Bubbles and Credit Constraints Jianjun Miaoy

Pengfei Wangz

October 14, 2015

Abstract We provide a theory of stock price bubbles in production economies with in…nitely lived agents. Firms meet stochastic investment opportunities and face endogenous credit constraints. They have limited commitment to repay debt. Credit constraints are derived from incentive constraints in optimal contracts which ensure default never occurs in equilibrium. A stock price bubble can emerge through a positive feedback loop mechanism. It commands a liquidity premium and improves investment e¢ ciency because it raises debt capacity by relaxing incentive constraints. We provide conditions under which bubbles can coexist with other types of assets. We show that the collapse of stock price bubbles leads to a recession and a stock market crash. We propose a government policy that can eliminate bubbles and achieve e¢ cient allocation. Keywords: Stock Price Bubbles, Credit Constraints, Limited Commitment, Q Theory, Liquidity, Multiple Equilibria JEL codes: E2, E44, G1

We thank Bruno Biais, Jess Benhabib, Toni Braun, Markus Brunnermeier, Henry Cao, Christophe Chamley, Tim Cogley, Russell Cooper, Douglas Gale, Jordi Gali, Mark Gertler, Simon Gilchrist, Christian Hellwig, Hugo Hopenhayn, Andreas Hornstein, Boyan Jovanovic, Bob King, Nobu Kiyotaki, Anton Korinek, Felix Kubler, Kevin Lansing, John Leahy, Eric Leeper, Zheng Liu, Gustavo Manso, Ramon Marimon, Erwan Morellec, Fabrizio Perri, Jean-Charles Rochet, Tom Sargent, Jean Tirole, Jon Willis, Mike Woodford, Tao Zha, Lin Zhang, and, especially, Wei Xiong and Yi Wen for helpful discussions. We have also bene…tted from comments from seminar and conference participants at the BU macro lunch workshop, Cheung Kong Graduate School of Business, European University Institute, Indiana University, New York University, Toulouse School of Economics, CREI at University of Pompeu Fabra, University of Mannheim, University of Lausanne, University of Southern Denmark, University of Zurich, Zhejiang University, the 2011 Econometric Society Summer Meeting, the 2011 International Workshop of Macroeconomics and Financial Economics at the Southwestern University of Finance and Economics, the Federal Reserve Banks of Atlanta, Boston, San Francisco, Richmond, and Kansas, the Theory Workshop on Corporate Finance and Financial Markets at Stanford, Shanghai University of Finance and Economics, the 2011 SED conference in Ghent, and the 7th Chinese Finance Annual Meeting. First version: December 2010. y Department of Economics, Boston University, 270 Bay State Road, Boston, MA 02215. Tel.: 617-353-6675. Email: [email protected]. Homepage: http://people.bu.edu/miaoj. z Department of Economics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong. Tel: (+852) 2358 7612. Email: [email protected]

1

Introduction

This paper provides a theory of stock price bubbles in the presence of endogenous credit constraints. Our theory is motivated by two observations. First, ‡uctuations in observable fundamentals cannot adequately explain stock market booms and busts (Shiller (2005)). Second, stock market booms are often accompanied by credit market booms. For example, overoptimism in the 1990s towards an “East Asian miracle” generated booms in the housing and stock markets in some east Asian countries which were accompanied by lending booms and a large expansion of domestic credit (Collyns and Senhadji (2002)). Jordà, Schularick, and Taylor (2014) document empirical evidence on the relation between credit booms and asset price booms in 17 developed countries since 1870. They …nd that leveraged bubbles are more dangerous to the macroeconomy than other types of bubbles, e.g., unleveraged “irrational exuberance” bubbles. To formalize our theory, we construct a tractable continuous-time model of a production economy in which identical households are in…nitely lived and trade …rm stocks. There is no aggregate uncertainty. In the baseline model households are risk neutral so that the rate of return on any stock is equal to the constant subjective discount rate. A continuum of …rms meet idiosyncratic stochastic investment opportunities as in Kiyotaki and Moore (1997, 2005, 2008). Firms use intratemporal debt to …nance investment because of a liquidity mismatch (Jermann and Quadrini (2012)): investment spending must be paid before the realization of investment returns.1 Firms cannot raise equity or sell a lump-sum amount of capital to …nance investment. This assumption re‡ects the fact that equity …nancing is more costly than debt …nancing due to commitment problems or information asymmetries not explicitly modeled in the paper. Another interpretation following Kiyotaki and Moore (2005, 2008) is that investment opportunities disappear so quickly that …rms do not have enough time to raise equity or sell a large amount of capital. Firms face endogenous credit constraints, which are modeled in a similar way to Bulow and Rogo¤ (1989), Kehoe and Levine (1993), Kiyotaki and Moore (1997), Alvarez and Jermann (2000), Albuquerque and Hopenhayn (2004), and Jermann and Quadrini (2012). The key idea is that borrowers (…rms) have limited commitment and debt repayments are imperfectly enforced. We consider the following lending contract to ensure the repayment of debt. A …rm pledges its physical assets (capital) as collateral. If the …rm does not repay its debt, then it loses its collateralized assets and the right to run the …rm all to the lender. Thus the collateral value to the lender is equal to the market value of the …rm with the collateralized assets. The lender and the …rm renegotiate the debt such that the debt is limited by this collateral value. The resulting credit constraint is 1 We de…ne liquidity as the amount of money that is quickly available for investment. Sometimes we also refer to liquidity as the degree to which an asset can be quickly turned into cash. See Kiyotaki and Moore (2008), Farhi and Tirole (2012), and Vayanos and Wang (2012) for related studies of liquidity.

1

endogenously derived from the incentive constraint in an optimal contracting problem. Unlike Kiyotaki and Moore (1997) who assume that the collateral value is equal to the liquidation value of the collateralized assets, we derive the collateral value from the incentive constraint as the going-concern value of the reorganized …rm. Because the going-concern value is priced in the stock market, it may contain a bubble component. If both the lender and the credit-constrained …rms optimistically believe that the collateral value is high possibly because of bubbles, the …rms will borrow more and the lender will not mind lending more. Thus …rms can …nance more investment and make more pro…ts, making their assets indeed more valuable. This positive feedback loop mechanism makes the lender’s and the borrower’s beliefs self-ful…lling and allows bubbles to exist in equilibrium. We refer to this type of equilibrium as the bubbly equilibrium. Our credit constraint is equivalent to that endogenously derived from the incentive constraint in Gertler and Kiyotaki (2010) and Gertler and Karadi (2011). Suppose that there is no collateral to borrow. A …rm can default on debt by stealing a fraction of its assets. The incentive constraint in an optimal contract ensures that the continuation value of not defaulting is not lower than the outside value of the stolen assets. The outside value does not contain a bubble. But a stock price bubble can relax the incentive constraint by raising the continuation value of not defaulting, thereby relaxing the credit constraint.2 The aforementioned mechanism still works. There is another type of equilibrium in which no one believes in bubbles and hence bubbles do not exist. We call this type the bubbleless equilibrium. We provide explicit conditions to determine which type of equilibrium can exist. We prove that the economy has two steady states: a bubbly one and a bubbleless one. Both steady states are ine¢ cient due to credit constraints and both are local saddle points. The equilibrium around the bubbly steady state is unique and bubbles persist in the long run along a stable manifold. But the equilibrium around the bubbleless steady state has indeterminacy of degree one and bubbles eventually burst along a stable manifold. Thus multiple equilibria in our model are not generated by indeterminacy with a unique steady state as in the literature surveyed by Benhabib and Farmer (1999) and Farmer (1999). We extend our baseline model to include other types of assets such as intertemporal bonds, assets with rents/dividends (e.g., trees), and assets without rents (pure bubbles, e.g., tulips). Suppose that …rms can trade one of these assets to …nance investment. We study the conditions under which stock price bubbles or …rm bubbles can coexist with other types of assets. If an asset can play the same role as a …rm bubble in helping …rms …nance investment, then this asset will generate additional dividends to the …rms, which are identical to the collateral yield. If, in addition, this asset delivers positive rents, then it dominates a bubble and hence the two cannot coexist in equilibrium. 2 See the online appendix for other types of credit constraints endogenously derived from optimal contracts that can generate a stock price bubble.

2

But if this asset is a pure bubble, then it is a perfect substitute for the …rm bubble. Only the total size of the bubble can be determined in equilibrium. For a …rm bubble to coexist with intertemporal bonds, the equilibrium interest rate on the bonds must be zero in the steady state. We also need to introduce market frictions such as short-sales constraints on the additional assets (Kocherlakota (1992, 2009) and Kiyotaki and Moore (2005, 2008)). Without market frictions, the economy would achieve the e¢ cient equilibrium and no bubble would exist. So far we have only considered deterministic bubbles. Following Blanchard and Watson (1982) and Weil (1987), we construct a third type of equilibrium with stochastic bubbles in the baseline model. In this equilibrium all agents believe that bubbles will burst at each date with a positive probability. When bubbles burst, they cannot reappear. We show that when all agents believe that the probability of bubble bursting is small enough, an equilibrium with stochastic bubbles exists. In contrast to Weil (1987), we show that after a bubble bursts, a recession occurs in that there is a credit crunch and consumption and output fall eventually. In addition, immediately after the bubble bursts, investment falls discontinuously and the stock market crashes. The recession and the stock market crash occur without any exogenous shock to the fundamentals of the economy. What is an appropriate government policy in the wake of a bubble collapse? The ine¢ ciency in our model comes from the …rms’ credit constraints. The collapse of bubbles tightens these constraints and impairs investment e¢ ciency. To overcome this ine¢ ciency, the government can issue public bonds backed by lump-sum taxes. Public bonds can provide liquidity to …rms and thus they can help relax credit constraints in the same way that bubbles do. A government policy of constantly retiring public bonds at the interest rate to maintain a …xed total bond value and levying lump-sum taxes to pay interest on these bonds will eliminate bubbles and allow the economy to achieve the e¢ cient equilibrium. Related literature Building on Samuelson (1958) and Diamond (1965), Tirole (1985) provides a theory of asset bubbles in the overlapping generations (OLG) framework. The key condition for the existence of a bubble is that the bubbleless equilibrium must be dynamically ine¢ cient. This condition is not supported by the empirical evidence reported by Abel et al. (1989). For a bubble to exist in dynamically e¢ cient economies, Caballero and Krishnamurthy (2006), Farhi and Tirole (2012), and Martin and Ventura (2012) introduce credit constraints to OLG models. Caballero and Krishnamurthy (2006) show that stochastic bubbles are bene…cial because they provide domestic stores of value, thereby reducing capital out‡ows while increasing investment. The downside is that they expose the country to bubble crashes and capital ‡ow reversals. Farhi and Tirole (2012) study the interplay between inside and outside liquidity by assuming that only a fraction of investment returns can be pledgeable. Bubbles and outside liquidity do not raise

3

debt capacity directly, but they can raise entrepreneurs’net worth enabling them to …nance more investment. The rise in investment then a¤ects debt capacity indirectly through a leverage e¤ect, which depends on the interest rate. The impact of bubbles on investment thus depends on the relative potency of a liquidity e¤ect and a leverage e¤ect. Martin and Ventura (2012) introduce investor sentiment shocks and study stochastic equilibria with bubble creation and destruction. They assume that productive investors cannot borrow from unproductive ones. When a market for pure bubbles opens, productive investors raise their net worth by selling bubbles to unproductive ones. Thus productive investors can make more investment, thereby raising investment e¢ ciency. Since pure bubbles (except for money, arts, etc.) are rarely traded in reality, Martin and Ventura (2012) reinterpret their benchmark model by allowing entrepreneurs to trade …rms in the stock market. Entrepreneurs can pledge an empty …rm as collateral. This modi…ed model is equivalent to their benchmark when the market value of the …rm is equal to the pure bubble value. In a similar model Martin and Ventura (2011) assume that entrepreneurs can start new …rms and non-entrepreneurs trade old …rms and give credit to new …rms. A …rm can borrow using a fraction of its investment returns and market value as collateral. A bubble in …rm value can emerge in addition to a fundamental component. Existing OLG models of bubbles are con…ned to two- or three-period lived agents. Although these models are tractable for delivering economic insights, they are not suitable for quantitative analysis because the time period in these models cannot be tied to the data frequency. Moreover, if each generation is equally altruistic toward the next generation with intergenerational transfers, then the entire dynasty behaves as a single in…nitely lived agent (Barro (1974)). Thus studying models with in…nitely lived agents is important and will deepen our understanding of asset bubbles by complementing OLG models. It is di¢ cult to generate rational bubbles for economies with in…nitely lived agents (Tirole (1982) and Santos and Woodford (1997)). A necessary condition for bubbles to exist is that their growth rate cannot exceed the growth rate of the economy. Otherwise, investors cannot a¤ord to buy into bubbles. In standard deterministic models bubbles on assets with exogenous payo¤s or on intrinsically useless assets must grow at the interest rate on these assets by the no-arbitrage principle using households’consumption Euler equations. Thus the interest rate cannot exceed the growth rate of the economy. For in…nitely lived agents, utility maximization implies a transversality condition for bubbles, which requires that the present value of bubbles be zero in the limit. This condition rules out bubbles in standard models with in…nitely lived agents. How can our model generate bubbles and how do we reconcile our result with that in Santos and Woodford (1997) or Tirole (1982)? The key is that stock price bubbles in our model are attached to productive assets (capital) with endogenous payo¤s. Our novel insight is that stock price bubbles

4

have real e¤ects and a¤ect …rm dividends. Although a consumption Euler equation holds for the representative risk-neutral household so that the stock return is equal to the subjective discount rate, the growth rate of stock price bubbles is not equal to this rate. Rather, it is equal to the discount rate minus a “collateral yield” or “liquidity premium.” The collateral yield comes from the fact that stock price bubbles help relax credit constraints in the …rms’ optimization problem and allow …rms to make more investment. Some studies (e.g., Scheinkman and Weiss (1986), Kocherlakota (1992, 2008), Santos and Woodford (1997) and Hellwig and Lorenzoni (2009)) …nd that in…nite-horizon models with borrowing constraints can generate rational bubbles. Unlike these studies which examine pure exchange economies, our paper analyzes a production economy with bubbles in stock prices whose payo¤s are endogenously determined by investment and a¤ected by bubbles.3 Our paper is closely related to some recent in…nite-horizon models of production economies (Woodford (1990), Kiyotaki and Moore (2008), Kocherlakota (2009), Wang and Wen (2012), and Hirano and Yanagawa (2013)). These models also contain the idea that bubbles can help relax credit constraints and raise investment. But our modeling of credit constraints is di¤erent from theirs in that our credit constraints are endogenously derived from the incentive constraints in the optimal contracts with limited commitment. Our novel insight is that certain types of credit constraints beyond the usual Kiyotaki-Moore collateral constraint can generate a stock price bubble because the bubble can raise …rm dividends by relaxing incentive constraints in optimal contracts and raising debt capacity. Credit constraints are not essential for the emergence of bubbles in the traditional OLG models, unlike in in…nite-horizon models. They simply allow bubbles to emerge in dynamically e¢ cient OLG economies in which the interest rate may be low. An important issue speci…c to in…nite-horizon models is that …rms may eventually overcome credit constraints by saving over time if they can buy su¢ ciently many (intertemporal) bonds from households. Thus one must impose borrowing constraints or short-sales constraints on households so that they cannot sell too many bonds. This generates a low interest rate on bonds. The spread between the stock return and the interest rate re‡ects the liquidity premium. By contrast, the stock return is equal to the interest rate in Farhi and Tirole (2011) and Martin and Ventura (2011, 2012). Rather than studying stock price bubbles, the extant literature typically studies pure bubbles on intrinsically useless assets (e.g., money) that can provide liquidity by raising the borrower’s net worth. For example, Kiyotaki and Moore (2008) assume that entrepreneurs can pledge a fraction of investment returns as collateral to borrow. Equity (capital) is illiquid and only a fraction can 3

See Scheinkman and Xiong (2003) for a model of bubbles based on heterogeneous beliefs and Adam, Marcet, and Nicolini (2015) for an asset-pricing model where agents have subjective beliefs about the pricing function. See Brunnermeier (2009) and Miao (2014) for surveys of various theories of bubbles.

5

be sold to …nance investment. They show that money as a pure bubble asset can circulate because it is more liquid than other assets and raises net worth. Building on their insights, Kocherlakota (2009) assumes that entrepreneurs use land as collateral. Land is traded as an intrinsically useless asset. Land bubbles can emerge and burst. Unlike these studies, our main contribution is to provide a theory of stock price bubbles in production economies. The distinction between a traded pure bubble and a stock price bubble is important because the latter is not directly observable or tradable, but can a¤ect dividends endogenously. We also show that …rm value consists of a fundamental component and a bubble component. Unlike the extant literature, we explicitly characterize the collateral yield provided by the bubble component and link the fundamental component to the Q theory of investment (Tobin (1969) and Hayashi (1982)). As in Hayashi (1982), …rms are in…nitely lived and make investment decisions that maximize their stock market values. Our framework of in…nite-horizon production economies with bubbles can be easily extended to incorporate many standard ingredients for both theoretical and quantitative analyses of asset prices, business cycles, and economic growth (Miao and Wang (2012, 2014, 2015), Miao, Wang, Xu (2014), Miao, Wang and Xu (2015), and Miao, Wang, and Zhou (2015)). In particular, Miao, Wang and Xu (2014) apply Bayesian estimation methods to study stock market bubbles and business cycles using our framework.

2

The Baseline Model

We consider an in…nite-horizon production economy, consisting of a continuum of identical households and a continuum of ex ante identical …rms indexed by j 2 [0; 1]. There is no aggregate uncertainty. Time is continuous and denoted by t

2.1

0:

Households

There is a continuum of identical households of a unit measure. The representative household is risk neutral and derives utility from a consumption stream fCt g according to the utility function R 1 rt Ct dt; where r is the subjective discount rate (or rate of time preference).4 Households 0 e

supply labor inelastically and aggregate labor supply is normalized to one. They trade …rm stocks without any trading frictions. The net supply of each …rm’s stock is normalized to one. The representative household faces the budget constraint Ct +

Z

where Vtj denotes …rm j’s stock price, 4

j

Vtj _ t dj = j t

Z

j t

j t dj

+ wt Nt ;

denotes holdings of …rm j’s stocks,

(1) j t

denotes …rm j’s

In Appendix B we will show that our key insights hold true for the case of risk-averse households.

6

dividends determined by its optimization problem, wt denotes the wage rate, and Nt denotes labor supply.5 Because there is no aggregate uncertainty, linear utility gives the …rst-order condition rVtj =

j t

+ V_ tj ;

(2)

for each …rm j: This equation says that the rate of return (or the discount rate) on each stock must be equal to r. Linear utility implies the transversality condition (see, e.g., Ekeland and Scheinkman (1986) and Acemoglu (2009)), lim e

rT

T !1

VTj

j T

= lim e

where we have used the market-clearing condition

2.2

rT

T !1 j T

VTj = 0;

(3)

= 1 for all T and all j:

Firms

Each …rm j 2 [0; 1] combines labor Ntj and capital Ktj to produce output according to the CobbDouglas production function Ytj = (Ktj ) (Ntj )1

;

2 (0; 1) : After solving the static labor choice

problem, we obtain the operating pro…ts Rt Ktj = max (Ktj ) (Ntj )1

wt Ntj ;

Ntj

(4)

where wt is the wage rate and Rt is given by Rt =

wt 1

1

:

(5)

We will show later that Rt is equal to the marginal product of capital in equilibrium. Following Kiyotaki and Moore (1997, 2005, 2008), we assume that each …rm j meets an opportunity to invest in capital over the small time interval [t; t + With probability 1

t] with Poisson probability

t.

t; no investment opportunity arrives. This assumption captures …rm-level

investment lumpiness and generates ex post …rm heterogeneity. Assume that the arrival of an investment opportunity is independent across …rms so that a law of large numbers can be applied for aggregation. This means that only a fraction

t of …rms have investment opportunities at

each time t. There is no insurance market against having an investment opportunity. Investment transforms one unit of consumption goods into one unit of capital. Firms can buy or sell capital at a ‡ow rate in a capital goods market at the price Qt at each time t. For simplicity, suppose that …rms …nance investment Itj using intratemporal loans (Jermann and We use X_ t to denote dXt =dt R for any variable Xt : Households’optimization problem must also satisfy a no-Ponzigame condition limT !1 e rT VTj jT dj 0 (see Acemoglu (2009)). 5

7

Quadrini (2012)) and cannot use equity …nancing. This assumption re‡ects the fact that equity …nancing is more costly than debt …nancing and that selling a lump-sum amount of capital stock takes time.6 Intratemporal loans have no interest and the credit market for these loans are operated among …rms. The interest rate on the intratemporal debt is zero and its price is one. Following Kiyotaki and Moore (2005, 2008) and Jermann and Quadrini (2012), we assume that there is a liquidity mismatch in that investment spending Itj must be paid immediately after the arrival of an investment opportunity at the beginning of time t and before the realization of investment returns Qt Itj . At the end of time t …rms repay their debt after the realization of the returns. Firms without investment opportunities are lenders. Assume that …rms cannot borrow or save by trading intertemporal debt. We will relax this assumption in Section 6.1. Let the ex ante market value of the …rm prior to the realization of an investment opportunity shock be Vt (Ktj ); where we suppress aggregate state variables in the argument. Management acts in the best interest of shareholders so that Vt (Ktj ) = Vtj satis…es the following Bellman equation by (2): rVt Ktj = max

_ j ; Ij K t t

Dtj + V_ t Ktj +

h

Ljt + Qt Itj

Itj + Ljt

i

(6)

subject to Dtj = Rt Ktj

Qt K_ tj + Ktj ;

and some additional constraints to be speci…ed shortly. Here

(7)

represents the depreciation rate of

capital and Dtj represents dividends excluding net investment returns.7 Firm j receives internal funds Rt Ktj and purchases (or sells) capital K_ tj at price Qt . When an investment opportunity arrives at the Poisson rate , the …rm borrows Ljt from other …rms that do not have investment opportunities, and makes investment Itj before receiving investment returns Qt Itj . The change in …rm value is equal to Ljt + Qt Itj

Itj + Ljt : This explains the last term in (6).

There is no jump in capital because we assume that the …rm cannot sell a lump-sum amount of capital immediately after the Poisson arrival of an investment opportunity. Instead, …rms can trade the capital good at a ‡ow rate continuously. Investment is subject to a …nancing constraint Itj

Ljt :

Loans Ljt are fully repaid after the realization of investment returns Qt Itj when Qt Itj

(8) Ljt :

The key assumption of our model is that loans are subject to credit constraints. In the baseline 6 In the online appendix we allow …rms to sell a fraction of their capital to …nance investment as in Kiyotaki and Moore (2005, 2008). We show that our results carry over except that is replaced with + : 7 We have jt = Dtj + Ljt + Qt Itj Itj + Ljt :

8

model we consider the following collateral constraint: Ljt

Vt ( Ktj ):

(9)

This constraint is endogenously derived from an incentive constraint in an optimal contract between …rm j and the lender with limited commitment (Jermann and Quadrini (2012)). To best understand it, we consider a discrete time approximation. In the time interval [t; t + investment Itj and loans Ljt at time t; and repayments Ljt at t + opportunity arrives with Poisson probability

t]; the contract speci…es

t; only when an investment

t: When no investment opportunity arrives, the

…rm does not invest and hence does not borrow. Firm j may default on its debt at t +

t. If it

defaults, then the …rm and the lender will renegotiate the loan repayment. In addition, the lender will have the right to reorganize the …rm. Because of default costs, the lender can only seize a fraction

j of capital Kt+ t : Alternatively, we may interpret

as an e¢ ciency parameter in that the

j lender may not be able to e¢ ciently use the …rm’s assets Kt+ t : The lender can run the …rm with j r t these assets at t + t and obtain …rm value e Vt+ t ( Kt+ t ) at time t. Or it can sell these j assets to a third party at the going-concern value e r t Vt+ t ( Kt+ t ) if the third party can run j the …rm using assets Kt+ t at t + t. This value is the threat value (or the collateral value) to

the lender at the end of period t. Following Jermann and Quadrini (2012), we assume that the …rm has all the bargaining power in the renegotiation through Nash bargaining and the lender obtains only the threat value. The key di¤erence from Jermann and Quadrini (2012) is that the threat value to the lender is the going-concern value in our model, while Jermann and Quadrini (2012) assume that the lender liquidates the …rm’s assets and obtains the liquidation value in the event of default.8 Enforcement requires that, after an investment opportunity arrives at date t; the continuation value to the …rm of not defaulting be no lower than the continuation value of defaulting, that is, Ljt + e

r t

j Vt+ t (Kt+ t)

e

r t

j Vt+ t (Kt+ t)

e

r t

j Vt+ t ( Kt+ t ):

(10)

This constraint ensures that there is no default in an optimal contract. Simplifying yields Ljt Taking the continuous-time limit as

e

r t

j Vt+ t ( Kt+ t ):

t ! 0; we obtain the credit constraint in (9).

8

U.S. bankruptcy law has recognized the need to preserve the going-concern value when reorganizing businesses in order to maximize recoveries by creditors and shareholders (see 11 U.S.C. 1101 et seq.). Bankruptcy laws seek to preserve the going-concern value whenever possible by promoting the reorganization, as opposed to the liquidation, of businesses. Bris, Welch and Zhu (2006) …nd empirical evidence that Chapter 11 reorganizations are less costly and more widely observed than Chapter 7 liquidations.

9

Note that our modeling of the collateral constraint is di¤erent from that of Kiyotaki and Moore (1997): Ljt

Qt Ktj :

(11)

where Qt Ktj is the liquidation value of the collateralized assets. We may reinterpret this constraint as an incentive constraint as in (9) where Vt ( Ktj ) is replaced with Qt Ktj : In Section 5 we will show that this type of collateral constraint will rule out stock price bubbles.9 By contrast, according to (9), we allow the collateralized assets to be valued in the stock market as the going-concern value when the …rm is reorganized and kept running using the collateralized assets after default. If both the …rm and the lender believe that the …rm’s assets are overvalued due to stock market bubbles, then these bubbles will help relax the collateral constraint, providing a positive feedback loop mechanism.

2.3

Competitive Equilibrium

Let Kt =

R1 0

Ktj dj; It =

R1 0

Itj dj; and Yt =

R1 0

Ytj dj denote the aggregate capital stock, average

investment of …rms with investment opportunities, the aggregate labor demand, and aggregate output, respectively. Then a competitive equilibrium is de…ned as sequences of fYt g ; fCt g ; fKt g, fIt g ; fNt g ; fwt g ; fRt g ; fVt (Ktj )g; fItj g; fKtj g; fNtj g such that households and …rms optimize and

markets clear in that j t

= 1; Nt =

Z

0

Ct + It = Yt ; K_ t =

3

1

Ntj dj = 1; Kt + It :

Equilibrium System

We …rst solve an individual …rm’s optimal contracting problem (6) subject to (7), (8), and (9) when the wage rate wt or Rt in (5) is taken as given. This problem does not give a contraction mapping and hence may admit multiple solutions. We conjecture and verify that the ex ante …rm value takes the following form: Vt (Ktj ) = Qt Ktj + Bt ;

(12)

where Bt is a nonpredetermined variable. Since Vt (Ktj ) must always be nonnegative due to limited liability, we must have Bt case we interpret

Qt Ktj

0: Note that Bt = 0 is a possible solution in general equilibrium. In this

as the fundamental value of the …rm. The fundamental value is proportional

9

In Chapter 14 of Tirole’s (2006) textbook, he shows that there may exist multiple equilibria in a simpli…ed variant of the Kiyotaki and Moore (1997) model. In contrast to ours, these equilibria are characterized by a one-dimensional nonlinear dynamical system. Some equilibria may exhibit cycles. We would like to thank Jean Tirole for a helpful discussion on this point.

10

to the …rm’s assets Ktj ; which has the same form as that in Hayashi (1982). Intuitively, the …rm has no fundamental value if it has no assets (Ktj = 0): There may be another solution in which Bt > 0 due to optimistic beliefs. In this case, we interpret Bt as a bubble component since the …rm is still valued at Bt even when there is no market fundamental, i.e., Ktj = 0: In Section 6.2 we will show that when an intrinsically useless asset is traded in the market, its price and Bt follow the same asset-pricing equation (i.e., they are perfect substitutes), further justifying our interpretation of Bt as a bubble component.10 The following result characterizes …rm j’s optimization problem and its proof along with proofs of other results in the paper is given in Appendix A. Proposition 1 Suppose that Qt > 1. Then the optimal investment level when an investment opportunity arrives is given by Itj = Qt Ktj + Bt ;

(13)

where B_ t = rBt

Bt (Qt

Q_ t = (r + ) Qt

Rt

1);

(14)

Qt (Qt

1);

(15)

and Rt is given by (5). Moreover, the following transversality conditions hold: lim e

T !1

rT

QT KT = 0,

lim e

T !1

rT

BT = 0:

(16)

To better understand the intuition behind this proposition, we substitute (12) and (7) into (6) and rewrite the …rm’s problem explicitly as rQt Ktj + rBt = max Rt Ktj _j Itj ;K t

Qt K_ tj + Ktj + (Qt

1) Itj

(17)

+Q_ t Ktj + Qt K_ tj + B_ t subject to Itj

Qt Ktj + Bt ;

(18)

where (18) follows from (8), (9), and (12). When Qt > 1; the …rm chooses to invest at the maximal level so that (18) binds. We then obtain (13). Substituting this equation into (17) and matching 10

According to the standard de…nition for exchange economies, a bubble is equal to the di¤erence between the market value of an asset and the present value of the asset’s exogenously given dividends. It is subtle to apply this de…nition to our model since dividends are endogenously generated through investment and production. Bubbles can help …rms make more investment and hence generate additional dividends. One criticism of the standard test for bubbles is that it is hard to separate bubbles from fundamentals in the data (see Gurkaynak (2008) and Galí and Gambetti (2013)). If one prefers not to use the term “bubbles,”one can call Bt a sunspot, self-ful…lling or speculative component without a¤ecting our results.

11

coe¢ cients, we obtain (14) and (15). In the special case where

= 0; (18) reduces to the credit constraint studied by Martin and

Ventura (2012) who show that …rm value is equal to the bubble in a bubbly equilibrium in their OLG model. By contrast, we will show in Section 5 that …rm value consists of a bubble component Bt and a fundamental component Qt Kt in a bubbly equilibrium. Given (12), we can reinterpret our credit constraint (9) following Gertler and Kiyotaki (2010) and Gertler and Karadi (2011). In particular, (9) is equivalent to Ljt + Vt (Ktj )

(1

) Qt Ktj :

(19)

The left-hand side is the continuation value of the …rm if it chooses to repay the debt Ljt . The right-hand side is the value if the …rm chooses to default by stealing a fraction 1

of its assets

and running away. But it cannot steal the bubble attached to the …rm. In an optimal contract, the preceding incentive constraint must hold. A notable feature of this incentive constraint is that the default value does not contain a bubble. But the stock price bubble Bt can still relax the incentive constraint. When an investment opportunity arrives, an additional unit of investment costs the …rm one unit of the consumption good, but generates an additional value of Qt . Since Qt = @Vt (Ktj )=@Ktj by (12), Qt represents the marginal value of the …rm following a unit increase in installed capital, i.e., Tobin’s marginal Q: If Qt > 1; the …rm will make the maximum possible level of investment so that the credit constraint (9) or (18) binds. If Qt = 1; the investment level is indeterminate. If Qt < 1; the …rm will make the minimum possible level of investment. This investment choice is similar to Tobin’s Q theory (Tobin (1969) and Hayashi (1982)). In what follows, we impose assumptions to ensure Qt > 1 in the neighborhood of the steady state equilibrium. We thus obtain the investment rule given in (13). Substituting this rule into the Bellman equation (17) and matching coe¢ cients, we obtain equations (14) and (15). Since Qt K_ j cancels itself out in (17) due to the constant-returns-to-scale technology, the amount t

of capital purchases K_ tj is indeterminate at the …rm level. Thus …rm dynamics are indeterminate. It is possible that some …rms grow slower and others grow faster. The …rm size is bounded by the aggregate capital stock. The indeterminacy of …rm dynamics at the micro-level will not a¤ect the aggregate equilibrium dynamics as shown in Proposition 2 below, which are our focus. Equation (15) is an asset-pricing equation for capital. It says that the return on a unit of capital rQt is equal to the sum of the marginal product of capital Rt , the additional value generated from new investment

Qt (Qt

1); and capital gains, minus the depreciation Qt : Equation (14) is an

asset-pricing equation for the bubble. Note that the stock price bubble is not attached to a traded intrinsically useless asset. Only stocks are directly traded in the market. Shareholders’ portfolio 12

choice problem studied in Section 2.1 delivers the no-arbitrage equation (2). Management acts in the best interest of shareholders and solves the dynamic programming problem (6) or (17). Both (14) and (15) are derived from this optimization problem. To interpret (14), we observe that the expectation of a higher …rm value due to a bubble Bt > 0 allows the borrowing constraint to be relaxed as revealed by (18). Thus, bubbles are accompanied by a credit boom, leading the …rm to make more investments by Bt . This raises …rm value by (Qt

1) Bt if Qt > 1 by (17), justifying the initial optimistic beliefs. We can interpret

(Qt

1)

as a “collateral yield” or a “liquidity premium.” The sum of the collateral yield and the capital gain of the bubble B_ t is equal to the total return rBt as shown in (14). We will provide a further discussion of equation (14) in Section 5. Although our model features a constant-returns-to-scale technology, marginal Q is not equal to average Q in the presence of bubbles, because average Q is equal to Bt Vt (Kt ) = Qt + for Bt > 0: Kt Kt Thus the existence of stock price bubbles invalidates Hayashi’s (1982) result. In the empirical investment literature, researchers typically use average Q to replace marginal Q under the constantreturns-to-scale assumption because marginal Q is not observable. Our analysis demonstrates that the existence of collateral constraints implies that stock prices may contain a bubble component that makes marginal Q not equal to average Q: Next we aggregate individual …rms’decision rules and impose market-clearing conditions. We then characterize a competitive equilibrium by a system of nonlinear di¤erential equations. Proposition 2 Suppose that Qt > 1: Then the equilibrium variables (Bt ; Qt ; Kt ) satisfy the following system of di¤ erential equations, (14), (15), and K_ t =

Kt + ( Qt Kt + Bt ); K0 given,

and the transversality conditions (16), where Rt = Kt

1

(20)

.

Equation (20) gives the law of motion for the aggregate capital stock derived from the marketclearing condition for capital. We use the market-clearing condition for labor and (5) to derive Rt = K t

1

: The system of di¤erential equations (14), (15), and (20) provides a tractable way to

analyze equilibrium. If we just focus on the …rm’s optimization problem in partial equilibrium taking Qt and wt as given, then Vt (Ktj ) = Qt Ktj + Bt with Bt > 0 gives the maximal …rm value. However, since Vt (Ktj ) is the stock price, it is prone to speculation in general equilibrium. We will show later that 13

both Bt = 0 and Bt > 0 can be supported in general equilibrium under some conditions. That is, our model has multiple equilibria. This re‡ects the usual notion of a competitive equilibrium: Given a price system, individuals optimize. If this price system also clears all markets, then it is an equilibrium system. There could be multiple equilibria with di¤erent price systems. And di¤erent price systems would generate di¤erent optimization problems with di¤erent sets of constraints. After obtaining the solution for (Bt ; Qt ; Kt ) ; we can derive the equilibrium wage rate wt = (1

) Kt ; aggregate output Yt = Kt ; aggregate investment It =

and aggregate consumption Ct = Yt

( Qt Kt + Bt ) ;

(21)

It : We focus on two types of equilibrium.11 The …rst type

is bubbleless, for which Bt = 0 for all t: In this case, the market value of …rm j is equal to its fundamental value, i.e., Vt (Ktj ) = Qt Ktj . The second type is bubbly, for which Bt > 0 for some t and Vt (Ktj ) = Qt Ktj + Bt : Both types can exist due to self-ful…lling beliefs. We next study these two types of equilibrium.

4

Bubbleless Equilibrium

In a bubbleless equilibrium Bt = 0 for all t: Equation (14) becomes an identity. We only need to focus on (Qt ; Kt ) as determined by the di¤erential equations (15) and (20) in which Bt = 0 for all t. We …rst analyze the steady state, in which all aggregate variables are constant over time so that Q_ t = K_ t = 0. We use X to denote the steady-state value of any variable Xt : We use a variable with an asterisk to denote its value in the bubbleless equilibrium. Proposition 3 (i) If ;

(22)

then there exists a unique bubbleless steady-state equilibrium with Q = QE where KE is the e¢ cient capital stock satisfying

(KE )

1

1 and K = KE ;

=r+ :

(ii) If 0
1. We can then apply Proposition 2 in this neighborhood. Proposition 3 also shows that the steady-state capital stock for the bubbleless equilibrium is less than the e¢ cient steady-state capital stock. This re‡ects the fact that not enough resources are transferred from savers to investors due to the collateral constraints. We can verify that R K > I = K so that …rms without investment opportunities have enough funds to lend to …rms with investment opportunities in the bubbleless steady state and hence in the neighborhood of the bubbleless steady state. More intuitively, in the small time interval [t; t +

t] ; the total funds needed to …nance investment for all …rms with investment

opportunities are It (1

t: The total cash owned by all …rms without investment opportunities is

t) Rt Kt t: In a neighborhood of the bubbleless steady state (1

for a su¢ ciently small

t) Rt Kt t > It

t

t:

For (23) to hold, the arrival rate

of investment opportunities must be su¢ ciently small,

holding everything else constant. The intuition is that if

is too high, then too many …rms will

have investment opportunities, which would make the accumulated aggregate capital stock so large as to lower the capital price Q to the e¢ cient level as shown in part (i) of Proposition 3. Condition (23) requires that technological constraints at the …rm level be su¢ ciently tight. To study the local dynamics around the bubbleless steady state (Q ; K ) ; we linearize the system of di¤erential equations (15) and (20) around (Q ; K ) for Bt = 0 for all t: In the online appendix we prove that the linearized system has a positive eigenvalue and a negative eigenvalue 15

so that (Q ; K ) is a saddle point. Thus, in the neighborhood of (Q ; K ) ; for any given initial value K0 ; there is a unique initial value Q0 such that (Qt ; Kt ) converges to the bubbleless steady state (Q ; K ) along a unique saddle path as t ! 1.

5

Bubbly Equilibrium

In this section we study the bubbly equilibrium in which Bt > 0 for all t: We will analyze the dynamic system for (Bt ; Qt ; Kt ) given in (14), (15), and (20). Before we conduct a formal analysis, we …rst explain why bubbles can exist in our in…nite-horizon model. The key lies in understanding equation (14), rewritten here as r=

B_ t + (Qt Bt

1); for Bt > 0:

(26)

The …rst term on the right-hand side is the rate of capital gains of bubbles. The second term represents the collateral yield. Equation (14) or (26) re‡ects a no-arbitrage relation in that the discount rate r is equal to the sum of the rate of capital gains and the collateral yield. Following Kiyotaki and Moore (2008), we can de…ne the return on the bubble as B_ t =Bt : It follows that the return on the bubble is equal to the discount rate minus the collateral yield. The existing literature on bubbles (e.g., Tirole (1985), and Weil (1987)) typically studies bubbles on zero-payo¤ assets or assets with exogenous payo¤s. In this literature the second term on the right-hand side of (26) vanishes and bubbles grow at the discount rate r. If we adopt the Kiyotaki and Moore (1997) collateral constraint (11), then it follows from (8) and (11) that optimal investment satis…es Itj = Qt Ktj when Qt > 1: Substituting this investment rule and (12) into (17), we deduce that bubbles grow at the rate r; i.e., rBt = B_ t . In this case there is no collateral yield. The transversality condition (16) implies that limT !0 e

rT B erT 0

= B0 = 0

and thus a stock price bubble cannot emerge. The transversality condition is not needed in OLG models and a bubble can emerge in dynamically ine¢ cient OLG economies (Tirole (1985)). By contrast, stock price bubbles in our model can in‡uence fundamentals (dividends) due to the positive feedback e¤ect through our collateral constraint (9) or (18). Speci…cally, a one-dollar bubble raises the collateral value by a dollar and allows the …rm to borrow an additional dollar. The …rm then makes one more dollar of investment when an investment opportunity arrives at the Poisson arrival rate . The investment raises …rm value by Qt > 1: Subtracting one dollar of costs, we deduce that the second term on the right-hand side of (26) represents the net increase in …rm value for each dollar of a bubble. This term makes the growth rate of bubbles lower than the discount rate r. Thus the transversality conditions cannot rule out bubbles in our model. We can also show that the bubbleless equilibrium is dynamically e¢ cient in our model. Speci…cally, the 16

golden rule capital stock is given by KGR = ( = )

1 1

: One can verify that K < KGR : Thus the

condition that the economy must be dynamically ine¢ cient for the OLG model of Tirole (1985) cannot ensure the existence of bubbles in our model. Next we will give our new conditions.

5.1

Steady State

We …rst study the existence of a bubbly steady state in which B > 0: We use a variable with a subscript b to denote this variable’s bubbly steady state value. Proposition 4 There exists a bubbly steady state satisfying B = Kb Qb = (Kb )

1

( r

r

+ 1) > 0;

+ 1 > 1;

= [(1

)r + ](

(27) (28)

r

+ 1);

(29)

if and only if the following condition holds: 0
Kb > K , (iii) CE > Cb > C ; and (iv) the bubble-asset ratio B=Kb decreases with : Condition (30) reveals that bubbles occur when

is su¢ ciently small, ceteris paribus. The

intuition is as follows. When the degree of pledgeability is su¢ ciently low, the credit constraint is too tight. A bubble can help relax this constraint and allows …rms to borrow more and invest more. If the collateral constraint is not tight enough, …rms would be able to borrow su¢ cient funds to …nance investment. In this case a bubble serves no function. Note that condition (30) implies condition (23). Thus, if condition (30) holds, then there exist two steady state equilibria: one bubbleless and the other bubbly. The bubbleless steady state is analyzed in Proposition 3. Propositions 3 and 4 reveal that the steady-state capital price is lower in the bubbly equilibrium than in the bubbleless equilibrium, i.e., Qb < Q . The intuition is as follows. Bubbles help relax credit constraints and induce …rms to make more investment than in the case without bubbles. The increased capital stock in the bubbly equilibrium lowers the marginal product of capital. Since the capital price partly re‡ects the present value of the marginal product of capital by (15), it is lower in the bubbly steady state than in the bubbleless steady state. We can verify that Rb Kb > Ib = Kb in the bubbly steady state. By a similar discussion in Section 4, we deduce that …rms without investment opportunities have enough funds to lend to 17

investing …rms to …nance investment in a neighborhood of the bubbly steady state. Do bubbles crowd out capital in the steady state? In Tirole’s (1985) OLG model, households may use part of their savings to buy bubble assets instead of accumulating capital. Thus bubbles crowd out capital in the steady state. In our model, bubbles are attached to productive assets. If the capital price were the same in the bubbly and bubbleless steady states, then bubbles would induce …rms to invest more and hence to accumulate more capital stock. On the other hand, there is a general equilibrium price feedback e¤ect as discussed earlier. The lower capital price in the bubbly steady state discourages …rms from accumulating more capital stock. The net e¤ect is that bubbles lead to higher capital accumulation, contrary to Tirole’s (1985) result. Note that the bubble improves resource allocation even if it does not bring the economy to the …rst-best allocation. As Proposition 4 shows, the bubbly steady-state capital stock Kb is higher than the bubbleless steady-state capital stock K ; but lower than the …rst-best steady-state capital stock KE ; which in turn is lower than the golden rule capital stock KGR : Moreover the existence of bubbles improves welfare in terms of consumption, i.e., CE > Cb > C : By contrast, bubbles overcome dynamic ine¢ ciency by crowding out capital in Tirole’s (1985) OLG model. Introducing credit constraints to Tirole’s (1985) model, Martin and Venture (2012) show that bubbles can also help improve resource allocation and this e¤ect dominates the crowding out e¤ect. How does the parameter

a¤ect the size of bubbles? Proposition 4 shows that a smaller

leads

to a larger bubble relative to capital in the steady state. This is intuitive. If …rms can only pledge a smaller amount of assets, they will face a tighter collateral constraint so that a larger bubble will emerge to relax this constraint.

5.2

Dynamics

Now we study the stability of the bubbleless and bubbly steady states and the local dynamics around them. We linearize the equilibrium system (14), (15), and (20) around the two steady states. We then compute the eigenvalues of the linearized system and compare the number of stable eigenvalues with the number of predetermined variables.12 The equilibrium system has only one predetermined variable (Kt ) and two nonpredetermined variables (Bt and Qt ): Proposition 5 Suppose that condition (30) holds. Then there exists a unique local equilibrium around the bubbly steady state (B; Qb ; Kb ) and the local equilibrium around the bubbleless steady state (0; Q ; K ) has indeterminacy of degree one. We prove that there is a unique stable eigenvalue for the linearized system around the bubbly R3+ of the bubbly steady state (B; Qb ; Kb ) and

steady state. Thus there is a neighborhood N 12

See Theorem 6.6 in Stokey, Lucas, and Prescott (1989, pp.147-148) for local dynamics of nonlinear systems in discrete time. See Coddington and Levinson (1955) for the analysis in continuous time.

18

a continuously di¤erentiable function solution (B0 ; Q0 ) to the equation

: N ! R2 such that given any K0 there exists a unique

(B0 ; Q0 ; K0 ) = 0 with (B0 ; Q0 ; K0 ) 2 N ; and (Bt ; Qt ; Kt )

converges to (B; Qb ; Kb ) starting at (B0 ; Q0 ; K0 ) as t approaches in…nity. The set of points (B; Q; K) satisfying the equation

(B; Q; K) = 0 is a one-dimensional stable manifold of the system. If the

initial value (B0 ; Q0 ; K0 ) is on the stable manifold, then the solution to the nonlinear system (14), (15), and (20) is also on the stable manifold and converges to (B; Qb ; Kb ) as t approaches in…nity. Although the bubbleless steady state (0; Q ; K ) is also a local saddle point, the local dynamics around this steady state are di¤erent. In Appendix A we prove that the stable manifold for the bubbleless steady state is two dimensional because there are two stable eigenvalues for the linearized system around the bubbleless steady state. Thus the local equilibrium has indeterminacy of degree one. Formally, there is a neighborhood N function the equation

R3+ of (0; Q ; K ) and a continuously di¤erentiable

: N ! R such that given K0 for any B0 > 0 there exists a unique solution Q0 to (B0 ; Q0 ; K0 ) = 0 with (B0 ; Q0 ; K0 ) 2 N ; and (Bt ; Qt ; Kt ) converges to (0; Q ; K )

starting at (B0 ; Q0 ; K0 ) as t approaches in…nity. Intuitively, along the two-dimensional stable manifold, the bubbly equilibrium is asymptotically bubbleless in that bubbles will burst eventually. There exist multiple bubbly equilibrium paths converging to the bubbleless steady state and the initial value B0 > 0 is indeterminate. This feature suggests that self-ful…lling beliefs can generate economic ‡uctuations without any shocks to economic fundamentals.

6

Additional Assets

So far we have assumed that households can trade stocks only without trading frictions. Firms can only take on intratemporal debt without interest among themselves subject to credit constraints. In this section we introduce other types of assets for both households and …rms to trade and study the conditions under which a bubble can exist in the presence of di¤erent types of assets. We …rst consider intertemporal private bonds with interest in Section 6.1. We then consider assets with rents and assets with a zero market fundamental in Section 6.2. The latter type of assets can be thought of as pieces of paper (money) or pure bubbles. We will show that both households and …rms must face trading constraints for a bubble to exist in an in…nite-horizon economy. Kocherlakota (1992) and Santos and Woodford (1997) make a similar point for pure-exchange economies.

6.1

Intertemporal Borrowing and Savings

Suppose that there is no intratemporal debt, but …rms can borrow or save by selling or buying intertemporal bonds. Firms can use these bonds to …nance investments subject to credit con-

19

straints.13 These bonds are in zero net supply. Households can also trade them, but are subject to short-sales constraints (Kocherlakota (1992, 2009) and Kiyotaki and Moore (2008)). One may interpret the bonds here as corporate bonds issued by …rms and households cannot borrow by issuing bonds. We will derive an equilibrium in which …rms with investment opportunities choose to borrow by selling bonds, …rms without investment opportunities choose to save by buying bonds, and households do not hold any bonds. We will also show that the equilibrium interest rate rf t on the bonds is lower than the subjective discount rate r because these bonds provide liquidity to the …rms and demand a liquidity premium. This is similar to the role of money studied by Kiyotaki and Moore (2008). Let Lht denote the representative household’s bond holdings. It faces the budget constraint Ct +

Z

j

Vtj _ t dj + L_ ht =

Z

j t

j t dj

+ wt Nt + rf t Lht ;

0 for all t: Let Ljt denote …rm j’s debt level. When Ljt < 0; R Ljt means savings by …rm j. The market-clearing condition for the bonds is Ljt dj = Lht : Let

and the short-sales constraint Lht

Vt (Ktj ; Ljt ) denote the ex ante equity value of a typical …rm j when its capital stock and debt level

at time t are Ktj and Ljt , respectively, prior to the realization of the Poisson shock. We suppress the aggregate state variables in the argument. Then Vt satis…es the following Bellman equation: rVt Ktj ; Ljt

=

max

j j D0t ;D1t ;Itj ;Lj1t

+

h

j + V_ t Ktj ; Ljt D0t

j + Vt Ktj ; Lj1t D1t

Vt Ktj ; Ljt

(31) i

subject to j L_ jt = rf t Ljt + D0t

Rt Ktj + Qt K_ tj + Ktj ;

j D1t = Qt Itj + Lj1t

Itj Vt (Ktj ; Lj1t )

Lj1t

Vt Ktj ; 0

Ljt

Itj ;

Ljt ;

(32) (33) (34)

Vt ( Ktj ; 0);

(35)

j j where D0t and D1t denote dividends and Lj1t represents the new debt level when an investment

opportunity arrives. The interpretation of the Bellman equation (31) is as follows. When no investment opportunity j arrives, …rm j pays out dividends D0t ; purchases new capital, and borrows or saves Ljt , resulting in 13

In the online appendix we show that our results do not change when …rms can use corporate bonds as collateral to borrow.

20

the ‡ow-of-funds constraint (32). When an investment opportunity arrives at Poisson arrival rate , …rm j borrows Lj1t

j 0; pays dividends D1t , and makes investments Itj . The last two terms in

(31) re‡ect the fact that equity value changes from Vt (Ktj ; Ljt ) to Vt (Ktj ; Lj1t ) when the debt level jumps from Ljt to Lj1t : In this case the ‡ow-of-funds constraint is (33) and investment is subject to the …nancing constraint (34). Because we have assumed that …rms cannot raise equity or sell a lump-sum amount of capital to …nance investment immediately after the arrival of an investment opportunity, there is no jump in the capital stock Ktj . Debt is subject to the credit constraint (35), which is interpreted in a similar way to (9). Consider the discrete time approximation. Suppose that at time t, …rm j pledges a fraction j of its capital Kt+

t

as collateral. It may default on debt Lj1t+

default, it obtains continuation value Vt+ the repayment Lj1t+

t

j j Kt+ t ; L1t+ t

t

t

at time t +

t. If it does not

: If it defaults, debt is renegotiated and

j is relieved. The lender can seize the collateralized assets Kt+

t

and keep

the …rm running with these assets by reorganizing the …rm. Thus the threat value to the lender j is Vt+ t ( Kt+ t ; 0): Assume that …rms have full bargaining power. We then have the incentive

constraint j j Vt+ t (Kt+ t ; L1t+ t )

Vt+

t

j Kt+ t; 0

j Vt+ t ( Kt+ t ; 0):

(36)

The expression on the right-hand side of (36) is the value to the …rm if it chooses to default. This constraint ensures that …rm j does not have an incentive to default. Its continuous-time limit as t ! 0 is (35). We conjecture and verify that equity value takes the following form: Vt Ktj ; Ljt = Qt Ktj

Ljt + Bt ;

(37)

and thus the credit constraint (35) becomes Lj1t where Bt

Qt Ktj + Bt ;

(38)

0 is the bubble component of equity value. Given the value function in (37), the credit

constraint (35) is equivalent to Vt (Ktj ; Lj1t )

(1

) Qt Ktj :

(39)

This constraint can be interpreted as in Gertler and Kiyotaki (2010) and Gertler and Karadi (2011). At each time t, …rm j can default on the debt Lj1t and run away by stealing a fraction (1 assets. The outside value on default is (1

) Qt Ktj :

has no incentive to default.

21

) of

The inequality above ensures that the …rm

Proposition 6 Suppose that Qt > 1. Then the equilibrium system for (Kt ; Qt ; Bt ; rf t ) is given by (14), (15), (20), and rf t = r where Rt = Kt

1

(Qt

1) < r;

(40)

; and the usual transversality conditions hold.

When Qt > 1; both (34) and (38) bind. Thus, with intertemporal bonds, …rms can use both savings when Ljt < 0 and new debt Lj1t = Qt Ktj + Bt to …nance investment. Because bonds are in zero net supply, to clear the bond market, some …rms must be borrowers and repay previous debt (Ljt > 0) before raising new debt to …nance investment. Thus aggregate investment is the same as that in the case of intratemporal debt studied in Section 2 and the equilibrium system for aggregate variables (Kt ; Qt ; Bt ) is also the same. Equation (40) shows that the interest rate rf t is equal to the subjective discount rate r minus a liquidity premium (Qt

1). The liquidity premium is not observable in the data but exists because

bonds can provide liquidity to investing …rms by relaxing their credit constraints. Firms without investment opportunities are willing to save and lend even though rf t < r because they anticipate that they will be credit constrained when a future investment opportunity arrives. Holding equity and lending to other …rms yield the same total return of r if we include the liquidity premium component. Because both the bubble and the bonds can help …rms …nance investment to the same extent, they command the same amount of the liquidity premium. Since r > rf t ; households prefer to borrow by selling bonds until their short-sales constraints bind. This property is similar to that for money studied by Kiyotaki and Moore (2008). Without a short-sales constraint, households would keep shorting these bonds (or e¤ectively borrowing) until rf t = r: In this case the liquidity premium would be zero so that Qt = 1 and the economy would reach the e¢ cient equilibrium and no bubble could exist. Next we characterize the steady state. Proposition 7 (i) If condition (23) holds, then there exists a unique bubbleless steady state (K ; Q ; rf ) given by (24), (25), and rf = r +

= . (ii) If condition (30) holds, then there exists a unique

bubbleless steady state (K ; Q ; rf ) given in part (i) and a unique bubbly steady state (Kb ; Qb ; B; rf ) given by (27), (28), (29), and rf = 0: Conditions (23) and (30) ensure that the assumption Qt > 1 in Proposition 6 is satis…ed in a neighborhood of both the bubbleless and bubbly steady states so that this proposition can be applied. Condition (30) is equivalent to the standard condition (Tirole (1985) and Santos and Woodford (1997)) that the interest rate on bonds in the bubbleless steady state be less than the rate of economic growth (rf < 0). Proposition 7 shows that the interest rate on bonds must be equal to zero in the bubbly steady state (rf = 0), because both bonds and bubbles can be used 22

to …nance investment. But because the bubble is intrinsically useless, if the steady-state interest rate on bonds is positive, then bonds dominate bubbles and a bubble cannot exist. This result is related to the rate-of-return-dominance puzzle in monetary economics. One may introduce economic growth to generate the coexistence of a bubble and a positiveinterest-rate bond as in Tirole (1985). Formally, let the production function be Ytj = (Ktj ) (At Ntj )1 where At =

egt

(g > 0) represents technical progress. We can then show that aggregate capital and

the …rm bubble grow at the same rate g: Proposition 6 still characterizes the equilibrium system except that now Rt =

(Kt =At )

1

. Assume that r > g so that Qt > 1 in the neighborhood of the

bubbly steady state. It follows from (14) and (40) that the steady-state interest rate on the bond is equal to g > 0:

6.2

Assets with or without Rents

We now follow Tirole (1985) and introduce an asset that brings a real rent (or dividend) to the baseline model in Section 2. For simplicity, suppose that the rent is given by a constant X

0 each

time. If X = 0; then this asset has no market fundamental and is called a pure bubble (e.g., tulip). If X > 0; we call it a “tree.”Firms can purchase the asset and sell it to …nance its investment when an investment opportunity arrives. We normalize the total supply of the asset to unity. We follow Kiyotaki and Moore (2008) and Kocherlakota (1992, 2009) and assume that both households and …rms face short-sales constraints. The representative household faces the budget constraint Ct +

Z

j

Vtj _ t dj + Pt M_ th =

and the short-sales constraint Mth

Z

j t

j t dj

+ XMth + wt Nt ;

0; where Mth denotes the household’s asset holdings. We will

show that households will sell the asset until the short-sales constraint binds in equilibrium. Since we have shown that intertemporal borrowing is not essential for the existence of a bubble, we will focus on the case of intratemporal loans for simplicity. Let Vt (Ktj ; Mtj ) denote the ex ante market value of a typical …rm j when its capital stock and asset holdings at time t are Ktj and Mtj

0; respectively. Let Pt denote the market price of the asset. Then Vt satis…es the following

Bellman equation: rVt Ktj ; Mtj

=

max

j j j D0t ;D1t ;M1t ;Itj ;Ljt

+

h

j D0t + V_ t Ktj ; Mtj

j j D1t + Vt Ktj ; M1t

23

Vt Ktj ; Mtj

(41) i

;

subject to j D0t = Rt Ktj + XMtj Pt M_ tj h j j D1t = Pt Mtj M1t + Ljt

Itj

Pt Mtj Ljt

j where Ljt and M1t

Qt K_ tj + Ktj ; i Itj + Qt Itj Ljt ;

j M1t + Ljt ;

(42)

Ktj ; 0 ;

Vt

(43)

0 denote the intratemporal debt and the new asset holdings when an investment

opportunity arrives. The interpretations of the Bellman equation and the constraints are similar to those in Sections 2 and 6.1. In particular, equation (42) is the …nancing constraint. Firm j can sell assets (Mtj

j ) M1t

and borrow Ljt to …nance investment. Equation (43) gives the credit constraints. Since …rm j can sell the asset directly to …nance investment when an investment opportunity arrives, it uses Ktj as collateral only. An alternative formulation is to assume that the …rm cannot sell the asset Mtj but can use it j : as collateral to borrow. The borrowed funds can …nance investment Itj and asset purchases M1t

The ‡ow-of-funds constraint, the …nancing constraint, and the credit constraint become j = Ljt D1t

Itj

j + Qt Itj Pt M1t

j Itj + Pt M1t

Ljt

Vt

Ljt ;

Ljt ;

Ktj ; Mtj :

(44) (45) (46)

By a similar analysis in Section 2.2, the right-hand side of (46) gives the threat value or the recovery value to the lender if the …rm defaults. In Appendix A we show that Vt takes the following form: Vt Ktj ; Mtj = Qt Ktj + Pt Mtj + Bt : Given this value function, we can verify that the preceding two ways of …nancing are equivalent and give an identical equilibrium outcome. Proposition 8 Suppose Qt > 1: Then the equilibrium system for (Kt ; Qt ; Bt ; Pt ) is given by (14),

24

(15), and K_ t =

Kt + (Qt Kt + Pt + Bt );

P_t = rPt where Rt = Kt

1

X

(Qt

1)Pt ;

(47) (48)

; and the usual transversality conditions hold.

j When Qt > 1; …rms with investment opportunities sell all their assets (M1t = 0) and take on

the maximum level of debt to …nance investments. Firms without investment opportunities buy the assets and lend. The interpretations of equations (14), (15), and (47) are similar to those in Section 2. We interpret (Qt

1) as the collateral yield or the liquidity premium. Equation (48)

is a pricing equation for the asset. We rewrite it as X P_t + =r Pt Pt

(Qt

1) for Pt > 0;

which implies that the return on the asset (P_t + X)=Pt is equal to the subjective discount rate r minus the liquidity premium. Since the return on the asset is lower than the subjective discount rate, households have no incentive to hold the asset and want to sell it until their short-sales constraints bind. If there were no household short-sales constraints, then no arbitrage would force the liquidity premium to vanish so that Qt = 1: In this case, there would be no investment friction and hence the economy would reach the e¢ cient equilibrium in which no bubble could exist. In the special case of X = 0; the asset is intrinsically useless and becomes a pure bubble with a zero market fundamental (e.g., money). Equation (48) reduces to P_t =r Pt

(Qt

1);

which implies that the return on the pure bubble P_t =Pt is less than the subjective discount rate when Qt > 1. A similar point for money is made by Kiyotaki and Moore (2008). Comparing the equation above with (14) reveals that the bubble value Pt and the component Bt in …rm value follow the same asset-pricing equation. By (47), we can determine the total value Pt + Bt only, but not Bt or Pt separately. That is, the bubble asset and the bubble component Bt in …rm value are perfect substitutes. This justi…es our interpretation of Bt as a bubble component. In this case equilibria can still be characterized as in Sections 4 and 5. The bubbly equilibrium determines the total size of the bubble Pt + Bt , but the decomposition of the total bubble is indeterminate. The equilibrium real allocation is independent of the decomposition. This result is analogous to that discussed in Section 5 of Tirole (1985). 25

In the case of X > 0; a bubbly equilibrium cannot exist because the tree with dividends dominates the bubble. To see this, suppose that a bubble exists in the steady state. Then (14) and (48) imply the steady-state relation 0=r

(Q

1) = X=P > 0;

which is a contradiction. The intuition is that the tree with positive dividends plays the same role as a …rm bubble in that both can be used to …nance investment. The tree dominates the bubble since it delivers positive dividends, but the bubble has a zero market fundamental. For a bubble and a tree to coexist, we may introduce economic growth to the model by allowing technical progress as in Section 6.1 (or Tirole (1985)). In this case Proposition 8 still applies except that Rt =

(Kt =At )

1

. We then obtain b_ t = (r

g)bt

(Qt

1)bt ;

p_t = (r

g)pt

(Qt

1)pt

X=egt ;

where bt = Bt =At and pt = Pt =At : Since the detrended dividend X=egt vanishes in the long run, the …rm bubble and the tree can coexist in the steady state. More generally, if dividends grow at a lower rate than the economy does, then dividends normalized by the trend disappears in the long run. In this case the tree can still coexist with the …rm bubble. One may wonder whether bubbles can still exist if rents on some assets (e.g., land) grow as fast as the economy. Tirole (1985) shows that this is possible in an OLG model by assuming that rents are not capitalized before their creation. In the online appendix we show that this is also possible in our in…nite-horizon model when we assume that the asset with dividends or rents (land) is less liquid than the bubble. Building on the present paper, Miao, Wang, and Zha (2014) endogenize rents, which grow as fast as the economy, and show that land bubbles can also exist in an in…nite-horizon DSGE model with endogenous credit constraints.

7

Stochastic Bubbles and Policy Implications

So far we have focused on deterministic bubbles. Following Blanchard and Watson (1982) and Weil (1987), we now introduce stochastic bubbles to the baseline model in Section 2 with intratemporal loans. Suppose that a bubble exists initially, i.e., B0 > 0. The bubble may burst in the future at Poisson arrival rate : Once it bursts, it will never have value again.14 14

If a bubble reemerged in the future, it would have value today by the no-arbitrage asset-pricing equation. To generate recurrent bubbles and crashes, Miao, Wang, and Xu (2014) introduce …rm entry and exit. See Martin and Ventura (2012) and Wang and Wen (2012) for other approaches.

26

7.1

Equilibrium with Stochastic Bubbles

First, we consider the case in which the bubble has collapsed. This corresponds to the bubbleless equilibrium studied in Section 4. We use a variable with an asterisk (except for Kt ) to denote its value in the bubbleless equilibrium. In particular, V (Ktj ) denotes …rm j’s value function. It is given by V (Ktj ; Qt ) = Qt Ktj ; where (Qt ; Kt ) satis…es the equilibrium system (15) and (20) with Bt = 0. We may express the solution for Qt in a feedback form, i.e., Qt = g (Kt ) for some function g: Next we consider the case in which the bubble has not collapsed. We assume that the debt contract is not contingent on extraneous beliefs or sunspots described earlier. Firms borrow only when an investment opportunity arrives. The credit constraint is still given by (9). We write …rm j’s dynamic programming problem as follows: rV Ktj ; Qt ; Bt

= max Rt Ktj Itj

+

h

V

Qt K_ t + Kt + V_ Ktj ; Qt ; Bt + (Qt

Ktj ; Qt

V Ktj ; Qt ; Bt

1) Itj

i

subject to (8) and (9). The expression on the second line re‡ects the fact that once the bubble bursts, …rm value changes from V (Ktj ; Qt ; Bt ) to V (Ktj ; Qt ): We can conjecture and verify that the value function takes the form V (Ktj ; Qt ; Bt ) = Qt Ktj +Bt : Proposition 9 Suppose Qt > 1: Before the bubble bursts, the equilibrium with stochastic bubbles (Bt ; Qt ; Kt ) satis…es the following system of di¤ erential equations: B_ t = (r + )Bt Q_ t = (r + + )Qt and (20), where Rt = Kt

1

Qt

(Qt

1)Bt ;

Rt

(Qt

(49) 1) Qt ;

(50)

and Qt = g (Kt ) is the capital price after the bubble bursts.

Equation (49) is an asset-pricing equation for the bubble and re‡ects the possibly of the collapse of the bubble. In general, it is di¢ cult to characterize the full set of equilibria with stochastic bubbles. In order to transparently illustrate the adverse impact of bubble bursting on the economy, we consider a simple type of equilibrium. Following Weil (1987) and Kocherlakota (2009), we study a stationary equilibrium with stochastic bubbles that has the following properties: The capital stock is constant at the value Ks over time before the bubble collapses. It continuously moves to the bubbleless steady-state value K after the bubble collapses. The bubble is also constant at the value Bs > 0 before it collapses. It jumps to zero and then stays at this value after collapsing. The capital price is constant at the value Qs before the bubble collapses. It jumps to the value 27

g (K) after the bubble collapses and then converges to the bubbleless steady-state value Q given in equation (24). Our objective is to show the existence of (Bs ; Qs ; Ks ) : By (49), we can show that Qs =

r+

+ 1:

(51)

Since Qs > 1, we can apply Proposition 9 in some neighborhood of Qs : Equation (50) implies that 0 = (r + + )Qs where R = K

1:

g (K)

R

(Qs

1) Qs ;

(52)

The solution to this equation gives Ks : Once we have obtained Ks and Qs ; we

use equation (20) to determine Bs : The di¢ cult part is to solve for Ks since g (K) is not an explicit function. To show the existence of Ks , we de…ne

as r+

That is,

+1=

=Q :

is the bursting probability such that the capital price in the stationary equilibrium with

stochastic bubbles is the same as that in the bubbleless equilibrium. Proposition 10 Let condition (30) hold. If 0
0:

(53)

By a similar analysis to that in Section 6.2, we can derive that the price of the government bond satis…es the asset-pricing equation: rPt = P_t + (Qt

1) Pt :

If the government bond is not backed by taxes (i.e., Tt = 0), then it is a pure bubble (Diamond (1965)) and it can coexist with …rm bubbles in equilibrium. As long as the government bond is backed by taxes (i.e., Tt > 0), then equation (53) permits D_ t = 0 and P_t > 0 in the steady state. It dominates …rm bubbles since B_ t = 0 in the steady state. The government can choose the amount of debt by adjusting the size of lump-sum taxes so that …rms can overcome credit constraints. Proposition 11 Suppose that assumption (30) holds. Let the government issue a constant value D of government debt given by Dt = D which is backed by lump-sum taxes Tt = T

KE

> 0;

(54)

rD for all t: Firms can trade government bonds or use

them as collateral to …nance investment. This credit policy will eliminate the bubble in …rm value and enable the economy to achieve e¢ cient allocation. This proposition indicates that the government can design a policy that eliminates bubbles and achieves e¢ cient allocation. The key intuition is that if the government can provide su¢ cient liquidity to …rms, then …rms would not need to rely on bubbles to relax credit constraints. The government plays the role of …nancial intermediaries by transferring funds from households to …rms directly so that …rms can overcome credit constraints. The government bond is a store of value and can also generate dividends for …rms. The liquidity premium comes from the net bene…t from new investment. Households prefer to sell as much bonds to …rms as possible until the short-sales constraints bind when the interest rate on the government bond is lower than the subjective discount 15

As an idealized benchmark, we ignore the issues of moral hazard and distortional taxes.

30

rate. Under the government policy in the proposition, the government can make the interest rate on the bond exactly equal to the subjective discount rate. Thus the liquidity premium is zero so that Tobin’s marginal Q is equal to 1. To implement the policy above, the government constantly retires the public bonds at the interest rate r in order to keep the total bond value constant.16 To back the government bonds, the government levies constant lump-sum taxes equal to the interest payments of bonds. Our discussion of government policy is related to that in Caballero and Krishnamurthy (2006) and Kocherlakota (2009). As in these studies, government bonds can serve as collateral to help relax credit constraints in our model. Unlike their proposed policies, our proposed policy enables the economy to achieve the e¢ cient allocation when government bonds are backed by lump-sum taxes. Unbacked public assets are intrinsically useless and may have positive value (a pure bubble). Although issuing unbacked public assets can boost the economy after the collapse of stock price bubbles, the real allocation is still ine¢ cient and the bubble on unbacked public assets can also burst and cause the economy to enter a recession again.

8

Conclusion

We have developed a theory of stock price bubbles in the presence of endogenous credit constraints in production economies with in…nitely lived agents. Bubbles emerge through a positive feedback loop mechanism in which credit constraints derived from optimal contracts with limited commitment play an essential role. Our analysis di¤ers from most studies in the literature that analyze bubbles on intrinsically useless assets or on assets with exogenously given rents or dividends in an endowment economy framework or an OLG framework. Our model can incorporate this type of bubbles and thus provides a uni…ed framework to study asset bubbles. In future research it would be interesting to study how bubbles can help explain asset pricing puzzles, how bubbles contribute to business cycles in a quantitative dynamic stochastic general equilibrium model (Miao, Wang and Xu (2014)), how bubbles a¤ect long-run growth (Caballero, Farhi and Hammour (2006), Martin and Venture (2012), Hirano and Yanagawa (2013) and Miao and Wang (2014)), and what the implications of asset price bubbles are for monetary policy (Galí (2014)).

16

This policy is analogous to the Friedman rule in monetary economics.

31

Appendices A

Proofs

Proof of Proposition 1: As described in the main text, we can rewrite the dynamic programming (6) as (17). Given the assumption Qt > 1; (8) and (18) bind. We then obtains (13). Substituting this equation back into (17) and matching coe¢ cients, we obtain (14) and (15). By the transversality condition (3) and the form of the value function, rT

lim e

T !1

We then obtain (16).

(QT KT + BT ) = 0:

Q.E.D.

Proof of Proposition 2: Using the optimal investment rule in (13), we can derive the aggregate capital accumulation equation (20) and the aggregate investment equation (21). The …rst-order condition for the static labor choice problem (4) gives wt = (1 and Ktj = Ntj (wt = (1

))1= : Thus the capital-labor ratio is identical for each …rm. Aggregating

yields Kt = Nt (wt = (1 (5) yields Rt = Kt Yt =

Z

) (Ktj =Ntj ) . We then obtain (5)

1

Nt1

))1= so that Ktj =Ntj = Kt =Nt for all j 2 [0; 1] : Substituting out wt in = Kt

(Ktj ) (Ntj )1

This completes the proof.

dj =

1

Z

since Nt = 1 in equilibrium. Aggregate output satis…es (Ktj =Ntj ) Ntj dj = (Ktj =Ntj )

Z

Ntj dj = Kt Nt1

:

Q.E.D.

Proof of Proposition 3: (i) The social planner solves the following problem: max It

Z

0

1

e

rt

(Kt

It ) dt;

subject to K_ t =

Kt + It ; K0 given,

where Kt is the aggregate capital stock and It is the investment level for a …rm with an investment opportunity. From this problem, we can derive the e¢ cient capital stock KE ; which satis…es (KE )

1

= r + : The e¢ cient output, investment and consumption levels are given by YE =

(KE ) ; IE = = KE ; and CE = (KE )

KE ; respectively.

Suppose that assumption (22) holds. We conjecture that Q = Qt = 1 in the steady state. In this case, …rm value is given by V Ktj = Ktj : The optimal investment rule for each …rm satis…es Rt = r +

=

Kt

1

: Thus Kt = KE : Given this constant capital stock for all …rms, we must 32

have Kt = It for all t: Let each …rm’s optimal investment level satisfy Itj = Ktj = : Then, when assumption (22) holds, the investment and credit constraints, Itj = Ktj =

Ktj = V

Ktj ; are

satis…ed for all t . We conclude that, under assumption (22), the solutions Qt = 1, Kt = KE ; and It =Kt = = give the bubbleless equilibrium, which also achieves the e¢ cient allocation. (ii) Suppose that (23) holds. Conjecture that Qt > 1 in some neighborhood of the bubbleless steady state in which Bt = 0 for all t. We can then apply Proposition 2 and derive the steady-state equations for (15) and (20) as Q_ = 0 = (r + ) Q K_ = 0 = where R =

K

1:

R

Q(Q

1);

K + ( QK);

(A.1) (A.2)

From these equations, we obtain the steady-state solutions Q and K in

(24) and (25), respectively. Assumption (23) implies that Q > 1: By continuity, Qt > 1 in some neighborhood of (Q ; K ) : This veri…es our conjecture.

Q.E.D.

Proof of Proposition 4: In the bubbly steady state, (14) and (20) imply that 0 = rB 0= where R =

K

1:

B (Q

1); and

(A.3)

K + [ QK + B] ;

(A.4)

Solving equations (A.1), (A.3), and (A.4) yields equations (27), (28), and

(29). By (27), B > 0 if and only if (30) holds. From (24) and (28), we deduce that Qb < Q : Using condition (30), it is straightforward to check that KGR > KE > Kb > K . By the resource constraint, steady-state consumption satis…es C = Y

I=K

K: Substituting the expressions

for KE ; Kb ; and K in Propositions 3 and 4, we can show that CE > Cb > C : From (27), it is also straightforward to verify that the bubble-asset ratio B=Kb decreases with : Q.E.D. Proof of Proposition 5: First, we consider the log-linearized system around the bubbly steady ^ t to denote the percentage deviation from the steady state value for state (B; Qb ; Kb ) : We use X ^ t = ln Xt any variable Xt , i.e., X

ln X: We can show that the log-linearized system is given by 2

3 2 ^t =dt dB 6 ^ 7 6 4 dQt =dt 5 = A 4 ^ t =dt dK

33

3 ^t B ^t 7 Q 5; ^t K

where

2

0

6 A=4

(r + )

0

+r

B=Kb

(2r + ) [(1

2

7 ) 5:

)r + ](1

(r + )

We denote this matrix by

3

0 B=Kb

0 a 0

3

7 c 5; d e f

6 A=4 0 b

where we deduce from (A.5) that a < 0, c > 0, d > 0; e > 0; and f < 0: Since b = (1

)r +

We observe that F (0) =

x3

(b + f )x2 + (bf

acd > 0 and F ( 1) =

above equation, denoted by

1

= x3

1 )(x

(

1

2 )(x

+

1 2

We consider two cases. (i) If

1 3

2

> 0 and

(ii) If either

2

+

2

2

or

3 3

1 2

+

2 3 )x

+

1 3

and

+

3

1 3

+

(A.6)

2

and

3:

We rewrite F (x) as

+(

1 2

2 3

+

1 2 3

= bf

1 3

+

2 3 )x

= acd < 0 and cd < 0:

< 0 and

2

2 3

(A.7)

1 2 3:

are two real roots, then it follows from 3

< 0. We then have

(A.8) 1 1 2

< 0 that > 0 and

> 0, which contradicts equation (A.8). Thus we

is complex, then the other must also be complex. Let = g + hi and

where g and h are some real numbers and i = 1 2

1

acd = 0:

> 0.

2

Since

, we have

3)

must have the same sign. Suppose

> 0. This implies that

must have

r+

1. Thus, there exists a negative root to the

Matching terms in equations (A.6) and (A.7) yields

3

ce)x

< 0. Let the other two roots be

F (x) = (x

and


0. The characteristic equation for the matrix A is F (x)

2

(A.5)

+

1 3

+

3

=g

hi;

p

1. We can show that

2 3

= 2g

1

+ g 2 + h2 :

< 0, the above equation and equation (A.8) imply that g > 0.

From the above analysis, we conclude that the matrix A has one negative eigenvalue and the other two eigenvalues are either positive real numbers or complex numbers with a positive real

34

part. As a result, the bubbly steady state is a local saddle point and the stable manifold is one dimensional. Next, we consider the local dynamics around the bubbleless steady state (0; Q ; K ). We linearize Bt around zero and log-linearize Qt and Kt and obtain the following linearized system: 3 2 r dBt =dt 7 6 6 ^ 4 dQt =dt 5 = 4 ^ t =dt dK 2

1) 0 0

32

R (1 Q

) > 0;

3 Bt 76 ^ 7 a b 54 Q t 5; ^t c d K

(Q 0 K

where a =

R Q

c =

Q ; b= Q > 0; d = 0:

Using a similar method for the bubbly steady state, we analyze the three eigenvalues of the matrix J. One eigenvalue, denoted by and

3;

1;

is equal to r

(Q

1) < 0 and the other two, denoted by

2

satisfy 2 3

It follows from (A.9) that

2

and

3

= ad

bc = 0

bc < 0:

(A.9)

must be two real numbers with opposite signs. We conclude

that the bubbleless steady state is a local saddle point and the stable manifold is two dimensional. Q.E.D. Proof of Proposition 6: Conjecture that the value function takes the form in (37). Then the credit constraint (35) becomes (38). Substituting (37), (32), and (33) into (31) yields r Qt Ktj

Ljt + Bt

=

max Rt Ktj

_ j ;L_ j K t t

rf t Ljt + L_ jt

Qt K_ tj + Ktj

+Q_ t Ktj + K_ tj Qt L_ jt + B_ t h + max Qt Itj + Lj1t Ljt Itj ;Lj1t

Itj

Lj1t

Ljt

i

:

subject to (34) and (38). Simplifying yields r Qt Ktj

Ljt + Bt = max Rt Ktj Lj1t ;Itj

Qt Ktj

rf t Ljt + Q_ t Ktj + B_ t + (Qt

1) Itj

(A.10)

subject to (34) and (38). When Qt > 1; (34) and (38) bind. Thus optimal investment is given by Itj = Qt Ktj + Bt

35

Ljt :

(A.11)

Substituting (A.11) back into (A.10) and matching coe¢ cients of Ktj ; Ljt ; and Bt , we can derive (14), (15),and (40). Because K_ j and L_ j cancel out in the preceding dynamic programming problem, t

t

these values are indeterminate at the …rm level. When Qt > 1; equation (40) implies that rf t < r so that households will not hold any bonds R Lht = 0: The market-clearing condition for the bonds become Ljt dj = 0: It follows from this condition and a law of large numbers that aggregate investment is given by

then obtain (20).

( Qt Kt + Bt ) : We

Q.E.D.

Proof of Proposition 7: The proof follows from that for Propositions 3 and 4.

Q.E.D.

Proof of Proposition 8: Let V Ktj ; Mtj ; Bt ; Qt ; Pt denote the value function. Conjecture that V Ktj ; Mtj ; Bt ; Qt ; Pt = Qt Ktj + Pt Mtj + Bt : Substituting this conjectured function and the ‡ow-of-funds constraints into the dynamic programming problem (41) yields r Qt Ktj + Pt Mtj + Bt =

max

j M1t ;Itj ;Ljt

Rt Kt + XMtj

Pt M_ tj

Qt K_ tj + Ktj

+Qt K_ tj + Ktj Q_ t + P_t Mtj + Pt M_ tj + B_ t h j + Ljt Itj + Qt Itj + Pt Mtj M1t

j Ljt + Pt M1t

Mtj

i

subject to (42) and Ljt

Qt Ktj + Bt :

j = 0 and the optimal investment level Under the assumption Qt > 1; we have M1t

Itj = Qt Ktj + Pt Mtj + Bt : Substituting this solution back into the preceding Bellman equation and matching coe¢ cients, we obtain equations (14), (15), and (48). It follows from (48) that rPt > P_t + X: Thus households will not hold the asset and their short-sales constraint binds. This means that the market-clearing condition for the asset is given R by Mtj dj = 1: By a law of large numbers, aggregate capital satis…es K_ t = Kt +

Qt Kt + Pt

36

Z

Mtj dj + Bt :

We then obtain (47).

Q.E.D.

Proof of Proposition 9: Substituting the conjectured value function V (Qt ; Ktj ; Bt ) = Qt Ktj +Bt into the dynamic programming problem yields r Qt Ktj + Bt

=

max Rt Ktj

Itj ;

_j K t

Qt K_ tj + Ktj + Q_ t Ktj + Qt K_ tj + B_ t h

1) Itj +

+ (Qt

Qt Ktj

Qt Ktj + Bt

i

subject to Itj

Qt Ktj + Bt :

When Qt > 1; optimal investment is given by Itj = Qt Ktj + Bt : Substituting this rule back into the preceding Bellman equation and matching coe¢ cients yield (49) and (50). Equation (20) follows from aggregation and the market-clearing condition. Q.E.D. Proof of Proposition 10: Let Q ( ) be the expression on the right-hand side of equation (51). We then use this equation to rewrite equation (52) as K

1

(r + + )Q( ) + g(K) + (r + ) Q( ) = 0:

De…ne the function F (K; ) as the expression on the left-hand side of the equation above. Notice that Q( ) = Q = g(K ) by de…nition and Q(0) = Qb where Qb is given in (28). Condition (30) ensures the existence of the bubbly steady-state value Qb and the bubbleless steady-state values Q and K . De…ne Kmax = max

(r + +

(r + ) )Q( )

Q

0

1 1

:

By (29), we can show that Kb = Thus we have Kmax

(r +

r )Q(0)

1 1

:

Kb and hence Kmax > K . We want to prove that F (K ; ) > 0; F (Kmax ; ) < 0;

for

2 (0;

) : If this is true, then it follows from the intermediate value theorem that there exists

a solution Ks to F (K; ) = 0 such that Ks 2 (K ; Kmax ) :

37

First, notice that F (K ; 0) = K and F (K ; 0
(1

)F (K ; 0) +

F (K ;

) > 0:

Next we can derive F (Kmax ; ) =

Kmax1

(r + + )Q( ) + g(Kmax ) + (r + ) Q( )


K

K is increasing for all K < KGR ; we deduce that

K : This implies that the consumption level before the bubble collapses is

higher than the consumption level in the steady state after the bubble collapses.

Q.E.D.

Proof of Proposition 11: The equilibrium with government debt can be characterized similarly to Proposition 8. In particular, (Kt ; Qt ; Bt ; Pt ) satis…es K_ t =

Kt + (Qt Kt + Pt Mt + Bt );

Q_ t = (r + )Qt

Rt

(Qt

1)Qt ;

(A.12) (A.13)

B_ t = rBt

(Qt

1)Bt ;

(A.14)

P_t = rPt

(Qt

1)Pt ;

(A.15)

38

and the usual transversality condition. The di¤erence from Proposition 8 is that (i) the total supply of the government bond is Mt instead of 1, and (ii) X = 0: Under the policy in the proposition, equation (53) implies that P_t = rPt : It follows from (A.15) that Qt = 1: Substituting Qt into equation (A.13) reveals that Rt = r + : This equation gives the e¢ cient capital stock KE for all time t after the collapse of the bubble. Let Kt = KE and Pt Mt = D in (A.12), where D is given by (54). We can show that Bt = 0 for all t: Q.E.D.

B

Risk-Averse Households

Suppose that the representative household has the following utility function: Z

1

1 t Ct

e

1

0

where

is the subjective discount rate and

dt;

is the risk aversion parameter. The household faces

the budget constraint (1) subject to a no-Ponzi-game condition similar to that in footnote 5. Then we can derive the consumption Euler equation 1 C_ t = (rt Ct

);

(B.1)

where rt is equal to the return on any stock j in the absence of aggregate uncertainty and is also called the discount rate. Equation (2) holds where r is replaced with rt : Firm j solves the following dynamic programming problem: rt Vt Ktj = max

Dtj ; Itj

Dtj + V_ t Ktj +

h

Ljt

Itj + Qt Itj

Ljt

i

(B.2)

subject to (7), (8), and (9). The aggregate state variables of the economy are Bt ; Qt ; and Kt ; where Bt represents the aggregate size of the bubble. The discount rate rt is a function of the aggregate state variables. Conjecture that Vt Ktj = Qt Ktj + Btj ; where Btj is the bubble component in …rm j’s stock price. Substituting this conjecture into the preceding dynamic programming problem yields rt Qt Ktj + rt Btj

= max Rt Ktj _j Itj ;K t

Qt K_ tj + Ktj + (Qt

+Q_ t Ktj + Qt K_ tj + B_ tj ;

39

1) Itj

(B.3)

subject to Itj

Qt Ktj + Btj :

(B.4)

When Qt > 1; the constraint (B.4) binds so that the optimal investment level is Itj = Qt Ktj + Btj . Substituting this rule back into the Bellman equation and matching coe¢ cients of Ktj ; we obtain Q_ t = (rt + ) Qt B_ tj = rt Btj

Rt

Qt (Qt

Btj (Qt

1);

1):

(B.5) (B.6)

The usual transversality conditions must hold. R Since Bt = Btj dj; it follows from (B.6) that the aggregate bubble satis…es B_ t = rt Bt

Bt (Qt

1):

(B.7)

The law of motion for aggregate capital still satis…es (20). The resource constraint is given by Ct + ( Qt Kt + Bt ) = Yt :

(B.8)

The equilibrium system consists of …ve equations (20), (B.1), (B.5), (B.7), and (B.8) for …ve aggregate variables (Ct ; rt ; Kt ; Qt ; Bt ) : The transversality condition also holds lim e

T !1

RT 0

rs ds

QT KT = 0,

lim e

T !1

RT 0

rs ds

BT = 0:

(B.9)

Note that an equilibrium only determines the aggregate size Bt of the bubble, but an individual …rm’s bubble size Btj is indeterminate. So it is possible that some …rms have no bubble and others have di¤erent sizes of bubbles. We use a variable without the time subscript to denote its steady state value. Then (B.1) implies r=

and hence the steady-state system is the same as that in the baseline model of Section 2. Our

analysis of steady states in Sections 4 and 5 still applies to the case of risk-averse households. We are unable to derive analytical results for local dynamics because the equilibrium system contains …ve equations, but it is straightforward to derive numerical solutions.

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