Bank Competition and Degree of Liquidity Risk
Taweewan Sidthidet∗and Hassan Benchekroun†
Department of Economics, McGill University, Montreal, Canada
Abstract
This paper investigates how the uncertainty in withdrawals from depositors, which creates liquidity shortages (risk), affects bank behavior in pre- and post-merger settings. We
use a model of horizontal mergers in banking industry, based on an inventory-theoretic
approach, for our study. Our paper differs from previous literature in that it explicitly
incorporates the degree of uncertainty in deposit withdrawals in the modeling framework.
Specifically, we demonstrate how the equilibrium decision variables - loan rate and (cash)
reserve holdings - as well as profits for pre- and post-merger banks behave with respect
to this degree of uncertainty. In this context, the behavior of reserve holdings is rather
surprising. As regards the effects of mergers on the values of decision variables/profits,
we show that, in absence of any associated cost benefits, mergers increase the loan rate
charged to the customers and profits of all banks involved; however, it can either increase
or decrease the reserve holdings. We also comment on how the results change if mergers
indeed provide cost benefits to the merged banks.
Keywords: Liquidity risk; Reserve management; Loan market competition; Bank mergers
JEL codes: D43, G21, G28, L13
∗
[email protected]
†
[email protected]
1
1
Introduction and Related Literature
Banks are exposed to liquidity risk because it is an integral part of their daily activities. Liquidity of a bank is its ability to meet obligations when the short-term liabilities, such as deposit
withdrawals, come due, without incurring unacceptable losses (Saunders et al 2006). Liquidity
risk, especially on the liability side, principally arises from unexpected withdrawals by depositors.1 Under normal situation and with appropriate planning, the deposit withdrawals do not
create significant liquidity problem if banks hold large amounts of cash reserves as liquid assets.
Unfortunately, liquidity management is not an easy task; holding too much liquid assets penalizes banks’ earnings because reserves provide no (or minimal) returns. On the other hand, banks
that hold too little liquid assets face increased risks of liquidity shortages. In reality, banks know
that normally only a small proportion of its depositors will be withdrawing on any given day.
However, major liquidity problems can occur if deposit withdrawals are abnormally large and
unexpected. Such sudden and unexpected surges in deposit withdrawals risk triggering a bank
run that could ultimately force a bank into insolvency (Saunders et al 2006).
Liquidity management becomes even more difficult in a competitive environment since competition significantly affects a bank’s decision-making. Over the last decade, the structure of banking
industry has noticeably changed due to merger activities. Mergers in the banking industry have
been reviewed in details in various official reports such as those from European Central Bank
(2000), Group of Ten (2001) and OECD (2000) as well as in academic papers (e.g., Berger et
al 1999; Hanweck and Shull 1999; Cornett et al 2006; DeYoung et al 2009). OECD (2000) suggests that many OECD countries have witnessed a substantial increase in the frequency of bank
mergers in recent times.
There has been previous academic research regarding the effects of liquidity risk on bank behav1
Liquidity risk can arise from either the liability side or the asset side. The liquidity risk from liability side
occurs when banks face early withdrawals from depositors. On the other hand, the liquidity risk from asset side
occurs from supplying off-balance-sheet loan commitments (see Saunders et al 2006, chapter 13, for details). In
this paper, we focus only on the liquidity risk from the liability side.
2
ior. Prisman et al (1986) incorporate liquidity risk in the Monti-Klein model2 of a monopolist
bank by assuming some randomness in the amount of loans. On the other hand, Freixas and
Rochet (2008) introduce liquidity risk based on the Monti-Klein model by assuming some randomness in the amount of deposits. The main contribution of the above two studies is the insight
that introduction of liquidity risk in the Monti-Klein model results in the loss of separability between optimal loan and deposit rates. That is, the lending and deposit rates are then linked
together through a refinancing cost which is affected by the probability of a liquidity shortage.
This refinancing cost depends on the amount of reserves held by bank (reserves = amount of
deposits collected - amount of loans provided). Analysis by Kashyap et al (2002) also supports
this idea that there is a synergy between deposit-taking and lending which makes banks different
from the other financial institutions.
As far as the effects of horizontal mergers are concerned, there is a large literature related to this
(for general industry) starting with the seminal work by Salant et al (1983). They show that
mergers in a Cournot game are not necessarily profitable (in absence of associated cost savings),
unless most of the firms in the industry decide to merge. Subsequently, a number of papers argue
that the results in Salant et al (1983) are sensitive to their assumptions. For example, Deneckere
and Davidson (1985) prove that, when firms produce differentiated products, mergers are always
profitable under Bertrand competition even in absence of cost savings. Some other studies have
shown that mergers are profitable under Cournot competition if there is sufficient cost savings
(e.g., Perry and Porter 1985; Farrell and Shapiro 1990). Recently, Leahy (2002) examines the
incentive for bilateral horizontal mergers in a setting that allows for general demand functions,
quantity or price competition and differentiated goods. The paper finds that mergers under
Cournot competition and strategic substitutes are profitable if demand is convex enough or if
commodities are differentiated enough, while mergers under Bertrand competition with strategic
complements are always profitable.
2
Monti-Klein model is one of the earliest models of bank behavior introduced by Klein (1971) and Monti
(1972). This is a prototype model of industrial organization approach to banking industry, which has been since
used as the basic framework by a number of researchers.
3
We are primarily interested in theoretical models related to bank mergers, especially those studying the effects of bank mergers on a bank’s liquidity risk/management. However, this issue has
not been addressed much in the literature. van Cayseele (2004) analyzes the effects of bank
mergers by using a model of localized banking competition. His analysis shows that it might
make sense for banks to merge in order to improve liquidity management, rather than to create
market power. The paper most relevant for us is a recent one by Carletti et al (2007) who
analyze a banking model with the objective of analyzing the effects of bank mergers on a bank’s
reserve and liquidity management decisions. The authors apply an industrial organization modeling approach to banking industry incorporating a differentiated oligopolistic loan market and
where banks hold reserves to protect against liquidity shocks. The main finding of their study
is that the probability of a liquidity shortage can either increase or decrease after the merger,
depending on the cost of refinancing relative to the cost of raising deposits. However, both van
Cayseele (2004) and Carletti et al (2007) do not include the degree of uncertainty in deposit
withdrawals in their analyses.3 As a result, their models cannot explain the effects of the degree
of liquidity risk on a bank’s decision-making behavior, both before and after mergers (we discuss
the difference of our model with that of Carletti et al (2007) in more details §2).
In this paper our main goal is to analyze the effects of the degree of uncertainty in liquidity risk
(i.e., demand for deposit withdrawals) on a bank’s decisions/profits, and how this is affected by
mergers. To do so, we examine a model of horizontal merger for banking industry, where the
objective of both the pre- and post-merger banks is profit maximization and which incorporates
liquidity management decisions and loan market competition. We use an inventory-theoretic
approach to describe a bank’s incentive to hold reserves and protect against liquidity shortages.
The main strategic variables in our price setting model are how much to charge customers
for loans and how much (cash) reserves to hold. That is, banks, facing uncertainty in deposit
withdrawals, must optimally allocate their deposits into loans and reserves so as to maximize their
profits. We show that, for both pre- and post-merger banks, the equilibrium loan rate increases
in the degree of uncertainty while profits decrease. However, surprisingly, the equilibrium reserve
3
They only model scenarios with and without uncertainty, but do not include the degree of uncertainty.
4
amount is either decreasing or unimodal (first increasing and then decreasing) with respect to the
degree of uncertainty. In absence of any associated cost benefits, mergers always result in higher
equilibrium profits and loan rates; while reserve amount increases for non-merged banks, rather
counter-intuitively, mergers result in merged banks keeping lower amount of reserves. In general,
the non-merged banks charge lower interest rates, keep higher reserves and makes higher profits
than merged ones. Obviously, some of the effects of mergers change when there are associate
cost benefits - we also comment on them.
The structure of the rest of the paper is as follows. In §2, we present our basic model framework.
We then analyze a model without any uncertainty in deposit withdrawals in §3 (as a base case) so
as to understand the effects of mergers on bank decisions in a deterministic setting. §4 analyzes
the pre- and post-merger models with a random liquidity shock. Lastly, the conclusions of the
study are presented in §5.
2
Model Framework
The basic model framework of our research involves an economy with N (> 3) banks, all of which
are risk-neutral. The primary activities undertaken by these banks are: i) accepting deposits
from depositors, and ii) providing loans to borrowers. We assume that the banks have no capital,
and so they can only use collected deposits to provide loans. The detailed timeline of the events
in our model setting is as follows.4
At time T = 0, bank i (i ∈ [1, .., N ]) collects deposits of amount Di from depositors, and then
uses it to provide loans of amount Li to borrowers. The usual term for deposits is 2 periods (i.e.,
the deposits must be withdrawn at time T = 2); however, depositors, if they want, can withdraw
early at time T = 1. So, the bank cannot lend the whole Di ; it keeps Ri (≤ Di ) as reserves
for depositors who want to withdraw at time T = 1. Reserves are liquid assets, and so do not
earn any interests. But if banks do not have enough reserves to satisfy early withdrawals (i.e., if
4
Our basic framework follows Carletti et al (2007); however, the way we model loan demand and randomness
of liquidity shock are quite different. We explain that issue later on in this section.
5
they face liquidity shortages), they need to borrow the shortfall by paying an interbank market
rate of rI . On the other hand, the depositors who withdraw on due date (i.e., T = 2) are paid
an interest on their deposits at the rate of rD . Following Carletti et al (2007), we assume that
rI > rD .5 Based on above, we can then write the balance sheet for each bank i as follows:
(1)
L i + Ri = Di ,
where Li and Ri appear on the asset side of the balance sheet for the bank, while Di is a liability.
Note that the above model is somewhat similar to inventory management models in operations
management literature (Silver et al 1998). In those models, retailers hold inventory to deal with
uncertainty in customer demand, and if the amount of inventory is not enough they need to pay
a penalty for unsatisfied demands. In our context, inventory is equivalent to reserves, customer
demand is demand for loan and the penalty cost is the excess interest rate paid by the banks to
borrow money in case of liquidity shortages.
Obviously, banks accrue their revenues from giving out loans. We assume that the loans provided
by each bank are somewhat differentiated, and the banks compete among themselves in terms of
loan rates in order to attract borrowers. We assume that if bank i offers a loan rate of riL , then
it will face the following linear demand Li for loans:
Li = l − riL + γ
N
X
rjL
(2)
j6=i,j=1
In the above direct demand equation, Li is the demand for bank i’s loans, riL is the loan rate
set by bank i, and rjL is the loan rate set by bank j. The parameter γ > 0 expresses the degree
of product (loans) differentiation, ranging from zero, when the goods are independent, to one
when the goods are perfect substitutes. That is, the lower (higher) the value of γ, the more
(less) differentiated are the loans offered by banks. When γ approaches one, we are closed to a
homogeneous market (see Singh and Vives 1984). Note that
5
PN
i=1
Li = N l−(1−(N −1)γ)
PN
L
i=1 ri .
This assumption is realistic. LIBOR (London Inter-Bank Offered Rate) is often used as a generic term for
all interbank rates (rI ). The deposit rate (rD ) is the cost to pay for demand deposits which have a high degree
of withdrawal risk. In practice, banks use deposits as a low-cost source of funds, and so the interbank rate is
(usually) significantly higher than the deposit rate (see Saunders et al 2006).
6
Throughout the paper we assume that 1 − (N − 1)γ > 0, which implies that any increase in loan
rate by bank i results in loss of total loan demand for the market. To be more specific, when
bank i increases its loan rate by one unit, it loses 1 unit of loan demand. Out of this 1 unit, γ
amount of loan demand goes to each of its (N − 1) competitors, and 1 − (N − 1)γ loan demand
exits market. In contrast to our loan demand, Carletti et al (2007) assume:
Li = l − γriL +
So, in their setting
PN
i=1
N
γ X
rL .
N j=1 j
(3)
Li = N l. That means any increase in loan rate by bank i does not result
in any loss of total loan demand for the market. When bank i increases its loan rate by an unit
amount, it loses γ(1 −
1
)
N
amount of loan demand. This is divided equally among its (N − 1)
competitors, and no loan demand exits the market. As we will show later on, this difference
in the loan demand function has significant effect in terms of how liquidity uncertainty affects
equilibrium loan amounts and profits.6
As indicated above, while deciding on how much of the deposits to give out as loans (i.e., what
should be the value of riL ), bank i also needs to consider how much to keep as reserves in order to
cope with early withdrawals. Before going further, it is important to model the liquidity shock
that bank i faces due to early withdrawals at time T = 1. We assume that a random fraction
δ of initial depositors Di decides to withdraw at T = 1, and represent this random amount of
early withdrawals by:
(4)
xi = δDi .
Another distinguishing feature of our framework compared that of Carletti et al (2007) is the
way we model δ. Specifically, Carletti et al (2007) assume δ to be uniformly distributed between
0 and 1, i.e., δ ∼ U (0, 1). In contrast, we assume δ ∼ U [(δ̄ − ²), (δ̄ + ²)], where δ̄ represents the
fraction of deposits that the bank expects to be withdrawn early (i.e., expected liquidity shock
= δ̄Di ), and ² is a measure of the uncertainty that the bank faces in terms of liquidity shock.
6
Our model approaches that of Carletti et al. as γ approaches
product market compared to them.
7
1
N −1 .
So, we model a more differentiated
So, in this paper, f (δ) =
1
,
2²
E(δ) = δ̄ and V ar(δ) =
²2 7
.
3
Since δ is a fraction, so δ̄ < 1 and
² ∈ (0, M in(1 − δ̄, δ̄)). The above difference in model setting allows us to explicitly analyze the
role that the liquidity shock uncertainty plays in shaping bank decisions/profits by performing
comparative statics with respect to ².
Note that based on above we can also calculate the following measures:
The probability of a liquidity shortage
φ = P rob(xi > Ri ) =
Z (δ̄+²)Di
Ri
f (xi )dxi
(5)
where Ri > (δ̄ − ²)Di , and the expected size of liquidity shortage
ω=
Z (δ̄+²)Di
Ri
(xi − Ri )f (xi )dxi
(6)
where Ri > (δ − ²)Di .
Before proceeding to the analysis of the random deposit withdrawal model, it is worthwhile to
analyze the case where the banks exactly know the amount of early deposit withdrawals.
3
Deterministic Liquidity Shock Models
In this section, we assume that each bank (both pre- and post-merger) knows that the amount
of early deposit withdrawals at time T = 1 is xi = δ̄Di .
3.1
Pre-merger Model
In this case there is an oligopolistic competition between N (> 3) banks. Since there is no uncertainty about how much demand will be there for early withdrawals (i.e., demand for liquidity),
banks know exactly how much reserves to hold: Ri = δ̄Di =
from using Li +Ri = Di ). This implies Di =
7
For Carletti et al (2007) f (δ) = 1, E(δ) =
1
2
1
L.
(1−δ̄) i
(the last equality follows
Obviously, banks in this case do not need to
and V ar(δ) =
out to be a special case of our model when δ̄ = 0.5 and ² = 0.5
8
δ̄
L
(1−δ̄) i
1
12 .
Carletti et al shock uncertainty indeed turns
borrow reserves from interbank market (and so there is no need to pay the interest rate rI ). The
only decision for each bank then is to set its loan rate riL (simultaneously) that would maximize
its profit (keeping in mind the loan rates of other banks):
πi = (riL − c)Li − rD Di (1 − δ̄),
(7)
where c is the unit operating cost associated with each unit of loaned amount and Li is as given
in (2). The equilibrium value of riL can then be uniquely determined, based on which we can
then also decide on the unique equilibrium values of loan amounts, reserve amounts and profits
(details are shown in Appendix).
3.2
Post-merger Model
In this section, we assume 2 banks out of N merge, and without loss of generality we denote the
merging banks as 1 and 2. As far as the post-merger case is concerned, we need to analyze two
separate cases - one for the insider banks (i.e., the ones who merge) and other for the outsider
ones (those who do not merge). We denote the decision variables/profits for the insider banks
by the subscript m, and the outsider ones by the subscript c (pre-merger ones have subscript i).
Insider Profit: The profits for an insider bank is given by:
πm = (r1L − βc)L1 + (r2L − βc)L2 − rD Dm (1 − δ̄),
(8)
where β is a measure of the gains in efficiency in terms of the operating cost due to merger
(lower the value of β, more is the gains in efficiency). Note that, in this case, Rm = δ̄Dm ,
Dm =
1
L ,
(1−δ̄) m
and Lm = L1 + L2 .
Outsider Profit: The profits for an outsider bank is given by:
πc = (rcL − c)Lc − rD Dc (1 − δ̄).
For the outsider banks, Rc = δ̄Dc and Dc =
(9)
1
L.
(1−δ̄) c
Note that both insider and outsider banks need to simultaneously decide on their loan rates
and Li is as given in (2). Based on the above profit expressions, we can then derive the unique
9
post-merger equilibrium decisions and profits for insider and outsider banks (see Appendix for
details).
Suppose that there are no cost-efficiency gains due to merger (i.e., β = 1). Comparing the
equilibrium pre-merger and post-merger values, we can then show that8 :
Proposition 1 When there is no randomness in liquidity shock, mergers result in (compared to
the pre-merger scenario):
1. Higher loan rates for both insider and outsider banks.
2. Lower reserves for insider but higher for outsider banks.
3. Lower total loan amounts for insider but higher for outsider banks.
4. Lower total deposit amounts for insider but higher for outsider banks.
5. Higher profits for both insider and outsider banks.
When β < 1 (i.e., the merged banks gain cost efficiency), the comparisons are more involved
and some of the results of Proposition 1 might not be true. For example, it might be the case
that the post-merger profits of outsider banks are less than those of pre-merger ones (if β is
sufficiently large). We discuss this issue in more details in §4.
4
Random Liquidity Shock Model
We now focus our attention on the case when there is a random liquidity shock. That is, the
amount of withdrawals at time T = 1 is given by xi = δDi , where δ ∼ [(δ̄ − ²), (δ̄ + ²)].
4.1
Pre-merger Model
In this case, each bank i simultaneously needs to make two decisions at time T = 0: i) what
loan rate riL to charge to the borrowers, and ii) how much liquid reserves Ri to hold for early
8
For β = 1, the symmetric post-merger equilibrium decisions and profits are shown in Appendix.
10
withdrawals, in order to maximize the following total expected profit given by:
πi = (riL − c)Li −
Z (δ̄+²)Di
Ri
rI (xi − Ri )f (xi )dxi − rD Di (1 − E(δ)),
(10)
where Li is as given in (2).
The first order conditions (FOCs) with respect to the decision variables (riL and Ri ) are as follows
∂πi
∂Li
= Li + (riL − c) L
L
∂ri
∂ri
I
r ((δ̄ + ²)(Li + Ri ))2 − Ri2
∂Li
−[
+ rD (1 − δ̄)] L = 0
2
2²
2(Li + Ri )
∂ri
∂πi
rI (δ̄ + ²)
Ri (2Li + Ri )
=
[(
− 1)(δ̄ + ²) +
] + rD (1 − δ̄) = 0
∂Ri
2²
2
2(Li + Ri )2
(11)
(12)
for i=1,...,N.
Analyzing the above FOCs we can show that:
Proposition 2 The symmetric pre-merger equilibrium decisions and profits for bank i in the
random liquidity shock model are given by:
1. riL =
l+c
B+1
+
q
1 rI
[
B+1 2²
A
rI
− (1 − (δ̄ + ²))],
where A = (1 − (δ̄ + ²))2 rI + 4(1 − δ̄)²rD and B = (1 − γ(N − 1)).
2. Li =
l
B+1
q
3. Ri = (
−
rI
A
B
c
B+1
−
q
B rI
[
B+1 2²
A
rI
− (1 − (δ̄ + ²))].
− 1)Li .
4. Di = Li + Ri .
5. Substitution of the above four equilibrium values in πi = (riL − c)Li −
²) +
Ri2
2Di
rI Di
[( 2 (δ̄
2²
+ ²) − Ri )(δ̄ +
] − rD Di (1 − δ̄) yields the equilibrium profit.
Remark: Note that the limiting values of the above equilibrium loan rates and reserves in the
random liquidity shock model coincide with those in the deterministic model as ² tends to 0 (see
Appendix for proof).
11
Next we focus on understanding the effects of the randomness of liquidity shock on the decision
variable values. For this we performed comparative statics of the equilibrium decision variables
with respect to ² ∈ (0, δ̄). For expositional convenience, we suppose that δ̄ = 0.5. Consequently,
² ∈ (0, 0.5) (recall that ² ∈ (0, M in(1 − δ̄, δ̄))).9 The main findings of the comparative statics
analysis can be summarized in the following proposition.
Proposition 3 The following are true:
• The equilibrium loan rate to be charged to the borrowers increases in ².
• The equilibrium reserve amount for early withdrawal behaves as follows in terms of ².
I
D
– if rI > 2rD , then the equilibrium reserve amount is increasing in ² for ² ∈ [0, (1−δ̄)(rrI −2r ) ]
I
D
and decreasing in ² thereafter (i.e., for ² ∈ ( (1−δ̄)(rrI −2r ) , 0.5]).
– if rD < rI < 2rD , then the equilibrium reserve amount is decreasing in ².
The above proposition clearly shows the behavior of the decision variables with respect to ², one
of the main objectives of this paper. The underlying reason as to why equilibrium loan rate
increases in ² (i.e., withdrawal risk) is as follows. As ² increases, the bank wants to reduce the
amount of loans given out so that it can keep more of the deposits for reserves. Higher loan rates
enable the bank to do so by reducing the amount of loans. Note that the equilibrium amount of
loans is also decreasing in ². This makes sense since as the environments become riskier, banks
become more wary of giving out loans.
The effects of ² on the equilibrium reserve amount against early withdrawals is quite counterintuitive. We see that this effect depends on the relative values of two costs: deposit rate (rD )
and interbank rate (rI ). When the interbank rate is relatively high (i.e., rI > 2rD ), the reserve
holdings are unimodal in ² - increasing for low values and decreasing for higher ones (refer to the
figures below). While the usual perception is that the amount to hold in reserves should increase
9
The analysis can easily be extended to a general δ̄ < 1, although the expressions are then more cumbersome.
The details are available from the authors on request.
12
with the randomness in shock, clearly, this is not necessarily true. Rather, for high values of ²,
any increase in randomness will result in the bank holding less reserves. The underlying reason
is that the bank has another lever to play with in this case - the loan rate. When ² is low, the
bank increases its reserves with increase in randomness of shock in order to make sure that it
does not have to pay the penalty rI . However, when ² becomes large, the bank feels that it
needs to keep ”too much” of its deposits in reserves, which takes away from the revenue it can
earn from giving those deposits as loans. In that case, it decides to sacrifice safety (in terms of
liquidity) so as to gain higher revenue by making more amount of deposits available for loans.
In fact, if the interbank rate is not too high relative to the deposit rate (rD < rI < 2rD ), the
banks actually always reduce their reserve amount as ² increases. Note that low interbank rates
imply that the shortage penalty is relatively low. In that case, it makes more sense for banks to
pay interbank rate cost if there is a liquidity shortage. This allows them to reduce their reserve
holdings, and use deposits for providing loans.
Since the amount of deposits is driven by the sum of loans and reserves, it can also be nonmonotone. Lastly, as regards profits, it is (concave) decreasing in ², i.e., more randomness
results in profits penalty for the bank.10
4.2
Post-merger Model
Like in the deterministic case, we again assume that only 2 out of N banks merge. Moreover,
we also assume that the merging banks face the same shock (δ) like the pre-merger ones (i.e.,
the shocks are perfectly correlated) that affects their total deposits (Dm ). The merging banks’
expected demand for liquidity is then as follows
xm = δDm ,
(13)
where Dm = D1 + D2 . As explained before, δ ∼ U [δ̄ − ², δ̄ + ²].
10
We can show that if Carletti et al’s (2007) loan demand model is used in our setup, both equilibrium profit
and loan amounts will be independent of the value of ². This does not make sense from a real-life and intuitive
viewpoint.
13
Based on above, the combined demand for loan of the merging banks is
Lm = L1 + L2 = [l − r1L + γ(r2L +
N
X
rcL )] + [l − r2L + γ(r1L +
c=3
N
X
rcL )],
(14)
c=3
and the balance sheet identity to be
Dm = L1 + L2 + Rm
(15)
The merging banks use their total reserves (Rm = R1 + R2 ) to pay depositors who decide to
withdraw early at time T = 1.
The demand for loan of an outsider bank is as follows:
Lc = [l − rcL + γ(r1L + r2L +
N
X
rc0 )].
(16)
c0 =4,c0 6=c
In the random liquidity shock case, the profits of merging banks and those of the outsiders are
as follows
πm = π1 + π2 = (r1L − βc)L1 + (r2L − βc)L2 −
Z (δ̄+²)Dm
Rm
rI (xm − Rm )f (xm )dxm − rD Dm (1 − E(δ));
(17)
πc =
(rcL
− c)Lc −
Z (δ̄+²)Dc
Rc
rI (xc − Rc )f (xc )dxc − rD Dc (1 − E(δ)).
(18)
The merging banks have two decisions to make: i) what loan rates - r1L and r2L - to charge to the
borrowers, and ii) how much reserves (Rm ) to keep for early withdrawals, in order to maximize
their combined profits. The FOCs for insider banks then are as follows:
∂L1
∂L2
∂πm
= Lh + (r1L − βc) L + (r2L − βc) L
L
∂rh
∂rh
∂rh
2
Rm
rI (N − 2) (δ̄ + ²)2
]
γ[
−
+
L − r L ))2
2² N
1
(Rm + 2l − 2γ(N − 2)(rm
c
(N − 2)
+rD γ(1 − δ̄)
=0
N
(19)
(20)
with h=1, 2, and
rI (δ̄ + ²)
∂πm
=
[(
− 1)(δ̄ + ²)
∂Rm
2²
2
L
− rcL )
Rm N (Rm N + 4N l − 4γ(N − 2)(rm
] + rD (1 − δ̄) = 0
−
L
L
2(Rm N + 2N l − 2γ(N − 2)(rm − rc )
14
(21)
Similarly, outsider banks also have to decide on their profit-maximizing loan rates (rcL ) and
reserves (Rc ) (the insider and outsider banks make their decisions simultaneously). The FOCs
for outsider banks are as follows:
∂πc
∂Lc
= Lc + (rcL − c) L
L
∂rc
∂rc
I
r ((δ̄ + ²)(Lc + Rc ))2 − Rc2
∂Lc
−[
+ rD (1 − δ̄)] L = 0
2
2²
2(Lc + Rc )
∂rc
rI (δ̄ + ²)
∂πc
Rc (2Lc + Rc )
=
[(
− 1)(δ̄ + ²) +
] + rD (1 − δ̄) = 0
2
∂Rc
2²
2
2(Lc + Rc )
(22)
(23)
for c = 3, ..., N.
As evident from the above expressions, the post-merger scenario for the random shock case is
quite involved. It is difficult to analytically establish the uniqueness of the decision variables as
well as to compare them to the pre-merger values or to understand the behavior of the decision
variables with respect to the degree of uncertainty (i.e., ²). Consequently, we resort to numerical
experiments for this purpose.
4.3
Numerical Study Results
In this section we use a detailed numerical study to address two issues: i) how do the equilibrium
post-merger decisions of banks behave with respect to the degree of uncertainty in early deposit
withdrawals (i.e., ²) and ii) how do the post-merger equilibrium decision variable/profit values
compare with those of the pre-merger scenario in the random shock environment. In order to
do so we perform a large set of numerical experiments. Due to lack of space we only provide a
subset of our results here (the behavior always remains the same).11 The basic parameter set
for the purpose of this paper, which we keep fixed throughout, is as follows: number of firms
(N ) = 10, cost of providing loans (c) = 0.2, deposit rate (rD ) = 0.01, expected fraction of early
deposit withdrawals (δ) = 0.5, the intercept of the loan demand function (l) = 0.1, and degree
of substitutability among loans offered by banks (γ) = 0.1. We then change ² in the range 0.01
11
Details are available from the authors on request.
15
to 0.5 (step size of 0.01). Moreover, in order to capture different risks of liquidity shortages and
efficiency gains due to merger, we use the following two scenarios for the interbank rate (rI ) and
cost-efficiency gains (β).
Parameter
Explanation
Value
rI
Interbank rate
High rI =0.2
Low rI =0.05
β
Cost-efficiency gains
With gains β=0.8
Without gains β=1
First of all, note that all our numerical experiments resulted in unique equilibrium decision
variable values for both loan rates and reserve holdings in pre- and post-merger settings. From
these we can then determine the other related variables like loan and deposit amounts, as well
as equilibrium profits. We plot the effects of the degree of uncertainty in deposit withdrawals
(²) on the two primary equilibrium decision variable values as well as profits for both scenarios
of the above table in Figures 1-6 (Figures 1 and 2 for equilibrium loan rates, Figures 3 and 4 for
equilibrium reserve amounts and Figures 5 and 6 for equilibrium profits).12 These figures also
show how the pre-merger decision variable values compare to those of post-merger ones (recall
that subscript i stands for a pre-merger bank, m for a post-merger insider bank and c for a postmerger outsider bank). We summarize the value comparisons in Tables 1 and 2. Specifically,
Table 1 shows how the decisions and profits for pre-merger banks compare individually with
decisions and profits for insider and outsider banks, while Table 2 shows the relative ordering of
the decisions and profits for pre-merger, insider and outsider banks.
OBSERVATIONS:
The tables and figures provide us with a number of interesting insights about the values and
behavior of the equilibrium decisions and profits. We discuss them below.
12
In Figures 1, 3 and 5 we change the value of rI (0.05 and 0.2) while keeping β constant at 1. On the other
hand, in Figures 2, 4 and 6 we change the value of β (1 and 0.8) while keeping rI constant at 0.2.
16
Behavior:
Note from the figures that the behavior of the equilibrium decisions for post-merger insider and
outsider banks with respect to ² are the same as that of pre-merger banks. Specifically, as in
Proposition 3, the equilibrium loan rate to be charged to the borrowers increases in ², while
the equilibrium reserve amount for early withdrawal is unimodal with respect to ² - it is first
increasing and then decreasing (since our rI > 2rD ).13 The behavior of the profits in the preand post-merger cases are also similar (decreasing in ²). Note that although we do not show here,
the behavior of equilibrium loans and deposits are also similar in both cases. The underlying
reasons for the above behavior are the same as discussed in §4.1.
As regards the effects of interbank rate (rI ) and cost efficiency due to mergers (β), we note the
following from the figures:
- Since higher interbank rates imply higher penalty cost for the bank in case of liquidity shortage,
this always results in higher amount of equilibrium reserves (so that there is less chance of
shortages), higher equilibrium loan rates (so as to reduce the loan demand and make more
deposits available for reserves) and lower equilibrium profits.
- On the other hand, as the cost efficiency due to mergers increases (i.e., β decreases), the
insider banks can take advantage of cost benefits to charge lower equilibrium loan rates (so
as to increase loan demand) and also keep higher amount of equilibrium reserves (to reduce
liquidity shortages), resulting in higher profits. But for the outsider banks, such a phenomenon
has the opposite effects on equilibrium reserve holdings and profits (although equilibrium loan
rate still decreases). Obviously, the cost-efficiency gains have no effect on the decisions/profits
of a pre-merger bank.
Values:
Our numerical analysis shows that the equilibrium decision variable values as well as the values of
equilibrium loans, deposits and profits are significantly different for pre- and post-merger banks.
Specifically, in presence of a randomness in liquidity shock and when there are no cost-efficiency
13
Indeed if rD < rI < 2rD , then the equilibrium reserve amount for both pre-merger and post-merger scenarios
are decreasing in ².
17
Table 1: Difference between Pre- and Post-Merger Decision Variable Values and Profits
No cost-efficiency gains
Cost-efficiency gains
(β = 1)
(β = 0.8)
Decision Variable
rI = 0.05
rI = 0.2
rI = 0.05
rI = 0.2
L
L
∆rm
= rm
− riL
+
+
−
−
∆rcL = rcL − riL
+
+
−
−
∆Rm = Rm /2 − Ri
−
−
+
+
∆Rc = Rc − Ri
+
+
−
−
∆Πm = Πm /2 − Πi
+
+
+
+
∆Πc = Πc − Πi
+
+
−
−
Table 2: Relative Ordering of Pre- and Post-Merger Decision Variable Values and Profits
No cost-efficiency gains
Cost-efficiency gains
(β = 1)
(β = 0.8)
Decision Variable
rI = 0.05
rI = 0.2
rI = 0.05
rI = 0.2
Loan Rate
L
rm
> rcL > riL
L
rm
> rcL > riL
L
riL > rcL > rm
L
rcL > riL > rm
Reserves
Rc > Ri > Rm
Rc > Ri > Rm
Rm > Ri > Rc
Rm > Ri > Rc
Profits
Πc > Πm > Πi
Πc > Πm > Πi
Πm > Πi > Πc
Πm > Πi > Πc
gains (i.e., β = 1), mergers result in (compared to pre-merger scenario) (refer to Table 1):14
1. Higher equilibrium loan rates for both insider and outsider banks.
2. Lower equilibrium reserve amount for insider banks, but higher amount for outsider banks.
3. Higher equilibrium profits for both insider and outsider banks.
Note that the above results follow those in Proposition 1 when there is no randomness in liquidity
shock. More importantly, when we compare the equilibrium decisions/profits of pre-merger,
insider and outsider banks (Table 2), we note that following interesting insights: i) while both
14
These results are consistent with the literature regarding the effects of mergers in general industry in a
Bertrand framework (e.g., Deneckere and Davidson 1985). That is, mergers are profitable for all firms and
post-merger firms charge higher prices.
18
insider and outsider banks charge higher equilibrium interest rates than pre-merger banks, the
merged (i.e., insider) banks can use their market power to charge even more than the non-merged
(i.e., outsider) ones; ii) when there is randomness in shock, the merged banks take advantage of
risk-pooling and keep less reserves than both pre-merger and non-merged banks; and iii) while
both insider and outsider banks make higher profits than pre-merger ones, the outsider banks
actually make even more profits than the insider ones.
Obviously, when there are cost-efficiency gains the equilibrium values for the insider banks are
affected significantly (the values for the outsider banks are affected slightly, while the pre-merger
ones are not affected). So, some of the above insights might change. For example, the merged
banks would then keep reserves higher than outsider ones, and will charge loan rates even lower
than pre-merger banks (since the merged banks are saving substantially in their operating costs
for low values of β, they can use that to charge low interest rates attracting a lot of loan
demand, and also counterbalance the loss in revenue from giving out loans by keeping high
reserves. However, high reserves enable them to save on payment to depositors who decide to
withdraw early at time T = 1). To capture some profits, the outsiders expand their market share
by lower their loan rates relative to pre-merger values either.
As indicated in §2, we can use the equilibrium decisions to determine two additional measures:
probability of liquidity shortage (φ) and the expected size of liquidity shortages (ω). We numerically investigate the effects of ² and mergers on these two measures. Keeping in mind the space
constraints, we do not go into the details of the analysis or do not present the results. However,
our analysis shows that:
1. Both φ and ω increase in the degree of uncertainty in deposit withdrawals (i.e., ²).
2. The probability of a liquidity shortage for pre-merger, merged and non-merged banks are
identical because we assume that the random fraction δm of early withdrawals for merged banks
are perfectly correlated.
3. The expected size of liquidity shortage of a merged bank is less than that of a non-merged
one (ωm < ωc ). However, if we compare the expected size of liquidity shortages of post-merger
banks to pre-merger ones, the result is ambiguous. Specifically, the size of the shortage can be
19
higher or lower depending on the values of interbank rate (rI ) and cost-efficiency gains (β).
5
Concluding Remarks
In this study, we have analyzed a model of banking industry with the goal of understanding how
the degree of liquidity risk affects decisions and profits for banks in pre- and post-merger settings.
Our basic model framework is based on an inventory-theoretic approach following Carletti et al
(2007). In our setting, the liquidity risk is in the form of early deposit withdrawals, while banks
hold reserves to deal with such withdrawals. Any liquidity shortages are handled by borrowing
at a relatively high interbank rate. The banks need to use their deposits for holding reserves
as well as for giving out loans. The strategic decision variables for banks are their loan rates
and reserve holdings and all the banks need to set these simultaneously. First, we analyze a
deterministic model where the banks know exactly how much reserves to hold; subsequently we
study a model with uncertainty in terms of early deposit withdrawals.
As far as the behavior of the decisions with respect to the degree of uncertainty in deposit
withdrawals (i.e., liquidity risk) is concerned, it is the same for pre- and post-merger (both
insider and outsider) banks. Specifically, we show that the equilibrium loan rate is increasing
in the degree of uncertainty, while equilibrium profit is decreasing. However, banks need to be
careful about their reserve management since the behavior of reserve amount is rather interesting.
It depends crucially on the relative values of interbank and deposit rates. If the interbank rate
is relatively higher, then the reserve amount is unimodal (first increasing and then decreasing)
in the degree of uncertainty, whereas if the interbank rate is not so high, then the equilibrium
reserve amount is decreasing in the degree of uncertainty. As regards the effects of mergers, we
find that, when there are no cost-efficiency gains, mergers result in higher loan rates and profits
for all banks involved, but once again the reserve holdings may increase or decrease. In the postmerger setting, the merged banks hold less reserves and set higher loan rates than non-merged
banks; interestingly, non-merged banks benefit more from mergers than merged ones. However,
when mergers are associated with cost savings for (only) merged banks, they might hold more
reserves and charge less loan rates than non-merged banks and earn higher profits. Mergers not
only affect banks’ reserves and liquidity management decision, they also affect their size and
20
probability of a liquidity shortage. In that context, we show that the size of a liquidity shortage
of a merged bank is lower than that of a non-merged one. Comparison of the sizes of liquidity
shortages of pre- and post-merger banks, however, is ambiguous - it depends on the interbank
rate and degree of cost-efficiency gains. One of the limitations of our study is that we assume
the random fraction of early withdrawals for merging banks to be perfectly correlated; extending
to the case when they are independent or partially correlated would be interesting. In spite of
this limitation, we think that our model makes significant contribution regarding explaining the
effects of bank mergers and degree of liquidity risk on the banking industry.
21
Figure 1: Effects of ² on the Equilibrium Loan Rate (rI = 0.05 and rI = 0.2, β = 1)
0.32
rLm(rI=0.2)
rLi(rI=0.2)
rLc(rI=0.2)
rLi(rI=0.05)
rLi(rI=0.2)
Loan Rate
0.31
rLm(rI=0.2,Beta=1)
rLc(rI=0.2,Beta=1)
rLm(rI=0.05,Beta=1)
0.30
rLc(rI=0.05,Beta=1)
rLm(rI=0.05)
rLc(rI=0.05)
rLi(rI=0.05)
0.29
0.28
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Epsilon
Figure 2: Effects of ² on the Equilibrium Loan Rate (β = 1 and β = 0.8, rI = 0.2)
0.32
rLi(rI=0.2)
rLm(rI=0.2,Beta=1)
Loan Rate
rLc(rI=0.2,Beta=1)
rLm(rI=0.2,Beta=0.8)
rLc(rI=0.2,Beta=0.8)
0.29
0.26
0.00
0.05
0.10
0.15
0.20
0.25
Epsilon
22
0.30
0.35
0.40
0.45
0.50
Figure 3: Effects of ² on the Equilibrium Reserves (rI = 0.05 and rI = 0.2, β = 1)
0.25
Ri(rI=0.2)
Rc(rI=0.2)
Ri(rI=0.2)
Rm/2(rI=0.2)
Ri(rI=0.05)
Reserves
Rm/2(rI=0.2,Beta=1)
0.20
Rc(rI=0.2,Beta=1)
Rm/2(rI=0.05,Beta=1)
Rc(rI=0.05,Beta=1)
0.15
0.10
Rc(rI=0.05)
Ri(rI=0.05)
Rm/2(rI=0.05)
0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Epsilon
Figure 4: Effects of ² on the Equilibrium Reserves (β = 1 and β = 0.8, rI = 0.2)
Ri(rI=0.2)
Rm/2(rI=0.2,Beta=1)
0.25
Rc(rI=0.2,Beta=1)
Reserves
Rm/2(rI=0.2,Beta=0.8)
Rc(rI=0.2,Beta=0.8)
0.15
0.05
0.00
0.05
0.10
0.15
0.20
0.25
Epsilon
23
0.30
0.35
0.40
0.45
0.50
Profits
Figure 5: Effects of ² on the Equilibrium Profits (rI = 0.05 and rI = 0.2, β = 1)
PROFc(rI=0.05)
PROFm/2(rI=0.05)
PROFi(rI=0.2)
PROFi(rI=0.05)
PROFi(rI=0.05)
PROFm/2(rI=0.2,Beta=1)
PROFc(rI=0.2,Beta=1)
0.005
0.00
PROFm/2(rI=0.05,Beta=1)
PROFc(rI=0.2)
PROFc(rI=0.05,Beta=1)
PROFm/2(rI=02)
PROFi(rI=0.2)
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Epsilon
Figure 6: Effects of ² on the Equilibrium Profits (β = 1 and β = 0.8, rI = 0.2)
Profits
PROFi(rI=0.2)
PROFm/2(rI=0.2,Beta=1)
0.007
PROFc(rI=0.2,Beta=1)
PROFm/2(rI=0.2,Beta=0.8)
PROFc(rI=0.2,Beta=0.8)
0.004
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Epsilon
24
0.35
0.40
0.45
0.50
Appendix
Deterministic Liquidity Shock Models
The symmetric pre-merger equilibrium decisions and profits for bank i in the deterministic deposit withdrawal shock model are given by:15
1. riL =
l+(rD +c)
.
2−γ(N −1)
2. Li =
l−(rD +c)(1−γ(N −1))
.
2−γ(N −1)
3. Di =
l−(rD +c)(1−γ(N −1))
.
(1−δ̄)(2−γ(N −1))
4. Ri =
δ̄ l−(rD +c)(1−γ(N −1))
.
2−γ(N −1)
(1−δ̄)
5. πi =
(l−(rD +c)(1−γ(N −1)))2
.
(2−γ(N −1))2
The above expressions can be derived as follows. Using equation (1) with equation (7) and
differentiating, we get the first order condition with respect to riL :
∂πi
∂Li
= Li + (riL − c − rD ) L = 0
L
∂ri
∂ri
(24)
Solving (24) for riL results in a symmetric equilibrium for i = 1, .., N . Substituting the equilibrium
riL in the demand for loan (2), we obtain the equilibrium Li . The equilibrium values of Di and
Ri then follow. Substituting all relevant equilibrium decision variable values in (6), we get the
expression for the equilibrium profits.
The symmetric post-merger equilibrium decisions and profits for the deterministic deposit withdrawal shock model are given by:
L
=
1. rm
rcL =
l+(1−γ)((βc+rD )γ+(c+rD ))
(2−γ(N −1+γ)
2. Lm =
Lc =
l(γ+2)+(1−γ)(2−γ(N −3))βc+(2−γ(1−γ(N −3))rD +(N −2)γc)
2(2−γ(N −1+γ))
.
(1−γ)(2+γ)(l−(1−(N −1)γ)rD )+(1−γ)c(γ(N −2)−β(2−((N −1)(γ+1)γ))
2−γ(N −1+γ)
l−(1−(N −1)γ)rD −(1−γ(N +γ−2)−(1−γ)γβ)c
.
2−γ(N −1+γ)
3. Rm =
δ̄
L
(1−δ̄) m
and Rc =
δ̄
L
(1−δ̄) c
.
4. Dm = Lm + Rm and Dc = Lc + Rc .
15
and
Note that 2 − γ(N − 1) > 0.
25
and
5. πm =
πc =
(1−γ)[(γ+2)(l−(1−(N −1)γ)rD )+c(γ(N −2)−β(2−γ(N −1)(γ+1)))]2
2[2−γ(N −1+γ)]2
[(l−(1−(N −1)γ)rD )+c((1−(N −1)γ)+γ(1−γ)(1−β))
[2−γ(N −1+γ)]2
and
.
The above expressions can be derived as follows. At the post-merger equilibrium, the two-merged
L
banks set r1L =r2L =rm
, and all outsider banks set their equilibrium rates at riL =rcL . Substituting
Dm =
1
L
1−δ m
into equation(8), the first order condition of the merged banks with respect to loan
rate is as follows:
∂L1
∂L2
∂πm
= Lh + (r1L − βc − rD ) L + (r2L − βc − rD ) L = 0
L
∂rh
∂rh
∂rh
1
L
1−δ c
where h = 1, 2. Substituting Dc =
(25)
into equation (9), the first order condition of the
outsider bank with respect to loan rate is as follows:
∂πc
∂Lc
= Lc + (rcL − c − rD ) L = 0
L
∂rc
∂rc
(26)
L
Solving (25) and (26) simultaneously, we get symmetric post-merger equilibrium loan rates rm
and rcL . Substituting these values into proper equations we can get all other relevant equilibrium
values - Rm , Rc , Lm , Lc , Dm , Dc , πm , and πc .
For β = 1, the symmetric post-merger equilibrium decisions and profits in the deterministic
deposit withdrawal shock model are given by (by substituting β = 1 in the above expressions):
L
1. rm
=
(2+γ)l+(2−γ(1−γ(N −3)))(rD +c)
2(2−γ(N −1+γ))
and rcL =
2. Lm =
(2−γ(1+γ))l−(1−γ)(2+γ)(1−γ(N −1))(rD +c)
2−γ(N −1+γ)
3. Rm =
δ̄
L
(1−δ̄) m
and Rc =
l+(1−γ)(1+γ)(rD +c)
.
2−γ(N −1+γ)
and Lc =
l−(1−γ(N −1))(rD +c)
.
2−γ(N −1+γ)
δ̄
L.
(1−δ̄) c
4. Dm = Lm + Rm and Dc = Lc + Rc .
5. πm =
(1−γ)[(2+γ)(l−(1−(N −1)γ)(rD +c)]2
2[2−γ(N −1+γ)]2
and πc =
[l−(1−(N −1)γ)(rD +c)]2
.
[2−γ(N −1+γ)]2
Proof of Proposition 1:
Based on the above pre-merger and post-merger (for β = 1) equilibrium expressions, we can
analyze the difference between them.
Recall that there are three conditions in the model, as follows
26
1) 1 − (N − 1)γ > 0 from an assumption of demand for loan. It follows that (2 − γ(N − 1)) is
also positive and γ is relatively small (0 < γ <
1
).
(N −1)
2) l − (rD + c)(1 − γ(N − 1)) > 0 from they symmetric pre-merger equilibrium of deterministic
model that amounts of equilibrium loans, deposits, reserves and profits must be positive.
3) N > 3.
i) Loan rates
L
L
Insider: ∆rm
= rm
− riL =
γ (2−γ(N −3)) (l−(rD +c)(1−γ(N −1))
2 (2−γ(N −1))
(2−γ(N −1)−γ 2 )
> 0.
(l−(rD +c)(1−γ(N −1))
γ2
(2−γ(N −1))
(2−γ(N −1)−γ 2 )
> 0.
Outsider: ∆rcL = rcL − riL =
ii) Reserves
δ̄ γ(γ(1+γ)(N −1)−2) (l−(rD +c)(1−γ(N −1))
(2−γ(N −1))
(2−γ(N −1)−γ 2 )
(1−δ̄)
Insider: ∆Rm = Rm − 2Ri =
(l−(rD +c)(1−γ(N −1))
γ2
δ̄
(2−γ(N −1)−γ 2 )
(1−δ̄) (2−γ(N −1))
Outsider: ∆Rc = Rc − Ri =
< 0.
> 0.
iii) Loans
γ(γ(1+γ)(N −1)−2) (l−(rD +c)(1−γ(N −1))
(2−γ(N −1))
(2−γ(N −1)−γ 2 )
Insider: ∆Lm = Lm − 2Li =
Outsider: ∆Lc = Lc − Li =
(l−(rD +c)(1−γ(N −1))
γ2
(2−γ(N −1))
(2−γ(N −1)−γ 2 )
< 0.
> 0.
iv) Deposits
Insider: ∆Dm = Dm − 2Di =
Outsider: ∆Dc = Dc − Di =
1 γ(γ(1+γ)(N −1)−2) (l−(rD +c)(1−γ(N −1))
(2−γ(N −1))
(2−γ(N −1)−γ 2 )
(1−δ̄)
(l−(rD +c)(1−γ(N −1))
γ2
1
(2−γ(N −1)−γ 2 )
(1−δ̄) (2−γ(N −1))
< 0.
> 0.
v) Profits
2
D
2
(l−(r +c)(1−γ(N −1))
γ
Insider: ∆πm = πm − 2πi = −Λ 2(2−γ(N
>0
−1))2
(2−γ(N −1)−γ 2 )2
where Λ = (−4 − 4γN + 8γ + 3γ 2 N 2 − 10γ 2 N + 11γ 2 + γ 3 N 2 − 2γ 3 N + γ 3 ) > 0
The shape of Λ is convex with respect to γ, that is
G=
∂2Λ
∂γ 2
= 6N 2 + 6N 2 − 12N γ − 20N + 6γ + 22
27
where G is positive with respect to γ, i.e.,
∂G
∂γ
= 6(N − 1)2 > 0.
Also, G is always positive within the range of 0 < γ <
1
,
(N −1)
i.e.,
limγ→0 G = 6N 2 − 20N + 22 > 0, and
limγ→
1
(N −1)
G = 6N 2 − 14N + 16 > 0.
Now, we can show that Λ is convex and always negative with respect to γ for N > 3, i.e.,
limγ→0 Λ = −4 < 0, and
limγ→
1
(N −1)
−1)(N −2)
Λ = − (5N(N
< 0.
−1)2
It follows that Λ is always negative with respect to γ for 0 < γ <
Outsider: ∆πc = πc − πi =
2(2−γ(N −1)−γ 2 )γ 2 (l−(rD +c)(1−γ(N −1))2
(2−γ(N −1))2
(2−γ(N −1)−γ 2 )2
1
.
(N −1)
> 0.
Proof of Proposition 2:
We first use (12) to solve for the pre-merger equilibrium reserves Ri in terms of Li . Substituting
the equilibrium reserves into (11) and using (2), we get the pre-merger equilibrium loan rate riL .
Substituting the equilibrium loan rate into the demand for loan in (2), gives us the equilibrium
amount of loans, and based on that we can then get the equilibrium reserves values. Substituting
the equilibrium reserves and loan rate into (1), we obtain the equilibrium amount of deposits
and then the pre-merger equilibrium profit.
The limiting values of equilibrium loan rates and reserves
The symmetric pre-merger equilibrium loan rate from Proposition 2 is as follows:
riL∗ =
l+c
(2−γ(N −1))
+
q
1
rI
[
(2−γ(N −1)) 2²
A
rI
− (1 − (δ̄ + ²))],
where A = (1 − (δ̄ + ²))2 rI + 4(1 − δ̄)²rD .
28
To evaluate limit of equilibrium loan rate, L0 H ôpital0 s rule is used.
q
Let Q =
Q1
Q2
=
A
rI
−(1−(δ̄+²))
²
, then riL =
l+c
(2−γ(N −1))
lim Q =
²→0
lim riL∗ =
²→0
∂Q1
∂²
∂Q2
∂²
+
=
rI
Q.
2(2−γ(N −1))
2rD
.
rI
l + rD + c
.
(2 − γ(N − 1))
(27)
The symmetric pre-merger equilibrium reserves from Proposition 2 is as follows:
Ri∗ = (
q
rI
A
q
− 1)Li = (
where riL∗ =
l+c
(2−γ(N −1))
rI
A
+
− 1)(l − riL∗ + γ(N − 1)riL∗ ),
q
1
rI
[
(2−γ(N −1)) 2²
A
rI
− (1 − (δ̄ + ²))].
To evaluate limit of equilibrium reserves, L0 H ôpital0 s rule is used.
s
lim Ri∗
²→0
rI
− 1) lim(l − riL∗ + γ(N − 1)riL∗ )
²→0
²→0
A
δ̄ (l − (rD + c)(1 − γ(N − 1)))
=
2 − γ(N − 1)
(1 − δ̄)
= lim(
(28)
Proof of Proposition 3:
i) Differentiation of the equilibrium loan rate (riL∗ )provided in Proposition 2 with respect to ²
and simplification yields:
(1 − δ̄)
∂riL∗
q Z.
=
∂²
2²2 rAI
√
where A = (1 − (δ̄ + ²))2 rI + 4(1 − δ̄)²rD > 0 and Z = ArI + ²(rI − 2rD ) − (1 − δ̄)rI .
Clearly, the sign of
∂riL∗
∂²
(29)
depends on the sign of Z. We investigate it below.
Note the following:
Z|²=0 = rI (1 − δ̄) −
29
q
(1 − δ̄)2 rI = 0.
(30)
∂Z
(1 − δ)(rI − 2rD )
q
|²=0 = rI − 2rD −
≥ 0.
∂²
(1 − δ̄)2
(31)
∂ 2Z
4rD (1 − δ̄)2 (rI − rD )
q
=
> 0.
∂²2
A AI
(32)
r
So, the values of Z and its first derivative are zero at ² = 0. Moreover, the second derivative is
positive, which implies that the first derivative is increasing. Since the first derivative is zero at
² = 0, so the first derivative is always non-negative. This in turn means that Z itself is increasing
(note that Z = 0 at ² = 0). The above discussion implies that Z is always non-negative. It then
follows that
∂riL
∂²
> 0.
ii) Differentiation of the equilibrium reserve holdings (Ri∗ ) provided in Proposition 2 with respect
to ² and simplification yields:
∂Ri∗
[(1 − (δ̄ + ²))rI − 2rD (1 − δ)]l
=
∂²
A
s
rI
.
A
(33)
where A = (1 − (δ̄ + ²))2 rI + 4(1 − δ̄)²rD > 0.
Now, note the following (recall that δ̄ = 0.5):
For rI > 2rD , the sign of
∂Ri∗
∂²
depends on [(1 − (δ̄ + ²))rI − 2rD (1 − δ̄)]. This term can indeed
be positive or negative; but it is linearly decreasing in terms of ².
∂Ri∗
(rI − 2rD )l
|²=0 = I
> 0.
∂²
r (1 − δ̄)2
(34)
∂Ri∗
rI
|²=δ̄,δ̄=0.5 = −( D )l < 0.
∂²
r
(35)
So, Ri∗ is initially increasing until ² = ²∗ then decreasing in ² at the end. Moreover, solving
[(1 − (δ̄ + ²))rI − 2rD (1 − δ̄)] = 0, we get that
∂Ri∗
∂²
= 0 for ²∗ =
(1−δ̄)(rI −2rD )
.
rI
It then follows that
Ri∗ is increasing for ² ∈ [0, ²∗ ), and decreasing for ² ∈ [²∗ , 0.5].
For rD < rI < 2rD , (1 − (δ̄ + ²))rI − 2rD (1 − δ̄) is negative for ² ∈ [0, 0.5] and so Ri∗ is decreasing
in ².
30
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