Is is optimal, because by consuming in period 2 patient depositors earn

Is is optimal, because by consuming in period 2 patient depositors earn the return (R > 1). The second restriction is the Q-VD-OPhMedChemExpress QVD-OPH resource constraint. For simplicity, we will denote c1?by c?and c2?by c?. The solution to this problem is characterized by 1 1 2 2 u0 ???Ru0 ?? 1 2 which implies R > c?> c?> 1: 2 1 The optimal allocation can be implemented by banks via a simple deposit contract. The depositors withdrawing in period 1 are given c?, while those who keep their funds in the bank 1 reap the benefit of the long-term investment and divide it STI-571 web equally among themselves. The consumption of those who wait depends on the mass of withdrawals in period 1 () and can be expressed in the following way: 8 ?> max 0; R? ?oc1 ?if 1 ?o > 0 < 1 ; ??c2 ??> : 0 if 1 ?o ?0 If = , then c2 ??c?. Nevertheless, if is high enough, then withdrawing in period 1 is a 2 better option for a patient depositor, than keeping the money deposited (provided the bank has money left to pay). There is a threshold value such that if the number of those who have withdrawn is over this threshold, then the period-2 consumption will be less than c?. 1 ??PLOS ONE | DOI:10.1371/journal.pone.0147268 April 1,7 /Correlated Observations, the Law of Small Numbers and Bank RunsLemma 1 There exists a po < 1 such that c2 ?< c?for any o < o 1 and c???c2 ?for any o ! o:Proof. The bank cannot pay in period 1 to all depositors c?> 1, since depositors have a unit 1 endowment and gross return upon withdrawing jir.2012.0140 in the first period is 1. Hence, for any c1?o;1 2 ?c2() = 0. On the other hand, c??c2 ?for = and 0 < @c@o for any c1?< o < p, so by continuity of c2() for 1 1 > 0 there is a unique o such that for any o o we have ?? c1 c2 ? whereas for any o < o we have c2 ?< c1 : Lemma 2 Assume that the utility function exhibits constant relative risk aversion (CRRA) and takes the following standard from: u i ??c1 i ; 11 ? d �p1R? c??and > 1, R > 1. In this case the optimal allocation is the following: c??1 c??1 ? d �p,Rd1 . The threshold value o is given by o ?c? ?? We have the following comparativeR ?statics results: @ o > 0, @ o < 0, and @o < 0. @p @d @R @c?1 @p @o @dProof. Since > 1, 1 < 0 and given R > 1 we have that R d < 0. It is straightforward to show that ?? @ o @c1 @c?@d1 d< 1. As a consequence,@c?1 @d@o @c?< 0, so1=d@o @p?? @ o @c1 @c?@p> 0. Similarly,> 0, therefore< 0.d? ? dR? . The sign of this derivative Regarding @ o , we can journal.pone.0158910 compute it as @o ? ? R d ?? R @R @Rdepends on the sign of f(R, )(R1/(( – 1)R + 1) – R). Some properties of f(R, ): > 0 if R > 1. Further, f(1, ) = 0, therefore f(R, )>0 if R > 1. From there we conclude that < 0 if R > 1. Patient depositors increase their optimal withdrawal threshold o when there are more impatient depositors in the economy since they are aware that more withdrawals are due to impatient and not patient depositors. The derivative of the threshold with respect to the relative risk aversion coefficient is negative (@ o < 0), so the more risk-averse are the depositors, the @d more they want to smooth the consumption. Consequently, higher risk aversion implies a smaller difference between c?and c2, so the threshold decreases. 1 As for @ o , an increase in the return has a double effect. On the one hand, it increases the con@R sumption given to those who withdraw in the first period (c?) which--ceteris paribus--lowers 1 @o the threshold (@c?< 0). On the other hand, a higher return increases also the period-2 con sumption which ha.Is is optimal, because by consuming in period 2 patient depositors earn the return (R > 1). The second restriction is the resource constraint. For simplicity, we will denote c1?by c?and c2?by c?. The solution to this problem is characterized by 1 1 2 2 u0 ???Ru0 ?? 1 2 which implies R > c?> c?> 1: 2 1 The optimal allocation can be implemented by banks via a simple deposit contract. The depositors withdrawing in period 1 are given c?, while those who keep their funds in the bank 1 reap the benefit of the long-term investment and divide it equally among themselves. The consumption of those who wait depends on the mass of withdrawals in period 1 () and can be expressed in the following way: 8 ?> max 0; R? ?oc1 ?if 1 ?o > 0 < 1 ; ??c2 ??> : 0 if 1 ?o ?0 If = , then c2 ??c?. Nevertheless, if is high enough, then withdrawing in period 1 is a 2 better option for a patient depositor, than keeping the money deposited (provided the bank has money left to pay). There is a threshold value such that if the number of those who have withdrawn is over this threshold, then the period-2 consumption will be less than c?. 1 ??PLOS ONE | DOI:10.1371/journal.pone.0147268 April 1,7 /Correlated Observations, the Law of Small Numbers and Bank RunsLemma 1 There exists a po < 1 such that c2 ?< c?for any o < o 1 and c???c2 ?for any o ! o:Proof. The bank cannot pay in period 1 to all depositors c?> 1, since depositors have a unit 1 endowment and gross return upon withdrawing jir.2012.0140 in the first period is 1. Hence, for any c1?o;1 2 ?c2() = 0. On the other hand, c??c2 ?for = and 0 < @c@o for any c1?< o < p, so by continuity of c2() for 1 1 > 0 there is a unique o such that for any o o we have ?? c1 c2 ? whereas for any o < o we have c2 ?< c1 : Lemma 2 Assume that the utility function exhibits constant relative risk aversion (CRRA) and takes the following standard from: u i ??c1 i ; 11 ? d �p1R? c??and > 1, R > 1. In this case the optimal allocation is the following: c??1 c??1 ? d �p,Rd1 . The threshold value o is given by o ?c? ?? We have the following comparativeR ?statics results: @ o > 0, @ o < 0, and @o < 0. @p @d @R @c?1 @p @o @dProof. Since > 1, 1 < 0 and given R > 1 we have that R d < 0. It is straightforward to show that ?? @ o @c1 @c?@d1 d< 1. As a consequence,@c?1 @d@o @c?< 0, so1=d@o @p?? @ o @c1 @c?@p> 0. Similarly,> 0, therefore< 0.d? ? dR? . The sign of this derivative Regarding @ o , we can journal.pone.0158910 compute it as @o ? ? R d ?? R @R @Rdepends on the sign of f(R, )(R1/(( – 1)R + 1) – R). Some properties of f(R, ): > 0 if R > 1. Further, f(1, ) = 0, therefore f(R, )>0 if R > 1. From there we conclude that < 0 if R > 1. Patient depositors increase their optimal withdrawal threshold o when there are more impatient depositors in the economy since they are aware that more withdrawals are due to impatient and not patient depositors. The derivative of the threshold with respect to the relative risk aversion coefficient is negative (@ o < 0), so the more risk-averse are the depositors, the @d more they want to smooth the consumption. Consequently, higher risk aversion implies a smaller difference between c?and c2, so the threshold decreases. 1 As for @ o , an increase in the return has a double effect. On the one hand, it increases the con@R sumption given to those who withdraw in the first period (c?) which--ceteris paribus--lowers 1 @o the threshold (@c?< 0). On the other hand, a higher return increases also the period-2 con sumption which ha.