Demand Deposit Contracts and the Probability of Bank Runs

Demand Deposit Contracts and the Probability of Bank Runs! Itay Goldstein* and Ady Pauzner** First version: August, 1999 This version: April, 2000 ABSTRACT We extend the Diamond and Dybvig model of bank runs by assuming that agents do not have common knowledge regarding the fundamentals of the economy, but rather receive slightly noisy signals. The new model has a unique equilibrium in which the fundamentals determine whether a bank run would occur. This lets us compute the probability of a bank run and relate it to the parameters of the demand deposit contract. We find that offering a higher return to agents demanding early withdrawal makes the bank more vulnerable to runs. Nonetheless, we show that even when this drawback is taken into account, demand deposit contracts still improve the welfare of agents. ! This paper is part of Itay Goldstein's Ph.D. dissertation, under the supervision of Elhanan Helpman. We thank Joshua Aizenman, Joseph Djivre, Elhanan Helpman, Zvi Hercowitz, Leonardo Leiderman, Nissan Liviatan, Stephen Morris, Assaf Razin, Dani Tsiddon, and Oved Yosha for helpful comments. * Research Department, Bank of Israel and the Eitan Berglas School of Economics, Tel Aviv University. E-mail: [email protected] ** The Eitan Berglas School of Economics, Tel Aviv University. E-mail: [email protected]. 1. Introduction: Banks often finance long-term investments with short-term liabilities. While this can have obvious welfare benefits, it makes banks vulnerable to ‘runs’. A run occurs when investors rush to withdraw their deposits, believing that the bank is bound to fail. As a result, the bank is forced to liquidate investments at a loss and indeed fails. Diamond and Dybvig (1983, henceforth D&D) provide the first coherent explanation as to why a bank might hold such a risky portfolio. By offering ‘demand deposit’ contracts, the bank enables investors who are afraid to face early liquidity needs to participate in profitable long-term investments. Since the bank deals with many investors, it can respond to their idiosyncratic liquidity needs and thereby provide a form of risk sharing which increases welfare. D&D show that their model has two equilibria. In the first, only investors who face liquidity needs demand early withdrawal and risk sharing is achieved. The second equilibrium, however, involves a ‘bank run’ in which all investors, including those with no liquidity need, demand early withdrawal. This equilibrium provides no risk sharing and is even inferior to the allocation agents can obtain without the bank. The D&D framework has a number of difficulties related to the multiplicity of equilibria. First, agents in the D&D model do not consider the possibility of a bank run at the first stage, when they choose to deposit money in the bank. This means that the bank run in the second stage is not part of a rational-expectations equilibrium of the two-stage model. Second, the D&D model does not explain what causes a particular equilibrium to be selected or under which circumstances a bank run is more likely to happen. Third, since the probability of runs is unknown, the model cannot even say whether the banking contract is desirable. For the same reason, the applicability of the model for evaluating policies intended to avoid bank runs is limited. In this paper we modify D&D's model by assuming that investors do not have common knowledge about the fundamentals (specifically, the return on long-term investments). Instead, they only obtain private signals that are very close to the real value. In many cases, this is more realistic than the assumption that all investors share precisely the same information and opinions. We show that the modified model has a unique ra- 1 tional-expectations equilibrium, in which every realization of the fundamentals uniquely determines whether there will be a bank run. Specifically, the model predicts that bank runs will occur if and only if the realization of fundamentals in the economy is lower than some critical value. This result is in line with empirical findings according to which most bank runs occur following negative real shocks.1 It is important to stress, however, that although in our model the fundamentals uniquely determine whether a bank run will occur, runs are still panic-based, that is, driven by bad expectations. In most scenarios, each investor would like to take the action she believes that others take – she demands early withdrawal just because she fears others would. The key point, however, is that the beliefs of investors are uniquely determined by the realization of the fundamentals. This in turn uniquely determines whether a bank run occurs. In other words, the fundamentals do not determine agents’ actions directly, but rather serve as a device that coordinates agents’ beliefs on a particular outcome. Knowing which equilibrium occurs at each realization of the fundamentals, we can compute the probability of a bank run. We find that this probability depends positively on the degree of risk sharing embodied in the banking contract. The intuition is simple: an agent’s incentive to demand early withdrawal is greater when she is offered a higher short-term return. This incentive is even further increased since the agent knows that other agents are also attracted by the higher return. Thus, a bank that offers more risk sharing becomes more vulnerable to runs. When the bank designs the demand deposit contract, it must take into account the positive impact that an increase in the short-term return has on the likelihood of runs. One may now wonder whether demand deposit contracts are still desirable when their destabilizing effect is considered. We answer this affirmatively: banks would still offer risk sharing, although this would cause runs to occur for some realizations of the fundamentals. Several authors have tackled the difficulties in D&D’s model. Postlewaite and Vives (1987) were the first to present an example in which bank runs occur as an equilibrium phenomenon. Cooper and Ross (1998) present a more comprehensive model, in which 1 See: Gorton (1988), Demirguc and Detragiache (1998) and Kaminsky and Reinhart (1999). 2 the probability of a run is taken into account before investment decisions are made. However, in both papers the probability of a run is exogenous and unaffected by the form of the banking contract.2 In Allen and Gale (1998), agents obtain exact information regarding the long term return and coordinate on an equilibrium in which they run only when running is efficient. Goldfajn and Valdes (1997) present a model with an endoge- nous probability of runs, but neglect the possibility of panic-based runs. Moreover, they do not analyze the relations between the banking contract and the likelihood of runs. Temzelides (1997) employs an evolutionary model for equilibrium selection. His model, however, does not deal with the probability of runs, and does not explain why agents deposit money at the bank when they know that a run must occur. A number of authors have studied models in which only some agents receive information concerning the prospects of the bank. Jacklin and Bhattacharya (1988) (see also Alonso (1996), Loewy (1998), and Bougheas (1999)) explain bank runs as an equilibrium phenomenon, but again neglect the possibility of panic-based runs. Chari and Jaganna- than (1988) and Chen (1999) study panic-based bank runs, but of a different kind. Here, runs occur when uninformed agents interpret the fact that others run as an indication that fundamentals are bad.3 The method used in our paper to obtain a unique equilibrium is based on Carlsson and van Damme (1993). They show how the introduction of noisy signals to 2x2 multiple-equilibria games leads to a contagion effect that generates a unique equilibrium.4 Morris and Shin (1998) extend this technique to a model with a continuum of agents. They show that episodes of currency attacks are uniquely determined by the fundamentals of the economy. 2 Chang and Velasco (1999) use the same method in an application of the D&D model to international financial crises. 3 For more developments of the D&D model see: Waldo (1985), Jacklin (1987), Bental and Eckstein and Peled (1988), Williamson (1988), Wallace (1988,1990), Eng(19W (1,19C.7(9)9.Char.4(198)9.998), 98e3-7.8(g)-6.5( )-1, 98 3 The remainder of this paper is organized as follows. Section 2 presents a benchmark model which is a slight modification of the D&D model. This model has multiple equilibria. In section 3 we introduce private signals into the model and obtain a unique equilibrium. Section 4 studies the relation between the banking contract and the likelihood of runs. Section 5 concludes. Proofs are relegated to the appendix. 2. The Benchmark Model: Multiple Equilibria The economy Our benchmark model is a slight variation of D&D’s model. There are three periods (0,1,2), one good, and a continuum [0,1] of agents. Each agent is born at period 0 with an endowment of 1. Consumption occurs only at periods 1 or 2 (c1 and c2 denote an agent’s consumption levels). Each agent can be of two types: with probability λ the agent is impatient and with probability 1-λ she is patient. Agents' are i.i.d.; we assume no aggre- gate uncertainty. Agents learn their types types (which are their private information) at the beginning of period 1. Impatient agents can consume only at period 1 and obtain + 5 utility of u(c1 ) . Patient agents can consume at either period; their utility is u(c1 c2 ) . u is twice continuously differentiable, increasing, and has a relative risk-aversion coeffi- cient, − cu''(c) /u'(c) , greater than 1.

Load more