Multiple equilibrium overnight rates in a dynamic interbank market game

Games and Economic Behavior 56 (2006) 350–370 www.elsevier.com/locate/geb

Multiple equilibrium overnight rates in a dynamic interbank market game ✩ Jens Tapking ∗ European Central Bank, Kaiserstrasse 29, D-60311 Frankfurt, Germany Received 27 March 2003 Available online 12 October 2005

Abstract We analyse a two-period model of the interbank market, i.e. the market where banks trade liquidity. We assume that banks do not take the interbank interest rate as given, but instead negotiate on interest rates and transaction volumes with each other. The solution concept applied is the Shapley value. We show that there are a multiplicity of average equilibrium interest rates of the first period so that the average interest rate in this period does not convey any information on the expected liquidity situation on the interbank market. As the banks control not only the transaction volumes, but also the interest rates, they can leave the interest rates constant and adjust the transaction volumes when, for example, a liquidity deficit becomes more likely. © 2005 Elsevier Inc. All rights reserved. JEL classification: C71; G21 Keywords: Interbank market; Market games; Shapley value

1. Introduction Central banks normally use the average overnight interbank market rate (the EONIA rate in European Monetary Union, the Fed funds rate in the US) as an indicator for the (expected) liquidity situation on the interbank market. If the rate is high, the market is assumed to expect a liquidity deficit, whereas if the rate is low, it is assumed to expect a liquidity surplus. This method of assessing the liquidity situation on the interbank market can easily be justified theoretically. Ho and Saunders (1985) and Spindt and Hoffmeister (1988), for example, discuss models of the ✩

The views expressed in this paper are my own and do not necessarily reflect the view of the European Central Bank.

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Fed funds market, while Välimäki (2001), Mendizábal and Quirós (2001) and Tapking (2002) analyse models of the European interbank market. In all of these models, the more likely a liquidity deficit is, the higher the equilibrium rate. However, most of these models are general equilibrium models and are thus based on the assumption that all banks take the interbank rate as given. At least in Europe, transactions on the interbank market are usually agreed on in direct negotiations between banks, often on the telephone. Sometimes brokers are involved to help banks find a transaction partner, but the terms of transactions, i.e. the interest rates and transaction volumes, are still a matter of negotiation between the banks.1 Therefore, one may question whether a general equilibrium model with interest rate taking banks is an appropriate model of the interbank market. Consequently, one may ask whether the overnight interbank market rate is still a good indicator for the liquidity situation on the interbank market, if banks do not take interest rates as given but determine interest rates and transaction volumes in negotiations. The latter question is exactly what we are going to address in this paper. Why may the overnight rate not be a good indicator of the liquidity situation on the interbank market if banks determine the terms of transactions in negotiations? Consider for example two banks that provisionally agree that the first will lend a certain amount of liquidity to the second at a certain interest rate. If the banks now receive new information that indicates that a liquidity deficit is more likely than previously expected, we might expect the banks to agree to adjust the terms of their transaction. But what kind of new agreement will they choose? They might agree to leave the transaction volume as it is, but raise the interest rate. However, they could also agree to leave the interest rate as it is, but adjust the transaction volume. In the latter case, changes in expectations would not lead to changes in interest rates. To put forward this idea in a precise and consistent way, we consider a model of an interbank market with institutional characteristics similar to those of the European interbank market.2 We look at only one so-called maintenance period which, lasts two days. At the beginning of the first day of the maintenance period, each bank starts with an initial endowment of liquidity (i.e. deposits on accounts with the central bank). At the beginning of the second day, banks face a random and exogenous influx or drain of liquidity. The central bank requires each bank to hold a certain amount of liquidity on a so-called minimum reserve account with the central bank. The banks are allowed to average their reserve holdings with the central bank over the maintenance period. For example, if a bank has a reserve requirement of, say, 100 per day, it does not have to hold reserves of exactly 100 with the central bank every day. It may instead hold 50 on the first and 150 on the second day, or 120 on the first and 80 on the second day. This assumption is in line with the practice of all (major) central banks that impose reserve requirements on banks. On the last day of the maintenance period, banks can lend liquidity to the central bank’s deposit facility, and can borrow liquidity from the central bank’s marginal lending facility. The interest rate paid for lending to the deposit facility is called the deposit rate, and the rate for borrowing from the marginal lending facility is the marginal lending rate. Both facility rates are fixed by the central bank. The marginal lending rate is higher than the deposit rate. On both days of the maintenance period, banks can borrow liquidity from and lend liquidity to other banks. Thus, each bank that has less liquidity at its disposal at the beginning of the second day than remaining reserve requirements on that day must borrow liquidity either from the central bank’s marginal 1 A detailed description of the European interbank market can be found in Hartmann et al. (2001). 2 A complete description of the ECB’s regulatory instruments related to the interbank market can be found in European

Central Bank (2003).

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lending facility or from other banks on the interbank market. All other banks can lend liquidity to the central bank’s deposit facility or to other banks. The objective of each bank is to maximise the (expected) liquidity it has at its disposal after the end of the maintenance period. The crucial assumption in our model is that no bank takes interest rates as given. Instead, banks negotiate interest rates and transaction volumes with each other on both days of the maintenance period. The negotiations are modelled as a cooperative game and the solution concept applied is the Shapley value. Our model can therefore be described as a dynamic cooperative game. On the last day of the maintenance period, negotiations take place in which the outcome of the negotiations of the first day are taken as given. In the negotiations on the first day, banks take the expected outcome of the negotiations of the second day into consideration. Why do we use a cooperative game approach instead of a non-cooperative one? A noncooperative approach would require us to assume a set of well-defined rules that govern the negotiations among banks. In particular, we would need to assume a certain set of strategies for each bank and a mapping that assigns to each strategy combination a payoff for each bank. However, it is not clear what the rules that govern negotiations on the interbank market are— or indeed whether such rules exist at all. For example, we know that, at least on the European interbank market, trades are often concluded in negotiations over the telephone. But we do not know whether there are any rules that determine the order in which banks call each other. We also do not know how often two banks talk to each other within a trading round. Traders may often first call several other traders to obtain information about the liquidity situation in the market and to exchange views on possible terms of transactions without concluding a trade. In many cases, only after a trader has gone through such an informative round of conversations, will he start calling again traders he had already called some minutes before—now in order to conclude a trade. Some banks may even occasionally conduct informal ascending auctions in which they call some other banks several times to ask them to submit new bids. As negotiations on the interbank market do not seem to be governed by a clear set of rules, any non-cooperative approach to model this market might be arbitrary. Cooperative game approaches have the advantage that they do not require assumptions as to the rules that govern the negotiations, but instead start with assumptions on plausible outcomes of these negotiations. The most prominent solution concepts for cooperative games are the Core and the Shapley value. Other well-known solution concepts are the bargaining set, the kernel and the nucleolus.3 We use the Shapley value because the Core can be either empty or contain more than a single allocation. Indeed, one can show that in our model, the Core is often an infinite set. Similarly, the bargaining set and the kernel typically contain more than a single allocation. The nucleolus, like the Shapley value, is unique and never empty. However, it is more difficult to compute than the Shapley value. Moreover, it has been shown that several very different non-cooperative bargaining games implement the Shapley value as an equilibrium. Gul (1989) and Pérez-Castrillo and Wettstein (2001) define non-cooperative games in which players negotiate bilaterally (and in the latter case sequentially) with each other. Hart and Mas-Colell (1996) describe a noncooperative multilateral bargaining game. In all these cases, the equilibrium outcome of the noncooperative game coincides with or converges to the Shapley value. It should be mentioned that our cooperative game is not necessarily convex so that there is no guarantee that the Shapley value is in the Core and could therefore be interpreted as potentially instable solution. However, the

3 See, for example, Myerson (1991).

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Core may well be considered as imposing too strong stability requirements as the development of the bargaining set, of which the Core is a subset, has shown.4 It is of special importance that the banks do not directly negotiate on interest rates and transaction volumes in our model. Instead, they negotiate on how to share the (maximum expected) joint liquidity of the banking industry after the end of the maintenance period. The direct outcome of the negotiations is therefore a number for each bank (the Shapley value of the bank), the bank’s expected final liquidity. The implicit interest rates and transaction volumes that are compatible with these numbers can then be derived. Finally, one can calculate the implicit average interest rate from these implicit interest rates and transaction volumes. The main results of our analysis are the following: as soon as the interest rates and transaction volumes of the first day of the maintenance period are given, a unique implicit average interest rate of the second day of the maintenance period can be derived. However, there are a multiplicity of implicit average interest rates on the first day. The average interest rate on that day can be any number between 0 and 1, no matter whether the market is expecting a liquidity deficit or a liquidity surplus. The average overnight interest rate on the first day of the maintenance period is therefore useless as an indicator of the expected liquidity situation on the interbank market. The reason for the multiplicity of average overnight rates is simple. Consider the first day of the maintenance period and a provisional situation with a certain constellation of interest rates and transaction volumes for each pair of banks. Now assume that all interest rates increase. If the transaction volumes remain fixed, borrowing banks will become worse off and lending banks will become better off. However, an adjustment of the transaction volumes in the right direction can compensate the borrowers. If for example interest rates were relatively low in the provisional situation and the increase in interest rates is moderate, an increase in transaction volumes would compensate borrowers. Alternatively, if interest rates were initially already relatively high, a decrease in the transaction volumes would compensate the borrowers. This implies that there are different combinations of overnight interest rates and transaction volumes that all lead to the same expected final amount of liquidity for each bank. Since both the overnight rates and the transaction volumes are directly determined by the banks, and all banks are indifferent between these combinations, they are all compatible with the assumptions of our model. Note that the multiplicity of rates on the first day is due to the fact that banks are allowed to average their reserve holdings with the central bank and therefore are relatively flexible when they choose transaction volumes on the first day. If a bank borrows less on the first day, it automatically holds less reserves on that day. But it can still fulfil its reserve requirements on the second day. The banks can therefore adjust transaction volumes on the first day when interest rates move up or down. However, the amount of reserves a bank has to hold on the second day and thus the liquidity this bank has to borrow or lend on that day is fixed at the end of the first 4 Note that the Shapley value has often been used as a solution concept for models of exchange economies. The Shapley value was introduced and justified axiomatically by Shapley (1953). A textbook treatment of the Shapley value can be found for example in Myerson (1991) and Eichberger (1993). The first applications of the Shapley value as an equilibrium concept for exchange economies with transferable utility are Shapley (1964) and Chapter 6 in Aumann and Shapley (1974). A recent discussion of Aumann and Shapley (1974) can be found in Butnariu and Klement (1996). In the wake of these contributions, many articles have been published which analyse the relation between the general equilibrium of an exchange economy and the Core and Shapley value of the related cooperative game, the so-called market game. This literature has clearly established that under quite general conditions, the utility allocation of the general equilibrium is the unique element of the Core of the related market game and coincides with its Shapley value, if no player has significant market power (infinite and non-atomic player sets). See Chapters A and B in Mertens and Sorin (1994) for an overview.

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day. Transaction volumes therefore can no longer be adjusted to offset the effect of rate changes. For this reason, there is a unique average overnight rate on the second day.5 In Section 2, we describe our model, which is analysed in Sections 3 and 4. The last day of the maintenance period is considered in Section 3. Going backwards, we deal with the first day of the maintenance period in Section 4. In Section 5, we discuss how realistic our model is. We basically argue that given the information we have about real interbank markets, it is difficult to assess whether or not our model is a good description of reality. Our model has some testable implications which could however only be tested with quite a substantial amount of data. Concluding remarks and an appendix with proofs follow. 2. The model In this section, we formally describe the main institutional assumptions used in our model. Our assumptions on how prices and transaction volumes are determined in negotiations among banks will be introduced in the following two sections. We consider a set B = {1, . . . , n} of n banks and only one maintenance period lasting for just two days t = 1 and t = 2. Moreover, we assume that all borrowing and lending of liquidity has a maturity of only one day. Finally, a day in our model will be thought of as a point of time rather than a period in time. We assume that the central bank imposes reserve requirements on the banks that have to be fulfilled on average over the maintenance period. As mentioned in the introduction, this assumption coincides with the practice of all (major) central banks that impose reserve requirements. Let mi be bank i’s aggregate minimum reserve requirement for the maintenance period under consideration. It is set by the central bank and therefore exogenous. Let Ai,t  0 be the reserves that bank i holds on its minimum reserve account with the central bank from t to t + 1 (t = 1, 2). Thus, bank i has to satisfy Ai,1 + Ai,2  mi . The central bank pays no interest on reserves, thus Ai,1 + Ai,2 = mi

(1) necessary.6

i.e. bank i holds no more reserves with the central bank than On day t = 1, bank i starts with an initial endowment Li of liquidity which may partly be received from a central bank open market operation. As central banks normally quite successfully make sure through open market operations that the banking system does not run into an aggregate  liquidity  deficit or surplus before the end of the maintenance period, we assume that 0 < ni=1 Li < ni=1 mi . On both days of the maintenance period, banks can go to the interbank j market to lend liquidity to or borrow liquidity from other banks. Let Fi,t be the liquidity bank j j

borrows from bank i from t to t + 1 and ri,t the corresponding interest rate, so that by definition j

j

i and r = r i . The terms of trade between banks i and j on the interbank we have Fi,t = −Fj,t j,t i,t j

j

market, given by Fi,t and ri,t , are a matter of negotiation as will be explained in the following j

sections. Thus, neither bank i nor bank j takes the interest rate ri,t as given. 5 Thus, if we were to assume that the banks are not allowed to average their reserve holdings, then there would be a unique average overnight rate on both days of the maintenance period. 6 If we were assuming that the banks are not allowed to average their reserve requirements over the maintenance period, then the banks would instead have to fulfil Ai,1  12 mi and Ai,2  12 mi . Our assumption that the central bank does not pay interest on reserves would now imply Ai,1 = Ai,2 = 12 mi and the average overnight rate would be unique as mentioned in the previous footnote.

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In t = 2, bank i realises a random and exogenous liquidity influx gi which may be positive or negative. The term gi is mainly driven by customers of bank i paying in or withdrawing money from their accounts. Moreover, gi may comprise for example liquidity drains because of dividend payments, payouts of factor income and real investments. In t = 2, the banks can lend a liquidity surplus to and borrow a liquidity deficit from the central bank’s standing facilities. The liquidity lent by bank i to the central bank via the deposit facility is Di , and the liquidity borrowed from the central bank via the marginal lending facility is Si . The deposit rate is rD and the marginal lending rate is rS (rS > rD ). We assume both rates to be exogenous and non-random. The banks have no access to standing facilities in t = 1. This assumption is of course not in accordance with the reality on the European interbank market, where banks can resort to the ECB’s standing facilities every day. However, it can be shown that banks would in our model use the standing facilities in t = 1, if available, only under extreme parameter constellations.7 The assumption that banks can resort to the standing facilities only on the last day of the maintenance period is thus quite harmless and simplifies our analysis significantly as it reduces the number of choice variables for the banks in t = 1. From the assumptions described so far, we can derive several identity equations. In t = 1, each bank can use its initial endowment either to fulfil reserve requirements or to lend it to other banks on the interbank market. We therefore have Li = Ai,1 +

n 

j

(2)

Fi,1 .

j =1 j =i

The liquidity bank i can dispose of in t = 2 is given by Li,2 = Ai,1 +

n   j  j 1 + ri,1 Fi,1 + gi

(3)

j =1 j =i

or, from the expenditure side, by Li,2 = Ai,2 + Di − Si +

n 

j

Fi,2 .

(4)

j =1 j =i

Finally, bank i’s liquidity after the end of the maintenance period, i.e. at day t = 3, is Li,3 = Ai,2 +

n   j  j 1 + ri,2 Fi,2 + (1 + rD )Di − (1 + rS )Si .

(5)

j =1 j =i

Note that Li,3 is the final value of bank i’s activity on the interbank market in the maintenance period under consideration. This is because we assume that no endogenous liquidity drains occur during the maintenance period, but that the whole liquidity surplus is reinvested according to 7 A recourse to the marginal lending facility would never occur on day t = 1 and a recourse to the deposit facility would occur only if it is almost certain that there will be an aggregate liquidity surplus on day t = 2. In this case, the banks can choose between (1) fulfilling reserve requirements in t = 1 and lending to the deposit facility in t = 2, and (2) lending to the deposit facility in t = 1 and fulfilling reserve requirements in t = 2. Option (2) is slightly more attractive as interest from the deposit facility is realised earlier than with option (1).

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Eqs. (3) and (4). Therefore, we assume that bank i’s objective at day t, t = 1, 2, is to maximise Et [Li,3 ], i.e. it is in t expected liquidity after the end of the maintenance period. In the next two sections, we analyse the banks’ behaviour on the interbank market. We consider the interaction between several banks as a cooperative game. Banks are not price-takers, but do negotiate with each other on interest rates and the amounts of liquidity lent and borrowed. The main analytical concept used is the Shapley value. To begin with, we consider the last day of the maintenance period. 3. The last day of the maintenance period On the last day of the maintenance period t = 2, each bank i has to satisfy Ai,2 = mi − Ai,1 , where Ai,1 is given from the first day. Thus, Ai,2 is given and each bank has to decide only on its recourse to the facilities (Di and Si ) and its activities on the interbank market. The terms of deals are determined in negotiations among banks. The payoff bank i obtains in these negotiations in t = 2 will be denoted by L∗i,3 . We will assume that L∗i,3 is equal to bank i’s Shapley value. As a cooperative game approach, the Shapley value is based on assumptions on formal characteristics of the outcome of negotiations. It is assumed that the outcome of the negotiations depends on the bargaining power of the various banks. The bargaining power is given by the socalled characteristic function. Let ρ = B 2 denote the set of all subsets of B. Each k ∈ ρ is called ∗ to each possible a possible coalition of banks. The characteristic function assigns a value vk,2 coalition k. This value is the maximum joint liquidity the members in k could obtain if they were allowed to trade only within the coalition. xi denotes bank i’s liquidity surplus in t = 2. It is defined by xi = Li,2 − (mi − Ai,1 ).

(6) j

j

To obtain the value of coalition k, we need to choose Di , Si , Fi,2 and ri,2 for all i ∈ k and j ∈ B to maximise  Li,3 vk,2 ≡ i∈k

subject to Di  0, Si  0, Di − Si +

n 

j

Fi,2 = xi

for all i ∈ k,

j =1 j =i j

/ k. Fi,2 = 0 for all i ∈ k, j ∈ ∗ of coalition k is the solution to the above problem, i.e. the maximum v . The value vk,2 k,2 Since rS > rD , recourse to both facilities on the same day is never optimal for a given coali  tion. The solution to the above problem therefore has to satisfy (1) x = D i i∈k i∈k i and    S = 0 for all i ∈ k if x  0 and (2) x = − S and D = 0 for all i ∈ k if i i i i,T i i∈k i∈k i∈k  x  0. With this, we obtain from Eq. (5) i i∈k   ∗ vk,2 = (mi − Ai,1 ) + Ik (1 + rD ) xi i∈k

+ (1 − Ik )(1 + rS )

 i∈k

i∈k

xi ,

(7)

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where

 1, if i∈k xi  0,  Ik = 0, if i∈k xi < 0,

357



(8)

as the obvious solution to the above problem. Equation (7) gives us the characteristic function ∗ − v∗ of our cooperative game in t = 2. The difference [vk,2 k\{i},2 ] can be interpreted as bank i’s contribution to coalition k and reflects i’s bargaining power in negotiations with the other members of k. The more the coalition would lose if bank i were to drop out of the coalition, the ∗ − v∗ higher is i’s bargaining power. If i ∈ / k, then we of course have [vk,2 k\{i},2 ] = 0. ∗ ∗ ∗ 2 First note that we have vk∪l,2  vk,2 + vl,2 for all k, l ∈ B with k ∩ l = ∅, i.e. the game is supperadditive. This is easy to demonstrate. Supperadditivity implies that it is advantageous for the banks to form large coalitions. We assume that the banks negotiate way,  in an efficient ∗ . Furi.e. the negotiations result in a joint payoff for the banking system of i∈B L∗i,3 = vk,2 thermore, we assume that two banks i, j with the same bargaining power achieve the same ∗ ∗ 2 = vk∪{j / k), then L∗i,3 = L∗j,3 . And we assume that payoff, i.e. if vk∪{i},2 },2 for all k ∈ B (i, j ∈ a bank that contributes its own value to each coalition can only achieve its own value, i.e. if ∗ − v∗ ∗ ∗ ∗ 2 [vk,2 k\{i},2 ] = v{i},2 for all k ∈ B (i ∈ k), then Li,3 = v{i},2 . It is well known that these assumptions, together with a somewhat more technical additivity assumption, imply that the payoff that bank i obtains from the negotiations in t = 2 is given by    ∗ ∗ (9) qk vk,2 − vk\{i},2 L∗i,3 = k∈ρ

with (#k − 1)!(n − #k)! (10) n! where #k is the number of banks in k. Equation (9) defines bank i’s Shapley value. The numbers qk are non-negative weights. Let ρi = {k ∈ ρ | i ∈ k}. It is easy to show that k∈ρi qk = 1. Bank i’s Shapley value is thus a weighted average of i’s contributions to the various possible coalitions. With Eq. (7), we obtain after some rearrangements

 ∗ qk Ik Li,3 = xi 1 + rS − (rS − rD ) qk =

k∈ρi

− (rS − rD )

n  j =1 j =i

+ mi − Ai,1

xj



qk (Ik − Ik\{i} )

k∈ρi,j

(11)

where ρi,j = {k ∈ ρ | i, j ∈ k} and ρi as defined above. We should examine Eq. (11) in a little more detail. Our model predicts that trade of liquidity on the interbank market in t = 2 and recourse to the central bank’s facilities in t = 2 will be such that the amount of liquidity bank i can dispose of on the first day after the maintenance period is L∗i,3 as given in Eq. (11). Note that the liquidity that bank i can dispose of on that day originates either from the required reserve holdings (Ai,2 = mi − Ai,1 ) or from the usage of xi . The last line in Eq. (11) obviously describes the liquidity influx stemming from the required reserve holdings. Thus, the first two lines give us the liquidity originating from the usage of xi . All parts of Eq. (11)

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that drop out if rS = rD describe the impact of negotiations on L∗i,3 , since interbank lending is useless if rS = rD . Consider the extreme case that Ik = 1 for all k ∈ ρ. It is easy to show that in this case we  have k∈ρi qk Ik = 1. The second line in Eq. (11) is obviously 0. Thus, the influx of liquidity to bank i in t = 3 originating from the usage of xi,T is xi (1 + rD ). This is because all banks have a liquidity surplus, so that no interbank trade can take place and bank i lends its whole surplus xi to the central bank via the deposit facility. Now let us take some bank i and assume that Ik = 1 for all k ∈ ρi and Ik = 0 for all k ∈ / ρi . The first line in Eq. (11) is still xi (1 + rD ), while the second line is now n  

xj −(rS − rD ) qk j =1 j =i

k∈ρi,j

 where 0 < k∈ρi,j qk < 1 and xj  0 for all j = i. Thus, the influx of liquidity to bank i in t = 3 originating from the usage of xi is higher than xi (1 + rD ). This proves that bank i can now lend parts of its liquidity surplus to other banks at a rate higher than rD . Finally, assume the other two extreme cases. Firstly, if Ik = 0 for all k ∈ ρ (i.e. xi < 0 for all i), bank i’s influx of liquidity stemming from the usage of xi is xi (1 + rS ), since all banks have a liquidity deficit. Borrowing from other banks is therefore not possible and bank i only borrows from the central bank. Secondly, if Ik = 0 for all k ∈ ρi and Ik = 1 for all k ∈ / ρi , bank i’s influx of liquidity stemming from the usage of xi is higher than xi (1 + rS ), since it can now borrow parts of its liquidity deficit from other banks at a rate less than rS . Equation (11) only gives us the equilibrium liquidity that bank i can dispose of in t = 3. We now want to determine the underlying activities of bank i in t = 2 which lead to L∗i,3 as described in Eq. (11). To do so, we need to solve ⎫ L∗i,3 = Li,3 , ⎪ ⎪ ⎪ n ⎪ j ⎪ ⎬ xi = Di − Si + j =1 Fi,2 , j =i (12) j i , r j = r i , D  0, S  0, ⎪ ⎪ ⎪ Fi,2 = −Fj,2 i i ⎪ j,2 i,2 ⎪ ⎭ i, j = 1, . . . , n, for Di , Si , Fi,3 and ri,3 for all i, j ∈ B. Here, L∗i,3 is given in Eq. (11), Li,3 is given in Eq. (5) and the second equation in (12) is derived from Eqs. (1), (4) and (6). Note that we do not assume j ri,2 = r2 for all i, j ∈ B and some number r2 . In a general equilibrium model, two different prices for a homogeneous good are impossible. However, our model is not a general equilibrium but a bargaining model, and we show now that different prices for the same good do not contradict our assumptions. First note that (12) has no unique solution. Consider the following case: n = 3, I{1} = 1, Ik = 0 for all k ∈ ρ\{1}. In this case featuring only three banks, bank 1 has a liquidity surplus in t = 2, while the other two banks have a liquidity deficit. Moreover, the liquidity deficit of both bank 2 and bank 3 is higher than the liquidity surplus of bank 1. It is easy to check that the j 2 = F 3 = 1 x , F 3 = 0, following is a solution of (12): ri,2 = 23 rS + 13 rD for i, j = 1, 2, 3, F1,2 1,2 2,2 2 1 1 1 S1 = Di = 0 for i = 1, 2, 3, S2 = −x2 − 2 x1 and S3 = −x3 − 2 x1 . In this solution, all interbank 2 = 1r + 2r , rates are equal. However, it is also easy to check that another solution to (12) is: r1,2 3 S 3 D j

j

J. Tapking / Games and Economic Behavior 56 (2006) 350–370

359

3 = 7 r + 2 r , F 2 = 1 x , F 3 = 3 x , F 3 = 0, S = D = 0 for i = 1, 2, 3, S = −x − 1 x r1,2 1 i 2 2 1,2 1,2 2,2 9 S 9 D 4 1 4 1 4 1 and S3 = −x3 − 34 x1 . This example shows that j

(1) there is not necessarily a unique set {ri,2 | i, j ∈ B, i = j } of equilibrium interbank rates, and j

(2) we do not necessarily have ri,2 = r2 for all i, j ∈ B and some number r2 in equilibrium.8 To continue, we make the following symmetry assumptions for notational convenience: j

(1) sign(xi ) = sign(xj ) implies Fi,2 = 0 for all i, j ∈ B, i.e. two banks do not trade with each other n if both have a liquidity deficit or if both have a liquidity surplus. (2) i=1 xi  0 and xi  0 implies Di = 0, i.e. banks with an individual liquidity deficit do not resort to the deposit facility when there is an aggregate liquidity surplus in the banking system. n (3) i=1 xi  0 and xi  0 implies Si = 0, i.e. banks with an individual liquidity surplus do not resort to the marginal lending facility when there is an aggregate liquidity deficit in the banking system. Now define B + = {i ∈ B | xi > 0}, B − = B\B + and   j j i∈B + j ∈B − ri,2 Fi,2 R2 =  .  j i∈B + j ∈B − Fi,2

(13)

Note that R2 is the interbank rate index, i.e. a weighted average of all interbank rates in t = 2. It is straightforward to show that (12) implies  n ∗  i∈B − [Li,3 − Li,2 ]  (a) 0  x i ⇒ R2 = , i∈B − xi i=1  n ∗  i∈B + [Li,3 − Li,2 ]  x i ⇒ R2 = . (14) (b) 0  i∈B + xi i=1

Note that L∗i,3 is uniquely given by Eq. (11). Thus, R2 is unique, thoughri,2 for some i, j ∈ B is not. However, note that there is no trade at all if in case (a) we have i∈B − xi = 0 or in (b)  i∈B + xi = 0. In both cases, R2 is not defined. We now briefly summarise the results of our cooperative game analysis of the interbank market on day t = 2. j

(1) There is not necessarily a unique equilibrium interbank rate for the trade of liquidity between two given banks. (2) However, the equilibrium interbank rate index R2 is unique. j l for (3) The interbank rate in one deal can differ from the rate in another deal, thus ri,2 = rk,2 i, j, k, l ∈ B (i = j = k = l) is possible. j

8 Note that (12) often has no solution with r = r for all i, j ∈ B and some number r . If for example n = 4, I = 1 k 2 2 i,2 for all k ∈ {{1}, {2}, {1, 2}, {1, 3}, {1, 4}, {1, 2, 3}, {1, 2, 4}} and Ik = 0 otherwise, then there is no solution with an equal interbank rate for all trades.

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4. The first day of the maintenance period We now consider day t = 1 and start with some introductory considerations to obtain some intuition as to what is happening on the first day of the maintenance period. With Eqs. (2), (3) and (6), we easily obtain xi = 2Li − mi −

n   j  j 1 − ri,1 Fi,1 + gi .

(15)

j =1 j =i j

If we assume that ri,1 < 1 for all i, j ∈ B, then bank i’s liquidity surplus in t = 2, i.e. xi is j

decreasing in Fi,1 for all j ∈ B. The reason is as follows: lending more liquidity to other banks in t = 1 means (1) holding less reserves in t = 1, i.e. more remaining reserve requirements and thus a lower liquidity surplus in t = 2, and (2) receiving more interest from other banks and thus a higher liquidity surplus in t = 2. Effect (1) is clearly stronger than effect (2).9 However, what is the effect of Fi,1 on L∗i,3 ? If we replace in Eq. (11) xi and xj for all j ∈ B\{i} by the right-hand side of Eq. (15) and Ai,1 by means of Eq. (2), we obtain ∂L∗i,3  j  = 1 − 1 − ri,1 1 + rS j ∂Fi,1 

   (16) +(rS − rD ) qk (Ik − Ik\{i} ) − qk Ik . j

k∈ρi,j

k∈ρi j

Lending to bank j is of course more profitable for bank i, the higher ri,1 is. However, the sign of j

Eq. (16) is not clear. It even depends on Fi,1 , because the indicator functions in Eq. (16) change when

j Fi,1

changes (L∗i,3 is piecewise linear in Fi,1 ). However, the higher the liquidity surplus j

j

in t = 2 is, the higher in principle the right-hand side of Eq. (16). For a given ri,1 , Eq. (16) is the highest if Ik = 1 for all k ∈ ρi , and the lowest if Ik = 0 for all k ∈ ρi . There is a simple reason for this: the more bank i lends to other banks in t = 1, the lower is Ai,1 , the higher is its reserve requirement in t = 2 and thus the lower is its liquidity surplus in t = 2. And having a low liquidity surplus is better the higher the liquidity surplus of the market. We now consider the cooperative game played in t = 1. As on day t = 2, all banks negotiate with each other on interbank rates and on amounts of liquidity borrowed by one bank from another. Again, we assume that the outcome of the negotiations is described by the Shapley value. ∗ for each possible coalition To determine the Shapley value, we need to determine the value vk,1 k ∈ ρ in t = 1, i.e. the characteristic function in t = 1. Summation of the right-hand side of Eq. (11) for all i ∈ k after substituting Ai,1 by means of Eqs. (1) and (2) and after rearranging using Eq. (10) gives: 9 If r j > 1, then effect (2) would be stronger than effect (1) as bank i would receive a very high interest payment from i,1

bank j in t = 2.

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361

  L∗i,3 = 1 + rS − (rS − rD )IB xi

i∈k

+



i∈k

mi − Li +



j

Fi,1

j =1 j =i

i∈k

+ (rS − rD )

n 

   xi ql (Il − Il\{j } ) i∈k

j ∈k / l∈ρi,j

i ∈k /

j ∈k l∈ρi,j

   − (rS − rD ) xi ql (Il − Il\{j } ) .

(17)

Note that replacing xi for all i ∈ B in Eq. (17) by the right-hand side of Eq. (15) leads us to a  formula for i∈k L∗i,3 which contains only variables chosen in t = 1 and variables exogenously given in t = 1. ∗ of coalition k, we need to choose F j and r j for all i ∈ k and j ∈ B To obtain the value vk,1 i,1 i,1 to maximise  ⎫ vk,1 ≡ E1 [ i∈k L∗i,3 ] ⎪ ⎪ ⎪ ⎪ ⎪ subject to ⎬ n j (18) 0  Li − j =1 Fi,1  mi , ⎪ ⎪ ⎪ j =i ⎪ ⎪ j ⎭ i for all i, j ∈ B, F j = 0 for all i ∈ k, j ∈ / k. Fi,1 = −Fj,1 i,1 Note that vk,1 depends on variables chosen by banks that are not members of coalition k. Thus, coalition k maximises vk,1 given the decisions of other coalitions. But what are the other coalitions? For simplicity we assume that if coalition k were formed in t = 1, there would be only one other coalition in t = 1, namely B\k. Thus, if coalition k were formed, coalition B\k would also be formed, and both coalitions would play a two-player Nash game in which coalition k’s payoff function is vk,1 as defined in problem (18), and its strategy set is described by the constraints ∗ of coalition k is its Nash equilibrium payoff in this presented in problem (18). The value vk,1 game. j i It is important to make sure that this game has an equilibrium. Firstly, note that Fi,1 = −Fj,1  n j for all i, j ∈ B implies that i∈k j =1,j =i Fi,1 = 0, i.e. we can simplify Eq. (17) accordingly. j

j

This implies that the only strategy variables for coalition k are (1 − ri,1 )Fi,1 for all i, j ∈ k. If we j

assume that the interest rates are restricted such that a  ri,1  b for some numbers a and b and all i, j ∈ k, then the strategy set of k is compact. Secondly, it is very easy to check that L∗i,3 as given in Eq. (11) is a continuous (and piecewise linear) function in xj for all j ∈ B. Thus, vk,1 j j is also a continuous function in xj for all j ∈ B. Since xi is continuous in (1 − ri,1 )Fi,1 for all j ∈ B, we know that coalition k’s payoff function vk,1 is continuous in the strategy variables. It is well known that a game with a compact strategy set and a continuous payoff function for all players has at least one equilibrium in mixed strategies.10 We can therefore be sure that the game described above has at least one equilibrium in mixed strategies. However, the payoff function 10

See for example Eichberger (1993, p. 95).

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is not always quasi-concave so that we cannot be sure that the game has an equilibrium in pure strategies. For k = B, the last two lines of Eq. (17) are 0 andthe first two lines do not depend on strategy ∗ =v ∗ variables. Thus, we always have vB,1 B,1 = E1 [ i∈k Li,3 ], i.e.

n n      ∗ (19) = mi − Li + E1 1 + rS − (rS − rD )IB xi . vB,1 i=1

i=1

∗ in general. Therefore Unfortunately, for any coalition k = B, it is very hard to determine vk,1 we assume from now on that there are only n = 3 banks. This is a very simple case indeed, since there is nothing a coalition of only one bank can do so that its opponent, a coalition of two banks, has a very simple problem to solve. Moreover, we assume that there are only two states of the world s ∈ {s1 , s2 } with probabilities p1 and p2 = 1 − p1 ( p1 , p2 > 0). The random parameters are determined by the states according to g1 (s1 ) = g2 (s1 ) = g3 (s2 ) = g and g1 (s2 ) = assume 2Li = mi for i = 1, 2, 3. With g2 (s2 ) = g3 (s1 ) = −g for some g > 0. Moreover, we  these assumptions it is clear that ni=1 xi (s1 ) = g and ni=1 xi (s2 ) = −g. Therefore, there is a liquidity surplus in state s1 and a liquidity deficit in state s2 . With this example, we immediately obtain from Eq. (19): 3    ∗ = p1 (1 + rD ) − p2 (1 + rS ) g + Li . v{1,2,3},1

(20)

i=1

Now assume that coalitions {1, 2} and {3} are formed. The following lemma is proved in Appendix A: 2 )F 2  g (or, because of symmetry, Lemma 1. In t = 1, coalition {1, 2} would choose (1 − r1,1 1,1 2 )F 2  −g). (1 − r1,1 1,1 j

The economic reason for this result is the following: if Fi,1 = 0 for all i, j ∈ B, bank 3 would have a very good bargaining position in t = 2, because it would have a deficit while the other two banks have a surplus, and a surplus if the other two banks have a deficit, i.e. both bank 1 and 2 )F 2  g and F j = 0 for bank 2 would need bank 3 in both states. However, if instead (1 − r1,1 1,1 3,1 j = 1, 2, the position of bank 3 in t = 2 would be less strong. For in s1 both banks 1 and 3 would have a deficit and bank 2 a surplus, while in s2 banks 2 and 3 would have a surplus and bank 1 a deficit. j With Lemma 1 and F3,1 = 0 for j = 1, 2, we obtain from (17) and (15):   1 ∗ v{1,2},1 = p1 (1 + rD ) − p2 (1 + rS ) 2g + (rS − rD ) g + L1 + L2 2

(21)

and   1 ∗ v{3},1 = p2 (1 + rS ) − p1 (1 + rD ) g − (rS − rD ) g + L3 . (22) 2 The situation occurring if the coalitions {1, 3} and {2} are formed is described by the following lemma: 3 )F 3 = g if p  p and Lemma 2. In t = 1, coalition {1, 3} would choose (1 − r1,1 1 2 1,1 3 )F 3 = −g if p  p . (1 − r1,1 1 2 1,1

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363

This strategy would imply that x1 = x3 = 0 in the state with the highest probability, i.e. neither bank 1 nor bank 3 would need to have recourse to the facilities in this state.11 j With Lemma 2 and F2,1 = 0 for j = 1, 3 we obtain 1 ∗ = −(rS − rD )g min{p1 , p2 } + L1 + L3 v{1,3},1 6

(23)

and ∗ v{2},1 = (1 + rS )g(p1 − p2 )   1 − (rS − rD )g p1 − min{p1 , p2 } + L2 . 6

(24)

Finally, because of symmetry we obtain 1 ∗ v{2,3},1 = −(rS − rD )g min{p1 , p2 } + L2 + L3 6

(25)

and ∗ v{1},1 = (1 + rS )g(p1 − p2 )   1 − (rS − rD )g p1 − min{p1 , p2 } + L1 . 6

(26)

∗ It is easy to show that the characteristic function for t = 1 is supperadditive, i.e. vk∪l,1  ∗ ∗ ∗∗ 2 vk,1 + vl,1 for all k, l ∈ B with k ∩ l = ∅. Bank i’s Shapley value is denoted by E1 [Li,3 ] and, analogously to Eq. (9), is defined by     ∗  ∗ (27) E1 L∗∗ qk vk,1 − vk\{i},1 i,3 = k∈ρ

where qk has been defined in Eq. (10) for all k ∈ ρ. With the characteristic function given by Eqs. (20)–(26), it is quite easy to determine the Shapley value for the cooperative game played in t = 1 as defined in Eq. (27):   E1 L∗∗ i,3 = Li + (1 + rS )g(p1 − p2 )   1 1 min{p1 , p2 } − (rS − rD )g p1 − − (28) 6 18 for i = 1, 2, and   E1 L∗∗ 3,3 = L3 + (1 + rS )g(p2 − p1 )   1 1 − p1 + min{p1 , p2 } . − (rS − rD )g 3 9

(29)

Our model implies that the in t = 1 expected liquidity that bank i (i = 1, 2, 3) can dispose of in t = 3 is given by Eqs. (28) and (29). As previously explained, these expectations are the result of 11 Note that a coalition {i, j }’s choice of (1 − r j )F j could lead to a situation in which, for example, E [L∗ ] is lower 1 i,3 i,1 i,1 j than it would be if Fi,1 = 0. Bank i would agree on such a deal only if i and j simultaneously agree that j pays a transfer

to i in t = 3 to offset the losses of bank i in a coalition with j . Thus, we have to assume that banks can agree in t = 1 on interbank payments in t = 3. However, in equilibrium such agreements will not be made.

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negotiations between the three banks in t = 1 and the rational expectations of the outcome of the negotiations in t = 2. The first question we want to address is whether there is an interbank trade in t = 1. Define j 0 0 v{i},1 = v{i},1 for Fh,1 = 0 for all j, h ∈ B. Thus, v{i},1 is the in t = 1 expected liquidity of bank i in t = 3, if no trade in t = 1 takes place. It is easy to verify that in the example above we have 0 0 ∗∗ v{i},1 < E1 [L∗∗ i,3 ] for i = 1 and i = 2 and v{3},1 > E1 [L3,3 ], if 0 < p1 < 1. Thus, we can tell j 0 and E1 [L∗∗ from the fact that v{i},1 i,3 ] are not equal, that a trade takes place in t = 1, i.e. Fi,1 = 0 for some i, j ∈ B. The trade increases banks’ 1 and 2 expected payoff, while decreasing that of bank 3. This is an interesting finding. In our model, banks are ultimately interested only in their liquidity position after the end of the second period. These liquidity positions are determined by the trades of the second period. Why then are there trades in the first period? The reason is simple: trades in the first period influence the bargaining positions of the players in the second period. The players try to negotiate deals in the first period to better their bargaining position in the second period. Trades in the first period do not affect the ultimate liquidity positions directly, but instead indirectly as they affect the bargaining power for the second period. We now want to derive from Eqs. (28) and (29) the interest rates and the credit volumes the banks have agreed on in the negotiations of day t = 1. Thus, we have to solve ⎫ E1 [L∗∗ ⎬ i,3 ] = v{i},1 , j i , rj = ri , Fi,1 = −Fj,1 i,1 j,1 ⎭ i, j = 1, . . . , n, j

(30)

j

for Fi,1 and ri,1 and all i, j ∈ B. It is not hard to verify that Eq. (30) has no unique solution. 3 = F 3 , F 2 = 0, 0 < From now on, we concentrate on solutions to Eq. (30) that satisfy F1,1 1,1 2,1 3 = r 3 < 1. The interbank rate index in t = 1 is thus R = r1,1 1 2,1

3 F 3 +r 3 F 3 r1,1 1,1 2,1 2,1 3 +F 3 F1,1 2,1

3 . The next = r1,1

proposition is the main result of this section: 3 = F3 , Proposition 3. For every ω ∈ ]0, 1[, there is at least one solution to Eq. (30) with F1,1 2,1 2 = 0 and r 3 = r 3 = ω, i.e. R = ω. F1,1 1 1,1 2,1

This proposition states that there is no unique equilibrium interbank rate index in t = 1. The equilibrium rate index can be any number between 1 and 2. There is for instance always an equilibrium with R1 = 12 (rS + rD ), i.e. the average equilibrium rate in t = 1 is exactly in the middle between the two facility rates. This is true no matter whether the market is expecting a liquidity surplus (p1 high) or a deficit (p1 low). And the interbank rate index may be either low or high if there is for example an expected liquidity deficit. This implies that in our model, the interbank rate index R1 does not convey any information about the expected liquidity situation in the market. The reason for this result is as follows: when negotiating with other banks in t = 1, each bank is only interested in maximising the expected liquidity it can dispose of in t = 3. Each bank is j indifferent between two different constellations of interbank rates ri,1 and amounts of liquidity j

borrowed and lent Fi,1 , i, j ∈ B, if both result in the same expected disposable liquidity in t = 3.

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Proposition 3 simply states that there are many such constellations that all lead to an expected 12 disposable liquidity of E1 [L∗∗ i,3 ] for all three banks i = 1, 2, 3. Finally, we take a closer look at the set of constellations that solve Eq. (30). To do so, we use a few assumptions to exclude some constellations that may appear less plausible. We still 3 = F 3 , F 2 = 0 and 0 < r 3 = r 3 < 1. Furthermore, we assume that no assume that F1,1 1,1 2,1 1,1 2,1 bank i lends in t = 1 so excessively that it would end up with a deficit (xi < 0) even if gi = g. And no bank i borrows in t = 1 so excessively that it would end up with a surplus (xi > 0) even 3 )F 3  − 1 g. The if gi = −g. It is easy to show that these two assumptions imply 12 g  (1 − r1,1 1,1 2 3 and r 3 that solve Eq. (30) under these assumptions is depicted in set of all combinations of F1,1 1,1 3 = F 3 > 0 and r 3 = r 3 > β. Fig. 1 as curves L and M. Points on L are characterised by F1,1 2,1 1,1 2,1 The equation for curve L is given in Appendix A as Eq. (A.2), while the number β is given in 3 = F 3 < 0 and r 3 = r 3 < γ . The equation Eq. (A.3). Points on M are characterised by F1,1 2,1 1,1 2,1 for curve M is given in Appendix A as Eq. (A.4), while the number γ is given in Eq. (A.5). 3 is positive if r 3 is high (higher than β), and negative if r 3 is low Firstly note that F1,1 1,1 1,1 (lower than γ ). Hence if the interest rate is relatively low, then banks 1 and 2 borrow from bank 3, whereas if the interest rate is relatively high, then bank 3 borrows from banks 1 and 2. This reflects the fact that in t = 1, banks 1 and 2 have a better bargaining position than bank 3, so that the trade in t = 1 makes banks 1 and 2 better off and bank 3 worse off as shown above. 3 and F 3 . First consider the case that F 3 > 0, i.e. We now study the relation between r1,1 1,1 1,1 3 is relatively high. Assume that the trading volume bank 3 borrows from banks 1 and 2 and r1,1 3 increases. In this case, the interest rate obviously has to decrease so that we move from F1,1 point a to point b in Fig. 1. The reason for this is simple. As the interest rate is relatively high, increasing the amount banks 1 and 2 lend to bank 3 and keeping the interest rate constant would reduce bank 3’s expected profit and would increase the expected profits of banks 1 and 2. To

Fig. 1. Transaction volumes and interest rates in t = 1. 12 Note that it can be shown that R depends on R , i.e. the uniqueness of R holds only if R has been fixed. 2 1 2 1

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3 has to decrease when F 3 increases. Now consider the case F 3 < 0 compensate bank 3, r1,1 1,1 1,1 3 | increases. The interest rate obviously has to rise now and assume that the trading volume |F1,1 so that we move from point c to point d. As the interest rate is relatively low, increasing the amount that banks 1 and 2 borrow from bank 3 and keeping the interest rate constant would reduce bank 3’s expected profit and would increase the expected profits of banks 1 and 2. To 3 has to increase along with |F 3 |. compensate bank 3, r1,1 1,1 Finally, it is interesting to consider changes in p1 and p2 . First assume that p1 > p2 , i.e. banks 1 and 2 will probably have a liquidity surplus while bank 3 will probably have a liquidity deficit. It may appear plausible that in this case, bank 3 will not lend liquidity to banks 1 and 2 in t = 1. We therefore concentrate on curve L, i.e. assume that bank 3 borrows from the other two banks in t = 1. It is easy to show using Eq. (A.2) that curve L moves down if p1 increases, i.e. 3 , the interest rate r 3 would decrease. This appears intuitive for a given transaction volume F1,1 1,1 as a liquidity surplus becomes increasingly probable when p1 increases. For a given interest rate 3 , the transaction volume F 3 would decrease. r1,1 1,1 Now assume that p1 < p2 and concentrate on curve M, i.e. assume that bank 3 lends to the other two banks in t = 1. Using Eq. (A.4), it is easy to show that curve M moves up if p2 3 , the interest rate r 3 would increase. This increases, i.e. for a given transaction volume F1,1 1,1 again appears intuitive as the probability of a liquidity deficit rises when p2 increases. For a 3 , the transaction volume |F 3 | would decrease. given interest rate r1,1 1,1 We again briefly summarise the results of our analysis of the interbank market on day t = 1.

(1) The equilibrium interbank rate index R1 is not unique. There is a multiplicity of combinations of trading volumes and interbank rate indices that all implement the same expected liquidity positions for all banks. (2) If interest rates are high (low), than an increase in the trading volumes would require a decrease (increase) in the interest rates to ensure that the expected liquidity positions for all banks remain unchanged. (3) If an aggregate liquidity surplus (deficit) becomes more likely, then trading volumes can adjust while interest rates may remain unchanged. 5. Empirical considerations In this section, we discuss whether the model presented in this paper describes a realistic situation. There are in principle two ways to compare a model with reality. One can look at the assumptions of the model, or at its implications. We will argue that on the basis of available information, it is difficult to assess whether or not our model is a good description of reality. Furthermore, we will show that the model has several testable implications, although testing these will require a substantial amount of data. The institutional assumptions laid down in Section 2 may appear appropriate for many interbank markets characterised by reserve requirements. However, our central behavioural assumption that banks negotiate on the terms of transactions and that the outcome of these negotiations is given by the Shapley value deserves a closer look. Firstly, it should be noted that the Shapley value is based on the assumption that every bank in principle negotiates (directly or indirectly) with each other bank within a given day. However, the European interbank market for example consists of several thousand banks. Of course, no bank can negotiate with all the other banks in the market, but there are indications that interbank markets are partially fragmented. For example, Cocco et al. (2003) report that banks in the Portuguese interbank market tend to trade

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with a relatively small number of other banks. If it could additionally be shown that the banks that a given bank tends to trade with also tend to trade frequently with each other, then we would have evidence that interbank markets may consist of relatively small segments. Our model may thus be appropriate as a model of one such segment. As explained in general in the introduction and more technically in Section 3, the Shapley value is not based on assumptions on the rules that govern the negotiations, but instead on assumptions on the characteristics of the outcome of these negotiations. This has the advantage that we do not need to specify the type of negotiations, for example bilateral or multilateral ones, that we have in mind. However, this also implies that we cannot directly compare our assumptions with reality. Instead, we have to turn to the implications of our model. Our model has several testable implications. In many cases however, a test would require a substantial amount of data. For example, we have shown in Section 3 that in our model, the interbank rate index R2 of the last day of the maintenance period is given by Eq. (14). This is a testable relation provided that data on the liquidity situation of banks on the last day of the maintenance period (xi and Li,2 ) and on their trading activities on that day (in order to calculate Li,3 according to Eq. (5)) are available. Another implication of our model is important as it contradicts one of the implications of standard general equilibrium models of the interbank market. We first introduce the following wording: a bank i has on a given day t a liquidity surplus if its liquidity position Li,t at the start of the day is greater than its remaining reserve requirements. On that day it has a liquidity deficit if Li,t < 0. In general equilibrium models, two banks do not trade with each other before the last day of the maintenance period unless one of them has a liquidity surplus or deficit.13 This is not the case in our model, as has been shown in Section 4.14 In our model, even banks that do not have to offset liquidity surpluses or deficits trade with each other before the end of the maintenance period. They do that in order to better their bargaining position on the last day of the maintenance period. This is clearly a testable implication of the present model. However, the test would require data about the liquidity situations of banks before the end of the maintenance period. Another result of our paper that may be empirically relevant has been mentioned in the last paragraph of Section 3. The interbank rate in one deal can differ from the rate in another deal j l for some i, j, k, l ∈ B). There is strong evidence that on the European and US inter(ri,2 = rk,2 bank markets, the law of one price is not valid.15 The standard explanation for this phenomenon is differences in the creditworthiness of borrowers. Furfine (2001) now shows empirically for the US federal funds market that interest rates partially, but not fully, reflect the borrowers’ degrees of credit risk. Our model may provide an interesting explanation for interest rate differences that cannot be explained by individual degrees of credit risk. Finally, the indetermination of interest rates in our model may provide an explanation for possible “strange” interest rate patterns. For example, it may be shown that the interest rate for trades between two banks i and j usually remains by and large unchanged before the last day of the maintenance period, although new information frequently arrives on the market. The banks S may for example have a habit of trading at an interest rate that is equal to the mid-point rD +r 2 of the facility rates. Such interest rate persistence could then be explained by our model as due 13 See for example Tapking (2002). 14 See paragraph before Eq. (30). 15 See for example Ewerhart et al. (2003) and Furfine (2001).

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to the habit of the two banks of adjusting transaction volumes instead of interest rates when new information arrives on the days before the last day of the maintenance period. Given that testing some of the implications of our model might require a substantial amount of data, the most economic way to test the model may be by means of well-designed experiments. Such experiments might be particularly helpful to check whether our model is really appropriate for a market where trades are concluded in negotiations over the telephone. 6. Concluding remarks We have discussed a model of the interbank market that is based on the assumption that banks do not take the interest rate as given, but negotiate on each day of a two-day maintenance period on interest rates and on the amount of liquidity one bank borrows from another. The theoretical concept we have used to model these negotiations is the Shapley value. We have shown that the implicit average interbank rate on the first day of the maintenance period is not unique: it is any number between 0 and 1, and therefore conveys no information on the expected liquidity situation in the market. A crucial question is of course whether our model is realistic. In this respect, we have argued that on the basis of available information, it is difficult to assess whether or not our model is a good description of reality. However, our model does have some testable implications, although to test these would require a substantial amount of data. For this reason, the best way to test our model may be by means of an experiment. From a theoretical point of view, it should be emphasised that we model a dynamic cooperative game. No theoretical foundation of the Shapley value as a solution concept for dynamic cooperative bargaining games exists. From this perspective, the usage of the Shapley value in this paper may appear somewhat ad hoc. However, there is at present no well-founded solution concept for dynamic cooperative bargaining games. Providing such a foundation is of course beyond the scope of this paper. However, the paper may provide some guidance for designing such a foundation as it provides an example of a concrete situation. For example, we have shown in Section 4 that banks trade on the first day of the maintenance period only to improve their bargaining positions on the second day. Our paper thus describes strategic considerations of players that may also be relevant in a general, well-founded model. Acknowledgments I would like to thank Ulrich Bindseil, Heinz Herrmann, Joachim Keller and two anonymous referees for helpful discussions and comments. Appendix A h = F i = 0, Proof of Lemmas 1 and 2. Consider two coalitions of form {i, h} and {j }. With Fj,1 j,1 we obtain  ∂v{h,i},1 1  = (rS − rD ) p1 I{i} (s1 ) − I{i,j } (s1 ) − I{h} (s1 ) + I{h,j } (s1 ) h h 6 ∂(1 − ri,1 )Fi,1   + p2 I{i} (s2 ) − I{i,j } (s2 ) − I{h} (s2 ) + I{h,j } (s2 ) . (A.1)  Note that v{h,i},1 is piecewise linear, i.e. Eq. (A.1) is only defined if i∈k xi (sz ) = 0 for k ∈ {h}, {h, j }, {i}, {i, j } and all z ∈ {1, 2}. However, it is a continuous function.

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369

2 )F 2 < g, we have I (s ) = I (s ) = Lemma 1. Let i = 1, h = 2. For 0 < (1 − r1,1 {1} 1 {2} 1 1,1 I{2,3} (s1 ) = I{2,3} (s2 ) = 1 and I{1} (s2 ) = I{2} (s2 ) = I{1,3} (s1 ) = I{1,3} (s2 ) = 0, i.e. the right-hand 2 )F 2 > g, side of Eq. (A.1) is (rS − rD ) 16 > 0 and v{1,2},1 is not maximised. For (1 − r1,1 1,1 we have I{2} (s1 ) = I{2} (s2 ) = I{2,3} (s1 ) = I{2,3} (s2 ) = 1 and I{1} (s1 ) = I{1} (s2 ) = I{1,3} (s1 ) = 2 )F 2 I{1,3} (s2 ) = 0, i.e. the right-hand side of Eq. (A.1) is 0. A further increase of (1 − r1,1 1,1 2 2 would no longer change v{1,2},1 , thus any (1 − r1,1 )F1,1  g or—because of symmetry— 2 )F 2  −g maximises v (1 − r1,1 {1,2},1 . 1,1 Lemma 2. Let h = 1 and i = 3. 3 )F 3 < g, we have I (s ) = I (s ) = I (1) For 0 < (1 − r1,1 {1} 1 {3} 2 {1,2} (s1 ) = I{2,3} (s1 ) = 1,1 I{2,3} (s2 ) = 1 and I{1} (s2 ) = I{3} (s1 ) = I{1,2} (s2 ) = 0, i.e. the right-hand side of Eq. (A.1) is (rS − rD )p1 61 > 0 and v{1,3},1 is not maximised. 3 )F 3 < 2g, we have I (s ) = I (s ) = I (2) For g < (1 − r1,1 {3} 1 {3} 2 {2,3} (s1 ) = I{2,3} (s2 ) = 1,1 I{1,2} (s1 ) = 1 and I{1} (s1 ) = I{1} (s2 ) = I{1,2} (s2 ) = 0, i.e. the right-hand side of Eq. (A.1) is 3 )F 3 = g is a local maximum. −(rS − rD )p1 61 < 0. Thus, (1 − r1,1 1,1 3 3 (3) For (1 − r1,1 )F1,1 > 2g, we have I{3} (s1 ) = I{3} (s2 ) = I{2,3} (s1 ) = I{2,3} (s2 ) = 1 and I{1} (s1 ) = I{1} (s2 ) = I{1,2} (s2 ) = I{1,2} (s1 ) = 0, i.e. the right-hand side of Eq. (A.1) is 0. A further 3 )F 3 would not change v increase of (1 − r1,1 {1,3},1 any further. 1,1 3 3 (4) For −g < (1 − r1,1 )F1,1 < 0, we have I{1} (s1 ) = I{3} (s2 ) = I{1,2} (s1 ) = 1 and I{1} (s2 ) = I{3} (s1 ) = I{1,2} (s2 ) = I{2,3} (s1 ) = I{2,3} (s2 ) = 0, i.e. the right-hand side of Eq. (A.1) is −(rS − rD )p2 61 > 0 and v{1,3},1 is not maximised. 3 )F 3 < −g, we have I (s ) = I (s ) = I (5) For −2g < (1 − r1,1 {1} 1 {1} 2 {1,2} (s1 ) = 1 and 1,1 I{3} (s1 ) = I{3} (s2 ) = I{1,2} (s2 ) = I{2,3} (s2 ) = I{1,2} (s1 ) = 0, i.e. the right-hand side of Eq. (A.1) is 3 )F 3 = −g is a local maximum. −(rS − rD )p2 61 > 0. Thus, (1 − r1,1 1,1 3 3 (6) For (1 − r1,1 )F1,1 < −2g, we have I{1} (s1 ) = I{1} (s2 ) = I{1,2} (s1 ) = I{1,2} (s2 ) = 1 and I{3} (s1 ) = I{3} (s2 ) = I{2,3} (s2 ) = I{2,3} (s1 ) = 0, i.e. the right-hand side of Eq. (A.1) is 0. A fur3 )F 3 would not change v ther decrease of (1 − r1,1 {1,3},1 any further. It is clear that the local 1,1 maximum (2) is higher (lower) than the local maximum (5) if and only if p1 > p2 (p1 < p2 ). 2 3 = F 3 , F 2 = 0, 0 < r 3 = Proof of Proposition 3. We assume as mentioned in the text F1,1 1,1 2,1 1,1 1 3 < 1. First consider cases with 0  (1 − r 3 )F 3  g, i.e. I (s ) = I (s ) = I (s ) = r2,1 {3} 1 {1} 2 {2} 2 1,1 1,1 2 I{1,2} (s2 ) = IB (s2 ) = 0 and Ik (sz ) = 1 for all other k and z. It is easy to show that in this case v{i},1 = E1 [L∗∗ i,3 ] is equivalent to 3 F1,1 =

1 min{p1 , p2 } (rS − rD )g 18

(A.2)

3 )[1 + r − (r − r )( 1 + 1 p )] 1 − (1 − r1,1 S S D 2 6 1

for all i ∈ {1, 2, 3}. To be consistent, we have to ensure that 0

1 3 ) (rS − rD )g 18 min{p1 , p2 }(1 − r1,1 3 )[1 + r 1 − (1 − r1,1 S

− (rS − rD )( 12

+

1 6 p1 )]

1  g. 2

It is easy to show that this is equivalent to 3 r1,1 

rS − (rS − rD )( 16 p1 +

1 2

1 + rS − (rS − rD )( 16 p1 +

− 19 min{p1 , p2 }) 1 2

− 19 min{p1 , p2 })

≡ β.

3 ∈ ]β, 1], there is a F 3 given by Eq. (A.2) that solves Eq. (30). Thus, for every r1,1 1,1

(A.3)

370

J. Tapking / Games and Economic Behavior 56 (2006) 350–370

3 )F 3  − 1 g, i.e. I (s ) = I (s ) = I Now consider cases where 0  (1 − r1,1 {1} 1 {2} 1 {1,2} (s1 ) = 1,1 2 IB (s1 ) = I{3} (s2 ) = 1 and Ik (sz ) = 0 for all other k and z. In this case, v{i},1 = E1 [L∗∗ i,3 ] is equivalent to 3 = F1,1

1 g min{p1 , p2 } (rS − rD ) 18

(A.4)

3 )[1 + r − (r − r )( 1 p + 1 )] 1 − (1 − r1,1 S S D 6 1 3

for all i ∈ {1, 2, 3}. To be consistent, we have to ensure that 0

1 3 ) (rS − rD ) 18 g min{p1 , p2 }(1 − r1,1 3 )[1 + r 1 − (1 − r1,1 S

− (rS − rD )( 16 p1

+

1 3 )]

1  − g. 2

This is equivalent to 3 r1,1 

rS − (rS − rD )( 16 p1 +

1 3

1 + rS − (rS − rD )( 16 p1 +

+ 19 min{p1 , p2 }) 1 3

+ 19 min{p1 , p2 })

≡ γ.

(A.5)

3 ∈ ]0, γ ], there is a F 3 given by Eq. (A.4) that solves Eq. (30). Thus, for every r1,1 1,1 It is easy to show that 0 < β < γ < 1. 2

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