubbly. Through this channel, a rise in the size of the bubble leads, ceteris paribus, to an increase in aggregate demand and output. On the other hand, log-linearization of (25) around a BGP yields the following difference equation describing fluctuations in the aggregate bubble. where q B = γ(β ΛΓυ) (1 βγ)(1 ΛΓυγ) 0. q B t = ΛΓE t { q B t+1} q B r t (28) By combining (26), (27), and (28) with the goods market clearing condition ŷ t = ĉ t the following version of a dynamic IS equation can be derived: ŷ t = ΛΓυ β E t{ŷ t+1 } + Φ q B t Υυ β r t (29) 21

23 where Φ (1 βγ)(1 υγ) βγ > 0 and Υ 1 + (1 βγ)(λγ 1) 1 ΛΓυγ. Thus, the equilibrium dynamics around a BGP are described by (28) and (29), together with the New Keynesian Phillips curve (derived above) π t = ΛΓE t {π t+1 } + κŷ t (30) and the assumed monetary policy rule î t = φ π π t + φ q q B t (31) with r t = î t E t {π t+1 }. Note that the previous dynamical system has a form similar to that describing the equilibrium of the standard New Keynesian model, but augmented now with equation (28) describing the evolution of the bubble, and with the latter variable entering the dynamic IS equation (29). 4.1 Monetary Policy and Equilibrium Uniqueness Next I turn to the analysis of the uniqueness of equilibria in a neighborhood of a given BGP and its connection to monetary policy. Using (31) to eliminate the interest rate in (29) and (28), we can rewrite the system as follows: A 0 x t = A 1 E t {x t+1 } where x t [ŷ t, π t, q B t ] and A 0 1 Υυ β φ π Φ + Υυ β φ q κ q B φ π 1 + q B φ q ; A 1 = ΛΓυ β Υυ β 0 0 ΛΓ 0 0 q B ΛΓ The solution to the previous dynamical system is locally unique if and only if the three eigenvalues of the companion matrix A A 1 0 A 1 lie inside the unit circle. Figures 3 and 4 display the uniqueness and indeterminacy regions on the (φ q, φ π ) plane, under alternative assumptions regarding the real interest rate in the underlying BGP, and given the baseline calibration introduced above. In each case, I report results for κ = 0.1 and κ = 0.01, two values of the slope of the New Keynesian Phillips curve which arguably span the range of values considered as plausible in the literature (theoretically and empirically). 22

24 Figure 3a displays the determinacy and indeterminacy regions around a bubbly BGP corresponding to the case of r = g = 0.004, which is associated (by construction) with a bubble of size q B = q B = 4 (relative to quarterly output). Note that under that calibration there is no bubble creation and ΛΓ = Υ = 1. The figure suggests that the conventional Taylor principle is at work, i.e. the equilibrium is locally unique as long as φ π > 1, independently of the value taken by φ q. Figures 3b illustrate the pervasiveness of indeterminacy when we consider the equilibrium dynamics around a bubbly BGP with r < g (i.e. a BGP with bubble creation). Figure 3b corresponds to a BGP with an interest rate "close to" the growth rate, namely, r = Now the possibility of (one-dimensional) local indeterminacy arises even for a range of φ π values well above one, whenever φ q > 0. Most interestingly, note that in this case the size of the indeterminacy region increases with φ q. Figures 3c and 3d illustrate the properties of equilibria around a bubbly BGP with an interest rate "further from" the growth rate, r = and r = Now that under such calibrations the equilibrium is locally indeterminate for all values of φ π and φ q considered. Note also that in both cases the possibility of indeterminacy of dimension 2 arises if φ π < 1. In the latter case that property emerges independently of the size of φ q The Case of Bubbleless Equilibria Here I consider the particular case of bubbleless equilibria, i.e. with qt B = 0, for all t. Using the fact that ΛΓυ = β around the bubbleless BGP, one can rewrite the New Keynesian Phillips curve (19) as: π t = (β/υ)e t {π t+1 } + κŷ t (32) which nests the "standard" specification in the limiting case of υ = 1. In general, however, the coeffi cient associated with expected inflation will be larger than β, and larger than one when υ < β. In the absence of bubbles, dynamic IS equation (29) collapses to its coun- 23

25 terpart in the standard model, namely: ŷ t = E t {ŷ t+1 } (î t E t {π t+1 }) (33) The interest rate rule can now be written: î t = φ π π t (34) Using (34) to substitute out the interest rate in (33), we can write the equilibrium conditions more compactly as: [ ] [ ŷt φ π β ] [ ] Et {ŷ = υ t+1 } π t 1 + κφ π κ κ + β (35) E υ t {π t+1 } Note that ŷ t = π t = 0 for all t is always a solution to the previous dynamical system and hence an admissible equilibrium. That solution and its associated equilibrium is locally unique if and only if the two eigenvalues of the companion matrix [ φ π β ] A υ 1 + κφ π κ κ + β υ lie within the unit circle. As shown in the appendix, the necessary and suffi cient condition for the local uniqueness of the solution to (35), is given by [ φ π > max 1, 1 ( )] β κ υ 1 (36) Thus, under that condition, and in the absence of exogenous shocks, the only (bubbleless) equilibrium that remains in a neighborhood of the steady state is given by ŷ t = π t = 0 for all t. Note that as long as υ β 1+κ the previous condition can be written as φ π > 1, which corresponds to the well known "Taylor β 1+κ, principle" in the basic New Keynesian model. On the other hand, when υ < ( ) the uniqueness condition takes the form φ π > 1 β κ υ 1 > 1, which can be thought of as a "reinforced Taylor principle," requiring that the central bank adjust the policy rate more aggressively than implied by the usual "more than one-for-one" condition in order to guarantee the uniqueness of the equilibrium. Note that a low value for υ reduces the average interest rate, making current inflation more sensitive to expectations, making stability of backward solutions 24

26 to the equilibrium dynamical system more likely, especially when κ is small, unless the central bank responds more aggressively to inflation. Figure 4a represents the regions of the parameter space for φ π and υ, consistent with a determinate and an indeterminate equilibrium, using the baseline calibration for β, and under the two alternative κ values considered above. Note that when inflation is less sensitive to output variations (κ = 0.01), the conventional Taylor principle breaks down for a larger υ value, making the indeterminacy larger. If (36) is not satisfied, there exist equilibria involving stationary sunspot fluctuations around the bubbleless BGP, driven by self-fulfilling revisions of expectations. Two possible types of sunspot equilibria may emerge depending on the size of φ π, and as shown in Figure 4b. If φ π < 1 then only one eigenvalue of A is larger than one, implying indeterminacy of dimension one. 25 On the ( ( )) other hand, if υ < β 1+κ and φ π 1, 1 β κ υ 1 both eigenvalues of A have moduli larger than one, with the resulting indeterminacy being two-dimensional a case that can be ruled out in the New Keynesian model with a representative agent. See the appendix for a discussion of the representation of such equilibria. The previous results point to an important dimension in which the introduction of finite lives and an OLG structure in an otherwise standard New Keynesian model affect the properties of a simple interest rule, even in a bubbleless equilibrium. 5 Bubble-Driven Fluctuations: An Example In the present section I provide an example of equilibrium fluctuations driven by a stochastic bubble in the economy described in the previous sections, and discuss the implications of "leaning against the bubble" monetary policies. I assume the following process for the aggregate bubble. If q B t = 0, then q B t+1 = u t+1 = 0 with probability ω (0, 1) and q B t+1 = u t+1 U[0, u] with probability 1 ω. On the other hand, if q B t > 0, then q B t+1 t = Rt δγ qb t 25 Such sunspot equilibria are similar in nature to those analyzed in, e.g. Clarida, Galí and Gertler (2000). and 25

27 u t+1 = 0 with probability δ (0, 1), and qt+1 B = 0 with probability 1 δ, where R t (1 + i t )E t {P t /P t+1 } is the gross real interest rate. It can be easily checked that the previous process satisfies the bubble s equilibrium condition, namely, qt B = E t {Λ t,t+1 Γ(qt+1 B u t+1 )} > 0 for all t. The remaining equilibrium conditions describing the economy s nonpolicy block are dynamic IS equation (29) and the New Keynesian Phillips curve (30), both evaluated at the bubbleless steady state: ŷ t = E t {ŷ t+1 } + Φqt B (î t E t {π t+1 }) π t = (β/υ)e t {π t+1 } + κŷ t The model is closed by assuming interest rate rule (31). I solve the model numerically using the collocation method. Figure 7 displays a simulation of the model s equilibrium, under the baseline calibration, supplemented with the assumptions φ π = 1.5, φ q = 0, ω = δ = 0.9 and u = Note that episodes characterized by a rise in the size of the bubble generate an output boom and the consequent inflationary pressures, which the central bank seeks to dampen through higher real rates. collapses, both output and inflation revert to their trend values. When the bubble Figure 8 shows the standard deviation of several macro variables as a function of φ q. Interestingly, the lowest volatility is attained for values of φ q in a neighborhood of zero. "Leaning against the bubble" policies, on the other hand, are shown to generate instability, which is increasing in φ q. The reason is straightforward: a tightening of monetary policy in response to a growing bubble increases the required expected growth of the bubble, as well as the realized growth conditional on the bubble not bursting. As a result, aggregate demand becomes more volatile, and so do output and inflation. 26 Together with some of the findings in previous sections, the previous result calls into question the effectiveness and desirability of a monetary policy that 26 In Galí (2014) "leaning against the bubble" policies generally increase the volatility of the bubble, but do not have an effect on aggregate output. Instead they induce unnecessary randomness in the allocation of consumption across generations, which is welfare reducing. 26

28 implies a systematic increase in the policy rate in response to a rising asset price bubble. As shown above, a "leaning against the bubble" strategy may in some cases make more likely the possibility of macro fluctuations unrelated to fundamentals and, hence, of unnecessary macro instability. 6 Concluding Comments The New Keynesian model has become the workhorse framework for monetary policy analysis, even though it is unsuitable in its standard formulation to address one of the key questions facing policy makers, namely, how monetary policy should respond to the emergence and growth of asset price bubbles. That shortcoming, however, is not tied to any key ingredient of the model (e.g. staggered price setting), but to the convenient (albeit unrealistic) assumption of an infinite-lived representative consumer. In the present paper I have developed an extension of the basic New Keynesian model featuring overlapping generations, finite lives and retirement. In contrast with the standard model, the proposed framework allows for the existence of rational expectations equilibria with asset price bubbles. Plausible calibrations of the model s parameters are consistent with the existence of bubbly balanced growth paths. Furthermore, the analysis of the equilibrium dynamics around such balanced growth paths suggests that "leaning against the bubble" rules, i.e. that call for a rise in the interest rate in response to an increase in the size of the bubble, may be destabilizing. Furthermore, the analysis above suggests that the introduction of an overlapping generation structure, even when one abstracts from the possibility of bubbly equilibria, may change the conditions under which simple interest rate rules will guarantee equilibrium uniqueness. In particular, and under some parameter configurations, the so-called Taylor principle has to be "reinforced," calling for a stronger anti-inflationary stance in order to avoid indeterminacy Two additional remarks, pointing to possible future research avenues are in order. The analysis of the equilibrium dynamics above has assumed "rationality" of asset price bubbles. That assumption underlies the equilibrium 27

29 conditions that individual and aggregate bubbles must satisfy, i.e. (15) and (17), respectively, and the implied log-linear approximation (28). However, the remaining equilibrium conditions, including the modified dynamic IS equations, are still valid even if the process followed by the aggregate bubble { q t B } is inconsistent with a rational bubble. That observation opens the door to analyses of the macroeconomic effects and policy implications of alternative specifications of the aggregate bubble s behavior. Secondly, the analyses of the equilibrium dynamics above has assumed that the central bank takes as given the BGP on which the economy settles and, hence, its associated real interest rate. That assumption is reflected in the implicit exogeneity of the real interest rate term embedded in the policy rule through the term î t i t r. But while that assumption is a natural one in the context of economies whose real interest rate along a BGP is uniquely pinned down (as it is the case in the standard New Keynesian model with a representative consumer), it is no longer so in an economy like the one analyzed above, in which a multiplicity of real interest rates may be consistent with a perfect foresight BGP. In future research I plan to explore the implications of relaxing the assumption of an exogenously given steady state real interest rate. 28

30 References Bernanke, Ben S. (2002): "Asset Price Bubbles and Monetary Policy," speech before the New York Chapter of the National Association of Business Economists. Bernanke, Ben S. and Mark Gertler (1999): "Monetary Policy and Asset Price Volatility," in New Challenges for Monetary Policy, Federal Reserve Bank of Kansas City, Bernanke, Ben S. and Mark Gertler (2001): "Should Central Banks Respond to Movements in Asset Prices?" American Economic Review 91(2), Blanchard, Olivier J. (1985): "Debt, Deficits, and Finite Horizons." Journal of Political Economy, 93(2), Borio, Claudio and Philip Lowe (2002): "Asset Prices, Financial and Monetary Stability: Exploring the Nexus," BIS Working Papers no Carvalho, Carlos, Andrea Ferrero and Fernanda Nechio (2016): "Demographics and real interest rates: Inspecting the Mechanism," European Economic Review, forthcoming. Cecchetti, Stephen G., Hans Gensberg, John Lipsky and Sushil Wadhwani (2000): Asset Prices and Central Bank Policy, Geneva Reports on the World Economy 2, CEPR. Del Negro, Marco, Marc Giannoni, and Christina Patterson (2015): "The Forward Guidance Puzzle," unpublished manuscript. Galí, Jordi (2014): "Monetary Policy and Rational Asset Price Bubbles," American Economic Review, vol. 104(3), Gertler, Mark (1999): "Government Debt and Social Security in a Life-Cycle Economy," Carnegie-Rochester Conference Series on Public Policy 50, Kohn, Donald L. (2006): "Monetary Policy and Asset Prices," speech at an ECB colloquium on "Monetary Policy: A Journey from Theory to Practice," held in honor of Otmar Issing. Kohn, Donald L. (2008): "Monetary Policy and Asset Prices Revisited," speech delivered at the Caton Institute s 26th Annual Monetary Policy Confer- 29

31 ence, Washington, D.C. Martín, Alberto, and Jaume Ventura (2012): "Economic Growth with Bubbles," American Economic Review, vol.102(6), Nisticò, Salvatore (2012): "Monetary Policy and Stock-Price Dynamics in a DSGE Framework," Journal of Macroeconomics 34, Piergallini, Alessandro (2006): "Real Balance Effects and Monetary Policy," Economic Inquiry 44(3), Samuelson, Paul A. (1958): "An exact consumption-loan model of interest with or without the social contrivance of money," Journal of Political Economy 66, Santos, Manuel S. and Michael Woodford (1997): "Rational Asset Pricing Bubbles," Econometrica 65(1), Schularick, Moritz and Alan M. Taylor (2012): "Credit Booms Gone Bust: Monetary Policy, Leverage Cycles and Financial Crises, " American Economic Review, 102(2), Taylor, John (2014): "The Role of Policy in the Great Recession and the Weak Recovery," American Economic Review 104(5), Tirole, Jean (1985): "Asset Bubbles and Overlapping Generations," Econometrica 53, Yaari, Menahem E. (1965): "Uncertain Lifetime, Life Insurance, and the Theory of the Consumer," The Review of Economic Studies, 32(2),

32 TECHNICAL APPENDIX 1. Log-linearized individual intertemporal budget constraint The intertemporal budget constraint for an individual born in period s and still active in period t s can be derived by iterating (3) forward to yield: γ k E t {Λ t,t+k C t+k s } = A a t s + 1 (γυ) k E t {Λ t,t+k W t+k N t+k } (37) α k=0 k=0 For retired individuals, the corresponding constraint is: γ k E t {Λ t,t+k C t+k s } = A r t s (38) k=0 Letting lowercase letters with a "ˆ" symbol denote the deviation of a variable from its value along a perfect foresight balanced growth path (BGP), the left hand side term of (37) and (38) can be approximated in a neighborhood of the BGP as: k=0 γ k E t {Λ t,t+k C t+k s } Γt C t s 1 βγ + Γt (ΛΓγ) k C t+k s E t {ĉ t+k s + λ t,t+k } k=0 = Γt C t s 1 βγ + Γt C t s 1 βγ ĉt s where C j denotes consumption of an individual aged j (normalized by current productivity) along a BGP, λ t,t+k = log(λ t,t+k /Λ k ), and where I have made use of the fact that C t+k s = [β(1 + r)/γ] k C t s and the fact that E t { λ t,t+k + ĉ t+k s } = ĉ t s (resulting from (4)). The second term on the right hand side of (37), which is relevant only for active individuals, can be approximated around the BGP as: 1 α (γυ) k E t {Λ t,t+k W t+k N t+k } k=0 Γ t WN α(1 ΛΓυγ) + ( Γ t ) WN (ΛΓυγ) k E t {ŵ t+k + n t+k + λ t,t+k } α k=0 where W is the wage along the BGP, normalized by productivity. 2. Aggregation and derivation of log-linearized aggregate consumption Euler equation 31

33 Let C denote aggregate consumption (normalized by current productivity), along the BGP. One can derive the approximate relations Cĉ t = (1 γ) t s= γt s C t s ĉ t s and the BGP relation C = (1 γ) t s= γt s C t s. Those relations can be used to aggregate the log-linearized individual consumption functions across all individuals to yield: Γ t C 1 βγ + Γt C 1 βγ ĉt = (Q F t +Q B t )+ Γt WN 1 ΛΓυγ +Γt WN Note also that in a neighborhood of the BGP, (ΛΓυγ) k E t {ŵ t+k + n t+k + λ t,t+k } k=0 Q F t = (υγ) k E t {Λ t,t+k D t+k } k=0 Γ t D 1 ΛΓυγ + Γt D (ΛΓυγ) k E t { d t+k + λ t,t+k } k=0 where D denotes aggregate dividends (normalized by productivity), along a BGP. Thus, using the approximation ŷ t = (WN/Y)(ŵ t + n t ) + (D/Y) d t, the BGP relation Γt C 1 βγ = QB + Γt (WN+D) 1 ΛΓυγ, and the goods market clearing condition Y = C, we obtain the log-linearized aggregate consumption function ĉ t = (1 βγ) [ q B t + ] (ΛΓυγ) k E t {ŷ t+k + λ t,t+k } k=0 k 1 = (1 βγ) q t B + (ΛΓυγ) k E t {ŷ t+k } (ΛΓυγ) k E t { r t+j } = (1 βγ) [ q B t + k=0 k=0 k=1 (ΛΓυγ) k E t {ŷ t+k } ΛΓυγ 1 ΛΓυγ j=0 ] (ΛΓυγ) k E t { r t+k } where q B t q B t q B, with q B t Q B t /(Γ t Y) and q B its corresponding value along a BGP, and r t î t E t {π t+1 }. Note that we can write the consumption function more compactly as: k=0 ĉ t = (1 βγ)( q B t + x t ) (39) 32

34 where x t k=0 (ΛΓυγ)k E t {ŷ t+k } ΛΓυγ 1 ΛΓυγ k=0 (ΛΓυγ)k E t { r t+k } the nonbubbly component of wealth, expressed in log deviations from its value along a BGP. It satisfies the recursive equation: x t = ΛΓυγE t { x t+1 } + ŷ t ΛΓυγ 1 ΛΓυγ r t Furthermore, log-linearization of (??) and (??) about a BGP yields q B t = ΛΓE t { q B t+1} q B r t (40) In a neighborhood of the bubbleless BGP q B = 0 and ΛΓυ = β, thus implying the approximate aggregate Euler equation: ĉ t = βγe t {ĉ t+1 } + (1 βγ)(1 υγ)q B t + (1 βγ)ŷ t βγ r t Imposing goods market clearing (ĉ t = ŷ t ) and rearranging terms, we obtain a dynamic IS equation: ŷ t = E t {ŷ t+1 } + Φq B t r t where Φ (1 βγ)(1 υγ)/βγ > 0 and {qt B } must satisfy qt B = (β/υ)e t {qt+1} B and qt B 0 for all t. Note that in a bubbleless equilibrium around the bubbleless BGP: ŷ t = E t {ŷ t+1 } r t which is identical to the standard dynamic IS equation of the representative consumer model. In any bubbly steady state, q B = 1 1 βγ 1 1 ΛΓυγ. Combining (39) and (40) yields the Euler equation: ĉ t = ΛΓυγE t {ĉ t+1 } + (1 βγ)(1 υγ) q t B + (1 βγ)ŷ t Υυγ r t ( ) ( ) where Υ 1 + (1 βγ)(λγ 1) 1 ΛΓυγ > 1 and Λ 1, β Γυ. Imposing goods market clearing: ŷ t = ΛΓυ β E t{ŷ t+1 } + Φ q B t Υυ β r t (41) 33

35 In the particular case of δ = 0, we have ΛΓ = Υ = 1 thus implying a dynamic IS equation of the form: ŷ t = υ β E t{ŷ t+1 } + Φ q B t υ β r t BGP. 3. Conditions for equilibrium uniqueness around the bubbleless The necessary and suffi cient conditions for A 0 to have two eigenvalues within the unit circle are: (i) det(a 0 ) < 1 and (ii) tr(a 0 ) < 1 + det(a 0 ) (LaSalle (1986)). Note that in the case of a bubbleless BGP, det(a 0 ) = β υ 1+κφ π while tr(a 0 ) = 1+κ+ β υ 1+κφ. Condition (ii) corresponds to φ π π > 1. Condition (i) requires ( φ π > 1 β κ υ ) Sunspot Fluctuations Let the equilibrium be described by the system of difference equations x t = AE t {x t+1 } where x t [ŷ t, π t, q B t ] are all non-predetermined variables. Let A have q 3 eigenvalues with modulus less than one. Consider the transformation x t = Qv t where QJQ 1 = A where J is the canonical Jordan matrix and Q [q (1), q (2), q (3) ] is the matrix of generalized eigenvectors, corresponding to the three eigenvalues. Thus, v t = JE t {v t+1 } [ ] Ju 0 where J = and v 0 J t = [vt u, vt s ]. s Consider the case where A has two eigenvalues with modulus less than one and one with modulus greater than one. For concreteness, [ assume λ] 1 < 1, λ1 0 λ 2 < 1, and λ 3 > 1. If all eigenvalues are real J u =, J 0 λ s = 2 λ 3, and q (k) corresponds to the eigenvector associated with eigenvalue k, for k = 1, 2, 3. Otherwise, if λ 1 = a + bi and λ 2 = a bi are complex conjugates, 34

36 [ a b J u = b a ], J s = λ 3 and q (1) and q (2) are, respectively, the imaginary and real components of the eigenvector associated with the complex eigenvalues. Thus, v u t = 0 all t. In addition v s t = λ 3 E t {v s t+1}, which has a stable solution: v s t = λ 1 3 v s t 1 + ξ t where ξ t is a martingale-difference (univariate) process. Thus, we have x t = q (3) v s t is the sunspot solution. The three variables in x t will be perfectly correlated (positively or negatively), and will display a firstorder autocorrelation λ 1 3. We can normalize the the third element of q (3) to unity. Consider next the case where A has one eigenvalue with modulus less than one and two with modulus greater than one. For concreteness, assume [ λ 1 ] < 1, λ2 0 λ 2 > 1, and λ 3 > 1. If all eigenvalues are real J u = λ 1, J s = and 0 λ 3 q (k) corresponds to the eigenvector associated with eigenvalue k, for k = 1, 2, 3. Otherwise, [ if λ 2 ] = a + bi and λ 3 = a bi are complex conjugates, J u = λ 1, a b J s = and q b a (2) and q (3) are, respectively, the imaginary and real components of the eigenvector associated with the complex eigenvalues, while q (1) is the eigenvector associated with λ 1. Thus, v u t = 0 all t. In addition v s t = J s E t {v s t+1}, which has a stable solution: vt s = J 1 s vt 1 s + ξ t where ξ t is a (bivariate) martingale-difference process. Thus, we have x t = [q (2), q (3) ]v s t is the sunspot solution. We can normalize the third element of q (2) and q (3) to unity. 35

37 Figure 1. Balanced Growth Paths r ρ + g g 0 β 1 υ -1 Bubbly without bubble creation Bubbly with bubble creation Bubbleless

38 Figure 2. Balanced Growth Paths: Bubble Size Bubbly without bubble creation Bubbly with bubble creation Bubbleless

39 Figure 3a Monetary Policy and Equilibrium Uniqueness: The Case of No Bubble Creation (r=g=0.004)

40 Figure 3b Monetary Policy and Equilibrium Uniqueness around a Bubbly BGP with Bubble Creation (r= )

41 Figure 3c Monetary Policy and Equilibrium Uniqueness around a Bubbly BGP with Bubble Creation (r= )

42 Figure 3d Monetary Policy and Equilibrium Uniqueness around a Bubbly BGP with Bubble Creation (r= )

43 Figure 4a Monetary Policy and Uniqueness of Bubbleless Equilibria