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3 Stability of an individual bank In this section we want to analyze the stability of an individual bank. It is important to understand how decisions in the bank will be taken because of their inﬂuence on the stability and the payoﬀs of the three stakeholder groups of the bank: bankers, capital owners, and depositors. The optimal decision concerning restructuring or continuing late projects depends on the particular interest rate r that occurs in date t = 1.14 Although the bank manager would always prefer to continue late projects, since only when continuing he earns a rent but gets nothing in case of restructuring, the capital owners will force the banker to maximize the net present value of the projects.15 The capital owners of the bank want to consume at date 1 and therefore they try to maximize the t1 -consumption goods available to the bank. This means they will force γC the banker to restructure a project if c1 and let she continue it otherwise, (1+k)r
γC γCi.e. if c1 ≤. We will denote this hurdle rate with r = ˜. The higher the (1+k)r (1+k)c1 interest rate for getting liquidity, the more valuable is restructuring because it generates liquidity immediately. But this restructuring decision is biased, because only part of late projects’ return is pledgable to outside ﬁnanciers of the bank. As long as γC kγC c1 + + (1 − γ) C, it is socially ineﬃcient to restructure late projects.
(1+k)r (1+k)ρ Turning to the decision of depositors, we already mentioned that it is individually rational for them to withdraw their funds whenever the net present value of the bank at date 1 is not enough to fulﬁll their claims. Consequently, a run on the particular bank is triggered whenever the sum of deposits exceeds the net present value of the bank at date 1: D ≥ V1.16 Therefore, given that capital owners force bankers to restructure late projects beIn the following analysis we have taken the banks’ date 0 portfolio decision concerning investment in storage and lending as given and analyze the case where the bank will not store but invest any funds in lending activity. We are sure this is the optimal decision when the probability p1 for the state where all the projects in both regions are early, is suﬃciently high.
We will use the terms banker and bank manager synonymously. The banker will continue the project despite having a strong preference for date 1 consumption. This means that even with a high discount rate of date 2 consumption the present value of the rent she can earn is positive.
Clearly, as in Diamond/Dybvig there exist two pure strategy equilibria in those cases where D V1 but D c1. Under these circumstances the individually rational decision of every depositor depend on his belief about the decision of all other depositors. As long as he expects the others to withdraw he also has an incentive to do so. But if he thinks the others will wait until t = 1 he is also inclined to withdraw not before t = 1. Here we assume that depositors will always wait until t = 1 as long as D ≤ V1.
4 Equilibrium in the liquidity market The gross liquidity produced in the economy is the return on early projects. But part of the liquidity goes to banks, which split it into rents to the banker, return to capital owners and repayment to depositors. Since we assume that bank managers, capital owners as well as depositors have a discount rate of t2 - consumption that exceeds any upper bound of the equilibrium interest rate, they will immediately consume this fraction of the liquidity. The other part of the liquidity produced by early projects are the rents of the entrepreneurs. Since they do not discount future consumption, they will supply their liquidity at the t1 -ﬁnancial market, as long as they get at least a return of 1. Given the overall fraction (α + α) of early projects in both regions, the
aggregate liquidity supply amounts to:
LS = (α + α) (1 − γ) C (6)
Because all the stake holders in the bank - bank manager, capital owner and depositors - have a strong preference for immediate consumption in t1, the bank manager will try to raise liquidity against the pledgable income of late projects, in order to repay deposits, pay the return on capital and consume his own rents.
Proposition 1 In the secondary ﬁnancial market banks try to borrow liquidity from early entrepreneurs against the pledgable return of late projects.
Obviously, given this aggregate liquidity demand three qualitatively very diﬀerent equilibria occur depending on the aggregate liquidity supply, which is given by the overall fraction of early projects in the economy.
Proposition 2 Depending on the aggregate fraction of late projects three types of ﬁnancial crises may emerge. 1) Slight liquidity crises, in which no bank collapses, 2) moderate liquidity crises, in which only weak banks are subject to a run and 3) severe liquidity squeezes, which also destabilize stronger banks.
Given that the overall fraction of late projects is rather limited, a slight liquidity crises occurs. This case is depicted in ﬁgure 1. Trying to attract new funds from the early entrepreneurs against the required mixture of deposits and capital banks bid up the interest rate only slightly to
higher the relation of pledgable to non-pledgable income of ﬁnished projects, since both determine the relative scarcity of liquidity in t1. Moreover, the interest rate is higher if the capital requirements are smaller, since capital requirements increase the rents of the banker and thereby reduce the returns of late project that can be promised to new depositors and capital owners in t1.
However, if the ”cash in the market”-constraint is more restrictive, i.e. the aggregate fraction of early projects smaller, the economy ends up in a moderate liquidity crises, in which part of the banking sector collapses. In that case, which is shown in ﬁgure 2, the lack of liquidity causes the equilibrium interest rate to climb up to
that is subject to a more or less idiosyncratic adverse liquidity shock will collapse.
The other part of the banking sector that does not face a severe idiosyncratic liquidity shock, because only a limited fraction of its projects turns out to be late, can ﬁnish all projects.
In contrast, if the aggregate fraction of late projects is even higher the economy ends up in a severe liquidity crisis. In this case the equilibrium interest rate will reach its upper bound r∗∗∗ = r ˜ (10) Obviously, at this interest rate level weak banks collapse. But what diﬀerentiates a moderate from a severe liquidity crisis is that in the latter even strong banks have to restructure part of their late projects. At the equilibrium interest rate r capital owners ˜ are indiﬀerent between restructuring and continuing late projects. However, the available liquidity is insuﬃcient to repay all depositors. Therefore, the bank manager, who only receives a rent if projects are ﬁnished, will restructure just enough late projects to produce suﬃcient liquidity to prevent a run. The fraction of late projects that can be continued in a severe liquidity crises is given in equilibrium by
projects relative to the fraction of late projects at strong banks, 2) the higher the non-pledgable returns of entrepreneurs in relation to the pledgable returns going to the banks and 3) the smaller the present value of the fraction of the banks’ returns that can credibly be promised to new capital owners and depositors at the given interest rate r. Inserting the equilibrium value for r into the last expression shows that this ˜ ˜ is just the relation between the pledgable return of late projects if continued to the return of these projects if restructured (see equation (11)). Consequently, if continuing late projects gives a higher return to banks relative to restructuring, a higher fraction of late projects will be ﬁnished even in a severe liquidity shortage.
To sum up, in a severe liquidity shortage it is not enough that weak banks fail and therefore stop demanding liquidity. If the aggregate fraction of late projects is too high, even those banks that have ﬁnanced a comparatively small fraction of projects that turn out to be late will not be able to raise enough liquidity at the ﬁnancial market. However, these liquidity rationed banks do not collapse, but they will have to restructure late projects to raise suﬃcient liquidity to repay deposits.
Having described the equilibrium in the ﬁnancial market it is straightforward to see which impact the particular type of the ﬁnancial system has on the equilibrium.
Obviously, the higher fraction of pledgable income (γ) in bank-dominated ﬁnancial systems shifts the entire liquidity demand to the upper right. Because the higher the pledgable income the higher the present value of late projects and the more aggressive banks can bid for funds in t1 in slight and moderate liquidity crises. In severe liquidity crises the higher return on late projects makes capital owners more willing to accept a continuation of late projects even for higher interest rates. On the supply side a higher fraction of pledgable income reduces the return of early entrepreneurs, thereby lowering the liquidity supply in the economy. All these eﬀects of a higher fraction of pledgable returns point in same direction: Fluctuations of the interest rate in case of a ﬁnancial crisis are higher in bank-dominated ﬁnancial systems than in market-oriented ﬁnancial systems. This is also reﬂected in the respective equations of the equilibrium interest rate (see equations (8), (9) and (10)) A lower return on restructured projects (c1 ), which we also characterized as being typical for a bank-dominated ﬁnancial system only inﬂuences the equilibrium interest rate in severe liquidity crises. The lower the returns from restructuring late projects the higher the interest rate up to which capital owners will accept a continuation of late projects of the bank manager. Thus, as can also be seen in equation (10), the interest rate ﬂuctuations in severe liquidity crises also increase with a lower c1 and are therefore higher in bank-dominated ﬁnancial systems.
It is interesting to note, that also the threshold level for the diﬀerent ﬁnancial crises with respect to a given liquidity supply depends on the type of the ﬁnancial system.
Inserting r into the liquidity demand one can derive the threshold level for aggregate ˜ liquidity supply between moderate and severe liquidity crises. This shows that if the aggregate liquidity supply falls short of (1 − α) · c1 the economy ends up in a severe crisis. While this threshold level obviously is not inﬂuenced by the fraction of pledgable returns, it rises the higher the returns on restructured projects. Thus, in marketoriented ﬁnancial systems, in which c1 is higher, the economy ends up more often in a severe liquidity crisis, while in bank-dominated ﬁnancial systems given a certain level of aggregate liquidity supply moderate liquidity crises are more likely. Similarly, the threshold level between slight and moderate liquidity crises can be derived by inserting r into the liquidity demand function showing that in bank-dominated ﬁnancial systems ˆ characterized by a high γ it is more likely to be in a moderate than in a slight liquidity crisis.
Proposition 3 In bank-dominated ﬁnancial systems interest rate ﬂuctuations are higher during ﬁnancial crises than in market-oriented ﬁnancial systems. Moderate liquidity crises are more likely in bank-dominated ﬁnancial systems, while in market-oriented ﬁnancial systems severe but also slight liquidity crises are more likely to occur.
5 Optimal LOLR-policy Restructuring late projects is always welfare reducing in this economy. If the interest rate is below r this is most obvious, since in that case the net present value of the ˜ pledgable income from late projects that can credibly be promised to capital owners and depositors of the bank is higher than the returns generated if the projects are restructured.