Financial Institutions I:
The Economics of Banking
Prof. Dr. Isabel Schnabel
Gutenberg School of Management and Economics
Johannes Gutenberg University Mainz
Summer term 2011
V5
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I. Introduction
II. Do Financial Institutions Matter?
III. Why Do Banks Exist?
IV. Credit Rationing
1. Motivation
2. Loan Rates as Screening Device
3. Loan Rates as Incentive Device
4. Solutions to Credit Rationing
5. Conclusion
V. Bank Runs and Systemic Risk
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I. Introduction
II. Do Financial Institutions Matter?
III. Why Do Banks Exist?
IV. Credit Rationing
1. Motivation
2. Loan Rates as Screening Device
3. Loan Rates as Incentive Device
4. Solutions to Credit Rationing
5. Conclusion
V. Bank Runs and Systemic Risk
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IV. Credit Rationing
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Reading material: Freixas/Rochet, p. 171–185; *Stiglitz/Weiss
(American Economic Review, 1981)
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Here we are using the notation from the original model
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Third very influential paper from the banking literature
Paper shows that credit markets may not always work
properly due to asymmetric information
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Adverse selection and moral hazard may lead to equilibria
with credit rationing
Interest rates are sticky and do not adjust to achieve market
clearing in loan markets
Why?
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1. Motivation
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Observation: Loan markets are characterized by credit
rationing → Loan demand > loan supply at the market rate,
no market clearing
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Even relatively good borrowers do not obtain a loan or do not
obtain the desired loan amount
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Loans are not even granted at higher interest rates
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Example: Credit crunch during the financial crisis
= credit rationing?
Questions:
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Why do banks not simply raise loan rates to reduce the excess
demand?
Why is rationing such a pervasive feature of credit markets?
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Basic Argument of the Model
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Financial relationships are especially prone to problems of
asymmetric information
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An increase in loan rates affects not only the quantity of
loans, but also the quality of loans
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Therefore, rationing can be optimal (instead of raising loan
rates) → Credit rationing as an equilibrium phenomenon
2 effects of a loan rate increase:
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“Good” (i. e. relatively safe) borrowers drop out of the market,
such that the average quality of borrowers deteriorates
→ Adverse selection
Borrowers have an incentive to choose riskier projects
→ Moral hazard
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Credit Rationing
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Definition of credit rationing: A borrower does not obtain a
loan or not the desired amount even if he is willing to pay the
market rates and provide the demanded collateral
2 types of rationing:
1. Type I: Partial or complete rationing of all borrowers within a
certain group
2. Type II: Within a group of homogenous borrowers (from the
viewpoint of lenders), some obtain a loan, others do not (here!)
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Credit Rationing
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No credit rationing:
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Legal upper bounds on loan rates: This is also credit rationing,
but it is not an equilibrium phenomenon
Credit rationing due to a prohibition of price discrimination
A borrower is rejected because the project return is not large
enough to repay the loan
A borrower is rejected because he cannot provide sufficient
collateral
A borrower does not obtain a higher loan amount at the same
conditions
Empirically, it is difficult to determine whether there exists
credit rationing according to our definition
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Asymmetric Information
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Note: In perfect and complete markets there is no credit
rationing → Everybody can take up a loan up to the present
discounted value of future returns
Stiglitz/Weiss (1981): Two alternative types of asymmetric
information:
1. Quality uncertainty: The bank cannot observed borrowers’
creditworthiness (their “types”) → Adverse selection
2. Behavioral uncertainty: The bank cannot observe how the
loan’s funds are employed → Moral hazard
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The modelling of these two types of information problems in
the paper by Stiglitz and Weiss (1981) has become standard
in the literature
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2. Loan Rates as Screening Device
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The bank cannot observe the “quality” of borrowers
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“Bad” borrowers are willing to pay higher loan rates because
they default on the loan with a high probability
⇒ An increase in the loan rate leads to a deterioration in
the pool of borrowers
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This is anticipated by banks
Loan rate can be used to influence the quality of borrowers
From the viewpoint of a bank, it is not necessarily optimal to
set a market-clearing loan rate because rate increases go along
with a lower average quality of borrowers
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If the effect of worsening quality overcompensates the effect of
higher (promised) interest revenues the bank will choose a
non-market-clearing loan rate
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Assumptions of the Model
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θ = risk of the project
→ Higher θ = higher risk = lower quality
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R = return of the project where R ≥ 0 (ex post observable!)
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F (R, θ) = cumulative distribution function of R depending on
θ
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All projects have the same expected value
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Bank cannot observe risk θ
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Assumptions of the Model
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Higher θ = higher risk according to a “mean preserving
spread” (as in Rothschild/Stiglitz 1970)
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Examples: Normal distribution with different variances θ,
two-point distribution with returns µ ± θ (graphs!)
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From the condition of a mean preserving spread, it follows
that the variance is higher for higher θ (reverse is not true)
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Consequence of mean preserving spread: If the expected
value of project returns is identical every risk-averse agent
prefers the investment with lower θ relative to an investment
with higher θ
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Assumptions of the Model
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Each firm obtains a loan amount B at a loan rate r̂ (identical
loan rate because firms are undistinguishable by the bank)
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Projects are indivisible (i. e., they can be carried out as a
whole or they cannot be carried out at all)
Each firm provides collateral C
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For the model, it is crucial that C does not vary across
borrowers (cf. later section with different amounts of collateral)
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Firms are subject to limited liability: The smallest possible
return is a complete loss of collateral (in case of bankruptcy)
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All agents are risk neutral
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Profits of Firms and the Bank
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Firms go bankrupt if the project return plus the collateral are
not sufficient to repay the loan (including interest and
principal), i. e. if
C + R ≤ (1 + r̂ ) B
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Then the full project returns and the collateral are passed on
to the bank
If project returns enable the firm to repay only part of the
loan, but there is sufficient collateral, part of the collateral is
passed on to the bank, but the firms do not go bankrupt
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Profits of Firms and the Bank
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Profits of firms (graph!):
π(R, r̂ ) = max{R − (1 + r̂ ) B; −C }
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Gross return of the bank (graph!):
ρ(R, r̂ ) = min{R + C ; (1 + r̂) B}
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Theorem (1)
For a given loan rate r̂ , there is a critical value θ̂ such that a firm
borrows from the bank if and only if θ > θ̂.
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Loans are relatively more attractive for riskier firms
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Reason: Convex profit function of firms
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When project returns are low, firms’ profits are bounded from
below due to limited liability
⇒ Part of the losses is shifted to the lender
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High returns benefit only the firms, not the lender
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Consequence: For a given expected value, expected profits of
firms are particularly high when project risk is high (i. e., when
very high and very low returns are likely)
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Later: Similar effect when firms can choose their risk (risk
shifting): A leveraged firm with limited liability prefers risky
projects
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Difference to risk shifting: Here the firm cannot affect project
risk θ!
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Theorem (2)
As the loan rate r̂ increases, the critical value of θ̂, below which
firms do not apply for loans, increases.
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Expected profit of each firm decreases when r̂ rises
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Some firms, for which borrowing was profitable before, drop
out of the market at the higher rate
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These are relatively “good” (safe) borrowers
Hence, there exists a critical loan rate for each firm above
which borrowing is no longer profitable
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The higher θ, the higher is the loan rate, at which borrowing is
still profitable
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Theorem (3)
The expected return on a loan to a bank is a decreasing function
of the riskiness of loan (ceteris paribus, i. e. at a given loan rate r̂ ):
∂E [ρ(R, r̂ )]
<0
∂θ
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Reason: Concave return function of banks
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When project returns are high, the bank’s return is bounded
from above (no participation in high returns)
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Low returns are borne by the bank
⇒ Risk is partly shifted to the bank
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The higher a loan’s risk, the higher are the firms’ expected
profits, but the lower are the bank’s expected returns
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Consequence: A loan rate increase has 2 effects on a bank’s
expected returns:
1. Direct effect: Increasing returns due to higher promised
interest payments
2. Indirect effect: Lower returns due to a deterioration of
average loan quality of the bank’s loan portfolio (increase in
average loan risk)
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Central message of the paper: The second effect can
overcompensate the first, such that it is not profitable for the
bank to raise loan rates even though there is an excess
demand for loans at the given rate
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Example with 2 types of firms:
1. θ = θ1 : “Safe” borrowers take up a loan if r̂ ≤ r1
2. θ = θ2 > θ1 : “Risky” borrowers take up a loan if r̂ ≤ r2 , where
r2 > r1
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For r̂ ≤ r1 , the average quality of the bank’s loan portfolio is
between θ1 and θ2
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When the loan rate exceeds r1 , the quality of borrowers drops
discretely to θ2
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This deterioration of average quality leads to a discrete drop
in the bank’s expected return (graph!)
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Theorem (4)
If there is a finite (discrete) number of potential borrower types θ,
the average return of the bank per borrower, ρ̄(r̂ ), will not be a
monotonic function r̂ . When single types drop out of the market,
there is a discrete fall in ρ̄(r̂ ).
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Consequence: It may be that there exists a loan rate below
the market-clearing loan rate at which the bank’s returns
are higher
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In this case, there will be credit rationing!
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3. Loan Rates as Incentive Device
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Now: Model with behavioral uncertainty: Bank cannot
observe the borrowers’ risk-taking
→ Possibility of risk-shifting
An increase in the loan rate leads again to a deterioration of
the borrower pool because borrowers have an incentive to
choose higher risk
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This is anticipated by banks
Loan rate can be used to influence the behavior of borrowers
and hence the quality of the borrower pool
A similar mechanism as before again implies credit rationing
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Assumptions of the Model
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Section II.B. in Stiglitz/Weiss (1981), here: Freixas/Rochet
(2008), Section 5.3.3 with slightly modified notation
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Now all firms are identical
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Simplification: No collateral (C = 0)
2 projects (technologies) with a required investment of 1:
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1. “Good” project yields a return of RG with probability πG and
zero otherwise
2. “Bad” project yields a return of RB with probability πB and
zero otherwise
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The bank cannot observe the project choice
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Risk choice of Firms
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The “good” project has a higher expected return:
πG · R G > πB · R B
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But: The “bad” project has a higher return in the good
state: RB > RG
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These two conditions imply that πG >> πB , i. e., the “good”
project is less risky
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The firm chooses the “good” (safe) project if and only if
πG [RG − (1 + r̂ )] ≥ πB [RB − (1 + r̂ )]
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Risk choice of Firms
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This condition yields a critical loan rate below which firms
choose the safe project:
1 + r̂ ≤ 1 + r̂ crit =
πG · R G − πB · R B
πG − πB
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Hence, higher loan rates increase the incentive of firms to
choose the risky project
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Reason: Limited liability
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The firm can shift part of its losses to the bank
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Expected Return of the Bank
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Expected (gross) return of the bank = πG (1 + r̂ ) as long as
r̂ ≤ r̂ crit
→ In this range, the return increases in r̂ !
At r̂ crit , the expected return of the bank drops discretely from
πG (1 + r̂ crit ) to πB (1 + r̂ crit ) (graph!)
Again the expected return of the bank is not a monotonic
function of r̂ ⇒ Possibility of credit rationing
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4. Solutions to Credit Rationing
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We now consider again the model with adverse selection
(Section 2.)
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Here we follow the presentation in Freixas/Rochet (2008),
Section 5.4 (model by Bester, American Economic Review
1985)
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Basic idea stems from the model by Rothschild and Stiglitz
(1976) (adverse selection in insurance markets), which is
applied to loan markets
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Collateral as a Screening Device
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Idea: The bank could try to sort borrowers by offering
different contracts with different loan rates and collateral
requirements
(cf. deductible in insurance contracts)
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Borrowers choose the contract that is most favorable for them
(self-selection)
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Assumptions of the Model
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2 types of firms: θ ∈ {θ1 , θ2 }, where θ1 < θ2
(as above)
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Firms own assets that can be used as collateral
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The assets are large enough such that any firm can provide
the required collateral
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The collateral is worth less for the bank than for the firm
(e. g. liquidation costs)
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Perfect competition in the banking sector
→ Banks earn zero profits (i. e., the expected interest
payments are equal to banks’ capital costs)
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Contracts
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Banks can offer different contracts: (r̂i , Ci ), i = 1, 2
2 possible Nash equilibria:
1. Separating equilibrium
2. Pooling equilibrium
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Definition of a separating equilibrium: The bank offers 2
contracts (r̂1 , C1 ) and (r̂2 , C2 ), such that
1. a firm with θ1 (low risk) chooses the first and a firm with θ2
(high risk) the second contract (self-selection),
2. no bank has an incentive to offer another contract, and
3. the zero-profit condition is satisfied for both contracts
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Separating Equilibrium
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Bester (1985) shows that a separating equilibrium exists
with the following properties:
1. The “good” type (θ1 ) obtains a contract (r̂1 , C1 ) with C1 > 0
2. The “bad” type (θ2 ) obtains a contract (r̂2 , C2 ) with r̂2 > r̂1
and C2 = 0
(r̂2 can be derived from the zero-profit condition)
3. r̂1 and C1 are chosen such that the “bad” type is just
indifferent between the two contracts (such that he does not
want to mimic the “good” type) and that the bank earns zero
profits on this contract as well
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Surprising result: “Good” types have to provide higher
collateral in equilibrium!
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Intuition
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Banks prefer higher loan rates and higher collateral
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Firms prefer lower loan rates and lower collateral
But: For “good” types, providing collateral is less expensive:
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“Good” types require only a small decrease in loan rates to
provide higher collateral
“Bad” types require a larger decrease in loan rates than “good
types” if they are asked to provide collateral
⇒ Single-crossing condition (= necessary condition for
screening)
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Intuition
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Note: Collateral reduces welfare ex post because it is worth
less for the bank than for the firm
→ In equilibrium, collateral requirements are chosen as low as
possible
Due to competition, banks have to offer the best possible
contract to both types:
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“Bad” types do not have to provide any collateral
“Good” types have to provide just as much collateral as is
necessary to prevent imitation by “bad” types
Central result: If a separating equilibrium exists, there is no
credit rationing!
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Pooling Equilibrium
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Definition of a pooling equilibrium:
1. Banks offer the same contract (r̂ , C ) to both types
2. This contract leads to zero profits for the banks
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Since collateral is not used here to sort different types, C will
be zero in the pooling contract
Bester (1985) shows that a pooling contract will never be an
equilibrium
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Pooling contract can only be an equilibrium if it is better than
a separating equilibrium for both types
However, it is always possible for a bank to offer a contract
that is strictly better for the “good” type, but not for the
“bad” type (cherry picking)
→ Pooling cannot be an equilibrium
Conclusion: There either exists an equilibrium without credit
rationing or there is no equilibrium
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Incentive Effects of Collateral
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It can be shown that collateral also has positive incentive
effects (model with moral hazard)
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The provision of collateral (or equity) raises the critical loan
rate below which the firm chooses the safe project
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This may prevent credit rationing because banks’ return
function is monotonic in the relevant range
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The same argument justifies the capital regulation of banks:
When banks are forced to hold more capital, they have a
lower incentive to extend risky loans (at the expense of
depositors or the deposit insurance)
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5. Conclusion
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Model by Stiglitz/Weiss (1981) shows that credit rationing
may occur in equilibrium
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Reason: An increase in loan rates implies a deterioration of
the borrower pool (either due to adverse selection or
moral hazard)
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Consequence: It can be optimal for banks not to raise loan
rates even though there is an excess demand for loans
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Critical Assessment
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Model shows that banks’ interest rate setting behavior has an
impact on the quality of their loan portfolios
→ Important idea that has been used in many other papers
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Model shows that markets can be in equilibrium without
market clearing
→ The market mechanism may fail under asymmetric
information
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But: Model is in many ways special and not robust
(Example: credit rationing disappears if one allows for
screening, Bester 1985)
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How relevant is credit rationing in reality?
→ Empirical evaluation is difficult, especially because loan
quality cannot easily be observed
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Program of Next Lecture
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V. Bank Runs and Systemic Risk
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2nd part of *Diamond/Dybvig (Journal of Political Economy,
1983)
Solutions to the bank run problem
Financial contagion
Too big to fail
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