Adverse Selection Problem Set Answers

1. a) π = -24r^2 + 6r - x

b)
solving for r, and picking the lesser interest rate,

r= [3 - √(9-24x)]/24

c)
Setting the quantity under the radical to zero, so that we obtain a unique real
root,

x=9/24

2. a) π = -24r^2 + 10r – x

b) r= [5 - √(25-24x)]/24

c)
x=25/24. Note that this is greater than
one, so a collapse is impossible.

3. a) π = -24r^2 + 2r – x

b) r= [1 - √(1-24x)]/24

c) x=0. The market
cannot support any background probability of default. So if we think at least a few borrowers will probably default, regardless of what the interest rate is, then the market will always collapse.

4, 5, 6.

Here is a general solution.

Assume perfect competition, so equilibrium profits are zero.

π = Revenue – Costs

Recall that R=P*Q and costs are both fixed and variable

So we can write π = Q*(P-VC) – FC

Recall that in banking, the interest rate is a price, and the bank’s cost of funds is the variable cost.

Losses due to bad borrowers are a fixed cost, and equal the number of bad loans * cost of funds (bad borrowers in this model default on the interest, but not the principle).

D = 4000* (.1 –r) (given; this is quantity demanded).

P-VC is the interest rates banks charge, r, minus the cost of funds, which I will call c.

Therefore, π = 4000*(.1-r)*(r-.c) – y*c

Note that r is a variable, and y and c are parameters of the model.

Using the quadratic equation to solve for r, and picking the lesser interest rate,

r = {100(c+.1) + √[1000(c+0.1)²-(y*c+400c)]}/200

Notice that stability of the market depends on the parameters y and c only.

So the maximum value of c for which the market does not collapse is a function of y

c = [-√(y²+400y)+y+200]/2000

c(1000) = .839%

c(2000) = .455%

c(3000) = .312%

So whether
at 1%, 2%, or 3% cost of funds, the market collapses.

If y=2000, then c=.455%, and banks will charge
the interest rate r=5.28%

If y=1000, then c=.839% and banks will charge
the interest rate r=5.44%

If y=3000, then c=.312%, and banks will charge
the interest rate r=5.23%

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