LSE Banking Slides 1

7/24/2019 LSE Banking Slides 1

1/62

Banking Theory and Regulation

The Industrial Organization of Banking

Rafael RepulloCEMFI, Madrid, Spain

FM403 Management and Regulation of Risk

2 November 2015


2/62

Introduction

Banks as intermediaries that take deposits and give loans

Banks face

Supply of deposits that depends on the deposit rate

Demand for loans that depend on the loan rate

Focus on determination of equilibrium deposit and loan rates

For different market structures (competition to monopoly)

Effect of having an interbank market

Effect of regulations such as reserve or capital requirements


3/62

Outline

Model of perfect competition

Reserve and capital requirements

Intermediation costs

Model of (local) monopoly banks

Model of Cournot competition

Model of Bertrand competition

Model of monopolistic competition (circular road)


4/62

Part 1

Perfect competition


5/62

Model setup (i)

Large number of banks that compete for deposits and loans

Banks take as given deposit rate rD and loan rate rL

Market supply of depositsD(rD), withD(rD) > 0

Market demand for loansL(rL), withL(rL) < 0

Interbank market where banks can borrow or lend at rate r

Monetary policy rate set by central bank


6/62

Model setup (ii)

Assumptions:

1. Bank loans are riskless

2. Banks have no equity capital

3. There are no intermediation costs


7/62

Banks balance sheet and profits

Balance sheet of an individual bank

l= d+ b

where ldenotes the amount of loans, dthe amount of deposits,

and b the amount of interbank borrowing (lending if b < 0)

Banks profits

= lrLdrDbr

where lrL are the revenues from lending, drD the costs of deposit

funding, and br the costs of interbank borrowing


8/62

Rewriting banks profits

Substituting b = ld into the expression for gives

= lrLdrD (ld)r= l(rLr) + d(rrD)

Banks profits can be decomposed into two terms

Profits from lending: l(rLr)

Profits from deposit taking: d(rrD)


9/62

Competitive equilibrium

Competitive equilibrium characterized byzero-profit conditions

rL = r and rD = r

Why rL = r? Profits from lending: l(rLr)

If rL < r no bank would want to lend If rL > rbanks would like to lend infinitely large amounts

Why rD = r? Profits from deposit taking: d(rrD)

If rD > r no bank would want to take deposits

If rD < rbanks would like to take infinite deposits


10/62

Equilibrium quantities of deposits and loans

Equilibrium deposits D(rD) =D(r)

Equilibrium loans L(rL) =L(r)

What happens when L(r) >D(r)?

Banks would be net borrowers in interbank market Central bank has to lendL(r) D(r) > 0 to banks

What happens when L(r)


11/62

The case of a structural liquidity deficit

r

( )D r

( )r

1r

1( )D r 1( )r

CB lending


12/62

The case of a structural liquidity surplus

r

( )D r

( )r

2r

2( )D r2( )r

CB borrowing


13/62

Separability of lending and deposit taking

Given the policy rate r

Loan quantities and loan rates only depend onL(rL)

Deposit quantities and deposit rates only depend onD(rD)

Some banks could specialize in lending Using interbank market to borrow required funds

Some banks could specialize in deposit taking

Using interbank market to place surplus funds


14/62

Part 2a

Reserve requirements


15/62

A non-remunerated reserve requirement

Suppose that banks are required to invest fraction of deposits

in a non-remunerated account at central bank

If interbank rate r> 0 there will be no excess reserves Why?


l+ d= d+ b

where dare the banks reserves , which implies

b = l(1 )d


16/62

Banks profits with the reserve requirement

Since reserves are non-remunerated, banks profits are the same

= lrLdrDbr

Substituting b = l (1 )d into this expression gives

= lrLdrD [l (1 )d]r= l(rLr) + d[(1 )rrD]


17/62



rL = r and rD = (1 )r

Given interbank rate r

No change in loan rate rL

or in loan quantities

Deposit rate rD is now a fraction 1 of interbank rate r

Reduction in the amount of deposits


18/62

Implications for the central bank

In case of a structural liquidity deficit

Deficit will increase by D(r) D((1 )r)

In case of structural liquidity surplus

Surplus will decrease by D(r) D((1 )r)


19/62


r( )D r

( )r

1r

1((1 ) )D r1( )r

CB lending

((1 ) )D r


20/62


r( )D r

( )r

2r

2((1 ) )D r2( )r

CBborrowing

((1 ) )D r

2( )D r


21/62

Part 2b

Capital requirements


22/62

Introducing bank capital

Suppose that banks are allowed to raise equity capital


l= d+ b + k

where kdenotes the banks capital

Suppose that shareholders require return on their capital

If > rbanks will have no capital Why?


23/62

Introducing a capital requirement

Suppose banks are required to fund a fraction of their loans

with capital (interbank lending is not subject to the requirement)

If > r the capital requirement will be binding Why?


l= d+ b + l

where l is the banks capital, which implies

b = (1 )ld


24/62

Banks profits with the capital requirement

Banks profits

= lrLdrDbrl

Substituting b = (1 )ld into this expression gives

= lrLdrD [(1 )ld]rl

= l[rL ( + (1 )r)] + d(rrD)


25/62



rL = + (1 )r and rD = r

Given interbank rate r

No change in deposit rate rD

or in deposit quantities

Loan rate rL is now weighted average of interbank rate r

(with weight 1 ) and cost of capital (with weight )

Reduction in the amount of loans


26/62

Implications for the central bank

In case of a structural liquidity deficit

Deficit will decrease by L(r) L( + (1 )r)

In case of structural liquidity surplus

Surplus will increase by L(r) L( + (1 )r)


27/62


r( )D r

( )r

1r

1( )D r1( (1 ) )r +

CBlending

( (1 ) )r +


28/62


r( )D r

( )r

2r

2( )D r

CB borrowing

( (1 ) )r +

1( (1 ) )r +


29/62

Part 2c

Intermediation costs


30/62

Introducing intermediation costs

Suppose that each bank is characterized by a cost function c(d,l)

that is increasing (a reasonable assumption) and convex (to getinterior solutions) in the amounts of deposits dand loans l

Banks profits

= lrLdrDbrc(d,l)

Substituting b = ld from balance sheet gives

= lrLdrD (ld)rc(d,l)= l(rLr) + d(rrD) c(d,l)


31/62

Bank behavior

Since banks objective function is concave, its behavior is

characterized by first-order conditions

Notice that in this model

1. Intermediation margins are positive (to cover costs)

2. Lending and deposit taking are not separable unless

( , ) ( , ) andL D

c d l c d l r r r r

l d

= =

2 ( , )0

c d l

d l

=


32/62


Solution to the first-order conditions gives

Individual loan supply function l(rL, rD, r) Individual deposit demand function d(rD, rL, r)

Given the policy rate r, equilibrium loan and deposit rates

are obtained by solving

where n is the number of (identical) banks in the market

( , , ) ( ) and ( , , ) ( )L D L D L Dnl r r r L r nd r r r D r = =


33/62

Part 3

Local monopoly banks


34/62

Local monopoly banks

Suppose that each bank in this economy is a local monopolist

characterized by Local supply of depositsD(rD), withD(rD) > 0

Local demand for loansL(rL), withL(rL) < 0

There is an economy-wide interbank market where banks can

borrow or lend at rate r


35/62

Bank behavior (i)

The banks problem is

Assuming concavity, its behavior is characterized by

first-order conditions

[ ]( , )max ( )( ) ( )( )L D L L D Dr r r r r D r r r +

( ) ( ) ( ) 0 and ( ) ( ) ( ) 0L L L D D Dr r L r L r r r D r D r + = =


36/62

Bank behavior (ii)

The first-order conditions may be written as

where is the elasticity of the local demand for loans

and is the elasticity of the local supply of deposits

Notice that in this model

1. Banks will be making positive profits (when )

2. When we converge to perfect competition

rL = r and rD = r

1 1 andL D

L L D D

r r r r

r r

= =

L

D

, 0L D >

,L D


37/62

Part 4

Cournot competition


38/62

Cournot competition

Suppose that there are n 2 identical banks that compete la

Cournot (i.e., they use quantities as their strategic variables)


In Cournot models it is convenient to work with Inverse supply of deposits rD (D)

Inverse demand for loans rL (L)


39/62

Inverse demand for loans

( )LL r

( )L

r

Lr

( )Lr L


40/62

Bank behavior (i)

The problem of bankj is

Note that banks are playing a game

rL depends onjs decision and that of the other n 1 banks

rD depends onjs decision and that of the other n 1 banks

( ) ( )( , )max ( ) ( )j j j L j i j i j D j i j id l l r l l r d r r d d + + +

Loans of

bank

Loans of

other banks

Deposits

of bank

Deposits of

other banks


41/62

Bank behavior (ii)

Assuming concavity, optimal behavior is characterized by

first-order conditions

where

( ) ( ) 0 and ( ) ( ) 0L j L D j Dr L r l r L r r D d r D + = =

1 1andn n

i i i il D d= == =


42/62

Cournot equilibrium

Setting lj =L/n and dj =D/n in first-order conditions gives

conditions that characterizesymmetric Cournot equilibrium

These conditions may be written as

Notice that when we converge to perfect competition

rL = r and rD = r

[ ] [ ]( ) ( ) 0 and ( ) ( ) 0L L D Dn r L r Lr L n r r D Dr D + = =

1 1 andL D

L L D D

r r r r

r n r n

= =

n


43/62

Special case

When deposit supply and loan demand functions are isoelastic

we can solve for Cournot equilibrium rates

rL equals the interbank rate rmultiplied by mark-up

rD equals the interbank rate rmultiplied by mark-down

( ) and ( )D L

D D L LD r r L r r

= =

1 11 and 1

1 11 1

L D

L D

L D

r rr r r r

n n

n n

= + = +


44/62

Discussion (i)

Use of quantities as strategic variables does not appear realistic

Prices seem to provide better approximation to real world

However, Cournot equilibrium could be obtained as outcome of

a two-stage game (Kreps and Scheinkman, 1983)

First stage: Banks choose capacity (branches, etc.)

Second stage: Banks compete in prices


45/62

Discussion (ii)

Problem with price competition in the case of banks

Two-sided competition for deposits and loans

When banks simultaneously choose deposit and loan rates

Balance sheet constraint lj = dj may not be satisfied

Game played by banks is not well defined

Problem is solved when there is competitive interbank market

Separation of lending and deposit taking

Alternatively, focus on competition for either deposits or loans


46/62

Part 5

Bertrand competition


47/62

Bertrand competition

Suppose that there are n 2 identical banks that compete la

Bertrand (i.e., they use prices as their strategic variables)


In Bertrand models it is convenient to work with Supply of depositsD(rD)

Demand for loansL(rL)

where rD is the highest deposit rate and rL is the lowest loan rate

in the market


48/62

Bertrand equilibrium

Thesymmetric Bertrand equilibrium is characterized by

zero profit conditions

rL = r and rD = r

Proof in four steps

Cannot have rL < r

Cannot have rL > r

Cannot have rD > r

Cannot have rD < r


49/62

Bertrand paradox

It is hard to believe that firms in industries with few firms never

succeed in manipulating the market price to make profits.

Jean Tirole (1988)

How can the Bertrand paradox be avoided?

Relax assumption that deposits (or loans) of the different

banks are perfect substitutes Bertrand competition with differentiated products


50/62

Part 6

Monopolistic competition


51/62

The circular road model

Model of competition in deposit market in which banks invest

in asset that pays an exogenous return r

There are n 2 banks located on circumference of unit length

Continuum of depositors distributed uniformly in circumference

Each depositor has unit wealth that wants to deposit in bank

Travelling to bank involves travel cost per unit of distance


52/62

The case n = 4

Bank 1

Bank 3

Bank 4 Bank 2


53/62

The case n = 4

Bank 1

Bank 3

Bank 4 Bank 2

Depositor i

Distance Costi i


54/62

Symmetric Nash equilibrium (i)

Deposits of bankj when if offers rDj and the other banks offer rD

Bankj has two effective competitors, banksj 1 andj + 1

Depositor at distance from bankj (and distance 1/n from

bankj

1) will be indifferent between the two banks if

Solving for in this equation gives

1Dj Dr r

n

=

1( , )

2 2

Dj D

Dj D

r rr r

n

= +


55/62

The case n = 4

Bank 1j

Bank 1j+

Bank

Marginal depositor

Distance to bankj

1Distance to bank 1

n


56/62

Symmetric Nash equilibrium (ii)

Taking into account symmetric market area betweenj andj + 1

Supply of deposits of bankj

Notice that1. is increasing in rDj and decreasing in rD

2. equal sharing of depositors

1( , ) 2 ( , )

Dj D

Dj D Dj D

r rd r r r r

n

= = +

( , )Dj Dd r r

( , ) 1/ forDj D Dj D

d r r n r r = =

S i N h ilib i (iii)


57/62

Symmetric Nash equilibrium (iii)

The problem of bankj is

The first-order condition is

Setting rDj = rD (symmetric equilibrium) gives deposit rate

1max ( , )( ) ( )

Dj

Dj D

Dj D Dj Djr

r r

d r r r r r r n

= +

10

Dj Dj Dr r r r

n

=

Dr rn

=

P ti f ilib i


58/62

Properties of equilibrium

Intermediation margin

Increasing in travel cost (greater market power)

Decreasing in number of banks n (greater competition) When/n 0 we converge to perfect competition rD = r

Equilibrium bank profits

D

r rn

=

2

1( ) ( )Dn r r

n n

= =

E ilib i ith f t


59/62

Equilibrium with free entry

Assume that banks can freely enter the market at a fixed costF

Number of banks in a free entry equilibrium

Increasing in travel cost (greater market power)

Decreasing in fixed cost of entryF

( )n F n

= =

n

O ti l b f b k (i)


60/62

Optimal number of banks (i)

Social welfare

Three terms

1. Return of banks unit investment

2. Banks entry costs

3. Depositors travel costs

1/2

0

( ) 2n

W n r nF n x dx=

1/2

02

4

n

n x dxn

= =

nF=

r=

Optimal n mber of banks (ii)


61/62

Optimal number of banks (ii)

Socially optimal number of banks

Excess entry in the free entry equilibrium

Twice as many banks as in the optimum

Rationale for restrictions in entry (or in branches)

* 1arg max ( ) 2 2

n

nn W n F

= = =

References


62/62

References

Freixas, X., and J.-C. Rochet (1997),Microeconomics of Banking, MIT

Press, Chapter 3.

Chiappori, P.-A., D. Perez-Castrillo, and T. Verdier (1995), Spatial

Competition in the Banking System: Localization, Cross Subsidies, and theRegulation of Deposit Rates,European Economic Review, 39, 889-918.

Klein, M. (1989), Theory of the Banking Firm,Journal of Money, Credit

and Banking, 3, 205-218.

Kreps, D., and J. Scheinkman (1983), Quantity Precommitment and

Bertrand Competition Yield Cournot Outcomes,Bell Journal of Economics,

14, 326-337.

Salop, S. C., (1979), Monopolistic Competition with Outside Goods,Bell

Journal of Economics, 10, 141-156.

Tirole, J. (1988), The Theory of Industrial Organization, MIT Press, Chapters

5 and 7.

Documents

LSE Banking Slides 1