Re not publicly observable and there is no aggregate uncertainty regarding the liquidity (S)-(-)-Blebbistatin custom synthesis preference of the depositors. At period 0 depositors form a bank by pooling their resources, so that they can share the risk of becoming impatient. Intuitively, the bank will liquidate at no cost some long-term investment in period 1 to pay a relatively high (that is, higher than their endowment) consumption to depositors who turn out to be impatient, while patient depositors may enjoy an even higher consumption in period 2. The bank offers this liquidity insurance through a simple deposit contract which is subject to a sequential service constraint. Hence, the bank commits to pay a fixed consumption (c1) to those who withdraw in period 1 (unless it runs out of funds) and a contingent payment in period 2 to those who keep deposited their money. The sequential service constraint means that the bank must pay c1 as a depositor claims her funds back in period 1. The depositors who keep the money deposited receive in period 2 a pro-rata share of the matured assets which have not been withdrawn during period 1. As usual in the Citarinostat web literature ([32]), in period 1, depositors are isolated and no trade can occur among them. The timing of events is as follows. At period 0 each depositor deposits in the bank her endowment which is invested in the technology. At the beginning of period 1 depositors learn their type privately and nature assigns a position in the line to each depositor. Depositors decide according to this exogenously determined sequence and we suppose that depositors do not know their position in the line and they assign a uniform probability of being at any possible position. This assumption follows [21]. Introducing (exact or approximate) knowledgePLOS ONE | DOI:10.1371/journal.pone.0147268 April 1,6 /Correlated Observations, the Law of Small Numbers and Bank Runsabout the position would complicate the analysis. Moreover, it seems rather unrealistic to assume that depositors have an accurate view about how many other depositors have already made a decision. Assume that before decision depositors observe a sample of previous actions which they use to form beliefs about the share of depositors who choose to withdraw. Based on these beliefs, they choose either to withdraw their funds from the bank or keep it deposited (that we also call waiting). We denote withdrawal as action zero (ai = 0) and keeping the money deposited as action 1 (ai = 1). Denote by (c1, c2) the consumption bundle of a depositor in the two periods. Consider journal.pone.0158910 the following utility function u 1 ; c2 ; yi ??u 1 ?yi c2 ? where i is a binomial random variable with support 0,1. After realization of the needs, if i = 0, then depositor i is impatient caring only about consumption in period 1, i = 1 represents a patient depositor. The utility function, u(.) is twice continuously differentiable, increasing, strictly concave, satisfies the Inada conditions and j.jecp.2014.02.009 the relative risk-aversion coefficient -cu00 (c)/u0 (c) > 1 for every c. If types were publicly observable in period 1, then the bank could calculate the optimal allocation based on types and independently of the position in the line. Denote by cy the consumpT tion of type in period T. Then the optimization problem takes the following form: max pu 0 ?c0 ??? ?p 1 ?c1 ?1 2 1 2 s:t: ??c0 ?c1 ?0 2 1 ??pc1 ? 1 ?p 2 =R ?1 1 2 The first restriction requires that impatient depositors consume in period 1 and patient depositors in period 2. Th.Re not publicly observable and there is no aggregate uncertainty regarding the liquidity preference of the depositors. At period 0 depositors form a bank by pooling their resources, so that they can share the risk of becoming impatient. Intuitively, the bank will liquidate at no cost some long-term investment in period 1 to pay a relatively high (that is, higher than their endowment) consumption to depositors who turn out to be impatient, while patient depositors may enjoy an even higher consumption in period 2. The bank offers this liquidity insurance through a simple deposit contract which is subject to a sequential service constraint. Hence, the bank commits to pay a fixed consumption (c1) to those who withdraw in period 1 (unless it runs out of funds) and a contingent payment in period 2 to those who keep deposited their money. The sequential service constraint means that the bank must pay c1 as a depositor claims her funds back in period 1. The depositors who keep the money deposited receive in period 2 a pro-rata share of the matured assets which have not been withdrawn during period 1. As usual in the literature ([32]), in period 1, depositors are isolated and no trade can occur among them. The timing of events is as follows. At period 0 each depositor deposits in the bank her endowment which is invested in the technology. At the beginning of period 1 depositors learn their type privately and nature assigns a position in the line to each depositor. Depositors decide according to this exogenously determined sequence and we suppose that depositors do not know their position in the line and they assign a uniform probability of being at any possible position. This assumption follows [21]. Introducing (exact or approximate) knowledgePLOS ONE | DOI:10.1371/journal.pone.0147268 April 1,6 /Correlated Observations, the Law of Small Numbers and Bank Runsabout the position would complicate the analysis. Moreover, it seems rather unrealistic to assume that depositors have an accurate view about how many other depositors have already made a decision. Assume that before decision depositors observe a sample of previous actions which they use to form beliefs about the share of depositors who choose to withdraw. Based on these beliefs, they choose either to withdraw their funds from the bank or keep it deposited (that we also call waiting). We denote withdrawal as action zero (ai = 0) and keeping the money deposited as action 1 (ai = 1). Denote by (c1, c2) the consumption bundle of a depositor in the two periods. Consider journal.pone.0158910 the following utility function u 1 ; c2 ; yi ??u 1 ?yi c2 ? where i is a binomial random variable with support 0,1. After realization of the needs, if i = 0, then depositor i is impatient caring only about consumption in period 1, i = 1 represents a patient depositor. The utility function, u(.) is twice continuously differentiable, increasing, strictly concave, satisfies the Inada conditions and j.jecp.2014.02.009 the relative risk-aversion coefficient -cu00 (c)/u0 (c) > 1 for every c. If types were publicly observable in period 1, then the bank could calculate the optimal allocation based on types and independently of the position in the line. Denote by cy the consumpT tion of type in period T. Then the optimization problem takes the following form: max pu 0 ?c0 ??? ?p 1 ?c1 ?1 2 1 2 s:t: ??c0 ?c1 ?0 2 1 ??pc1 ? 1 ?p 2 =R ?1 1 2 The first restriction requires that impatient depositors consume in period 1 and patient depositors in period 2. Th.