Lean Against Bubbles versus Clean Up After. Bubbles Collapse in a Rational-Bubble Model

Lean Against Bubbles versus Clean Up After Bubbles Collapse in a Rational-Bubble Model Tomohiro Hirano y Jun Aoyagi Masaru Inaba First Version, April 2013 This Version, April 2015 Very Preliminary Abstract This paper analyzes lean against bubble versus clean up after bubblecrashes in a rational-bubble model. The main results are as follows. Firstly, macro-prudential regulation can be justi ed in the case of over-sized bubbles which are more likely to occur when the quality ofthe nancial system is relatively high. Secondly, although macro-prudential regulation reduces the over-sized bubbles, it may end up increasing boom-bust cycles in real variables. Thirdly, when the degree of externality (i.e., interconnectedness in production) is large, bailout policy can improve taxpayer s welfare, but it creates a time-inconsistency problem if government cannot commit to it, thereby generating welfare loss. Macro-prudential regulation can mitigate the welfare We have bene ted greatly from continuous discussions with Joseph Stiglitz. We also thank Jose Scheinkman, Nobuhiro Kiyotaki, Michihiro Kandori, Kiminori Matsuyama, Katsuhito Iwai, Bruce Greenwald, Ricardo Reis, and seminar participants at McGill University, Bank of England, INET seminar at Columbia University, and U of Tokyo. y Corresponding Author (University of Tokyo) : tomohih@gmail.com 1

loss due to commitment problem. Under some conditions, macro-prudential regulation can function as a commitment device. Moreover, even if government can commit to future bailout policy, macro-prudential regulation can mitigate welfare loss associated with commitment equilibrium, thus improving taxpayer s welfare. These ndings provide a theoretical foundation of the case for leaning against bubble policy as well as for clean up policy after the collapse of bubbles. Key Words: Over-Sized Bubbles, Externality, Commitment Problem, Time Inconsistency, 2

1 Introduction There are two di erent views on how the government should react to asset price bubbles. One view is that it is better for government (central bank or nancial services agency) to clean up crises after bubbles break rather than to interfere in a nancial market before. This view is maily supported by Alan Greenspan, or many economists in central banks. The other view is that government should interfere in a nancial market to lean against bubbles. This view is maily supported by BIS economists. The main purpose of this paper is to investigate which one of these views is more desirable from a welfare perspective in a dynamic macroeconomic model with rational bubbles. We also provide results on under what conditions lean against bubble policy can be justi ed. The main results are as follows. Firstly, macro-prudential regulation can be justi- ed in the case of over-sized bubbles which are more likely to occur when the quality ofthe nancial system is relatively high. Secondly, although macro-prudential regulation reduces the over-sized bubbles, it may end up increasing boom-bust cycles in real variables. Thirdly, when the degree of externality (i.e., interconnectedness in production) is large, bailout policy can improve taxpayer s welfare, but it creates a time-inconsistency problem if government cannot commit to it, thereby generating welfare losses. Macro-prudential regulation can mitigate the welfare losses arising from the commitment problem. Under some conditions, macro-prudential regulation can function as a commitment device. Moreover, even if government can commit to future bailout policy, macro-prudential regulation can mitigate welfare loss associated with commitment equilibrium, thus increasing taxpayer s welfare. These ndings provide a theoretical foundation of the case for leaning against bubble policy as well as for clean up policy after the collapse of bubbles. 3

2 The Model 2.1 Framework Consider a discrete-time economy with one homogeneous good and a continuum of entrepreneurs and workers. A typical entrepreneur and a representative worker have the following expected discounted utility, " 1 # X (1) E 0 t log c i t ; t=0 where i is the index for each entrepreneur, and c i t is the consumption of him/her at date t. 2 (0; 1) is the subjective discount factor, and E 0 [a] is the expected value of a conditional on information at date 0. Let us start with the entrepreneurs. At each date, each entrepreneur meets high-productivity investment projects (hereinafter H-projects) with probability p, and low productivity ones (L-projects) with probability 1 p. The investment projects produce capital. The investment technologies are as follows: (2) k i t+1 = i tz i t; where z i t( 0) is the investment level at date t; and k i t+1 is the capital at date t + 1 produced by the investment. i t is the marginal productivity of investment at date t. i t = H if the entrepreneur has H-projects, and i t = L if he/she has L-projects. We assume H > L. For simplicity, we assume that capital fully depreciates in one period. 1 The probability p is exogenous, and independent across entrepreneurs and over time. The entrepreneur knows his/her own type at date t, whether he/she 1 As in Kocherlakota (2009), we can consider a case where only a fraction of capital depreciates, and consumption goods can be converted one-for-one into capital, and vice-versa. In this setting, we can also obtain the same results as in the present paper. 4

has H-projects or L-projects. Assuming that the initial population measure of each type is p and 1 p at date 0, the population measure of each type after date 1 is p and 1 p, respectively. Throughout this paper, we call the entrepreneurs with H-projects H-types and the entrepreneurs with L-projects L-types. We assume that because of frictions in a nancial market, the entrepreneur can pledge at most a fraction of the future return from his/her investment to creditors as in Kiyotaki and Moore (1997). In such a situation, in order for debt contracts to be credible, debt repayment cannot exceed the pledgeable value. That is, the borrowing constraint becomes: (3) r t b i t q t+1 i tz i t; where q t+1 is the relative price of capital to consumption goods at date t+1. 2 r t and b i t are the gross interest rate and the amount of borrowing at date t. The parameter 2 (0; 1], which is assumed to be exogenous, can be naturally taken to be the degree of imperfection of the nancial market, capturing the quality of the nancial system. In this economy, there are bubble assets denoted by x. The aggregate supply of bubble assets is assumed to be constant over time X: As in Tirole (1985), we de ne bubble assets as those assets that do not generate any payo or dividend. However, under some conditions, the prices of bubble assets become positive, which means that bubbles arise in equilibrium. Here, following Weil (1987), we consider stochastic bubbles, in the sense that they may collapse. In each period, bubble prices become zero (i.e., bubbles burst) at a probability of 1 conditional on survival in the previous period. When is lower, the bursting probability is higher. Once 2 On an equilibrium path, q t+1 is not a ected by the collapse of bubbles. Hence, there is no uncertainty with regard to q t+1 : 5

bubbles collapse, they do not arise again unless agents change their expectations about their formation through, for example, unexpected shocks. Reappearance of bubbles is not expected. Let P x t be the per unit price of bubble assets at date t in terms of consumption goods. P x t = P t > 0 if bubbles survive at date t with probability, and P x t = 0 if they collapse at date t with probability 1 : As we will show below, P t is determined endogenously in equilibrium. The entrepreneur s ow of funds constraint is given by (4) c i t + (1 s t)z i t + (1 + )P x t x i t = q t i t 1z i t 1 r t 1 b i t 1 + b i t + P x t x i t 1 + m i t: where x i t is the level of bubble assets purchased by a type i entrepreneur at date t. The left hand side of (4) is expenditure on consumption, investment, and the purchase of bubble assets. The right hand side is the available funds at date t, which is the return from investment in the previous period minus debts repayment, plus new borrowing, the return from selling bubble assets, and bailout money, m i t. is a tax on bubble assets, and s t is a subsidy to investments. Following Greenwald and Stiglitz (1986) and the recent literature on macro-prudential regulation (Gertler et al. 2012; Korinek, 2012; Lorenzoni, 2008), we consider macro-prudential regulation as a tax/subsidy policy. When bubbles collapse at the beginning of date t, all the wealth invested in bubble assets is wiped out. This decreased wealth of entrepreneurs lead to contractions during the bursting of bubbles. To mitigate these contractions, government bails out entrepreneurs, but, not all entrepreneurs are necessarily rescued. Following Hirano et al. (2012), we assume that only a certain proportion 2 [0; 1] of the entrepreneurs who su er losses from bubble investments are rescued. = 0 means that no-entrepreneurs are rescued, while = 1 means that all are rescued. A rise in 6

means expansions in the government s nancial safety net. This bailout scheme suggests that from an ex-ante perspective, each entrepreneur anticipates government bailouts with a probability : When entrepreneur i is rescued, we assume that the government guarantees bubble investments against losses and that the bailout is proportional to the entrepreneur s holdings of bubble assets: (5) m i t = ~ d i tx i t 1: Here, we speci cally consider bailouts that fully guarantee bubble investments against losses. In other words, government fully ensures the rate of return on bubble assets, but only a certain proportion of entrepreneurs are rescued. Hence, ~ d i t = P t > 0 if the agent i is rescued when bubbles collapse at t: Otherwise m i t = ~ d i t = 0: 3 The main reason for our approach is analytical tractability. In our setting, we can solve dynamics analytically and derive analytical solutions explicitly. We de ne the net worth of the entrepreneur at date t as e i t q t i t 1z i t 1 r t 1 b i t 1 + P x t x i t 1 + m i t: We also impose the short sale constraint on bubble assets: (6) x i t 0: Let us now turn to the maximization problem of workers. There are workers with a unit measure. Each worker is endowed with one unit of labor endowment in each period, which is supplied inelastically in labor markets, and earns wage income, w t. Workers do not have investment opportunities, and cannot borrow against their 3 As discussed in Hirano et al. (2012), we examine this type of partial bailouts, but we may be able to consider other types of partial bailouts. We can easily imagine, for example, a bailout policy in which government guarantees only a part of bubble investments against losses for all bubble holders. The main reason for our approach is analytical tractability. In our setting, we can solve dynamics analytically and derive analytical solutions explicitly. Even if we consider more general bailout policies, our qualitative results which will be explained would be una ected. 7

future labor incomes. The ow of funds constraint, and the short sale constraint for them are given by (7) c u t + (1 + )P x t x u t = w t r t 1 b u t 1 + b u t + P x t x u t 1 T u t ; (8) x u t 0; where u stands for workers. When bubbles collapse, government levies a tax T u t (lump sum tax or labor income tax) on workers and transfers those funds to entrepreneurs who su er losses from bubble investments. 4 This means that workers are taxpayers and incur the direct costs of the bubble bursts. Thus, T u t bubbles collapse with probability 1, while T u t = T t > 0 when = 0 if they survive with probability. As in Farhi and Tirole (2012a), the aim of this transfer policy (i.e., bailout policy) is to boost the net worth of entrepreneurs. In our model, this increased net worth can mitigate the adverse e ects of the collapse of bubbles. 5 In the section? and in the Appendix?, we also consider cases where government nances bailout by using government debt or by taxing entrepreneurs as well. Lastly, we explain the nal production technology. There are competitive rms which produce nal consumption goods using capital and labor. The production function of each rm is (9) y t = kt n 1 t k t ; 4 Since we assume that workers supply one unit of labor inelastically in each period, labor income tax is equivalent to a lump sum tax. 5 Let us mention the main reason behind the transfer policy from workers to entrepreneurs. In our model, as long as the government transfers resources among entrepreneurs, the aggregate wealth of entrepreneurs does not increase. As a result, economic contractions following the collapse of bubbles are not mitigated. The transfer policy from workers to entrepreneurs, however, increases the aggregate wealth of entrepreneurs and mitigates such contractions. See Appendix for details. 8

where y t is output of each rm, and k t K t =N t is an average capital stock in the economy, capturing externality, where K t and N t are aggregate capital stock and aggregate labor input. This production function suggests that production of other rms a ects my own production. If production level of other rms is high (low), this has positive (negative) spillover e ect on my own production. 6 We take this production function as a reduced form to capture interconnectedness in production among rms, for example, production network (see Acemoglu et al. 2010, 2014a, 2014b for microfoundation). < 1 is the degree of externality, i.e., if is large (small), externality is large (small). 7 Considering the equilibrium where k t = k t holds, factors of production are paid their marginal product: (10) q t = K + 1 t and w t = (1 )K + t ; where K t is the aggregate capital stock at date t. 2.2 Equilibrium Let us denote the aggregate consumption of H-and L-types and workers at date t as P i2h t c i t C H t, P i2l t c i t C L t, C u t ; where H t and L t mean a family of H-and L-types at date t. Similarly, let Y t ; P i2h t z i t Z H t ; P i2l t z i t Z L t ; P i2h t b i t B H t ; P i2l t b i t B L t ; B u t ; ( P i2h t[l t x i t + P x u t ) X t be the aggregate output, the aggregate investments of each type, the aggregate borrowing of each type, and the aggregate demand for bubble assets. Then, the market clearing condition for goods, 6 For example, during booms (recessions), average production level of the economy is high (low), which has a positive (negative) spillover e ect on my own production. 7 When = 1 ; the model is endogenous growth model. For simplicity, we exclude this case in our analyses. 9

credit, capital, labor, and bubble assets are (11) C H t + C L t + C u t + Z H t + Z L t = Y t ; (12) B H t + B L t + B u t = 0; (13) K t = X i2h t[l t k i t; (14) N t = 1; (15) X X x i t + x u t = X: i2h t[l t The competitive equilibrium is de ned as a set of prices fr t ; w t ; Pt x ; q t g 1 t=0 and quantities Ct H ; Ct L ; Ct u ; Bt H ; Bt L ; Bt u ; Zt H ; Zt L 1 ; X t ; K t+1 ; Y t ; N t, such that (i) the mar- t=0 ket clearing conditions, (11)-(15), are satis ed in each period, and (ii) each entrepreneur chooses consumption, borrowing, investment, and the amount of bubble assets, fc i t; b i t; zt; i x i tg 1 t=0 ; to maximize his/her expected discounted utility (1) under the constraints (2)-(6), taking into consideration the bursting probability of bubbles and the bailout probability, and (iii) each worker chooses consumption, borrowing, and the amount of bubble assets, fc u t ; b u t ; x u t g 1 t=0 ; to maximize his/her expected discounted utility (1) under the constraints (7)-(8), taking the bursting probability into consideration, and (iv) government budget for macro-prudential policy is balanced 10

in each period: (16) P x t X = s t(z H t + Z L t ) where the left hand side is tax revenues, and the right hand side is expenditures on investment subsidy. 8 Also, government budget for bailout policy is balanced, i.e., T u t = P i2h t[l t P x t x i t 1: 2.3 Optimal Behavior of Entrepreneurs and Workers We now characterize the equilibrium behavior of entrepreneurs and workers. In the bubble economy, we focus on the equilibrium where q t+1 L =(1 s t) r t < q t+1 H =(1 s t): In equilibrium, interest rate must be at least as high as q t+1 L =(1 s t), since nobody lends to the projects if r t < q t+1 L =(1 s t). Moreover, if the interest rate is higher than the rate of return of H-projects, nobody borrows. Since the utility function is log-linear, each entrepreneur consumes a fraction 1 of the net worth in each period, that is, c i t = (1 )e i t. 9 For H-types at date t, the borrowing constraint (3) is binding since r t < q t+1 H =(1 s t) and the investment in bubbles is not attractive, that is, (6) is also binding. We will verify this result in the Technical Appendix. Then, by using (3), (4), and (6), the investment function of H-types at date t can be written as 8 We may be able to think another tax/subsidy policy. For example, tax revenues can be used as a lump sum transfer to workers. But, this generates negative e ects on H-investments. MORE DETAILS 9 See, for example, chapter 1.7 of Sargent (1988). 11

(17) zt i = (q t i t 1zt i 1 r t 1 b i t 1 + Pt x x i t 1 + m i t) : 1 s q t+1 H t From this investment function, we understand that for the entrepreneurs who purchased bubble assets in the previous period, they are able to sell those assets at the time they encounter H-projects. By selling bubble assets, their net worth increases, which boosts their investments. Thus, bubbles generate a crowd-in e ect on productive investments. In our model, the entrepreneurs buy bubble assets for speculative purpose. They buy bubble assets when they have L-projects, and sell those assets to L-types when they have opportunities to invest in H-projects. r t For L-types at date t, since c i t = (1 )e i t; the budget constraint (4) becomes (18) (1 s t)z i t + (1 + )P x t x i t + ( b i t) = e i t: Each L-type allocates his/her savings, e i t; into three assets, i.e., (1 s t)z i t; (1 + )P x t x i t; and ( b i t): Each L-type chooses optimal amounts of b i t; x i t; and z i t so that the expected marginal utility from investing in three assets is equalized. By solving the utility maximization problem explained in the Technical Appendix, we can explicitly derive the demand function for bubble assets of a L-type: (19) P t x i t = 1 1 + () P t+1 P t (1 + )r t P t+1 e i t; P t (1 + )r t where () + (1 ): From (19), we learn that an entrepreneur s portfolio decision depends on macro-prudential regulation (), and expectations about government bailouts (). A rise in reduces entrepreneur s demand for bubble assets, 12

while a rise in encourages entrepreneur s risk-taking to buy bubble assets. The remaining fraction of savings is split across (1 s t)z i t and ( b i t) : (1 s t)zt i + ( b i t) = [1 ()] P t+1 P t P t+1 e i t: P t (1 + )r t Since investing in L-projects (z i t) and secured lending to other entrepreneurs ( b i t) are both safe assets, z i t 0 if r t = q t+1 L =(1 s t); and z i t = 0 if r t > q t+1 L =(1 s t): Moreover, when r t = q t+1 L =(1 s t); investing in L-projects and secured lending to other entrepreneurs are indi erent for L-types, aggregate investment level of L- types, Zt L ; is determined from (11). Next, regarding the optimal behavior of workers, we restrict our analysis to the equilibrium where workers do not save. We verify in the Technical Appendix that there exist parameter values where workers indeed do not save. 10 That is, c u t = w t T u t : 2.4 Dynamics We are now in a position to derive aggregate dynamics in the bubble economy. Since we assume that rational bubbles are stochastic, that is, bubbles persist with probability (< 1), here, we focus on the dynamics until bubbles collapse, i.e., 10 Since the equilibrium interest rate becomes relatively low because of the borrowing constraint, saving through lending or buying bubble assets is not an attractive behavior for workers. Thus, we can prove that they consume all the wage income in each period unless there is no bailout policy. On the other hand, workers might save to smooth their consumption if government uses a bailout policy. This is because if bubbles collapse, workers have to pay a tax to rescue entrepreneurs, which lowers their consumption, while if bubbles do not collapse, they do not have to pay it. So, consumption will be more volatile compared to the case without a bailout policy. Thus, workers may save if there is a bailout policy. In this paper, however, we restrict our attention to an equilibrium where workers do not save. We verify in the Technical Appendix that there exist parameter values where workers indeed do not save even if there is a bailout policy. Also, this behavior of workers holds true under reasonable parameter values in our numerical examples. 13

P x t = P t > 0. By using aggregate consumption function, C H t + C L t = (1 )A t ; and the government budget constraint, (16), (11) can be written as (20) Z H t + Z L t + P t X = A t : Then, we have the evolution of aggregate capital stock: (21) 8 >< K t+1 = H 1 s t pa t + L pa t A t 1 s L t H H L P t X! if r t = q t+1 L =(1 s t); >: H [A t P t X] if r t > q t+1 L =(1 s t): where A t q t K t + P t X is the aggregate wealth of entrepreneurs at date t in the bubble economy, and P i2h t e i t pa t is the aggregate wealth of H-types at date t. (More details about aggregation of each variable will be explained in the Technical Appendix). When r t = q t+1 L =(1 s t); both H-and L-types may invest. The rst term and the second term of the rst line represent the capital stock at date t + 1 produced by H-and L-types, respectively. When r t > q t+1 L =(1 s t); only H-types invest. From (20), we know Z H t = A t P t X: ( P t X) in (21) captures a traditional crowd-out e ect of bubbles analyzed in Tirole (1985), i.e., the presence of bubble assets crowds savings away from investments. As long as r t > q t+1 L =(1 s t); the interest rate is determined by the credit market clearing condition (12), which can be written as pa t 1 s t q t+1 H r t + P t X = A t : 14

That is, the aggregate savings of entrepreneurs, A t ; ow to aggregate H-investments and bubbles. By de ning t P t X=A t as the size of bubbles and by using the relation s t = t =(1 t ); 11 we can rewrite the above relation as r t = q t+1 H (1 t ) 1 p (1 + ) t : It follows that r t increases with t, re ecting the tightness of the credit markets. Thus, the equilibrium interest rate is determined as L (22) r t = q t+1 Max 1 s t H (1 ; t ) 1 p (1 + ) t : In other words, r t = q t+1 L =(1 s t) if t < (), and r t = H (1 t ) 1 p (1+) t if t (), where () L (1 p) H (1+)( L H ) : 3 Dynamics of Rational Bubbles Next, we examine the dynamics of rational bubbles. From the de nition of t P t X=A t, t evolves over time as (23) t+1 = P t+1 P t A t+1 A t t : The evolution of the size of bubbles depends on the relation between the growth rate of wealth and the growth rate of bubbles. When we aggregate (19), and solve 11 Rewriting (20), we have Zt H + Zt L = (1 t )A t : Moreover, we have P t X = t 1 K + t t and A t = 1 1 K + t t : By using these relations, we can derive s t = t =(1 t ) from (16). 15

for P t+1 =P t ; then we obtain the required rate of return on bubble assets: (24) P t+1 P t = r t(1 + ) [1 p (1 + ) t ] ()(1 p) (1 + ) t ; where (1+)(1 p t )=[()(1 p) t ] is the risk premium on bubble assets. We learn that the required rate of return increases with higher : By using (22), (24), and the de nition of aggregate wealth of entrepreneurs, (23) can be written as (25) 8 (1 + ) [1 p (1 + ) t ] >< t+1 = 1 + H L ()(1 p) (1 + ) t L p (1 + )[1 ()](1 p) + H ()(1 p) (1 + ) t t t if t (); >: (1 + ) 1 ()(1 p) (1 )(1 + ) t t if t > (): Using this (25), we examine the sustainable dynamics of t. In order for bubbles to be sustainable, the following condition must be satis ed for any t: t < 1: Violation of this condition means explosion of bubbles. As examined in the literature (Tirole 1985; Weil 1989; Farhi and Tirole 2012b), dynamics of bubbles take three patterns. The rst one is that bubbles become too large and explode to t 1. This dynamic path cannot be sustained by this economy and thus, bubbles cannot exist in this pattern. The second pattern is that t becomes smaller over time and converges to zero. This path is called asymptotically bubbleless. In this dynamic path, the e ects of bubbles converge to zero. Hence, we exclude this path from our consideration as usual in the literature. The third pattern is that t converges to a positive value as long as the bubbles survive. This dynamic path is a saddle path which converges to a stochastic stationary-state with positive bubbles where all variables (K t ; A t ; q t ; r t ; w t ; P t ; t ) become constant over 16

time. In this paper, we concentrate on this saddle path equilibrium as usual in the literature (Tirole 1985; Weil 1989; Farhi and Tirole 2012b). By using t, (21) can be written as (26) 8 >< K t+1 = >: 1 t 1 (1 + ) t h (1+ H L L H p) L L (1 + ) t i K + t 1 t H [1 t ] K + t 1 t if t (); if t > (): As long as bubbles exist, the dynamics of K t+1 =K + t = K t+1 =Y t is an increasing function of t as long as t < (); and it becomes a decreasing function of t if t (). Intuitively, as long as the size of bubbles is small, i.e., t < (); both H-and L-types invest. An increase in the size of bubbles crowds L-projects out and crowds in H-projects, thus increasing capital stock. In t < (); the crowd-in e ect dominates the crowd-out e ect at the margin. We call this region small-sized bubble region. When the size of bubbles equals t = (); all L- projects are completely crowded out, and only H-types invest: If the size of bubbles becomes larger, i.e., t > (), even H-projects are crowded out, thus reducing capital stock. Large size bubble is harmful to production. In t > (); the crowdout e ect dominates the crowd-in e ect at the margin. Thus, we call this region large-sized bubble region. The dynamic system of this economy is characterized by (26) and (25). However, (25) is independent from K t and the dynamics of t is derived only by (25). From (25), we can derive that t must be constant over time unless t is asymptotically bubbleless. This means that on the saddle path equilibrium, wealth of entrepreneurs and bubbles grow at the same rate. More precisely, t = on the saddle path 17

equilibrium as long as bubbles persist and can be written as (27) 8 >< (; ) = 1 1 + () 1 + ( H L L )p H 1 + ( H L L H )p (1 + ) (1 p) if (); (1 p)[1 ()] (1 + ) >: ()(1 p) (1 + ) (1 )(1 + ) if (): It follows that decreases with ; and increases with : That is, equilibrium bubble size is reduced by macro-prudential regulation, and is increased by expectations about government bailouts. Also, if equilibrium bubble size gets larger (smaller), then bailout money in relation to output also increases (decreases) 12. By using (27), we can derive the dynamics of K as a simple formulation: (28) K t+1 = H(; )K + t ; with 8 >< H(; ) = >: 1 t 1 (1+) h(1+ H L i L H p)l L (1+) 1 ; if (); H [1 ] 1 ; if (): and is given (27). Given an initial K 0 is given, the bubble economy runs according to (28) and converges toward the stochastic stationary state as long as bubbles persist. An 12 When we solve for aggregate bailout money in relation to output (i.e., GDP ratio), we have T u t Y t = P tx (; ) = Y t 1 (; ) : It follows that the ratio is an increasing function of ; and a decreasing function of : 18

important point is that H is independent of time t. From this property, we can characterize the dynamics of K simply. We see that bubbly dynamics depends on equilibrium bubble size, which in turn depends on macro-prudential regulation, ; and expectations about government bailouts, : 4 Laissez-Faire Economy Here let us examine what happens if there is no government policy, i.e., = = s t = 0 for all t. We rst characterize the existence condition of stochastic bubbles. In other words, we investigate whether a dynamic path with bubbles does not explode. Mathematically, we check whether the dynamic system (25) has a non-negative steady-state, t =. As we show below, the nancial market condition,, is crucial to the existence condition of bubbles. (Hereafter, proofs of all Propositions are given in Appendix). Proposition 1 In the laissez-faire economy, i.e., = = s t = 0 for all t, stochastic bubbles with survival probability can exist if and only if satis es the following condition, L [ L + ( H L )p] max ; 0 < < (1 p): H (1 ) From this Proposition, we can understand that bubbles tend to exist when the degree of nancial imperfection,, is in the middle range. In other words, improving nancial market conditions might enhance the possibility of bubbles when the initial condition of is low. 13 Intuitively, if is low, enough resources cannot be transferred 13 This result is similar to the result in Hirano and Yanagawa (2010) who characterize the existence condition of bubbles in an endogenous growth framework. 19

to productive sector, because the borrowing constraint is su ciently tight. As a result, economic growth rate with bubbles becomes low. On the other hand, interest rate can not be lower than the rate of return on L-projects and bubbles grow faster than interest rate because of the risk premium. As a result, under a very low, bubbles growth rate becomes higher than the economic growth rate, and the economy cannot support growing bubbles. Hence, the bubbles are unlikely to arise under a very low. 14 Moreover, within the bubble regions, there are two di erent regions, i.e., smallsized bubble region and large-sized bubble region. As Hirano et al. (2012) showed, the relation between the bubble size and capital stock is non-monotonic like Figure 1. In other words, there is the bubble size, = ( = 0); that maximizes capital stock and output for any t (before and after the collapse of bubbles). An important point is that in the laissez-faire economy, the bubble size on the saddle path equilibrium may be larger or smaller than ( = 0): More speci cally, when we solve for the equilibrium bubble size without government policy, we have (29) 8 >< () = 1 + ( H L L )p 1 1 H + ( H L L )p (1 p)(1 ) 1 H (1 p) < ( = 0) if < < ; >: (1 p) (1 ) > ( = 0) if < < ; 14 Here let us explain why the externality parameter does not a ect the existence condition of asset bubbles. As usual in the literature (Tirole 1985; Weil 1989; Farhi and Tirole 2012b), whether bubbles can arise depends on the relation between the economy s growth rate and bubbles growth rate (or interest rate in the case of deterministic bubbles). In our model, since a ects both growth rates through q t+1 in the same direction, the e ects on on both growth rates are cancelled out. That s why the externality parameter does not a ect the existence condition. 20

I.e, equilibirum bubble size depends on the quality of the nancial system, where is given in Appendix. We summarize (29) in Proposition 2. Proposition 2 (i) In the laissez-faire economy, i.e., = = s t = 0 for all t; if < < ; equilibrium bubble size is smaller than ( = 0): In other words, within the bubble regions and if the quality of the nancial system is relatively low, small-sized bubbles arise. (ii) In the laissez-faire economy, i.e., = = s t = 0 for all t, if < < ; equilibrium bubble size is larger than ( = 0): In other words, within the bubble regions and if the quality of the nancial system is relatively high, large-sized bubbles arise. Proposition 2 suggests that in advanced economies where the quality of the nancial system tends to be better than that of emerging enomomies, large-sized bubbles are more likely to arise, while in emerging economies, small-sized bubbles are more likely to emerge. Intuitively, within the bubble regions and if is relatively high, enough savings can be transferred to H-projects even without bubbles, because the borrowing constraint is relatively loose. In this situation, when bubbles appear, bubbles crowd out even H-projects at the margin as well as crowd out L-projects completely. On the other hand, within the bubble regions and if is relatively low, L-types cannot lend all of their savings to H-types even with bubbles, because the borrowing constraint is su ciently tight. As a result, L-types hold a lot of idle savings, but they don t want to invest all of their idle savings into bubble assets, because bubbles are risky and they are risk-averse agents. In equilibrium, they end up investing in their own safe projects with low returns for risk-hedge. This means that in aggregate, some of the savings in the economy ow to L-projects. 21

Moreover, together with Proposition 2, we can characterize whether bubbles increase output compared to output in the bubbleless economy, and how those e ects are related to the quality of the nancial system. Proposition 3 In the laissez-faire economy, i.e., = = s t = 0 for all t, (i) if < < ; small-sized bubbles arise and given an initial Y 0 ; output in the bubble economy is higher than that in the bubbleless economy for any t 1; (ii) if < < 1 ; large-sized bubbles arise and given an initial Y 0 ; output in the bubble economy is higher than that in the bubbleless economy for any t 1, and (iii) if 1 < < ; large-sized bubbles arise and given an initial Y 0 ; output in the bubble economy is lower than that in the bubbleless economy for any t 1. Figure 2 illustrates Proposition 2 and 3. 15 In Figure 2, we compare output in the stochastic stationary state of the bubble economy with output in the steady-state of the bubbleless economy. In the Appendix, we provide a full characterization of the bubbleless economy and derive 1. Proposition 3 shows that in < < 1 ; bubbles increase output compared to output in the bubbleless economy, while in 1 < < ; bubbles decrease output. Intuitively, if the quality of the nancial system is relatively low and if there is no bubble, enough savings cannot be transferred to H-types and even L-types end up producing in equilibrium. In this situation, when bubbles arise, bubbles crowd out L-projects and crowds in H-projects, thereby increasing output. On the other hand, if the quality of the nancial system is relatively high, even without bubbles, the nancial system can allocate enough funds to H-projects. In this situation, bubbles end up with crowding out H-projects largely, thus reducing output. 15 Hirano and Yanagawa (2010) provide a full characterization on the relation between bubbles and long-run economic growth rate in an endogenous growth framework. 22

In the rest of our paper, we restrict our analyses to < < 1 where bubbles increase output compared to output in the bubbleless economy. 5 Welfare E ects for Taxpayers In this section, we conduct a full welfare analysis including transitional dynamics, and derive optimal policy for workers (i.e., taxpayers). 5.1 Ex-ante welfare Let us rst consider welfare e ects of bailouts. There are two competing e ects, i.e., positive and negative e ects. The rst negative e ect is that when bubbles collapse, workers have to pay tax to rescue entrepreneurs, which lowers their consumption. On the other hand, because of the bailouts, the net worth of the rescued entrepreneurs increases and their investments expand compared to the no-bailout case. This thereby increases wage income and workers consumption by expanding output. These are the ex-post e ects of bailouts. Moreover, expectations about bailouts increase equilibrium bubble size and enhance the crowding-out e ects of bubbles on productive investments, thereby reducing capital stock. This lowers workers lifetime wage income, because the decrease in capital stock during the bubble period has persistent e ects on wage income, i.e., not just before the bubble bursts, but also after the bursts. Whether bailouts are good for taxpayers from an ex-ante perspective depends on these competing e ects. In this paper, ex-ante means before the bubble breaks and ex-post means after the bubble breaks. Let us compute ex-ante welfare for taxpayers. Let Vt BB (K t ) be the value function of taxpayers at date t in the bubble economy. Let Vt BL (K t ) be the value function 23

of taxpayers at date t when bubbles collapse at date t. Given the optimal decision rules, the Bellman equation can be written as Vt BB (K t ) = log c t + Vt+1 BB (K t+1 ) + (1 )Vt+1 BL (K t+1 ) ; where c t = w t as long as bubbles persist, and V BL t follows Vt BL (K t ) = log c t + Vt+1 BL (K t+1 ); with c t = w t T u t when bubble collapses at date t; c t = w t after date t + 1: T u t = P t X = (;) 1 (;) K+ t is bailout money per unit of workers. Solving the value function yields (see the Technical Appendix for derivation.) (30) V BB t (K t ) = 1 1 ( + ) 1 ( + ) log H(; ) + 1 + log(1 ) 1 (1 ) ( + ) M(; ) + 1 1 ( + ) log K t; with H(; ) = H [1 (; )] ; 1 (; ) and (; ) M(; ) = log 1 1 (; ) ( + ) + 1 ( + ) log (; ) 1 + 1 (; ) ( + ) 1 + 1 1 ( + ) 1 log + H L L p L + log(1 ): H 1 24

Given the date-t capital stock, K t ; taxpayers welfare is composed of a weighted average of two e ects. H(; ) and (1 ) in the rst line of (30) capture welfare e ects from capital stock before the bubble bursts. M(; ) in the second line is welfare e ect after the bursts. Since we compute expected discounted welfare, both two welfare e ects are weighted by the survival rate of bubbles and time preference rate. More precisely, a change in H(; ) a ects productive investments and capital stock during the bubble period, which in turn a ects workers wage income not just before the collapse of bubbles, but also overall welfare after the collapse of bubbles through wage income and bailout money. The rst term in the rst line includes all of these e ects. On the other hand, the rst term in M(; ) captures the welfare e ect from paying bailout money, while the second term captures the welfare e ect of bailouts on wage income after the bursts. The remaining terms in M(; ) capture welfare e ect from wage income independent of the e ects of bailouts. We should mention that (30) includes transitional dynamics, given an initial capital stock (for example, the initial capital stock can be set to the capital stock level in the steady-state of the bubbleless economy). Thus, by using (30), we can compare taxpayers welfare under di erent policies, given the same initial capital stock K 0 : An important point is that both H(; ) and M(; ) are a ected by and ; i.e., macro-prudential policy and expectations about bailouts a ect taxpayers welfare through H(; ) and M(; ): When both H(; ) and M(; ) are the highest level, then taxpayers welfare also reaches the highest level. In this paper, we call H(; ) ex-ante welfare e ect and M(; ) ex-post welfare e ect. 25

5.2 Ex-ante Optimum: when government can commit to future bailout policy Suppose the government can fully commit to future bailout policy. Then, the objective of the government is to maximize V BB t by choosing : If the government can commit, both entrepreneurs and workers rationally expect the government chooses. Di erentiating (30) with respect to yields (31) @V BB t (K t ) @ = 1 1 ( + ) @ log H(; ) + 1 ( + ) @ (1 ) 1 @M(; ) : @ (31) says that whether bailout expansions increase taxpayers welfare depends on the marginal expected ex-ante welfare e ect of bailouts (the rst term) and the marginal expected ex-post welfare e ect of bailouts (the second term). Firstly, @ log H(;) @ < 0; i.e., expectations about bailouts increase the crowd-out e ect of bubbles on capital stock and lower wage income. Secondly, @M(; ) sgn @ = sgn ( + ) (; ) 1 (; ) ; where (; ) is given in (??). The sign of @M(;) @ depends on two competing expost e ects of bailouts, i.e., the marginal increase in wage income (bene ts) and the marginal increase in bailout money (costs). The increase in wage income which corresponds to ( + ) becomes larger with externality, because the spillover e ect in production is larger. The amount of bailout money which corresponds to + (;) increases with bailout level ; because a rise in increases equilibrium 1 (;) bubble size and proportion of entreprenuers rescued by the government. Whether 26

M(; ) increases with depends on : The following Lemma summarizes it. 16 Lemma 1 (a) If 0 ( 1 < 0 for 0: (b) If ( 1 ) < < @ min ( 1 ) + (1 p) ; 1 ; there is a critical value of = ^ 2 (0; 1) that 1 (1 p) ); @M(;) maximizes M(; = 0): That is, @M(;=0) @ ^ < 1: > 0 in 0 < ^; and @M(;=0) @ < 0 in Let us remark one thing here. Since our main focus is to examine why macroprudential regulation is needed, in all Propositions and Lemmas until subsection 6.5, we rst examine the e ects of bailouts under = 0 if the degree of externality is large, i.e., macro-prudential regulation is not initially undertaken, and show that without macro-prudential regulation, bailout policy generates welfare loss. And then, in subsection 6.5, we show that macro-prudential regulation can solve the welfare loss. If the degree of externality is small, we characterize the e ects of bailouts under 0; because all results remain unchanged for 0: Lemma 1 and @ log H(;) @ < 0 lead to the following Proposition. Proposition 4 Let c denote the value of which maximizes ex-ante taxpayers welfare when government can commit to future bailout policy. Then: (a) If 0 ( 1 ); @V BB t (K t) @ < 0 for 0: Thus, c = 0: That is, if the degree of externality is su ciently small, no-bailout is optimal for taxpayers. (b) If ( 1 @V BB ) < < and = 0; t (K t) < 0 in ^ h < 1: Thus, @ c 2 0; ^ : That is, if the degree of externality is large and when macro-prudential regulation is not undertaken, partial bailout can be optimal for taxpayers. 16 Results in the case where the degree of externality is su ciently large, i.e., ( 1 ) + (1 p) 1 (1 p) < 1 ; are presented in the Appendix. 27

Intuitively, when the degree of externality is su ciently small, even if the government does not rescue any entrepreneurs after the collapse of bubbles, output decrease is small, because externality in production is small. In this case, the bene ts of government bailouts are relatively small compared to the costs of bailouts, i.e., @M(;) @ < 0 for > 0: Moreover, expectations about bailouts lead to larger bubbles, which crowds out productive investments, lowering wage income. That is, @ log H(;) @ < 0: As a result, no-bailout is optimal from an ex-ante perspective when the degree of externality is su ciently small. If the degree of externality is large, the bene ts of bailouts become large enough after the bursts, because output and wage income decrease largely without bailouts. Thus, bailouts can increase taxpayers welfare after the bursts up to ^, i.e., @M(;=0) @ > 0 in 0 < ^: On the other hand, @ log H(;) @ < 0; i.e., expectations about bailouts crowd out productive investments and lower wage income. Two competing e ects operate in 0 < ^: If the former e ect dominates the latter e ect, then bailouts increase welfare, and thus partial-bailout can be optimal. Figure 4 presents a numerical example in case (b) on the relation between government bailouts and ex-ante taxpayers welfare. Parameter values are set as follows: H = 1:5; L = 1:0; = 0:98; = 0:35; = 0:36; p = 0:35; = 0:2; = s t = 0: Figure 4 shows that under these parameter values, no-bailout is not optimal, and full-bailout is not optimal, neither. Partial bailout is optimal. Thus, if government can fully commit to future bailout policy, then government chooses c. We should mention that c is time-invariant. 28

5.3 Ex-post optimum: when government cannot commit to future bailout policy In the previous subsection, we assumed that government can fully commit to future bailout policy. In that case, the government can achieve c ; and both entrepreneurs and workers expect c rationally. However, from an ex-post perspective (i.e., after the collapse of bubbles), the optimal bailout level may become di erent from c : If that is the case, then it may be di cult for government to commit to c, because after the bubble breaks, the government may renege on precommited policy c : A time-inconsistency problem arises. In this subsection, we examine the case where the government cannot commit. For this purpose, let us consider the optimal bailout from an ex-post persepective. Suppose that at date s (for any s); the bubble collapses (i.e., after date s; the economy is bubbleless). Let ex post denote the value of which maximizes taxpayers welfare at date s: Also, let e denote the bailout level both entrepreneurs and workers expected until date s 1: When we compute ex post at date s, we need to take ( e ; ) as given. This is because, after the bubble collapses, the purchase price of bubble assets is predetermined, in other words, the size of bubbles is predetermined. Given the purchase price of bubble assets, (i.e., given ( e ; )), we re considering how many proportions of entrepreneurs should be rescued at date s. Then, the objective of the government at date s is to maximize V BL s by choosing : (32) Max Vs BL ( + ) (K s ) = M() + 1 ( + ) log K s; (33) subject to 0 ex post 1, given ( e ; ) in M() 29

Di erentiating (32) with respect to yields depends on dv BL s (K s) d = dm() d ; and the sign of dm() d sgn dm() d = sgn ( + ) (e ; ) 1 ( e : ; ) Whether bailouts can increase taxpayers welfare at date s (i.e., from an ex-post perspective) depends on ( + ) and + (e ;) ; which are the marginal 1 ( e ;) increase in wage income and the marginal increase in bailout money, respectively. 17 Proposition 5 Optimal bailout from an ex-post perspective: (a) If 0 ( 1 ); ex post = 0 for 0; i.e., if the degree of externality is su ciently small, no-bailout is optimal for taxpayers after the collapse of bubbles. (b) If ( 1 ) < < and = 0; ex post 2 (0; 1] ; i.e., if the degree of externality is large, bailouts can increase taxpayers welfare after the collapse of bubbles. An important point is that ex post does not necessarily become identical to e. If ex post 6= e ; this means that the government reneges on precommited policy after the bubble breaks. A time-inconsistency problem arises. In such a situation, unless government has commitment technology, government commitment (or annoucement) is not credible. Figure 4 presents an example of this situation. Figure 4 shows that from an ex-ante perspective, c is desirable. But, after the bubble breaks, bailing out entrepreneurs up to ex post ( c ) > c is desirable. Thus, without commitment technology, c is not credible. 17 Let us mention the relation to Bernanke s statement. When Bernanke was asked about the reasons Fed rescued wall street, Bernanke said "I care about the wall street for one reason only. Because what happed in the wall street matters for the main street". If we interpret taxpayers who are non-bubble holders as main street people, and bubble-holders as wall street people, then Bernanke s statement can be justi ed if and only if the degree of externality is large in our model. 30

5.4 Credible equilibrium Here we consider credible equilibrium. Let nc denote credible equilibrium: Credible equilibrium satis es ex post ( nc ) = nc = e, i.e., government has no incentive to renege on precommited policy even after the bubble breaks, and entrepreneurs and workers expectation about bailouts is consistent with rational expectation. For example, suppose that bubbles arise and persisted until date s 1 under the expectation with e = nc ; and then bubbles collapse at date s. Under the credible equilibrium, the optimal policy at date s is indeed equivalent to nc. The following Proposition characterizes the credible equilibirum: Proposition 6 (a) If 0 ( 1 ); nc = 0 for 0: That is, if the degree of externality is su ciently small, no-bailout is credible policy. (b) If ( 1 the following two equations: ) < < and = 0; there exists a unique nc 2 (0; 1) that satis es (34) ( + ) = ex post ( e ; = 0) 1 ( e ; = 0) and ex post = e ; where nc is the positive root of the above quadratic equation. The rst equation of (34) is government optimal reaction at the time bubbles collapse, i.e., given e (i.e., given ( e ; = 0)); government determines ex post to satisfy the rst equation. The second equation is the condition of rational expectation equilibrium. Credible equilibrium nc is obtained as a xed point of the two equations. Intuitively, if the degree of externality is su ciently small, no-bailout is optimal after the bubble bursts from Proposition 5. Moreover, as Proposition 4 shows, no-bailout is optimal even from an ex-ante perspective. Thus, there is no timeinconsistency problem and no-bailout is credible policy. 31

On the other hand, if the degree of externality is large, unless = nc 2 (0; 1) ; the government always has incentives to renege on precommited policy after the bubble breaks. In other words, unless = nc 2 (0; 1) ; a time-inconsistency problem always arises. Thus, = nc 2 (0; 1) is the only credible equilibrium consistent with rational expectation. Moreover, when ( 1 can show This means ) < < and under no macro-prudential regulation, we @V BB t @ =nc =0 > 0: Vt BB ( c ( = 0); = 0) > Vt BB ( nc ( = 0); = 0): I.e., without macro-prudential policy, the credible nc is an excessive bailout from an ex-ante perspective. In other words, too large bubble is created under the credible bailout policy. The di erence between Vt BB ( c ( = 0); = 0) and Vt BB ( nc ( = 0); = 0) is welfare loss arising from government s inability to commit to future bailout policy. This is one of the reasons macro-prudential regulation is needed. That is, macro-prudential regulation can mitigate the welfare loss. To put it another way, under the credible equilibrium without macro-prudential regulation, although the welfare e ect after the bursts, M( nc ( = 0); = 0); is maximized, the government cannot internalize the welfare e ect before the bursts, H( nc ( = 0); = 0). The credible bailout policy generates distortions in H( nc ( = 0); = 0) by creating large bubbles. As a result, welfare loss arises. By using macroprudential policy, the government can indeed internalize the welfare e ect before the bursts, and can correct the distortions produced by a large bubble, thereby improving welfare. We will show this point clearly in the next subsection. 32