Define (x
1
, x
2
) as the state-contingent consumption vector of a customer where x
1
(with loss) = W – L + d – ρ · d and x
2
(without loss) = W – ρ · d
W = wealth, L = loss, d = coverage, ρ = rate premium
Define (y where y
1
1
, y
2
) as the state-contingent profit vector of the insurer
(with loss)= ρ·d - d= W – L x
1 and y
2
(without loss) = ρ·d = W x
2
The insurance policy (d, ρ) corresponds to state-contingent consumption vectors (x
1
, x
2
) for customers and (y
1
, y
2
) for the insurer.

Risk neutral insurer’s problem
Assume that an insurer cannot distinguish between high-risk and low-risk customers, with loss probabilities π
H and π
L
, and provides the same amount of insurance (D) to both customers under the policy (d, ρ).
The insurer’s optimization problem is max ( π·(ρ 1)·D + (1π)· ρ ·D = ( ρ-π ) ·D
A “risk-neutral insurer” will supply any amount of insurance if ρ=π.
That is, in equilibrium the price of insurance must equal the probability of loss. We can show this graphically in a statecontingent consumption diagram.

Since y
1
= ( ρ-1)·D and y
2
= ρ·D the choice of D for a given premium ρ corresponds with the restriction that y
2
= [( ρ-1) / ρ] · y
1
But y
1
= W – L – x
1 and y
2
= W – x
2
So, the restriction is, x
2
= [Wρ·L]/(1-ρ) – [ρ / (1- ρ)] · x
1 which gives the slope of the insurer’s offer curve = ρ / (1- ρ)
The expected profit of the insurer can be written as
G = π·y
1
+ (1 π)· y
2
= π·(W - L - x
1
) + (1 – π)·(W - x
2
)
By fixing G, the iso-expected profit lines for the insurer are x
2
= [W π·L – G] / (1 - π) – [π / (1- π)] · x
1 which gives the slope of the iso-expected profit curves = π / (1- π)

Define the customer’s endowment point as (W-L,W). For the insurer the more profitable contracts lie closer to the origin.

Consider the demand for insurance of customers whose loss probabilities are π
H and π
L
.
The customer will choose policy (d, ρ ) to maximize his expected utility. But that maximization problem can also be written as a function of the choice variables x
1 and x
2
.

The “full insurance” solution is d = L, x
1
= x
2 and MRS = π / (1 – π)
(this would be the preferred point if ρ ≤ π)

Optimal insurance of two customers - high risk ( broken curves), low risk ( solid curves)
The equilibrium insurance premium, ρ*, will result in an expected loss to the insurer, since π
H
> ρ* = θ·π
H
+ (1θ)·π
L
> π
L
.

Under asymmetric information low-risk customers (low loss probability) will demand incomplete, or no, insurance and high-risk customers (high loss probability) will demand full insurance.
Anonymity breaks down as customers reveal their types to the insurer by the level of their demands for insurance.
To induce self-selection the insurer will offer different policies, (d, ρ), with different price-quantity pairs.

The insurer maximizes an objective function subject to the following constraints:
• feasible (the insurer can provide sufficient funds to support the contract),
• individually rational (allocations below the indifference curve through the endowment point are ruled out), and
• incentive-compatible (allocations must not provide incentives for agents to misrepresent their private information, truth-telling).

Incentive compatibility (IC
L
) requires that (x
1
L , x
2
L ) lie on or above the indifference curve of the low-risk customer through (x
1
H ,x
2
H ).

Contracts (x
1
H ,x
2
H ) that are IC
H given (x
1
L , x
2
L ) and that improve the expected profit from the high-risk customer contract

Contracts that are IR
L and yield the insurer a higher expected profit, if IC
H is binding & the high-risk customer is fully insured

2. We want to show that differences in incentives of borrowers and lenders may lead to nonprice credit rationing
Assume a project return function, f(s), and repayment, R. Then we can write the state-dependent expressions,
π(s, R) = max {f(s)-R, 0}
ρ(s, R) = min {R, f(s)}
(return of the entrepreneur)
(repayment to the lender) and,
π(s, R) + ρ(s, R) = f(s) (neglecting losses due to monitoring costs)

Conflict of interest - the borrower prefers riskier project f(s)
A and the lender prefers safer project f(s)
B

Credit rationing (Stiglitz & Weiss, 1981)
• Assume a bank faces N loan applicants that are of two types, t = 1,
2 (distinguished by their project riskiness).
• The return distribution of type-2 applicant project returns (x 2 ) is a
“mean preserving spread” of the return distribution of type-1 applicant project returns (x 1 ).
• There are two states of project returns, high (H) and low (L). The probability of each state = 0.5
• Half of the applicants are of each type, but the bank cannot distinguish between applicants according to their type due to asymmetric information.
• The bank can raise funds through deposits at an interest rate
I = (1+ i) with linear supply of funds L(I) = α · I, α>0.
• Competition between banks forces expected profit to equal zero, so
E[ ρ( )] = I.
• Banks can recover amount C from a bankrupt customer.

Expected return for a loan applicant

Expected return for a loan applicant: positive for R ≤ R t (loan demand is positive) negative otherwise (loan demand is zero).
If R t is the interest rate at which expected profit = 0, then R t = x
H t is a critical value for the exit of customer t. Also, R 2 > R 1 .
– C

Expected return to the bank, E[ ρ t (s, R, C)], is a concave function of the loan interest rate, R

The bank’s expected return is the weighted average of the expected returns. Therefore, banks will not simply raise interest rates when facing excess demand for loans
The return function drops whenever a customer leaves the market.
Low risk customers leave the market first.

•
The bank’s loan supply and demand functions
The bank has no incentive to increase interest rates above R 1 because the expected return unambiguously falls and at R 1 credit rationing occurs.

The expected revenue for lenders in the SW adverse selection model is not globally hump-shaped as a function of the loan rate. Rather,
• if there is credit rationing, there must be two equilibrium loan rates,
• there is rationing at the low rate, but the credit market clears at the high rate as long as customers are willing to pay the higher rate.
Model:
• Lenders earn expected revenue per loan V(R) with gross loan rate
R.
• All borrowers (with type t) have the same mean project return μ, but they vary in risk. So, the gross project return is y(t) = μ + z(t) where return (z) is random.
• Project payoff to a successful loan applicant is written u(z, R) = Max ( μ + z – R, -C) where C is collateral

The expected payoff of borrower type t is
The average expected profit of all loan applicants is where θ(R) is the lowest borrower type in the market with gross loan rate R.

Since all projects have the same mean return we have E[U(R)] = μ - R and the lender’s expected revenue on a loan is, V(R) = μ – E[U(t, R)]
• For small R there is no bankruptcy and U(t, R) = μ – R for all t.
• Let R be the rate at which the riskiest borrower just loses his collateral if a bad state occurs. The probability of bankruptcy is positive if R > R.
• At R = μ the probability of bankruptcy of the borrower is strictly positive. So, adverse selection starts at some interest rate R a
> μ.
• Let R* be the interest rate at which only the riskiest projects break even, U(R*) = 0. Then, the expected revenue of the lender
= V(R*) = μ.
Then, there are two possibilities: V(R) is everywhere increasing, or
V(R) has at least two turning points.

Expected lender revenue function, V(R), with turning points R
1 and R
2
.

The market for loanable funds has no single equilibrium loan rate
The demand curve segments are in bold. At R
1 there is excess demand and at R
3 there is excess supply. There is a two interest rate
(separating) equilibrium.

4. Extensions of SW:
“Risk rationing” - - Boucher et al.(2008)
Information asymmetry and enforcement costs make some credit contracts infeasible if they are conditional on borrower behavior.
• This restricts the set of available contracts by eliminating as incentive incompatible those loans that carry high interest rates and low collateral requirements.
• The result of this “contraction of contract space” is quantity rationing in credit markets.
“Risk rationing” is where so much contractual risk is shifted to the borrower that the borrower voluntarily withdraws from the market even though s/he has sufficient collateral for the loan contract.
• Nonprice (quantity) rationing: borrowers who are willing to pay the market interest rate, but do not qualify for a loan.
• Nonprice (risk) rationing: borrowers are willing to pay the market interest rate and qualify for the loan, but afraid to take a loan because they are afraid of losing their collateral.

Risk rationed farms appear similar to quantity rationed farms

In a competitive loan market, the optimal contract maximizes the borrower’s expected utility subject to the lender’s participation constraint (3) and the borrower’s incentive compatibility constraint (4) where e = borrower effort, p t
T = land holdings, x = gross revenues, s = project state dependent payoffs, Ф = state probability, d( ) is a disutility of effort function

g
b
The incentive compatible boundary is an upward-sloping function as long as marginal utility of consumption, u′ (C) > 0.

First best contract is at point A. If at point B (bad state) the IC constraint is binding. π ( s j
|H) is the lender’s zero expected profit curve. Point B lies below the indifference curve through point C, so a contract at point B is not IC.

Risk rationing and activity choice (numerical analysis)