Speculative Bubbles and Financial Crises

184

American Economic Journal: Macroeconomics 2012, 4(3): 184–221http://dx.doi.org/10.1257/mac.4.3.184

It is widely believed that the current worldwide financial crisis was caused by the bursting of a housing market bubble in the United States. However, a situation

like this is not new. History has too often witnessed the rise and collapse of similar asset “bubbles.” The first recorded such bubble was the “tulip mania, ”a period in Dutch history during which contract prices for tulip bulbs reached extraordinarily high levels and then suddenly collapsed. At the peak of the tulip mania in February 1637, tulip contracts sold for more than 10 times the annual income of a skilled craftsman, which was more than the value of a luxury house in seventeenth-century Amsterdam.1 Figure 1 shows the tulip price index during the 1636–1937 period.2

According to Mackay (1841, 107), during the tulip mania, people sold their other possessions to speculate in the tulip market:

… [T]he population, even to its lowest dregs, embarked in the tulip trade. … Many individuals grew suddenly rich. A golden bait hung temptingly out before the people, and, one after the other, they rushed to the tulip marts, like flies around a honey-pot. Every one imagined that the passion for tulips would last for ever, and that the wealthy from every part of the world would send to Holland, and pay whatever prices were asked for

1 See, for example, Mackay (1841).2 Source: Wikipedia (http://en.wikipedia.org/wiki/Tulip_mania) and Thompson (2007).

Speculative Bubbles and Financial Crises†

By Pengfei Wang and Yi Wen*

Are asset prices unduly volatile and often detached from their fun-damentals? Does the bursting of financial bubbles depress the real economy? This paper addresses these issues by constructing a DSGE model with speculative bubbles. We characterize conditions under which storable goods, regardless of their intrinsic values, can carry bubbles, and agents are willing to invest in such bubbles despite their positive probability of bursting. The results show that systemic risk, commonly perceived changes in the bubble’s probability of bursting, can generate boom-bust cycles with hump-shaped output dynamics and produce asset price movements many times more volatile than the economy’s fundamentals. (JEL E13, E23, E32, E44, G01, G12).

* Wang: Department of Economics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China (e-mail: [email protected]); Wen: Research Department, Federal Reserve Bank of St. Louis, PO Box 442, St. Louis, MO 63166, and School of Economics and Management, Tsinghua University, Beijing, China (e-mail: [email protected]). We thank Nobu Kiyotaki, Richard Rogerson, two anonymous referees, and seminar partici-pants at Tsinghua University and the Chinese University of Hong Kong for comments. We also thank Mingyu Chen and Luke Shimek for research assistance, and Judy Ahlers for editorial assistance. Pengfei Wang acknowledges the financial support from the Hong Kong Research Grants Council (project #643908). The usual disclaimer applies.

† To comment on this article in the online discussion forum, or to view additional materials, visit the article page at http://dx.doi.org/10.1257/mac.4.3.184.

Speculative Bubbles and Financial CrisesI. The Benchmark ModelA. FirmsB. HouseholdC. Competitive EquilibriumD. Decision RulesE. General EquilibriumII. Systemic Risk and Asset Price VolatilityA. Aggregation and General EquilibriumB. Stationary Sunspot EquilibriaC. Calibration and Impulse ResponsesIII. ConclusionAppendix I: Proof of Proposition 1Appendix II: Proof of Proposition 2Appendix III: Proof of Proposition 3Appendix IV: Proof of Proposition 4Appendix V: Proof of Proposition 5References

http://dx.doi.org/10.1257/mac.4.3.184

http://en.wikipedia.org/wiki/Tulip_mania

mailto:[email protected]

mailto:[email protected]


VoL. 4 No. 3 185WANG AND WEN: SpEcuLATiVE BuBBLES

them. The riches of Europe would be concentrated on the shores of the Zuyder Zee, and poverty banished from the favoured clime of Holland. Nobles, citizens, farmers, mechanics, seamen, footmen, maidservants, even chimney-sweeps and old clotheswomen, dabbled in tulips.

People were buying tulips at higher and higher prices, intending to resell them for a profit. However, such a scheme could not last because tulip prices were growing faster than income. As a result, the demand for tulips eventually collapsed and the bubble burst. The Dutch economy went into a deep recession in 1637.

Although historians and economists continue to debate whether the tulip mania was indeed a bubble caused by what Mackay (1841) termed “extraordinary popular delusions and the madness of crowds” (see, e.g., Garber 1989, 1990; Dash 1999; and Thompson 2007), many observers believe that bubbles are important elements of real-world asset markets. Yet, despite the widespread belief in the existence of bubbles in the real world, it has been known, at least since the work of Scheinkman (1977, 1988), that it is difficult to construct infinite-horizon model economies in which asset price bubbles, as an intertemporal equilibrium, exist. Santos and Woodford (1997) characterize conditions for the existence of asset price bubbles in a general class of intertemporal equilibrium models and show that the conditions under which bubbles are possible are quite fragile.3

Despite this fragility, it may nonetheless be worthwhile to investigate if rational bubbles can help explain the business cycle. This paper constructs asset price bubbles in an infinite-horizon model with incomplete financial markets and short-sale con-straints. It is shown that genuine asset bubbles with prices far exceeding the assets’ fundamental values, and with movements similar to Figure 1, can be constructed. In

3 For more recent discussions on this literature, see Kocherlakota (2008).

Tulip price index 1636–37

Dec. 12

Dec. 1Nov. 25

Nov. 12May 1

Feb. 9

Feb. 5

Feb. 3200

150

100

50

Figure 1. The Tulip Mania Bubble

186 AMEricAN EcoNoMic JourNAL: MAcroEcoNoMicS JuLy 2012

the model, infinitely lived agents are willing to invest in bubbles even though they may burst at any moment. The reason is that with incomplete financial markets and borrowing constraints, bubbles provide liquidity by serving as liquid assets. Our results demonstrate that the bursting of such bubbles can generate recessions, and perceived changes in the probability of the bubbles’ bursting can cause asset price movements many times more volatile than aggregate output.

People invest in bubbles for many reasons. The idea that infinitely lived rational agents are willing to hold bubbles with no intrinsic value to smooth consumption can be traced back at least to Bewley (1980) and Lucas (1980).4 This idea has been applied more recently in general equilibrium models by Kiyotaki and Moore (2008) and Kocherlakota (2009) to study economic fluctuations, where heterogeneous firms use intrinsically worthless assets to improve resource allocation and invest-ment efficiency when financial markets are incomplete. The related literature also includes works of Tirole (1985); Scheinkman and Weiss (1986); Woodford (1986); Kocherlakota (1992); Araujo, Pascoa, and Torres-Martinez (2005); Caballero and Krishnamurthy (2006); Angeletos (2007); Farhi and Tirole (2008); Kocherlakota (2008); and Hellwig and Lorenzoni (2009); among others. This literature focuses on asset price bubbles and financial market frictions and differ from the indeterminacy literature of Benhabib and Farmer (1994) and Wang and Wen (2008, 2009a). For the earlier literature on sunspots, see Azariadis (1981) and Cass and Shell (1983).

This paper builds on this literature to study asset price volatility and rational bubbles that may grow based on storable goods with intrinsic value. This extension is not trivial because sunspot equilibrium may disappear in the Kiyotaki-Moore-Kocherlakota model once the object supporting the bubble (e.g., land) is allowed to have positive fundamental value (e.g., as a factor of production or a source of util-ity). Bubbles are often believed to exist in goods with positive fundamental value, such as antiques, bottles of wine, paintings, flower bulbs, rare stamps, houses, land, and so on. More importantly, the model is applied to explaining business-cycle fluc-tuations in aggregate output and asset prices quantitatively, which may be consid-ered as the main contribution of this paper.

We use a DSGE model to characterize conditions for the existence of rational bubbles in goods with fundamental value. We show that any inelastically supplied storable good,5 regardless of its intrinsic value, can support a bubble with the fol-lowing features: the market price of the goods exceed their fundamental value, and the market price can collapse to the fundamental value so the fundamental value is itself a possible equilibrium.6

4 It can be traced further back to Samuelson’s (1958) overlapping generations model of money. Bewley (1980) and Lucas (1980) show in infinite-horizon economies that people are willing to hold money as a store of value to self-insure against idiosyncratic shocks despite money having no intrinsic value. The major difference between this literature (Bewley 1980; Lucas 1980) and our paper is that we assume a different source of idiosyncratic uncertainty and we use a production economy instead of an endowment economy.

5 Goods can be producible yet at the same time inelastically supplied. For example, antiques and bottles of wine are produced goods, but their dates of production make them unique and nonsubstitutable by newly produced ones.

6 If the fundamental value is not an equilibrium, it may be argued that bubbles do not exist (because it is dif-ficult to know empirically whether there is a bubble if it never bursts). Also, the existence of multiple fundamental equilibria does not imply bubbles because asset values never exceed fundamentals in a fundamental equilibrium.


The basic structure of our model closely resembles those of Kiyotaki and Moore (2008) and Kocherlakota (2009) wherein firms, instead of households, invest in bubbles.7 The main differences between our model and the literature include the following:

• In addition to characterizing general equilibrium conditions for bubbles todevelop on objects with positive fundamental value, in our model the prob-ability of capital investment is endogenously determined by firms rather than exogenously fixed. That is, the fraction of firms optimally choosing to invest in fixed capital during each period is endogenous to the model. So, at equilibrium, the number of firms investing can respond to aggregate shocks and monetary policy. This extensive margin has been missing in previous work.

• There aremultiple assets in themodel.Thismultiple asset approach allowsus to construct stochastic sunspot equilibrium featuring systemic risk (i.e., the public’s perception of the bursting probability of bubbles) and conduct impulse response analyses and time-series simulations.

• We focus on asset price volatility and show that the model can generatehump-shaped output dynamics and match the asset-price movements in the US economy.8

• Weprovideananalyticallytractablemethodtosolvethegeneral-equilibriumpaths of the model (without resorting to numerical computational techniques) despite assuming a continuum of heterogeneous agents with irreversible invest-ment and borrowing constraints.9

The rest of the paper is organized as follows. Section I presents the basic model and characterizes conditions under which bubbles can be supported by goods with intrinsic value. Section II introduces sunspot shocks to a version of the basic model (by allowing the perceived probability of the bubble’s bursting to be stochastic) and calibrates the model to match the US business cycles and asset price volatility. Section III concludes the paper.

I. The Benchmark Model

Consider an infinite horizon economy. Time is discrete and indexed by t = 0, 1, 2, … . There is a unit mass of heterogeneous firms indexed by i ∈ [0, 1]. Each firm is subject to an aggregate labor-augmenting productivity shock and an idiosyncratic investment-specific productivity shock. Households are identical and they trade firms’ shares.

7 See Wen (2009) for a DSGE model where households invest in bubbles.8 Pintus and Wen (2008) show that collateralized borrowing constraints can lead to overinvestment and boom-

bust credit cycles. Their model requires only independently and identically distributed shocks to trigger hump-shaped dynamics and the overshooting phenomenon in the recessions. However, they do not study bubbles where asset prices exceed fundamental values.

9 Our solution method follows that of Wang and Wen (2009b). As far as we know, no other work has shown how to solve discrete-time models with irreversible investment and borrowing constraints analytically.


Assume that at t = 0 there exists one unit of a divisible good endowed from nature and equally distributed among the firms. We call the good “tulips” throughout the paper. Each tulip can be converted into f units of consumption goods and paid to households (firm owners) as dividends. Assume that households do not have the technology to store tulips but firms do, and there exists a fixed storage cost, ζ ≥ 0, per tulip per period for firms. Hence, f is the fundamental value of a tulip.10

Denote the market price of tulips by q t . Obviously, if q t < f, then the demand for tulips is infinite and no firm will ever want to sell tulips (i.e., the aggregate market supply is zero). So if an equilibrium price for tulips exists, it must satisfy q t ≥ f. One interesting question is: Do firms have an incentive to hold and invest in tulips when q t > f ? In other words, can q t > f be a competitive (bubble) equilibrium in the economy other than the fundamental equilibrium, q t * = f ?

In what follows, we characterize conditions under which a bubble equilibrium with the following features exists: the market price of tulips exceeds f , and the mar-ket price of tulips can collapse to the fundamental value f.11

A. Firms

Each firm i maximizes its discounted future dividends, E 0 ∑ t=0 ∞ β t ( Λ t / Λ 0 ) d t (i),

where d t denotes the dividend at time t, Λ t a representative household’s marginal utility that firms take as given (and which may be stochastic), and β ∈ (0, 1) the time-discounting factor. The production technology of firm i is described by

(1) y t (i) = A t k t (i ) α n t (i ) 1−α , α ∈ (0, 1),

where A t is an index of aggregate total factor productivity (TFP), k t (i) is the firm’s capital stock, and n t (i) is its employment. The capital stock is accumulated accord-ing to the law of motion, k t+1 (i) = (1 − δ) k t (i) + ( i t (i)/ ε t (i)), where investment is irreversible, i t (i) ≥ 0, and is subject to an idiosyncratic rate of return (cost) shock, ε t (i), with support, [ ε _ ,

_ ε ] ∈ r+ ∪ {+∞}, and the cumulative distribution function,

F(ε). Firms make investment decisions in each period after observing ε t (i).Firms pay dividends to their owners. A firm’s dividend is

(2) d t (i) = A t k t (i ) α n t (i ) 1−α + q t h t (i) − i t (i) − ( q t + ζ) h t+1 (i) − w t n t (i),

where w t is the real wage, h t+1 the quantity (or shares) of tulips purchased at the beginning of period t as a store of value, and ζ h t+1 the total fixed storage costs paid for storing the tulips within period t. In addition, we impose the following constraints: d t (i) ≥ 0, and h t+1 (i) ≥ 0. That is, firms can neither pay negative divi-dends nor hold negative amounts of tulips. These assumptions imply that firms are financially constrained and the asset markets are incomplete. Such constraints plus

10 For simplicity, assume that tulips cannot be used as a factor of production.11 This definition of a bubble equilibrium requires that the fundamental value itself be an equilibrium so that the

bursting of the bubble is possible.


investment irreversibility may give rise to speculative (precautionary) motives for investing in tulips.

To characterize a firm’s optimization program, the following steps simplify our analysis. Using the firm’s optimal labor demand schedule,

(3) (1 − α) A t k t (i ) α n t (i ) −α = w t ,

the labor demand can be expressed as a linear function of the capital stock, k t (i),

(4) n t (i) = [ (1 − α) A t _ w t ] 1 _ α k t (i).

Accordingly, output y t (i) is also a linear function of k t (i),

(5) y t (i) = A t [ (1 − α) A t _ w t ] 1−α _ α k t (i).

These linear relations imply that aggregate output and employment may depend only on the aggregate capital stock. Thus, we do not need to track the distribution of k t (i) to study aggregate dynamics. Defining r t ≡ α A t [(1 − α) A t / w t ] (1−α)/α , the firm’s net revenue is given by

(6) y t (i) − w t n t (i) = r t k t (i),

which is also linear in the capital stock.Using the definition of r t , the firm’s problem is to solve

(7) V 0 (i) = max E 0 [ ∑ t=0

∞

β t Λ t _ Λ 0 d t (i)],

where d t (i) = [ r t k t (i) − i t (i) + q t h t (i) − ( q t + ζ) h t+1 (i)], subject to the following constraints:

(8) d t (i) ≥ 0

(9) h t+1 (i) ≥ 0

(10) i t (i) ≥ 0

(11) k t+1 (i) = (1 − δ) k t (i) + i t (i) _ ε t (i) ,

given k 0 (i) > 0 and h 0 (i) = 1. Here, β t Λ t / Λ 0 is the stochastic discount factor between period 0 and t. We will show later that Λ t is the representative household’s marginal utility in period t.


B. Household

All households are assumed identical. A representative household chooses con-sumption c t , labor supply N t , and firm shares s t+1 (i) to maximize its life-time utility,

(12) max E 0 ∑ t=0

∞

β t (log c t − A n N t 1+ γ n

_ 1 + γ n

), β ∈ (0, 1);

subject to the budget constraint

(13) c t + ∫ 0 1

s t+1 (i) [ V t (i) − d t (i)] di = ∫ 0 1

s t (i) V t (i) di + w t N t ,

where V t (i) denotes firm i’s market value (stock price). In addition, households can-not short tulips. Defining Λ t as the Lagrangian multiplier of equation (13), the first-order conditions for { s t+1 (i), c t , N t } are given, respectively, by

(14) V t (i) = β E t { Λ t+1 _ Λ t

V t+1 (i)} + d t (i),

1/ c t = Λ t , and A n N t γ n = Λ t w t . Equation (14) implies that the stock price V t (i) is given by the discounted present value of dividends as in equation (7). Also, Λ t is the marginal utility of consumption.

C. competitive Equilibrium

A competitive equilibrium is defined as the sequences of quantities, { i t (i), n t (i), k t+1 (i), y t (i), h t (i), s t+1 (i) } t≥0 and { c t , N t } t≥0 , and prices { w t , q t , V t (i) } t≥0 for i ∈ [0, 1], such that:

•Givenprices{ w t , q t } t≥0 , the sequence of quantities { i t (i), n t (i), k t+1 (i), h t+1 (i), x t (i) } t≥0 solves each firm i’s problem (7) subject to the constraints (8) through (11).

• Givenprices { w t , V t (i)} t≥0 , the sequence { c t , N t , s t (i) } t≥0 maximizes household utility (12) subject to the budget constraint (13).

• Allmarketsclear:

(15) s t (i) = 1 for all i ∈ [0, 1]

(16) N t = ∫ 0 1

n t (i) di

(17) c t + ∫ 0 1

i t (i) _ ε t (i) di + ζ ∫

0 1

h t+1 (i) di = ∫ 0 1

y t (i) di + f x t ,

where the first equation pertains to the shares market, the second to the labor market, and the third to the aggregate goods market. The third term on the left-hand side


(LHS) of equation (17) is the total storage costs and the second term on the right-hand side (RHS) of equation (17), x t ≡ [ ∫

0 1 h t (i) di − ∫

0 1 h t+1 (i) di], is the total

number of tulips taken out of circulation from the tulip market and converted into consumption goods in period t.

D. Decision rules

This subsection shows how to derive firms’ optimal decision rules in closed form. It should be noted that the main simplification comes from the linearity of firms’ objective functions and constraints and the fact that firms can take the household’s marginal utilities and other market prices as given.

PROPOSITION 1: A firm’s optimal decision rule for fixed investment is given by

(18) r t k t (i) + q t h t (i) ε t (i) ≤ ε t * i t (i) = { , 0 ε t (i) > ε t *

where ε t * is a time-varying cutoff that is independent of i and is determined by the following Euler equation:

(19) ε t * = β E t Λ t+1 _ Λ t

[ r t+1 Q ( ε t+1 * ) + (1 − δ) ε t+1 * ],

where Q(⋅) > 1 captures the option value (liquidity premium) of one unit of net revenue (or cash flow) and this option value is determined by

(20) Q ( ε t * ) = ∫

m ax {1, ε t * _ ε t (i) } dF (ε) > 1.

When the aggregate demand for tulips Ω t+1 ≡ ∫0 1 h t+1 (i) di > 0, the equilibrium

tulip price is determined by the following asset pricing equation:

(21) q t + ζ = β E t [ Λ t+1 _ Λ t q t+1 Q ( ε t+1 * )].

PROOF: See Appendix I.The intuition behind proposition 1 is as follows. First, the marginal cost of invest-

ment is 1 and the marginal investment return is 1 _

ε t (i) × λ t , where λ t is the market

value of one unit of newly installed capital. Thus, the firm-specific Tobin’s q is λ t _

ε t (i)

and the firm will invest if and only if λ t _

ε t (i) ≥ 1 or ε t (i) ≤ λ t . Therefore, the cutoff is

given by ε t * ≡ λ t . Since the market value of newly installed capital is determined by

its expected future marginal productivity, the cutoff is independent of i because ε t (i) is independently and identically distributed. Note that constant returns and


flexible labor imply that the value of installed capital is equal across firms and that if a firm invests it invests all of its revenue.

The value of one unit of cash flow ( Q t ) is greater than 1 because of the option of waiting. When the cost of capital investment is low in period t ( ε t (i) < ε t * ), one dollar of cash flow yields ε * _ ε(i) > 1 dollars of new capital through investment. When the cost is high in period t ( ε t (i) > ε t * ), firms can hold on to the cash and the value is simply 1. This explains why the option value of cash ( Q t+1 ) enters the RHS of equations (19) and (21) as part of future returns to reflect a liquidity premium on cash.

Equation (19) is the Euler equation for capital investment. The LHS is the market price of one unit of newly installed capital. The RHS is the expected marginal gain from having one unit of newly installed capital, which includes two terms: one unit of new capital can generate r units of output in the next period with option value rQ; and it has a residual market value (1 − δ) ε t+1 * in the next period after depreciation. A firm will invest up to the point where the LHS equals the RHS. This equation also determines the optimal cutoff ( ε t * ) in the model.

Equation (21) is the Euler equation for tulip investment. The LHS is the marginal cost of holding one tulip, which includes the market price q and storage cost ζ. The RHS is the expected next-period gain from having one tulip in hand. Because there is a liquidity premium on holding cash (Q > 1), holding tulips can improve a firm’s cash position through liquidation when cash is short. Thus, the effective rate of return on tulip assets in the next period is q t+1 Q t+1 . Equation (21) is thus the asset pricing equation for tulips.

Intuitively, because tulips are storable for firms, they allow a firm to self-insure against idiosyncratic shocks by serving as a store of value (i.e., liquidity). If the cost shock ε t (i) is large (or the rate of return on capital investment is low), firms may opt to invest in tulips so as to have liquidity available in the future when the next period cost of capital investment may be low. On the other hand, if the rate of return on capital investment is high ( ε t (i) is small), firms may opt to liquidate (sell) their tulips and make more liquidity available to purchase fixed capital and expand production capacity. Such behavior may be rational despite the fact that the tulip price exceeds its fundamental value and the tulip bubble has a positive probability of bursting.

E. General Equilibrium

Define aggregate variables N t = ∫0 1 n t (i) di, i t = ∫

0 1 i t (i) di, K t = ∫

0 1 k t (i) di,

and y t = ∫0 1 y t (i) di. The aggregate demand for tulips is denoted by Ω t+1

≡ ∫0 1 h t+1 (i) di. Given that k t (i) is a state variable, by the factor demand functions

of firms, we have N t = [ (1 − α) A t _ w t ]

1 _ α K t and y t = A t [

(1 − α) A t _ w t ]

1−α _ α K t . These two equa-tions imply that aggregate output can be written as a simple function of aggregate labor and capital, y t = A t K t α N t 1−α . Hence, the real wage w t = (1 − α) y t / N t and r t = α ( y t / K t ), which turns out to be the aggregate marginal product of capital.


PROPOSITION 2: The general equilibrium paths of the model are characterized by nine aggregate variables, { c t , i t , N t , y t , K t+1 , q t , ε t * , Ω t+1 , ϕ t * }, which are fully deter-mined by the following system of eight nonlinear dynamic equations:12

(22) y t = A t K t α N t 1−α

(23) c t + i t = y t + f ( Ω t − Ω t+1 ) − ζ Ω t+1

(24) (1 − α) y t _ c t = A n N t 1+ γ n

(25) q t + ζ _ c t

= β E t { q t+1 _ c t+1

Q t+1 } + ϕ t * _ c t

(26) i t = [α y t + q t Ω t ] F ( ε t * )

(27) ε t * _ c t = β E t { ε t+1 * _ c t+1

[α y t+1 _ K t+1

Q t+1 _ ε t+1 * + 1 − δ]} (28) K t+1 = (1 − δ) K t + ω ( ε t * ) i t

(29) Ω t+1 ϕ t * = 0,

where the coefficient ω( ε t * ) ≡ ( ∫ε≤ ε t *

ε −1 dF)/F( ε t * ) > 1/ ε t * measures the marginal

efficiency of aggregate investment.

PROOF: See Appendix II.Equation (22) is the aggregate production function. Equation (23) is the aggre-

gate resource constraint derived from equation (17). Equation (24) pertains to the household’s optimal labor supply decisions. Equation (25) is the Euler equation for optimal tulip investment based on equation (21). Equation (26) characterizes the level of aggregate investment based on equation (18). Equation (27) is the Euler equation for optimal capital investment based on equation (19). Equation (28) expresses the law of motion of aggregate capital accumulation, and equation (29) is a complementarity condition that determines whether tulips are traded or not in general equilibrium. That is, Ω t+1 > 0 only if ϕ t * = 0, and ϕ t * > 0 only if Ω t+1 = 0.

Equation (26) shows that tulip assets affect aggregate investment through two channels. First, they directly increase all firms’ cash flows through the liquidation value, q t Ω t . Second, as will be shown in Proposition 3, they influence the cutoff value, ε * , thus affecting the number of active firms making fixed investments and consequently the marginal efficiency of aggregate investment. This second channel

12 The complementarity equation (29) actually consists of two equations: Ω t+1 > 0 only if ϕ t * = 0, and ϕ t * > 0 only if Ω t+1 = 0.


plays a critical role in the model’s dynamics, but it is absent in the models of Kiyotaki and Moore (2008) and Kocherlakota (2009).

The model has three steady states, which can be grouped in two types. In one type of steady state, tulips are traded and their market price is greater than or equal to their fundamental value—q ≥ f, Ω ∈ (0, 1], and ϕ * = 0. In the other type of steady state, the market price equals the fundamental value and tulips are not traded—q = f, Ω = 0, and ϕ * ≥ 0. Let us now further characterize these steady states.

Steady State A: q > f, Ω = 1, and ϕ * = 0. In this situation, all tulips are traded. Equation (23) and equations (25) through (28) become

(30) c + i = y − ζ

(31) q + ζ = β q Q ( ε * )

(32) i = (αy + q) F ( ε * )

(33) 1 − β (1 − δ) = βα y _ K

Q ( ε * ) _ ε *

(34) δK = ω ( ε * ) i.

Given parameters {β, α, δ, ζ, f } and the distribution of ε, these five equations plus equations (22) and (24) determine seven aggregate variables, {y, c, i, K, N, q, ε * }. For example, given the cutoff ε * , equations (33) and (34) determine the capital-to-output ratio (K/y) and the saving rate (i/y). Equations (32), (31), and (30) then determine the ratios {q/y, ζ/y, c/y }, respectively. Equation (24) and the production function then determine the levels of aggregate output and employment, and, hence, the lev-els of consumption and investment. Finally, using the identity, ζ = ζ _ y ( ε * ) × y( ε * ), the cutoff ( ε * ) can then be pinned down. Notice that equation (31) suggests Q( ε * ) > 1. An interior solution for the cutoff ε * ∈ [ ε _ ,

_ ε ] then exists provided that the storage

cost ζ is not too high.13

Steady State B: q = f, Ω = 0, and ϕ * ≥ 0. In this situation, no firm will invest in tulips regardless of the value of ζ. Denoting x b as the value of x in steady state B, the first-order conditions (23) and (25) through (28) become

(35) c b + i b = y b

(36) f + ζ = β f Q ( ε b * ) + ϕ b *

(37) i b = α y b F ( ε b * )

13 Since Q( ε * ) is bounded above by Q( _ ε ), if ζ is too high, equation (31) becomes an inequality, q + ζ > βqQ( _ ε ), and no firm will hold tulips.


(38) 1 − β (1 − δ) = β α y b _ K b

Q ( ε b * ) _ ε b *

(39) δ K b = ω ( ε b * ) i b .

Equations (37) through (39) imply that

(40) 1 − β + β δ = β δ Q ( ε b * ) _ ε b * ∫

ε≤ ε b *

ε −1 dF ,

which uniquely solves for the cutoff ε b * . To see that we have an interior solution for the cutoff, ε _ < ε b * < _ ε , notice that the RHS of equation (40) equals infinity at ε b * = ε _ and equals βδ at ε b * = _ ε . Hence, a unique interior solution exists as long as β < 1. Given ε b * , we can then solve for { y b , c b , K b , N b , ϕ b * } according to the follow-ing. Equation (37) determines the i b / y b ratio. Equation (35) determines the c b / y b ratio. The labor market condition (24) determines N b . The production function, in conjunction with the capital-output ratio, then determines the levels of capital stock and output.

To ensure that ϕ b * ≥ 0, equation (36) implies that ζ ≥ f [ βQ( ε b * ) − 1]. That is, the storage cost must be large enough for steady state B to be an equilibrium. This suggests that when ζ = 0 and f > 0, steady state B may not be possible (see Proposition 4 below).

Steady State C: q = f, Ω ∈ (0, 1] and ϕ * = 0. In this situation, because tulip price equals the fundamental value, firms are indifferent between holding tulips and converting them into consumption goods. So in equilibrium, only a fraction of tulips (0 < Ω ≤ 1) are traded. However, because the quantity of tulips traded determines tulip price q, the equilibrium value of Ω is unique at steady state c if f > 0.14 Equations (30)–(34) then become

(41) c c + i c = y c − ζ Ω

(42) f + ζ = β f Q ( ε c * )

(43) i c = (α y c + f Ω) F ( ε c * )

(44) 1 − β (1 − δ) = βα y c _ K c

Q ( ε c * ) _ ε c *

(45) δ K c = ω ( ε c * ) i c .

To determine steady state c (when f > 0), we can redefine variables by set-ting ζ = ζ Ω and q = f Ω. So equations (41)–(43) become c c + i c = y c − ζ ,

14 In other words, it is not possible to have more than one value of Ω satisfying q = f in steady state c. However, if f = 0, then it is possible for Ω to be indeterminate in steady state c. But this case is less interesting because tulips do not affect resource allocations when their liquidation value is zero.


q + ζ = β q Q( ε c * ), and i c = (α y c + q ) F( ε c * ), which are indistinguishable from equations (30)–(32). Steady state c can then be solved the same way as in solving steady state A (given the values of ζ and Ω). In particular, given ζ> 0 and Ω ≤ 1, we can solve for q , which can then be used to determine the required value, f = q /Ω, for steady state c to hold.15

We call steady state A a bubble steady state and steady state B a fundamental steady state. Steady state c can be referred to as a special bubble steady state because Ω > 0 (even though q = f ). It is called a special bubble steady state because it has alloca-tions similar to those at steady state A since tulips can improve firms’ investment efficiency even when their price equals their fundamental value. A bubble may or may not coexist with a fundamental equilibrium, depending on the parameter values.

The following propositions characterize the properties and conditions necessary for these steady states to (co)exist. To simplify discussions and notations, we ignore the special case, steady state c. However, most properties and conditions of steady state A apply to steady state c as well. Readers will be alerted whenever this is not the case.

PROPOSITION 3: The cutoff value ( ε * ) is lower in a bubble steady state than in a fundamental steady state. consequently, there are fewer firms investing in fixed capital in a bubble steady state than in a fundamental steady state, but the marginal efficiency of aggregate investment is higher in a bubble equilibrium than in a no-bubble equilibrium. As a result, the aggregate capital-stock-to-output ratio is higher in a bubble equilibrium than in a no-bubble equilibrium. However, the aggregate investment-to-output ratio is not necessarily higher in a bubble equilibrium.

PROOF: See Appendix III.This proposition also applies to steady state c when f > 0 and the proof is simi-

lar. The intuition behind proposition 3 is as follows. The first-best allocation would be to have only the most productive firm (i.e., the firm with ε t (i) = ε _ ) investing in fixed capital and all other firms lending to the most efficient firm. However, due to market incompleteness, and the inability to share risk across firms, less efficient firms will also opt to invest in fixed capital despite the high costs (i.e., high realiza-tions of ε t (i)). Because tulips provide a self-insurance device that enables firms to intertemporally transfer funds and thus reduce borrowing constraints, a bubble equi-librium reduces the need for less efficient firms to invest in fixed capital (because the alternative of investing in tulips may yield higher expected returns when ε t (i) is sufficiently high). So the fraction of firms investing in fixed capital is lower in a bubble steady state than in a no-bubble steady state. Since the number of firms investing is determined by the cutoff (i.e., a firm will invest if and only if ε t (i) ≤ ε * ), the existence of a tulip bubble also lowers the cutoff. However, although the number of capital-investing firms is smaller at a bubble equilibrium, the average efficiency

15 Alternatively, we can solve steady state c by taking the values of ζ and f as given, and then solving for the cutoff using equation (42), which is well-defined when f > 0. Once the cutoff is determined, we can then solve for the rest of the variables (including Ω).


of investment is improved (because inefficient firms drop out). This improvement in aggregate investment efficiency enables the economy to have both higher consump-tion (c) and a larger capital stock (K) with the same amount of total investment expenditure (i). Nonetheless, the total investment-to-output ratio could be either lower or higher in a bubble equilibrium, depending on the relative strength of the income effect (due to investment efficiency) and the substitution effect (since invest-ment crowds out consumption). These in turn depend on the model’s parameter values and the distribution of ε t (i). Impulse response analysis reveals, however, that aggregate investment expenditure ( i t ) and the investment-to-output ratio ( i t / y t ) always increase along the transitional path toward steady state when more bubbles are injected into the economy (see, e.g., Figure 3 in the next section).

PROPOSITION 4:

(i) if ζ = 0 and f = 0, then a fundamental equilibrium exists where q = 0, Ω = 0, and ϕ * ≥ 0. in addition, a bubble equilibrium also coexists where q > 0, Ω = 1, and ϕ * = 0 provided that β is sufficiently large.

(ii) if ζ = 0 and f > 0, then only one equilibrium exists. This equilibrium is a fundamental equilibrium with q = f, Ω = 0 and ϕ * ≥ 0 if agents are suf-ficiently impatient (i.e., β is small enough); or a bubble equilibrium with q > f, Ω = 1, and ϕ * = 0, if β is large enough. in other words, given β (and other parameters), a fundamental equilibrium and a bubble equilibrium can-not coexist if f > 0 and ζ = 0.

(iii) if ζ > 0 and f ≥ 0, then a fundamental equilibrium may exist where q = f, Ω = 0, and ϕ * ≥ 0 ; and a bubble equilibrium may also coexist where q > f, Ω = 1, and ϕ * = 0, provided that β is large enough.

PROOF:See Appendix IV.Cases (i) and (ii) of Proposition 4 apply to steady state c as well, but case (iii)

does not. Case (i) of Proposition 4 states that multiple equilibria are possible if tulip assets have no fundamental value and agents are sufficiently patient. This is a stan-dard result in monetary theory. Case (ii) of Proposition 4 states that when the storage cost ζ = 0 and the fundamental value f > 0, if the model’s structural parameters are such that q > f and Ω = 1 constitute a possible equilibrium, then q = f and Ω = 0 cannot be an equilibrium. In other words, bubbles will never burst if tulips (or money) have intrinsic value. On the other hand, if the parameters are such that q = f and Ω = 0 is an equilibrium, then there cannot be a bubble equilibrium. This suggests that a sunspot equilibrium is not possible in the models of Kiyotaki and Moore (2008) and Kocherlakota (2009) if land has intrinsic value but there is zero (or negligible) storage cost, regardless of the value of β.

This result is a bit counterintuitive, so we provide some explanations here. When a storable good has fundamental value (e.g., f > 0), then eating it or holding it as a store of value involves a trade-off and this trade-off depends on the expected future


value of the good, β q t+1 Q t+1 , where Q is the option value of cash. Since f > 0, the expected future value of tulips is strictly positive for any firm even if the firm does not expect other firms to hold the good as an asset in the next period (that is, even if the asset is only worth f in the future). Thus, if the good is an effective device for self insurance (i.e., the option value Q is large enough so that the rate of return βQ ≥ 1), all firms will opt to hold the good as an asset, and this is the only equilibrium because the behavior of other firms no longer matters in each firm’s decision making. Hence, the aggregate demand for the asset can never be zero (i.e., Ω > 0). On the other hand, if β is small enough, then βQ < 1 regardless of how other firms behave, so eating the asset is the only equilibrium. Therefore, multiple equilibria are not pos-sible if ζ = 0 and f > 0.

An alternative explanation is to note that when the storage cost ζ = 0, equation (31) and equation (36) cannot be satisfied simultaneously. In a no-bubble steady state, βQ( ε b * ) = 1 − ϕ b * /f ≤ 1 from equation (36). On the other hand, Proposition 3 shows that the cutoff in a bubble steady state must satisfy ε * < ε b * . This suggests that βQ( ε * ) < 1 because Q decreases with ε * , which is inconsistent with equation (31). So if Ω = 0 is an equilibrium, then Ω > 0 cannot be an equilibrium, and vice versa.

Therefore, the only situation in which a bubble steady state and a fundamental steady state can coexist when f > 0 is for the storage cost ζ to be sufficiently large but still below a threshold level.16 This explains case (iii) in Proposition 4. The intu-ition is as follows. First, if ζ is sufficiently large and q = f, no firm has incentives to deviate from the fundamental equilibrium by investing in tulips because the stor-age cost is too high to yield a large enough expected asset return E t β

f _

f + ζ Q t+1 < 1. On the other hand, if firms expect other firms to hold tulips and the market price of tulips is sufficiently high relative to the fundamental value f and the storage cost ζ, then it may also be in their own interest to hold tulips because the expected future return, E t β

q _

q + ζ Q t+1 > E t β f _

f + ζ Q t+1 , is large enough (due to a high enough equilib-rium tulip price q). This also suggests that steady state c (where q = f ) can never coexist with steady state B if f > 0. That is, case (iii) in Proposition 4 does not apply to steady state c.

COROLLARY 1: Steady state c cannot coexist with steady state B if f > 0.

As a numerical illustration of the possible equilibria and the associated alloca-tions, suppose we calibrate the model’s structural parameters as follows. Assume that ε(i) follows the Pareto distribution, F(ε) = 1 − ε −θ , with the shape param eter θ = 1.5 and the support (1, ∞).17 With this distribution, we have Q = θ _

1 + θ ε * + 1

_ 1 + θ ε

*−θ and ω = ( ∫ε≤ ε *

ε −1 dF)/F = θ _ 1 + θ

(1 − ε *−θ−1 ) _

1 − ε *−θ . Normalize the steady-

state values of A = A n = 1 and set the capital’s income share α = 0.4, the time-discounting factor β = 0.99, the capital depreciation rate δ = 0.025, and the inverse elasticity of labor supply γ n = 0 (indivisible labor).

16 Infinitely large storage costs will eliminate the bubble equilibrium because no firms can afford keeping tulips.17 The results are not sensitive to the values of θ. For example, θ = 3 yields similar results.


Under these parameter values, the cutoff at steady state B is ε b * = 1.923. Suppose the storage cost is ζ = 0.01 (i.e., case (iii) in Proposition 4). The critical value of ζ for steady state B to exist is then given by ζ ≥ f [βQ( ε b * ) − 1] = 0.2905f, or equiva-lently, f ≤

ζ _

0.2905 . Then there exist two steady states (A and B) if 0 < f ≤ 0.0344,

and there exists only one steady state (A or c or B) if f > 0.0344. Table 1 reports the equilibrium allocations of these different steady states.

The table shows several things. First, there are two equilibria when 0 < f ≤ 0.0344, confirming case (iii) in Proposition 4. Second, steady state A has a lower cutoff ( ε * = 1.139) than steady state B ( ε * = 1.923), confirming Proposition 3 that there are more firms investing in steady state B than in steady state A. However, steady state A has a higher capital stock and output (K = 39 and y = 3.859) than steady state B (K = 24 and y = 3.127) because of higher investment efficiency. Third, there exists only one equilibrium when f = 0.1 > 0.0344, which turns out to be steady state A, not steady state B. Fourth, it is possible for steady state c to exist if the fundamental value f increases significantly while the storage cost remains at ζ = 0.01, as shown in the last row of the table. Notice that the allocation at steady state c is similar to that at steady state A even though only half of the tulips are traded at steady state c. Nonetheless, the market price of tulips is significantly higher (q = f = 9.291) when there are fewer tulips on the market, which generates high liquidation values for tulips and thus improves investment efficiency.

II. Systemic Risk and Asset Price Volatility

The benchmark model can be extended to explain asset price volatility in the US economy by allowing the possibility for bubbles to burst (as in Kocherlakota 2009). We introduce multiple assets and stochastic sunspot shocks which affect the burst probability. The idea of multiple bubble assets is akin to those of Kareken and Wallace (1981) and King, Wallace, and Weber (1992). Although the steps for deriving equilibrium conditions are similar and analogous to those in the benchmark model, we will detail some of the equations for the sake of completeness.

Multiple assets allow the possibility of recurrent bubbles. In the benchmark model, a bubble never comes back once it bursts. If the same bubble could come back, it would be rational to hold the asset forever, so the bubble would never burst in the first place. If we allow new bubbles to emerge in each period, the recurrent bursting of bubbles becomes possible. Assume that new bubbles differ from the

Table 1—Steady State Allocations for Case iii (ζ = 0.01, f > 0)

y c i K N q ε * Multiple Equilibria ( f ≤ 0.0344) A (Ω = 1.0) 3.859 2.809 1.040 39.10 0.824 4.326 1.139 B (Ω = 0) 3.127 2.345 0.781 24.16 0.800 f 1.923

Unique Equilibrium ( f > 0.0344) A (Ω = 1, f = 0.1) 3.859 2.809 1.040 39.10 0.824 4.326 1.139 C (Ω = 0.5, f = 9.291) 3.870 2.820 1.046 39.42 0.824 9.291 1.131

Note: y = output, c = consumption, i = investment, K = capital, N = hours, q = tulip price, ε * = cutoff.


old ones only by color, and that producing new bubbles costs no social resources. Except colors, the new bubbles are otherwise identical to the old ones.

Assume there is a continuum of types of “tulips” indexed by a spectrum of colors j ∈ R+. To simplify the analysis, tulips are assumed to be perfectly storable with no storage costs (ζ = 0), to differ only in their colors (types), and to have no intrinsic value ( f = 0).18 Thus, according to Proposition 4, each type of tulip can be the subject of a bubble with the following properties: its equilibrium price is zero if no firms in the economy expect other firms to invest in it; and the market price is strictly positive if all agents expect others to hold it.19

In each period a constant measure z of new colors of tulips is born (issued) by nature and equally distributed across firms.20 The newborn tulips are distributed equally to all agents (firms) as endowments, and issuing (producing) new tulips does not cost any social resources. The supply of each color (variety) of tulips is normal-ized to 1 and each tulip type has a unique color. Also, all tulips have the same prob-ability p t of disappearing (dying or being destroyed) in each period regardless of color, but these events are independent of each other. In other words, different types of assets decay independently but the probability of decaying is a common shock that hits all types of assets. This assumption captures the concept of systemic risk.

Let q t j denote the price of a tulip with color j and h t+1 j (i) the quantity of the tulip

j demanded by firm i ∈ [0, 1]. The aggregate number (stock) of tulips evolves over time according to the law of motion:

(46) Ω t+1 = (1 − p t ) Ω t + z,

where Ω is the measure of the stock of tulips in the entire economy. The market clearing condition for tulips with color j is

(47) ∫ i=0

1

h t+1 j (i) di = 1.

As in the benchmark model, firms have the same constant returns to scale produc-tion technologies and are hit by idiosyncratic shocks to the marginal efficiency of investment ε(i). A firm’s problem is to determine a portfolio of tulips to maximize discounted future dividends. Its resource constraint is

(48) d t (i) + i t (i) + ∫ j∈ Ω t+1

q t j h t+1 j (i) dj + w t n t (i)

= A t k t (i ) α n t (i ) 1−α + ∫ j∈ Ω t

q t j h t j (i) 1 t j dj + ∫ j∈z

q t j dj,

18 These assumptions reduce the number of parameters and simplify our calibration analysis.19 We ignore steady state c in the analysis for simplicity.20 We can also allow z to be a stochastic endowment shock.


where w is the real wage, Ω is the set of available colors of tulips (this is abusing notation slightly because Ω was used as the measure of total supply of tulips), and the index variable 1 t j satisfies

(49) 1 with probability 1 − p t 1 t j = { 0 with probability p t .

So each tulip bought in period t − 1 may lose its value completely with probability p t at the beginning of period t. As in the basic model, we impose the following con-straints: i t (i) ≥ 0, d t (i) ≥ 0, and h t+1 j

(i) ≥ 0 for all j ∈ Ω.Using the same definition of r as in the benchmark model, the firm’s problem is

to solve

(50) max E 0 ∑ t=0

∞

β t Λ t [ r t k t (i) − i t (i) + ∫ j∈ Ω t

q t j h t j (i) 1 t j dj

+ ∫ j∈z

q t j dj − ∫ j∈ Ω t+1

q t j h t+1 j (i) dj],

subject to

(51) d t (i) ≥ 0

(52) h t+1 j (i) ≥ 0 for all j

(53) i t (i) ≥ 0

(54) k t+1 (i) = (1 − δ) k t (i) + i t (i) _ ε t (i) ,

where the fourth term in the objective function, ∫j∈z

q t j dj, captures tulip injection by

nature in each period.

PROPOSITION 5: The firm-level investment decision rule is given by

(55) r t k t (i) + ∫ j∈ Ω t

q t j h t j (i) 1 t j dj + ∫ j∈z

q t j dj ε t (i) ≤ ε t * i t (i) = { 0 ε t (i) > ε t * .

The equilibrium asset price for tulips with color j is determined by

(56) q t j = β E t Λ t+1 _ Λ t

q t+1 j 1 t+1 j

Q t+1 .


PROOF: See Appendix V.The RHS of the asset-pricing equation (56) is the expected rate of return for tulip

j. This equation shows that if p t = 1 (i.e., 1 t j = 0 with probability 1), then tulip j ’s equilibrium price is given by q t j = 0 for all t because the demand for such an asset is zero when it has no market value in the next period. More importantly, even if p < 1, q t j = 0 for all t is still an equilibrium because no firms will hold tulip j if they do not expect others to hold it.

A. Aggregation and General Equilibrium

As in the benchmark model, at the aggregate level we have N t = [ (1 − α) A t _ w t ]

1 _ α K t , y t

= A t [ (1 − α) A t

_ w t ] 1−α _ α K t , y t = A t K t α N t 1−α , w t = (1 − α) y t / N t , and r t = α y t / K t . Consider a symmetric equilibrium21 where the prices of tulips of all colors are the same:

(57) q t with probability 1 − p t q t j 1 t j = { 0 with probability p t ,

where q t is the market price of tulips of all colors. Define x t (i) ≡ ∫j∈ Ω t

q t j h t j (i) 1 t j dj + ∫

j∈z

q t j dj and X t ≡ ∫0 1 x (i) di. By the law of large numbers, X t = (1 − p t ) Ω t q t + z q t .

Hence, aggregate capital investment is given by i t = (α y t + [(1 − p t ) Ω t + z] q t ) F( ε t * ).The households remain as in the benchmark model with f = ζ = 0. The first-

order conditions for the households are thus the same as before. The equilibrium paths of the model can be characterized by eight variables, {c, i, N, y, K′, q, ε * , Ω′ }, which are linked through the following system of eight nonlinear equations:

(58) y t = A t K t α N t 1−α

(59) c t + i t = y t

(60) (1 − α) y t _ c t = A n N t 1+ γ n

(61) Ω t+1 = (1 − p t ) Ω t + z

(62) q t _ c t

= β E t { q t+1 _ c t+1

(1 − p t+1 ) Q t+1 } (63) i t = [α y t + ((1 − p t ) Ω t + z) q t ] F ( ε * )

(64) ε t * _ c t = β E t { ε t+1 * _ c t+1

[α y t+1 _ K t+1

Q t+1 _ ε t+1 * + 1 − δ]} 21 A symmetric equilibrium exists because of arbitrage among tulips of different colors.


(65) K t+1 = (1 − δ) K t + i t ∫ε≤ ε *

ε −1 dF

_ F ( ε t * )

.

This system of equations is saddle-path stable, so the number of stable eigenvalues equals the number of predetermined state variables. The model has a bubble steady state and a fundamental steady state. The equilibrium dynamics of the model can be solved by log-linearizing the system of equations around the bubble steady state.

B. Stationary Sunspot Equilibria

We call tulip j a bubble asset if q t j > 0. When the bubble bursts, q t j = 0. Arbitrage ensures that after a bubble bursts, its value must remain zero permanently, otherwise people may opt to hold it indefinitely for speculation. In each period, a fraction p t of the bubbles burst and z new bubbles are born. Changes in p t are driven completely by exogenous action, call it sunspots, which can be any stochastic process. In what follows, we focus on stationary sunspot equilibria with positive and bounded asset prices ( q t > 0 for all t).

The sunspot equilibrium condition, q > 0, puts some restrictions on the values of p. With q > 0, equation (62) implies that 1 = β(1 − p)Q( ε * ) at steady state. Equation (64) implies that Q( ε * ) = 1 − β(1 − δ)

_ βr ε * . Together we have

(66) 1 − p = r __ 1 − β (1 − δ)

1 _ ε * .

This equation determines the cutoff value ε * (p) as a function of p. It is clearly satis-fied when p = 0. Proposition 3 shows that the cutoff in the bubble steady state is less than the cutoff in the no-bubble steady state ( ε b * ). So an interior solution for the cutoff requires that ε * (p) < ε b * . Thus, it must be true that

(67) 1 − p > r __ 1 − β (1 − δ)

1 _ ε b *

.

Notice that r _

1 − β(1 − δ) = ∑ j=0 ∞ β j (1 − δ) j α y _ K is the present value of the marginal

products of capital and 1/ε is the marginal efficiency of investment. So equation (67) states that the real expected rate of return on tulips (i.e., the survival prob-ability of a speculative bubble) must be comparable to that of capital investment (Tobin’s q) to induce people to hold both capital and bubble assets simultaneously.

Since li m ε * → ε _ r _

1 − β(1 − δ) (1/ ε * ) = li m ε * → ε _ 1 _ βQ( ε * ) = 1/β, the larger the spread

between the lower and upper bounds of support [ ε _ , _ ε ], the larger is the permissible

region for the value of p. This simply restates the finding that uninsured idiosyn-cratic uncertainty in the expected rate of return on capital investment (or Tobin’s q) is the fundamental reason for people to invest in bubbles. When idiosyncratic assessments of risks converge (e.g., ε _ = _ ε = ε * ), it becomes impossible for bubbles to arise (i.e., the measure of sunspot equilibria becomes zero).


At steady state, equations (62) through (65) become

(68) 1 = β (1 − p) Q ( ε * )

(69) i = (αy + Ωq) F ( ε * )

(70) 1 − β (1 − δ) = βα y _ K

Q ( ε * ) _

ε *

(71) δK = i ∫ε≤ ε *

ε −1 dF

_ F ( ε * ) .

Equation (68) solves for the cutoff value ε * . Equation (70) implies that the capital-to-output ratio K _ y = βα

_ 1 − β (1 − δ)

Q _ ε * . Equations (71) and (70) give the household sav-

ings rate, i _ y = αβδ _

1 − β(1 − δ) FQ _

Q −1 + F , where Q − 1 + F = ε * ∫ε≤ ε *

ε −1 dF. Equation (69)

defines the asset-to-output ratio as a function of the savings rate, Ωq

_ y = 1 _

F( ε * ) i _ y − α.

As in the basic model, to ensure that Ωq

_ y > 0, we must have i _ y > αF, which implies δ > ( β −1 − 1 + F) (1 − β(1 − δ)).

C. calibration and impulse responses

Assume that ε(i) follows the Pareto distribution, F(ε) = 1 − ε −θ , with the shape parameter θ = 1.5 and the support (1, ∞).22 With this distribution, Q = θ _

1 + θ ε * +

1 _

1 + θ ε *−θ . Normalize the steady-state values of z and A to 1 and calibrate the struc-

tural parameters of the model as follows: the time period is a quarter, the capital’s income share α = 0.4, the time-discounting factor β = 0.99, the capital deprecia-tion rate δ = 0.025, and the inverse elasticity of labor supply γ n = 0 (indivisible labor as in Hansen 1985).

With these parameter values the aggregate investment-to-output ratio (i/y) increases with the quantity of tulips. In other words, the ratio is a decreasing func-tion of the steady-state probability of bubbles bursting ( _ p ) or the asset-to-output ratio (Ω/y). For example, when

_ p = 0.1, then Ωq/y = 0.147 and i/y = 0.253; and

when _ p = 0.22, then Ωq/y = 0.0034 and i/y = 0.249.

The processes driving the model are assumed to be AR(1) processes:

(72) ln p t = ρ p ln p t−1 + (1 − ρ p ) ln _ p + ε pt

(73) ln A t = ρ A ln A t−1 + ε At ,

where the steady-state probability of bubbles bursting is set to _ p = 0.1. Set ρ p

= ρ A = 0.9. Figure 2 plots the impulse responses of the model economy to a

22 The results are not sensitive to the values of θ. For example, θ = 3 yields similar results.


10 percent increase in the probability of bubbles bursting ( ε pt ) and compares them with the responses to a 1 percent decrease in total productivity ε At (see Figure 3). A positive shock to ε p is akin to a financial crisis because it implies a higher systemic financial risk.

The solid lines in Figure 2 show that a persistent increase in systemic risk (i.e., the burst probability p t , see the first window in the upper panel) generates a prolonged recession in aggregate output (the second window in the upper panel) and a dramatic drop in asset prices (the first window in the lower panel). When the perceived rate of return on tulips decreases (or the financial risk increases), agents rationally decrease their demand for tulips anticipating that the demand for tulips will be persistently low in the future, leading to a sharp fall in the asset’s price. Because tulips provide liquidity for firms, “fire” sales of tulips reduce firms’ net worth and working capital, leading to significant declines in output, employment, and capital investment. The decline in output is U-shaped because tulip assets enter firms’ investment decision rules (equation (63)) directly as a stock variable—less tulip investment today reduces a firm’s cash position and capital investment for many periods to come and thus propagates the impact of risk shock over time. Thus, a persistent AR(1) risk shock will tend to induce an AR(2) dynamic structure in output and other variables. Such hump-shaped output dynamics suggest that asset prices lead output over the business cycle. This behavior provides a key test of business cycle models (see Cogley and Nason 1995), and this model passes this test with flying colors.

Importantly, asset prices are far more volatile than the fundamentals. For exam-ple, the initial drop in asset prices is more than 25 times larger than that in output,

Shock to probability Output Employment Investment

Asset price Consumption Liquidity premium Interest rate

10

9

8

7

6

5

4

3

2

1

0

0.02

−0.02

−0.06

−0.10

−0.14

−0.18

−0.22

−0.26

0.05

0.00

−0.05

−0.10

−0.15

−0.20

−0.25

−0.30

−0.35

0.1

−0.1

−0.3

−0.5

−0.7

−0.9

−1.1

−1.3

1.00.50.0

−1.0

−2.0

−3.0

−4.0

−5.0

−6.0

0.16

0.12

0.08

0.04

0.00

−0.04

−0.08

−0.12

0.22

0.18

0.14

0.10

0.06

0.02

0.3

0.2

0.1

0.0

−0.1

−0.2

−0.3

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

p = 0.1

p = 0.22

p = 0.1

p = 0.22

p = 0.1

p = 0.22

p = 0.1

p = 0.22

p = 0.1

p = 0.22

p = 0.1

p = 0.22

p = 0.1

p = 0.22

Figure 2. Impulse Responses to Risk Shock


resembling a typical stock market crash.23 The figure suggests that a 50 percent fall in asset prices can cause more than 2 percent decrease in output. This is similar to the magnitude of the recent US experience during the subprime mortgage crisis.24 In the meantime, aggregate employment (the third window in the upper panel) and investment (the fourth window in the upper panel) also decrease sharply after the shock, with investment dropping nearly six times more than output. Aggregate con-sumption (the second window in the lower panel) significantly lags output. The reason that consumption responds to the shock positively in the initial period is because of the higher dividends paid to households after a sharp reduction in fixed investment. But consumption eventually decreases because of the persistently lower aggregate income.

The liquidity premium ( Q t ) is countercyclical in this model (see the third window in the lower panel). Equation (62) indicates that the expected asset return is deter-mined by (1 − p) Q. So a higher burst probability requires a higher liquidity pre-mium to induce firms to hold the bubble assets at equilibrium. However, the increase in the liquidity premium will be smaller in absolute magnitude than the increase in p because the anticipated growth in asset prices ( E t

q t+1 _ q t ) lowers the required liquid-

ity premium along the transitional path. In the meantime, the real interest rate (the fourth window in the lower panel) declines initially and then rises in the middle of the deep recession because of a large reduction in the capital stock due to the lack of investment.

Proposition 3 states that the investment-to-output ratio may not necessarily be higher in a bubble steady state than in a no-bubble steady state. But the impulse response analysis in Figure 2 shows that along the transitional path the investment-to-output ratio increases with the quantity of bubble assets because investment is more volatile than output.

To show that bubbles amplify the business cycle, Figure 2 also plots the impulse responses of the model to a systemic risk shock (of the same magnitude) when there are less tulips in the economy to start with (i.e., the calibrated steady-state value of

_ p increases from 0.1 to 0.22, suggesting a lower steady-state asset-to-output

ratio Ωq/y). The results are presented in Figure 2 using dashed lines in each win-dow. For example, the windows in the upper panel show that output, employment, and capital investment become much less volatile when the steady-state quantity of bubble assets is reduced. This suggests that the existence of bubbles amplifies the impact of financial shocks. In particular, as the steady-state quantity of bubble assets approaches zero (i.e., as

_ p approaches one), systemic risk ceases to have any impact

on the economy (except on the asset price q t ).25

Figure 3 (solid lines) shows the impulse responses of the model to a negative 1 percent technology shock. An adverse shock to aggregate TFP (the first window

23 The dynamic movements in the number of firms investing are quantitatively important to our results. For example, if the idiosyncratic shocks follow a binary distribution (as in Kiyotaki and Moore 2008), the relative volatility of asset prices would be reduced by more than 20 percent.

24 For example, between May 2008 and March 2009, real GDP in the United States declined by 2.63 percent and the S&P 500 index dropped by 44 percent. This was a drop of more than 50 percent from the peak in 2007.

25 Unlike the other economic variables, asset price q t is more volatile relative to its steady-state value when there are fewer tulips (see the dashed line in the first window in the lower panel of Figure 2).


in the upper panel) can generate a fall in asset prices (solid line in the first window in the lower panel). The fall in asset prices is twice as large as the fall in output (the second window in the upper panel). Asset prices fall because firms reduce tulip demand when their investment demand declines after a reduction in TFP. However, the impulse response of output is not hump-shaped and the asset-price volatility relative to output volatility is not big enough to match the US data. In addition, the liquidity premium drops along with asset prices (3rd window in the lower panel). The reason is that the supply of tulips has not changed but the demand for them has declined. This excess supply of tulips leads to an immediate drop in their current price but also an anticipated rise in their future price (namely, E t [ q t+1 − q t ] > 0). So the equilibrium liquidity premium declines accordingly, which is inconsistent with data. Therefore, our analysis suggests that to understand the excessive asset market volatilities and movements of liquidity premium over the business cycle, systemic financial risk shocks are more important than shocks to TFP.

The dashed lines in Figure 3 show that bubbles amplify the effects of TFP shocks too, although the effects are not as significant as those under financial risk shocks. For example, when the steady-state asset-to-output ratio is low (i.e.,

_ p = 0.22 and

Ωq/y = 0.0034), the declines in aggregate output, employment, and investment (dashed lines in the upper windows of Figure 4) are less than when the steady-state asset-to-output ratio is high (solid lines in the upper windows where

_ p = 0.1 and

Ωq/y = 0.147).To test whether the model can match US time-series data quantitatively,

the technology shocks process { A t } is calibrated using the Solow residual, and

Shock to technology Output Employment Investment

Asset price Consumption Liquidity premium Interest rate

0.0

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

−0.7

−0.8

−0.9

−1.00 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

0.0

−0.2

−0.4

−0.6

−0.8

−1.0

−1.2

−1.4

0.1

0.0

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

0.2

−0.2

−0.6

−1.0

−1.4

−1.8

−2.2

−2.6

−3.0

3.0

2.5

2.0

1.5

1.0

0.5

0.0

−0.5

−1.0

−1.5

−2.0

−2.5

−3.0

0.0

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

−0.7

−0.8

−0.9

−1.0

0.1

0.0

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

−0.7

0.30.20.10.0

−0.2

−0.4

−0.6

−0.8

−1.0

−1.2

−1.4

p = 0.1

p = 0.22

p = 0.1

p = 0.22

p = 0.1

p = 0.22

p = 0.1

p = 0.22

p = 0.1

p = 0.22

p = 0.1

p = 0.22

p = 0.1

p = 0.22

Figure 3. Impulse Responses to TFP Shock


the parameters { ρ p , σ εp } for the driving process { p t } are chosen so that the model-predicted asset price volatility ( σ q ) relative to output volatility ( σ y ) matches the empirical counterpart ( σ q / σ y ) for the US economy. More specifically, we follow King and Rebelo (1999) by setting ρ A = 0.98 and σ εA = 0.0072 for technology shocks. Since many asset prices share similar business-cycle properties with stock prices, we use real S&P 500 index data (deflated with the GDP deflator) for the period (1947:1–2009:1) as a proxy for q t . The estimated standard deviation of the S&P 500 index is σ Sp = 0.099, which is 6 times the standard deviation of real GDP ( σ y = 0.017). Like King and Rebelo (1999), the Hodrick-Prescott filter is applied to both the US data and the model-generated time series prior to moment estima-tion. That is, we apply the HP filter to the logged series when estimating the sec-ond moments. With the specified driving process for { A t , p t }, setting ρ p = 0.9 and σ εp = 12 × 0.0072 = 0.086 4 in the model generates a ratio of σ q / σ y = 6.0. So the model is able to exactly match the relative volatility of asset prices in the US data by properly choosing the two parameters of the driving process { p t }. The calibrated parameter values are summarized in Table 2.

To check whether the calibrated values for { ρ p , σ εp } are reasonable, we estimate a univariate AR(1) model using the HP filtered S&P 500 data and obtain an AR(1) coefficient of 0.85 and a standard deviation of the innovation of 0.056, which are close in magnitude to the chosen values of { ρ p , σ εp }. So we believe the calibrated val-ues of { ρ p , σ εp } are empirically reasonable.26 Using the parameter values in Table 2, we generate time series samples from the model, apply the HP filter to the artificial data, and estimate the model’s second moments.

Table 3 reports the second moments predicted from the model and their counter-parts in the data (where Δq ≡ E t q t+1 / q t − 1 denotes real asset returns).27 According to Table 3, the model’s predictions are broadly consistent with the US data. For exam-ple, in terms of standard deviations (top-left panel in Table 3), the model is able to explain 70 percent of the output fluctuations and 75 percent of the stock price volatility in the data. In terms of relative volatilities with respect to output, the model predicts that consumption is about 40 percent less volatile, investment about 2.6 times more

26 We admit that using the S&P as a proxy for tulip prices in our model is not perfect and perhaps even prob-lematic, but we used it because it is the most readily available data for asset prices with high data quality. In addi-tion, setting f > 0 in our model does not change our results (see Table 1 for examples where q > f > 0) because we are investigating movements around the steady state where the value of f is irrelevant as long as it satisfies the constraints discussed in the previous sections. Also, we matched the standard deviation of the S&P using two driv-ing forces in our model: technology shocks and sunspot shocks. So part of the business cycle movements in q t will reflect changes in technology fundamentals, suggesting that q t is not driven entirely by non-fundamentals.

27 The data are quarterly and include real GDP (y), nondurable goods consumption (c), total fixed investment (i), total private employment by establishment (N), and the S&P 500 price index normalized by the GDP deflator (q). The sample covers the period 1947:1—2009:1.

Table 2—Parameter Values

β α δ γ n θ0.99 0.4 0.025 0 1.5

_ p ρ A σ εA ρ p σ εp

0.1 0.98 0.0072 0.9 0.0864


volatile, and asset prices 6 times more volatile than output (this value is calibrated). These predictions are broadly consistent with the US data. The correlation between stock prices and output at the business cycle frequencies is 0.46 in the data. This value is 0.73 in the model, qualitatively matching the data. The model can also generate strong autocorrelations in output, consumption, investment, labor, and asset prices that are close to the data. As in the data, the model also predicts a negative correla-tion between expected asset returns and output (see the last column in the lower-right panel). The gap between the predictions and the data is most significant in the relative volatility of asset returns and employment with respect to output. The model explains less than 15 percent of the volatility in asset returns (changes in asset prices) and less than half of the volatility in employment (even with indivisible labor).

As mentioned at the start of this paper, a key feature of the tulip mania is that tulip prices grew faster than income before the collapse of the bubble. We can simulate a tulip bubble with similar characteristics (as shown in Figure 1) using the model. For example, assuming the time period to be a quarter and letting the probability of bursting follow a moving average process

(74) p t = _ p + ∑ j=0

T−1

α j ε pt−j ,

where ε pt represents zero mean independently and identically distributed innova-tions. Suppose T = 9,

_ p = 0.1, and the probability weight vector α = 1

_ 100 [1.25, 3, 1, 1, 1, 1, 0.25, −2, −6.5]. The simulated tulip bubble is graphed in Figure 4 (top panel).28 The lower panel of Figure 4 shows the one-period ahead forecast of output

28 Partly because of the caveat in footnote 26, and partly because we have used the tulip mania to motivate our paper, we are interested in checking if our model is able to generate a pure bubble just like the tulip mania. However, since the tulip bubble only lasted for several quarters, we cannot use an AR(1) process for p t to drive our model.

Table 3—Selected Business Cycle Moments

Standard deviation ( σ x )x y c i N q ΔqData 0.017 0.010 0.050 0.016 0.099 0.061Model 0.012 0.007 0.031 0.007 0.074 0.009

SD relative to y ( σ x / σ y )x y c i N q ΔqData 1 0.60 2.98 0.98 5.93 3.67Model 1 0.57 2.56 0.55 6.0 0.75

corr(x, y) x y c i N q ΔqData 1 0.78 0.77 0.83 0.46 − 0.15Model 1 0.89 0.95 0.88 0.73 − 0.47

corr( x t , x t−1 ) x y c i N q ΔqData 0.83 0.81 0.84 0.90 0.80 0.39Model 0.75 0.75 0.76 0.77 0.71 0.70

Note: y: output, c: consumption, i: investment, N: hours, q: asset price, Δq: asset returns.


growth. Output growth follows a path similar to that of asset price but with a mag-nitude a couple of orders smaller.

The vector α has zero mean and determines the bubble’s shape. The intuition behind the values of α is as follows. Because agents are forward-looking, they react to good financial news by buying tulips now when they perceive that the burst prob-ability of bubbles will be low several periods from now. Tulip prices thus increase immediately. To prevent a big jump in current tulip prices, there must be enough bad news today so that investors are cautious about entering the tulip market. This is why α takes positive values initially, so that the bubble grows only gradually. Because of the internal propagation mechanism in the model to transmit current shocks into the future, the asset price overshoots its steady-state value from above after a crash and does not return to the steady state immediately.

III. Conclusion

This study has developed an infinite-horizon DSGE model with incomplete finan-cial markets to explain asset bubbles and asset price volatility over the business cycle. It characterized conditions under which bubbles may arise. The results show that rational agents are willing to invest in bubbles despite their positive probability of bursting and that changes in the perceived systemic risk in the asset market can

An AR(1) process is an infinite-period moving average process, whereas Figure 1 shows that the tulip price index is a finite moving average process. So we choose to use a finite-order moving average process for p t to drive out the model. Thus, Figure 4 shows the response of asset price in the model to multi-period independently and identically distributed shocks, where the value of α j indicates the sign and size of the shock in period j.

Figure 4. Simulated Tulip Bubble

Quarters

For

war

d G

DP

gro

wth

(%

)A

sset

pric

e (%

)

0.4

0.3

0.2

0.1

0.0

−0.1

−1 0 1 2 3 4 5 6 7 8 9 10 11

−1 0 1 2 3 4 5 6 7 8 9 10 11

9080706050403020100

−20


trigger boom-bust cycles and asset price collapse. Calibration exercises confirm that the model has the potential to quantitatively explain the US business cycle, espe-cially its hump-shaped output dynamics and asset price volatility.

However, as Santos and Woodford (1997) have pointed out, bubbles are frag-ile in the sense that if there exist interest-bearing assets that are as liquid as the tulips, there will be no bubbles in the model. It would be interesting in future research to consider welfare and optimal policies in such bubble situations. Another interesting avenue of research would be to study bubbles in assets with nonstationary prices.

Appendix I: Proof of Proposition 1

PROOF: Let {μ(i), ϕ t (i), π(i), λ(i)} denote the Lagrangian multipliers of constraints (8)

through (11), respectively. The first-order conditions for { i t (i), k t+1 (i), h t+1 (i)} are given, respectively, by

(75) 1 + μ t (i) = λ t (i) _ ε t (i) + π t (i)

(76) λ t (i) = β E t Λ t+1 _ Λ t

{[1 + μ t+1 (i)] r t+1 + (1 − δ) λ t+1 (i)}

(77) [1 + μ t (i)] ( q t + ζ) = β E t Λ t+1 _ Λ t

{ q t+1 [1 + μ t+1 (i)]} + ϕ t (i),

plus the complementary slackness conditions,

(78) π t (i) i t (i) = 0

(79) ϕ t (i) h t+1 (i) = 0

(80) [1 + μ t (i)] [ r t k t (i) − i t (i) + q t h t (i) − ( q t + ζ) h t+1 (i)] = 0.

Notice that equation (76) implies that the value of λ t (i) is the same across firms because ε(i) is independently and identically distributed and is orthogonal to aggre-gate shocks.

We derive the optimal decision rules of firms using a guess-and-varify strategy. Given that ε t (i) is independently and identically distributed, we conjecture that the Lagrangian multipliers, { μ t (i), ϕ t (i), π t (i)}, depend only on idiosyncratic shocks and the aggregate states in period t. Thus, by the orthogonality assumption between aggregate and idiosyncratic shocks, and by the law of iterated expectations, the first-order conditions (75)–(77) can be rewritten as

(81) 1 + μ t (i) = λ t (i) _ ε t (i) + π t (i)


(82) λ t (i) = β E t Λ t+1 _ Λ t

{[1 + _ μ t+1 ] r t+1 + (1 − δ) _ λ t+1 }

(83) [1 + μ t (i)] ( q t + ζ) = β E t Λ t+1 _ Λ t

{ q t+1 [1 + _ μ t+1 ]} + ϕ t (i),

where _ μ t ≡ ∫

μ t (ε) dF (ε), _ ϕ t ≡ ∫

ϕ t (ε) dF (ε), and _ λ t = ∫

λ t (ε) dF (ε) denote expected values.

The decision rules at the firm level are characterized by a cutoff strategy. Notice that equation (82) implies that λ t (i) = λ t is independent of i. Define the cutoff ε t * such that π t ( ε t * ) = μ t ( ε t * ) = 0. Then, by equation (A7), ε t * = λ t . Thus, by equation (82),

(84) ε t * ≡ β E t Λ t+1 _ Λ t

{[1 + _ μ t+1 ] r t+1 + (1 − δ) _ λ t+1 },

which determines the cutoff. The intuition for the cutoff to equal λ t is that λ t is the marginal value of installing one unit of capital, but to generate one unit of capital a firm needs to invest ε t (i) units of goods. Thus, investment is profitable if and only if ε t (i) ≤ λ t . Hence, λ t should be the cutoff ε t * . Also define ϕ t * = ϕ t ( ε t * ), then according to (83), we have

(85) ( q t + ζ) = β E t Λ t+1 _ Λ t

{ q t+1 [1 + _ μ t+1 ]} + ϕ t * .

Now, consider two possibilities:

Case A: ε t (i) ≤ ε t * . In this case, the cost of capital investment is low. Suppose i t (i) > 0; accordingly we have π t (i) = 0. Equations (81) and (82) imply

(86) ε t (i) [1 + μ t (i)] = β E t Λ t+1 _ Λ t

{[1 + _ μ t+1 ] r t+1 + (1 − δ) _ λ t+1 }.

Given that μ t (i) ≥ 0, it must be true that ε t (i) ≤ β E t Λ t+1

_ Λ t

{[1 + μ t+1 (i)] r t+1 + (1 − δ) λ t+1 (i)}, which is precisely the cutoff defined in equation (A10). Equa-tion (A7) then becomes

(87) 1 + μ t (i) = ε t * _ ε t (i) .

Hence, whenever ε t (i) < ε t * , we must have μ t (i) = ε t * _ ε t (i)

− 1 > 0 and d t (i) = 0. Equation (83) then becomes

(88) ε t * _ ε t (i) ( q t + ζ) = β E t

Λ t+1 _ Λ t { q t+1 [1 + _ μ t+1 ]} + ϕ t (i).


Given the definition of ϕ t * in equation (85), equation (88) implies ϕ t (i) > ϕ t * when-ever ε t (i) < ε t * . Given that ϕ t * ≥ 0 (because ϕ t (i) ≥ 0 for all ε t (i)), the fact that ϕ t (i) > ϕ t * under Case A yields

(89) ϕ t (i) > 0.

That is, if ε t (i) < ε t * , we must have

(90) h t+1 (i) = 0

and

(91) i t (i) = r t k t (i) + q t h t (i).

This suggests that firms opt to liquidate all financial assets to maximize investment in fixed capital when the cost of fixed investment is low.

Case B: ε t (i) > ε t * . In this case, the cost of investing in fixed capital is high. Suppose d t (i) > 0 and μ t (i) = 0. Then equations (81) and (82), and the definition of the cutoff ε t * , imply π t (i) = 1 − ε t *

_ ε t (i)

> 0. Hence, we have i t (i) = 0. In such a case, firms opt not to invest in fixed capital and instead pay shareholders a posi-tive dividend. Given that μ t (i) = 0, equation (83) implies ϕ t (i) = ϕ t * ≥ 0. That is, the Lagrangian multiplier ϕ(i) is the same across firms in Case B because ϕ t * is independent of i. However, depending on the liquidation value of tulips in the next period, there are two possible choices (outcomes) for tulip investment under Case B: (B1) ∫

0 1 h t+1 (i) di > 0 and (B2) ∫

0 1 h t+1 (i) di = 0. The first outcome (B1)

implies a positive aggregate demand for tulips (i.e., tulips are held as a store of value in the economy) because firms expect other firms to accept tulips in the future and the liquidation value is high enough to cover storage costs, so we must have ϕ t * = 0. The second outcome (B2) implies that tulips are not traded and all existing tulips are consumed (i.e., paid to households as dividends). Hence, we must have ϕ t * ≥ 0 and h t+1 (i) = 0 for all i. Given outcome (B2), we must also have q t = f.

Thus, whether a positive demand exists for tulips in Case B depends on firms’ expectations about the liquidation value of tulips in the future (i.e., about whether or not tulips will be traded in the next period). Denoting

(92) Ω t+1 ≡ ∫ 0 1

h t+1 (i) di

as the aggregate demand for tulips in period t, the two possible outcomes (B1 and B2) imply the equilibrium complementary slackness condition

(93) Ω t+1 ϕ t * = 0.

Combining Cases A and B, the decision rule for capital investment is given by equa-tion (18). The rate of return on tulips depends on the expected marginal value of


liquidity (cash flow), which is greater than 1 because of the option of waiting. This option value is denoted by

(94) Q( ε * ) ≡ E [1 + μ (i)] = ∫

m ax {1, ε * _ ε (i) } dF (ε) > 1.

When the cost of capital investment is low in period t (Case A), one tulip yields ε *

_ ε(i) > 1 units of new capital through investment by liquidating the tulip asset at market price q t . When the cost is high in period t (Case B), firms opt to hold on to the liquid asset as inventory and the rate of return is simply 1.

Using equations (88) and (85), the value of the Lagrangian multiplier for the non-negativity constraint (9) is determined by

(95) ( ε * _ ε (i) − 1) ( q t + ζ) + ϕ t * ε (i) ≤ ε * ϕ t (i) = { . ϕ t * ε (i) > ε *

This suggests that the cross-firm average shadow value of relaxing the borrowing constraint (9) by purchasing one additional tulip is

(96) ∫ 0 1

ϕ (i) di = (q + ζ) ∫ ε≤ ε *

( ε * _ ε − 1) dF (ε) + ϕ t *

= (q + ζ) (Q( ε * ) − 1) + ϕ * ,

which is independent of i but positively related to the tulip price q. Equation (85) becomes equation (21).

Appendix II: Proof of Proposition 2

PROOF: Because r t = α y t

_ K t , equation (19) can be written as ε t * = β E t { Λ t+1

_ Λ t

ε t+1 * [α y t+1

_ K t+1 Q( ε t+1 * ) _

ε t+1 * + 1 − δ]}. Also, the effective aggregate investment is given by

∫0 1 i(i) _ ε(i) di = ω( ε t * ) i t , where the coefficient ω( ε t * ) ≡ ( ∫ε≤ ε t *

ε −1 dF)/F( ε t * ) > 1/ ε t *

measures the marginal efficiency of aggregate investment. By the law of large num-bers, the firm-level investment decision rule in equation (18) implies the aggregate investment function in equation (26). Equation (22) is simply the aggregate produc-tion function, equation (23) is the aggregate resource constraint derived from equa-tion (17), equation (24) pertains to the household’s optimal labor supply decisions, equation (25) is the Euler equation for optimal tulip investment based on equation (21), equation (28) expresses the law of motion of aggregate capital accumulation, and equation (29) is the complementarity condition that determines whether tulips are traded or not in general equilibrium.


Appendix III: Proof of Proposition 3

PROOF: In steady state B, by equation (40), we have

(97) 1 − β (1 − δ) = β δ Q( ε b * ) __ ε b * ∫

ε≤ ε b *

ε −1 dF = β δ [1 + 1 − F( ε b * ) __

ε b * ∫ε≤ ε b *

ε −1 dF

];

whereas in steady state A, we have i/y > αF( ε * ) by equation (32). Hence, equa-tions (33) and (34) imply that

(98) 1 − β(1 − δ) = β δα y _ i

Q( ε * )F( ε * ) _ ε * ∫ε≤ ε *

ε −1 dF

< β δ [1 + 1 − F ( ε * ) _ ε * ∫ε≤ ε *

ε −1 dF

].

Equations (97) and(98) then imply that

(99) 1 − F ( ε * ) _

ε * ∫ε≤ ε *

ε −1 dF > 1 − F ( ε b * ) __

ε b * ∫ε≤ ε b *

ε −1 dF

,

or ε * < ε b * (with equality only when q = f = 0). That is, there are fewer firms invest-ing in fixed capital in the bubble equilibrium because the optimal cutoff ε * is lower

in steady state A. The marginal efficiency of aggregate investment is given by ω( ε * ) ≡

∫ε≤ ε *

ε −1 dF _

F( ε * ) in equation (28), which is decreasing in ε * , implying that investment

efficiency is higher in the bubble equilibrium than in the fundamental equilibrium. Also, the capital-to-output ratio is decreasing in the cutoff by equations (33) and (38). Thus, K/y > K b / y b .

To show that the aggregate investment-to-output ratio is not necessarily higher in a bubble equilibrium, it suffices to show that this ratio is not a monotonic function of the cutoff. Note first that the aggregate investment-to-output ratio in the model is strictly less than its counterpart in a frictionless economy with complete markets:

i _ y

= i _ K

K _ y

= βα __ 1 − β (1 − δ)

δ _ ω ( ε * ) Q ( ε * ) _ ε *

= βα __ 1 − β (1 − δ)

∫ ε min ε *

f (ε) dε __

∫ε≤ ε *

ε −1 f (ε) dε ∫ ε min

ε * ε

* _ ε f (ε) dε + ∫ ε * ε max f (ε) dε ___ ε *

= βα __ 1 − β (1 − δ) F ( ε * )[1 + 1 − F ( ε * ) __

ε * ∫ε≤ ε *

ε −1 f (ε) dε ],


where βα _

1 − β(1 − δ) is the investment to output ratio in a frictionless economy. Notice

that the coefficient term Δ( ε * ) ≡ F ( ε * )[1 + 1 − F ( ε * ) _

ε * ∫ε≤ ε *

ε −1 f (ε) dε

] is bounded above by 1

because

F ( ε * )[1 + 1 − F ( ε * ) __ ε * ∫ε≤ ε *

ε −1 f (ε) dε

] ≤ F ( ε * )[1 + 1 − F ( ε * ) _ F ( ε * ) ] = 1.

Because ε * ∫ε≤ ε *

ε −1 f (ε) dε > ∫ε≤ ε *

f (ε) dε = F( ε * ) for any interior value ε * ∈ ( ε _ , _ ε ),

we have Δ( ε * ) < 1 for any interior value ε * ∈ ( ε _ , _ ε ). This suggests that the invest-

ment-to-output ratio is the highest in a frictionless economy. Second, notice that Δ( ε _ ) = 1 (by L’Hospital’s rule) and Δ( _ ε ) = 1. Hence, the function Δ( ε * ) must be

U-shaped with a negative slope for ε * < ε and a positive slope for ε * > ε . Therefore,

whether ∂Δ( ε * ) _

∂ ε * is positive, negative, or zero depends on the value of ε * .

Appendix IV: Proof of Proposition 4

PROOF: The proof of the proposition can be demonstrated case by case.(i) Suppose ζ = 0 and f = 0. Then equation (36) is clearly satisfied if ϕ * = 0.

In such a case, Ω = 0 and q = 0 is an equilibrium because no firm has incentives to deviate by holding tulips when the liquidation value is zero. Hence, a fundamen-tal equilibrium with q = f = 0 exists. To prove the bubble equilibrium, suppose q > 0. Equation (31) implies that Q( ε * ) = 1/β, which solves for the cutoff value ε * as an interior point in ε ∈ [ ε _ , ε b * ] because 1/β > 1, provided ε b * is large enough. Given this, we have 0 < F( ε * ) < 1. Equation (33) implies that the capital-to- output ratio

(100) K _ y

= α __ 1 − β (1 − δ)

1 _ ε * .

This ratio in conjunction with equation (34) gives the aggregate saving rate

(101) i _ y

= αδ __ 1 − β (1 − δ)

F _ Q − 1 + F

,

where Q − 1 + F = ε * ∫ε≤ ε *

ε −1 dF. Equation (32) then implies the asset value-to-output ratio as a function of the saving rate,

(102) q _

y = [ i _

y 1 _

F ( ε * ) − α].To ensure that q > 0 (i.e., q/y > 0), we must have i/y > αF( ε * ), which implies the following restriction on the parameters:

(103) δ > ( β −1 − 1 + F) (1 − β (1 − δ)).


This condition is clearly satisfied if the household is sufficiently patient (i.e., β is close to 1). In this case, Q( ε * ) is close to 1, ε * is close to its lower bound ε _ , and F( ε _ ) is close to 0. Hence, q > 0, Ω = 1, ϕ * = 0, and firms have incentives to hold bubble assets. This is the case analyzed by Kiyotaki and Moore (2008), and Kocherlakota (2009).

(ii) Suppose ζ = 0 and f > 0. In this case, q < f is clearly not an equilibrium because the demand for tulips will rise to infinity. So let q ≥ f. First, if ζ = 0 and f > 0, steady state A and steady state B cannot coexist because equation (31) and equation (36) cannot hold simultaneously because Proposition 3 shows that the cutoff in steady state A must be lower than the cutoff in steady state B: ε * < ε b * , which implies that Q( ε * ) < Q( ε b * ). But equations (31) and (36) imply Q( ε * ) ≥ equilibrium (i.e., 1 = βQ( ε * ) by equation (31)) because an interior solution for ε * ∈ (ε, ε b * ) exists according to equation (103) in the previous analysis. That is, fol-lowing similar steps in case (i), equation (102) implies that q > f is equivalent to the following condition:

(104) αδ __ 1 − β (1 − δ)

1 __ β −1 − 1 + F

− α > f _ y

.

There exists a unique steady-state equilibrium whenever this condition is satis-fied. For example, the above condition is satisfied when β → 1. Third, if β is small enough, then steady state B is an equilibrium because equation (40) indicates that an unique interior solution for ε b * exists.

(iii) Suppose ζ > 0 and f ≥ 0. In this case, q = f, Ω = 0 and ϕ * ≥ 0 is an equi-librium if ζ is sufficiently large, ζ ≥ f [β Q( ε b * ) − 1], because firms do not have incentives to deviate from the fundamental equilibrium by investing in tulips if the storage cost is too high. Now consider whether q > f is also a possible equilibrium.

In this case, equation (31) implies βQ( ε * ) = q + ζ _ q ≡

_ κ > 1. Substituting this defini-

tion of _ κ into equation (33) gives K/y = α _

1 − β (1 − δ) _ κ _ ε *

. Equation (34) gives

(105) i _ y

= αδ __ 1 − β (1 − δ)

_ κ F __

β −1 κ − 1 + F .

Because equation (32) implies (102), the requirement q > f then implies

(106) αδ __ 1 − β (1 − δ)

_ κ __

β −1 _ κ − 1 + F

− α > f _ y

.

This condition can be easily satisfied if β → 1 and y is large enough (e.g., when TFP is high).

Notice that the left-hand side of condition (104) approaches infinity as β → 1 (because in this case Q( ε * ) → 1 and F( ε * ) → 0). Hence, assets with any intrinsic value can always support bubbles as long as agents are sufficiently patient. However, given β, the larger the fundamental value of an asset, the more difficult it is for bubbles to develop when f is too high, because the benefit of using tulips as a store of value does not outweigh the marginal utility of consumption.


Similarly, case (iii) (i.e., equation 106) states that the bubble-to-fundamental-value ratio, q/f, can be made arbitrarily large if β is sufficiently close to 1 and if the economy is sufficiently productive (i.e., the output level is sufficiently high due to a high TFP). Case (iii) also indicates that multiple equilibria are possible when f > 0 if and only if the storage cost ζ is strictly positive (i.e.,

_ κ ≡ q + ζ

_ q > 1) but not too large.

Appendix V: Proof of Proposition 5

PROOF: Let {μ(i), ϕ t j (i), λ(i), π(i)} denote the Lagrangian multipliers of constraints (51)

through (53), respectively. The first-order conditions for { i t (i), k t+1 (i), h t+1 j (i)} are

similar to those in the benchmark model:

(107) 1 + μ t (i) = λ t (i) _ ε t (i) + π t (i)

(108) λ t (i) = β E t Λ t+1 _ Λ t

{[1 + _ μ t+1 ] r t+1 + (1 − δ) _ λ t+1 }

(109) [1 + μ t (i)] q t j = β E t Λ t+1 _ Λ t

{ q t+1 j 1 t+1 j

[1 + _ μ t+1 ]} + ϕ t j (i),

plus the complementary slackness conditions,

(110) π t (i) i t (i) = 0

(111) ϕ t j (i) h t+1 j (i) = 0 for all j

(112) [1 + μ t (i)] [ r t k t (i) − i t (i) + ∫ j∈ Ω t

q t j h t j (i) 1 t j dj

+ ∫ j∈z

q t j dj − ∫ j∈ Ω t+1

q t j h t+1 j (i) dj] = 0.

As in the benchmark model, the decision rules at the firm level are characterized by a cutoff strategy. The following steps are analogous to those in the benchmark model. Consider two possibilities:

Case A: ε t (i) ≤ ε t * . In this case, the cost of capital investment is low. Suppose i t (i) > 0, then π t (i) = 0. Equations (107) and (108) imply that

(113) ε t (i) [1 + μ t (i)] = β E t Λ t+1 _ Λ t

{[1 + _ μ t+1 ] r t+1 + (1 − δ) _ λ t+1 }.


Given that μ t (i) ≥ 0, it must be true that ε t (i) ≤ β E t Λ t+1

_ Λ t

{[1 + _ μ t+1 ] r t+1 + (1 − δ)

_ λ t+1 }, which defines the cutoff value, ε t * :

(114) ε t * ≡ β E t Λ t+1 _ Λ t

{[1 + _ μ t+1 ] r t+1 + (1 − δ) _ λ t+1 }.

Equation (107) then becomes 1 + μ t (i) = ε t * _

ε t (i) . Hence, whenever ε t (i) < ε t * , we

must have μ t (i) = ε t * _

ε t (i) − 1 > 0 and d t (i) = 0. Equation (109) becomes

(115) ε t * _ ε t (i) q t j = β E t

Λ t+1 _ Λ t { q t+1 j

1 t+1 j [1 + _ μ t+1 ]} + ϕ t j (i).

Defining ϕ t * as the cutoff value of ϕ t j (i) for firms with ε t (i) = ε t * , equation (116) implies

(116) q t j = β E t Λ t+1 _ Λ t

{ q t+1 j 1 t+1 j

[1 + _ μ t+1 ]} + ϕ t * .

Given that ϕ t * ≥ 0, the fact that ϕ t j (i) > ϕ t * under Case A yields ϕ t j (i) > 0. That is, for any ε t (i) < ε t * , we must have h t+1 j

(i) = 0 for all j ∈ Ω t+1 and i t (i) = r t k t (i) + ∫j∈ Ω t

q t j h t j (i) 1 t j dj + ∫

j∈z

q t j dj. This suggests that firms opt to liquidate all tulip assets to maximize investment in fixed capital when the cost of fixed investment is low.

Case B: ε t (i) > ε t * . In this case, the cost of investing in fixed capital is high. Suppose d t (i) > 0 and μ t (i) = 0. Then equations (107) and (108) and the defini-tion of the cutoff ε * imply that π t (i) = 1 − ε t *

_ ε t (i)

> 0. Hence, we have i t (i) = 0. In such a case, firms opt not to invest in fixed capital and, instead, pay the sharehold-ers positive dividends. Because the market clearing condition for each tulip j is ∫

h t+1 j (i) di = 1, we must also have h t+1 j

(i) > 0 and ϕ t j (i) = 0 for all j ∈ Ω t+1 under Case B. Thus, equations (109) and (116) then imply that ϕ t * = 0.

Combining these two cases gives the decision rule for capital invest-ment in equation (55). The option value of liquidity is again defined by Q( ε * ) ≡ E[1 + μ(i)] = ∫

m ax {1, ε * _ ε(i) } dF(ε). Using equations (115) and (116), the Lagrangian multiplier for the nonnegativity constraint (52) is given by

( ε * _ ε (i)

− 1) q t j ε(i) ≤ ε *

(117) ϕ t j (i) = {

0 ε(i) > ε * ,

and the average shadow value of ϕ j is ∫ ϕ j (i) di = q j ∫

ε≤ ε *

( ε * _ ε − 1) dF(ε)

= q j (Q − 1), which is independent of i but proportional to a tulip’s price, q j . Inte-

gra ting equation (109) over i and rearranging gives equation (56).


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Speculative Bubbles and Financial Crises