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Berkeley ECON 202A - Financial Instability

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Economics 202ALecture Outline #13 (version 2.0)Maurice ObstfeldFinancial InstabilityIn this lecture I describe the Diamond-Dybvig model of bank runs, fromJournal of Political Economy, June 1983. In this model, bank-like nancialintermediaries promote risk sharing among individuals, but they are subjectto arbitrary panics.The modelThere are three periods, T = 0; 1; 2:There are two possible technologies on date 0, short and long.Investment of 1 unit of output in the short technology at T = 0 yields 1unit of output in period 1 and 0 in period 2.Investment of 1 unit of output in the long technology at T = 0 yields 0units of output in period 1 and R > 1 units in period 2.Individuals need not specify the technology they are choosing ex ante.They opt for the short or long technology simply by \harvesting" the yieldeither on date 1 or 2, respectively.The idea is that more roundabout technologies are more productive.At time 0, a depositor does not know his/her \type," patient or impatient.Depositors are indexed by the unit interval, [0; 1]. at the start of period 1,a fraction p is revealed to be of type 1, or impatient. The rest (of measure1  p) are of type 2, patient. An agent has an endowment 1 in period 0 andconsumes in period 1 and/or 2. The utility functions of types 1 and 2 areU(c1; c2; 1) = u(c1);U(c1; c2; 2) = u(c1+ c2);where limc!0u0(c) = 1, limc!1u0(c) = 0, and cu00(c)=u0(c) > 1.1First consider an autarkic individual. That person will pick c1= 1 ifhe/she turns out to be impatient, c2= R if patient. That person's ex anteexpected utility is an average over the utilities of the two types:EU = pu(1) + (1  p)u(R):People can do better than this, however, if there are nancial intermedi-aries.Social optimumA benevolent and omnipotent planner would withdraw an amount 1 x from investment on T = 1 so as to maximize the expected utility of arepresentative individualpuc11+ (1  p)u(c21+ c22)subject to the aggregate resource constraintspc11+ (1  p)c21= 1  x;(1  p)c22= Rx:Here, cijis the amount type i consumes in period j. Of course, it is alwaysoptimal that c21= 0:So we are left with the simpler problem:maxc11;c22puc11+ (1  p)u(c22)subject topc11+ (1  p)c22R= 1:If  is the Lagrange multiplier on the resource constraint, the rst-orderconditions for a maximum areu0(c11) = u0(c22) = =R)=)u0c11=u0c22= R:2This social optimum implies that an impatient person gets to consumemore than c11= 1, the autarky value. Why? The budget constraint of theplanner isc22=R1  ppR1  pc11:At the autarky allocation, however, because relative risk aversion exceeds 1,the absolute-value slope of the social indierence curve satisespu0(1)(1  p)u0(R)>pR1  p;which means that it exceeds the absolute-value slope of the planner's budgetline. (Please refer to the diagram on the next page so that you can visualizethis.) For example, if u(c) = c1=(1  ), this condition is u0(1) > Ru0(R);or 1 > R1; which holds for  > 1 (because R > 1). In this case of highrisk aversion, the social optimum \insures" agents against being impatientand ending up with relatively low consumption. I denote the social optimumconsumption levels by c1 1and c2 2. Observe that c1 1must be stricly less thanc2 2(as is also indicated in the diagram).1Banks and bank runsTo make the model interesting, assume that an individual's type andconsumption cannot be veried. Imagine there were contracts that wouldinsure people upon learning they were impatient. The payments would haveto come from patient types liquidating part of their investment.Such contracts would never work. You would have an incentive to pre-tend to be impatient, reaping an insurance payment, say x, that you could1The slope of an indierence curve U at the consumption pairc11; c22isdc22dc11U= pu0c11(1  p)u0(c22):This implies that where c11= c22, the absolute-value slope of any indierence curve isp=(1  p). Because R > 1, that slope is strictly b e low the absolute-value slope of theplanner's budget line, pR=(1  p). As a result, c1 1is stricly less than c2 23consume in period 1 (making c21= x). Then you could leave your investmentin place and still consume c22= R in period 2.So consider instead a bank contract. Everyone deposits their resources inthe bank at time 0. Patient types can withdraw r1> 1 in perio d 1 | withtheir withdrawals monitored by the bank. Patient depositors get their prorata share of what is left after period 1 withdrawals.Banks have the potential to implement the optimum. If r1= c1 1and afraction p of the population (the impatient) withdraws deposits on date 1,then each of the patient consumes his or her pro rata share of the balance,R(1  pc1 1)=(1  p) = c2 2: Because c2 2> c1 1, as observed above, no patientdepositor has an incentive to withdraw early. So this setup clearly yields anequilibrium for periods 1 and 2. Furthermore, if on at T = 0 agents expectthis equilibrium to prevail with probability 1, each of them, knowing thatthe expected utility from signing the contract exceeds the autarky level, willindeed sign and deposit his or her resources in the bank.Things can go wrong however, because the preceding equilibrium for pe-riods 1 and 2 is not the only one. To capture the reality of banking, themodel assumes a sequential service constraint: essentially, this means thatthe bank services customers' claims, in the order in which they arrive, untilits resources run out. Let V1be the payo you get (depending on your placein line) is you withdraw in period 1, and V2the payo you get in period 2 ifyou do not withdraw in period 1. If fjdenotes the number of depositors ser-viced before depositor j on date 1, and f is the total number of withdrawalson date 1, thenV1(fj; r1) =(r1if fjr1< 10 if fjr1 1andV2(f; r1) = max(R(1  r1f)1  f; 0):In the rst-best equilibrium, f = p and soV2(f; r1) = V2(f; c1 1) =R(1  pc1 1)1  p= c2 2:4Alas, if r1were equal to 1, then we would have V2(f; 1) = max fR; 0g = R,and patient types would never have an incentive to withdraw in period 1.But then, banks would be no better than autarky. To do better, we needr1> 1, and in that case, there can be a depositor panic | a run on the bank.For example, suppose you turn out to be patient but think that f will be1=r1. In that case, you expect depositors to withdraw all the bank's resourcesat T = 1, making V21r1; r1= 0: So it is


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