Bank demand for central bank liquidity and its impact on interbank markets

Abstract

We develop a model of monetary policy implementation in which banks bid for liquidity provided by the central bank in fixed rate auctions, considering liquidity injections and extractions as well as the impact of a subsequent interbank market. We derive the equilibrium demands of banks. We also investigate the impact the central bank auction has on the subsequent interbank market and find that while lending in the interbank market is reduced, the interest rates are moving in the desired direction. In the context of the interbank network, the impact of monetary policy on banks depends on their network locations, which may give rise to the prospects of distributional effects of monetary policy.

Appendices

Appendix: Proofs

1.1 Proof of Lemma 1

Based on reservation prices in equations (4) and (5), we easily get $\lim _{Q\rightarrow 0} r_i^a\left( Q_i\right) =r_i^a(0)=r^f+\frac{\theta _i}{1-\theta _i}\frac{\left( 1+r_i^E\right) {\mathbf {E}}_i\left( {\mathbf {D}}_i+{\mathbf {B}}_i-{\mathbf {R}}_i\right) }{{\mathbf {R}}_i\left( {\mathbf {D}}_i+{\mathbf {B}}_i\right) }$ and $\lim _{Q\rightarrow 0} r_i^b\left( Q_i\right) =r_i^b(0)=r^f+\frac{\theta _i}{1-\theta _i}\frac{\left( 1+r_i^E\right) {\mathbf {E}}_i}{{\mathbf {R}}_i}$. From this, we easily obtain that $r_i^b(0)-r_i^a(0)=\frac{\theta _i}{1-\theta _i}\frac{\left( 1+r^E_i\right) {\mathbf {E}}_i}{{\mathbf {R}}_i}\frac{{\mathbf {R}}_i}{\left( {\mathbf {D}}_i+{\mathbf {B}}_i\right) }>0.$

1.2 Proof of Proposition 1

Let us first consider the case of liquidity injection, i. e. $Q_i>0$. The optimization problem (6) has a unique solution as $U_i$ is a concave function of $Q_i$ whose second derivative is

$$\begin{aligned} \frac{d^2 U_i}{d Q_i^2}= & {} -2\frac{{\mathbf {D}}_i+{\mathbf {B}}_i-{\mathbf {R}}_i}{\left( {\mathbf {D}}_i+{\mathbf {B}}_i+Q_i\right) ^3}U_1+\left( \frac{{\mathbf {D}}_i+{\mathbf {B}}_i-{\mathbf {R}}_i}{\left( {\mathbf {D}}_i+{\mathbf {B}}_i+Q_i\right) ^2}\right) ^2U_{11}\\&+\frac{{\mathbf {D}}_i+{\mathbf {B}}_i-{\mathbf {R}}_i}{\left( {\mathbf {D}}_i+{\mathbf {B}}_i+Q_i\right) ^2}\frac{r^f-r^{CB}}{{\mathbf {E}}_i}\left( U_{12}+U_{21}\right) +\left( \frac{r^f-r^{CB}}{{\mathbf {E}}_i}\right) ^2U_{22}<0, \end{aligned}$$

where $U_1=\frac{\partial U_i}{\partial \rho _i}>0$, $U_{11}=\frac{\partial ^2 U_i}{\partial \rho _i^2}<0$, $U_{12}=U_{21}=\frac{\partial ^2 U_i}{\partial \rho _i \partial {r^E_i}}>0$, $U_{22}=\frac{\partial ^2 U_i}{\partial {r^E_i}^2}\,{<}\,0$.

Therefore, the solution to the problem in equation (6) is either at the boundary if one of the constraints is binding or at the local maximum of $U_i$. Solving for $\frac{\partial U_i}{\partial Q_i}=0$ gives the local maximum. The first constraint cannot be binding as insolvency gives zero utility. Also note the last constraint is binding when bank i’s valuation for borrowing is already lower than $r^{CB}_f$ at $Q_i=0$, that is $r_i^a(0)<r^{CB}_f$. Therefore, we can write the solution to problem in equation (6) as,

$$\begin{aligned} Q_i^f\left( r^{CB}_f\right) =\left\{ \begin{array}{lcl} \min \left\{ {\overline{Q}}_i,-\psi _i+\varphi ^{\frac{1}{2}}\right\} &{}\quad \text {if} &{} r^f<r^{CB}_f<r_i^a(0) \\ 0 &{}\quad \text {if} &{} r_i^a(0)\le r^{CB}_f \\ \end{array}\right. , \end{aligned}$$

where $\varphi =\psi _i^2-\left( {\mathbf {D}}_i+{\mathbf {B}}_i\right) {\mathbf {R}}_i\frac{r^{CB}-r_i^a(0)}{r^{CB}-r^f}$.

Secondly, consider liquidity extraction, i. e. $Q_i<0$. Similarly, this problem has a unique solution because $U_i$ is a concave function of $Q_i$ whose second derivative is

$$\begin{aligned} \frac{d^2 U_i}{d Q_i^2}=\frac{U_{11}}{\left( {\mathbf {D}}_i+{\mathbf {B}}_i\right) ^2}+\frac{U_{12}+U_{21}}{{\mathbf {D}}_i+{\mathbf {B}}_i}\frac{r^f-r^{CB}}{{\mathbf {E}}_i}+\left( \frac{r^f-r^{CB}}{{\mathbf {E}}_i}\right) ^2U_{22} <0. \end{aligned}$$

The second constraint is binding when bank i’s valuation for lending is already higher than $r^{CB}_f$ at $Q_i=0$, or $r^{CB}_f>r_i^b(0)$. Solving for the local maximum by letting $\frac{d U_i}{d Q_i}=0$ completes the solution.

$$\begin{aligned} Q_i^f\left( r^{CB}_f\right) =\left\{ \begin{array}{lcl} 0 &{}\quad \text {if} &{} r^f<r^{CB}_f\le r_i^b(0) \\ \theta _i\frac{\left( 1+r_i^E\right) {\mathbf {E}}_i}{r^{CB}-r^f}-\left( 1-\theta _i\right) {\mathbf {R}}_i &{}\quad \text {if} &{} r_i^b(0)<r^{CB}_f \end{array}\right. . \end{aligned}$$

Combining these two results gives us the result shown in the proposition.

Dropping the constraint that $Q_i\le {\overline{Q}}_i$ as it does not affect the sign of the derivative of $Q_i^f(r)$, we obtain that

$$\begin{aligned} \frac{\partial Q_i^f}{\partial r_f^{CB}}=\frac{1}{2}\varphi ^{-\frac{1}{2}}\left( \left( {\mathbf {D}}_i+{\mathbf {B}}_i\right) {\mathbf {R}}_i\frac{r^f-r_i^a(0)}{\left( r_f^{CB}-r^f\right) ^2}\right) <0 \end{aligned}$$

in the case of liquidity injection and

$$\begin{aligned} \frac{\partial Q_i^f}{\partial r_f^{CB}}=-\theta _i\frac{\left( 1+r_i^E\right) {\mathbf {E}}_i}{\left( r_f^{CB}-r^f\right) ^2}<0 \end{aligned}$$

(9)

in the case of liquidity extraction.

1.3 Proof of Lemma 2

The proof is trivial from inverting the equilibrium bid schedule in Proposition 1.

1.4 Proof of Lemma 3

We prove the individual parts in turn:

• 1. By inserting $Q_i^f=0$ into the inverse bid schedule given in Lemma 2 we instantly see that these are identical to the reservation prices defined in Lemma 1.

• 2. By inserting $Q_i^f=0$ into the inverse bid schedule given in Lemma 2 we instantly see that these are identical to the reservation prices defined in Lemma 1.

• 3. $r_i^a(0)-r_i^b(0)=\frac{\theta _i}{1-\theta _i}\frac{\left( 1+r_i^E\right) {\mathbf {E}}_i}{{\mathbf {D}}_i+{\mathbf {B}}_i}>0$ which in combination with claims 1 and 2 of this lemma completes this proof.

• 4. Suppose there is a $Q_i^f<0$ such that $r^{CB}\left( Q_i^f\right) <r_i^b\left( Q_i^f\right)$. As the reservation prices are determined such that upon making a deposit of $Q_i^f$, the utility level does not change from the situation of not making a deposit. Receiving an amount less than $r_i^b\left( Q_i^f\right)$ will reduce the utility level of bank i, contradicting the requirement that $r^{CB}\left( Q_i^f\right)$ maximizes the utility.

• 5. Suppose there is a $Q_i^f>0$ such that $r^{CB}\left( Q_i^f\right) >r_i^a\left( Q_i^f\right)$. As the reservation prices are determined such that upon taking a loan from the central bank of $Q_i^f$, the utility level does not change from the situation of not taking a loan. Paying an amount more than $r_i^a\left( Q_i^f\right)$ will reduce the utility level of bank i, contradicting the requirement that $r^{CB}\left( Q_i^f\right)$ maximizes the utility.

1.5 Proof of Lemma 4

The marginal prices here are bank i’s marginal valuation for liquidity. Thus, for $Q_i>0$, ${\widetilde{r}}_i^{a}= \frac{\partial Q_ir_i^a}{\partial Q_i}$, while for $Q_i<0$, ${\widetilde{r}}_i^{b}= \frac{\partial Q_ir_i^b}{\partial Q_i}$, where $r_i^a$ and $r_i^b$ are the reservation prices given determined in equations (4) and (5).

1.6 Proof of Proposition 3

We prove both claims in this proposition in turn, commencing with the case of liquidity injection. Let us consider an arbitrary bank i and denote the equilibrium demand schedule of any bank by $Q_i^v(r)$. If all banks, apart from bank i submit their optimal demand schedules and the total supply of liquidity by the central bank is $Q^{CB}$, the residual demand schedule this bank faces, considering the constraint on the amount it can bid for, is given by

$$\begin{aligned} Q_i^c(r)=\min \left\{ {\bar{Q}}_i,Q^{CB}-\sum _{j \ne i} Q_j^v(r)\right\} , \end{aligned}$$

where we require that $r > r^f$. Assume now that $Q_i^c$ is on the optimal demand curve for bank i at an interest rate $r^c$. This rate $r^c$ would be the lowest possible rate at which the bank can submit its bid and still obtain the requested amount. Due to discriminatory pricing in variable rate auctions, any bid higher than $r^c$ would result in a utility loss as the bank pays more than it has to. Thus, for any rate $r>r^c$ the submitted demand is zero. On the other hand, at a rate $r<r^c$ a bid would not be successful as it is too low; hence, what price or amount is submitted becomes irrelevant. Hence, the only possible equilibrium would be for a bank to submit a bid at exactly $r^c$ for the quantity it requires at that rate.

In the following, we show that in equilibrium a bank will submit a bid schedule as indicated in the proposition.

If $r^a_i(0)\le {\widetilde{r}}$ the reservation price of not submitting a bid, or equivalently a bid of zero, is optimal as exceeding your reservation price will result in a loss of utility.

In all other cases, we now show that alternative points on the residual demand schedule give the bank a lower utility and can thus not be an equilibrium. Let us now consider another equilibrium ${\widehat{r}}\ne {\widetilde{r}}$. If we have that $r^a_i(0)>{\widehat{r}}>{\widetilde{r}}$, we find that $Q^c_i={\overline{Q}}_i$ as can be easily seen by inserting the expressions for $Q_j^v$ into $Q_i^c$ above.

In the case of ${\widehat{r}}\le {\widetilde{r}}$ we compare ${\widetilde{Q}}_i\left( {\widehat{r}}\right)$ and $Q_i^c\left( {\widehat{r}}\right)$. By construction ${\widetilde{Q}}_i\left( {\widehat{r}}\right)$ gives the same utility level at ${\widehat{r}}$ as $Q_i^f\left( {\widetilde{r}}\right)$ at ${\widetilde{r}}$, i.e. it lies on the same indifference curve as the optimal demand schedule. If ${\widetilde{Q}}_i\left( {\widehat{r}}\right) \ge Q_i^c\left( {\widehat{r}}\right)$ then $Q_i^c$ would give the bank less cash than ${\widetilde{Q}}_i$ at the same price; given that banks prefer more cash, this would lead to a lower utility level and would thus not be optimal.

We now show that ${\widetilde{Q}}_i\left( {\widehat{r}}\right) \ge Q_i^c\left( {\widehat{r}}\right)$ as follows:

$$\begin{aligned} {\widetilde{Q}}_i({\widehat{r}})-Q_i^c({\widehat{r}})= & {} {\widetilde{Q}}_i({\widehat{r}})- \min \left\{ {\bar{Q}}_i,Q^{CB}-\sum _{i \ne j} Q_j^v({\widehat{r}})\right\} \\= & {} \max \left\{ {\widetilde{Q}}_i({\widehat{r}})-Q_i,{\widetilde{Q}}_i({\widehat{r}})-Q^{CB}+\sum _{j \ne i} Q_j^v({\widehat{r}})\right\} \\= & {} \max \left\{ {\widetilde{Q}}_i({\widehat{r}})-{\bar{Q}}_i,{\widetilde{Q}}_i({\widehat{r}})-Q^{CB}\right. \\&\left. + \sum _{j \ne i}\min \left\{ {{\bar{Q}}}_j,Q_{j}^f\left( {\widetilde{r}} \right) +\max _{k=1,\ldots ,N} \left( Q_{k}^f\left( {\widetilde{r}} \right) -{\widetilde{Q}}_k({\widehat{r}})\right) \right\} \right\} \\= & {} \max \left\{ {\widetilde{Q}}_i({\widehat{r}})-{\bar{Q}}_i,{\widetilde{Q}}_i({\widehat{r}})-Q^{CB}\right. \\&+\left. \sum _{j \ne i}Q_{j}^f\left( {\widetilde{r}}\right) + \sum _{j \ne i}\min \left\{ {{\bar{Q}}}_j-Q_{j}^f\left( {\widetilde{r}}\right) ,+\max _{k=1,\ldots ,N} \left( Q_{k}^f\left( {\widetilde{r}} \right) -{\widetilde{Q}}_k({\widehat{r}})\right) \right\} \right\} \\= & {} \max \left\{ {\widetilde{Q}}_i({\widehat{r}})-{\bar{Q}}_i,{\widetilde{Q}}_i({\widehat{r}})-Q_{i}^f\left( {\widetilde{r}}\right) \right. \\&\left. +\sum _{j \ne i}\min \left\{ {{\bar{Q}}}_j-Q_{j}^f\left( {\widetilde{r}}\right) ,\max _{k=1,\ldots ,N} \left( Q_{k}^f\left( {\widetilde{r}} \right) -{\widetilde{Q}}_k({\widehat{r}})\right) \right\} \right\} \\= & {} {\widetilde{Q}}_i({\widehat{r}})-Q_{i}^f\left( {\widetilde{r}}\right) +\sum _{j \ne i}\min \left\{ {{\bar{Q}}}_j-Q_{j}^f\left( {\widetilde{r}}\right) ,\max _{k=1,\ldots ,N} \left( Q_{k}^f\left( {\widetilde{r}} \right) -{\widetilde{Q}}_k({\widehat{r}})\right) \right\} , \end{aligned}$$

where the last step is obtained as $Q_{i}^f\left( {\widetilde{r}}\right) \le {{\bar{Q}}}_i$ and ${{\bar{Q}}}_j-Q_{j}^f\left( {\widetilde{r}}\right)$ and $Q_{k}^f\left( {\widetilde{r}}\right) -{\widetilde{Q}}_k({\widehat{r}})$ are always non-negative. If there exists one $j \ne i$, such that ${{\bar{Q}}}_j-Q_{j}^f\left( {\widetilde{r}}\right) \ge \max _{k=1,\ldots ,N} \left( Q_{k}^f\left( {\widetilde{r}}\right) -{\widetilde{Q}}_k({\widehat{r}})\right)$, we have,

$$\begin{aligned} {\widetilde{Q}}_i({\widehat{r}})-Q_i^c({\widehat{r}}) \ge {\widetilde{Q}}_i({\widehat{r}})-Q_{i}^f\left( {\widetilde{r}}\right) +\max _{k=1,\ldots ,N} \left( Q_{k}^f\left( {\widetilde{r}} \right) -{\widetilde{Q}}_k({\widehat{r}})\right) \ge 0. \end{aligned}$$

Otherwise we have

$$\begin{aligned} {\widetilde{Q}}_i({\widehat{r}})-Q_i^c({\widehat{r}})= & {} {\widetilde{Q}}_i({\widehat{r}})-Q_{i}^f\left( {\widetilde{r}}\right) +\sum _{j \ne i}\left( {{\bar{Q}}}_j-Q_{j}^f\left( {\widetilde{r}}\right) \right) \\= & {} {\widetilde{Q}}_i({\widehat{r}})+\sum _{j \ne i}{{\bar{Q}}}_j-Q^{CB}\\= & {} {\widetilde{Q}}_i({\widehat{r}})+(N-1){{\bar{Q}}}_i-Q^{CB}\\\ge & {} 0. \end{aligned}$$

In the case of liquidity extraction the same steps are followed as above. The possible demand by bank i given the demand by all other banks is determined as

$$\begin{aligned} Q_i^c(r)=\max \left\{ -{\mathbf {R}}_i,Q^{CB}-\sum _{j \ne i} Q_j^v(r)\right\} , \end{aligned}$$

where we take into account that banks cannot deposit more than their cash reserves and require that $r>r^f$. With the same arguments made before, for any $r<r^c$, the optimal interest rate, the bank does not receive sufficient interest on their deposits with the central bank and thus will bid an amount of zero. Furthermore, if $r_i^b(0)\ge {\widetilde{r}}$, the reservation price is too high for the bank to bid for depositing cash with the central bank, and thus will also bid an amount of zero.

For the case of $r_i^b(0)< {\widehat{r}}<{\widetilde{r}}$, we can easily show that $Q_i^c(r)=\max \left( -{\mathbf {R}}_i,Q^{CB}\right)$ by inserting for $Q_j^v(r)$. In the case that ${\widehat{r}}\ge {\widetilde{r}}$ we follow the same arguments as in the case of liquidity injection and need to show that ${\overline{Q}}_i({\widehat{r}})\le Q_i^c({\widehat{r}})$. We obtain

$$\begin{aligned} {\overline{Q}}_i({\widehat{r}})-Q_i^c({\widehat{r}})= & {} Q_i({\widehat{r}})- \max \left\{ -{\mathbf {R}}_i,Q^{CB}-\sum _{j \ne i} Q_j^v({\widehat{r}})\right\} \\= & {} \min \left\{ {\overline{Q}}_i({\widehat{r}})+{\mathbf {R}}_i,{\overline{Q}}_i({\widehat{r}})-Q^{CB}+\sum _{j \ne i} Q_j^v({\widehat{r}})\right\} \\= & {} \min \left\{ {\overline{Q}}_i({\widehat{r}})+{\mathbf {R}}_i,{\overline{Q}}_i({\widehat{r}})-Q^{CB}+ \sum _{j \ne i}\left( Q_{j}^f\left( {\widetilde{r}} \right) -\max _{k=1,\ldots ,N} \left( {\overline{Q}}_k({\widehat{r}})-Q_{k}^f\left( {\widetilde{r}} \right) \right) \right) \right\} \\= & {} \min \left\{ {\overline{Q}}_i({\widehat{r}})+{\mathbf {R}}_i,{\overline{Q}}_i({\widehat{r}})-Q_{i}^f\left( {\widetilde{r}} \right) - (N-1)\left( \max _{k=1,\ldots ,N} \left( {\overline{Q}}_k({\widehat{r}})-Q_{k}^f\left( {\widetilde{r}}\right) \right) \right) \right\} \\= & {} {\overline{Q}}_i({\widehat{r}})-Q_{i}^f\left( {\widetilde{r}}\right) - (N-1)\left( \max _{k=1,\ldots ,N} \left( {\overline{Q}}_k({\widehat{r}})-Q_{k}^f\left( {\widetilde{r}}\right) \right) \right) \\\le & {} 0 \end{aligned}$$

The penultimate step arises as $Q_{i}^f\left( {\widetilde{r}}\right) \ge -{\mathbf {R}}_i$ and ${\overline{Q}}_k({\widehat{r}})\ge Q_{k}^f\left( {\widetilde{r}}\right)$.

1.7 Proof of Lemma 5

This lemma follows from the definition of the inverse bid schedule, $r_i^v(Q_i)=\inf \left\{ r>r^f| Q_i^v(r)\le Q_i\right\}$. It is also obvious that $r_i^v(Q_i)$ is a non-increasing function of $Q_i$. To show this, we only need to show that $R^{-1}(Q_i)$ is a non-increasing function of $Q_i$. This is true as when r increases, both $\min \Big \{{{\bar{Q}}}_i,Q_{i}^f\left( {\widetilde{r}} \right) +\max _{j=1,\ldots ,N} \left( Q_{j}^f\left( {\widetilde{r}} \right) -{\widetilde{Q}}_j(r)\right) \Big \}$ and $Q_{i}^f\left( \widetilde{{\widetilde{r}}} \right) -\max _{j=1,\ldots ,N} \left( \widetilde{{\widetilde{Q}}}_j(r)- Q_{j}^f\left( \widetilde{{\widetilde{r}}} \right) \right)$ are non-increasing as can easily be seen.

1.8 Proof of Proposition 4

We need to show the tuple $\left( r^{CB}_f,Q_1^f,\ldots ,Q_N^f\right)$ also clears the market when bid schedules are as described in Proposition 3. First consider the case of $Q_i^f\ge 0$, when the amount of operation is $Q^{CB}=\sum _{i=1}^N Q_i^f=\sum _{i=1}^N \max \left( 0,Q_i^f\left( r^{CB}_f\right) \right)$, obviously $r^{CB}_f \in \left\{ r\ge r^f|\sum _i^N \max \left( 0,Q_i^f(r)\right) = Q^{CB}\right\}$. Note that $Q_i^f(r)$ is strictly decreasing before reaching limit ${{\bar{Q}}}_i$. Therefore, $\sum _i^N \max \left( 0,Q_i^f(r)\right)$ is strictly decreasing as $0<Q^{CB}<(N-1){{\bar{Q}}}_i$ and has thus a unique solution. Therefore, ${\widetilde{r}}$ defined in Proposition 3 equals $r^{CB}_f$, since $Q_{i}^v\left( r^*\right) =Q_{i}^f\left( r^*\right)$ (if ${\widetilde{r}}>r_i^a(0)$ this also holds as both are zero), what remains to be shown is $r^{CB}_v={\widetilde{r}}$. This is obvious as $\sum _{i=1}^N Q_{i}^v\left( {\widetilde{r}}\right) =Q^{CB}$ clears the market while any $r<{\widetilde{r}}$ cannot.

The proof for $Q_i^f\le 0$ follows exactly the same process.

1.9 Proof of Proposition 2

It is easy to verify that $\left( {\widehat{r}}^{IB}, \lambda Q_1^I\left( {\widehat{r}}^{IB}\right) ,\ldots , \lambda Q_1^I\left( {\widehat{r}}^{IB}\right) \right)$ clears the market. We show here that a bank i cannot gain higher expected utility after the interbank market by deviating from the bid schedule proposed here. As shown in Proposition 3, it is optimal for any bank i to pay no more than the clearing rate of the primary market, so in the following we only consider bid schedules where a bank demands zero if the interest rate charged is larger than expected clearing rate in the interbank market. We show the case for liquidity injection here as for liquidity extraction the argument can be made in exactly the same way.

Let us consider bank i having any alternative bid schedule where it bids $Q_i>0$ at a rate $r_i^{CB}$ and $Q_i=0$ at some rate $r>r_i^{CB}$. Firstly, if $r_i^{CB}<{\widehat{r}}^{IB}$, bank i does not participate in the primary market but only the interbank market. This does not change the clearing rate as the reduced allocation to bank i is compensated by increased allocation to other banks. Consequently, in the interbank market, bank i demands more funds, while other banks are expected to demand less due to their increased allocation by the central bank; hence, the aggregate amount is unchanged and so is the expected interbank rate. Here, bank i only shifts part of its demand from central bank funds to the interbank market and its expected overall utility increase is the same; hence, it is not better off.

Secondly, if $r_i^{CB}={\widehat{r}}^{IB}$, but $Q_i^f\left( {\widehat{r}}^{IB}\right) \ge Q_i\ne Q_i^I\left( {\widehat{r}}^{IB}\right)$, this results in bank i borrowing less from the central bank as the rate is less favourable and this has a similar effect as in the case where $r_i^{CB}<{\widehat{r}}^{IB}$. If $r_i^{CB}={\widehat{r}}^{IB}$, but $Q_i^f\left( {\widehat{r}}^{IB}\right) <Q_i \ne Q_i^f\left( {\widehat{r}}^{IB}\right)$, bank i could be worse off. Its allocation could exceed $Q_i^f\left( {\widehat{r}}^{IB}\right)$ which is the optimal amount that maximize i’s utility or by over-reporting its demand bank i also makes the demand for liquidity to be greater in subsequent interbank markets and thus raises the expected interbank rate for all banks, including itself.

Thirdly, if $r_i^{CB}>{\widehat{r}}^{IB}$ and if $Q_i\ge Q_i^I\left( {\widehat{r}}^{IB}\right)$ bank i would be strictly worse off because it pays more for liquidity from the central bank as well as the interbank market. The former is obvious, and the latter is because over-reporting bank i’s demand raises expected interbank rates as discussed above. On the other hand, if $Q_i < Q_i^I\left( {\widehat{r}}^{IB}\right)$, bank i would still be worse off. Suppose in this case, bank i gets an allocation of $Q_i^{CB}$ from the central bank and demands $Q_i^{IB}$ in the interbank market. Obviously, for $Q_i^{CB}$, bank i pays more than ${\widehat{r}}^{IB}$ which reduces its utility. For $Q_i^{IB}$, there is a chance bank i pays less than ${\widehat{r}}^{IB}$, even so, this is not enough to compensate for i’s utility loss from central bank funds. Suppose the opposite is true, that bank i pays in the interbank market $r_i^{IB}<{\widehat{r}}^{IB}$, keeping its utility the same as before. Consider all combinations of rate and quantity (r, Q) in the interbank market that gives the same utility as originally, i. e. $\Big \{(r,Q)| U_i\left( \rho _i(Q),r_i^E(r, Q)\right) =U_i\left( \rho _i\left( Q_{i}^I\left( {\widehat{r}}^{IB}\right) \right) ,r_i^E\left( {\widehat{r}}^{IB},Q_{i}^I\left( {\widehat{r}}^{IB}\right) \right) \right) , {\widehat{r}}^{IB}>r >r^f\Big \}.$ If the collateral constraint is not binding for bank i, it has to demand $Q_i^{IB}$ in order to maximize its utility, corresponding to the maximum of r, but $r<{\widehat{r}}^{IB}$ implies $Q_i^{IB}+Q_i^{CB}\ge Q_i^I \left( {\widehat{r}}^{IB}\right)$. If the collateral is binding, equality holds here.

Overall, this implies interbank markets cannot clear as banks’ aggregate demand must be more than the supply due to a drop in interbank market rate and the fact bank i is also demanding more than or equal as before. Therefore, bank i cannot reach the same level of utility as originally.

Appendix: Variable rate tenders

In variable rate tenders, the central bank exogenously fixes the total amount of liquidity extracted or injected at $Q^{CB}$. The interest rate is set such that only those banks can participate that have submitted the highest (lowest) demand schedules for borrowing (depositing) until the total amount is reached. The interest rate is thus set such that $\sum _{i=1}^N Q_i\le Q^{CB}$. Each bank pays the interest rate at which it has submitted its bids, i.e. pricing is discriminatory and banks will not pay the same price, but according to their bid schedule. Therefore, the reservation price of a bank will be the marginal value of any amount obtained. The following lemma determines these marginal prices:

Lemma 4

In variable rate tenders, the marginal prices are given by

$$\begin{aligned} {\widetilde{r}}_i^{a}(Q_i)= r^f+\left( 1+r^E_i\right) {\mathbf {E}}_i\frac{\theta _i}{1-\theta _i}\left( \frac{{\mathbf {R}}_i}{{\mathbf {R}}_i+Q_i}\frac{{\mathbf {D}}_i+{\mathbf {B}}_i+Q_i}{{\mathbf {D}}_i+{\mathbf {B}}_i}\right) ^{\frac{\theta _i}{1-\theta _i}}\frac{\left( {\mathbf {D}}_i+{\mathbf {B}}_i-{\mathbf {R}}_i\right) }{\left( {\mathbf {R}}_i+Q_i\right) \left( {\mathbf {D}}_i+{\mathbf {B}}_i+Q_i\right) } \end{aligned}$$

for $Q_i>0$ and

$$\begin{aligned} {\widetilde{r}}_i^{b}(Q_i)=r^f+\left( 1+r^E_i\right) {\mathbf {E}}_i\frac{\theta _i}{1-\theta _i}\left( \frac{{\mathbf {R}}_i}{{\mathbf {R}}_i+Q_i}\right) ^{\frac{\theta _i}{1-\theta _i}}\frac{1}{{\mathbf {R}}_i+Q_i} \end{aligned}$$

for $Q_i<0$.

As before for fixed rate tenders, in order to maximize utility, banks will not submit their marginal prices, but act strategically. Proposition 3 shows the characterization of one such equilibrium, where we assume that banks know each other’s liquidity positions.

Proposition 3

Let ${\widetilde{r}} \in \left\{ r\ge r^f\right| \left. \sum _{i=1}^N \max \left( 0,Q_i^f(r)\right) = Q^{CB}\right\}$, $\widetilde{{\widetilde{r}}} \in \left\{ r\ge r^f\right| \left. \sum _{i=1}^N \min \left( 0,Q_i^f(r)\right) = Q^{CB}\right\}$, ${\widetilde{Q}}_i(r)\in \left\{ Q_i\le Q_{i}^f\left( {\widetilde{r}} \right) \right| \left. U_i\left( \rho _i(Q_i),r_i^E(r, Q_i)\right) \right.$$\left. =U_i\left( \rho _i\left( Q_{i}^f\left( {\widetilde{r}}\right) \right) ,r_i^E\left( {\widetilde{r}},Q_{i}^f\left( {\widetilde{r}}\right) \right) \right) , r>r^f\right\}$, and $\widetilde{{\widetilde{Q}}}_i(r)\in \left\{ Q_i\ge Q_{i}^f\left( {\widetilde{r}} \right) \right| \left. \right.$$\left. U_i\left( \rho _i(Q_i),r_i^E(r, Q_i)\right) =U_i\left( \rho _i\left( Q_{i}^f\left( {\widetilde{r}}\right) \right) ,r_i^E\left( {\widetilde{r}},Q_{i}^f\left( {\widetilde{r}}\right) \right) \right) , r>r^f\right\}$.

One Nash equilibrium bid schedule is then determined as follows:

• If $0<Q^{CB}<(N-1){\bar{Q}}_i$ the demand schedule is given by

  $$\begin{aligned} Q_{i}^v(r)=\left\{ \begin{array}{lll} \min \left\{ {{\bar{Q}}}_i,Q_{i}^f\left( {\widetilde{r}} \right) +\max _{j=1,\ldots ,N} \left( Q_{j}^f\left( {\widetilde{r}} \right) -{\widetilde{Q}}_j(r)\right) \right\} &{}\quad { if} &{} r^f \le r < {\widetilde{r}}\\ Q_{i}^f\left( {\widetilde{r}} \right) &{}\quad { if} &{} r = {\widetilde{r}} \\ 0 &{}\quad { if} &{} r > {\widetilde{r}} \text {~or~} r_i^a(0)\le {\widetilde{r}} \\ \end{array} \right. . \end{aligned}$$

• If $-\min _j\left( \sum _{i\ne j}^N{\mathbf {R}}_i\right)<Q^{CB}<0$, the demand schedule is given by

  $$\begin{aligned} Q_{i}^v(r)=\left\{ \begin{array}{lll} 0 &{}\quad { if} &{} r^f \le r < \widetilde{{\widetilde{r}}} \text {~or~} r_i^b(0)\ge \widetilde{{\widetilde{r}}}\\ Q_{i}^f\left( \widetilde{{\widetilde{r}}} \right) &{}\quad { if} &{} r= \widetilde{{\widetilde{r}}} \\ Q_{i}^f\left( \widetilde{{\widetilde{r}}} \right) -\max _{j=1,\ldots ,N} \left( \widetilde{{\widetilde{Q}}}_j(r)- Q_{j}^f\left( \widetilde{{\widetilde{r}}} \right) \right) &{}\quad { if} &{} r > \widetilde{{\widetilde{r}}} \\ \end{array} \right. \end{aligned}$$

The equilibrium is then trivially determined such that $\sum _{i=1}^NQ_i^v\left( r\right) =Q^{CB}$. Due to the full information banks have of each other’s liquidity position, they can fully anticipate the respective demands and submit bids that ensure this equilibrium to be reached.

This demand schedule is not easily interpreted and comparable to the result obtained in the fixed rate tender. Hence, we illustrate the equilibrium in Fig. 3. We see that the bids submitted by the banks are entirely flat at ${\widetilde{r}}$ and $\widetilde{{\widetilde{r}}}$, respectively, until the quantity bid reaches $Q_i^f$. For larger quantities beyond this threshold the rate acceptable would be lower for liquidity injections and higher for liquidity extractions, the exact shape depending on the liquidity shocks and preferences of the banks. This area of the demand schedule has no unique solution for the same allocations and interest rates. Proposition 3 provides one such bid schedule explicitly.

This inverse bid schedule is given more formally in the following lemma.

Lemma 5

The inverse bid schedule in variable rate tenders is given by

$$\begin{aligned} r_v^{CB}\left( Q_i\right) =\left\{ \begin{array}{lll} R^{-1}(Q_i) &{} if &{} Q_{i}^f\left( \widetilde{{\widetilde{r}}}\right) \ge Q_i \\ \widetilde{{\widetilde{r}}} &{} if &{} 0>Q_i > Q_{i}^f\left( \widetilde{{\widetilde{r}}}\right) \\ {\widetilde{r}} &{} if &{} 0< Q_i < Q_{i}^f\left( {\widetilde{r}}\right) \\ R^{-1}(Q_i) &{} if &{} Q_{i}^f\left( {\widetilde{r}}\right) \le Q_i \le {\bar{Q}}_i \\ \end{array} \right. , \end{aligned}$$

where $R^{-1}(Q_i)$ denotes the inverse function of $Q_{i}^v(r)$ as defines in Proposition 3.

Fig. 3

Equilibrium bid schedules in variable rate tenders

As in the case of fixed rate tenders we observe bid shading by banks, which can easily be verified by comparing the equilibrium in Proposition 4 with the marginal prices in Lemma 4.

Even though the bid schedules in the two considered tender mechanisms are very different, we can show that the allocation the central bank achieves can be identical in both cases, i.e. each bank obtains the same amount and the same interest rate.

Proposition 4

An equilibrium exists with $Q_i^f=Q_i^v$ and $r^{CB}_f=r^{CB}_v$.

From a central bank perspective, the two tender formats generate the same revenue. Similar results have been found in single unit auctions with risk neutral participants and a private value framework such as in Vickrey (1961), Holt Jr (1980), or Harris and Raviv (1981). For multi-unit auctions, as in our case, such a result is not generally valid. Which auction mechanism gives the higher revenue can be ambiguous and is also quite sensitive to assumptions about the auction as shown in Ausubel et al. (2014). In our model, using the assumption that information of other banks’ reservation prices is known leads to not only a tractable equilibrium but also the revenue equivalence of the two auction mechanisms.