Bank Competition and Financial Stability:
A General Equilibrium Exposition
Gianni De Nicolò
International Monetary Fund and CESifo
Marcella Lucchetta
University of Venice, Department of Economics
The views expressed in this paper are those of the authors and do not necessarily
represent those of the IMF.
Motivation


The issue of whether bank competition should be
restrained has a long history in the bank regulatory
debate, and has resurfaced in the aftermath of the
recent financial crisis
The existing theoretical banking literature does
not offer much guidance to this debate, since:
a) it is based on partial equilibrium
b) focuses on the relationship between banks’ risk of
failure and competitive conditions
The inconclusive results of the partial
equilibrium literature (1)



When banks are modeled as limited liability firms
raising funds from insured depositors, choose the risk
of their investment, and this choice is unobservable
(moral hazard), then the standard risk-shifting argument
applies:
More competition for funds implies a higher risk of
bank failure
Many contributions: Keeley (1990), Hellmann Murdock
and Stiglitz, (2000), etc….
The inconclusive results of the partial
equilibrium literature (2)

When banks are modeled a la Cournot as limited
liability firms raising funds from insured depositors and
lending to entrepreneurs under moral hazard, then

More competition (an increase in the number of banks)
implies a LOWER risk of bank failure if project risks are
perfectly correlated (Boyd and De Nicolò, 2005), or a Ushaped relationship between the number of banks and bank
risk of failure (Martinez-Meira and Repullo, 2010) if project
risks are not perfectly correlated
General equilibrium




Restraining bank competition seems at odds with the
implications of some general equilibrium models:
Allen and Gale (2004): perfect competition is Pareto optimal
under complete markets, and constrained Pareto optimal under
incomplete markets, with “financial instability” as a necessary
condition for optimality
Boyd, De Nicolò and Smith (2004): analogous results in a
general equilibrium monetary economy with aggregate liquidity
risk under low inflation.
Yet, these papers do not model bank risk choices under moral
hazard, which we introduce in general equilibrium.
The key unanswered normative question is:



Is there a trade-off between bank competition and
financial stability?
Is a lower level of risk of bank failure necessarily
desirable in a welfare sense?
In this paper we address the following question:
What is the welfare-maximizing level of competition
and the associated optimal level of bank risk of
failure?
Key features of our model



We introduce all features of partial equilibrium models
that find that competition increases bank’s risk of
failure
Agents “occupational” choices between being bankers
or depositors determine endogenously the allocation of
resources to productive activity and bank
intermediation
We consider no deposit insurance and deposit
insurance : How the presence of deposit insurance
affects the welfare ranking of competitive conditions?
Key Result
Perfect bank competition is
constrained Pareto optimal.



This result holds without and with deposit insurance
It holds even though competitive bank’s risk of failure is higher
than banks enjoying monopoly rents
It holds under any social cost function associated with costs of
banks’ failures not internalized by banks that are consistent with
essential bank intermediation.
There is no trade-off between bank
competition and financial stability
WHY?
The key resource re-allocation mechanism at work:
The general equilibrium effect of bank competition



As bank competition for funds increases, the relative
return of intermediated investment (deposits) relative to
shares of bank ownership increases.
This causes a shift of resources from investment in
costly bank intermediation to investment in productive
assets intermediated by banks.
The resulting increase in expected output (net of
monitoring and production costs) is large enough to
offset any reduction in the expected return on
investment due to the higher risk of failure of banks
operating under more intense competition.
Plan







The model
Equilibrium
Definition of competition
Deposit insurance
Welfare
Welfare implications
Conclusion
The Model:
Time, Endowments and Preferences





There are 2 dates: 0 and 1
A continuum of risk neutral agents on [0, A]
Every agent has an endowment of 1 of date 0 goods
All agents have access to a risk-free technology which
yields  per unit invested.
At date 0 agents decide either to become bankers or
depositors.
Banks (1)




If an agent chooses to become a banker, she forgoes
her initial endowment in exchange of the ability to form
coalitions, called banks, which operate a risky project
The project is indexed by the probability of success P [P,1]
An investment z in a risky project yields Xz with
probability P, and 0 otherwise
Banks choose P and operating capacity z (or demand
for funds) at an effort cost.
Banks (2)

Bank effort cost function is given by
1 2
C(P) 
P z
2

The transformation of effort into productive
capacity is simply a standard production
technology. The effort cost of setting up
operating capacity z is
1 2
c(z) 
2
z
Competition





Agents who have chosen to be bankers can move at
no cost to one of two unconnected locations, labeled
M and C
Each bank in M acts as a monopolist and in C as a
perfect competitive bank
there is free entry in the monopolistic and competitive
banking sectors
project risks are independent across locations, but
perfectly correlated within locations. Denote with
and P the risk choices P
M
C
If an agent chooses to be a depositor, he will move to
location C with probability  (switching costs)
Deposit insurance



Deposit insurance (DI) is pre-funded by taxation of
initial resources A
The tax revenues are invested in the safe technology
that yields 
Let  denote the tax rate. The total “end-of-period”
assets of the deposit insurance fund (DIF) are equal to
 A

Denote with ZC and Z M total investment (deposits)
and g (0,1]the guarantee per unit of deposits
Contracts and sequence of
decisions



Depositors finance the bank with simple debt
contracts that pay a fixed amount R per unit invested
Moral hazard is introduced by assuming that bank
choices of P are not observable by depositors
Denote with x the fraction of bankers in C, with AB
the measure of bankers, with ni the number of banks,
with z i bank size (capacity), and with Ri the deposit
rates, for i {C, M}
Sequence of decisions and
determined variables
Ti me Age n tsÕ se qu e n ce
f deoci si on s
t=0
t=1
If g (0,1] , the deposit insurance
fund (DIF) is established by taxing
initial resources
Agents choose t o become bankers or
deposit ors
Bankers choose t o locate in M or in C
Deposit orsÕlocate in M or C
according t o their location draw
The number of banks and debt
equilibrium are determined
Banks choose capacity (fund demand)
Debt contract terms between the bank
and deposit ors are determined.
Banks choose risk.
De termi n e dvari able s
x(1 x) : fraction of bankers in C (M)
AB : measure of bankers
A  AB : measure of deposit ors
 : fraction of deposit ors in C
nC ,n M : number of banks in C and M
ZC , Z M : t otal supply of funds (deposits) in
the competitive and monopolistic sect ors
zC , z M
RC , RM : deposit rates in the competitive and
monopolistic sect ors
ZC , Z M : t otal investment in the competitive
P rojectsÕoutput is realized and agentsÕ and monopolistic sect ors
consumption follows.
PC , PM : risk choices in the competitive and
The DIF pays out deposit ors (if
monopolistic sect ors
necessary) and distributes remaining
funds in necessary
Bank Problems



We solve backward, starting with the competitive and
monopolistic bank problems
Competitive banks choose P to maximize:
1 2
1 2 (1)
PC ( X  RC )zC 
PC zC 
zC
2
2
With solution:
PC   ( X  RC )
*
(2)
Competitive banks


Bertrand competition implies that RC
maximizes depositors’ expected return
PC* RC  (1 PC* )g   ( X  RC )(RC  g)  g
(3)
Then:
Xg
RC 
2
*
PC *   (
Xg
)
2
2
(
X

g)
1 2
C
 (zC )  
zC 
zC
8
2
Monopolistic banks
The representative monopolistic bank chooses
(PM , RM ) to maximize expected profits:
1
1
  (P ( X  R ) 
P )z 
z
(9)
2
2
 subject to the depositors’ participation
constraint
PM * RM  (1 PM * )g  
(10)

M
2
M
M
M
2
M
M
Monopolistic banks

where
PM * argmaxM
is given by:
PM*   ( X  RM )
(11)
And
2
1
1
X

g

X

4



g(g

4

 2X )
*
RM 
2
(14)
Monopolistic banks

The optimal risk choice of the monopolistic
bank is thus:
X  g  X  4   g(g  2X  4 )
P  ( X  R )  
(15)
2
Using (14) and (15), the expected profits of the
monopolistic bank are:
( X  g  X  4   g(g  2 X  4 ))
1
 
z 
z
(16)
8
2
2
*
M

1
1
*
M
2
1
1
2
M
2
M
M
Comparing bank optimal choices
Lemma 1 For all g [0,1], PC*  PM *
Lemma 2 The risk of failure of the competitivebank increasesm onotonicallywith deposi
insurance coverage.
Lemma 3 The risk of failure of the monopolisti c bank declines monotonically with deposit
insurance cov erage.
Equilibrium (1)
Equality between the demand for funds of all banks in each sector to the supply of
funds:
ni zi  Zi
for
i {C, M}
(20i)
Free entry implies that the returns of shares of bank ownership in the two sect ors are
equalized:
nCC
nM  M

xAB
(1 x) AB
(21)
Equalization of the return of shares of bank ownership with the expected return of
deposits, we obtain the fraction of agents who decide t o become bankers:
nC  C
xAB
  ( PC RC  (1 PC )g)  (1  )( PM RM  (1 PM )g) 
 PC RC  (1  )  g[ (1 PC )  (1   )(1  PM ))  r( , g)
(22)
Equilibrium (2)
Equilibrium conditions on the supply of funds in the two sect ors:
ZC   ( A(1  )  AB )
(23)
Z M  (1   )( A(1  )  AB ) (24)
Finally, the tax rate charged to set up the deposit insurance fund (DIF):
 A  g(ZC  Z M )   
Lemma 4
For all g [0,1],
Z
0

gZ
 A  gZ
 1  
A
A
(25)
Welfare
The welfare function can be written as:
Y( , g)  [(PC X 
1 2 1
1 2 1
PC   C )  (PM X 
PM   M )(1  )  g]Z( , g)
2
2
2
2
(32)
Proposition1
For all g [0,1] ,
Y
 0 : perfect com petition(   1) is (constrained)Pareto optim al

Social costs of bank failures
Consider a welfare function augmented with social costs:
W ( , g)  Y( , g)  SC( , g)
(35)
Where:
W ( , g)  Y ( , g)  SC( , g)   A for all  [0,1] and g [0,1]
(36)
A social cost function that satisfies (35) is called adm issible.
Proposi ti on2
For any adm issiblesocial cost functionthat is increasing and convexin investm ent
(deposits), for all g [0,1] , and if bank interm ediationis essential, then
com petition 
(  1) is (constrained)Pareto optim al.
W
 0 : perfect

Conclusion

General equilibrium modeling of
intermediation appear an essential tools to
throw light on the desirable level of
systemic risk in the economy, and how it
could be attained