Runs, panics and bubbles: Diamond–Dybvig and Morris–Shin reconsidered

Abstract

The basic two-noncooperative-equilibrium-point model of Diamond and Dybvig is considered along with the work of Morris and Shin utilizing the possibility of outside noise to select a unique equilibrium point. Both of these approaches are essentially nondynamic. We add an explicit replicator dynamic from evolutionary game theory to provide for a sensitivity analysis that encompasses both models and contains the results of both depending on parameter settings.

Appendices

Appendix 1: Proof of existence of interior threshold strategies

Lemma

If the utility satisfies $r u^{\prime } \! \left( r \right)$ monotone decreasing on $r \ge 1$, then the expected utility (11) along the subgame-perfect $\left( r , {\theta }^{*} \right)$ contour has an interior maximum.

Proof

Existence follows from the signs of derivatives of $\left\langle u \right\rangle$ at $r = 1$ and $r = R$.

For $r = 1$, there is never an incentive to run, while for all $\theta < 1$ there is an incentive to wait, so the subgame-perfect ${\left. {\theta }^{*} \right| }_{r \rightarrow 1} > 1$ and $f_{\theta } = \theta$ everywhere. Then the expected utility becomes

$$\begin{aligned} { \left. \left\langle u \right\rangle \right| }_{r = 1} = \left\langle \theta \right\rangle u \! \left( 1 \right) + \left( 1 - \left\langle \theta \right\rangle \right) u \! \left( R \right) \!, \end{aligned}$$

(13)

and the marginal expected utility may be computed and shown to be

$$\begin{aligned} { \left. \frac{d \! \left\langle u \right\rangle }{dr} \right| }_{r = 1} = \left\langle \theta \right\rangle \left[ u^{\prime } \! \left( 1 \right) - R u^{\prime } \! \left( R \right) \right] \!. \end{aligned}$$

(14)

As long as $R > 1$ and $r u^{\prime } \! \left( r \right)$ is monotone decreasing on $r \ge 1$, this quantity is always positive.¹⁰

Conversely, if $r = R$, The payout $R \left( 1 - f_{\theta } r \right) / \left( 1 - f_{\theta } \right) < R$ for all $\theta$, so there is never an incentive to wait, making ${\left. {\theta }^{*} \right| }_{r \rightarrow R} < 0$ and $f_{\theta } \rightarrow 1$ everywhere. The utility therefore becomes

$$\begin{aligned} { \left. \left\langle u \right\rangle \right| }_{r = R} = \left( 1 - \frac{1}{R} \right) u \! \left( 0 \right) + \frac{1}{R} u \! \left( R \right) , \end{aligned}$$

(15)

and the marginal expected utility becomes

$$\begin{aligned} { \left. \frac{d \! \left\langle u \right\rangle }{dr} \right| }_{r = R} = - \frac{ \left[ u \! \left( R \right) - u \! \left( 0 \right) \right] - R u^{\prime } \! \left( R \right) }{ R^2 }. \end{aligned}$$

(16)

For any concave utility, $\left[ u \! \left( r \right) - u \! \left( 0 \right) \right] - r u^{\prime } \! \left( r \right) > 0$, so the derivative (16) is negative and $\left\langle u \right\rangle$ has an interior maximum.

Appendix 2: Efficiency of threshold equilibria in relation to competitive equilibria

To illustrate the inefficiency of the threshold equilibria without reference to the detailed form of the utility, we consider the risk-minimizing equilibrium. Readers of Diamond (2007) will recognize that the competitive equilibrium for utilities of our form with $\alpha = 1$ are recovered in the following results by replacing $R \rightarrow \sqrt{R}$, and the same efficiency arguments go through.

In the risk-minimizing competitive equilibrium of Sect. 1.2, Type-2 agents never ran and the payout to those waiting until period-two to withdraw, in any population state $\theta$, was

$$\begin{aligned} r_{\text{ Competitive }} = \frac{ R }{ 1 + \left( R - 1 \right) \theta }. \end{aligned}$$

(17)

In contrast, for the threshold strategy used here, the inflection point of the function $f^{\left( {\theta }^{*} \right) }_{\theta }$ occurs at $\theta = {\theta }^{*}$, where $f = \left( 1 + {\theta }^{*} \right) / 2$ (shown as the line with slope $1/2$ in the left panel of Fig. 3). When this inflection point closely approximates the subgame-perfect threshold ${\theta }^{*}$, as it does in the numerical examples, the payout to agents who withdraw early is

$$\begin{aligned} r_{\text{ Threshold }} = \frac{ R }{ 1 + \left( R - 1 \right) \left( 1 + {\theta }^{*} \right) / 2 }. \end{aligned}$$

(18)

For the sake of comparison, considering a narrow distribution where ${\theta }^{*} \approx \left\langle \theta \right\rangle \approx \theta$ for almost-all samples $\theta$, the threshold equilibrium is less efficient than the competitive equilibrium by

$$\begin{aligned} \frac{1}{r_{\text{ Threshold }}} - \frac{1}{r_{\text{ Competitive }}} = \left( 1 - \frac{1}{R} \right) \frac{1 - \left\langle \theta \right\rangle }{2}. \end{aligned}$$

(19)

One half of Type-2 agents withdraw “needlessly” to implement the mechanism of regulation, reducing their own payouts because they decrease the total production relative to a competitive equilibrium.