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But besides the fact that parts of the returns a ﬁnished investment project generates can not be passed on by entrepreneurs and bank manager, which distorts the decision of bank owners to continue late projects, what contributes to the ineﬃcient termination of late project is the bank’s reﬁnancing through deposits. What is in general the advantage of demand deposits - the threat of a coordination failure among depositors that allows bankers to credibly commit to repay - turns out to be a serious drawback in a liquidity crises particularly for weak banks. Banks are not able to bargain on the repayment of deposits in a crises situation to ﬁnish late projects.
A LOLR can provide banks with additional liquidity. To keep the analysis as simple as possible, we assume that the LOLR can raise the liquidity by taxing t1 consumption. This can be interpreted as a shortcut for an inﬂation tax: The central bank as the LOLR increases the currency in circulation by providing additional means of payments to the banks to enable them to settle their nominal obligations. Since this increases the money supply without changing the contemporaneous provision of goods, it simply reduces the real value of money in terms of t1 -consumption goods. It therefore resembles a taxation of any t1 -consumption in the economy.18 However, the provision of liquidity by the LOLR is associated with a cost. An inﬂation tax just like any other tax (apart from per capita taxes) brings about ineﬃFor a more detailed discussion of this argument see Allen and Gale (1998).
ciencies in the economy that cause welfare losses. For simplicity we take these welfare losses (W L) as an exogenous cost, that increases proportional with the volume of the liquidity assistance (LA): W L = β · LA.
There are two distinct policies the LOLR can follow in providing the liquidity to the banking sector in a crisis. The ﬁrst option, which captures the basic features of Bagehot’s suggestions, is to supply liquidity to the market by buying ﬁnancial assets, i.e. bank equity or deposits. In doing so the LOLR can stabilize the interest rate and prevent the banks from restructuring late projects. The second option, which reﬂects a more discrete policy, is to provide liquidity assistance to individual banks. Applying this policy the LOLR can supply liquidity at diﬀerent terms to diﬀerent banks.
In a slight liquidity crisis there is no need for a LOLR-intervention. All late projects are continued in spite of the liquidity shortage. The interest rate increase due to the slight liquidity squeeze only raises the consumption of early entrepreneurs at the expense of bank managers and bank capital owners. Therefore, a slight liquidity crisis only causes a reallocation of resources, that does not bring about any ineﬃciencies.
Proposition 4 In a slight liquidity crisis there is no need for a lender of last resort, since all late projects are continued anyway.
In a moderate liquidity crisis weak banks are threatened by a run in which depositors would seize the assets and restructure the late projects. Therefore, a liquidity assistance to prevent this could be beneﬁcial.
If the LOLR decided to supply the weak banks with the funds to repay the deposits through an individual assistance (IA), the amount of liquidity the LOLR has to provide is given by deposits less the liquidity available to the bank from the returns on early
LAIA = D − α · γ · C (12) m The LOLR oﬀers the liquidity assistance at the interest rate r against the future ˆ income of late projects that can be promised to outside ﬁnanciers of the bank. So in t1 there is just enough liquidity available to the bank to repay depositors. Therefore, the LOLR-assistance enables depositors to collect the full value of their deposits (D) from late projects not just the return generated by restructuring (c1 ). Using the LOLR assistance even bank managers and bank capital owners gain since they can at least realize their rents from late projects ( 2·k·γ·C ). However, since these rents are realized in 1+k t2 they have to be discounted with the rather high discount factor ρ of bank managers and capital owners.19 In addition, the LOLR-assistance enabling the continuation of late projects also preserves the rents of late entrepreneurs. In sum, an individual liquidity assistance in a moderate liquidity crisis can generate welfare gains that amount
1. the larger the fraction of late projects at strong banks because
a) on the one hand this increases the additional liquidity demand of strong banks Remember that we assumed a discount rate for these agents that always exceeds the equilibrium interest rate. Therefore: ρ r.
b) on the other hand this reduces the supplied liquidity by early entrepreneurs in the economy, thereby increasing the liquidity that has to be supplied additionally to strong banks,
2. the larger the fraction of late projects at weak banks, because
a) on the one hand this reduces the liquidity supplied by early entrepreneurs, too, and
b) on the other hand this reduces the threshold level to which the LOLR has to bring down the interest rate to prevent a run on these banks,
3. the smaller the capital requirements, which is also mainly due to the reduction of liquidity demand by raising capital requirements and
4. the lower the fraction of non-pledgable income (the higher the pledgable return on late projects), also because a higher pledgable return increases additional liquidity demand of strong banks.
Consequently, in bank-dominated ﬁnancial systems, which are particularly characterized by comparatively high levels of pledgable income, the ineﬃciencies of market interventions are more severe, whereas they are relatively limited in market-oriented systems.
Proposition 5 If a LOLR-intervention is beneﬁcial at all in a moderate liquidity shortages an individual liquidity assistance is always preferable over a market-intervention.
However, the eﬃciency loss of a market intervention is higher in bank-dominated ﬁnancial systems.
In a severe liquidity crisis not only late projects at weak banks but also some of the delayed projects at strong banks would be restructured without an additional liquidity supply by a LOLR.
Applying individual liquidity assistance in a severe liquidity squeeze, the LOLR would have to supply to weak banks the same amount of liquidity as in moderate crises. In order to prevent the ineﬃcient restructuring of late projects at weak banks the LOLR has to provide the additional liquidity that weak banks need to repay depositors at the threshold level r. But in addition to prevent the ineﬃcient restructuring at ˆ strong banks the LOLR has to supply them with the funds needed to ﬁnish their late projects, too. However, at strong banks it is not a potential run that could bring about the restructuring of late projects. At these banks it is the capital owners that do not allow the manager to pay higher interest rates than r on funds allowing to continue ˜ late projects. Bank managers can use only the liquidity they get at r, to ﬁnish late ˜ projects, while they have to restructure the remaining delayed projects. Therefore, the LOLR simply has to supply the additional liquidity that strong banks need to continue all late projects at r. Thus, given the fraction of restructured late projects at strong ˜ banks without a LOLR-intervention (1 − µ∗∗∗ ) the overall liquidity the LOLR has to
provide to the banking system amounts to:
1. the larger the fraction of late projects at strong banks, because this raises the ineﬃcient additional liquidity demand of strong banks,
2. the larger the fraction of late projects at weak banks, because an increase in the fraction of late projects at weak banks reduces the threshold level to which the LOLR has to bring down the interest rate to prevent a run on these banks,
3. the smaller the capital requirements, which is also mainly due to the reduction of liquidity demand by increasing capital requirements,
4. the higher the pledgable return on late projects, also because this increases additional liquidity demand of strong banks and and
Proposition 6 In a severe liquidity crisis an individual liquidity assistance is always preferable over a market-intervention, too. Again, the eﬃciency loss of a market intervention is higher in bank-dominated ﬁnancial systems than in a market-based ﬁnancial system.
To sum up, in all kinds of liquidity crises in which a LOLR-intervention is beneﬁcial an individual liquidity assistance is strictly preferable. However, the welfare gains of an individual liquidity assistance compared to a market intervention vary with the particular parameter setting of the respective economy. Most interestingly, an individual liquidity assistance is in general more preferable the more the parametrization of the economy resembles a bank-dominated ﬁnancial system. For instance, in both moderate as well as severe liquidity crises a high relation of pledgable to non pledgable income in ﬁnancial relations between ﬁrms and banks (a higher γ), which is due to the relationship lending most characteristic for bank-dominated ﬁnancial systems, makes an individual liquidity assistance more preferable. Moreover, relatively low returns from restructured projects (c1 ), which is also typical for bank-dominated ﬁnancial system compared to market oriented ﬁnancial systems, make an individual assistance more beneﬁcial, too.
So far we did not take into account the diﬀerent informational requirements of the LOLR-policies. However, it is obvious that an individual liquidity assistance requires much more information to be eﬀective than a market intervention.
To pursue an individual liquidity assistance the LOLR has to collect precise information about the liquidity needs of every single bank.20 Besides the administrative costs, this takes time and may cause an ineﬃcient delay of the LOLR-intervention.
This is particularly true, since banks do not have an incentive to honestly report their liquidity needs to the LOLR. By overstating the fraction of late projects bank managers could increase the individual liquidity assistance and at the same time reduce the interest rate the LOLR demands on the provided liquidity. Both increases their rents.
In contrast, if the LOLR applies market interventions, the LOLR only has to keep the interest rate in the money market at the threshold level r. Given that the lower ˆ bound (1 − α) of the distribution of the fraction of late project is public information, there is no information on individual banks required by the LOLR.
In order to take these considerations into account but keep the analysis tractable we assume that there are some ﬁxed informational costs associated with a policy of individual liquidity assistance.