I will make some additional remarks to my lecture on Monday. I think the main

Transcription

1 Jon Vislie; august 04 Hand out EON 4335 Economics of Banking Sulement to the the lecture on the Diamond Dybvig model I will make some additional remarks to my lecture on onday. I think the main results of this model are rovided by Tirole in The Theory of ororate Finance ; chat.. However, there are still some issues that might be a bit confusing. (I dro the details related to bank runs see Tirole.) There is no need going through the model here. The assumtions and the structure of the model should be rather obvious, but for later reference, some main results are briefly outlined: The main oint of the model is that ex ante (t = 0) no agent (among a continuum of agents) knows what tye she will be assigned at t = ; either an early consumer (with robability ) or a late consumer (with robability - ). Each agent is endowed with one unit of wealth that can be consumed or invested. (No consumtion takes lace at t = 0.) The robability distribution is common knowledge. The tye of an agent is i.i.d. or ideosyncratic (diverifiable, only unsystematic risk no macro risk), but cannot be verified by a third arty. (No insurance comany will ex ante offer insurance contracts based on such information.) Being an early consumer is considered as a liquidity shock, or the agent is exosed to an early liquidity need or demand for cash. The objective of any agent is to maximize exected (ex ante) utility, without discounting, U = u( ) + (- ) u( ), where u ( ) is an ordinary VN utility function, with as consumtion er caita, increasing, strictly concave (risk aversion) with u (0) =, and a coefficient of relative risk aversion (or the intertemoral elasticity of substitution) -u ( ) > U ( ) d or [ u ( )] < 0. (This will d get an agent to refer a comressed consumtion rofile.) lso we have that with

2 d u ( ) these assumtions, RS - = d ( - ) u ( ) will be decreasing; with normal indifference mas. Ex ante there are two investment oortunities: One short term roject with no net return after one eriod (a ure storage technology each unit invested in this roject will leave you with one unit after one eriod), or a long term roject with a gross return reaed after two eriods; the gross return is R >. However, this long term roject can, if necessary, be stoed, interruted or liquidated at t =, with a gross return l < er unit invested. Hence there is a real loss caused by liquidation, as a fraction - l will disaear. The trade off is therefore between early need for cash and the benefit from investing in the long term roject. Let I Î é0,ù ê ë ú û be the fraction of one s wealth that is invested in the long term roject. n otimal allocation can be characterized as follows: t the ex ante stage a lanner will solve the following risk sharing/investment rogram, when relying on the law of large number i.e. the lanner knows the fraction of early consumers, but not exacly which erson will be an early or a late consumer. Because of identical agents ex ante, the lanner may solve the roblem of a reresentative agent: ì ï íaxu = u ( ) + (-) u ( ) st.. ïî = -I ü ï ý ( - ) = RI ïþ The exected consumtion in the first eriod is and is made u of short term investment - I, whereas exected consumtion in eriod is ( - ) = RI, being equal to the gross return from the long term investment. ombining these to constraints, we get: - d R I - = - = - + = = R R d -

3 3 This is an efficiency locus or the frontier of the oortunity set. Hence the first best solution is found by maximizing exected utility subject to this constraint. It is a straightforward otimization rogramme that can easily be illustrated as follows: In the otimal solution the sloe of an indifference curve is equal to the sloe of the efficiency locus; i.e., we must have: u ( ) u ( ) u ( ) = = - = R- ( - ) u ( ) u ( ) u ( ) R R - where the last equality says that the required rate of return from saving should be equal to the net return from long term investment. Hence we have: u ( ) = Ru ( ). R - (, ) = R l d Because we have ( u ( )) < 0, and because R >,we must have d u () > R u ( R), where ( =, = R) is a feasible allocation on the efficiency locus hence u () > R R > > >. Because R > and u < 0, we have u ( R) u ( ) > u ( ) <. This will also make the solution so called incentive

4 4 efficient in the sense that no late agent will be motivated to act as an early consumer. late consumer can always retend to be imatient; take out invest this amount in a short term roject with a gross return =. However, this behaviour will incur a utility loss. at t =, and to be consumed at t Under autarky or self insurance each agent must rovide for her own needs by hoarding liquidity to meet a liquidity shock. If it turns out that the agent becomes an early consumer, liquidity is rovided by liquidating the long term investment along with consuming the return from the short term roject. On the other hand, if she faces no liquidity shock, the short run investment is rolled over another eriod, while also reaing the return from the long term roject. In both cases, not knowing one s tye will be costly ex ost. For some given long term investment, an agent having an early liquidity need, will have a consumtion caacity = - I + li = -(- l) I ; i.e. the sum of the short run (liquid) investment and the liqidated value of the long term investment. If she is not exosed to a shock, then her consumtion caacity at t =, is = - I + RI = + ( R- ) I ³ ; the short term investment rolled over (without any net return) and the full return from the illiquid long term roject. Note that, equal to one only if I = 0 =. If I =, then = l < and = R. The bold line in the diagram above, between (,) and ( l, R), shows that autarky goes with a social loss; hence we have + ( - ) < because in ( l, R) R we have l + ( - ) < and in(,) we have - + < + - =. R The otimal self insurance is found from solving:

5 5 I Îé0,ù ê ë ú û ìï ï = -(-l) I ( R ) I ïî + - í ï = + - ax u( ) ( ) u( ) s. t. Let vi ( ) = u( -( - l) I) + ( - ) u( + ( R- ) I) v () I = u ( )(-(- l)) + (-) u ( )( R- ) v (0) =-u ()( - l) + ( -) u ()( R- ) = u () é( -)( R-) -( -l) ù êë úû which is less than or equal to zero only if R - I = 0, which is the case if R is -l - small and l small. In that case; invest only in the short term roject. On the other hand, if v () =-u ( l)( - l) + ( - ) u ( R) ³ 0, then I = u ( ) R - n interior solution is charactrized by: =. ( - ) u ( ) - l Note that ex ost, the investment decision is always inefficient. t t =, I = 0 is the otimal investment choice by an imatient agent, whereas I = is the best one for a atient one. The third institutional arrangement is the following: nnounce at t = 0, that we will oen a financial market at at t =, once we have learnt the tye, and then oen u for borrowing and lending or exchange between agents of different characteristics, by trading bonds. Will the first best be achieved? bond market is oened at t =. Suose that having one unit of a bond will entitle you to one unit of consumtion at t =. Then one can go around the direct liquidation of long term investment if it turns out that an agent is exosed to a liquidity shock, because we now will have no loss due to liquidation. n agent being exosed to a liquidity shock at t =, will want to borrow now so as to be able to consume early, by selling bonds in a number equal to the full return from

6 6 her long term roject. (Note that selling bonds is the mirror image of demanding or buying goods.) This means that an agent with early consumtion needs can convert future returns into current consumtion by selling bonds or borrowing, while the buyers of these bonds (the lenders that suly goods at t =, because they have have a surlus at t = ) are entitled to the return from the long term investment made by early consumers by giving away or lending the return from their short term investment. The late agents will have a surlus at t =, equal the return from the short term investment, which can be consumed by those having early consumtion needs. This tye of trade takes lace by buying and selling bonds. Let us look at the financial market equilbrium. s said above; one unit of bond bought at t = from an early (or tye ) agent entitles the owner to one unit of late consumtion. Now there is no loss from liquidation; we will end u on the efficiency locus, but the risk sharing will not be otimal because ex ante there cannot be a full set of contingent (rrow Debreu) markets. Let be the rice of a bond, which measures the number of eriod consumtion units er unit consumtion in eriod. Hence is the relative rice of late consumtion (in units of early consumtion), whereas the inverse,, measures the number of eriod consumtion units er unit consumtion in t =, which is like a rate of gross return from having one unit of a bond. n agent being exosed to a liquidity shock (having early consumtion needs) will like to borrow or selling RI number of bonds (equal the future income or the return from her long term investment that is reaed at t = ), each at a rice, converting the future income to eriod consumtion, so that the buyer (a late agent) will get this return later. n early agent will then have the following consumtion

7 7 oortunity: While reaing the return from my short term investment, I can sulement this oortunity by the amount I borrow from the late agents equal to the number of bonds I sell, evaluated today, as each unit bond sold entitles me to units of consumtion at t = : Hence an early consumer will have: = - I + RI. late agent, buying bonds will sell away her short term return or surlus of goods at t =, by transforming this surlus to a consumtion stream at t = : The oortunity for a late consumer is therefore: = RI + é Iù é I RIù êë - úû = - + êë úû. late agent will have her own return at t =, but by selling away, through buying bonds at t =, the surlus at t =, which is - I units of early consumtion, will be converted to - I units of eriod consumtion in the financial market by trading bonds. We observe that =. We guess that or ³. If ³ or < R, then the rate of return from having a bond is below the rate of return from investing in the illiquid long term roject. rational investor will then want to choose I = because the rate of return from this tye of investment is higher than the alternative. That means that at t =, there is an excess demand for goods or excess suly of bonds. This cannot constitute a market equilibrium. If on the other hand, R >, the return from having a bond exceeds the return from long term roject. rational investor will choose I = 0, creating an excess suly of goods or excess demand for bonds at t =. Not an equilibrium. The return from having a bond exceeds the return from investing in the illiquid roject. Therefore in equilibrium we must have R =. (Note that the market clearing condition in the bond market is: - I RI = ( - ).

8 8 Total number of bonds sulied is equal to total demand for bonds, at t =.) From the ersective of one agent, the individual ex ante choice within this arrangement is given by: ì ax ï ü u( I RI ) ( ) u( I RI ) I Îé0,ù ê ë ú í é - + ù ï ý û êë úû ïî ïþ which must obey the allocation: ìï > 0 if R - > 0 I = u ( )( R - ) + (-) u ( ) ( R - ) ï 0 if R 0 I 0, í= - = Î é ù êë úû ï < 0if R - < 0 I = 0 ïî Because we have = R, we get: I = ( -)( -I) I = -. We then have: = and = R and I = - <. Hence in this market equilbrium we have (as indicated in the revious diagram): = R > > > = and I I >. It should not come as a surrise that the market equilbrium Pareto dominates the autarky equilibrium because there is no liquidation in equilibrium: = ³ and = R ³. However, the market equilbrium is not ex ante Pareto efficient because we have not a comlete set of risk or insurance markets ex ante. Liquidity risk can therefore not be roerly allocated. The market equilibrium shows higher volatility in consumtion than the first best allocation, and because risk is not roerly shared, eole will take more cautious actions by investing more in the long term roject than what is otimal.

9 9 The last institutional arrangement is banks having sufficient thrust or confidence in the oulation. Banks then announce: Deosit your endowment in my bank, I (the bank) will undertake the required asset transformation by investing (lending) in the long term roject, while offering demand deosits to all agents, each one to be allowed to withdraw units if she needs early liquidity (or are romised a short run rate of interest), or allowed to withdraw at t = (or being offered a higher rate of interest). In a cometitive banking equilibrium, banks comete for deosits. If one bank should offer a deosit contract that differs from the efficient one, another bank will overturn this offer by offering one that is closer to the one that maximizes exected utility. With free entry in the banking industry, bank rofits will be zero and equilibrium is characterized by each bank offering the efficient contract. Hence, introducing banks will imlement the efficient allocation! Banks can bridge the ga or eliminate the mismatch between the maturity structure and the oulation s need for liquidity. free entry banking equilibrium is characterized by banks offering a contract or consumtion rofile to agents so that: ax [ u( ) + (-) u( ) s. t. (- ) = (- ) R ] (, ) (If not the exected utility maximizing rofile is offered, another bank will gain by offering the one that is referred by deositors.) The zero rofit condition says that what is left for late deositors is equal to the long term return from investment (lending), after having ut aside, as liquid reserves, an amount equal to what early consumers are exected to withdraw; hence an amount - is being lent out at t = 0, with a gross return equal to R. Note that the zero rofit condition is identical to

10 0 our efficiency locus. Hence, the cometitive equilibrium deosit contract will coincide with the efficient consumtion rofile. When agents are exosed to idiosyncratic (diversifiable) liquidity shocks, the allocation of the financial market equilibrium can be imroved uon by having banks offering deosits contracts. Banks or financial interemdiaries act like insurance comanies and eliminate the cost of maturity mismatch by undertaking the otimal long term investment. The balance sheet will then look like: ssets Liabilities/Debt Liquid reserves= Deosits = Loans = I = - In this equilbrium there is no need for equity: ll risk can be fully diversified and there is no credit risk; the borrowers will reay with robability one. But, and this is the story with bank runs, the good roerties of this solution is highly deendent on the fact that only a fraction of deositors withdraw early. (The efficient deosit contract constitutes a good Nash equilibrium. If any late deositor exects none of the remaining late deositors to withdraw early, no late deositor will have any incentive to deviate; i.e. to withdraw early.) On the other hand, as shown by Tirole, there is also a bad Nash equilbrium where all deositors withdraw early under a system first come first serve. In that case the romised original (efficient contract) cannot be honoured.