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Econ-266-A1 · revision r1 · on Econ-266

GPT 6 Astra Ultra attempt on Econ-266

Exploration · LLM-generated · gpt 6 astra ultra ·

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Contents
  1. Bank-run forecast bounds and robust intervention with coordination
  2. Question and an explicit selection problem
  3. Sharp equilibrium-selection bounds
  4. An exactly solvable policy problem
  5. Prediction and the remaining empirical gap
  6. Completion Estimate
  7. References

Bank-run forecast bounds and robust intervention with coordination

Problem ID: G21-262. Status: Unresolved. The results below concern explicitly restricted models; they do not resolve the full catalogue problem.

Question and an explicit selection problem

The catalogue asks how to forecast and prevent bank runs while distinguishing fundamentals, coordination, network effects, and policy costs. The foundational liquidity model [DD], equilibrium-selection analysis [GP], and depositor-network evidence [IP] motivate those distinct ingredients. We analyze a finite strategic-complementarity model with sharp forecast bounds and solve a two-depositor robust policy problem. The model does not identify an equilibrium selector from fundamentals alone.

There are $N$ depositors choosing withdrawal $w_i\in\{0,1\}$. For fundamentals $z$ and policy $u$, the payoff advantage of withdrawing is \[ \theta_i(z)-u_i+\sum_{j\ne i}a_{ij}w_j,\qquad a_{ij}\ge0. \tag{1} \] Policies increase the relative payoff to waiting. Their real resource cost must be specified separately. Assume the conditional shock law, given public information, assigns probability zero to all payoff ties for each of the finitely many action profiles at any policy under consideration. A run is a declared increasing event, for example $\sum_iD_iw_i\ge q$ for fixed deposit weights $D_i>0$ and threshold $q$. Ordinary individual liquidity withdrawals can be included in fundamentals; the threshold specifies which collective event is called a run.

Sharp equilibrium-selection bounds

For fixed $z,u$, let $T(w)$ be the best-response vector from (1), with a fixed convention at zero. The tie assumption makes the convention irrelevant almost surely.
Theorem (Computable sharp bounds without a selector): Starting from the all-zero vector and repeatedly applying $T$ reaches a least equilibrium $w^-$; starting from the all-one vector reaches a greatest equilibrium $w^+$. Each sequence stabilizes after at most $N$ strict vector changes. Every pure equilibrium lies coordinatewise between them. For any increasing run event $A$, every measurable pure-equilibrium selector has run probability between \[ p^-(u)={\mathbb{P}}\{w^-(z,u)\in A\},\qquad p^+(u)={\mathbb{P}}\{w^+(z,u)\in A\}. \tag{2} \] Both bounds are attained, and every intermediate probability is attained if an independent public randomization of selectors is permitted. Increasing any component of $u$ weakly lowers both bounds.
Proof: Nonnegative interactions make $T$ order preserving. Its iterates from zero form an increasing sequence: the first step is at least zero, and induction uses monotonicity. Each strict change turns at least one zero coordinate permanently to one, so there are at most $N$ such changes. The terminal vector is a fixed point and hence an equilibrium. If $w$ is any fixed point, induction gives $T^k(0)\le w$, so the limit is least. The decreasing iteration from one proves the symmetric assertion.

An increasing event respects the resulting pointwise bounds, giving (2) after taking expectations. Finite iterations of measurable comparisons make the extreme selectors measurable. Selecting them themselves attains the endpoints. Choosing the greatest selector with an independent fixed probability $r$ and the least otherwise gives $(1-r)p^-+rp^+$, filling the interval. A larger $u$ weakly decreases the best-response map at every argument. Induction couples the corresponding lower and upper iterations to prove the final monotonicity. ∎

The result concerns pure equilibria; mixed equilibria are outside the selector class. For a forecast to be a single number within the bounds, one must specify or identify a selection mechanism. The interval is economically informative when its endpoints are close and correctly signals coordination uncertainty when they are not.

An exactly solvable policy problem

Consider two equal depositors with common $z\sim\operatorname{Uniform}[0,1]$, symmetric interaction $\lambda\in[0,1]$, and common policy $u\in[0,1+\lambda]$. Each withdrawal advantage is \[ z-u+\lambda w_{-i}. \] A run means both withdraw. Let $F$ be the uniform distribution function extended by zero to negative arguments and one above one.
Lemma (Selection interval): Apart from zero-probability ties, all equilibria are either $(0,0)$ or $(1,1)$, and the sharp run-probability interval is \[ [1-F(u),\ 1-F(u-\lambda)]. \tag{3} \]
Proof: For $(0,0)$ to be an equilibrium requires $z<u$, whereas $(1,1)$ requires $z>u-\lambda$. If exactly one depositor withdrew, its best response would require $z>u$ and the other's waiting best response would require $z<u-\lambda$, impossible for $\lambda\ge0$. The least equilibrium therefore runs only when $z>u$; the greatest runs whenever $z>u-\lambda$. Integrating the uniform shock gives (3). ∎
Suppose a run causes real loss $H>0$, and strengthening waiting incentives has real cost $cu^2/2$, $c>0$. These are primitive welfare costs, not gross deposit repayments counted as social losses. For a policymaker robust to the pure-equilibrium selector, expected loss is \[ J(u)=\frac c2u^2+H[1-F(u-\lambda)]. \tag{4} \]
Theorem (A discontinuous robust intervention threshold): Write $x=H/c$. If $x<2\lambda$, the unique minimizer of (4) is $u=0$. If $x>2\lambda$, the unique minimizer is $u=\min\{x,1+\lambda\}$. At $x=2\lambda>0$, exactly two policies minimize loss: $u=0$ and $u=2\lambda$. In contrast, under commitment to the least-equilibrium selector, the unique optimal policy is $\min\{x,1\}$.
Proof: On $[0,\lambda]$, the worst-selector run probability is one, so the best policy in that interval is zero, with loss $H$. On $[\lambda,1+\lambda]$, \[ J(u)=\frac c2u^2+H(1+\lambda-u), \] a strictly convex quadratic whose constrained minimizer is $v=\min\{1+\lambda,\max\{\lambda,x\}\}$. If $x\le\lambda$ and $\lambda>0$, its value at $v=\lambda$ exceeds $H$ by $c\lambda^2/2$. If $\lambda<x<1+\lambda$, then $v=x$ and \[ J(x)-H=cx(\lambda-x/2). \] Thus it beats zero exactly when $x>2\lambda$, and ties at equality. Finally, if $x\ge1+\lambda$, the endpoint $v=1+\lambda$ has difference $c(1+\lambda)^2/2-cx$ from $H$. Since $0\le\lambda\le1$, this is nonpositive in the stated regime, strictly negative except at $\lambda=1,x=2$. These cases give the theorem, including that endpoint tie. The case $\lambda=0$ has $x>0$ and gives $\min\{x,1\}$ directly. Strict convexity on the second interval and strict increase on the first away from zero show there are no additional minimizers.

For the least selector, run probability is $1-F(u)$. On $[0,1]$ the objective is $cu^2/2+H(1-u)$, whose minimizer is $\min\{x,1\}$. For $u>1$, the run probability is zero and the cost strictly increases, so no other minimizer occurs. ∎

There is an initial range of policies that reduces withdrawals in the least equilibrium but does not change the worst-selector run probability. Paying to cross that range creates the robust policy's threshold. This is an effect of declared coordination uncertainty and policy technology, not a universal prescription for deposit insurance.

Prediction and the remaining empirical gap

If a fixed public-information cell and fixed policy produce independent run indicators $R_1,\ldots,R_m$ under a stable actual selector, their empirical mean estimates that selector's run probability. The elementary bounded-variable exponential bound gives \[ {\mathbb{P}}\{|m^{-1}\sum R_i-p|>\sqrt{\log(2/\alpha)/(2m)}\}\le\alpha. \] This estimates the selected probability at that policy. It does not identify how the selector changes under intervention; nor does it estimate the fundamental shock law or interaction coefficients from withdrawal frequencies alone. Observational equivalence between worse fundamentals and selection toward the run equilibrium remains possible.

The finite model leaves bank asset choice, hidden risk, insurance financing, liquidity facilities, mixed strategies, and strategic early information acquisition outside the theorem. To include moral hazard one must specify how the intervention changes bank incentives and include that response in welfare. Assuming the cost in (4) already captures every such response would be an unsupported empirical claim. The original joint forecasting-and-prevention problem remains unresolved beyond this explicitly solved restriction.

Completion Estimate

40%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt proves sharp pure-equilibrium bank-run probability bounds in a finite complementarity network and solves a two-depositor robust intervention threshold. Prospective forecasting and implementable prevention still require identified fundamentals and selection responses, systemic liquidation losses, bank risk incentives and fiscal costs of actual insurance or liquidity facilities.

References

  • [DD] D. W. Diamond and P. H. Dybvig (1983), "Bank Runs, Deposit Insurance, and Liquidity," Journal of Political Economy 91(3), 401--419. doi:10.1086/261155.
  • [GP] I. Goldstein and A. Pauzner (2005), "Demand--Deposit Contracts and the Probability of Bank Runs," Journal of Finance 60(3), 1293--1327. doi:10.1111/j.1540-6261.2005.00762.x.
  • [IP] R. Iyer and M. Puri (2012), "Understanding Bank Runs: The Importance of Depositor-Bank Relationships and Networks," American Economic Review 102(4), 1414--1445. doi:10.1257/aer.102.4.1414.