Toxic asset bubbles

Abstract

We develop an overlapping generations model with leveraged investment in speculative asset bubbles. Financial intermediaries use borrowed funds to speculate on a risky asset bubble, which promises high returns as long as it does not collapse. They can, however, default on their debt and shift the losses to lenders when the bubble collapses. This risk shifting leads to welfare-reducing (or “toxic”) rational asset bubbles. We then analyze a set of often discussed policy interventions: pricking bubbles, macroprudential regulations, and leverage restriction.

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Fig. 1

Notes

  1. 1.

    See Galati and Moessner (2013) for a review of the literature.

  2. 2.

    For a related near-rational bubble model of bubbles in equity price, see Lansing (2012).

  3. 3.

    “Microfoundation for standard debt contract” in Appendix provides a microfoundation for this assumption by using costly state verification, as in Townsend (1979).

  4. 4.

    Formally, following Diamond (1965), we consider a benevolent planner who allocates consumption and capital investment to maximize steady-state consumption, subject to resource constraint \(c+gk=f(k)\). The optimization problem is: \(\max _{k\ge 0}f(k)-gk\). The solution, which is known as the “golden rule,” is \(k_\mathrm{gold}=(\alpha /g)^{\frac{1}{1-\alpha }}=k_{b}\) and \(c_\mathrm{gold}=c_{b}\).

  5. 5.

    If \(\alpha =0.4\) instead as in the previous numerical examples, bubbles do not exist because the existence condition in this case is \(\alpha <1/3\).

  6. 6.

    The equilibrium quantities \(k_{b},R_{b}^{k},w_{b},p_{b}\) in Proposition 3 are unaffected, while the expected consumption is slightly different and given by:

    $$\begin{aligned} c_{b,t}^{e}=[1-\lambda (p_{b}k_{b}^{\alpha }g^{t})](1-\alpha )\left( \frac{\alpha }{g}\right) ^{\frac{\alpha }{1-\alpha }}+\lambda (p_{b}k_{b}^{\alpha }g^{t})\alpha \left( \frac{\alpha }{g}\right) ^{\frac{\alpha }{1-\alpha }}. \end{aligned}$$

    The condition for toxic bubbles, \(\lambda >\bar{\lambda }\), is equivalent to \(p_{b}k_{b}^{\alpha }g^{t}>\lambda ^{(-1)}(\bar{\lambda })\), where \(\lambda ^{(-1)}(x)\) is the inverse function of \(\lambda (x)\).

  7. 7.

    Such a bubble pricking policy includes outright banning the trading of the bubble asset and a tax on trading the bubble asset. For the latter policy, see Proposition 7, which shows that high tax effectively eliminates bubble equilibria.

  8. 8.

    See, e.g., Shin (2011) on Basel III framework.

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Acknowledgments

We thank Jianjun Miao and an anonymous referee for useful comments. We wish to thank our colleagues Gadi Barlevy, Bob Barsky, Craig Burnside, Jeff Campbell, Bill Keech, and Tomohiro Hirano for their suggestions. We also thank the seminar and workshop participants at Duke University, the Federal Reserve Bank of Chicago, the Bank of Japan, the University of Tokyo, the University of Montreal, the Asia Pacific Conference on Economic Dynamics, and the 7th Annual Workshops of the Asian Research Network for their helpful comments. The views expressed in this paper are those of the authors. They should not be interpreted as reflecting the views of the Bank of Japan.

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Correspondence to Toan Phan.

Appendix

Appendix

Proofs

Proof of Lemma 1

First, in the no-bubble economy, the law of motion for capital is given by \(k_{t+1}=[(1-\alpha )/g]k_{t}^{\alpha }\) and consumption is given by \(c_{t}=\alpha k_{t}^{\alpha }.\) These equations imply the no-bubble steady-state values in part 1 of Lemma 1.

Next, we show the existence of bubble equilibria. Given \(k_{0}>0\), Eqs. (3), (5), and (6) imply the following equilibrium dynamics for \(k_{t+1}\) and \(p_{t}\):

$$\begin{aligned} \alpha k_{t+1}^{\alpha -1}= & {} (1-\lambda )g\frac{p_{t+1}}{p_{t}},\\ p_{t}+gk_{t+1}= & {} (1-\alpha )k_{t}^{\alpha }. \end{aligned}$$

Let \(p_{t}^{*}\) denote a bubble–output ratio: \(p_{t}^{*}\equiv p_{t}/k_{t}^{\alpha }\). Then, combining these two equations, we obtain the law of motion for \(p_{t}^{*}\):

$$\begin{aligned} p_{t+1}^{*}=\frac{1}{1-\lambda }\frac{\alpha p_{t}^{*}}{1-\alpha -p_{t}^{*}}. \end{aligned}$$

Suppose that condition (1) does not hold. Then, the law of motion for \(p_{t}^{*}\) implies that for any \(\lambda \in [0,1), p^{*}<0\) in steady state, and this cannot be an equilibrium. This shows the necessity of condition (1) for the existence of bubble equilibria. Next, suppose that condition (1) holds. Then, the law of motion for \(p_{t}^{*}\) implies that as long as \(\lambda \) is less than \(\underline{\lambda }\equiv \frac{1-2\alpha }{1-\alpha }\), there exists a unique stochastic steady state \(p_\mathrm{self}^{*}>0,\) given by:

$$\begin{aligned} p_\mathrm{self}^{*}=1-\alpha -\frac{\alpha }{1-\lambda }. \end{aligned}$$

The law of motion also implies that \((p_{t+1}^{*}/p_{t}^{*})\vert _{p_{t}^{*}>p_\mathrm{self}^{*}}>1\) and \(0<(p_{t+1}^{*}/p_{t}^{*})\vert _{p_{t}^{*}<p_\mathrm{self}^{*}}<1\). Thus, any initial bubble that satisfies \(0<p_{0}^{*}\le p_\mathrm{self}^{*}\) constitutes an equilibrium. In particular, an initial bubble \(0<p_{0}^{*}<p_\mathrm{self}^{*}\) is a transitory bubble as it converges to zero (as long as it does not collapse), and an initial bubble \(p_{0}^{*}=p_\mathrm{self}^{*}\) constitutes an asymptotic bubble as it remains at \(p_{t}^{*}=p_\mathrm{self}^{*}\) (as long as it does not collapse). For \(p_{0}^{*}>p_\mathrm{self}^{*}, p_{t}^{*}\) diverges to infinity (as long as it does not collapse), which cannot be an equilibrium. This shows the sufficiency of condition (1) for the existence of bubbles. This also shows that a asymptotic bubble equilibrium is unique and coincides with the stochastic bubble steady state.

The allocation and prices in the stochastic bubble steady state are calculated as follows. From (3) and (5), the return on capital is given by \(R_\mathrm{self}^{k}=\left( 1-\lambda \right) g\). From (3), the capital stock is given by \(k_\mathrm{self}=\left\{ \alpha /\left[ \left( 1-\lambda \right) g\right] \right\} ^{\frac{1}{1-\alpha }}.\) Because the bubble–output ratio is \(p_\mathrm{self}^{*}=1-\alpha -\alpha /\left( 1-\lambda \right) \), the bubble is given by \(p_\mathrm{self}=p_\mathrm{self}^{*}k_\mathrm{self}=\left[ 1-\alpha -\alpha /\left( 1-\lambda \right) \right] k_\mathrm{self}.\) With \(k_\mathrm{self}\) and \(p_\mathrm{self}\) at hand, the wage and the consumptions are given by (2) and (6), respectively. Finally, the expected consumption \(c_\mathrm{self}^{e}\) is given by \(c_\mathrm{self}^{e}=(1-\lambda )p_\mathrm{self}+R_\mathrm{self}^{k}k_\mathrm{self}\) because the consumption in the next period is \(p_\mathrm{self}+R_\mathrm{self}^{k}k_\mathrm{self}\) if the bubble sustains, while it is \(R_\mathrm{self}^{k}k_\mathrm{self}\) if the bubble bursts. The exact expression for \(c_\mathrm{self}^{e}\) is obtained after substituting out \(p_\mathrm{self}\) and \(k_\mathrm{self}\):

$$\begin{aligned} c_\mathrm{self}^{e}= & {} (1-\lambda )(1-\alpha )\left( \frac{\alpha }{(1- \lambda )g}\right) ^{\frac{\alpha }{1-\alpha }}. \end{aligned}$$

Proof of Lemma 2

The existence condition, \(\lambda <\underline{\lambda }\equiv (1-2\alpha )/(1-\alpha )\), implies:

$$\begin{aligned} c_\mathrm{self}^{e}&=(1-\lambda )^{\frac{1-2\alpha }{1-\alpha }}(1-\alpha )\alpha ^{\frac{\alpha }{1-\alpha }}g^{-\frac{\alpha }{1-\alpha }}\\&>\left( 1-\frac{1-2\alpha }{1-\alpha }\right) ^{\frac{1-2\alpha }{1-\alpha }}(1-\alpha )\alpha ^{\frac{\alpha }{1-\alpha }}g^{-\frac{\alpha }{1-\alpha }}\\&=\alpha \left( \frac{1-\alpha }{g}\right) ^{\frac{\alpha }{1-\alpha }}=c_{nb}. \end{aligned}$$

Proof of Proposition 3

The bubble steady state exists if and only if \(p_{b}=(1-\alpha )(k_{b})^{\alpha }-gk_{b}>0\). The arbitrage condition (13) implies \(R_{b}^{k}=g\). From (3), the capital stock is given by \(k_{b}=(\alpha /g)^{1/(1-\alpha )}\). Substituting the expression for \(k_{b}\) into the condition of \(p_{b}>0\) yields the result: \(p_{b}>0\) if and only if \(\alpha <1/2\), which is the dynamic inefficiency condition (1). Because the consumption in the next period is \(p_{b}+R_{b}^{k}k_{b}\) if the bubble sustains, while it is \(R_{b}^{k}k_{b}\) if the bubble bursts, the expected consumption is given by:

$$\begin{aligned} c_{b}^{e}=(1-\lambda )\underbrace{(1-\alpha )\left( \frac{\alpha }{g}\right) ^{\frac{\alpha }{1-\alpha }}}_{\text {cons if bubble survives}}+\,\,\lambda \underbrace{\alpha \left( \frac{\alpha }{g}\right) ^{\frac{\alpha }{1-\alpha }}}_{\text {cons if bubble collapses}} \end{aligned}$$

Proof of Corollary 4

Immediate from Lemma 1 and Proposition 3.

Proof of Proposition 5

First, we show \(c_{b}^{e}<c_{nb}\) if and only if \(\lambda >\bar{\lambda }\). From Eq. (14) and part 1 of Lemma 1, we know that:

$$\begin{aligned} \frac{c_{b}^{e}}{c_{nb}}=(1-\lambda )\left( \frac{1-\alpha }{\alpha }\right) ^{\frac{1-2\alpha }{1-\alpha }}+\lambda \frac{\alpha }{1-\alpha } \end{aligned}$$

Thus, \(c_{b}^{e}/c_{nb}<1\) if and only if \(\lambda >\bar{\lambda }\), where \(\bar{\lambda }\) is given by (15). The threshold \(\bar{\lambda }\) satisfies \(0<\overline{\lambda }<1\):

$$\begin{aligned} \bar{\lambda }= & {} \frac{1-\alpha -\alpha ^{\frac{1-2\alpha }{1-\alpha }}(1-\alpha )^{\frac{\alpha }{1-\alpha }}}{1-2\alpha }>\frac{1-\alpha -(1-\alpha )^{\frac{1-2\alpha }{1-\alpha }}(1-\alpha )^{\frac{\alpha }{1-\alpha }}}{1-2\alpha }=0,\\ \bar{\lambda }= & {} \frac{1-\alpha -\alpha ^{\frac{1-2\alpha }{1-\alpha }}(1-\alpha )^{\frac{\alpha }{1-\alpha }}}{1-2\alpha }<\frac{1-\alpha -\alpha ^{\frac{1-2\alpha }{1-\alpha }}(\alpha )^{\frac{\alpha }{1-\alpha }}}{1-2\alpha }=1, \end{aligned}$$

where assumption (1) has been used in deriving the above inequalities.

Next, we shall show that \(\bar{\lambda }\) is greater than \(\underline{\lambda }\). The difference between \(\bar{\lambda }\) and \(\underline{\lambda }\) is:

$$\begin{aligned} \bar{\lambda }-\underline{\lambda }= & {} \frac{1-\alpha -\alpha ^{\frac{1-2\alpha }{1-\alpha }}(1-\alpha )^{\frac{\alpha }{1-\alpha }}}{1-2\alpha }-\frac{1-2\alpha }{1-\alpha }\\= & {} \frac{(1-\alpha )^{2}-\alpha ^{\frac{1-2\alpha }{1-\alpha }}(1-\alpha )^{\frac{1}{1-\alpha }}-(1-2\alpha )^{2}}{(1-2\alpha )(1-\alpha )}. \end{aligned}$$

The denominator is positive because of assumption (1: \(\alpha <1/2\)). The numerator is:

$$\begin{aligned} (1-\alpha )^{2}-\alpha ^{\frac{1-2\alpha }{1-\alpha }}(1-\alpha )^{\frac{1}{1-\alpha }}-(1-2\alpha )^{2}= & {} 2\alpha -3\alpha ^{2}-\alpha ^{\frac{1-2\alpha }{1-\alpha }}(1-\alpha )^{\frac{1}{1-\alpha }}\\= & {} \alpha \left[ 2-3\alpha -\alpha ^{-\frac{\alpha }{1-\alpha }}(1-\alpha )^{\frac{1}{1-\alpha }}\right] \\= & {} \frac{\alpha }{1-\alpha }\left[ 2-\left( \frac{1-\alpha }{\alpha }\right) ^{\frac{\alpha }{1-\alpha }}-\frac{\alpha }{1-\alpha }\right] \end{aligned}$$

Let us denote \(\delta \equiv \alpha /(1-\alpha )\). Because \(0<\alpha <1/2\), we have \(0<\delta <1\). The above expression now can be written as

$$\begin{aligned} \frac{\alpha }{1-\alpha }\left[ 2-\left( \frac{1-\alpha }{\alpha }\right) ^{\frac{\alpha }{1-\alpha }}-\frac{\alpha }{1-\alpha }\right]= & {} \delta \left( 2-\delta ^{-\delta }-\delta \right) \\= & {} \delta \left[ 2-\delta \left( 1+\delta ^{1-\delta }\right) \right] >0. \end{aligned}$$

The strict inequality follows from \(\delta ^{1-\delta }<1\) because \(0<\delta <1\). Therefore, \(\bar{\lambda }-\underline{\lambda }>0\).

Proof of Proposition 6

The expected utility of households born in period T (in which the bubble collapses) is given by \(c_{T+1}=\alpha k_{T+1}^{\alpha }\), where \(k_{T+1}=[\left( 1-\alpha \right) /g]k_{b}^{\alpha }\). Note that \(c_{T+1}\) can be expressed as \(c_{T+1}=\alpha (c_\mathrm{gold}/g)^{\alpha }\), while \(c_{b}^{e}\) is expressed as \(c_{b}^{e}=\left[ 1-\lambda +\lambda \alpha / \left( 1-\alpha \right) \right] c_\mathrm{gold}\), where \(c_\mathrm{gold}\) is the first-best (golden rule) consumption defined in Footnote 4. On the other hand, the expected utility of households born in period \(t>T\) comes from the dynamics in the no-bubble equilibrium. Their expected utility is greater than that of the generation born in period T, because from period T onward, the economy converges from the post-collapse levels of capital and consumption to the levels in the no-bubble steady state. Hence, the expected utility of households who are born in periods \(t\ge T\) is greater than the expected utility in the bubble steady state \(c_{b}^{e}\) if and only if condition 16 holds. This proves the first part of the Proposition.

Next, consider the second part of the Proposition. Old households (the generation born in period \(T-1\)) are not hurt by the policy if and only if the consumption when the bubble is pricked is equal or greater than that when the bubble is sustained: \(c_{T}=\alpha k_{b}^{\alpha }+\theta \ge c_{b}^{e}\). Setting \(\theta \) such that the old households are indifferent between the two events, \(c_{T}=c_{b}^{e}\), yields \(\theta =\alpha k_{b}^{\alpha }-c_{b}^{e}\). With such a transfer, the net income of young households in period T changes to \(w_{T}-\theta \) when a bubble is pricked. The young households’ welfare is given by its consumption in the next period, or, \(c_{T+1}=\alpha (w_{T}-\theta )^{\alpha }\). Solving \(c_{T+1}\ge c_{b}^{e}\) for \(\lambda \) yields the following condition:

$$\begin{aligned} \lambda \ge \hat{\lambda }\equiv \frac{1-\alpha }{2-\alpha }\left[ \frac{\left( \frac{1-\alpha }{\alpha }\right) ^{1-\alpha }-\frac{\alpha }{1-\alpha }}{1+\left( \frac{1-\alpha }{\alpha }\right) ^{1-\alpha }}\right] \in (0,1). \end{aligned}$$

It is straightforward to algebraically verify that the threshold \(\hat{\lambda }\) is greater than \(\check{\lambda }\).

Proof of Proposition 7

In a stochastic bubble equilibrium with bubble speculation tax \(\tau \), the first-order conditions of bankers imply the following no-arbitrage condition:

$$\begin{aligned} R_{t+1}^{k}=R_{t+1}^{d}=\frac{(1-\tau )P_{t+1}}{P_{t}}. \end{aligned}$$

Hence, in stochastic bubble steady state:

$$\begin{aligned} R^{k}=R^{d}=(1-\tau )g. \end{aligned}$$

Thus, the capital stock in stochastic bubble steady state is:

$$\begin{aligned} k=\left( \frac{\alpha }{(1-\tau )g}\right) ^{\frac{1}{1-\alpha }}. \end{aligned}$$

The resource constraint in steady state is the same as before:

$$\begin{aligned} p+gk=w=(1-\alpha )k^{\alpha }. \end{aligned}$$

Hence,

$$\begin{aligned} p= & {} (1-\alpha )k^{\alpha }-gk\\= & {} \left[ (\frac{1-\alpha }{\alpha })(1-\tau )-1\right] \left( \frac{\alpha }{(1-\tau )g}\right) ^{\frac{1}{1-\alpha }}g. \end{aligned}$$

Thus, \(p>0\) if and only if \((\frac{1-\alpha }{\alpha })(1-\tau )-1>0\), or

$$\begin{aligned} \tau <\frac{1-2\alpha }{1-\alpha }. \end{aligned}$$
(18)

In other words, there can be a stochastic bubble steady state if and only if \(\tau <\frac{1-2\alpha }{1-\alpha }.\)

Recall that all bubble tax in each period is redistributed to old households. Then, the expected consumption in the stochastic bubble steady state is given by

$$\begin{aligned} E(c)= & {} (1-\lambda )p+R^{k}k. \end{aligned}$$

Substituting values for k and p into this equation, we obtain:

$$\begin{aligned} E(c)=\left( 1+(1-\tau )\frac{1-\alpha }{\alpha }\right) (1-\tau )^{1-\frac{1}{1-\alpha }}-(1-\lambda )(1-\tau )^{-\frac{1}{1-\alpha }}. \end{aligned}$$

By taking the first-order condition with respect to \(\tau \), we find that the local optimum is:

$$\begin{aligned} \tau =\frac{\alpha \lambda }{\alpha +(1-\lambda )(1-\alpha )}. \end{aligned}$$
(19)

Combining (18) and (19), we conclude that the optimal bubble tax is:

$$\begin{aligned} \tau ^{*}=\min \left\{ \frac{1-2\alpha }{1-\alpha },\ \frac{\alpha \lambda }{\alpha +(1-\lambda )(1-\alpha )}\right\} . \end{aligned}$$

Proof of Proposition 8

We assume that bankers default if the bubble bursts and then later verify that this is the case in equilibrium. Then, the Lagrangian associated with bankers’ optimization problem can be written as:

$$\begin{aligned} (1-\lambda )(R_{t+1}^{k}K_{t+1}+P_{t+1}b_{t}-R_{t+1}^{d}K_{t+1}-R_{t+1}^{d}P_{t}b_{t})+\mu _{t}(\kappa K_{t+1}-P_{t}b_{t}) \end{aligned}$$

where \(\mu _{t}\ge 0\) is the Lagrange multiplier associated with constraint (17). The first-order conditions are:

$$\begin{aligned} (1-\lambda )(R_{t+1}^{k}-R_{t+1}^{d})+\mu _{t}\kappa= & {} 0\\ (1-\lambda )(P_{t+1}-R_{t+1}^{d}P_{t})-\mu _{t}P_{t}= & {} 0. \end{aligned}$$

These conditions imply:

$$\begin{aligned} R_{t+1}^{k}= & {} R_{t+1}^{d}-\frac{\mu _{t}\kappa }{1-\lambda }\end{aligned}$$
(20)
$$\begin{aligned} \frac{(1-\lambda )P_{t+1}}{P_{t}}= & {} (1-\lambda )R_{t+1}^{k}+(1+n)\mu _{t}. \end{aligned}$$
(21)

Equation (20) implies that \(R_{t+1}^{k}\le R_{t+1}^{d}\). Hence, when the bubble bursts, the profit if bankers do not default is:

$$\begin{aligned} R_{t+1}^{k}K_{t+1}-R_{t+1}^{d}(K_{t+1}+P_{t}b_{t})<0. \end{aligned}$$

Therefore, it is in fact optimal for bankers to default if the bubble bursts. Equation (21) implies that in the stochastic bubble steady state:

$$\begin{aligned} R^{k}=g-\frac{1+\kappa }{1-\lambda }\mu . \end{aligned}$$

Therefore, equations that determine the stochastic bubble steady state are:

$$\begin{aligned} R^{k}= & {} g-\frac{1+\kappa }{1-\lambda }\mu \\ p+gk= & {} (1-\alpha )k^{\alpha }\\ p\le & {} \kappa gk\\ \mu (\kappa gk-p)= & {} 0\text { (complementary slackness)}\\ \mu\ge & {} 0. \end{aligned}$$

We assume that \(\kappa \) is sufficiently small so that the constraint is binding (otherwise, the regulation has no effect on welfare). Then, the Lagrange multiplier in the stochastic bubble steady state is:

$$\begin{aligned} \mu =g(1-\lambda )\left( \frac{1}{1+\kappa }-\frac{\alpha }{1-\alpha }\right) . \end{aligned}$$

Thus, the constraint is strictly binding, i.e., \(\mu >0\), when \(\kappa <(1-2\alpha )/\alpha \). The steady-state bubble is:

$$\begin{aligned} p= & {} \left( \frac{1-\alpha }{\alpha }R^{k}-g\right) k\\= & {} \left[ \frac{1-\alpha }{\alpha }\left( g-\frac{1+\kappa }{1-\lambda }\mu \right) -g\right] k \end{aligned}$$

Thus, \(p>0\) if and only if:

$$\begin{aligned} \frac{1-2\alpha }{\alpha }g> & {} \frac{1-\alpha }{\alpha }\frac{1+\kappa }{1-\lambda }\mu \\= & {} \frac{1-\alpha }{\alpha }\frac{1+\kappa }{1-\lambda }g(1-\lambda )\left( \frac{1}{1+\kappa }-\frac{\alpha }{1-\alpha }\right) \\= & {} \frac{1-\alpha }{\alpha }g\left[ 1-(1+\kappa )\frac{\alpha }{1-\alpha }\right] \end{aligned}$$

or equivalently:

$$\begin{aligned} \frac{1-2\alpha }{1-\alpha }>1-(1+\kappa )\frac{\alpha }{1-\alpha } \end{aligned}$$

or \(\kappa >0\), which is always true. Hence, given that the regulation constraint is binding, there is always a stochastic bubble steady state.

Welfare, or the expected consumption in the stochastic bubble steady state, is:

$$\begin{aligned} E(c)= & {} (1-\lambda )p+\alpha k^{\alpha }\\= & {} (1-\lambda )[(1-\alpha )k^{\alpha }-gk]+\alpha k^{\alpha }\\= & {} [(1-\lambda )(1-\alpha )+\alpha ]k^{\alpha }-(1-\lambda )gk\\= & {} g^{1-\frac{1}{1-\alpha }}(1-\alpha )^{\frac{1}{1-\alpha }}\\&\left[ \left( 1-\lambda +\frac{\alpha }{1-\alpha }\right) (1+\kappa )^{1-\frac{1}{1-\alpha }}-(1-\lambda )(1+\kappa )^{-\frac{1}{1-\alpha }}\right] . \end{aligned}$$

From the first-order condition with respect to \(\kappa \), we find that the local optimum is:

$$\begin{aligned} \kappa =\frac{(1-\lambda )/\alpha }{1-\lambda +\frac{\alpha }{1-\alpha }}-1. \end{aligned}$$
(22)

Note that \(\kappa <(1-2\alpha )/\alpha \) when \(\lambda >0\) so that the optimal regulation is always binding if the risk of bubble burst is positive. Also, Eq. (22) implies \(\kappa <0\) when \(\lambda =1\). Therefore, the optimal regulation is:

$$\begin{aligned} \kappa ^{*}=\max \left\{ \frac{(1-\lambda )/\alpha }{1-\lambda +\frac{\alpha }{1-\alpha }}-1,\ 0\right\} . \end{aligned}$$

Proof of Proposition 9 and Welfare Implications

We focus on a case in which a bank defaults when a bubble bursts. Because the leverage restriction is binding, the flow budget constraint of the bank implies \(p_{t}+gk_{t+1}^{b}=(1+\phi )\epsilon (1-\alpha )k_{t}^{\alpha }\), where the variables are detrended and \(b_{t}=1\) is imposed. The aggregate capital is given by \(k_{t+1}=k_{t+1}^{b}+k_{t+1}^{w},\) where \(k_{t+1}^{w}\) is the amount of detrended capital invested by the household in period t. The household has \((1-\epsilon )w_{t}\), lends \(\phi \epsilon w_{t}\) to the bank, and invests the remaining amount \([1-\epsilon (1+\phi )]w_{t}\) in capital, where \(\epsilon (1+\phi )<1\) is assumed. Because the household incurs the cost \(\xi \) per unit of capital investment, \(k_{t+1}^{w}\) is given by \(gk_{t+1}^{w}=(1-\xi )[1-\epsilon (1+\phi )]w_{t}.\) Thus, the capital invested by the bank is given by \(k_{t+1}^{b}=k_{t+1}-k_{t+1}^{w}=k_{t+1}-g^{-1}(1-\xi )[1-\epsilon (1+\phi )](1-\alpha )k_{t}^{\alpha }.\) Substituting this expression into the flow budget constraint yields:

$$\begin{aligned} p_{t}=[1-\xi (1-\epsilon (1+\phi ))](1-\alpha )k_{t}^{\alpha }-gk_{t+1}. \end{aligned}$$

In an asymptotic bubble equilibrium, \(R_{b}^{k}=g\), \(k_{b}=(\alpha /g)^{\frac{1}{1-\alpha }},\) and \(p_{b}=\{[1-\xi (1-\epsilon (1+\phi ))](1-\alpha )-\alpha \}k_{b}^{\alpha }.\) Thus, \(p_{b}>0\) if and only if the leverage restriction is not tight enough to satisfy:

$$\begin{aligned} \phi >\frac{\frac{\alpha }{1-\alpha }-(1-\xi )}{\epsilon \xi }-1. \end{aligned}$$

When the bubble persists, the households consume \(R_{b}^{k}[1-\epsilon -\xi (1-\epsilon (1+\phi ))]w_{b}/g.\) When the bubble bursts, the household consumes \(R_{b}^{k}k_{b}\). Thus, the expected consumption is given by:

$$\begin{aligned} c_{b}^{e}=(1-\lambda )[1-\epsilon -\xi (1-\epsilon (1+\phi ))](1-\alpha )\left( \frac{\alpha }{g}\right) ^{\frac{\alpha }{1-\alpha }}+\lambda \alpha \left( \frac{\alpha }{g}\right) ^{\frac{\alpha }{1-\alpha }}. \end{aligned}$$

In the bubble-less equilibrium, the law of motion for capital is given by

$$\begin{aligned} gk_{t+1}=[1-\xi (1-\epsilon (1+\phi ))](1-\alpha )k_{t}^{\alpha }. \end{aligned}$$

In steady state, \(k_{nb}=\{[1-\xi (1-\epsilon (1+\phi ))](1-\alpha )/g\}^{\frac{1}{1-\alpha }}.\) The consumption is given by:

$$\begin{aligned} c_{nb}=\left[ 1-\epsilon -\xi (1-\epsilon (1+\phi )) \right] \left[ 1-\xi (1-\epsilon (1+\phi ))\right] ^{-\frac{1-2\alpha }{1-\alpha }} \alpha \left( \frac{1-\alpha }{g}\right) ^{\frac{\alpha }{1-\alpha }}. \end{aligned}$$

Thus, the bubble is toxic, i.e., \(c_{b}^{e}<c_{nb}\) if and only if

$$\begin{aligned} \lambda >\frac{(1-\alpha )-\left[ 1-\xi (1-\epsilon (1+\phi ))\right] ^{-\frac{1-2\alpha }{1-\alpha }}\alpha ^{\frac{1-2\alpha }{1-\alpha }}\left( 1-\alpha \right) ^{\frac{\alpha }{1-\alpha }}}{(1-\alpha )-\frac{\alpha }{1-\epsilon -\xi (1-\epsilon (1+\phi ))}}. \end{aligned}$$

Suppose that a regulator sets a leverage restriction so that a bubble does not emerge. Then, the welfare in the bubble-less steady state is greater than the welfare in the asymptotic bubble equilibrium in which there is no leverage restriction if and only if

$$\begin{aligned} \lambda >\frac{(1-\alpha )-\frac{1-\epsilon -\xi (1-\epsilon (1+\phi ))}{1-\epsilon }\left[ 1-\xi (1-\epsilon (1+\phi ))\right] ^{-\frac{1-2\alpha }{1-\alpha }}\alpha ^{\frac{1-2\alpha }{1-\alpha }}\left( 1-\alpha \right) ^{\frac{\alpha }{1-\alpha }}}{1-\alpha -\frac{\alpha }{1-\epsilon }}, \end{aligned}$$

as desired.

Extensions and robustness checks

This appendix provides the details of the series of robustness check exercises in Sect. 4.

n-period overlapping generations

Assume each household lives for n periods. The expected lifetime utility of a household born in period t is:

$$\begin{aligned} E_{t}(c_{1,t}+\beta c_{2,t+1}+\cdots +\beta ^{n-1}c_{n,t+n-1}). \end{aligned}$$

With linear utility, the savings decision is either indeterminate or at a corner solution. To avoid this, we adopt an assumption in the classic Solow growth model that households save an exogenous fraction s of their wealth and consume the remaining fraction in each period of their life, except for the last period in which they consume everything. Also, for simplicity, we assume that each household, with its size given by \(1/(n-1)\), supplies one unit of labor inelastically in each of the earlier \(n-1\) periods of their life and does not work in the last period of their life. The aggregate labor is 1 and wage rate earned by each household in period t is \(W_{t}/(n-1)\). Thus, the aggregate savings in each period is:

$$\begin{aligned} S_{t}=sW_{t}. \end{aligned}$$

These aggregate savings are deposited in bankers, whose optimization problem is as in Sect. 3 in the main text. In equilibrium, the no-arbitrage condition of bankers is the same as Eq. (13). The steady-state bubble is:

$$\begin{aligned} p_{b}=s(1-\alpha )(k_{b})^{\alpha }-gk_{b}, \end{aligned}$$

while the capital stock is given by \(k_{b}=(\alpha /g)^{1/(1-\alpha )}\). Hence, \(p_{b}>0\) if and only if \(s(1-\alpha )>\alpha \), which is the dynamic inefficiency condition in this environment. This existence condition of bubbles is again independent of the risk of bursting of the bubble. Thus, the intuition that risk shifting enables the existence of excessively risky bubbles applies. The rest of the arguments about toxic asset bubbles similar to those in Propositions 3 and 5 apply.

Risk-averse households in both periods of life

Assume lifetime utility of a household is \(\log (C_{t}^{y})+\beta E_{t}\log (C_{t+1}^{o})\). Young households decide how much of their wage income \(W_{t}\) to save and how much to consume. Let \(S_{t}\) denote an individual household’s savings, and let \(\overline{S}_{t}\) denote the aggregate economy’s savings. Households deposit their savings with bankers, who then invest in a portfolio consisting of capital and bubbles. As in the main text, when a bank defaults, all of its assets are seized and distributed equally among its depositors. Hence, the optimal saving decision of a young household in period t solves:

$$\begin{aligned} \max _{\{C_{t}^{y},C_{t+1}^{o},S_{t}\}}\log (C_{t}^{y})+\beta E_{t}\log (C_{t+1}^{o}) \end{aligned}$$

subject to:

$$\begin{aligned} C_{t}^{y}= & {} W_{t}-S_{t}\\ C_{t+1}^{o}= & {} {\left\{ \begin{array}{ll} R_{t+1}S_{t} &{} \text { if no default}\\ (R_{t+1}K_{t+1}+\tilde{P}_{t+1}b_{t})\frac{S_{t}}{\overline{S}_{t}} &{} \text { if the bank defaults} \end{array}\right. }. \end{aligned}$$

Thus, the equilibrium \(S_{t}\) solves:

$$\begin{aligned} \max _{S_{t}\ge 0}\log (W_{t}-S_{t})+\beta \left[ (1-\lambda )\log (R_{t+1}S_{t})+\lambda \log ((R_{t+1}K_{t+1}+\tilde{P}_{t+1}b_{t})\frac{S_{t}}{\overline{S}_{t}})\right] \end{aligned}$$

The solution is \(S_{t}=\frac{\beta }{1+\beta }W_{t}\). Hence, the steady-state bubble is given by

$$\begin{aligned} p_{b}=\frac{\beta }{1+\beta }(1-\alpha )(k_{b})^{\alpha }-gk_{b}, \end{aligned}$$

while the capital stock is given by \(k_{b}=(\alpha /g)^{1/(1-\alpha )}\). Thus, \(p_{b}>0\) if and only if

$$\begin{aligned} \alpha <\frac{\beta }{1+2\beta }, \end{aligned}$$

which is the dynamic inefficiency condition in this environment. This existence condition of bubbles is again independent of the risk of bursting of the bubble, as in Proposition 3. The rest of the arguments follows through. The expected lifetime utility (welfare) is given by:

$$\begin{aligned} V_{b}\equiv & {} \log \left( \frac{1}{1+\beta }(1-\alpha )\left( \frac{\alpha }{g}\right) ^{\frac{\alpha }{1-\alpha }}\right) \\&+\beta (1-\lambda )\log \left( g\frac{\beta }{1+\beta }(1-\alpha )\left( \frac{\alpha }{g}\right) ^{\frac{\alpha }{1-\alpha }}\right) +\beta \lambda \log \left( g\alpha \left( \frac{\alpha }{g}\right) ^{\frac{\alpha }{1-\alpha }}\right) \end{aligned}$$

In the bubble-less equilibrium, the equilibrium dynamics with log–log preferences are:

$$\begin{aligned} K_{t+1}= & {} \frac{\beta }{1+\beta }W_{t}=\frac{\beta }{1+\beta }(1-\alpha )A_{t}^{1-\alpha }K_{t}^{\alpha }\\ C_{t}^{y}= & {} \frac{1}{1+\beta }W_{t}\\ C_{t+1}^{o}= & {} R_{t+1}^{k}K_{t+1}. \end{aligned}$$

Thus, the steady-state capital stock and marginal product of capital are:

$$\begin{aligned} k_{nb}= & {} \left( \frac{\beta }{1+\beta }\frac{1-\alpha }{g}\right) ^{\frac{1}{1-\alpha }}\\ R_{nb}= & {} \frac{1+\beta }{\beta }\frac{\alpha }{1-\alpha }g \end{aligned}$$

and thus, the lifetime utility in steady state is:

$$\begin{aligned} V_{nb}\equiv \log \left( \frac{1}{1+\beta }(1-\alpha )\left( \frac{\beta }{1+\beta }\frac{1-\alpha }{g}\right) ^{\frac{\alpha }{1-\alpha }}\right) +\beta \log \left( g\alpha \left( \frac{\beta }{1+\beta }\frac{1-\alpha }{g}\right) ^{\frac{\alpha }{1-\alpha }}\right) \end{aligned}$$

Hence, the expected lifetime utility is worse in the bubble steady state if and only if:

$$\begin{aligned} V_{nb}-V_{b}>0. \end{aligned}$$

Algebraic manipulations show that this inequality is equivalent to \(\lambda >\overline{\lambda }\) where \(\overline{\lambda }\equiv 1-\frac{1+\beta }{\beta }\frac{\alpha }{1-\alpha }>0\). In summary, Propositions 3 and 5 are robust when households have log–log preferences.

Aggregate shocks

A representative household works and earns \((1-\epsilon )W_{t}\) in the first period of their lives. A representative banker works and earns \(\epsilon W_{t}\) in the first period of their lives and combines it with borrowing from households to invest in capital and an bubble asset. With aggregate TFP shocks, the marginal product of capital is \(R_{t+1}^{k}=A_{t+1}\alpha K_{t+1}^{\alpha -1}\). Since \(E_{t}(A_{t+1})=E_{t}(a_{t+1})g^{t+1}=g^{t+1}\) and the banker is risk neutral, the portfolio problem of the banker is essentially unchanged from (10). Apply the same guess and verify method as in the main text: We guess that a banker defaults if and only if the bubble bursts. Then, the portfolio optimization problem is the same as (12). Thus, we have the same set of first-order conditions. In an asymptotic bubble equilibrium, the bank defaults when a bubble bursts if and only if \(R_{b}^{k}gk_{b}-R_{b}^{d}d_{b}<0\) for all \(a_{t+1}\), i.e., \(\bar{a}<1+\frac{p_{b}-\epsilon w_{b}}{gk_{b}}=\frac{(1-\alpha )(1-\epsilon )}{gk_{b}^{1-\alpha }}\). The bank does not default when a bubble persists if and only if \(R_{b}^{k}(gk_{b}+p_{b})-R_{b}^{d}d_{b}\ge 0,\) i.e., \(\underline{a}\ge 1-\frac{\epsilon w_{b}}{gk_{b}}=1-\frac{\epsilon (1-\alpha )}{gk_{b}^{1-\alpha }}.\) Then the rest of the arguments in the proofs of Propositions 3 and 5 carry through in a straightforward manner.

Microfoundation for standard debt contract

We provide a microfoundation of a debt contract assumed in the main text. The microfoundation is standard and is based on asymmetric information and costly state verification à la Townsend (1979). In this setting, the environment of bankers remains the same. In a bubble equilibrium, the bankers invest in both capital and a bubble asset. Bankers return is high or low depending on the event of bubble burst. The environment of a households sector differs from that in the main text. In particular, households do not observe bankers ex-post return without a cost. The households can observe the return only when they conduct costly auditing. No stochastic auditing is allowed. In addition, the households cannot make a contract which specifies the portfolio of capital and a bubble asset. The assumption of asymmetric information implies that the households do not observe the event of bubble burst when they receive the return.

In the model, there are only two states: h and l, where h denotes a high return (when a bubble sustains) and l denotes a low return (when a bubble bursts). The households do not observe the state without conducting costly auditing, but they know the probability of low return, \(\lambda \,\). Without loss of generality, we restrict our attention to a truth-telling contract in which bankers truthfully reveal the state \(s\in \left\{ h,l\right\} \). In this setting, the households decide three objects which depend on state s. First, they make an auditing decision, \(\delta \left( s\right) \in \left\{ 0,1\right\} \), where 0 indicates no auditing and 1 indicates auditing. Second, they choose the amount of repayment from bankers per unit of deposit when they audited bankers, \(R_{a}^{k}\left( s\right) \). Third, they choose the amount of repayment from bankers per unit of deposit when they did not audit bankers, \(r\left( s\right) \). The households’ objective is to maximize the expected repayment per unit of deposit:

$$\begin{aligned}&\left( 1-\lambda \right) \left\{ \delta \left( h\right) \left[ R_{a}^{k}\left( h\right) -\epsilon \right] +\left[ 1-\delta \left( h\right) \right] r\left( h\right) \right\} +\lambda \left\{ \delta \left( l\right) \left[ R_{a}^{k}\left( l\right) -\epsilon \right] \right. \nonumber \\&\quad \left. +\left[ 1-\delta \left( l\right) \right] r\left( l\right) \right\} , \end{aligned}$$
(23)

where \(\epsilon >0\) denotes the auditing cost per unit of loan. Bankers are competitive and protected by a limited-liability law. The resulting participation constraints of bankers are: For \(s\in \left\{ h,l\right\} \)

$$\begin{aligned} s-r\left( s\right) \ge 0,\quad s-R_{a}^{k}\left( s\right) \ge 0, \end{aligned}$$
(24)

where the left-hand side in each inequality denotes the profit of bankers per unit of deposit in case of no-monitoring and monitoring, respectively. Two incentive constraints are required to make bankers reveal a state truthfully. First, if the households do not audit bankers in the both states, the repayment has to be the same:

$$\begin{aligned} r\left( h\right) =r\left( l\right) \quad \text { if }\delta \left( h\right) =\delta \left( l\right) =0. \end{aligned}$$
(25)

Otherwise, bankers will always report a state with lower repayment. Second, if the households audit bankers in a low state but not in a high state, the repayment in a low state is equal or less than that in a high state:

$$\begin{aligned} R_{a}^{k}\left( l\right) \le r\left( h\right) \quad \text { if }\delta \left( l\right) =1 \quad \text { and} \quad \delta \left( h\right) =0. \end{aligned}$$
(26)

Otherwise, bankers would report a high state and pay less when they are in a low state.

A contact that maximizes the return received by the households has two features. First, the households audit only when bankers report a low state: \(\delta (h)=0\) and \(\delta (l)=1\). This auditing is enough to prevent bankers to fake a state. If bankers in a high state faked to be in a low state, the households would audit bankers and confiscate all bankers’ assets. Thus, bankers have no incentive to fake when they are in a high state. If the households did not audit when bankers report a low state, bankers in a high state would fake to be in a low state and thus the households return would be lower. Auditing in a high state as well would not change the repayment, but the return would be low because of an additional auditing cost. Second, the participation constraints (24) are binding: \(r(s)=s\) if \(\delta (s)=0\) and \(R_{a}^{k}(s)=s\) if \(\delta (s)=1\). Otherwise, the households can increase the return by raising the repayment.

From (23) the households expected return under the contract is given by

$$\begin{aligned} (1-\lambda )h+\lambda (l-\epsilon ). \end{aligned}$$
(27)

So far, the returns, h and l, have been taken as given. In the model presented in the main text, h and l are endogenously determined by bankers portfolio choice. In particular, in the model, h and l are corresponding to:

$$\begin{aligned} h=\frac{P_{t+1}b_{t}+R_{t+1}^{k}K_{t+1}}{D_{t}},l=\frac{R_{t+1}^{k}K_{t+1}}{D_{t}}. \end{aligned}$$

By assumption, the households cannot write a contract which depends on bankers portfolio choice between capital and a bubble asset, \(\{K_{t+1},b_{t}\}\). The households arrange the contact so as to make bankers to choose the portfolio to maximize (27). Under the contract such that the households confiscate all bankers assets in case of auditing, however, bankers do not take into account the earning in a low state. Thus, the households arrange the contract to maximize bankers earning in a high state. Given that bankers are competitive, one way to maximize h is to offer a debt contract with interest rate \(R_{t+1}^{d}\). The resulting financial arrangement is exactly the same as in the main text except the presence of auditing costs \(\epsilon \). The model in the main text corresponds to a limiting case where \(\epsilon \rightarrow 0\).

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Ikeda, D., Phan, T. Toxic asset bubbles. Econ Theory 61, 241–271 (2016). https://doi.org/10.1007/s00199-015-0928-1

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Keywords

  • Rational bubbles
  • Risk shifting
  • Financial crises

JEL Classification

  • F32
  • F41
  • F44