Inside Money, Procyclical Leverage, and Banking Catastrophes

We explore a model of the interaction between banks and outside investors in which the ability of banks to issue inside money (short-term liabilities believed to be convertible into currency at par) can generate a collapse in asset prices and widespread bank insolvency. The banks and investors share a common belief about the future value of certain long-term assets, but they have different objective functions; changes to this common belief result in portfolio adjustments and trade. Positive belief shocks induce banks to buy risky assets from investors, and the banks finance those purchases by issuing new short-term liabilities. Negative belief shocks induce banks to sell assets in order to reduce their chance of insolvency to a tolerably low level, and they supply more assets at lower prices, which can result in multiple market-clearing prices. A sufficiently severe negative shock causes the set of equilibrium prices to contract (in a manner given by a cusp catastrophe), causing prices to plummet discontinuously and banks to become insolvent. Successive positive and negative shocks of equal magnitude do not cancel; rather, a banking catastrophe can occur even if beliefs simply return to their initial state. Capital requirements can prevent crises by curtailing the expansion of balance sheets when beliefs become more optimistic, but they can also force larger price declines. Emergency asset price supports can be understood as attempts by a central bank to coordinate expectations on an equilibrium with solvency.


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The continuity of u and u (by Assumption S2) and of f t+1 (by Assumption S3) enable us to use the Leibniz integration rule twice to move the derivative inside the integral sign to compute the first two Because u is strictly concave (by Assumption S1) and twice differentiable (by Assumption S2), and because its domain (0, ∞) is convex, we know that u < 0 [1, Sec. 3.1.4, page 71]. Also, we know that (v − π) 2 ≥ 0 with equality if and only if v = π. Combining these two conclusions with Eq. (S2b) gives Because ∂ 2 g/∂d 2 < 0 and because the domain of g is convex, we know that g(d) is strictly concave [1, Sec.
3.1.3, page 69]. Thus, any solution d * to the first-order (necessary) condition for the maximization in Eq. (S1), in which case g(d) has a unique maximum at y t /[π(1 − µ)]. Thus, the investor demand function (S1) is 34 single-valued.

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Furthermore, the first derivative of the objective function evaluated at the lower constraint Because u > 0 (by Assumptions S1-S2), we know that Equations (S5) and (S3) imply that D i (π) = −z t if and only if π ≥ E V t+1 , which completes the proof.

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The next lemma achieves stronger results for the particular utility function u(·) that exhibits constant 37 relative risk aversion. Specifically, the lemma establishes the price at which the investors first begin to 38 buy less than the maximum that they can afford.

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Lemma S2 (Investor demand less than they can afford iff π > π t+1 ). Suppose that the investors have 40 CRRA utility with parameter λ and that the belief follows a Beta distribution with parameters α t , β t . Then Proof. We will show that the first derivative with respect to d of the objective function g(d; z t , y t , c t , π) 44 of the maximization in D i (π) evaluated at the upper constraint d = y t /[π(1 − µ)] is negative for π < π t+1 45 and positive for π > π t+1 . By Lemma S1, we know that ∂ 2 g(d; z t , y t , c t , π)/∂d 2 < 0 for all −z t ≤ d ≤ 46 y t /[π(1 − µ)]. Because of this negative second derivative, we know that which proves the claim.

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Using u (w) = w −λ in Eq. (S2a) and evaluating at d To evaluate the arg max in Eq. (4), note that the marginal change in the banks' expected equity due to a infinitesimal increase in demand d is Thus, the expected equity E e t+1 is linear in d with slope E V t+1 − π, subject to the constraint (S6).
The sign of E V t+1 − π therefore determines whether the bank demand is the lower or upper constraint in (S6), and the sign of π − v t+1 determines whether to use constraint (S6a) or (S6b). Note that if E V t+1 < π < v t+1 , then the banks' demand is the lower constraint in (S6b), which simplifies to −x t under the assumption that πx t + r t − y t ≥ 0. In summary, the banks' demand function can be written Finally, considering whether the banks comply with their insolvency constraint (5) immediately after the 54 shock leads to Eq. (7).

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Appendix S3 Effect of a capital requirement on the region of 56 beliefs giving rise to three equilibria 57 Here we explain why, after implementing a capital requirement, the region of beliefs giving rise to three 58 equilibria is reduced from above (and not from below), as illustrated in Fig. 10. Recall that the banks 59 are insolvent if and only if the bond price π ≤ (y t − r t )/x t . Also recall [from the banks' demand in the 60 event of a negative shock, illustrated in Fig. 2(B)] that below the price (y t − r t )/x t the banks are forced 61 to sell all their bonds; by contrast, above the price (y t − r t )/x t and below E V t+1 , the banks' demand 62 increases with the price. That is, the kink in the bank demand occurs at the price (y t − r t )/x t .

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Implementing a capital requirement does not affect the location of this kink because the capital reserves r t (as typically occurs in practice), then C cap. req. (π) is increasing in the price π for π ≥ (y t − 66 r t )/x t , so a kink still occurs at the price (y t − r t )/x t . On the other hand, if reserves r t exceed deposits 67 y t (which rarely occurs in practice), then C cap. req. (π) > 0 for all prices π, so the capital requirement 68 does not bind for any price because the bank's demand is negative for a negative shock, and so the 69 bank demand still has a kink at the price (y t − r t )/x t . In summary, the position of the first saddle-node 70 bifurcation (at which two new equilibrium prices appear) is (y t −r t )/x t for any minimum capital-to-assets 71 ratio γ min t+1 ; that is, the lower boundary of the region of three equilibria [such as in Fig. 4(E) and in Fig. 10] 6 is independent of the capital requirement.

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Although a capital requirement does not move the location of the kink in the bank demand, a capital 74 requirement can bind (and hence reduce the banks' demand) for prices just above that price where the 75 kink occurs, (y t − r t )/x t , as illustrated in Fig. 9(A). Consequently, the left-hand side of the "hump" in 76 the total demand function [depicted in Fig. 4(A)-(C)] is truncated; for an illustration, see Fig. 9(A).

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Thus, a less severe negative shock causes the two larger equilibrium prices to disappear, as illustrated in 78 Fig. 9(B). In summary, the reduction in the banks' demand just above the price (y t − r t )/x t explains why 79 the region of three equilibria is truncated from above in Fig. 10 and hence why the capital requirement 80 can force a decline in the bond price and bank insolvency.