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Bubbly Recessions WP 18-05R Siddhartha Biswas University of North Carolina at Chapel Hill Andrew Hanson University of North Carolina at Chapel Hill Toan Phan Federal Reserve Bank of Richmond Bubbly Recessions Siddhartha Biswas Andrew Hanson Toan Phan∗ January 2, 2019 Working Paper No. 18-05R Abstract We develop a tractable bubbles model with financial frictions and downward wage rigidity. Competitive speculation in risky bubbles can result in excessive investment booms that precede inefficient busts, where post-bubble aggregate economic activities collapse below the pre-bubble trend. Risky bubbles can reduce ex-ante social welfare, and leaning-against-the-bubble policies that balance the boom-bust trade-off can be warranted. We further show that the collapse of a bubble can push the economy into a “secular stagnation” equilibrium, where the zero lower bound and the nominal wage rigidity constraint bind, leading to a persistent recession, such as the Japanese “lost decades.” 1 Introduction In the recent decades, many countries in the world, including Japan, the U.S., and several European economies, have experienced episodes of rapid speculative booms and busts in asset prices, followed by declines in economic activities and in some cases persistent recessions. More generally, throughout history, the collapse of large asset and credit booms tend to ∗ Biswas and Hanson are affiliated with the University of North Carolina at Chapel Hill; Phan is affiliated with the Federal Reserve Bank of Richmond. Emails: sbiswas1@live.unc.edu, ashanson@live.unc.edu and toanvphan@gmail.com (corresponding author). We thank the editor Simon Gilchrist and two anonymous referees for helpful suggestions. We also thank Kosuke Aoki, Vladimir Asriyan, Gadi Barlevy, Julien Bengui, Ippei Fujiwara, Tomohiro Hirano, Daisuke Ikeda, Lisa Kenney, Riccardo Masolo, Jianjun Miao, Juan Pablo Nicolini, Stan Rabinovich, Ricardo Reis, Etsuro Shioji, Nico Trachter, Pengfei Wang, Alex Wolman, Russell Wong, and the attendants at the LAEF Conference on Bubbles at the University of California Santa Barbara, the Canon Institute for Global Studies Conference on Macroeconomic Theory and Policy 2017, the University of Virginia and Federal Reserve Bank of Richmond Jamboree 2017, the ASSA 2018 meeting, the SEA 2018 meeting, the Northwestern University Alumni Conference 2018, the Conference in Honor of Carlstrom and Fuerst at the University of Notre Dame, Osaka University, and the Institute for Social and Economic Research for helpful comments. The views expressed herein are those of the authors and not those of the Federal Reserve Bank of Richmond or the Federal Reserve System. DOI: https://doi.org/10.21144/wp18-05 1 precede recessions and crises (e.g., Kindleberger and O’Keefe 2001; Jordà et al. 2015). These experiences have led policymakers to be increasingly aware of the potential risks of asset price bubbles, leading to discussions of macroprudential regulations such as “leaning-against-thewind” policies – preventive measures to curb the booms in asset prices in order to mitigate the eventual busts. However, despite the recent developments in the macroeconomic literature on asset bubbles, relatively little theoretical framework has analyzed the potential efficiency trade-off between the booms and busts of risky bubbly episodes and whether preventive policies are warranted. In particular, in most rational bubble models – the workhorse models to study the macroeconomic effects of bubbles in general equilibrium – private agents correctly perceive the risk of speculating in a bubbly asset and bubbles generally improve the efficiency of the financial system (e.g., see the literature surveys in Barlevy 2007, 2012, 2018; Miao 2014; Martin and Ventura 2017). In this paper, we develop a tractable general equilibrium model to address the question of when and how risky rational bubbles can lead to inefficiencies and evaluate the welfare tradeoff. We focus on the combination of financial frictions and downward wage rigidities during bubbly episodes. We posit an economy where entrepreneurial agents with heterogeneous productivity accumulate capital and, due to limited commitment, face financial frictions that constrain their ability to borrow from each other (e.g., Kiyotaki et al. 1997; Carlstrom and Fuerst 1997; Buera and Shin 2013). If the credit and capital markets cannot satisfy the demand for savings, speculative bubbles may arise. A rational bubble is an asset that is traded above its fundamental value; an agent purchases the overvalued asset because he or she expects to be able to sell it later. We assume bubbles are stochastic in the sense that in each period the price of the bubbly asset can collapse to the fundamental value with an exogenous probability (e.g., Blanchard and Watson 1982; Weil 1987). The possibility of trading the bubbly asset facilitates the reallocation of resources across time, because the bubbly asset can act as a savings vehicle. Trading also facilitates reallocation across agents, because the bubbly asset increases entrepreneurs’ net worth and hence their ability to borrow. Thus, the boom in the price of a bubbly asset leads to a boom in entrepreneurial net worth, credit, investment, output, wages, and consumption. When the boom finally turns into a bust, the economy simply converges back to the pre-bubble economy. Therefore, with financial frictions alone, the model so far implies that speculative bubbles help to crowd in productive investment and improve the overall efficiency of the economy, as implied in most existing expansionary bubble frameworks (e.g., Hirano et al. 2015; Miao and Wang 2018). However, the implications change with downward wage rigidities. When an expansionary 2 bubble collapses, the net worth of entrepreneurial agents also falls, leading to contractions in credit and investment. Thus, the demand for labor from firms also contracts. In a flexible labor market, wages will fall to clear the labor market. However, we assume that (real) wages are downwardly rigid. Then there will be rationing in the labor market, resulting in involuntary unemployment. An increase in unemployment can in turn lead to an endogenous and protracted recession by eroding the intertemporal allocation of resources. This is because the drop in employment reduces the return to capital investment, which then lowers entrepreneurs’ net worth. This further leads to a contraction in capital investment, since entrepreneurs’ ability to borrow and invest depends critically on their net worth. Therefore, the future capital stock will decline, causing further downward pressure on labor demand and wages, thus reducing future capital accumulation. The vicious cycle repeats and only stops when the capital stock has fallen enough, often undershooting the bubbleless steady-state level. In short, our theory identifies the booms and busts of speculative bubbly episodes as an important source of shocks that can potentially trigger a deep and persistent recession, such as the lost decades in Japan or the Great Depression and Great Recession in the U.S.1 We further show that when the bubble is sufficiently risky and the labor market is sufficiently rigid, society’s welfare can be better off without bubbles. Our model thus provides a step toward bridging the views of policymakers and theoretical models of bubbles (Barlevy, 2018). In particular, our theory naturally implies that a “leaning-against-the-bubble” type of macroprudential policy intervention is warranted for excessively large bubbles. The source of inefficiencies is a form of “bubbly pecuniary externality,” as individual investors do not internalize the effect of their portfolio choices in driving a large bubbly boom, which will lead to a large bust. Finally, we extend the real model to an environment where nominal wages are downwardly rigid (Schmitt-Grohé and Uribe 2016; Schmitt-Grohé and Uribe 2017) and the central bank sets the nominal interest rate according to a Taylor rule that is subject to the zero lower bound (ZLB). We then show that the collapse of a large expansionary bubble triggers a sharp drop in the real interest rate, pushing the nominal interest rate against the ZLB. The intuition is as follows. By crowding in capital investment, the bubble leads to an investment boom. Thus, after the bubble collapses, the economy enters the post-bubble phase with a capital stock above the steady state, a situation that has been referred to as an “investment hangover” (Rognlie et al. 2014). The high capital stock implies a low marginal 1 Other sources of shocks that have been highlighted in the literature, including but not limited to deleveraging shocks (e.g., Eggertsson and Krugman, 2012; Korinek and Simsek, 2016; Buera and Nicolini 2017), shocks to inflation expectations (e.g., Schmitt-Grohé and Uribe, 2017), or idiosyncratic risk shocks (e.g., Christiano et al., 2014; Acharya and Dogra, 2017). 3 product of capital and a low real interest rate. The collapse of a sufficiently large bubble can thus push the real interest rate so low that the ZLB binds. We show that, under certain conditions, the post-bubble economy may fall into a liquidity trap steady state, or “secular stagnation,” where employment and investment are persistently and inefficiently low and inflation is below target. A vicious cycle can arise from the interaction between (i) a low interest rate environment, which constrains the monetary authority from raising inflation, exacerbating the nominal wage rigidity and unemployment problem and (ii) inefficient unemployment that lowers the marginal product of capital, which in turn lowers the interest rates. In the absence of other shocks, this cycle can keep the economy in a persistent slump. Related literature. To the best of our knowledge, our paper is the first to construct a stochastic bubble model where the collapse of bubbles can trigger a collapse of aggregate economic activities below their bubbleless trend and hence provide a reason for why the economy may be better off without stochastic bubbles. Our paper is related to several strands of the literature. First, we help formalize the popular notion among policymakers that the collapse of risky bubbles can trigger inefficient recessions. A large number of papers emphasize the positive aspect of bubbles in reducing dynamic inefficiencies (e.g., Samuelson 1958; Diamond 1965; Tirole 1985) or reducing intratemporal inefficiencies in the allocation of resources (e.g., Farhi and Tirole 2011; Miao and Wang 2012, 2018; Martin and Ventura 2012; Graczyk and Phan 2016; Ikeda and Phan 2018). Other papers emphasize potential ex-ante inefficiencies of speculative bubble investment in diverting resources away from productive investment (e.g., Saint-Paul 1992; Grossman and Yanagawa 1993; King and Ferguson 1993; Hirano et al. 2015), generating excessive allocations of resources in certain sectors (e.g., Cahuc and Challe 2012; Miao et al. 2014), generating excessive volatility (Caballero and Krishnamurthy 2006; Ikeda and Phan 2016), or generating excessive default (Kocherlakota 2009; Barlevy 2014; Bengui and Phan 2018). Our paper complements this literature and highlights the ex-post inefficiency of bubbles by showing that their collapse can cause persistent involuntary unemployment.2 By embedding New Keynesian elements of rigidities into a rational bubbles framework, our paper is related to our earlier work, Hanson and Phan (2017). There, we developed a simple overlapping generations model based on Tirole (1985). A limitation of the overlapping generations framework in Hanson and Phan (2017) is that a period represents twenty or thirty years, making the model less appropriate for policy analyses at the business cycles frequency. 2 For a complementary approach to modeling post-bubble unemployment using a search and matching model à-la Diamond-Mortensen-Pissarides, see Kocherlakota (2011) and Miao et al. (2016). Also see Domeij and Ellingsen (2018) and Illing et al. (2018). 4 Furthermore, the previous paper does not study the welfare effects of policies or the ZLB. Our paper is also related to the overlapping-generations model of Asriyan et al. (2016) and provides a complementary approach to explaining post-bubble liquidity traps.3 In their model, inefficiency arises in the liquidity trap as the holding of cash crowds out productive investment. In contrast, the inefficiency arises in our model because of the aforementioned bubbly pecuniary externality. Moreover, while they introduce a new form of nominal rigidity through the assumption that expectations about the future values of bubbly assets are set in nominal terms, we assume a nominal wage rigidity that is relatively standard in the recent New Keynesian literature. Second, our paper is related to a large literature that investigates possible sources of shocks that trigger long recessions and liquidity traps in environments with New Keynesian frictions. Many papers have emphasized demand shocks driven by household deleveraging or tightening borrowing constraints (Eggertsson and Krugman 2012; Christiano et al. 2015; Korinek and Simsek 2016; Schmitt-Grohé and Uribe 2016), long-run factors such as aging demographics or safe asset shortages (Summers 2013; Eggertsson et al. 2016; Caballero and Farhi 2017), or overinvestment of capital (Rognlie et al. 2014). By highlighting the role of rational asset bubbles, our analysis offers a complementary narrative to those in the literature. Furthermore, in our model, the collapse of bubbles reduces the net worth of borrowers and an endogenous tightening of borrowing constraints in equilibrium, thus giving a possible microfoundation for the deleverage shocks in, e.g., Eggertsson and Krugman (2012) and Korinek and Simsek (2016). Similarly, in our model, expansionary bubbles lead to an endogenous boom in capital investment, thus giving a microfoundation to the investment overhang in Rognlie et al. (2014). Finally, by providing a normative analysis with macroprudential policies on speculative bubble investment, our paper complements the literature on macroprudential policies in environments with financial frictions or aggregate demand externalities (e.g., Lorenzoni 2008; Olivier and Korinek 2010; He and Krishnamurthy 2011; Bianchi 2011; Eberly and Krishnamurthy 2014; Farhi and Werning 2016; Bianchi and Mendoza 2018). The plan for the paper is as follows. Section 2 describes the main real model. Section 3 studies the bubbleless equilibrium and steady states, while Section 4 analyzes the bubbly equilibrium and steady states. Section 5 provides a welfare analysis. Section 6 provides an extension with nominal rigidity and the zero lower bound. Section 7 concludes. Detailed derivations and proofs are in the appendix. 3 For a related and emerging body of literature that analyzes the effects of monetary policies on rational bubbles, see Gali (2014, 2016), Ikeda (2016), Dong et al. (2017), and Hirano et al. (2017). 5 2 Model Consider an economy with two types of goods: a perishable consumption good and a capital good. Time is infinite and discrete. Firms are competitive, and there exist two types of agents, called entrepreneurs and workers, each with constant unit population. Entrepreneurs and workers have the same preferences over a consumption process {ct }∞ t=0 , given by E0 ∞ X ! β t u(ct ) t=0 where the period utility function is u(c) = log(c), β ∈ (0, 1) is the subjective discount factor, and E0 (·) is the expected value conditional on information in period 0. 2.1 Entrepreneurs Entrepreneurs are the only producers of the capital good. They rent the capital produced to firms through a competitive capital rental market. Entrepreneurs face idiosyncratic productivity shocks: in each period, an entrepreneur receives a random productivity shock a, where a is independently and identically distributed (i.i.d.) according to a continuous distribution with a cumulative distribution function (CDF) denoted by F .4 For tractability, we assume that the distribution is Pareto over [1, ∞) with shape parameter σ > 1. We denote the set of entrepreneurs by J ≡ [0, 1]. After knowing the idiosyncratic productivity shock the beginning of each period, an entrepreneur j ∈ J produces the capital good according to the following technology: j kt+1 = ajt Itj , j where Itj is the investment in units of the consumption good in period t, kt+1 is the amount j of the capital good produced in the subsequent period, and at is the productivity of the entrepreneur. For tractability, we assume capital depreciates completely after being used in each period (we relax this assumption in Appendix A.1.6 and in the numerical analysis). Following the literature (e.g., Tirole 1985), we introduce a (pure) bubbly asset, which is a durable and perfectly divisible asset in fixed unit supply that does not generate any dividend but can be traded at positive equilibrium prices under some conditions. To model a stochastic bubble, we follow the literature (e.g., Weil 1987) and assume that in each period the bubble 4 As is well known among models with heterogeneous productivity shocks (e.g., Carlstrom and Fuerst 1997; Bernanke et al. 1999; Kocherlakota 2009; Liu and Wang 2014), the i.i.d. assumption helps keep the model tractable. A model with persistent idiosyncratic shocks can only be solved numerically. 6 persists with a probability ρ ∈ (0, 1) and permanently collapses5 with the complementary probability 1 − ρ, where a lower ρ means a riskier bubble. The two sources of uncertainty in the model are thus the idiosyncratic productivity shock and the aggregate bubble shock that leads to the collapse of the bubble. Let bjt denote a share of a bubbly asset held by entrepreneur j and pbt be the price per unit of the bubbly asset. Then the entrepreneur’s flow budget constraint is written as cjt + Itj + pbt bjt = Rtk ktj + djt − Rt−1,t djt−1 + pbt bjt−1 , (2.1) where Rt−1,t is the gross interest rate between t − 1 and t, dit is the amount of net borrowing in period t, and Rtk is the rental rate of capital in t. The left-hand side of this budget constraint consists of expenditure on consumption, capital investment, and the purchase of bubbly assets. The right-hand side is the available funds at date t, which consists of the return from capital investment in the previous period, new net borrowing minus the net debt repayment, and the return from selling bubbly assets. As is standard in the literature (e.g., Martin and Ventura 2012; Hirano and Yanagawa 2017; Miao and Wang 2018), we assume agents cannot invest a negative amount in the capital stock or the bubbly asset, i.e.,6 Itj , bjt ≥ 0, ∀t. (2.2) The entrepreneur’s net worth at the beginning of period t is: j ejt ≡ Rtk ajt−1 It−1 − Rt−1,t djt−1 + pbt bjt−1 . (2.3) Financial frictions: In a frictionless world, less productive entrepreneurs would like to lend and thus delegate investment to highly productive entrepreneurs. Agents can borrow and lend through one-period debt contracts. However, we assume there are frictions in the financial market so entrepreneurs face a leverage constraint: djt ≤ θtj ejt , (2.4) which states that each entrepreneur’s borrowing is limited by her net worth ejt . The limit θtj ≥ 0 places a constraint on the entrepreneur’s leverage ratio. In general, a larger θ can be interpreted as representing an environment with less financial friction. This formulation of credit market friction is sufficiently general to encompass several 5 That is, once collapsed, bubbles are not expected to remerge. As in Guerron-Quintana et al. (2018), the model can be extended to relax this assumption and allow for recurring bubbles. 6 As otherwise, the ability to short sell would let agents borrow and bypass leverage constraint (2.4). 7 types of credit constraints considered in the financial friction literature. For example, if one k θa ¯ j Rt+1 ¯ assumes θtj ≡ Rt+1 −R k θa ¯ j , where θ ∈ [0, 1] is a constant, then the constraint maps to a stant+1 ¯ k Kt+1 (a), which can arise when entrepreneurs dard collateral constraint: Rt,t+1 Dt (a) ≤ θR t+1 can only pledge to repay in the next period at most a fraction θ¯ of the value of their asset (e.g., Kiyotaki et al. 1997). Alternatively, by assuming θtj ≡ θ, where θ ≥ 0 is a constant, the constraint maps to an analytically convenient form of credit constraint that has been used extensively in the recent literature of general equilibrium with heterogeneous agents (e.g., Banerjee and Moll 2010; Buera and Shin 2013; Moll 2014; Ikeda and Phan forthcoming).7 For tractability, we impose this assumption throughout the paper. While not essential for our results, the assumption of this simple leverage constraint, along with the assumption of a Pareto distribution, allows us to get analytical solutions to the model. These assumptions are relaxed in the general analysis in Appendix A.1.6. The optimization problem of each entrepreneur j is as follows. In each period after knowing her productivity shock ajt , the entrepreneur chooses consumption cjt , capital investment Itj , net debt position djt (where a negative djt means lending), and net purchase of the bubbly P j s asset bjt −bjt−1 . Her objective is to maximize the lifetime expected utility Et β log c , t+s s≥0 subject to budget constraint (2.1), nonnegativity constraint (2.2), and leverage constraint (2.4). 2.2 Workers Workers do not have access to capital production technologies. For simplicity, we assume workers are hand to mouth, i.e., cw (2.5) t = wt lt , where wt is the wage rate and lt is the employment level per worker.8 7 For a microfoundation of this constraint, see, e.g., Korinek (2010). Alternatively, we can assume workers cannot borrow against future labor income. Thus the optiPtheir ∞ mization problem of workers is to maximize lifetime utility E0 ( t=0 β t ln cw t ) subject to: 8 b w w w b w cw t + pt bt = wt lt + dt − Rt dt−1 + pt bt−1 , w and dw t ≤ 0 and bt ≥ 0. In equilibrium, it can be shown that workers will be effectively hand to mouth, i.e., w ct = wt lt . Intuitively, due to financial friction, the interest rate (and the returns from bubble speculation) will be too low relative to the discount factor, and thereby it will be suboptimal for workers to save or to buy the bubbly asset (see Hirano et al. 2015 for more details). 8 2.3 Firms In each period, there is a continuum of competitive firms that produce the consumption good from hiring labor from workers and renting capital from entrepreneurs. They employ a standard production function: yti = (kti )α (lti )1−α , 0 < α < 1, where kti and lti are capital and labor inputs of a representative firm i. Competitive factor prices are given by the marginal products of capital and labor: Rtk =α Lt Kt 1−α (2.6) wt = (1 − α) Kt Lt α , (2.7) where Kt and Lt are the aggregate capital stock and employment. 2.4 Downward wage rigidity (DWR) We assume that real wages are downwardly rigid: wt ≥ γwt−1 , ∀t ≥ 1, (2.8) where γ ∈ [0, 1] is a constant parameter that governs the degree of rigidity. The condition states that the real wage cannot fall below a certain fraction of the real wage in last period.9 The presence of rigid wages implies that the labor market does not necessarily clear. In each period, even though each worker inelastically supplies one unit of labor, the realized employment Lt per worker in equilibrium is determined by two conditions: feasibility constraint Lt ≤ 1, (2.9) and complementary-slackness condition (1 − Lt ) (wt − γwt−1 ) = 0. (2.10) These equations state that involuntary unemployment (Lt < 1) must be accompanied by a binding wage rigidity (2.8). Conversely, when (2.8) is slack, the economy must be in full 9 For empirical evidence of real wage rigidity, see, e.g., Holden and Wulfsberg (2009) and Babeck` y et al. (2010). 9 employment (Lt = 1). For simplicity, we also assume that in the initial period t = 0 the legacy wage w−1 is sufficiently small so that the labor market clears in t = 0. 2.5 Equilibrium Definition. Given initial k0j = K0 , dj0 = 0, bj0 = 1, pb0 , a competitive equilibrium consists of j , cjt }j∈J , cw prices {wt , Rtk , Rt,t+1 , pbt }t≥0 and quantities {{Itj , kt+1 t , Kt+1 , Lt }t≥0 such that: • Entrepreneurs and firms optimize, • The consumption of a representative worker is given by (2.5), • The credit market clears: R1 • The bubble market clears: 0 djt dj = 0, R1 0 bjt dj = 1 if pbt > 0, • The consumption good market clears: • The capital market clears: Kt = R1 0 R1 0 α 1−α (cjt + Itj )dj + cw , t = Kt L t ktj dj, • Labor market conditions (2.8), (2.9), and (2.10) hold As usual, a steady state is an equilibrium where quantities, prices (in units of the consumption good), and inflation are time invariant. 3 Bubbleless benchmark Let us first analyze the bubbleless equilibrium, where the price of the bubbly asset is equal to its fundamental value of zero throughout, i.e., pbt = 0 for all t. Omitted details of the derivations are relegated to Appendix A.1.1. 3.1 3.1.1 Equilibrium dynamics Optimal decisions of individual entrepreneurs In each period t, given the realization of her productivity shock ajt , each entrepreneur j chooses cjt , Itj , and djt . Since the period utility function is logarithmic, the optimal action for the entrepreneur is to consume a fraction 1 − β of her net worth ejt : cjt = (1 − β)ejt , 10 (3.1) and invest/save the remaining fraction β: Itj + (−djt ) = βejt . (3.2) In the bubbleless benchmark, net worth (as defined in (2.3)) is simply capital income minus net debt repayment: j − Rt−1,t djt−1 . ejt = Rtk ajt−1 It−1 Both the savings options of investing in capital and lending in the credit market are riskless. Hence, the entrepreneur will simply choose the option that offers the highest rate of return. Lending yields a rate of return Rt,t+1 , which is the same for all entrepreneurs. Capital k investment yields a rate of return ajt Rt+1 , which varies according to each entrepreneur’s j productivity at . Hence, in equilibrium, there is a cutoff productivity threshold a ¯t in each j period such that: all entrepreneurs with at < a ¯t will only lend and not invest in capital (i.e., j the constraint It ≥ 0 binds), while those with ajt > a ¯t will only invest in capital and borrow as much as possible (i.e., leverage constraint (2.4) binds). Entrepreneurs with ajt = a ¯t (the “marginal investors”) will be indifferent between lending and investing in capital, and their djt and Itj are indeterminate. The indifference condition yields a mapping between the interest rate and the marginal product of capital: k Rt,t+1 = a ¯t Rt+1 . (3.3) In summary, entrepreneurs’ leverage ratio is given by: −β if ajt < a ¯t j dt = ∈ [−β, θ] if ajt = a ¯t . ejt θ if ajt > a ¯t (3.4) Then, entrepreneurs’ capital investment is given by: 0 if ajt < a ¯t j j j It = ∈ [0, (β + θ)et ] if at = a ¯t , (β + θ)ej if ajt > a ¯t t 11 (3.5) and the amount of capital produced by each entrepreneur in t + 1 is given by: j kt+1 0 if ajt < a ¯t = ∈ [0, ajt (β + θ)ejt ] if ajt = a ¯t . aj (β + θ)ej if ajt > a ¯t t t (3.6) Finally, note that since the distribution of the productivity shocks is continuous, the measure of marginal investors is zero and thus their individual asset positions will not affect aggregation. 3.1.2 Aggregation Given the decisions of individual entrepreneurs, we can characterize the aggregate variables of the economy. The aggregate net worth of entrepreneurs is equal to the aggregate capital income: Z 1 ejt dj = αKtα L1−α . t et = (3.7) 0 R1 The cutoff threshold a ¯t is determined by the credit market clearing condition 0 djt dj = 0. By incorporating equations (3.3), (3.4), (3.5), and the assumption of i.i.d. productivity shocks, this condition can be rewritten as (detailed derivations in Appendix A.1.1): a )) · et , F (¯ a ) · βe = θ · (1 − F (¯ {z t } | | t{z }t agg. credit supply (3.8) agg. credit demand where the left-hand side is the aggregate supply of credit (from less productive entrepreneurs, ajt < a ¯t ) and the right-hand side is the aggregate demand of credit (from more productive entrepreneurs, ajt > a ¯t ). By canceling the et term on both sides, we get a simple equation that determines a ¯t = a ¯n , which is the time-invariant solution to the following equation: βF (¯ an ) = θ · (1 − F (¯ an )) . (3.9) Given that F is the CDF of a Pareto distribution over [1, ∞) with shape parameter σ, this equation gives a closed-form solution for a ¯n : a ¯n = β+θ β 1/σ . (3.10) The cutoff threshold is the key endogenous variable that characterizes the bubbleless equilibrium. 12 Given the cutoff threshold, the evolution of the aggregate capital stock can be derived from (3.6), (3.7), and (3.9) as: Z Kt+1 = 1 j kt+1 dj Z adF (a) · αKtα Lt1−α . = (β + θ) 0 (3.11) a ¯n The interest rate is then given by: k α−1 Rt,t+1 = a ¯n Rt+1 =a ¯n αKt+1 . Finally, the aggregate employment and equilibrium wage are determined by labor market conditions (2.8), (2.9), and (2.10). 3.2 Bubbleless steady state Given the equilibrium dynamics in Section 3.1 above, the steady state with no bubbles can be derived as follows. Because of the assumption that the rigidity parameter is a constant γ ≤ 1, the downward wage rigidity condition (2.8) does not bind in steady state, leading to full employment: Ln = 1. (3.12) Then, from (3.11) and (3.12), the aggregate capital stock can be expressed a function of a ¯n : 1 Kn = (An α) 1−α . (3.13) where for convenience we define: Z An ≡ (β + θ) adF (a). (3.14) a ¯n From (3.3) and (3.13), the interest rate can also be expressed as a function of a ¯n : Rn = a ¯n , An (3.15) In summary, equations (3.10), (3.12), (3.13), and (3.15) uniquely determine the bubbleless steady state. 13 4 Bubbly equilibrium We now analyze a stochastic bubbly equilibrium, where the bubble price conditional on persistence pbt is positive for all t. We focus on the relevant parameter range in which the DWR is slack as long as the bubble persists. Detailed derivations are relegated to Appendix A.1.2. 4.1 4.1.1 Equilibrium dynamics while the bubble persists Optimal decisions of individual entrepreneurs Suppose the bubble persists in t, i.e., pbt > 0. Then, while the optimal consumption of each entrepreneur is still a fraction 1 − β of net worth as in equation (3.1), her portfolio optimization will include a new decision of speculating on the bubbly asset: Itj + (−djt ) + pbt bjt = βejt . On the one hand, the savings options of investing in capital and lending yield riskless k and Rt,t+1 , respectively. As in the bubbleless benchmark, the bubbly returns of ajt Rt+1 equilibrium will feature a cutoff productivity threshold a ¯t in each period such that: all entrepreneurs with productivity shocks below this threshold will not invest in capital (i.e., the constraint Itj ≥ 0 binds), and all those with productivity shocks above it will only invest in capital, sell all of their assets, and borrow as much as possible (i.e., the leverage constraint binds). Thus, the entrepreneurial capital investment decision and the amount of capital produced are given by equations (3.5) and (3.6), respectively, as in the bubbleless benchmark. On the other hand, the speculative investment in the bubbly asset yields a risky return that is zero with probability 1 − ρ. In the bubbly equilibrium, the less productive entrepreneurs must be willing to both lend and purchase the bubbly assets, and so must be indifferent between the two options. Thus, the net debt position of entrepreneurs is given by: −βejt + pbt bjt if ajt < a ¯t djt = ∈ [−βejt + pbt bjt , θejt ] if ajt = a (4.1) ¯t , θej if aj > a ¯ t t 14 t the bubbly investment is given by: βej + djt if ajt < a ¯t t j pbt bt = ∈ [0, βejt + djt ] if ajt = a ¯t , 0 if ajt > a ¯t (4.2) and the indifference condition, which determines the growth rate of the price of the bubbly asset, is given by: b pt+1 0 j Et u (ct+1 ) − Rt,t+1 = 0, if ajt < a ¯t . (4.3) pbt In addition, the marginal investors are indifferent between lending and investing in capital: k Et u0 (cjt+1 ) ajt Rt+1 − Rt,t+1 = 0, if ajt = a ¯t . (4.4) The net worth of each entrepreneur now also contains the value of the bubbly assets purchased from last period: j . ejt = Rtk ajt−1 It−1 − Rt−1,t djt−1 + pbt bjt−1 | {z } bubbly component Intuitively, the bubbly asset provides an additional investment vehicle for entrepreneurs. When they are less productive, they can invest in the bubbly asset. Then when they become more productive, they sell the asset in order to make more capital investment. Note that at the individual level, because of their indifference, entrepreneurs may not see an advantage of having a bubble. However, at the aggregate level, the buying and selling of the bubbly asset allows for more resources to be transferred from less productive to more productive entrepreneurs in general equilibrium. In particular, more productive entrepreneurs can sell their holding of the bubbly asset to less productive ones. Thus, like money, the bubbly asset can be used as a storage of value that allows productive entrepreneurs to raise more resources for capital production. We now proceed to analyze the aggregate conditions in general equilibrium. Remark 1. The optimal decisions of each entrepreneur necessarily satisfy the transversality condition limt→∞ E0 β t u0 (cjt )pbt bjt = 0. In a representative agent model, this condition can be used to rule out the possibility of bubbles (e.g., Kamihigashi 2001). Intuitively, the condition imposes that the present discounted value of the individual investment in the bubbly asset pbt bjt must be zero. In a representative model, because bjt = bt = 1, this condition implies that the present discounted value of the total value of the bubbly asset pbt must be zero. However, with heterogeneous entrepreneurs and occasionally binding leverage constraints, 15 the individual bubbly investment is not the same as the total value of the bubbly asset (pbt bjt 6= pbt ), because entrepreneurs have heterogeneous bubbly investment positions (recall equation (4.2)). Therefore, in a heterogeneous agent model with incomplete markets like ours (or Kocherlakota 2009 and Hirano and Yanagawa 2017), the individual transversality condition does not rule out the possibility of bubbles in equilibrium (see Kocherlakota 1992 for a more general exposition of this point). 4.1.2 Aggregation Aggregate variables of the bubbly economy evolve as follows. The aggregate net worth of entrepreneurs now consists of not only capital income, but also the value of the bubbly asset: Z et = 1 ejt dj = αKtα L1−α + pbt . t (4.5) 0 The right-hand side of this equation highlights the bubble’s crowd-in effect: the bubble resale value pbt helps increase the net worth of entrepreneurs in equilibrium. R1 The cutoff threshold a ¯t is determined by the credit market clearing condition 0 djt dj = 0, or equivalently (see detailed derivations in Appendix A.1.2): a )) · et . F (¯ a ) · βe − pbt = θ · (1 − F (¯ } | {z t } | t {z t agg. credit supply agg. credit demand The left-hand side of this equation highlights the bubble’s crowd-out effect: the aggregate speculative investment in the bubbly asset (pbt ) crowds out the flow from aggregate savings (G(¯ at )βet ) into the supply side of the credit market. By canceling et on both sides and defining the bubble (over savings) ratio as φt ≡ pbt , βet the equation above can be rewritten as: β (F (¯ at ) − φt ) = θ · (1 − F (¯ at )) . (4.6) Note that φt necessarily lies in (0, 1). From (4.5) and (4.6), the aggregate capital stock evolves according to: Z Kt+1 = 1 j kt+1 dj Z = (β + θ) · 0 a ¯t 16 adF (a) · αKtα L1−α + pbt . t (4.7) Furthermore, indifference conditions (4.3) and (4.4) determine the interest rate and the growth of the bubbly asset, which are derived in Appendix A.1.2 as: and α−1 k =a ¯t αKt+1 . Rt,t+1 = a ¯t Rt+1 (4.8) at ) − φt φt+1 (1 − βφt+1 ) a ¯t F (¯ R = . φt at ) − φt (β + θ) a¯t adF (a) ρF (¯ (4.9) Finally, the aggregate employment and equilibrium wage are determined by labor market conditions (2.8), (2.9), and (2.10). 4.2 Stochastic bubbly steady state Given the equilibrium dynamics in the previous section, the stochastic bubbly steady state is characterized as follows. The steady-state version of credit-clearing condition (4.6) yields an equation that defines the bubble ratio φ as a function of a ¯b : β (F (¯ ab ) − φ) = θ · (1 − F (¯ ab )) . (4.10) The only difference between equation (4.10) and its counterpart (3.9) in the bubbleless benchmark is the presence of φ on the left-hand side, representing the fact that in the bubbly economy, relatively less productive entrepreneurs have the bubbly asset as an additional investment vehicle besides lending in the credit market. Since φ > 0 (the bubble ratio is necessary positive in any bubbly equilibrium), these two equations also imply that a ¯b > a ¯n . Intuitively, as the bubbly asset provides a new investment opportunity, some entrepreneurs find it optimal to switch from investing in capital to investing in the bubbly asset, causing the productivity threshold to rise from a ¯n to a ¯b .10 Given that F is the CDF of a Pareto distribution over [1, ∞) with shape parameter σ, from (4.10) we can get a closed-form expression for a ¯b : 1/σ 1/σ β+θ β+θ >a ¯n = . (4.11) a ¯b = β(1 − φ) β As in the bubbleless steady state, given the assumption that the rigidity parameter is a constant γ ≤ 1, the downward wage rigidity condition (2.8) does not bind in steady state, leading to: Lb = 1. (4.12) 10 As a consequence, the average entrepreneurial productivity is higher during a bubbly episode: it rises when the bubble arises and falls when the bubble collapses. Miao and Wang (2012) argue that this is consistent with empirical observations. 17 Then, from (4.7) and (4.12), the aggregate capital stock can be expressed as a function of a ¯b and φ: 1 1−α Z (β + θ)α Kb = adF (a) . (4.13) 1 − βφ a¯b From (4.4) and (4.13), the interest rate can also be expressed as a function of a ¯b and φ: Rb = (1 − βφ)¯ ab R . (β + θ) a¯b adF (a) (4.14) Finally, from indifference condition (4.9) and from (4.14), the steady-state bubble ratio also has a closed-form solution: ρ − Rb F (¯ ab ) 1 − Rb θ 1 − (1 − βρ)σ = . β βσ(1 − ρ) + θ φ= (4.15) Equations (4.11) to (4.15) characterize the endogenous variables in the stochastic bubbly steady state. Given (4.15), the condition for the existence of a bubbly steady state can be characterized as follows: Lemma 2. A bubbly steady state exists if and only if: σ−1 < ρ, βσ (4.16) Proof. Appendix A.2.1. The condition implies that for a stochastic bubble to exist, the probability that the bubble persists ρ has to be sufficiently high (as otherwise agents in the economy would deem the bubble to be too risky as an investment vehicle). Another direct corollary of (4.15) and (4.16) is that φ is strictly increasing in θ, implying that a more relaxed leverage constraint is associated with a larger bubble size in equilibrium. For the rest of the paper, we will impose the bubble existence condition (4.16). Furthermore, as in the recent literature, we will focus on the relevant range of parameters in which the bubble is expansionary (the crowd-in effect dominates the crowd-out effect in steady state), that is, Kb > Kn , (4.17) where the stochastic bubbly steady-state capital stock Kb is given by (4.13) and the bubble- 18 less steady-state capital Kn is given by (3.13).11 4.3 Post-bubble dynamics We now study the effect of the collapse of the bubble on the economy, which is the main focus of the paper. Suppose the bubble collapses at a certain period T , i.e., pbT +s = 0, ∀s ≥ 0. As the expansionary effect of the bubble ends, the post-bubble capital stock and wage will decline toward the bubbleless steady state levels. However, if the downward wage rigidity constraint binds, then the wage cannot flexibly fall to clear the labor market. Instead, employment is determined by the demand of firms. The rigidly high wage thus leads to involuntary unemployment. The contraction in employment has two effects on the intertemporal equilibrium dynamics: it reduces the return from capital, and it reduces entrepreneurs’ net worth. Both of these effects in turn reduce entrepreneurs’ accumulation of capital. The wage rigidity thus amplifies and propagates the shock of bursting bubbles. Let s∗ ≡ min{s ∈ N|LT +s = 1}, where N ≡ {0, 1, 2, . . . }. In other words, T + s∗ is the first post-bubble period when full employment is recovered. If s∗ > 0, then we say the economy is in a slump between T + 1 and T + s∗ − 1, as during this period there is involuntary unemployment: Lt < 1 for all T + 1 ≤ t ≤ T + s∗ . Given the tractability of the model, we can analytically characterize the post-bubble dynamics, including the depth and duration of the post-bubble unemployment episode. Intuitively, the economy escapes the slump when the equilibrium wage has fallen enough that the downward wage rigidity no longer binds, and the economy regains full employment and recovers toward the initial steady state. Proposition 3. [Post-bubble slump] Suppose the bubble collapses in period T . Then the economy enters a slump for s∗ periods. The post-bubble equilibrium dynamics are given by: s+1 α−1 s(s+1) KT +s+1 = An αKTα−1 γ α 2 KT 1 1−α α LT +s = Kt+s wT +s (4.18) ∀0 ≤ s < s∗ , wT +s = γ s wT , 11 Written in exogenous parameters, this assumption is equivalent to (1 − φ) by (4.15). 19 σ−1 σ > 1 − βφ, where φ is given and the duration of the slump is given by: KT 1+α s = max 0, 2α log 1 − γ Kn 1−α ∗ (4.19) where the ceiling function dxe denotes the least integer greater than or equal to x. The economy regains full employment and follows the dynamics of Section 3 for t ≥ T + s∗ . Proof. Appendix A.2.2. 4.3.1 Numerical illustration In this section, we conduct a simple calibrated numerical exercise to illustrate the equilibrium dynamics. Since the model is intentionally designed to be stylized and parsimonious, this exercise should not be viewed as a full-fledged quantitative analysis but rather a suggestive quantitative illustration of the model’s predictions. In this section, we also make two basic extensions to improve the mapping of the model to data: first, we assume the economy grows at an exogenous rate g ≥ 0; second, we assume capital partially depreciates at rate δ ∈ [0, 1] (see Appendix A.1.6 for details). We then calibrate the model to Japanese data as follows. We will choose parameters to match the pre-bubble phase (1970-1986) and the boom phase (the bubble period of 19871991) and let the model predict the bust phase (post-1991). There are two sets of parameters, the first of which can be set using relatively standard values from the literature. Specifically, we set a period to be a year, the capital share to be α = 0.33, the discount factor to be β = 0.96, the capital depreciation rate to be δ = 0.09, and the exogenous growth rate to be g = 0.03. Following Schmitt-Grohé and Uribe (2016), the downward wage rigidity parameter is set to be close to one: γ = 0.97. The second set of parameters, consisting of shape parameter σ of the productivity distribution, the financial friction parameter θ, and the bubble persistence parameter ρ, are less standard and will be calibrated. In particular, b n , and KbY+p to three moments: the average real interest we choose σ, θ, and ρ to match Rn , K Yn b rate of 1.02 in Japan in the pre-bubble phase, the wealth over income ratios in the pre-bubble phase and in the bubble phase of 3.67 and 5.18, respectively.12 The calibrated parameter values are σ = 15.14, θ = 0.05, and ρ = 0.998. Figure 1 illustrates a simulated equilibrium path for detrended aggregate variables under this parametrization. On this path, we set the economy at the stochastic bubbly steady state in the initial period, and then the bubble collapses in t = 10 (in the simulation, agents 12 Data for the wealth over income ratios come from Piketty and Zucman (2014); data for GDP and real interest rate come from the World Bank; the dating of the bubble period is according to Shioji (2013). 20 rationally expect that the bubble is stochastic and can burst in any period). Equilibrium variables are plotted with the solid lines, and for comparison, the bubbleless steady state counterparts are plotted with dashed lines. As seen in the figure, as long as the bubble lasts, the economy experiences a boom in entrepreneurial net worth (relative to the bubbleless steady state), which leads to a boom in aggregate credit to entrepreneurs, consequently an increase in aggregated capital accumulation, output, wage, and consumption (both across entrepreneurs and workers).13 Since the boom in the capital stock and bubble value is larger than that in output, the wealth over output ratio also increases during the bubbly episode. Then, after the bubble collapses, the economy begins a contraction. Without nominal rigidities, the labor market would be flexible and the equilibrium wage, along with other aggregate variables in the post-bubble economy, would simply converge back to the bubbleless steady state with full employment. However, with downward wage rigidity, the post-bubble equilibrium wage may not flexibly fall to clear the labor market, leading to involuntary unemployment. The drop in employment not only reduces the economy’s output, but also has important intertemporal effects. On the one hand, it reduces the net worth of entrepreneurs. On the other hand, it reduces the return rate on capital. Both of these effects depress capital accumulation, explaining the contractions of aggregate economic activities during the slump with involuntary unemployment. As a consequence, aggregate output, net worth, capital, credit, and consumption can undershoot (i.e., drop below) the pre-bubble trend. The figure highlights the boom-bust trade-off: the bubble leads to a boom of about 3.4% in output (relative to the bubbleless steady state) as long as it persists, but its collapse leads to a recession, where the aggregate output drops as much as 2.1% below the bubbleless steady state. The economy experiences long “lost decades”: about 20 years of declining output, which only recovers to its bubbleless trend after about 40 years.14 5 Welfare and policy analysis We will now investigate the welfare effects of stochastic bubbles. We will focus on the expected welfare in steady state, which can be computed in closed forms. The welfare functions are defined as the lifetime expected utility in the corresponding steady state. 13 Note that the boom in consumption is more pronounced for entrepreneurs, implying that entrepreneurs tend to gain more from the bubble than workers (as the increase in net worth allows entrepreneurs to increase their investment). This asymmetry could lead to interesting political economy implications, which are absent from this model and are left for future research. 14 We reiterate that these numbers should be taken with caution, as the model is highly stylized and does not take into account other important phenomena for Japan, such as demographics. 21 0.26 10.22 Bubble Size t Bubble Equilibrium Bubbleless SS Net Worth 8.51 0 6.8 0.38 Rt-1,t Credit 0.357 1 0.338 1.98 Output Capital 7.29 1.89 6.81 1.33 1.84 1 Wage Labor Equilibrium Unconstrained 1.29 1.25 1.47 0.96 Entrepreneurs Workers C/Cn 5.19 (K + P)/Y 4.43 1 0 20 40 60 0 20 40 3.67 60 Figure 1: Equilibrium dynamics with bubble boom-bust. Solid lines represent detrended equilibrium variable values; gray dashed horizontal lines represent the corresponding bubbleless steady state values. 5.1 Workers We start with the welfare of workers, which is more straightforward. The lifetime expected 1 utility of a representative worker in the bubbleless steady state is simply 1−β log(cw n ), where α cw n = wn = (1 − α)Kn . Thus, the welfare of workers in the bubbleless steady state, denoted by Wn , is given by: log [(1 − α)Knα ] Wn = , (5.1) 1−β where recall that Kn is given by (3.13). The welfare of workers in the stochastic bubbly steady state features a boom-bust tradeoff. As long as the bubble persists, their consumption is larger than that in the bubbleless α w steady state: cw b = (1 − α)Kb > cn . However, after the bubble collapses, the economy enters a slump for s∗ periods, during which workers suffer from involuntary unemployment. The lifetime expected utility of a representative worker will be a weighted sum of the utility before and after the bubble collapses, with the weights depending on the bubble’s risk of bursting 1 − ρ. In Appendix A.1.3, we show that the welfare of workers in the stochastic 22 bubbly steady state, denoted by Wb , is given by: Wb = log [(1 − α)Kbα ] 1 − ρβ | {z } + expected utility when bubble persists β(1 − ρ) [Γ0 (s∗ ) + Γ1 (s∗ ) log Kb ], 1 − ρβ | {z } (5.2) expected utility after bubble collapses where the expressions for Γ0 and Γ1 are provided in Appendix A.1.3. It is clear that Wb depends on the bubble’s risk of bursting 1 − ρ and on the duration of the post-bubble slump s∗ , which is itself a function of the degree of wage rigidity γ (recall (4.19)). It is straightforward to see that the slump length s∗ is increasing in the degree of rigidity γ, and consequently, Wb is decreasing in γ. Similarly, Wb is increasing in the persistence probability ρ, i.e., a safer bubble yields a higher payoff. 5.2 Entrepreneurs The welfare functions of entrepreneurs are more complex, due to their heterogeneity and portfolio optimization. In the bubbleless steady state, the lifetime expected utility of an entrepreneur j that starts the period with a net worth ej , denoted by Vn (ej ) satisfies the following equation: j j Vn (e ) = log (1 − β)e +β {z } | current period utility j F (¯ a )Vn (Rn βe ) +β } | n {z continuation value if aj ≤¯ an Z Vn aRnk (β + θ)ej dF (a) . {z } | a¯n (5.3) continuation value if aj >¯ an Appendix A.1.3 provides an analytical solution to this equation. To streamline the analysis, let us assume that each entrepreneur starts the bubbleless steady state with an equal net worth, leading to ej = αKnα . Then the bubbleless steady state entrepreneurial welfare is simply given by: Vn ≡ Vn (αKnα ). Similarly, in the bubbly steady state, lifetime expected utility of an entrepreneur j that 23 starts the period with a net worth ej , denoted by Vb (ej ) satisfies the following equation: Vb (ej ) = log (1 − β)ej {z } | current period utility Z F (¯ ab ) − φ k j k j +ρβ F (¯ ab )Vb ρ Vb aRb (β + θ)e dF (a) a ¯b Rb βe + ρF (¯ ab ) − φ a ¯b | {z } if bubble persists Z F (¯ ab ) − φ j k j k Vburst aRb (β + θ)e dF (a) , a ¯b Rb βe + +(1 − ρ)β F (¯ ab )Vburst F (¯ ab ) a ¯b | {z } if bubble bursts (5.4) where Vburst (·) denotes the continuation value after the bubble bursts. Appendix A.1.3 provide analytical solutions to Vb (·) and Vburst (·). As in the bubbleless case, we assume for simplicity that each entrepreneur starts the αKbα . Then the bubbly steady bubbly steady state with an equal net worth, leading to ej = 1−βφ state entrepreneurial welfare is given by: Vb ≡ Vb αKb1−α 1 − βφ . From the analyses in Sections 5.1 and 5.2, we can show the following proposition, which states that if the bubble is sufficiently risky and if there is sufficient wage rigidity, then agents in the economy are better off if there were no stochastically bursting bubbles. Proposition 4. [Welfare-reducing stochastic bubble] There exists γ¯ ∈ (0, 1) such that if there 2 ),15 is sufficient wage rigidity (γ > γ¯ ) and the bubble is sufficiently risky (ρ < ρ¯ ≡ 1 − α(1−β) β(β−α) then the bubble reduces steady-state welfare for both workers and entrepreneurs: Wb < Wn , Vb < Vn . Proof. Appendix A.2.3. 5.3 Leaning-against-the-bubble policy We have established that the boom and bust of a bubbly episode can push the economy into a recession with involuntary unemployment. The fundamental source of inefficiencies in this 15 As a simple numerical illustration, assuming α = 0.33, β = 0.96, then the proposition implies that if γ = 1 (wages cannot decline) and ρ < ρ¯ = 0.999, then agents in the economy will be better off without bubbles. 24 environment is a form of “bubbly pecuniary externality”: individual entrepreneurs do not internalize the effect of their investment portfolio choices in driving a large bubbly boom, which will lead to a large bust due to the downward wage rigidity. Under this context, policy responses are warranted. We will focus on a macroprudential policy of taxing bubble speculation, so that private agents internalize the pecuniary externality of the speculative bubble’s boom and bust. As we will show, this policy has an effect of reducing the bubble size and is thus akin to the kind of “leaning-against-the-wind” policies that have been extensively discussed in the policy circle (e.g., Barlevy 2012, 2018) and is similar to the type of tax policies often considered in the macroprudential literature (e.g., Lorenzoni 2008; Gertler et al. 2012; Jeanne and Korinek 2013).16 Formally, consider a benevolent constrained policymaker who cares about the welfare of both workers and entrepreneurs. The policymaker is constrained in the sense that she cannot undo the frictions in the credit market (e.g., via redistribution) or frictions in the labor market. However, she can levy a macroprudential tax τ on the return from bubble speculation. As our focus is on steady-state welfare, we will assume for simplicity that the tax rate is constant. Then, under the policy, the budget constraint (2.1) becomes: cjt + Itj + pbt bjt = Rtk ktj + djt − Rt−1,t djt−1 + (1 − τ )pbt bjt−1 + Ttj . The after-tax return on bubble speculation for the entrepreneur is then (1 − τ )pbt+1 /pbt . The policymaker rebates the tax revenue back to the entrepreneur through a lump-sum transfer: Ttj = τ pbt bjt−1 , which the entrepreneur takes as given. Consistent with the aforementioned notion of constrained policymaking, this specification of tax and transfer implies that the policymaker cannot redistribute resources across entrepreneurs. Appendix A.1.4 derives the bubbly equilibrium dynamics and steady state with the tax. A key result is that the steady-state bubble size is a decreasing function of the macroprudential tax τ : φ(τ ) = θ(1 − σ(1 − βρ(1 − τ ))) ≤ φ. β(β(1 − ρ)(1 − τ )σ + θ(1 − στ )) (5.5) When τ = 0, the bubble size collapses to φ(0) = φ, which is the laissez-faire size as derived 16 As in most of the literature, we implicitly assume that policymakers can observe the bubble. Of course, this is a strong assumption. Alternatively, one can interpret the macroprudential policy as imposing a tax on speculative investments in broad classes of assets that are ex ante perceived to be likely to experience bubbles, such as real estate or stocks of certain types of companies. 25 0.26 10.22 Bubble Size t Net Worth Benchmark Tax Implemented Bubbleless SS 8.47 0 0.38 6.73 Rt-1,t Credit 0.36 1 0.34 1.98 Output Capital 7.29 1.89 6.81 1.33 1.84 1 Labor Wage 1.29 0.98 1.25 1.47 0.96 1.04 Worker C/Cn Entre C/C n 1 1 0 5.19 20 40 60 (K + P)/Y 3.67 0 20 40 60 Figure 2: Equilibrium dynamics with bubble boom-bust. Solid and dashed lines represent detrended equilibrium variable values with and without tax, respectively; gray dashed horizontal lines represent the corresponding bubbleless steady state values. in Section 4. Hence, the tax not only makes the bubble smaller, but also makes the bubble harder to arise. Specifically, under the tax, a bubbly steady state exists (φ(τ ) > 0) if and only if σ−1 < ρ(1 − τ ), βσ which is more stringent than the laissez-faire existence condition (4.16). Thus, by setting τ ≥ τ¯ ≡ 1 − σ−1 , the policymaker can effectively rule out the possibility of a bubbly βσρ 17 equilibrium. Without loss of generality, we can thus focus on τ ≤ τ¯. Figure 2 illustrates an equilibrium path with and without the tax. The dashed lines represent the laissez-faire equilibrium path (exactly as plotted in Figure 1), while the solid lines represent the economy under a macroprudential tax of τ = 1%. As shown in the figure, the tax effectively reduces the bubble size. There is a boom-bust trade-off: the policy mitigates the effects of a collapsing bubble (the slump is shorter and less severe), but it also reduces the boom in aggregate economic activities while the bubble lasts. To evaluate the welfare effects of the tax, Appendix A.1.4 also derives the bubbly steadystate welfare expressions for both workers and entrepreneurs under the tax, denoted by 17 For the parameter values used in Section 4.3.1, the associated value for τ¯ is 2.54%. 26 Wb (τ ) and Vb (τ ), respectively. Assume the policymaker assigns a Pareto weight λ ∈ [0, 1] on the welfare of workers (and 1 − λ on the welfare of entrepreneurs). A constrained optimal policy is a macroprudential tax τ that maximizes the Pareto-weighted bubbly steady-state welfare:18 max λWb (τ ) + (1 − λ)Vb (τ ). τ ≤¯ τ Due to the highly nonlinear behaviors of Wb and Vb , in general the optimal tax can only be solved for with numerical methods. However, an interesting implication of our previous analysis is that when the conditions of Proposition 4 are met, an optimal policy is to rule out the possibility of the bubble altogether by setting τ = τ¯. Formally: Corollary 5. There exists γ¯ ∈ (0, 1) such that if there is sufficient wage rigidity (γ ≥ 2 γ¯ ) and the bubble is sufficiently risky (ρ < ρ¯ ≡ 1 − α(1−β) ), then a constrained-optimal β(β−α) macroprudential tax is to set τ = τ¯, which effectively rules out the possibility of a bubble. Proof. Appendix A.2.4. 6 Zero lower bound We now extend the real model by introducing downward nominal wage rigidity (DWNR) and a nominal interest rule that is constrained by the zero lower bound (ZLB). We will show that the collapse of a large bubble can push the economy into a “secular stagnation” equilibrium, where the ZLB on the nominal interest rate constrains the monetary authority from achieving the inflation target and the DNWR binds, leading to involuntary unemployment. Interestingly, under certain conditions, because of the interaction between the ZLB and the DNWR, the post-bubble economy may never exit from the liquidity trap and instead may converge to a bad bubbleless steady state with persistent unemployment. DNWR: Formally, let Pt denote the price level of the consumption good in period t in unit of a currency, and let wt continue to denote the real wage. Instead of the real wage rigidity condition (2.8), we impose the following assumption on nominal wages (à-la Schmitt-Grohé and Uribe, 2017): Pt wt ≥ γ(Lt )Pt−1 wt−1 , ∀t ≥ 1, 18 It is straightforward to show that a constrained-optimal policy implements a constrained-efficient allocation, where the notion of constrained efficiency is defined in Section A.1.5 of the Appendix, following the macroprudential literature. 27 where the degree of rigidity γ is now a function of Lt , which for simplicity is assumed to be: γ(L) ≡ γ0 Lγ1 , γ0 , γ1 > 0. The fact that γ is increasing in L implies that nominal wages are more flexible as unemployment increases and more rigid as employment increases. Furthermore, as we will show, the assumption γ1 > 0 implies that there could exist a “secular stagnation” bubbleless steady state that features involuntary unemployment. The nominal wage rigidity condition can be rewritten as: γ(Lt ) wt−1 , (6.1) wt ≥ Πt−1,t t where Πt−1,t ≡ PPt−1 is the gross inflation rate between t − 1 and t. ZLB: To close the model, we need to specify how price levels are determined. As is standard in the literature, we assume that the entrepreneurs can trade nominal government bonds, which yield an interest rate 1 + it,t+1 and are available in net zero supply.19 A monetary authority sets the nominal interest rate 1 + it,t+1 between each period t and t + 1 according to a Taylor rule subject to a ZLB on it,t+1 : 1 + it,t+1 n o ζ f ∗ 1−ζ , = max 1, Rt,t+1 (Πt−1,t ) (Π ) (6.2) f where Rt,t+1 is the real interest rate that would prevail with full employment in t + 1 (i.e., Lt+1 = 1), Π∗ > 0 is an inflation target, and ζ > 1 is a constant. As is standard, the rule implies that if the monetary authority were not constrained by the ZLB, inflation would be stabilized at the target Π∗ .20 The definition of an equilibrium is similar to before, except that we have an additional endogenous variable Pt in each period for the price level, the previous complementary-slackness condition for the labor market is now replaced with γ(Lt ) wt−1 = 0, (1 − Lt ) wt − Πt−1,t and the monetary policy rule (6.2) holds. 19 For algebraic simplicity, we have abstracted away from explicitly introducing the buying and selling of nominal bonds that are in net zero supply into the budget constraints of entrepreneurs. 20 We do not model optimal monetary policy explicitly here. This is because in our model, an increase in the inflation rate always weakly improves welfare by mitigating the wage rigidity. Thus, setting a very high inflation target to avoid involuntary unemployment and the ZLB will be optimal. Realistically, there are costs of inflation, such as the costs associated with nominal price rigidities, that are not modeled explicitly here. Also, in practice, central banks tend to follow similar Taylor rules with inflation targets. 28 6.1 Bubbleless equilibrium and multiple steady states Let us first characterize the bubbleless equilibrium. Detailed derivations are relegated to Appendix A.1. In the absence of bubbles, as in section3, the cutoff threshold and the evolution of the capital stock are again given by (3.9) and (3.11). The equilibrium wage wt and employment Lt depend on whether the DNWR binds or not. Similarly, the inflation rate depends on whether the ZLB on the nominal interest rate binds or not. Formally, wt−1 α wt = max (1 − α)Kt , γ(Lt ) Πt−1,t and n o f max 1, Rt,t+1 (Πt−1,t )ζ (Π∗ )1−ζ Πt,t+1 = Rt,t+1 . For the rest of the paper, we assume: Π∗ > γ0 > 1 , Rn (6.3) where Rn is the bubbleless steady state interest rate, as given by (3.15). Under this assumption, because of the kink in the Taylor rule and the fact that the degree of wage rigidity is a function of employment, there are two possible bubbleless steady states. In the “good” steady state (which will continue to be denoted with a subscript n), there is full employment (Ln = 1), the ZLB is slack, the inflation is at the target Π∗ , the capital stock is given by Kn as in (3.13), and the real interest rate is given by Rn as in (3.15). There is another “bad” steady state, where the ZLB binds (i = 0) and inflation is below target, and there is involuntary unemployment (L < 1), leading to a lower capital stock: K = Kn L < Kn . The real interest rate is given by the indifference condition of the marginal investor: R = a ¯n αK α−1 L1−α = Rn . The inflation rate is determined by the Fisher equation RΠ = 1, or equivalently R (β + θ) a¯n adF (a) Π= , a ¯n which is smaller than the target Π∗ under assumption (6.3). The employment level L is determined by the binding DNWR condition 1 = γ(L) , which gives: Π 1 L = (Π/γ0 ) γ1 . 29 Assumption (6.3) guarantees that there is involuntary unemployment in this steady state: L < 1, and the ZLB does indeed bind: RΠζ (Π∗ )1−ζ < 1. 6.2 Bubbly equilibrium We now analyze the bubbly economy. We focus on the relevant parameter range in which the DNWR and the ZLB are slack as long as the bubble persists.21 Then as inflation is stabilized at the target, the bubbly equilibrium dynamics are as characterized in Section 4, and the steady state is as characterized in Section 4.2. The post-bubble dynamics will however be different. Suppose the economy reaches the bubbly steady state and then bubble collapses at period T (i.e., pbT +s = 0, ∀s ≥ 0). The collapse of the bubble exerts downward pressure on the real interest rate through two channels. First, after the bubble collapses, the productivity of the marginal investor decreases from a ¯b to a ¯n . Thus, instead of the identity RT,T +1 = a ¯b RTk +1 that would have prevailed if the bubble ¯n < a ¯b . Second, did not collapse in T , the real interest is given by RT,T +1 = a ¯n RTk +1 , with a as the bubble has an expansionary effect on capital accumulation, the post-bubble economy will follow the bubbleless dynamics as specified in the previous section but with an initial capital stock Kb , which is larger than that in the good steady state Kn . A high capital stock leads to a low marginal product of capital and thus a low interest rate. The combination of these two mechanisms exerts a downward pressure on the real interest rate and thus the nominal interest rate. If the bubble leads to sufficient large accumulation of capital stock, its collapse can push the interest rate against the ZLB. Formally: Proposition 6. [Effect of bubble’s collapse on real interest rate] Suppose the economy has reached the steady state with a large expansionary bubble and then the bubble collapses in a period denoted by T . If the bubbly steady state Kb is sufficiently large such that 1 ¯ ≡ (¯ Kb > K an An Π∗ ) α(1−α) Kn , then the Taylor rule (6.2) is constrained by the ZLB: ζ f ∗ 1−ζ 1 + iT,T +1 = 1 > RT,T . +1 (ΠT −1,T ) (Π ) (6.4) Proof. Appendix A.2.5. Remark 7. One could think of this as corresponding to a situation of “investment hangover,” or capital overinvestment, at the end of an economic boom (Rognlie et al., 2014). The differThis is the case when the initial capital stock K0 and the initial bubble value pb0 are below the bubbly steady state levels, and Rb > 1/Π∗ , where Rb is given by (4.14). 21 30 ence between our paper and Rognlie et al. (2014) is that the overinvestment is endogenous in our framework, while it is imposed exogenously in theirs. The next result shows that in the post-bubble economy, whenever the ZLB binds, the the DNWR must also bind: Lemma 8. [ZLB implies DNWR] For any t ≥ T + 1, if it−1,t = 0 then Lt < 1. Proof. Appendix A.2.6. We say that the economy is in a liquidity trap in period t if the ZLB binds (implying it−1,t = 0) and the DNWR binds (implying Lt < 1). We now show a stark result that, under certain conditions, the post-bubble economy may never escape from the liquidity trap.22 Specifically, we will construct a post-bubble equilibrium path where Lt < 1 and it−1,t = 0 for all t ≥ T + 1. The laws of motion of equilibrium quantities and prices Kt , Lt , and Πt−1,t are given by the bubbleless law of motion of capital (as derived in Section 6.1): Z Kt = (β + θ) α 1−α adF (a) · αKt−1 Lt−1 , (6.5) a ¯n the binding DNWR: γ(Lt ) (Kt /Lt )α = , (Kt−1 /Lt−1 ) Πt−1,t (6.6) and a Fisher equation, which states that the marginal investor is indifferent between lending in the credit market and buying the nominal government bonds: a ¯ αKtα−1 Lt1−α Πt−1,t = 1. {z } |n (6.7) Rt−1,t For the prices and quantities to indeed constitute an equilibrium, a necessary and sufficient f condition is that the ZLB must bind, i.e., Rt−1,t (Πt−2,t−1 )ζ (Π∗ )1−ζ < 1 for all t, where the f real interest rate with full employment is given by Rt−1,t =a ¯n αKtα−1 . This inequality holds if and only if Kt is sufficiently large for all t, and the equilibrium dynamics above can be solved for in closed form. The following proposition shows that, under certain conditions, the collapse of a bubble can push the economy into a permanent liquidity trap: 22 In reality, there can be shocks (not modeled here) that pull the economy out of the liquidity trap, such as a good technology shock or another bubbly episode. 31 Proposition 9. [Post-bubble secular stagnation] Let {KT +t , LT +t , ΠT +t−1,T +t } be defined by the following closed-form expressions: 1−αt KT +t = (An α) 1−α (KT )α t An γ0 a ¯n 1−α γ 1 1−((1+γ1 )α)t−1 1−αt−1 − 1−α (1−(1+γ1 )α)(1+γ1 )t−1 (1+γ1 )t −1 An γ1 (1+γ1 )t = γ0 a ¯n 1−α 1 KT +t = . α¯ an LT +t LT +t ΠT +t−1,T +t 1. These values constitute a post-bubble equilibrium path if and only if 1 ζ 1−α Πt−2,t−1 for all t ≥ T + 1. Kt > α¯ an Π∗ Π∗ 2. On this equilibrium path, the economy experiences involuntary unemployment: LT +t < 1 for all t > 0, and the economy converges to the bad bubbleless steady state with involuntary unemployment and below-target inflation described in Section 6.1. Proof. Appendix A.2.7. Figure 3 plots a simulated equilibrium path, in a manner similar to the simulation in Figure 1 (the dashed horizontal lines represent the good bubbleless steady state).23 As seen in the figure, the collapse causes the real and nominal interest rate to fall sharply, and the nominal interest rate hits the ZLB. After the collapse, entrepreneurs cut down their investment, leading to a decline in the capital stock. The decline in the capital stock in turn causes a decline in the marginal product of labor. Wage would thus need to fall in order to clear the labor market. However, the wage floor creates a wedge that prevents labor market clearing, leading to involuntary unemployment. The economy gradually converges to the bad bubbleless steady state. Why does the post-bubble economy converge to the bad bubbleless steady state? As explained previously, a large expansionary bubble can lead to a large overinvestment of capital (relative to the good bubbleless steady state). The bubble’s eventual collapse will necessarily cause a sharp adjustment in market-clearing wages and a sharp decline in the real and nominal interest rates. A large drop in the interest rates can push the economy into a liquidity trap. The liquidity trap perpetuates as long as the monetary authority is constrained by the ZLB, leading to an inflation that is below the target. Low inflation in turn 23 Again the simulation is for a model with exogenous TFP growth rate g and partial capital depreciation rate δ. Parameter values are γ0 = 0.99, γ1 = 0.099, Π∗ = 1.02, and ζ = 1.5 (following Schmitt-Grohé and Uribe 2017), while the rest are as in Section 4.3.1. 32 0.26 Bubble Size 10.22 Bubble Equilibrium Bubbleless SS t Net Worth 7.99 0 5.75 Rt-1,t Credit 0.36 1 0.29 1.07 1.05 1 + it-1,t t-1,t 1.03 1.02 0.99 8.07 1 2.15 Capital Output 1.89 6.92 5.77 1.41 1.54 1 Wage 1.3 Equilibrium Unconstrained Labor Entrepreneurs Workers (K + P)/Y 0.9 1.19 1.47 0.81 C/C n 5.19 4.43 1 3.67 0 20 40 60 0 20 40 60 Figure 3: Persistent post-bubble liquidity trap. Solid lines represent detrended equilibrium variable values; gray dashed horizontal lines represent the corresponding good bubbleless steady state values. exacerbates the DNWR, leading to lower employment. Finally, lower employment further reduces the marginal product of capital and the interest rates, creating a vicious cycle that perpetuates the liquidity trap. 7 Conclusion We have developed a tractable rational bubbles model with downward wage rigidity. We show that expansionary bubbles could boost economic activities, but their collapse can push the economy into a persistent slump with involuntary unemployment, and investment, output, and consumption depressed below the pre-bubble levels. Under certain conditions, the economy is better off without stochastic bubbles altogether. The model’s predictions are consistent with stylized features of recent bubbly episodes. The model highlights the tradeoff between the economic gains during the boom due to the bubble and the loss from the bust. A macroprudential leaning-against-the-bubble policy of taxing speculative investment can help balance this boom-bust trade-off. The model has several limitations. For instance, the model predicts that, even though stochastic bubbles can reduce welfare, a perfectly safe bubble (or a bubble without wage 33 frictions) is desirable, as it helps mitigate financial frictions without any of the downside risk of an inefficient slump. Specifically, there is nothing in our model to inherently prevent a bubble from sustaining forever. Thus, the model cannot address the concern of policymakers that some rapid increases in asset prices are unsustainable. Incorporating elements from models with information frictions may help address this issue (see Brunnermeier and Oehmke 2013 or Barlevy 2018 for a survey). The model also features no equilibrium default and hence cannot address the fact that corporate and household bankruptcy rates rose sharply after the collapse of the Japanese or U.S. housing bubble. This drawback can potentially be addressed by incorporating an agency problem (e.g., Allen et al. 2017; Bengui and Phan 2018) into our framework. We leave these as potential avenues for future research. References Acharya, S. and Dogra, K. (2017). The side effects of safe asset creation. Working Paper. Allen, F., Barlevy, G., Gale, D., et al. (2017). On interest rate policy and asset bubbles. Federal Reserve Bank of Chicago Working Paper. Asriyan, V., Fornaro, L., Martin, A., and Ventura, J. (2016). Monetary policy for a bubbly world. NBER Working Paper. Babeck` y, J., Du Caju, P., Kosma, T., Lawless, M., Messina, J., and Rõõm, T. (2010). Downward nominal and real wage rigidity: Survey evidence from european firms. The Scandinavian Journal of Economics, 112(4):884–910. Banerjee, A. V. and Moll, B. (2010). Why does misallocation persist? American Economic Journal: Macroeconomics, 2(1):189–206. Barlevy, G. (2007). Economic theory and asset bubbles. Economic Perspectives, (Q III):44– 59. Barlevy, G. (2012). Rethinking theoretical models of bubbles. New Perspectives on Asset Price Bubbles. Barlevy, G. (2014). A leverage-based model of speculative bubbles. Journal of Economic Theory, 153:459–505. Barlevy, G. (2018). Bridging between policymakers’ and economists’ views on bubbles. Economic Perspectives, 42(4). 34 Bengui, J. and Phan, T. (2018). Asset pledgeability and endogenously leveraged bubbles. Journal of Economic Theory, 177:280 – 314. Bernanke, B. S., Gertler, M., and Gilchrist, S. (1999). The financial accelerator in a quantitative business cycle framework. Handbook of macroeconomics, 1:1341–1393. Bianchi, J. (2011). Overborrowing and systemic externalities in the business cycle. The American Economic Review, 101(7):3400. Bianchi, J. and Mendoza, E. G. (2018). Optimal time-consistent macroprudential policy. Journal of Political Economy, 126(2):588–634. Blanchard, O. J. and Watson, M. W. (1982). Bubbles, rational expectations and financial markets. Brunnermeier, M. K. and Oehmke, M. (2013). Bubbles, financial crises, and systemic risk. In Handbook of the Economics of Finance, volume 2, pages 1221–1288. Elsevier. Buera, F. and Nicolini, J. (2017). Liquidity traps and monetary policy: Managing a credit crunch. Working paper. Buera, F. J. and Shin, Y. (2013). Financial frictions and the persistence of history: A quantitative exploration. Journal of Political Economy, 121(2):221–272. Caballero, R. and Farhi, E. (2017). The safety trap. The Review of Economic Studies, page rdx013. Caballero, R. J. and Krishnamurthy, A. (2006). Bubbles and capital flow volatility: Causes and risk management. Journal of Monetary Economics, 53(1):35–53. Cahuc, P. and Challe, E. (2012). Produce or speculate? asset bubbles, occupational choice, and efficiency. International Economic Review, 53(4):1105–1131. Carlstrom, C. T. and Fuerst, T. S. (1997). Agency costs, net worth, and business fluctuations: A computable general equilibrium analysis. The American Economic Review, pages 893– 910. Christiano, L. J., Eichenbaum, M. S., and Trabandt, M. (2015). Understanding the Great Recession. American Economic Journal: Macroeconomics, 7(1):110–167. Christiano, L. J., Motto, R., and Rostagno, M. (2014). Risk shocks. American Economic Review, 104(1):27–65. 35 Diamond, P. (1965). National Debt in a Neoclassical Growth Model. American Economic Review, 55(5):1126–1150. Domeij, D. and Ellingsen, T. (2018). Bond-boom recessions. Working Paper. Dong, F., Miao, J., and Wang, P. (2017). Asset bubbles and monetary policy. Working Paper. Eberly, J. and Krishnamurthy, A. (2014). Efficient credit policies in a housing debt crisis. Brookings Papers on Economic Activity, 2014(2):73–136. Eggertsson, G. B. and Krugman, P. (2012). Debt, Deleveraging, and the Liquidity Trap: A Fisher-Minsky-Koo Approach. Quarterly Journal of Economics, 127(3):1469–1513. Eggertsson, G. B., Mehrotra, N. R., Singh, S. R., and Summers, L. H. (2016). A contagious malady? open economy dimensions of secular stagnation. IMF Economic Review, 64(4):581–634. Farhi, E. and Tirole, J. (2011). Bubbly liquidity. The Review of Economic Studies, page rdr039. Farhi, E. and Werning, I. (2016). A theory of macroprudential policies in the presence of nominal rigidities. Econometrica, 84(5):1645–1704. Gali, J. (2014). Monetary policy and rational asset price bubbles. American Economic Review, 104(3):721–752. Gali, J. (2016). Asset price bubbles and monetary policy in a New Keynesian model with overlapping generations. Working Paper. Gertler, M., Kiyotaki, N., and Queralto, A. (2012). Financial crises, bank risk exposure and government financial policy. Journal of Monetary Economics, 59:S17–S34. Graczyk, A. and Phan, T. (2016). Regressive welfare effects of housing bubbles. Working Paper. Grossman, G. M. and Yanagawa, N. (1993). Asset bubbles and endogenous growth. Journal of Monetary Economics, 31(1):3–19. Guerron-Quintana, P. A., Hirano, T., and Jinnai, R. (2018). Recurrent bubbles, economic fluctuations, and growth. Technical report, Bank of Japan. 36 Hanson, A. and Phan, T. (2017). Bubbles, wage rigidity, and persistent slumps. Economics Letters, 151:66–70. He, Z. and Krishnamurthy, A. (2011). A model of capital and crises. The Review of Economic Studies, page rdr036. Hirano, T., Ikeda, D., and Phan, T. (2017). Risky bubbles, public debt and monetary policies. Working Paper. Hirano, T., Inaba, M., and Yanagawa, N. (2015). Asset bubbles and bailouts. Journal of Monetary Economics, 76:S71–S89. Hirano, T. and Yanagawa, N. (2017). Asset bubbles, endogenous growth, and financial frictions. Review of Economic Studies, 84:406–443. Holden, S. and Wulfsberg, F. (2009). How strong is the macroeconomic case for downward real wage rigidity? Journal of monetary Economics, 56(4):605–615. Ikeda, D. (2016). Monetary policy, inflation and rational asset bubbles. Bank of Japan Working Paper. Ikeda, D. and Phan, T. (2016). Toxic asset bubbles. Economic Theory, 61(2):241–271. Ikeda, D. and Phan, T. (2018). Asset bubbles and global imbalances. American Economic Journal: Macroeconomics (forthcoming). Illing, G., Ono, Y., and Schlegl, M. (2018). Credit booms, debt overhang and secular stagnation. European Economic Review. Jeanne, O. and Korinek, A. (2013). Macroprudential regulation versus mopping up after the crash. National Bureau of Economic Research working paper. Jordà, Ò., Schularick, M., and Taylor, A. M. (2015). Leveraged bubbles. Journal of Monetary Economics. Kamihigashi, T. (2001). Necessity of transversality conditions for infinite horizon problems. Econometrica, 69(4):995–1012. Kindleberger, C. P. and O’Keefe, R. (2001). Manias, panics and crashes. Springer. King, I. and Ferguson, D. (1993). Dynamic inefficiency, endogenous growth, and Ponzi games. Journal of Monetary Economics, 32(1):79–104. 37 Kiyotaki, N., Moore, J., et al. (1997). 105(21):211–248. Credit chains. Journal of Political Economy, Kocherlakota, N. (2009). Bursting bubbles: Consequences and cures. manuscript, Federal Reserve Bank of Minneapolis. Unpublished Kocherlakota, N. (2011). Bubbles and unemployment. Federal Reserve Bank of Minneapolis. March. Kocherlakota, N. R. (1992). Bubbles and constraints on debt accumulation. Journal of Economic Theory, 57(1):245–256. Korinek, A. (2010). Excessive dollar borrowing in emerging markets: Balance sheet effects and macroeconomic externalities. Korinek, A. and Simsek, A. (2016). Liquidity Trap and Excessive Leverage. American Economic Review, 106(3):699–738. Liu, Z. and Wang, P. (2014). Credit constraints and self-fulfilling business cycles. American Economic Journal: Macroeconomics, 6(1):32–69. Lorenzoni, G. (2008). Inefficient credit booms. The Review of Economic Studies, 75(3):809– 833. Martin, A. and Ventura, J. (2012). Economic growth with bubbles. American Economic Review, 102(6):3033–3058. Martin, A. and Ventura, J. (2017). The macroeconomics of rational bubbles: a user’s guide. Technical report. Miao, J. (2014). Introduction to economic theory of bubbles. Journal of Mathematical Economics. Miao, J. and Wang, P. (2012). Bubbles and Total Factor Productivity. The American Economic Review, 102(3):82–87. Miao, J. and Wang, P. (2018). Asset bubbles and credit constraints. Miao, J., Wang, P., and Xu, L. (2016). Stock market bubbles and unemployment. Economic Theory, 61(2):273–307. Miao, J., Wang, P., and Zhou, J. (2014). Housing bubbles and policy analysis. Working Paper. 38 Moll, B. (2014). Productivity losses from financial frictions: Can self-financing undo capital misallocation? American Economic Review, 104(10):3186–3221. Olivier, J. and Korinek, A. (2010). Managing credit booms and busts: A pigouvian taxation approach. NBER Working Paper, page 16377. Piketty, T. and Zucman, G. (2014). Capital is back: Wealth-income ratios in rich countries 1700–2010. The Quarterly Journal of Economics, 129(3):1255–1310. Rognlie, M., Shleifer, A., and Simsek, A. (2014). Investment hangover and the great recession. NBER Working Paper. Saint-Paul, G. (1992). Fiscal policy in an endogenous growth model. The Quarterly Journal of Economics, 107(4):1243–1259. Samuelson, P. A. (1958). An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money. Journal of Political Economy, 66(6):467–482. Schmitt-Grohé, S. and Uribe, M. (2016). Downward nominal wage rigidity, currency pegs, and involuntary unemployment. The Journal of Political Economy, 124(5):1466–1514. Schmitt-Grohé, S. and Uribe, M. (2017). Liquidity traps and jobless recoveries. American Economic Journal: Macroeconomics, 9(1):165–204. Shioji, E. (2013). The bubble burst and stagnation of Japan. Routledge Handbook of Major Events in Economic History, page 316. Summers, L. (2013). Why Stagnation Might Prove to be the New Normal. Tirole, J. (1985). Asset Bubbles and Overlapping Generations. Econometrica, 53(6):1499– 1528. Weil, P. (1987). Confidence and the real value of money in an overlapping generations economy. The Quarterly Journal of Economics, pages 1–22. 39 A A.1 A.1.1 Appendix Derivations Bubbleless equilibrium The credit market clearing condition is R1 0 djt dj = 0. By substituting (3.4), we get Z F (¯ at )βet = θ 1 1aj >¯at ejt dj, 0 where 1 is the indicator function. Since net worth ejt is a function of past productivity ajt−1 and the productivity shocks are i.i.d., the above equality can be rewritten as Z 1 Z 1aj >¯at dj × F (¯ at )βet = θ 0 ejt dj , | {z } et which yields (3.8), as stated in the main text. Recall that the CDF for the Pareto distribution over [1, ∞) with shape parameter σ is F (a) = 1 − a−σ . Thus, the solution to equation (3.9) for the bubbleless cutoff threshold is given by (3.10), i.e.: a ¯n = β+θ β 1/σ . 1−σ σ n A closed-form expression for An is An = (β + θ) a¯σ−1 . Thus, a closed-form expression for the interest rate in the bubbleless steady state is Rn = σ−1 , βσ (A.1) and for the capital stock is: Kn = ασβ σ−1 σ (β + θ)1/σ σ−1 40 1 ! 1−α . A.1.2 Bubbly equilibrium Suppose the bubble persists in period t. By substituting (4.1) into credit market clearing R1 condition 0 djt dj = 0, we get: Z β(F (¯ at ) − φt )et = θ 1 1aj >¯at ejt dj. 0 R1 R As the productivity shocks are i.i.d., the right-hand side again reduces to θ 0 1aj >¯at dj× ejt dj. 1/σ β+θ , leading We thus arrive at (4.6). The closed-form solution to this equation is a ¯t = β(1−φ t) to the steady-state value of a ¯b as in (4.10). The capital accumulation equation is then given by (4.7). k . From Proposition 3, we know that the slump only We now determine Rt,t+1 and Rt+1 begins one period after the bubble collapses. That is, even if the bubble collapses in t + 1, the slump only begins in t + 2 and the labor market still clears in t + 1, leading to Lt+1 = 1. α−1 1−α α−1 k = αKt+1 Hence, the rental rate of capital is given by Rt+1 Lt+1 = αKt+1 in period t + 1, regardless of whether the bubble collapses or persists in t + 1. Thus, from the perspective of the marginal investors, both the options of lending and investing in capital are safe. As a consequence, their indifference condition (4.4) reduces to k α−1 Rt,t+1 = a ¯t Rt+1 =a ¯t αKt+1 . (A.2) We now derive the bubble growth. Indifference condition (4.3) gives: 1 pbt+1 ρ j,ρ b = ct+1 pt ρ 1 cj,ρ t+1 + (1 − ρ) ! 1 cj,1−ρ t+1 Rt,t+1 j,1−ρ for all j such that aj < a ¯t , where cj,ρ t+1 and ct+1 denote the consumption of entrepreneur j when the bubble persists and when the bubble collapses in t+1, respectively. By substituting out the consumption values, this equation can be algebraically simplified to ρ pbt+1 pbt − Rt,t+1 1−ρ = pbt+1 bjt βejt − pbt bjt , for all j such that aj < a ¯t . Furthermore, recall the following fact from algebra: if 41 (A.3) x y = x0 , y0 then x y = x0 y0 = x+x0 . y+y 0 ρ pbt+1 pbt Thus, − Rt,t+1 1−ρ = pbt+1 bjt βejt − pbt bjt = 1aj <¯at bjt dj R . 1aj <¯at ejt dj − pbt 1aj <¯at bjt dj pbt+1 β R R Because of the bubble market clearing conditions, we then get: ρ pbt+1 pbt − Rt,t+1 1−ρ By applying the definition that φt ≡ pbt βet pbt+1 = . βF (¯ at )et − pbt , we get: et+1 − Rt,t+1 ρ φφt+1 et t 1−ρ = φt+1 . F (¯ at ) − φ t (A.4) The growth of net worth is given by: Rk Kt+1 + pbt+1 Rk Kt+1 et+1 et+1 = t+1 = t+1 + φt+1 β . et et et et Combined with (4.7) and (A.2), the equation above yields: R k (β + θ) a¯t adF (a) Rt+1 et+1 = . et 1 − βφt+1 (A.5) Combining (A.4) and (A.5) yields equation (4.9). Finally, from the analysis above of the equilibrium dynamics, the bubbly steady state can be straightforwardly characterized as in the main text. A.1.3 Welfare functions The expected lifetime utility of a representative worker in the bubbly equilibrium in each period t is given by: Wb (Kt , φt ) = log (cw t ) + βρWb (Kt+1 , φt+1 ) + β(1 − ρ)Wburst (Kt+1 ) where the first term is the instantaneous utility, with α cw t = wt = (1 − α)Kt , 42 and the second term is the continuation value conditional on the bubble persisting, and the last term is the continuation value conditional on the bubble collapsing in t + 1. From Proposition 3, we can calculate this last term to be: Wburst (Kt+1 ) = Γ0 (s∗ ) + Γ1 (s∗ ) log Kt+1 , where Γ0 (s∗ ) = 1 log (1 − α) 1−β s∗ −1 1 − α X β s s (s + 1) ! ∗ α β s s∗ (s∗ − 1) − + log γ α 2 1 − β 2 s=0 ! s∗ X 1 − α − β ∗ log Kn + (1 − α) β s s − β s s∗ 1 − β s=0 ∗ Γ1 (s ) = ∗ −1 sX ∗ β s (α − (1 − α) s) + s=0 β s α (1 − (1 − α) s∗ ) . 1−β Thus, the worker’s welfare in the bubbly steady state is as given by (5.2). We now calculate the welfare of entrepreneurs in the bubbleless steady state. Recall from the main text that in the bubbleless steady state, the lifetime expected utility of an entrepreneur j that starts the period with a net worth ej , denoted by Vn (ej ), satisfies equation (5.3). We solve Vn (·) by the guess and verify method. We conjecture that Vn (ej ) = g0 + g1 log Kn + g2 log ej . By plugging into the equation above and solving for g0 , g1 , g2 , we get: β log (1 − β) + g0 = 1−β (1 − β)2 2 1−α β log Kn − 1−β 1 − βα 1 − α βα g1 = − 1 − β 1 − βα 1 g2 = . 1−β a (β + θ) log dF (a) a ¯n β a ¯n Z log (¯ an βα) + 43 Thus Z β a (β + θ) log (1 − β) + log (α¯ an β) + log dF (a) Vn (e ) = 1−β a ¯n β (1 − β)2 a ¯n β (1 − α) 1 − log ejt . 2 log Kn + 1−β (1 − β) j Under the definition Vn ≡ Vn (αKnα ), we then get: Z log α (1 − β) β a (β + θ) β−α Vn = + an β) + log dF (a) − log Kn . 2 log (α¯ 1−β a ¯n β (1 − β) (1 − β)2 a ¯n (A.6) Finally, we compute the welfare of entrepreneurs in the bubbly steady state. Recall that the lifetime expected utility of an entrepreneur j that starts the period with a net worth ej , denoted by Vb (ej ), satisfies equation (5.4). From Proposition 3, we can calculate the post-bubble continuation value as given by: Vburst (ej ) = κ0 (s∗ , a ¯n ) − κ1 (s∗ ) log Kb + log ((1 − β)ej ) , 1−β (A.7) where: # " 2 s∗ 1 − β 1 − α β ∗ log (1 − β) + β s Γn (¯ log Kn κ0 (s∗ , a ¯n ) = an ) − 1−β 1−β 1 − βα Z ∗ β − β s +1 a(β + θ) + log (α¯ an β) + log dF (a) a ¯n β (1 − β)2 a ¯n Z ∗ β s +1 s∗ α (1 − α) − log (β + θ)α adF (a) 1 − β 1 − βα a ¯n " s∗ −1 ! # s∗ ∗ 2 ∗ 2 X (1 − α) β s 1 − βα + s (1 − 2βα + βα ) − β s s (s + 1) + log γ 2α (1 − β) 1 − βα s=0 "s∗ −1 # ∗ 2 s∗ X β βα + s (1 − 2βα + βα ) κ1 (s∗ ) = (1 − α) β ss + . 1 − β 1 − βα s=0 Again, by applying the guess and verify method to equation (5.4), we get the following solution for Vb : β Vb (e ) = Γb (s ) − 1 − βρ j ∗ 1−α 1 ∗ + (1 − ρ) κ1 (s ) log Kb + log ej , 1−β 1−β 44 (A.8) where β (1 − ρ) 1 log (1 − β) + κ0 (s∗ , a ¯n ) 1 − βρ (1 − βρ) Z aj (β + θ) β log + log (α¯ ab β) + dF (a) (1 − βρ) (1 − β) a ¯b β a ¯b βF (¯ ab ) θ (1 − F (¯ ab )) θ (1 − F (¯ ab )) + ρ log ρ + (1 − ρ) log . (1 − βρ) (1 − β) θ − (θ + β (1 − ρ)) F (¯ ab ) βF (¯ ab ) Γb (s∗ ) = αK α b Under the definition Vb ≡ Vb ( 1−βφ ), we then get: β Vb = Γb (s ) − 1 − βρ ∗ A.1.4 1−α 1 αKbα ∗ + (1 − ρ) κ1 (s ) log Kb + log . 1−β 1−β 1 − βφ (A.9) Bubble dynamics and steady state with macroprudential tax In the presence of a macroprudential tax, the equilibrium dynamics are similar to that of the bubble equilibrium, except that τ will affect the first-order condition with respect to bubbly investment of entrepreneurs. The indifference condition between investing in the bubbly asset and lending for entrepreneurs with productivity shock below a ¯t is now given by: (1 − τ )pbt+1 = Et [u0 (¯ ct+1 )Rt,t+1 ] , Et u (¯ ct+1 ) pbt 0 if ajt < a ¯t , which can be reduced to: ρ (1−τ )pbt+1 pbt − Rt,t+1 = 1−ρ pbt+1 bjt βejt − pbt bjt . By integrating across aj < a ¯t , and with more algebraic manipulations, we then get an aggregate expression: (1 − τ ) φt+1 (1 − βφt+1 ) a ¯t F (¯ at ) − φt R = . φt at ) − φt (β + θ) a¯t adF (a) ρF (¯ This equation gives a new expression that determines the bubble size in the bubbly steady state: (1 − βφ) a ¯b F (¯ ab ) − φ R 1−τ = . ab ) − φ (β + θ) a¯b adF (a) ρF (¯ | {z } Rb 45 The closed-form expression for the bubble ratio is: φ(τ ) = θ(1 − σ(1 − βρ(1 − τ ))) β(β(1 − ρ)(1 − τ )σ + θ(1 − στ )) Note how φ is a decreasing function of τ . The bubble exists (φ > 0) if and only if τ < τ¯ ≡ . The capital stock in the bubbly steady state is then given by: 1 − σ−1 βσρ Kb (τ ) = β+θ α 1 − βφ(τ ) β+θ β(1−φ(τ )) 1 1−α Z adF (a) , a ¯b (τ ) 1/σ where the cutoff threshold is a ¯b (τ ) = . Similar to the analysis in the main text and Section A.1.3, the duration of the post-bubble slump is given by (4.19) and the bubbly steady-state welfare expression for workers is given by (5.2), where we note that Kb = Kb (τ ) is now a function of τ . The bubbly steady-state αKbα ), where Vb (ej ) is as defined welfare expression for entrepreneurs is given by Vb ≡ Vb ( 1−βφ(τ ) in (A.8), except that Γb (s∗ ) is now defined as: Γb (s∗ ) 1 β (1 − ρ) log (1 − β) + κ0 (s∗ , a ¯n ) 1 − βρ (1 − βρ) Z β a(β + θ) + log log (α¯ ab β) + dF (a) (1 − βρ) (1 − β) a ¯b β a ¯b F (¯ ab )−φ (1 − τ ) (F (¯ a ) − φ) + τ φ b F (¯ ab ) F (¯ ab ) − φ βF (¯ ab ) ρ log ρ + (1 − ρ) log . + (1 − βρ) (1 − β) (1 − τ ) ρF (¯ ab ) − (1 − τ ρ) φ F (¯ ab ) = A.1.5 Constrained efficiency Following the macroprudential literature (e.g., Bianchi 2011; Bianchi and Mendoza 2018), we define constrained efficiency as follows. Consider a benevolent social planner who cares about both entrepreneurs and workers. The planner has restricted planning abilities: it chooses allocations subject to the resource, implementability, and leverage constraints, but allows the markets to clear competitively. Its policy instrument is limited to a constant tax on bubbly speculation. Since prices remain market-determined, the first-order conditions for private agents enter the planner’s problem as implementability constraints. The key difference between the planner’s problem and private agents’ problems is that the planner internalizes how its decisions affect prices. The implementability constraints for the planner will come from the first-order conditions 46 of individual entrepreneurs. Recall that with the constant tax τ , the optimization problem of each entrepreneur j is: ∞ X max E0 β t u(cjt ) j j j j {ct ,dt ,bt ,It } t=0 subject to j cjt + Itj + pbt bjt = Rtk ajt−1 It−1 + (1 − τ )pbt bjt−1 − Rt−1,t djt + djt + Ttj Itj , bjt ≥ 0 j djt ≤ θ · (Rtk ajt−1 It−1 + pbt bjt−1 − Rt−1,t djt ). It is more convenient to rewrite this problem in the following equivalent form: instead of choosing the amount of the bubbly asset bjt , each entrepreneur chooses the amount of (beforetax) bubbly investment Btj ≡ pbt bjt . Then the problem becomes: max {cjt ,djt ,Btj ,Itj } E0 ∞ X β t u(cjt ) t=0 subject to j j cjt + Itj + Btj = FK (Kt , Lt )ajt−1 It−1 + (1 − τ )Rtb Bt−1 − Rt−1,t djt + djt + Ttj Itj , bjt ≥ 0 j djt ≤ θ · (Rtk ajt−1 It−1 + pbt bjt−1 − Rt−1,t djt ), pb where Rtb ≡ pb t denotes the return on the bubbly asset. In the budget constraint above, we t−1 have also replaced Rtk with the marginal product of capital. The entrepreneurs’ first-order conditions are: k Et λjd,t+1 u0 (cjt ) = βajt FK (Kt+1 , Lt+1 )Et u0 (cjt+1 ) + λjI,t βθajt Rt,t+1 b,ρ 0 j,ρ u0 (cjt ) = βρ(1 − τ )Rt+1 u (ct+1 ) + λjb,t /pbt + βθEt Rt,t+1 λjd,t+1 (A.10) (A.11) u0 (cjt ) = βRt,t+1 · Et u0 (cjt+1 ) + λjd,t + βθRt,t+1 Et λjd,t+1 , (A.12) j j + (1 − τ )pbt bjt−1 − Rt−1,t djt ) − dt = 0, µjI,t Itj = µjb,t bjt = µjd,t θ(Rtk ajt−1 It−1 (A.13) where Rtb,ρ and cj,ρ t are the return on the bubbly asset and consumption conditional on the state that the bubble persisting in t, µjI,t , µjb,t , µjd,t are the Lagrange multipliers associated with the nonnegativity constraints on Itj and bjt and the leverage constraint, respectively. We can now define the planner’s problem: 47 Definition 10. The constrained central planner’s problem is as follows: max {cjt ,Btj ≥0,Itj ≥0,djt ,Rt−1,t ,Rtb+ ,µjI,t ,µjb,t ,µjd,t ,τ } E0 ∞ X Z w β λ · u(ct ) + (1 − λ) · t u(cjt ) , j∈J t=0 where λ ∈ [0, 1] is the Pareto weight the planner assigns to the representative worker and 1 − λ is that for entrepreneurs, subject to the following individual budget constraints: cw t = FL (Kt , Lt )Lt j j cjt = FK (Kt , Lt )ajt−1 It−1 + ((1 − τ )Rtb Bt−1 − Btj ) − Rt−1,t djt−1 + djt − Itj + Ttj , the planner’s transfer being given by: Ttj = τ Btj , ∀j ∈ J, leverage constraint: j j djt ≤ θ · (FK (Kt , Lt )ajt−1 It−1 + Rtb Bt−1 − Rt−1,t djt−1 ), | {z } ejt implementability constraints (A.10-A.13), and the labor market conditions: FL (Kt , Lt ) ≥ γFL (Kt−1 , Lt−1 ) (A.14) Lt ≤ 1 (1 − Lt )(FL (Kt , Lt ) − γFL (Kt−1 , Lt−1 )) = 0. As in the main text, we focus on the planner’s problem in the bubbly steady state and assume that each entrepreneur begins the steady state with the same net worth. Note that the planner takes as given the exogenous sunspot process of bubble bursting. The key thing to notice is that unlike individual agents, the planner internalizes the downward wage rigidity condition (A.14) in its optimization problem. As a consequence, the first-order conditions of the planner will contain a Lagrange multiplier associated with this constraint, which would be otherwise absent in the laissez-faire first-order conditions of individual entrepreneurs. This formalizes the notion that the competitive equilibrium allocations are constrained inefficient. 48 A.1.6 Extension: a generalized model We now extend the model in several directions. First, we allow for exogenous growth. Specifically, assume that the production technology is given by: Yt = Ktα · (At Lt )1−α , where the technology term At grows at an exogenous growth rate g ≥ 0: At ≡ (1 + g)t . Throughout, we will use a lower case letter to denote a detrended variable, for example: yt ≡ Kt Wt Yt , kt ≡ , wt ≡ ... t t (1 + g) (1 + g) (1 + g)t The downward wage rigidity condition is growth-adjusted, meaning: wt ≥ (1 + g)γwt−1 . Second, we allow for partial capital depreciation. Specifically, we assume that the capital stock depreciates at a rate δ ∈ [0, 1]. As in Kocherlakota (2009), we assume that the capital good can be converted back one-to-one to the consumption good. Third, we allow for a more general continuous productivity distribution F over a subset of (0, ∞). Fourth, we allow for a more flexible leverage constraint: djt ≤ θt (ajt ) · ejt , where θt (·) is a (possibly time-varying) measurable function of entrepreneur j’s productivity shock ajt . Remark 11. The main model in Section 2 is a special case of this extended model (where g = 0, δ = 1, θt (a) = θ, and F is Pareto over [1, ∞)). Note further that the calibration exercise in Section 4.3.1 uses only the first two extended assumptions (i.e., g > 0 and δ < 1). The changes compared to the main model are as follows. We first consider the bubbleless equilibrium. The time-invariant cutoff threshold is implicitly determined by the creditmarket clearing condition: Z βF (¯ an ) = θ(a)dF (a). a ¯n 49 The detrended capital stock evolves according to the following new law of motion: Z (β + θt (a))a dF (a) αktα L1−α + (1 − δ) kt . t (1 + g) kt+1 = a ¯n The bubbleless steady-state capital stock is thus: kn = α R (β + θ(a))a dF (a) R (1 + g) − (1 − δ) a¯n (β + θ(a))a dF (a) 1 ! 1−α a ¯n . The bubbleless steady-state interest rate is: Rn = a ¯n αknα−1 + 1 − δ =R (1 + g)¯ an . (β + θ(a))a dF (a) a ¯n We now consider the bubbly equilibrium. The cutoff threshold is implicitly determined by the credit-market clearing condition: Z β (F (¯ at ) − φt ) = θ(a)dF (a). a ¯t The detrended capital stock evolves according to the following new law of motion: Z (β + θ(a))a dF (a) · (1 + g) kt+1 = a ¯t αktα Lt1−α + (1 − δ) kt 1 − βφ . The new law of motion of the bubble price is given by: (1 + g) pbt+1 F (¯ at ) − φt = Rt,t+1 . b ρF (¯ at ) − φt pt The bubbly steady state is characterized by the following new expression for the (detrended) capital stock: kb = α R (β + θ(a))a dF (a) a ¯b R (1 + g) (1 − βφ) − (1 − δ) a¯b (β + θ(a))a dF (a) and the following new expression for the interest rate: Rb = R (1 + g) (1 − βφ) a ¯b , (β + θ(a))a dF (a) a ¯b 50 1 ! 1−α , where the cutoff threshold a ¯b as given in the main model, and the following new equation that determines the bubble ratio: φ= (1 + g)ρ − Rb F (¯ ab ). (1 + g) − Rb After the bubble collapses, the economy evolves according to the bubbleless equilibrium dynamics. A.2 A.2.1 Proofs Proof of Lemma 2 Proof. For the variables characterized by equations (4.11) to (4.15) to constitute a bubbly steady state, a necessary and sufficient condition is φ ∈ (0, 1). From (4.15), φ > 0 equivalent to 1 − (1 − βρ)σ > 0, i.e., (4.16). And once (4.16) is satisfied, it is immediately true that φ < 1. Finally, note that from (A.1), condition (4.16) is equivalent to Rn < ρ. A.2.2 Proof of Proposition 3 Proof. By the labor market clearing conditions during the slump, the wage rigidity must bind in all periods for which LT +s < 1. Thus, defining s ∈ (0, s∗ ) as a period for which LT +s < 1, the wage must follow: wT +s = γwT +s−1 = γ s wT . (A.15) Recall from the first-order conditions of firms that for all t: Lt = 1−α wt α1 Kt . (A.16) By (3.11), (A.15) and (A.16), the capital law of motion after the burst is: KT +s+1 = An α = An α = An α KT +s Lt+s α−1 wT +s 1−α α−1 α γ s wT 1−α α−1 α 51 KT +s KT +s KT +s . Thus, by recursion, we get: " KT +s+1 = An α wT 1−α #s+1 α−1 α γ α−1 s(s+1) α 2 KT . Finally, by substituting in (A.16) and LT = 1 for wT , we get: s+1 α−1 s(s+1) KT +s+1 = An αKTα−1 γ α 2 KT , as desired. Finally, we determine the duration s∗ of the slump. Recall: s∗ ≡ min {s ∈ N | LT +s = 1} n o f = min s ∈ N | wT +s ≥ γwT +s−1 , where wTf +s = (1 − α) KTα+s represents the wage level consistent with full employment. Then we can rewrite s∗ as: s∗ = min s ∈ N | (1 − α) KTα+s ≥ γ s wT iα o n h s α−1 s(s−1) = min s ∈ N | An αKTα−1 γ α 2 KT ≥ γ s KTα . Algebraic manipulation yields (4.19). A.2.3 Proof of Proposition 4 Proof. We first focus on workers. Define ∆W ≡ Wb − Wn . Then from (5.1) and (5.2): ∆W = log ((1 − α)Kbα ) log ((1 − α)Knα ) − 1 − ρβ 1−β β (1 − ρ) + (Γ0 (s∗ ) + Γ1 (s∗ ) log Kb ) 1 − ρβ Recall from (4.19) that limγ→1 s∗ = ∞. Thus log (1 − α) β (1 − α) + log Kn γ→1 1−β (1 − β)2 α−β lim Γ1 (s∗ ) = γ→1 (1 − β)2 lim Γ0 (s∗ ) = 52 and: log ((1 − α)Kbα ) log ((1 − α)Knα ) − 1 − ρβ 1−β β (1 − ρ) 1 β (1 − α) α−β + log (1 − α) + log Kn + log Kb 1 − ρβ 1−β (1 − β)2 (1 − β)2 α (1 − β)2 + β (1 − ρ) (α − β) = (log Kb − log Kn ) . (1 − ρβ) (1 − β)2 lim ∆W = γ→1 Recall that the assume bubble is expansionary (Kb > Kn ). Therefore, if the bubble is sufficiently risky so that α (1 − β)2 ρ < ρ¯ ≡ 1 − , β (β − α) then the last expression is negative, implying limγ→1 ∆W < 0. Thus, there exists γw < 1 such that if γ > γ1 and ρ < ρ¯, then Wb < Wn . We now focus on entrepreneurs. Similarly, we define ∆V ≡ Vb − Vn . Then from (A.6) and (A.9) and by taking γ → 1, after algebraic manipulations, we can derive the following limit: lim ∆V =G(φ) (A.17) γ→1 β(βφ + θ) (1 − ρ) log ≡ − θ(1−φ) βφ+θ + ρ log θρ(1−φ) θρ−φ(β(1−ρ)+θ) (1 − β)(1 − βρ)(β + θ) 1 β 2 φ log β+θ + β σ − (1 − β)(1 − βρ)(β + θ) 1 ((2 − α)β − 1) log 1−βφ (1 − α)(β − 1)2 + (σ−1)(α−β) α−1 + (β−1)β βρ−1 (β − 1)2 σ log 1 1−φ . Note that G is a decreasing function of φ and furthermore G(0) = 0. Because in equilibrium φ > 0, it follows that limγ→1 ∆V < 0. Thus, there exists γe < 1 such that if γ > γe , then Vb < Vn . The proof is complete by letting γ¯ = max{γw , γe }. A.2.4 Proof of Corollary 5 Proof. First, we show that there exists γe < 1 such that if γ > γe , then Vb (τ ) ≤ Vn for all τ ≤ τ¯, with equality if and only if τ = τ¯. Fix any τ < τ¯. By applying the same algebraic ˜ ˜ is defined manipulations as in Section A.2.3, we have limγ→1 Vb (τ ) − Vn = G(φ(τ )), where G 53 as ˜ G(φ(τ )) ≡ θ(1−φ(τ )) (1−τ )θ(1−φ(τ ))+τ φ(τ )(β+θ) θ+βφ(τ ) θ(1−φ(τ )) β(βφ(τ ) + θ) (1 − ρ) log βφ(τ )+θ + ρ log (1−τ )ρθ−(1−τ ρ)φ(τ )(β(1−(1−τ )ρ)+θ) ρ (1 − β)(1 − βρ)(β + θ) β 2 φ log − − β+θ β + 1 σ (1 − β)(1 − βρ)(β + θ) 1 ((2 − α)β − 1) log 1−βφ(τ ) (1 − α)(β − 1)2 + (σ−1)(α−β) α−1 + (β−1)β βρ−1 (β − 1)2 σ log 1 1 − φ(τ ) . ˜ is a decreasing function of φ (τ ) and G(0) ˜ Note that G = 0. Because in equilibrium φ (τ ) > 0, ˜ it follows that limγ→1 Vb (τ ) − Vn = G(φ(τ )) < 0. Thus, for all > 0, there exists γ() < 1 ˜ ˜ such that Vb (τ ) − Vn < G(φ(τ )) + whenever γ > γ(). By letting = −G(φ(τ )) and ˜ γe = γ(−G(φ(τ ))), we then get Vb (τ ) < Vn for all γ > γe , as desired. Finally, note that when τ = τ¯, the bubble disappears, i.e., φ(¯ τ ) = 0 and thus Vb (¯ τ ) = Vn . Similarly, we show that there exists γw < 1 such that if γ > γw and ρ < ρ¯, then Wb (τ ) ≤ Wn for all τ ≤ τ¯, with equality if and only if τ = τ¯. Fix any τ < τ¯. If Kb (τ ) ≤ Kn , i.e., the taxed bubble is contractionary, then it is obvious that the welfare of workers cannot be better off with the bubble than without, as the equilibrium wage in the bubbly equilibrium path would always be smaller than wn – the wage in the bubbleless steady state. Thus, we can focus on the expansionary case of Kb (τ ) > Kn . By applying the same algebraic manipulations as in Section A.2.3, we get α (1 − β)2 + β (1 − ρ) (α − β) (log Kb (τ ) − log Kn ) , lim Wb (τ ) − Wn = H(τ ) ≡ γ→1 (1 − ρβ) (1 − β)2 where the right-hand side H(τ ) is strictly negative whenever ρ < ρ¯. Thus, for all > 0, there exists γ() such that Wb (τ ) − Wn < H(τ ) + . By letting = −H(τ ) and γw = γ(−H(τ )), we then get Wb (τ ) < Wn for all γ > γw , as desired. Finally, when τ = τ¯, the bubble disappears and thus trivially Wb (¯ τ ) = Wn . The proof is complete by letting γ¯ = max{γw , γe }, as it is immediate from the results above that arg maxτ ≤¯τ λWb (τ ) + (1 − λ)Vb (τ ) = τ¯ when γ > γ¯ and ρ < ρ¯. A.2.5 Proof of Proposition 6 First, we show that the inflation is at the target and there is full employment in the immediate aftermath of the bubble’s collapse: Lemma 12. ΠT −1,T = Π∗ and LT = 1. 54 Proof. To see this, recall that, by assumption, the economy is still in the bubbly steady state in T − 1, and therefore the nominal interest rate iT −1,T is determined by the unconstrained Taylor rule 1 + iT −1,T = Rb Π∗ . Furthermore, recall the Fisher equation that equates the expected return from nominal bond holding and real lending for entrepreneurs below the threshold a ¯b between period T − 1 and T : ρu0 (cLb )Rb Π∗ + (1 − ρ)u0 (cLT )RT −1,T ΠT −1,T , 1 + iT −1,T = ρu0 (cLb ) + (1 − ρ)u0 (cLT ) where the superscript L denotes entrepreneurs with productivity below a ¯b . Here, we have used the fact that in the good state that the bubble persists in period T (which happens with probability ρ from the information set at T − 1), the economy continues to be in the bubbly steady state with consumption level cLb for the L-type, the real interest rate is Rb and inflation is Π∗ . Thus the indifference condition above can be rewritten as: Rb Π∗ = ρu0 (cLb )Rb Π∗ + (1 − ρ)u0 (cLT )RT −1,T ΠT −1,T , ρu0 (cLb ) + (1 − ρ)u0 (cLT ) or equivalently Rb Π∗ = RT −1,T ΠT −1,T . In addition, recall that the real interest rate between T − 1 and T is given by: RT −1,T = a ¯b α LT KT 1−α = Rb L1−α T . Thus the equation above reduces to: Π∗ = ΠT −1,T LT1−α . (A.18) Now suppose on the contrary that LT < 1. Then the DNWR must bind at T , or γ(LT ) wT = . wT −1 ΠT −1,T By substituting the first-order condition of firms (2.7), we then get: L−α T = γ(LT ) . ΠT −1,T 55 (A.19) Equations (A.18) and (A.19) then imply 1 Π∗ = γ0 L1+γ < γ0 . T However, this violates assumption (6.3). Therefore, it must be that LT = 1. Equation (A.18) then implies ΠT −1,T = 1. Now the proof for the proposition follows straightforwardly: Proof. Given ΠT −1,T = Π∗ , it is immediate that (6.4) is equivalent to f ∗ RT,T +1 Π < 1. As the bubble has collapsed in T + 1, the real interest rate with full employment is simply given by: f RT,T ¯n αKTα−1 +1 = a +1 , as the post-bubble economy follows the bubbleless dynamics. In addition, from the law of f ∗ motion of capital, we have KT +1 = αAn KTα LT1−α = αAn Kbα . Therefore, RT,T +1 Π < 1 if and only if a ¯n α (αAn Kbα )α−1 < Π1∗ , which is equivalent to (6.4). A.2.6 Proof of Lemma 8 Proof. Suppose on the contrary that it−1,t = 0 but Lt = 1. The DNWR constraint is slack, wtf γ0 ≥ Πt−1,t , or equivalently, inflation must be sufficiently high: implying that wt−1 γ0 Ktα ≥ . α (Kt−1 /Lt−1 ) Πt−1,t (A.20) However, the inflation rate is determined by the Fisher equation 1 + it−1,t = Rt−1,t Πt−1,t , or 1= a ¯ αK α−1 Πt−1,t . | n {z t } (A.21) Rt−1,t with Lt =1 Substitute (A.21) into (A.20), we get a condition that the real interest rate and thus the marginal product of capital must be sufficiently low: γ0 a ¯n αKtα−1 ≤ Ktα , (Kt−1 /Lt−1 )α 56 or equivalently, the capital stock must be sufficiently high: Kt ≥ γ0 a ¯n α(Kt−1 /Lt−1 )α . 1−α α Substituting the law of motion of capital: Kt = An αKt−1 Lt−1 into the inequality above yields γ0 aL An Lt−1 ≥ However, as 1 ≥ Lt−1 , it then follows that 1 ≥ A.2.7 γ0 aL , An which contradicts assumption (6.3). Proof of Proposition 9 Proof. For notation simplicity, let us normalize the period when the bubble bursts to be period 0; that is, T = 0. Then, on the unemployment path L1 , L2 , · · · < 1 and i0,1 = i1,2 = · · · = 0. The unemployment path can be characterized as follows. The flow of capital is given by α 1−α Kt = An αKt−1 Lt−1 . Binding DNWR and ZLB provide the following two equations, respectively: (Kt /Lt )α = γ(1 − Lt ) Πt−1,t (Kt−1 /Lt−1 )α a ¯ αKtα−1 Lt1−α Πt−1,t = 1. |n {z } Rt−1,t Combining the two above equations yields (Kt /Lt )α =a ¯n αKtα−1 Lt1−α γ(1 − Lt ). α (Kt−1 /Lt−1 ) Rewriting the above equation by utilizing the parameterization of γ(·), Kt Lt−1 Kt−1 α 1 =a ¯n αγ0 L1+γ . t 57 By substituting in the flow of capital, we find a recursive form for labor: Lt = An Lt−1 γ0 a ¯n 1 1+γ 1 . Similarly, inflation can be expressed as a function of last period’s labor and capital: Πt−1,t = 1 Rt−1,t 1 = a ¯n α Kt Lt 1−α γ1 −α(1+γ1 ) 1+γ1 1 = L a α α Lt−1 γ0 a ¯n αKt−1 An γ0 a ¯n γ1 !1−α 1+γ 1 . These expressions can be further simplified by recursively plugging in for Lt−1 , Lt−2, . . . , L1 . Therefore, labor, Lt , can be written as a function of L0 = 1 (as shown in Appendix A.2.5, there is full employment in the period the bubble bursts) and t: 1 1+γ 1 An Lt = Lt−1 γ0 a ¯n 1 1+γ s t−1 1 Pt−1 1 1 s=0 1+γ1 An 1+γ1 L0 = | {z } γ0 a ¯n =1 = An γ0 a ¯n (1+γ1 )t −1 γ1 (1+γ1 )t . Similarly, using the flow of capital equation and working backward, Kt can be written as a function of K0 , t, and all past Lt : α Kt = An αKt−1 L1−α t−1 = (An α) Pt−1 s s=0 α αt K0 t−1 Y !1−α αs−1 Lt−s s=1 1−αt t = (An α) 1−α K0α An γ0 a ¯n 1−α γ 1 1−(α(1+γ1 ))t−1 1−αt−1 − (1−α) (1−α(1+γ1 ))(1+γ1 )t−1 . For these values to constitute an equilibrium path after the collapse of the bubble in period T , the necessary and sufficient conditions are that the DNWR and the ZLB do indeed bind. From Proposition 8, we know it is sufficient to show that the ZLB binds, i.e., f Rt−1,t (Πt−2,t−1 )ζ (Π∗ )1−ζ < 1 for all t, where the real interest rate with full employment is 58 given by f Rt−1,t =a ¯n αKtα−1 . This inequality holds if and only if Kt > a ¯n α Πt−2,t−1 Π∗ ζ Π∗ 1 1−α for all t, as stated in the proposition. Finally, it is algebraically straightforward to show that limt→∞ KT +t = K, limt→∞ LT +t = L and limt→∞ ΠT +t−1,T +t = Π, where K, L, and Π are the capital, labor, and inflation in the bad bubbleless steady state as established in Section 3. 59
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