Asset Bubbles and Monetary Policy

Asset Bubbles and Monetary Policy Abstract We provide a model of rational bubbles in a DNK framework. Entrepreneurs are heterogeneous in investment efficiency and face credit constraints. They can trade bubble assets to raise their net worth. The bubble assets command a liquidity premium and can have a positive value. Monetary policy affects the conditions for the existence of a bubble, its steady-state size, and its dynamics including the initial size. The leaning-against-the-wind interest rate policy reduces bubble volatility, but could raise inflation volatility. Whether monetary policy should respond to asset bubbles depends on the particular interest rate rule and exogenous shocks. 1 “We need to explicitly integrate bubbles, a combination of rational and nonrational intuitive human responses, and other aspects of behavioral economics into our monetary policy models.” — Greenspan (2015) 1 Introduction The booms and busts of asset prices in stock markets and real estate markets around the world have typically been associated with business cycles in the macroeconomy (Borio, Kennedy, and Prowse (1994) and Jordà, Schularick, and Taylor (2015)). The general public, policy makers, and academic researchers often attribute the large movements of asset prices to the emergence and collapse of bubbles because asset price volatility cannot be explained entirely by fundamentals (Shiller (1981)). How do asset bubbles affect the real economy? How does monetary policy affect asset bubbles? Should monetary policy respond to asset bubbles? The goal of our paper is to provide a theoretical framework to address these questions. These questions have been the subject of a heated debate in the literature. Two views are dominant (Galı́ (2014)).1 The first view is that central banks should view price stability and financial stability as highly complementary and mutually consistent objectives. Even if asset prices can amplify and propagate shocks, including asset prices in monetary policy rules may result in only slight gains (Bernanke and Gertler (1999, 2001)). Moreover, asset bubbles driven by nonfundamental shocks are highly unpredictable. Thus monetary policy should not respond to asset prices. The second view is that central banks should act preemptively to prevent bubbles from forming, by raising interest rates or decreasing money supply to appropriate levels.2 Such a policy, often referred to as “leaning against the wind,” may call for a change in the inflation target. One reason for the debate is that the literature has yet to agree on a theoretical framework for understanding the formation of asset bubbles and the mechanism of how asset bubbles interact with the macroeconomy and monetary policy. The debate stems from the model of Bernanke and Gertler (1999) who introduce an irrational bubble to the model of Bernanke, Gertler, and Gilchrist (1999) (henceforth BGG). This model cannot address the question of how and why a bubble can emerge and burst under rational expectations. We contribute to the literature by providing an infinite-horizon model of rational asset bubbles in a dynamic new Keynesian (DNK) framework.3 The key feature of our model is that entrepreneurs (or firms) are heterogenous in their investment efficiencies and face credit constraints. In a frictionless Arrow-Debreu economy, rational bubbles cannot emerge and movements of asset prices reflect changes in the underlying economic 1 See Schularick and Taylor (2012) and Brunnermeier and Schnabel (2015) for empirical evidence. See, e.g., Borio and Lowe (2002), Cecchetti et al. (2000), Issing (2009), ECB (2010), and Blanchard et al. (2012). 3 By rational bubbles, we mean that they are consistent with individual optimality, rational expectations, and market clearing. See Brunnermeier and Oehmek (2013) for a survey of the literature on rational and irrational bubbles. 2 2 fundamentals. In this case central banks would not have to worry about asset prices. By contrast, due to credit constraints, an intrinsically useless bubble asset can provide liquidity and command a liquidity premium.4 If all agents believe the bubble asset is valuable, this belief can be self-fulfilling and the bubble asset can raise its owners’ net worth. Efficient entrepreneurs sell the bubble asset to inefficient ones to finance investment, who want to buy the bubble asset for precautionary reasons because they may become efficient in the future. Thus the bubble asset can be traded at a positive price and has an intensive margin effect in that it raises an entrepreneur’s net worth and hence his investment. It also has an extensive margin effect in that inefficient entrepreneurs must hold the bubble and will not make investment. The net effect on aggregate investment is typically positive. To introduce money and monetary policy, we incorporate a banking system with legal restrictions in the sense that banks must meet reserve requirements. Reserve requirements generate a spread between the lending rate and the deposit rate. Households and entrepreneurs can save by making deposits in banks and banks can lend to investing entrepreneurs. The central bank changes the money supply by changing reserves (high-powered money or monetary base). Monetary policy is conducted by following an interest rate rule. In this case money supply is endogenous. To allow monetary policy to have a large impact on the real economy, we introduce monopolistic competition and sticky prices as in Calvo (1983). Our main results can be summarized as follows. First, monetary policy can affect the conditions for the existence of a bubble. High inflation erodes entrepreneurs’ real balance and hence their net worth, generating a large liquidity premium for holding a bubble asset. When the liquidity premium is sufficiently high, a bubble can emerge. Thus raising the inflation target by raising the money supply permanently can fuel a bubble. The higher the inflation target, the more likely a bubble will emerge. On the other hand, if the economy already has a bubble, cutting the inflation target by decreasing the money supply permanently can prick the bubble. Second, monetary policy can affect the steady-state size and dynamics of the asset bubble including its initial size. In particular, a higher inflation target is associated with a higher steadystate size of the bubble. An expansionary monetary policy by cutting the interest rate or raising the money supply can raise the initial size of the asset bubble, which in turn generates a large amplification effect of monetary policy. Moreover, the coefficients in the interest rate rule affect the dynamics of the asset bubble in response to exogenous shocks. A higher interest rate response to asset bubbles reduces bubble volatility, but may raise inflation volatility. Third, whether monetary policy should respond to asset bubbles depends on the particular interest rate rule adopted by the central bank and on the type of exogenous shocks hitting the economy. Following Bernanke and Gertler (1999, 2001) and Gilchrist and Leahy (2002), we consider 4 Introducing dividends or rents will complicate our analysis without changing our key insights. See Miao and Wang (2015, 2018), Miao, Wang, and Zha (2014), and Miao, Wang, and Xu (2015) for models of rational bubbles attached to assets with dividends or rents. 3 two types of interest rate rules: (i) a Taylor rule, which responds to inflation and the gap between the actual output and steady-state output, and (ii) an inflation targeting rule, which responds to expected inflation, but not the output gap. We consider two types of shocks: (i) a fundamental TFP shock, and (ii) a non-fundamental sentiment shock to the bubble. We include a weight on the asset bubble in the interest rate rules and search for an optimal weight to maximize household utility. Our calibrated model shows that the central bank should cut interest rates in response to either a positive TFP shock or a positive sentiment shock. The welfare gains are relatively larger conditional on sentiment shocks, and are generally small for all cases. In our model the benefit of an asset bubble is to improve investment efficiency and its cost is to increase business cycle volatility. In response to a positive TFP shock, an efficient real business cycles model implies positive comovements of consumption, investment, output, and labor. But labor drops under the Taylor rule in a DNK model (Galı́ (1999)). Cutting interest rates when the asset bubble is expanding can raise aggregate demand and hence labor. In response to a positive sentiment shock, the asset bubble expands, but consumption falls on impact under the standard Taylor rule because the nominal and real interest rates rise too much. Cutting interest rates when the asset bubble is expanding can also boost consumption and benefit households. Under the inflation targeting rule, the optimal coefficients on the asset bubble are generally very small negative numbers. The welfare gains are also quite small as in Bernanke and Gertler (1999) and Gilchrist and Leahy (2002). As Bernanke and Gertler (1999) argue, to the extent that asset bubbles tend to be positively correlated with movements in output and inflation, policies based on these two variables subsume most of the gains from reacting to asset bubbles. In fact, in response to a TFP shock or a sentiment shock, the simple inflation targeting rule can generate the right comovements of macroeconomic quantities, giving very small welfare gains from reacting to asset bubbles. 2 Basic Intuition and Related Literature Our model features a standard aggregate supply block as in the DNK literature. The key part of our model is about aggregate demand and asset pricing equations. In the case without aggregate uncertainty, our model implies that deposits (or bonds) and the bubble asset satisfy the following two asset pricing equations Rt (1 + LIQt+1 ) , Πt+1   = SDFt+1 pht+1 1 + LIQht+1 , 1 = SDFt+1 pht (1) (2) where SDFt+1 , Rt , and Πt+1 denote the stochastic discount factor (SDF), the nominal interest rate, and the inflation rate between periods t and t + 1, respectively, pht denotes the real price of the 4 bubble asset, and LIQt+1 and LIQht+1 are the liquidity premiums for the deposits and the bubble asset, respectively.5 To understand the above two pricing equations, we consider a simple real model consisting of a continuum of risk-neutral entrepreneurs (or firms) in [0, 1] . Each firm j’s optimization problem is given by max ∞ X t=0 β t Djt , β ∈ (0, 1) subject to Bj,t+1 ≥ 0, Hj,t+1 ≥ 0, Ijt ≥ 0, Djt ≥ 0, r Djt + Ijt + pht Hj,t+1 + Bj,t+1 = Rkt Kjt + Bjt Rt−1 + pht Hjt , Kj,t+1 = (1 − δ)Kjt + εjt Ijt . (3) In each period t, the firm pays dividends Djt , makes real investment Ijt , buys (Hj,t+1 − Hjt ) units of the bubble asset at price pht , and buys Bj,t+1 units of riskfree bonds. The real capital return is r . Suppose that investment is irreversible and that the firm Rkt , and the real interest rate is Rt−1 cannot borrow, issue new equity, or short the bubble asset. Capital Kjt evolves according to (3) where εjt is independently and identically drawn from distribution F across firms and time. Firm j’s value function satisfies the Bellman equation Vt (Kjt , Bjt , Hjt , εjt ) = max Djt + βEVt+1 (Kj,t+1 , Bj,t+1 , Hj,t+1 , εj,t+1 ) . Conjecture that the value function takes the following form: Vt (Kjt , Bjt , Hjt , εjt ) = φkt (εjt )Kjt + φbt (εjt )Bjt + φht (εjt )Hjt . Let qtk denote Tobin’s marginal Q, qtk ≡ β R φkt+1 (ε) dF (ε) . Using the conjectured value function, we can rewrite the Bellman equation as φkt (εjt )Kjt + φbt (εjt )Bjt + φht (εjt )Hjt = max Rkt Kjt + Bjt Rt−1 + pht Hjt − Ijt − pht Hj,t+1 − Bj,t+1 Z k +qt [(1 − δ)Kjt + εjt Ijt ] + β φbt+1 (ε) dF (ε) Bj,t+1 Z +β φht+1 (ε) dF (ε) Hj,t+1 subject to Bj,t+1 ≥ 0, Hj,t+1 ≥ 0, and r 0 ≤ Ijt ≤ Rkt Kjt + Bjt Rt−1 + pht Hjt − pht Hj,t+1 − Bj,t+1 . 5 Equations (1) and (2) follow from (30) and (48) with δh = 0. 5 (4) Clearly, if qtk εjt > 1, making real investment is profitable. In this case firm j sells all its holdings of the bubble asset (Hj,t+1 = 0) and exhausts its borrowing limit (Bj,t+1 = 0) to reach the maximum investment level r Ijt = Rkt Kjt + Bjt Rt−1 + pht Hjt . Substituting these decision rules for εjt > 1/qtk into (4) and matching coefficients yield φkt (εjt ) = qtk εjt Rkt + qtk (1 − δ), r φbt (εjt ) = qtk εjt Rt−1 , φht (εjt ) = qtk εjt pht . (5) If εjt < 1/qtk , firm j does not make real investment (Ijt = 0), and is a marginal trader of the bubble asset and bonds. The first-order conditions for Hj,t+1 and Bj,t+1 are given by Z Z h h pt = β φt+1 (ε) dF (ε) , 1 = β φbt+1 (ε) dF (ε) . (6) Then matching coefficients in (4) for εjt < 1/qtk yields r φkt (εjt ) = Rkt + qtk (1 − δ), φbt (εjt ) = Rt−1 , φht (εjt ) = pht . Substituting (5) and (7) into (6) yields " Z pht = β 1 + # k 1/qt+1 " 1 = βRtr 1 + (7) k (qt+1 ε − 1)dF (ε) pht+1 , (8) # Z k 1/qt+1 k (qt+1 ε − 1)dF (ε) . We then obtain equations (1) and (2) by noticing that Rtr = Rt /Πt+1 . The liquidity premiums for both bonds and the bubble asset are equal to the integral terms in the above two equations in k this simple model. The intuition is that a sufficiently efficient firm with εjt+1 > 1/qt+1 can sell the bubble asset to inefficient firms at a positive price pht+1 to raise its net worth, which can fund the maximum investment level. This makes extra profits to the efficient firm, generating gains of the bubble asset in addition to its resale value. Inefficient firms are willing to buy the bubble asset at a positive price because they may become efficient and sell the bubble asset to finance investment in the future. Equation (8) shows that the marginal cost of buying a unit of the bubble asset is equal to the associated marginal benefit. The intuition for the bond pricing equation is similar. In this simple model bonds and the bubble asset are perfect substitutes. The bubble asset is intrinsically useless in the sense that it does not pay any dividend and does not enter utility or technology. Its fundamental value is zero as the equilibrium with pht = 0 for all t satisfies (2). There may exist a bubbly equilibrium in which pht > 0 for all t. In a steady state with 6 an infinitely lived representative agent, the SDF is equal to his subjective discount factor β ∈ (0, 1) . For a bubble pht = pht+1 > 0 to exist in the steady state, equation (2) becomes   1 = β 1 + LIQh . This equation cannot hold in a frictionless model without liquidity premium LIQh = 0, but it can hold if there is a positive liquidity premium LIQh > 0.6 The liquidity premium provides a benefit for holding a bubble asset, even though the bubble asset does not provide any dividends, such that discounting by β cannot eliminate its value in the steady state. We have elaborated on this point in our previous studies in real models without monetary policy (e.g., Miao and Wang (2012, 2014, 2015, 2018), Miao, Wang, and Xu (2015), Miao, Wang, and Xu (2016), and Miao, Wang, and Zhou (2015, 2016)). Our analysis of the conditions for the existence of a bubble revolves around the existence of a solution to the preceding equation and how monetary policy affects LIQh . In the special case of LIQt+1 = LIQht+1 , equations (1) and (2) imply that the growth rate of the bubble is equal to the real interest rate pht = pht+1 . Rt /Πt+1 (9) This equation also holds in overlapping generations (OLG) models in which a bubble can exist without a liquidity premium because the SDF is not equal to β in the steady state of OLG models. In particular, equations (1) and (9) imply that SDF = R/Π = 1 in the steady state. The existence conditions are then about whether there is a solution to this equation. Our model is consistent with the conventional wisdom that monetary policy can have an impact on asset price bubbles and that it can fuel or prick a bubble. Galı́ (2014) challenges this wisdom using an OLG model. The OLG framework naturally incorporates household heterogeneity and incomplete market participation, which can allow a bubble to emerge without any other frictions in dynamically inefficient economies (Samuelson (1958), Diamond (1965), and Tirole (1985)). Our infinite-horizon model features financial frictions and has the advantage that it can be integrated into the dynamic stochastic general equilibrium framework or the DNK framework and hence has the potential to be quantified (see Ikeda (2013) and Miao, Wang, Xu (2015)). After all, asset bubbles are used to describe large price movements and a theory of bubbles would be vacuous if it cannot be quantified. Galı́ (2014) sets up an elegant simple model in which equilibrium dynamics can be summarized by a unidimensional system based on (9). He shows that the conditions for the existence of bubbles are independent of monetary policy. He also shows that his unidimensional equilibrium system has a continuum of stable bubbly steady states and a continuum of unstable bubbly steady states. 6 More technically, the transversality condition will rule out bubbles without liquidity premium. This condition is not needed in OLG models with finitely lived agents. However, Giglio, Maggiori, and Stroebel (2016) find no evidence of bubbles that violate the transversality condition in the UK and Singapore housing markets. 7 He focuses on a stable bubbly steady state and assumes that the bubble pht is a backward-looking predetermined variable.7 In this case the initial size of the bubble ph0 is exogenously given and independent of monetary policy. Equation (9) then implies that a larger real interest rate generates a higher growth rate of the bubble. By contrast, the asset bubble in our model is a nonpredetermined forward-looking variable like any asset prices. Our equilibrium system is multi-dimensional and also features a continuum of bubbly steady states. But any bubbly steady state is a local saddle point.8 In response to an exogenous shock, the initial size of the asset bubble jumps and provides a powerful amplification mechanism. In our model monetary policy affects the initial size of the bubble and its dynamics through interest rates and liquidity premium. Like Galı́ (2014), we find that the “leaning against the wind” policy may not be optimal, but for a different reason. Our model shows that this policy can lower bubble volatility at the expense of raising inflation volatility. Galı́’s model shows that this policy may raise bubble volatility and reduce dividend volatility depending on the size of the bubble. Inflation does not cause a welfare loss in his model. Caballero and Krishnamurthy (2006), Farhi and Tirole (2012), Martin and Ventura (2012, 2015), and Asriyan et al. (2016) introduce credit constraints to OLG models of asset bubbles. Only Asriyan et al. (2016) introduce money and monetary policy. None of these papers derives asset pricing equations like (1) and (2). The role of the liquidity premium is emphasized by Kiyotaki and Moore (2008) in their infinite-horizon model with credit constraints. They focus on fiat money, which is also a pure bubble. Our model borrows some insights from theirs but differs from theirs in many ways: (i) the bubble asset is privately supplied, (ii) we have a DNK framework with sticky prices, and (iii) we have a banking system with reserve requirements. The reserve requirements generate a real balance effect, which is critical for the non-superneutrality of money and for monetary policy to affect the conditions for the existence of a bubble.9 Our paper is also related to infinite-horizon real models of rational bubbles with credit constraints (Santos and Woodford (1997), Kocherlakota (2009), Wang and Wen (2012), Miao and Wang (2012, 2018), Hirano and Yanagawa (2017), Miao, Wang, and Xu (2015), Miao, Wang, and Zhou (2015), and Aoki and Nikolov (2015)).10 Ikeda (2013) introduces monetary policy and wage rigidities to the model of Miao, Wang, and Xu (2015). Based on Bayesian estimations, he shows that the optimal monetary policy calls for monetary tightening to restrain the boom at the cost 7 Miao, Shen, and Wang (2019) show that Gali’s results will be overturned if one focuses on an unstable bubbly steady state. Moreover, a stable bubbly steady state and the equilibrium around that state are not learnable (i.e., not E-stable). 8 This is a numerical result because our equilibrium system is too complicated to permit a theoretical result. We have obtained this result numerically in Miao, Wang, and Xu (2015) and formally proved this result theoretically in simpler models of Miao and Wang (2012, 2018) and Miao, Wang, and Zhou (2015). 9 As long as entrepreneurs hold money for whatever reasons, there will be a real balance effect even if there is no reserve requirement. 10 See Miao (2014) for a recent survey. 8 of greater inflation volatility. Biswas, Hanson, and Phan (2018) introduce nominal wage rigidities and a zero lower bound on interest rates to the model of Hirano and Yanagawa (2017). They study how the macro-prudential learning-against-the-wind policy balances between the tradeoff between the booms and busts of bubbles. More broadly, our model is related to the large literature on the relationship between monetary policy and asset prices (see Gilchrist and Leahy (2002) for a survey). Our model shares many insights with Kiyotaki and Moore (1997), BGG (1999), and Bernanke and Gertler (1999, 2001), in which the financial accelerator, bank lending, net worth, and balance sheet channels are important transmission and propagation mechanisms. Our result that a strong inflation targeting rule does not call for the need to respond to asset bubbles because the welfare gain is too small confirms the early finding of Bernanke and Gertler (1999), albeit asset bubbles are rational in our model. 3 The Model Consider an infinite-horizon economy consisting of households, firms, retailers, financial intermediaries (banks), and a government (monetary authority). Following BGG (1999), we assume that retailers are monopolistically competitive and their role is to introduce nominal price rigidities. 3.1 Households There is a continuum of identical households of measure unity. The representative household is an extended family consisting of workers, entrepreneurs, and bankers. Each entrepreneur runs a firm and workers supply labor to firms. Bankers are identical and each banker manages a bank. The family and firms can save by making deposits in banks which in turn extend loans to borrowers. Entrepreneurs, bankers, and retailers hand in their dividends to households who are shareholders. Each household chooses consumption {Ct }, labor supply {Nt } , and deposits {Sa,t+1 } to maximize utility max {Ct ,Sa,t+1 ,Nt } subject to Ct + E0 ∞ X t=0 β t (ln Ct − ψNt ) , Sa,t+1 Rt−1 Sat = wt Nt + Dt + + Xt , Pt Pt (10) (11) where β is the subjective discount factor, wt is the real wage, Pt is the price level, Dt is the total dividends from bankers, entrepreneurs, and retailers, Rt−1 is the nominal interest rate (deposit rate) between periods t − 1 and t, and Xt denotes the sum of money transfers and lump-sum taxes/transfers from the government. Suppose that the household cannot borrow so that Sa,t+1 ≥ 0. 9 The first-order conditions imply that ψ , Λt   βΛt+1 Rt , with equality when Sat+1 > 0, 1 ≥ Et Λt Πt+1 wt = (12) (13) where Λt = 1/Ct denotes the household marginal utility and Πt+1 = Pt+1 /Pt denotes the (gross) inflation rate between periods t and t + 1. 3.2 Banks Bankers are identical of measure unity. At each time t, the representative bank receives total deposits St+1 and chooses loans Lt+1 and reserves Mt+1 . For simplicity, suppose that there is no bank net worth and that there is no interest on reserves. Reserves are often called high-powered money or monetary base. We do not consider currency in circulation or inter-bank loan markets in this model. The balance sheet equation is given by Lt+1 + Mt+1 = St+1 . (14) The bank is also subject to the reserve requirement Mt+1 ≥ λSt+1 , (15) where λ ∈ (0, 1) denotes the required reserve ratio. The bank’s objective is to maximize profits   βΛt+1 Lt+1 Rlt + Mt+1 − St+1 Rt max Et , (16) Mt+1 ,Lt+1 Λt Pt+1 subject to (14) and (15), where Rlt denotes the loan rate between periods t and t + 1. We can show that, as long as Rlt > Rt , the constraint (15) binds so that Mt+1 = λSt+1 , (17) Rt − λ . 1−λ (18) and Rlt = This equation implies that the equilibrium deposit rate must satisfy Rt > 1 in order for Rlt > Rt to hold. It also shows that the bank makes zero profit and that the lending rate Rlt increases with the deposit rate in order for the bank to break even because reserves do not bear any interest. 10 3.3 Firms Each entrepreneur j ∈ [0, 1] runs a firm that combines labor Njt and capital Kjt to produce an intermediate (wholesale) good j according to the technology α 1−α Yjt = At Kjt Njt , α ∈ (0, 1) , where At denotes TFP, which follows an AR(1) process ln At = ρa ln At−1 + εat , where {εat } is a white noise process. The entrepreneur sells wholesale goods to retailers at the real price pwt . The static profit maximization problem yields α 1−α Rkt Kjt = max pwt At Kjt Njt − wt Njt , Njt where  Rkt = α 1−α wt  1−α α 1 (pwt At ) α . (19) and the first-order condition gives labor demand α −α wt = (1 − α) pwt At Kjt Njt . (20) At the beginning of period t, the entrepreneur faces idiosyncratic investment-specific shock εjt and makes investment Ijt to increase his capital stock so that the law of motion for capital follows Kjt+1 = (1 − δ)Kjt + εjt Ijt , (21) where δ ∈ (0, 1) represents the depreciation rate. Suppose that the cumulative distribution function of εjt is F and the density function is f on [εmin , εmax ] ⊂ [0, ∞) and εjt is independently and identically distributed across firms and over time. Assume that there is no insurance market against the idiosyncratic investment-specific shock and that investment is irreversible at the firm level so that Ijt ≥ 0. The entrepreneur chooses to save Sjt+1 ≥ 0 in banks at the deposit rate Rt and borrow Ljt+1 ≥ 0 at the lending rate Rlt . He is endowed with δh ∈ (0, 1) units of an intrinsically useless bubble asset h , which is nonnegative by at the beginning of each period t. Its nominal price is denoted by Pt|t free disposal. In each period a fraction δh of each vintage of the bubble assets is assumed to lose its value, due to depreciation for example. This implies that the total amount of bubble assets outstanding remains constant and is normalized to one. Our modeling of such recurrent bubbles is related to Martin and Ventura (2012), Galı́ (2014), and Miao, Wang, and Xu (2015). The purpose is to introduce a non-fundamental sentiment shock that drives movements of the asset bubbles. 11 Entrepreneurs can trade each vintage of the bubble asset. Let Hj,t+1|t−k represent the period t choice of the bubble asset endowed in period t − k. By convention, we set Hj,t|t = δh / (1 − δh ) . h Let Pt|t−k denote the period t nominal price of the bubble asset at vintage t − k. Assume that entrepreneurs cannot short the bubble asset so that Hj,t+1|t−k ≥ 0, t, k = 0, 1, 2, .... (22) Assume that entrepreneurs face borrowing constraints due to imperfect contract enforcement. We impose 0 ≤ Ljt+1 ≤ µPt Kjt , (23) where µ ∈ (0, 1) . The interpretation of this constraint is that the entrepreneur’s borrowing is limited by the collateral value of a fraction µ of his existing capital.11 We write the flow-of-funds constraints as Djt + Ijt + ∞ X pht|t−k Hj,t+1|t−k + k=0 1 (Sjt+1 + Ljt Rlt−1 ) Pt ∞ X 1 (Sjt Rt−1 + Ljt+1 ) + (1 − δh ) pht|t−k Hj,t|t−k , = Rkt Kjt + Pt (24) k=0 h where Djt denotes real dividends and pht|t−k = Pt|t−k /Pt denotes the real price of the bubble asset. Suppose that equity finance is so costly that the firm does not issue any new equity.12 Thus we impose Djt ≥ 0. (25) The entrepreneur’s objective is to maximize the discounted present value of dividends. We can write his decision problem using dynamic programming    ∞ Vt Kjt , Sjt , Ljt , Hj,t|t−k k=0 , εjt   Λt+1 Vt+1 Kjt+1 , Sjt+1 , Ljt+1 , Hj,t+1|t−k = max Djt + βEt Λt {Ijt ,Sjt+1 ,Ljt+1 ,Hjt+1 } (26) ∞ , εjt+1 k=0  , subject to (21), (22), (23), (24), and (25), where we have used the household’s intertemporal marginal rate of substitution as the stochastic discount factor. Here Vt (·) denotes the value function. Define Tobin’s (marginal) Q as qtk ≡   ∂ βΛt+1 Et Vt+1 Kjt+1 , Sjt+1 , Ljt+1 , Hj,t|t−k ∂Kjt+1 Λt ∞ , εjt+1 k=0  . The following proposition characterizes the entrepreneur’s optimal decisions. 11 Unlike Kiyotaki and Moore (1997), we do not use future capital as collateral. Using future capital as collateral will complicate algebra significantly without changing our key insights. See Caballero and Krishnamurthy (2006), Miao and Wang (2018), and Miao, Wang and Zhou (2015) for related discussions. 12 Our key insights will not change as long as new equity issues are sufficiently limited (see Miao and Wang (2008) and Miao, Wang, and Xu (2015)). 12 Proposition 1 In an equilibrium with Rlt > Rt > 1, there exist two cut-off values ε∗t and ε∗∗ t such that the decision rules are given by Ijt = 0 if εjt < ε∗t and 1 (Sjt Rt−1 − Sjt+1 + Ljt+1 − Ljt Rlt−1 ) Pt ∞ X + (1 − δh ) pht|t−k Hj,t|t−k if εjt ≥ ε∗t , Ijt = Rkt Kjt + k=0 ( 0, if εjt ≥ ε∗t Sjt+1 = , indeterminate, otherwise ( 0, if εjt < ε∗∗ Ljt+1 t = , Pt µKjt , otherwise ( 0, if εjt ≥ ε∗t Hj,t+1|t−k = , indeterminate, otherwise where the cut-off values satisfy 1 , qtk (27) Rlt ∗ ε , Rt t (28) ε∗t = ε∗∗ t = and qtk , Rt , and pht|t−k satisfy13 qtk # " Z εmax   Λt+1 k qt+1 ε − 1 dF (ε) = βEt Rkt+1 1 + Λt ε∗t+1  Z εmax  Λt+1 k Rlt Λt+1 k +βEt qt+1 ε − q (1 − δ) + βEt µ dF (ε) , Λt t+1 Λt Rt ε∗∗ t+1 pht|t−k # " Z εmax   Λt+1 Rt k qt+1 ε − 1 dF (ε) , 1+ 1 = βEt Λt Πt+1 ε∗t+1 " # Z εmax   Λt+1 h k = (1 − δh ) βEt p qt+1 ε − 1 dF (ε) . 1+ Λt t+1|t−k ε∗t+1 (29) (30) (31) This proposition shows that there are two cutoff values ε∗t and ε∗∗ t such that the firm makes investment if and only if εjt > ε∗t = 1/qtk . This is consistent with Tobin’s rule. Making one unit of investment costs one unit of consumption goods, but yields εjt units of capital, and hence the marginal benefit is qtk εjt . When the marginal benefit is higher than the marginal cost, the firm invests. When investing, it sells the bubble asset as much as possible to finance investment. It does not save, but borrows from banks if and only if investment is sufficiently profitable (i.e., investment ∗ efficiency exceeds the cutoff ε∗∗ t > εt ), since the lending rate Rlt is higher than the deposit rate Rt . 13 The usual transversality conditions must also hold. Moreover, for ease of exposition, we assume that parameter values are such that ε∗t and ε∗∗ t are in the interior of (εmin , εmax ) . 13 ∗ The ratio of these two cutoffs ε∗∗ t /εt is equal to the ratio of the lending rate to the deposit rate, which is interpreted as the external finance premium as in BGG (1999). When the firm decides to borrow, it does so up to the credit limit. Equations (29), (30), and (31) are the asset pricing equations for capital, deposits, and the bubble asset of vintage t − k. In addition to the usual terms in these equations, two integral terms deserve discussion. Both terms represent liquidity premium due to financial frictions. The first term represents the liquidity premium from internal funds, savings (deposits), and the bubble asset. The entrepreneur sells all the bubble asset when his investment efficiency exceeds ε∗t+1 , which appears k ε in period t + 1. Thus the preceding equation gives in equation (31). The marginal benefit is qt+1 the expected profits from one unit of borrowing. The multiplicative factor (1 − δh ) represents the undepreciated value because a fraction δh of the bubble asset loses its value. The factor µ appears in equation (29) because only µ dollars can be borrowed using one dollar of capital as collateral. 3.4 Retailers Retailers are monopolistically competitive. In each period t they buy intermediate goods from entrepreneurs at the real price pwt and sell good j at the nominal price Pjt . Intermediate goods are transformed into final goods according to the CES aggregator Z Yt = 1 Yjt σ−1 σ σ  σ−1 , σ > 1. dj (32) 0 Thus retailers face demand given by  Yjt = Pjt Pt −σ Yt , (33) where the price index is given by Pt ≡ 1 Z Pjt 1−σ 1  1−σ di . (34) 0 To introduce price stickiness, we assume that each retailer is free to change its price in any period only with probability 1 − ξ, following Calvo (1983). Following Erceg, Henderson, and Levin (2000), we also assume that whenever the retailer is not allowed to reset its price, its price is automatically increased at the steady-state inflation rate. The retailer selling good j chooses the nominal price Pt∗ in period t to maximize the discounted present value of profits max ∗ Pt ∞ X k=0 k ξ Et  β k Λt+k Λt    Πk Pt∗ ∗ (1 + τ ) − pw,t+k Yjt+k , Pt+k subject to the demand curve ∗ Yjt+k  = Πk Pt∗ Pt+k 14 (35) −σ Yt+k , (36) where τ denotes the output subsidy and Π denotes the steady-state inflation target. We use the household intertemporal marginal rate of substitution as the stochastic discount factor because retailers must hand in all profits to households who are the shareholders. The first-order condition gives the pricing rule P∞ k σ k −σ E σ 1 t k=0 (βξ) Λt+k pw,t+k Pt+k Yt+k (Π ) ∗ Pt = . P∞ σ−1 1 + τ σ − 1 Et k=0 (βξ)k Λt+k Pt+k (Πk )1−σ Yt+k (37) We set 1 + τ = σ/(σ − 1) to completely remove the distortion due to monopolistic competition. Let p∗t = Pt∗ /Pt . We can then write the pricing rule in a recursive form as follows p∗t = Γat , Γbt (38) where  Πt+1 σ a Γt+1 , Π   Πt+1 σ−1 b = Λt Yt + βξEt Γt+1 . Π Γat = Λt pwt Yt + βξEt Γbt  (39) (40) The aggregate price level satisfies h 1−σ Pt = ξ (ΠPt−1 ) or 3.5 + (1 − ξ) (Pt∗ )1−σ i 1 1−σ , "   # 1 1−σ Π 1−σ ∗1−σ 1= ξ . + (1 − ξ) pt Πt (41) Monetary Policy To close the model, we consider two types of interest rate rules for monetary policy. First, the central bank sets the nominal deposit rate in response to the current inflation, output, and asset prices:   ln Rt = ln R + φπ ln (Πt /Π) + φy ln (Yt /Y ) + φp ln pht /ph + vt , (42) where pht denotes the aggregate bubble index to be defined later, ph is its steady state value, Π is the target inflation rate, R is the steady-state nominal deposit rate, and Y is the steady-state output level. Assume that {vt } follows an AR(1) process vt = ρv vt−1 + εvt , where ρv ∈ (0, 1) and {εvt } is a white noise process. In the DNK literature one often focuses on the zero-inflation steady state by setting Π = 1. But we will study the case with Π > 1 because a positive inflation rate is important for the emergence of a bubble. 15 Second, we consider the inflation targeting rule following Bernanke and Getter (1999) and Gilchrist and Leahy (2002):   ln Rt = ln R + φπ Et ln (Πt+1 /Π) + φp ln pht /ph + vt . (43) As a baseline rule, we set φp = 0 in (42) and (43) and call the former the Taylor rule. In Section 6 we will search for the optimal value of φp to maximize household utility with full commitment. The interest rate rule policy implies that money supply is endogenous. Let reserves follow the dynamics Mt+1 = Mt exp (gt ) , (44) where gt is the endogenous exponential growth rate of money. Assume that the increased money is transferred to households in a lump-sum manner. If one assumes that gt is exogenous, then the nominal interest rate will be endogenous. In our analysis below we will follow the tradition of the DNK framework to adopt the interest rate rule. 3.6 Equilibrium System The market-clearing conditions for bank loans, deposits, and the bubble asset are given by Z Z Z Ljt dj = Lt , Sjt dj + Sat = St , Hj,t+1|t−k dj = δh (1 − δh )k , k = 0, 1, 2... (45) for all t. Equation (13) and (30) imply that Sa,t+1 = 0 because the deposit rate is too low.14 Assuming Sa0 = 0, we have Sat = 0 for all t ≥ 0. R Define aggregate capital, aggregate investment, and aggregate labor as Kt = Kjt dj, It = R R Ijt dj, and Nt = Njt dj. By the labor demand condition (20), we can show that the capital-labor ratio is independent of j and hence we have wt = (1 − α) pwt At Ktα Nt−α . (46) Plugging (46) into (19), we can show that Rkt is equal to the marginal revenue product of capital Rkt = αpwt At Ktα−1 Nt1−α . (47) Define an index for pre-existing bubbles as pE t = ∞ X k=1 pht|t−k (1 − δh )k δh . h The size of the total new bubble is denoted by pN t = δh pt|t . Define the economy’s aggregate bubble N as pht = pE t + pt . Using these definitions, we can rewrite equation (31) as " # Z εmax   Λ t+1 E k pht = βEt p 1+ qt+1 ε − 1 dF (ε) . Λt t+1 ε∗t+1 14 (48) Similarly, households will not hold any bubble asset by equation (31), even though they are allowed to trade it. 16 This equation is analogous to equation (14) in Galı́ (2014). Note that pE t+1 appears on the right-hand side of (48) because all bubbles today become old tomorrow. By Proposition 1, firms with εjt ≥ ε∗∗ t take loans from banks up to the borrowing limit in (23). Aggregating the borrowing limits from all lending banks yields Lt+1 = µKt (1 − F (ε∗∗ t )) . Pt Using (14) and (17), we can rewrite the preceding equation as mt+1 = λ µKt (1 − F (ε∗∗ t )) , 1−λ (49) where we have used mt+1 ≡ Mt+1 /Pt to denote the real balance. This equation gives money demand. Equation (44) gives the money supply mt+1 = mt exp (gt ) . Πt (50) By Proposition 1, equations (14), (17), (18), and the market-clearing conditions above, we can show that It = and   mt + pht (1 − F (ε∗t )) + µKt (1 − F (ε∗∗ Rkt Kt + t )) , Πt  Kt+1 mt = (1 − δ) Kt + Rkt Kt + + pht Πt Z εmax Z εdF (ε) + µKt ε∗t (51) εmax ε∗∗ t εdF (ε) . (52) Equation (51) shows that aggregate investment is financed by internal funds Rkt Kt , the real balance mt /Πt (or returns on net savings), asset sales pht , and bank loans µKt . The two equations above show that asset bubbles have intensive and extensive margin effects on aggregate investment and capital accumulation. The two cutoffs ε∗t and ε∗∗ t reflect the extensive margin effect. The term 1 − F (ε∗t ) represents the mass of firms that sell the bubble asset to finance investment and the term 1 − F (ε∗∗ t ) represents the mass of firms that take loans to finance investment. Aggregating (33) over all firms yields Yt = where ∆t = R  Pjt −σ Pt At α 1−α , K N ∆t t t (53) dj is the price dispersion, which satisfies the recursive equation ∆t = (1 − ξ) p∗−σ t  +ξ Π Πt −σ ∆t−1 , (54) where ∆−1 is exogenously given. The resource constraint is given by Ct + It = Yt . 17 (55) In summary, we have shown that the equilibrium system in a neighborhood of the steady state with Rt > 1 consists of 21 equations (12), (18), (27), (28), (29), (30), (38), (39), (40), (41), (46), (47), (48), (49), (50), (51), (52), (53), (54), and (55), plus a monetary policy equation (42) or (43), for k ∗ a b h E 22 variables {wt , ε∗t , ε∗∗ t , qt , Rt , Rlt , pt , Γt , Γt , Πt , pwt , Nt , Rkt , pt , pt , mt , gt , It , Kt , Yt , ∆t , Ct }, where Λt = 1/Ct . Here Kt , mt , and ∆t are endogenously predetermined variables in that K0 , m0 and ∆−1 are exogenously given. The usual transversality conditions must also hold. As in Galı́ (2014), the sizes of pre-existing bubbles and new bubbles cannot be independently determined in equilibrium. That is, there could exist a continuum of bubbly equilibria. Galı́ then assumes that new bubbles follow an exogenous IID process, which is interpreted as a bubble shock. For ease of computations, we follow Miao, Wang, and Xu (2015) and assume that new bubbles h account for a stochastic fraction of aggregate bubbles so that pE t = st pt , where st ∈ (0, 1) . Suppose that ln st = (1 − ρs ) ln s + ρs ln st−1 + εst , where ρs ∈ (0, 1) , s ∈ (0, 1) is the non-stochastic steady-state value of st , and {εst } is a white noise process. We can interpret st as an exogenously given bubble shock or sentiment shock that drives the movements of aggregate bubbles. 4 Steady-State Analysis There are two types of steady state equilibria: bubbleless and bubbly. We remove the time subscript from a variable to indicate its steady state value. Whenever necessary, we also use a variable with subscript f or b to indicate its bubbleless or bubbly steady state value, respectively. In both types of steady states we have p∗ = pw = ∆ = 1 and the inflation rate is equal to the growth rate of money, Π = exp (g). Due to the full price indexation assumed earlier, both the flexible price equilibrium and the sticky price equilibrium have the same steady states.15 We first derive some common equations that hold in both the bubbleless and bubbly steady states. By equation (30), we can derive the nominal deposit rate as R= 1+ R εmax ε∗ Πβ −1
≡ R(ε∗ ). ε − 1 dF (ε) ∗ ε (56) It is straightforward to show that R (ε∗ ) increases with ε∗ and R/Π < 1/β. Since the steady-state real interest rate R/Π in both the bubbly and bubbleless steady states is too low, households will not hold the bubble asset (even if they are allowed to trade it) and will not save in a neighborhood of either steady state so that Sa,t+1 = 0. 15 Without full indexation, since price dispersion increases with trend inflation, there will be non-superneutrality of money (i.e., a negative relationship between long-run output and inflation) in the steady state even in a standard DNK model without financial frictions (Ascari and Sbordone (2014)). 18 By equation (18) and (28), we have ε∗∗ = 1 − λ/R(ε∗ ) ∗ ε ≡ ε (ε∗ ) . 1−λ (57) We can easily check that ε∗∗ increases with ε∗ . Using equation (29) and (57), we can derive 1 Rk = ∗ ε 1 β R εmax − 1 + δ − µ ε∗∗ (ε − ε∗∗ ) dF (ε)
R εmax ε ≡ Rk (ε∗ ). − 1 dF (ε) 1 + ε∗ ε∗ (58) We will impose assumptions below such that R > 1 so that Rl > R and ε∗∗ > ε∗ . The critical step of establishing the existence of steady state equilibria is to show the existence of a cutoff ε∗ . Once this cutoff is obtained, other steady-state values can be easily derived. 4.1 Bubbleless Steady State In a bubbleless steady state ph = 0 and hence we can ignore equation (31). To show the existence of a bubbleless steady state with Rf > 1, we impose the following: Assumption 1 Let βE [ε] > Πεmin and Π > β. The first inequality holds for any distribution with εmin = 0 and the second is needed because it is implied by (56) for R > 1 in the steady state. Lemma 1 Under assumption 1, there exists a unique ε ∈ (εmin , εmax ) such that  Z εmax  ε 1+ − 1 dF (ε) = Πβ −1 . ε ε This lemma implies that R (ε) = 1. Since R (ε∗ ) increases with ε∗ , we need to find an ε∗ > ε so that R = R (ε∗ ) > 1 in the steady state. Proposition 2 Suppose that assumption 1 holds and µ satisfies the assumptions in Appendix A.   Then there exists a unique bubbleless steady state with Rf = R ε∗f > 1, where ε∗f ∈ (ε, εmax ) is the unique solution to the equation   Z εmax Z εmax λµ 1 − F (ε (ε∗ )) ∗ δ = Rk (ε ) + εdF (ε) + µ εdF (ε) , 1−λ Π ε∗ ε(ε∗ ) (59) where ε (ε∗ ) is given in (57). Equation (59) is derived from the steady-state version of equation (52). The left-hand side of (59) is the depreciation rate and the right-hand side is the ratio of investment in efficiency units to the capital stock. Investment (in consumption units) is financed by internal funds Rk (ε∗ )K and the real balance of reserves (i.e., savings minus loan repayments) by equation (49) m λµK 1 − F (ε (ε∗ )) = . Π 1−λ Π 19 (60) In addition, firms with efficiency levels higher than ε (ε∗ ) borrow from banks. Since the bubble asset has no value in a bubbleless equilibrium, firms can only borrow against a fraction µ of their capital. This explains the last term in (59). ∗ After determining the investment threshold ε∗f , we then derive the other threshold ε∗∗ f = ε(εf ) using (57). The other equilibrium variables can also be easily determined as shown in the proof of Proposition 3 in Appendix A. From the analysis above, our model clearly implies the nonsuperneutrality of money in the sense that the steady-state levels of the capital-labor ratio, consumption, and output all depend on the inflation rate. Proposition 3 Suppose that the assumptions in Proposition 2 hold. Then in the bubbleless steady state ε∗f , Kf /Nf , wf , Yf /Nf , and Cf all decrease with Π. The intuition is that the inflation rate Π affects entrepreneurs’ net worth and hence their investment behavior. In particular, it affects the investment cutoff ε∗f according to (59). The impact on ε∗f in turn affects the mass of investing entrepreneurs and the liquidity premium and hence the real economy. Note that the assumption of the reserve requirement λ ∈ (0, 1) generates a credit spread between the lending and deposit rates and hence a real balance effect, which is crucial for the non-superneutrality of money and also for inflation to affect the bubbly steady state analyzed in the next subsection. 4.2 Bubbly Steady State We now turn to the bubbly steady state. As in Galı́ (2014), there could exist a continuum of bubbly steady states because the sizes of pre-existing bubbles and new bubbles cannot be independently determined. Proposition 4 Suppose that the assumptions in Proposition 2 hold so that there exists a bubbleless   steady state with Rf = R ε∗f > 1, where ε∗f is the investment threshold. Then there exists a continuum of bubbly steady states indexed by s ∈ (0, 1) with Rb = R (ε∗b ) > Rf > 1 if and only if " ! # Z εmax ε βs 1 + − 1 dF (ε) > 1, (61) ε∗f ε∗f   where the cutoff ε∗b ∈ ε∗f , εmax is the unique solution to the equation  Z εmax βs 1 + ε∗  ε  − 1 dF (ε) = 1. ε∗ (62) Equation (62) follows from the steady-state version of the asset pricing equation (48) for the h bubble asset when pE t = st pt > 0 is constant over time. The interpretation of condition (61) is as follows: The left-hand side of (61) represents the benefit of buying one unit of the bubble 20 asset today and the right-hand side represents the associated cost. In addition to its resale value, the bubble asset commands a liquidity premium due to credit constraints. The liquidity premium emerges because an entrepreneur can sell all his bubble asset to finance investment if his investment efficiency exceeds ε∗f . This explains the integral terms in (61). When the benefit exceeds the cost in the bubbleless steady state, entrepreneurs have incentives to trade the bubble asset at a positive price, thereby creating a bubble. Once the cutoff ε∗b is determined, we can follow a similar procedure to that described in the previous subsection to derive other steady state variables. The details can be found in the proof of Proposition 4 in Appendix A. How do we relate our existence condition (61) to the traditional condition that the bubbleless steady-state interest rate must be lower than the rate of economic growth (Tirole (1985))? Unlike the OLG model of Tirole (1985), our model features credit constraints in the infinite-horizon DNK framework. We have the following result. Proposition 5 Suppose that the assumptions in Proposition 2 hold so that there exists a bubbleless   steady state with Rf = R ε∗f > 1, where ε∗f is the investment threshold. A necessary condition for the existence of a bubbly steady state with Rb > 1 is   1 Π > max , Rf > 1. s This proposition shows that for a bubbly steady state to exist, the net inflation rate must be positive and sufficiently high and the gross real interest rate Rf /Π must be less than the gross growth rate of the economy (which is 1). But these conditions are not sufficient as shown in the previous proposition. Does monetary policy affect the existence condition for a bubbly steady state? As shown in Proposition 3, an increase in the rate of inflation reduces an entrepreneur’s real balance and hence his net worth and investment. To maintain the aggregate investment rate at the same level as the depreciation rate in the bubbleless steady state, there must be more firms to make investment. Thus the investment threshold must fall or Tobin’s Q must rise, causing the liquidity premium to rise in the bubbleless steady state. This raises the benefit of trading the bubble asset so that condition (61) is more likely to be satisfied for higher inflation. It is possible that (61) does not hold for low inflation so that only a bubbleless equilibrium can exist initially. The central bank can conduct an expansionary monetary policy by increasing money supply such that inflation rises to a permanently higher level. At this higher level, condition (61) holds so that a bubble can emerge. On the other hand, suppose that the economy is initially in the bubbly steady state. When the central bank reduces money supply such that inflation decreases to a permanently lower level at which condition (61) fails to hold, the asset bubble will collapse. This 21 result is consistent with the conventional wisdom that an expansionary monetary policy can fuel a bubble and a contractionary monetary policy can prick a bubble. The key intuition comes from the balance sheet channel. Inflation caused by expansionary monetary policy reduces an entrepreneur’s net worth and raises the liquidity premium, thereby raising his demand for the bubble asset. We now conduct a comparative statics analysis based on the parameter values given in Section 5.1. We study the impact of the inflation target on the bubbly steady state when condition (61) holds. Figure 1 presents the result, which shows that higher inflation has a negative effect on the economy in the long run. Higher inflation raises the size of the asset bubble, but lowers output, consumption, capital, and labor. The intuition is that higher inflation lowers the real balance and hence entrepreneurs’ net worth, leading to lower investment. On the other hand, the creditconstrained entrepreneurs will demand more bubble assets to finance investment. In the meantime, higher inflation is associated with higher money supply, which can support a larger size of bubbles and inflate asset bubbles. As discussed earlier, a larger size of the asset bubble has both intensive and extensive margin effects. It allows entrepreneurs to borrow more to make investment. But as the borrowing cutoff ε∗∗ t rises (see Figure 1), fewer firms will borrow and invest. The net effect leads to reduced aggregate investment and capital accumulation. 5 Dynamic Responses to Shocks In this section we study the dynamic responses of the model economy to various shocks. We shall focus on a bubbly steady state and local dynamics around the bubbly steady state. To provide quantitative experiments, we need to assign parameter values. 5.1 Calibration We calibrate our model at quarterly frequency so that a particular bubbly steady state is roughly consistent with the long-run behavior of the US economy. We assume that the investment-efficiency shock follows a Pareto distribution with F (ε) = 1 − (ε/εmin ) − η1 . We set εmin = 1 − η so that the unconditional mean is 1. As is standard in the business cycle literature, we set α = 0.33, β = 0.99, and δ = 0.025. We choose η = 0.28 to match the aggregate investment-to-output ratio of 20% as in the data. We choose the utility weight on labor ψ = 3.39 so that the average number of hours worked in the bubbly steady state equals 25% of the total time endowment. We set Π = 1.01 so that the steady state annual inflation rate is 4%, which is consistent with the average inflation rate during the period between 1975 and 2016. We set λ = 0.1 so that the required reserve ratio is 10%. As in the DNK literature (e.g., BGG (1999) and Gilchrist and Leahy (2002)), we set ξ = 0.75 and σ = 11, implying that the duration of price adjustments is four quarters and the steady-state markup is σ/ (σ − 1) = 1.1. We set the pledgeability parameter for capital µ = 0.2, which is 22 consistent with the estimates reported in Liu, Wang, and Zha (2013) and Miao, Wang, and Xu (2015). We pick a bubbly steady state with s = 99.22% so that ph /Y = 20%. It is difficult to measure the size of the bubble asset in the data. Some people argue that cryptocurrencies are pure bubbles. The cryptocurrency market is very volatile and its current market capitalization is around $570 billion, which is around 20% of the US quarterly GDP in 2017. For robustness, we consider two other calibrations with ph /Y = 10% and 40%. We find our results do not change significantly. Now we assign parameter values for the shocks and the interest rate rules. As in the DNK literature (e.g., Galı́ (2008)), we set φπ = 1.5, φy = 0.5/4, and φp = 0 in the Taylor rule (42). For the inflation targeting rule (43), we set φp = 0, φπ = 1.1 (weak inflation targeting), and φπ = 2 (strong inflation targeting), as in Bernanke and Gertler (1999) and Gilchrist and Leahy (2002). As baseline values, we set the persistence parameters for the monetary policy shock and the technology shock as ρv = 0.5 and ρa = 0.9. Following Angeletos, Collard and Dellas (2014), we set the persistence and the volatility of the sentiment shock as ρs = 0.833 and σs = 0.613%. Similar to Galı́ (2014), our model implies a continuum of bubbly steady states indexed by s. In his unidimensional equilibrium system, Galı́ (2014) shows that there are two types of bubbly steady states: one is stable and the other is unstable. We are unable to derive this result theoretically for our multi-dimensional equilibrium system. However, we have verified numerically that the bubbly steady state is a saddle point for many values of s. Galı́ (2014) restricts his analysis to the stable steady state and views the bubble as a backward-looking predetermined variable that converges to the steady state starting at any initial value. He also introduces a sunspot shock to drive the dynamics of the bubble. By contrast, when the bubbly steady state is a saddle point, the bubble is a forward-looking nonpredetermined variable and its initial value must be solved endogenously. 5.2 Technology Shocks We start by analyzing the impact of a positive technology shock that raises εat by 1% initially. Figure 2 plots the impulse responses of some key variables and shows that consumption, investment, and output all rise, whereas inflation and nominal interest rate both decline on impact. The positive comovement of consumption, investment, and output is easy to understand as in the real business cycles literature. The negative responses of inflation and nominal interest rate follow the usual intuition in the DNK literature (e.g., Woodford (2003) and Galı́ (2008)). To see this, we loglinearize the equilibrium system around the non-stochastic bubbly steady state and derive the following New Keynesian Phillips curve16 1 Π̂t = (1 − ξ)(1 − βξ)p̂wt + βEt Π̂t+1 , ξ 16 We use hatted variable to denote the log deviation from the deterministic steady state. Appendix B presents the complete log-linearized equilibrium system. 23 where the marginal cost p̂wt (the relative price of wholesale goods) satisfies p̂wt = αR̂kt + (1 − α) ŵt − Ât . Following a positive technology shock, the marginal cost goes down or the markup (the inverse of the marginal cost) goes up. Thus inflation declines. The central bank’s interest rate rule calls for a decline of the nominal interest rate by increasing the money supply. Hours worked also decline due to price rigidities under the Taylor rule and the weak inflation targeting rule. Firms that do not adjust prices to meet their output demand have to reduce hours when facing a positive technology shock (see, e.g., Galı́ (1999, 2008)). This negative effect can dominate depending on the parameter values in preferences, technology, and the interest rate rule. For the strong inflation targeting rule, labor actually rises because the large fall of interest rates stimulates aggregate demand. This monetary policy rule causes consumption, investment, and output to have the largest rise on impact. A special feature of our model is that the asset bubble responds on impact even though the asset has no fundamental payoffs. To see the intuition, we can derive the log-linearized equation for the asset bubble Z   p̂ht = Et p̂ht+1 + Ĉt − Et Ĉt+1 + ŝt − sβ εmax ε∗ ε dF (ε) Et ε̂∗t+1 . ε∗ (63) Three factors affect the asset bubble: the SDF Ĉt − Et Ĉt+1 , the sentiment shock ŝt , and the liquidity premium. Figure 2 shows that the two cutoffs ε̂∗t and ε̂∗∗ t fall on impact in response to a positive technology shock. Thus the liquidity premium rises. The SDF also rises because consumption growth declines. Both components cause the asset bubble to expand on impact. Note that the bubble pht or p̂ht is a forward-looking variable and its initial value must be endogenously determined, just like any asset prices. The fall of ε∗t is due to the rise of the capital price qtk as ε∗t = 1/qtk , which in turn is driven by the rise in the marginal revenue product of capital Rkt following a positive technology shock. The fall of ε∗∗ t is due to the fall of the lending rate Rlt . When the nominal deposit rate Rt falls, Rlt must fall by equation (18), which is derived from the zero-profit condition of banks. Thus real bank lending Lt+1 /Pt rises to enable more investment. Note that money supply (gt ) increases in the short run in order to support the lower nominal deposit rate and higher bank lending. 5.3 Monetary Policy Shocks Next we consider the impact of an expansionary monetary policy shock when vt or εvt drops by 25 basis points initially in equation (42) and (43). This corresponds to a one percentage point drop of the annual nominal rate on impact holding other variables fixed. Figure 3 displays the impulse responses. We find that the (quarterly) nominal interest rate Rt initially drops by less than 25 24 basis points because the interest rate endogenously responds to changes in output and inflation by the Taylor rule (42) and to expected inflation by the inflation targeting rule (43). The traditional transmission mechanism of monetary policy is through the interest rate channel. When prices are sticky, the real rate falls and thus consumption rises following an expansionary monetary policy shock. This can be seen from the log-linearized equation of (30):   Rβ Z εmax ε dF (ε) Et ε̂∗t+1 . Ĉt = Et Ĉt+1 − R̂t − Et Π̂t+1 + Π ε∗ ε∗ (64) This equation is analogous to the dynamic IS curve in the traditional DNK model, with an additional liquidity premium term. The fall of the real rate also causes the increase in investment and hence output. Thus monetary policy is not neutral in the DNK framework. In addition to the interest rate channel, our model features a lending channel and an asset price channel. An expansionary monetary policy causes money supply to increase in the short run and hence bank lending rises. The lending rate Rlt falls with the deposit rate Rt . This in turn raises investment. A unique feature of our model is that asset prices rise for the following two reasons. First, the SDF rises on impact because consumption rises initially before gradually declining. Second, both components of the liquidity premium rise on impact. The intuition is the following. Since consumption, investment, and output rise as discussed earlier, labor demand must also rise on impact as capital is predetermined so that the real wage rate rises. Thus, the marginal cost or the relative price of wholesale goods pwt must go up, causing inflation to rise. Moreover, it follows from equation (47) that the marginal revenue product of capital Rkt must rise, pushing up the capital price qtk . As a result, the liquidity premium rises and the investment cutoff ε∗t falls, and hence the lending cutoff ε∗∗ t falls with Rt by (18) and (28). The increased asset price raises entrepreneurs’ net worth, allowing them to finance more investment. Both the intensive and extensive margin effects work in the same direction so that the asset price channel provides a large amplification effect on investment. As shown in Figure 3, investment rises by about 1-2% on impact in response to a cut of 25 basis points in the nominal deposit rate. Our result that an expansionary monetary policy by cutting interest rates drives asset bubbles on impact is different from that in Galı́ (2014). Galı́ focuses on the stable steady state in his unidimensional equilibrium system and assumes that the asset bubble is predetermined and hence its initial size is independent of monetary policy. By contrast, the asset bubble in our paper is a forward-looking nonpredetermined variable so that both its initial size and growth rate are determined by the real interest rate. The initial jump of the asset bubble provides an important amplification mechanism for monetary policy. 25 5.4 Sentiment Shocks Suppose that agents suddenly become more optimistic in that there is a positive sentiment shock that raises εst by one standard deviation 0.613% initially. As Figure 4 shows, this shock immediately raises the asset bubble by the asset pricing equation, thereby raising entrepreneurs’ net worth. Thus bank lending and investment rise, causing output to follow suit. Labor must rise on impact because output rises and capital is predetermined. The increased bank lending causes money supply and inflation to go up. The initial rise of inflation is quantitatively small. The nominal deposit rate must rise due to the interest rate rules. Since prices are sticky, the real deposit rate rises and hence households have an incentive to save instead of consuming. For the Taylor rule in (42), the real interest rate rises so much so that consumption falls on impact. Thus there is no comovement between consumption and investment in response to a positive sentiment shock. This may seem counterintuitive at first because asset prices do not have a wealth effect on households, but it is in fact a special feature of our model in which households do not hold the bubble in equilibrium. Miao, Wang, and Xu (2015) show that introducing endogenous capital utilization can amplify the initial impact on output and hence allow consumption to rise. For the inflation targeting rule in (43), the nominal interest rate does not respond to output directly so that the rise of the real interest rate is smaller, causing consumption to rise on impact. This effect is stronger for the strong inflation targeting rule. Increased investment generates more capital accumulation and hence the capital price qtk falls on impact. Consequently the investment cutoff ε∗t = 1/qtk rises as does the lending cutoff ε∗∗ t . This implies that fewer firms make investment. But this negative extensive margin effect is dominated by the positive intensive margin effect so that aggregate investment rises. The net effect on investment is relatively small: a 3.7% increase in the asset bubble only raises aggregate investment by about 0.1% on impact. In the meantime, the liquidity premium falls on impact, but this negative effect on the asset bubble is dominated by the direct effect of the positive sentiment shock. Note that Figure 4 shows that the asset bubble contracts over time, but the real interest rate is still positive. This is because the initial sentiment shock dies out over time so that the path of the asset bubble follows that of the shock by (63). The initial jump of the asset bubble roughly reflects the cumulative effect of the initial sentiment shock, εs0 / (1 − ρs ) = 3.7%. In summary, the non-fundamental sentiment shock can drive large movements of the asset bubble due to its forward-looking nature. But it has a small positive impact on the real economy and generates mild inflation. The main reason is that the opposite intensive and extensive margin effects partially offset each other and mitigate the impact of a sentiment shock on aggregate demand and the real economy. 26 Monetary Policy Rules R̂t = 1.5Π̂t + 0.125Ŷt + φp p̂ht R̂t = 1.1Et Π̂t+1 + φp p̂ht R̂t = 2Et Π̂t+1 + φp p̂ht Sentiment weight φp -0.07 -0.0009 -0.04 Shock gain Ω (%) 1.82 0.03 0.10 TFP shock weight φp -0.07 -0.01 -0.04 gain Ω (%) 0.09 0.09 0.03 Table 1: Optimal weights on asset bubbles and welfare gains. (ph /Y = 0.2) 6 Should Monetary Policy Respond to Asset Bubbles? In the previous section we have shown how asset bubbles and other macroeconomic variables respond to various exogenous shocks when monetary policy does not respond to asset bubbles. In this section we address the question whether monetary policy should respond to asset bubbles. We consider the Taylor rule in (42) by fixing the parameters φπ = 1.5 and φy = 0.125. We also consider the weak inflation targeting rule and strong inflation targeting rule by fixing φπ = 1.1 and φπ = 2 in (43). We turn off the monetary policy shock by setting vt = 0. We search for the weight φp that maximizes the unconditional mean of household utility in (10) in response to each of the two shocks to sentiment and TFP.17 We set the persistence parameters for these shocks to ρs = 0.833 and ρa = 0.9, and set the standard deviations to σs = 0.613% and σa = 1%. Table 1 presents the optimal weights φp on asset bubbles and the welfare gains from optimally responding to asset bubbles. The welfare gains are computed as follows. Let {Ct } and {Nt } denote the equilibrium consump- tion and labor processes for φp = 0. Let V and V ∗ denote the unconditional means of household utility when monetary policy does not respond and when it does respond to asset bubbles, respectively. Then the welfare gains Ω in terms of the increase in consumption satisfies the equation E ∞ X t=0 β t [ln(1 + Ω)Ct − ψNt ] = V ∗ . Solving yields Ω = exp ((1 − β) (V ∗ − V )) − 1. Table 1 shows that the welfare gain is equal to a 1.82% increase in consumption for monetary policy to respond to asset bubbles under the interest rate rule in (42), when only the sentiment shock hits. The weight on asset bubbles is equal to −0.07, meaning that the central bank under the Taylor rule should cut the nominal interest rate by about 0. 28% per annum when the asset bubble expands by 1% from the steady state. The welfare gains are small for all other cases (around 0.1% increase in consumption). To see the intuition, we present the impulse responses to a positive TFP shock in Figure 5 for three cases under the interest rate rule in (42): (i) monetary policy optimally responds to asset 17 We use the second-order approximation method to compute the welfare gains implemented by Dynare (Adjemian et al (2011)). We emphasize that our exercise is to find the optimal simple rule instead of the optimal Ramsey policy. 27 bubbles; (ii) monetary policy does not respond to asset bubbles; and (iii) prices are flexible with φp = 0. In the flexible price equilibrium, consumption, investment, output, and labor all comove in response to a positive technology shock. But when prices are sticky and φp = 0, labor falls on impact. To correct this inefficiency, monetary policy should respond to the expanding asset bubble by setting φp = −0.07 to cut the nominal interest rate . In this way the asset price can increase more so that bank lending rises more, mitigating the fall of inflation. The Taylor-type interest rate rule also calls for cutting the nominal interest rate in response to a decline of inflation. Thus the real interest rate falls. In the meantime, the asset price and bank lending channels cause investment and output to rise more. This allows consumption to rise more and labor to also rise instead of falling. Overall, the impulse responses of consumption, investment, output, and labor under the optimal simple rule are closer to those in the flexible price equilibrium. Figure 6 presents impulse responses to a positive sentiment shock under the interest rate rule in (42). We find that consumption, investment, output, and labor all increase in response to a positive sentiment shock under the optimal simple rule (φp = −0.07). Without responding to the asset bubble, consumption falls because the real interest rate rises (also see Figure 4). Moreover, the impact of the sentiment shock on the real economy and inflation is very small for both the sticky price and flexible price equilibria. Under the optimal simple rule, the central bank should cut the nominal interest rate to reduce the real interest rate and raise consumption, thereby raising household utility. In the meantime, the asset bubble rises more on impact. As Figures 5 and 6 show, our model is different from Galı́’s (2014) in which the asset price does not move on impact. In our model the initial rise of the bubble improves allocation efficiency by raising aggregate investment and output. Figure 7 shows that, under the Taylor-type rule (42), inflation volatility increases with φp conditional on the TFP shock and is a U-shaped function of φp conditional on the sentiment shock. For both cases, bubble volatility decreases with φp . Thus the ‘leaning against the wind’ interest rate policy will reduce bubble volatility, but at the possible cost of raising inflation volatility. This is in sharp contrast to Galı́’s (2014) result that such a policy may raise bubble volatility conditional on the sentiment shock. The intuition is that a larger φp reduces the initial response of the bubble to either a TFP shock or a sentiment shock as the bubble is a forward-looking variable. This reduces aggregate demand more and causes a larger drop of inflation from the target in response to a positive TFP shock, so that inflation volatility is higher conditional on the TFP shock. In response to a positive sentiment shock, a larger negative φp lowers nominal interest rates more and raises aggregate demand more, thereby generating a larger rise of inflation relative to the target. But a larger positive φp raises nominal interest rates more and reduces aggregate demand more, thereby generating a larger drop of inflation relative to the target. In both cases inflation volatility is higher conditional on the sentiment shock. 28 Will the preceding results change if the central bank adopts the inflation targeting rule in (43)? Table 1 shows that the optimal weights on the asset bubble are generally small under the inflation targeting rule. Under the inflation targeting rule, the nominal interest rate does not respond to output changes and hence is more accommodative to economic growth. Thus consumption can rise in response to a positive sentiment shock due to the small change in the real interest rate and labor can rise in response to a positive TFP shock due to the fall of the real interest rate, as shown in Figures 2 and 4. This means that the distortion caused by price stickiness is less severe. Thus the welfare gains are small for the inflation targeting rule.18 Our results confirm the early findings of Bernanke and Gertler (1999). 7 Discussions In our model deposits are money (or M1) and households do not hold money in equilibrium because the return is too low around the steady state. Moreover, there is no cash in our model. We can adopt several approaches in monetary economics (e.g., Chari, Christiano, and Eichenbaum (1995), Einarsson and Marquis (2001), and Galı́ (2014)) to allow households to hold money including deposits and cash in equilibrium. For example, we could introduce cash-in-advance constraints and deposit-in-advance constraints. We could also simply introduce cash or deposits in the utility function because they provide some services other than “storage of wealth.” For simplicity we ignore cash and introduce deposits in the utility function as follows E0 ∞ X t=0    Sa,t+1 β ln Ct − ψNt + χ ln , Pt t where χ > 0. In this case aggregate investment satisfies   mt sat Rt−1 h It = Rkt Kt + − + pt (1 − F (ε∗t )) + µKt (1 − F (ε∗∗ t )) , Πt Πt (65) (66) where sat ≡ Sa,t /Pt−1 denotes household real deposits. Since banks have to pay interest on house- hold deposits, the funds available for aggregate investment is reduced by sat Rt−1 /Πt relative to equation (51). The conditions for the existence of bubbleless and bubbly steady states are too complicated to permit a complete characterization. For example, one has to subtract sa R/Π in the first parenthesis of equation (59). Thus the bubbleless steady-state cutoff becomes smaller. But we need additional equations to pin down this cutoff because sa is endogenous. For the bubbly steady state, we can still prove that there is a unique solution for the bubbly steady-state cutoff in equation (62) if and only if condition (61) holds as in Proposition 4. But this is not sufficient to pin down the bubbly steady state for the reason discussed above. We 18 Regarding the impact of φp on the inflation and bubble volatilities under the inflation targeting rule (43), we find a figure similar to Figure 7 and hence will not report it here. 29 need numerical methods to solve the model. We use numerical examples to illustrate the intuition by varying χ > 0 and fixing other parameter values as in Section 5.1. We find that the bubble size is larger when χ > 0 is larger. In this case, households prefer to save more in banks, banks can extend more loans, and repayments to households are also higher. Inefficient firms have more funds to buy bubble assets and efficient firms can sell bubble assets at a higher price to repay loans. The increased bubble size is used partly to repay household deposits so that real investment, consumption, and output barely change. We also find that impulse responses are qualitatively similar for χ > 0 and χ = 0. The key difference is that household saving sa,t+1 responds to shocks. In particular, sa,t+1 falls in response to a positive TFP shock or an expansionary monetary policy shock because the real interest rate falls. But sa,t+1 rises in response to a sentiment shock because the real interest rate rises. The movements of sa,t+1 can amplify the shocks. For another extension, it is straightforward to introduce long-run growth to our model as in Tirole (1985), Galı́ (2014), Miao and Wang (2018), and Miao, Wang, and Xu (2015). With a positive economic growth rate, the bubbly and bubbleless steady-state real interest rates can be positive. Since such an analysis is standard, we leave out the details. In our model asset bubbles improves investment efficiency, but generate excessive volatility. Trading off between the benefit and the cost, optimal monetary policy encourages asset bubbles, even if the welfare gains could be small. One can identify other sources of benefits and costs of asset bubbles. For example, Miao and Wang (2014) and Miao, Wang, and Zhou (2015) show that asset bubbles can generate resource misallocation. Ikeda and Phan (2016) show that risk shifting can lead to welfare-reducing bubbles. Introducing these costs to our model may generate a different type of optimal monetary policy. 8 Conclusion We have presented an infinite-horizon model of rational asset bubbles in the DNK framework. We have studied how monetary policy affects asset bubbles in the steady state and their dynamics in response to fundamental and non-fundamental shocks. We have also studied whether central banks should respond to asset bubbles. We find that the real balance effect together with credit constraints is important for our results. Our model can deepen our understanding of the relation between monetary policy and asset bubbles. Many of our results are different from those in Galı́’s (2014) OLG model. We believe that modeling the interaction between financial frictions and asset bubbles and the role of the bubbles are critical for understanding the differences. Moreover, whether asset bubbles are treated as a forward-looking or backward-looking variable is important for understanding their dynamics. Some of our results confirm the early findings of Bernanke and Gertler (1999, 2001) and provide a theoretical foundation for the conventional wisdom on the role 30 of monetary policy in managing asset bubbles. 31