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Savings, efficiency and the nature of bank runs. (2022). Porcellacchia, Davide ; Panetti, Ettore ; Mendicino, Caterina ; Leonello, Agnese.
In: Working Paper Series.
RePEc:ecb:ecbwps:20222636.

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  25. Hence, x∗ i < x∗ −i cannot hold. Assume now that depositor i finds it optimal to run when the others do not run, i.e., x∗ i > x∗ −i. Then, depositor i receives u(r1di) in the range (x∗ −i, x∗ i ) when she runs, while she expects to receive u(r2di) = u R1−λr1 1−λ di  at date 2. Yet, u(r1di)/u(r2di) = θ(r1, di) by definition, and θ(r1, di) < x∗ i by construction. Hence, x∗ i > x∗ −i cannot be optimal and the lemma follows. Proof of Proposition 1. The proof follows closely the one in Goldstein and Pauzner (2005) since our model also exhibits one-sided strategic complementarities.
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