银行业务模型(PDF7)(一)

1、a model of bankingthe diamond-dybvng (1983) model marks an important advance in the theory of banking because it provides a niicroeconoinic account of banking as a distinct activity. it goes beyond the simple j)ortfolio choice model, which is c-oinmoii to all financial institutions, and addresses th

2、e issues of maturity matching and insurance against liquidity shocks.matcliing maturitiesit is a truism that banks have ikpiicl liabilities and illkpiid assets. in other words, they bonow short and lend long. this makes banks vulnerable to sudden deiiiaiids for liquidity (bank runs), but more on tli

3、is later. this maturity niismatch reflects the uiiderlyiiig structure of the economy: individuals have a j)reference for liquidity but the most profitable investment opportunities take a long time to pay off. banks aie an eflicient way of bridging the gap between technology and liquidity preference.

4、 what follows is a simple exam- ple. loosely based on the dianioncl-dybvig (1983) model, that shows how all the pieces fit together.there are three dates f = 0, 1.2. tliere is a single good tliat can be used for coiismiiption and investnieiit and serves as a nunieraire. there arc two tyj)es of asset

5、s: the liquid asset (also called the short asset) is a constant returns to scale tedinology that takes one unit of the good at date t and converts it into one unit of the good at date t + 1, where f = 0,1; the illiquid asset (also called the long is a constant retmiis to scale technology that takes

6、one unit of the good at date 0 and transforms it into r 1 units of the good at date 2; if the long asset is li(piidated prematurely at date 1 then it pays 0 r 0 denotes consumption at date t = 1.2 and u(-) is a neoclassical utility function (increasing, strictly concave, twice continuously different

7、iable). we assume that there is no aggregate uncertainty: in every state of natiue, the fraction of early coiisumers is a and the fraction of late consumers is 1 a.market eqiiilibriimithe agents could all invest their individual endownieiits in the long and short assets to provide for coiisuinptioii

8、 at dates 1 and 2. but this provides no insurance against the intertemporal preference shock. if they invest in the short asset to provide consumption at date 1, they miss out on the higher returns from the long asset. if they invest in the long asset to provide consumption at date 2. they run the r

9、isk of having to sell the asset at a low i)rice to provide consumption at date 1.suppose that an individual invests in a portfolio (y) consisting of x rniits of the long asset and y units of the short asset. his budget constraint at date 0 isat date 1 he discovers whether he is an earlv or late cons

10、umer. if he is an early consumer, he will liquidate his portfolio and consume the proceeds. suppose the j)rice of the long asset is p. then his consuinj)tion isci ?/ + px.if he is a late consumer, he will want to rebalance his portfolio. we nnist have p /? in equilibriiun, becaiise otherwise no one

11、will be willing to hold the long asset. in that case, it will be optimal for the late consulner to hold only the long asset, in which case liis coiismnptioji will be0 ( + y/p)r-then the agent s decision problem ismax au(ci) + (1 a)u(c2)s.t. % + y w 1.ci y + pxc2 1 then the long asset dominates the s

12、hort asset and no one will hold the short asset. in tliat case, early consumers will be offering the long asset for sale but there will be no buyers at date 1. then the price must fall to p = 0, a contradiction. on the other hand, if f p; if he is a late consumer, he buys the long asset and earns a

13、return of 7?/p r. then no one hokls the long asset at date 0 anti when late consumers try to buy it at date 1 the price will be bid up to p = oo, another contradiction.with p = i the two assets are effectively the same and the portfolio choice becomes iinmaterial. the agents consuinj)tion is ci =/ +

14、 /= 1 at date 1 and c2 = (x 4- y)r = b at date 2. the ecpiilibrium utility isau(l) + (l-a)u(z?).this serves as a benclmiark against which to measure the value of having a banking system.the banking solutiona bank, by pooling the depositors investments, can provide iiisinance against the jneference s

15、hock and allow early consumers to share the higher returns of the long aset. the bank takes one unit of the good fioin each agent at. date 0 and invests it in a portfolio (x, y) consisting of x units of the long asset and y units of the short asset. because there is no aggregate iiiicertainty. the b

16、ank can offer each consumer a non-stochcistic consumption profile (cnc2)5 where c is the consumption of an early consumer and c2 is the consumption of a late consumer. we can interpret (c15c2) as a deposit contract luider which the depositor has the right to withdraw either( at date 1 or c2 at date

17、2, but not both.there is assumed to be free entry into the banking sector. coinpetitioii among the banks forces them to maximize the ex ante expected utility of the typical depositor subject to a zero-profit (feasibility) constraint. formally, the bank s problem ismax au (ci) + (1 s.t. x y i aci y(1

18、 a)c2 rxthe objective function is the ex ante expected utility of the typical depositor and the three constraints correspond to the banks budget constraint at each of the dates 0,1,2.the first-order conditions for the inaxinnnn (assuming an interior solution) are:mo = ipo = 42a u(q) = u(c2)= 2where

19、/f is the lagrange iniiltiplier of the budget constraint at date t. simplifying these equations, wc have3 = /?(c2)or c1 c2. tliis is an important property, because it ensures that the late consumers never want to imitate the early consumers. an early consumer has no choice but to withdraw q at date

20、1. the late consumer, on the other hand, can either wait imtil date 2 and withdraw c2 or pretend to be an early consumer, withdraw c1, and use the short asset to save it until the final period. however, as long as ci 1 and c2 = 9)/(1 a) a sufficient condition is that r?/(c) be decreasing in c. which

21、 is equivalent to saying that the relative risk aversion is greater than one:if this inecpiality is reversed and t/(c) 1, early consumers get less( = y/a r than in the benchmark case.bank rmissuppose that (q,c2)is the optimal deposit contract and (x. y) is the optimal portfolio for the bank. in the

22、absence of aggregate uncertainty, the j)ortfolio (x. y) provides just the right amount of liquidity at each date assuming that the early consumers are the only ones to withdraw at date 1 and the late consumers all withdraw at date 2. this is an equilibrium in the sense that the bank is iirixirnizing

23、 its objective, the welfare of the typical depositor, and the early and late consumers are timing their withdrawals to niaximize their coiisumption.there is another equilibrium, however, if vve assume that the bank is required to liquidate whatever assets it has in order to meet the demands of the c

24、onsumers who withdraw at date 1. to see this, suppose that all depositors, whether they are early or late consuniers, decide to withdraw at date 1. the liquidated value of the l)ank s assets at date 1 isri y 1, the bank is insolvent and will be able to pay only a fraction of the promised amount. mor

25、e importantly, all the bank s assets will be used np at date 1 in the attempt to meet the demands of the early withclrawers. anyone who waits until the last period will get nothing. tinis, given that a late consumer thinks everyone else will withdraw at elate 1 it is optimal for a late consumer to w

26、ithdraw at date 1 and save the proceeds until date 2. tinis. bank rmis are ail e(piilibriuin phenoineiion. the following payoff matrix illustrates the two equilibria of this coordination game. the rows correspond to the decision of a distinguished late consumer and the columns to the decision of the

27、 typical late consumer. (note: this is not a 2 x 2 game; the choice of column represents the actions of all but one late consumer). the ordered j)airs are the payoff for the distinguished late consumer (the first element) and the typical late consumer (the second element).runno runrunno run(rx + y,rx + y)(0, rx + y)(ci,c2)4,3)it is clear that if0 rx + ?/ cj c2then (run, run) is an equilibrium and (no run. no run) is also an equilib- rimn.the analysis of the bank run is predicated on the bank li(pndating all of its assets in order to meet the demand for liquidity at date 1. this may be