Financial Intermediation

1
Financial Intermediation

• Lecture 4
• Bank Behavior Theory and Competition

2
Up to now

• We discussed (the history of) financial systems,
  corporate governance, and the nature of financial
  intermediation
• In this lecture we discuss basic banking models
  what does a bank do?
• We do so under various assumptions monopoly
  power and regulation

3
Are Banks Special?

• Banks deal with private contracts and securities
• Banks assets and liabilities differ in nature
• Banks fulfill a role in the payments system
• Banks are money creating institutions
• Banks are faced with systematic risk (bank runs
  and bank panics)

4
Why do banks exist?

• In an Arrow-Debreu economy we have no banks, FIs
  or money (as such)
• We need some kind of imperfections to be able to
  explain the existence of banks and/or money see
  the previous lecture for the Diamond delegated
  monitoring model
• What kind of imperfections? Information problems,
  missing markets, price stickiness (the latter not
  so handy in banking models!), or a lack of
  competition

5
This lecture

• Simple banking models (1) perfect competition,
  (2) monopolistic bank (the Klein-Monti model, (3)
  Cournot, (4) spatial competition (Salop)
• Next we discuss one of the core ideas of banking,
  fractional reserve banking, in more detail

6
A model of perfect competition in the banking
sector

• We have N different banks 1,, N.
• A bank has a cost function C(D,L) D deposits,
  L loans. The function is convex and twice
  differentiable. In most cases we assume a
  uniform cost function C(D,L)
• The assets are R reserves, L loans. Deposits
  D are the single liabilities
• R CR M, CR a D reserves at the central bank
  (no interest), M money market reserves

7
Banks as price takers (1)

• The loan rate rL, the deposit rate rD and the
  money market interest rate r are given
• A bank maximizes ? rLL rM rDD - C(D,L)
• M (1 - a) D - L, so ?(D,L) (rL - r)L
  (r (1 - a) - rD) D - C(D,L). We
  optimize L and D
• FOC1 (rL - r) CL(D,L)
• FOC2 (r(1 - a) - rD) CD(D,L)
• So with cL CL(D,L) and cD CD(D,L) we get
  rL r cL and rD (1-a)r cD

8
Banks as price takers (2)

• So an increase in rD will decrease the banks
  activity in deposits. An increase in rL will
  increase the supply of loans.
• The cross-effects (important) depend on CDL(D,L).
  If CDL gt 0 an increase in rL will decrease D
• Economies of scope CLD lt 0 an increase in L
  decreases the marginal costs of deposits for
  CLD gt 0 we have diseconomies of scope

9
A monopolistic bank the Klein-Monti model

• Downward-sloping demand for loans L(rL)
• Upward sloping demand for deposits D(rD)
• We will use the inverse functions rL(L) and rD(D)
• Max ?(L,D) rL(L)L rM - rD(D)D - C(D,L) There
  is e.g. an impact of L on rL
• M (1 - a) D - L, so
  ?(L,D) (rL(L) - r)L (r(1-a)
  - rD(D))D - C(D,L)

10
Klein-Monti (2)

• Assume the profit function is concave
• FOC1 rL(L)L rL r - CL(D,L)0
• FOC2 -rD(D)D r(1 - a) rD - CD(D,L)0
• Define the following elasticities
  eL -rLL(rL) / L(rL) gt 0 and
  eD rDD(rD) / D(rD) gt 0

11
Klein-Monti (3)

• We get a famous result a monopolistic bank will
  set its volume of loans and deposits such that
  Lerner indices equal inverse elasticities
• rL - (r CL) / rL 1/eL(rL)
• r(1 - a) CD - rD / rD 1/eD(rD)

12
Cournot competition

• Finite number of competing banks N
• The first-order conditions change into (no
  proof)
• rL - (r CL) / rL 1/N eL(rL)
• r (1-a) CD - rD / rD 1/N eD(rD)
• For N 1 we get the Klein-Monti and for N
  infinite we get perfect competition!

13
Double Bertrand competition

• Banks compete using interest rates.
• Bertrand competition prices are central, but
  with two banks we get perfect competition
• Idea banks compete on the markets for output
  (loans) and inputs (deposits)
• Assume constant marginal costs (normalized to
  zero)
• L(rL) demand for loans, D(rD) supply of deposits

14
Double Bertrand competition (2)

• Reserve requirements ignored
• Walrasian equilibrium rL rD rt, where rt is
  simply the solution of L(r) D(r)
• Suppose now that banks first compete for deposits
  and next to that operate on the loans market.
  Offering a slightly higher rD rt e creates a
  monopoly on the loans market

15
Double Bertrand competition (3)

• With two banks and a relatively high loan demand
  elasticity we get (see Stahl, 1988)
• Only one bank is active.The interest margin is
  positive rL gt rD. Both banks have zero profit
  though. The active bank has idle reserves
• The loan rate rL is the one that maximizes
  (1 rL) L(rL).
• The deposit rate is determined by
  (1 rD)D(rD) (1 rL) L(rL)
• There is excess supply of deposits L(rL) lt D(rD)

16
The Salop model (1)

• The Salop model is a model of spatial
  competition banks all are on a unit circle
• Depositors have to get to the bank to get a
  high(er) return
• It is a model of monopolistic competition
  product differentiation is generated by
  transportation costs
• Is free banking optimal?

17
The Salop Circle
Bank (i1)
1/n - xi
Depositor
xi
Bank i
18
The Salop model (2)

• n banks 1,.,n located on the circle of length 1
• Banks get deposits and invest in a riskless
  technology with r return. Depositors cannot
  invest in this technology
• Depositor deposit in a bank at a cost ax,
  proportional to x, the distance between depositor
  and the bank.
• There is a D mass of depositors

19
The Salop model (3)

• Depositors are uniformly distributed
• With n banks the maximal distance is 1/2n
• Sum of all depositors transportation costs is 2n
  ?0½n ax D dx aD/4n
• A setup cost of a bank is F
• The optimal number of banks minimizes nF
  aD/4n, which is n ½?aD/F

20
The Salop model (4)

• Free entry n banks settle on the circle and set
  deposit rate ri. How much deposits are in each
  bank?
• See the Salop circle the marginal depositor,
  indifferent to going to bank i or i 1
• ri axi ri1- a (1/n - xi). Remember that the
  distance between banks is 1/n
• So xi 1/2n (ri - ri1)/2a

21
The Salop model (5)

• Now we integrate over the distance to get the
  total amount of deposits per bank
• Di D 1/n (2ri - ri-1 - ri1)/2a
• Profit bank i D(r - ri) 1/n (2ri - ri-1 -
  ri1)/2a
• Equilibrium if for all i ri maximizes profits
• r ri a/n (2ri - ri1 - ri-1)/2
• This leads to r1 rn r - a/n. So profits
  are all aD/n2. With costs F this leads to n
  ?aD/F
• So free competition leads to too many banks
  there is scope for public intervention

22
Competition on markets macro tests

• Structure-Conduct-Performance paradigm market
  structure influences performance of banks (more
  concentrated markets facilitate collusive
  agreements, increase market power and so
  profitability)
• Relative-Efficiency hypothesis efficient firms
  earn relatively high profits and thus increase
  market share concentration of profitable firms

23
Competition on markets Panzar-Rosse test

• Panzar-Rosse formulate a revenue function per
  bank (firm) as function of input prices. The
  Panzar-Rosse H is the sum of elasticities of
  gross revenue with respect to input prices
• If H gt 0 any form of imperfect competition is
  rejected

24
Competition test Bresnahan-Lau

• Conjectural variation method estimate the banks
  anticipated response of its rivals to an output
  change
• Let qi the output of bank i and Q the aggregate
  output.
• The conjectural variation elasticity
  ki dQ/Q / dqi/qi
• ki 0 for perfect competition, perfect collusion
  ki 1. 0 lt ki lt 1 for oligopoly

25
Regulation in a model of Fractional Reserve
Banking

• Evolution of worthless monies. Use gold to
  (partially) back tradable receipts and use the
  other funds to supply (more profitable) loans.
• Deposits can be seen as the pool of liquidity
• Loans are illiquid and lead to possible
  instability of the fractional reserve system
• This demands regulation deposit insurance

26
A model

• A depositor earns y, consumes c and saves s
  y-c, which are safer at the FI. ? is the fee that
  the FI charges. Some of the depositors (?)
  deposit for one period, and so get s - ? back.
  The implicit (negative) interest rate is -?/s
  (officially (s - ? - s)/s), the
  other (1 - ?) hold the deposit for two periods
  (see hereafter)
• There are n depositors in total there is no
  discount rate

27
A model (2)

• The sequence of events t 0, n depositors
  deposit s each. The bank invests in loans M to
  merchants, which pay back at t 2 at a rate r
• At t 1 the bank pays out ?n (s - ?) in the
  aggregate
• At t 2 bank pays out (1 - ?) n (s - ?)
• A merchant (1) needs M gt s, (2) earns
  K gt M(1 r), but is unreliable in paying
  back and needs to be monitored at cost b. For a
  moment we assume that the earnings K is certain

28
A model (3)

• A merchant can approach individual depositors (we
  assume he needs more than 1, since M/s gt 1).
  Depositors will charge the monitoring costs, so
  profits are K - (1r) M - bM/s (the merchant
  approaches M/s depositors directly)
• A merchant can go to the bank and can earn
  K - (1r) M - b. This is more profitable, since
  M/s gt 1. This is the Diamond delegated monitoring
  result

29
A model (4)

• The bank needs to be sure that it can pay back
  early withdrawals. Merchants pay their
  obligations only at t 2
• Suppose there are m merchants. ns is the size of
  total deposits, mM is the total loan size, we
  assume that ns gt mM
• ns - mM must be guarded (mM is with the
  merchants!) at a rate ?/s mM must be monitored
  at a cost b
• The bank promises to depositors
  ns - (ns - mM) ?/s

30
A model (5)

• The bank repays depositors ns - (ns - mM)?/s,
  which is higher than without lending, since
  lending is profitable
• Early withdrawal requires an amount of
  ?ns - (ns - mM)?/s (label this term A), so
  the bank needs to determine m such that it has
  enough funds
• Funds available at t 1 are ns-mM (available
  funds) - (ns - mM)?/s (security funds) - mb
  (monitoring costs) (label this B). So determine m
  such that A B

31
A model (6)

• ?ns-(ns-mM)?/s ns-mM-(ns-mM)?/s-mb
• So m (1-?) ns (s - ?) /M s-(1-?)? bs
• If the bank has precisely this amount of
  borrowers it will not be a risk that withdrawals
  at t 1 get too big

32
A model (7)

• Banking business charge monitoring costs to
  borrowers, charge safeguarding costs to
  depositors and keep profits rmM for yourself
• The deposit rate ns - (ns - mM)?/s exceeds the
  deposit rate where the assets were idle -?/s ,
  but is still negative
• Suppose that the depositors are also owners of
  the bank, the get a return rmM/ns. If this return
  is high enough, there is a positive deposit rate

33
A model (8)

• Now the depositor gets
  ns - (ns - mM) ?/s mMr / ns, which
  equalsmMr - (ns - mM) ?/s / ns
• If r is high enough we get a positive interest
  rate on deposits
• So banks help in solving the social moral hazard
  model of theft and they are efficient in lending

34
A model (9)

• Problem in this model is the sure K. If K is
  unsure and liquidation values of merchants are
  low, there is a large probability of early
  withdrawal.
• Solutions? (1) narrow banking hold as many
  reserves as deposits, or (2) deposit insurance.
• But deposit insurance leads to risk taking by
  banks (moral hazard)

35
Background literature

• Greenbaum Thakor, Contemporary Financial
  Intermediation, Chapter 3
• Freixas Rochet, Microeconomics of Banking,
  Chapter 3