ec 410 problem set 4

ec 410 question, give me the answer in 3 days ,thank youProblem
Set 4
MSU EC 410
Prof. Ahlin due 11/24/15

1. Imagine
there are two types of potential borrowers in a village, those with pi=0.7
and those with pi=0.9. As in
the model discussed in class, these borrowers succeed with probability pi
and fail with probability (1?pi). If they fail, they get no payoff and can pay
nothing to the bank (there is no
collateral). If they succeed, they
pay back (1+r)L to the bank.
Assume L = $100.
Assume the expected gross
payoff (i.e. without accounting for payment to the bank) to borrowing is $200
for all borrowers.
Assume all potential borrowers have a non-borrowing option
they can choose instead of borrowing: working for a subsistence wage of $70.
First consider individual
(not group) lending.
a.
Write down the expected payment made to the bank by a
borrower with pi=0.9. Write
down the expected payment made to the bank by a borrower with pi=0.7. Leave both answers in terms of r.
b.
Assume the bank can observe the individuals’
risk-types, because of credit history for example, and can charge different
borrowers different interest rates. If
the bank must earn an expected return of 10% on its loans, what interest rate
would it charge a borrower with pi=0.9? with pi=0.7? Justify your answer.
c.
For both borrowers, what is their expected net payoff to borrowing if the bank
charges the interest rates of part b.?
Which types will borrow (comparing the payoff to borrowing to the
outside payoff)? Justify your answer.
d.
Now assume the bank cannot observe the individuals’
risk-types and must charge everyone the same interest rate. Assume the interest rate is the one charged
to the risky type (with pi = 0.7) in part b. (If you did not get part b., assume an
interest rate of r = 57%.) Write down
the expected payment made to the bank by a borrower with pi=0.9. Write down the expected payment made to the
bank by a borrower with pi=0.7.
e.
For both borrowers, what is their expected net payoff to borrowing? Which types will borrow (comparing the payoff
to borrowing to the outside payoff)?
Justify your answer.
f.
In words, what is the adverse selection problem, when
does it appear, and why?

Now consider group lending.
Borrowers form pairs homogeneously.
If a borrower succeeds, he pays (1+r)L.
In addition, if he succeeds and his partner fails, he makes an extra
payment of cL. Assume projects succeed
or fail independently of each other. Now
assume r = 30% and c = 90%, that is, 90% of the partner’s loan principal must
be paid by a borrower who succeeds and whose partner fails.
g.
Find the effective interest rate for a borrower with pi=0.9. Find the effective interest rate for a
borrower with pi=0.7. For
each, how does this interest rate compare with that of parts b. and d.?
h.
For both borrowers, what is their expected net payoff to borrowing? [Hint: expected repayment to the bank is just
pi(1+ ~ri
)L, where ~ri
is the effective interest rate for borrower i.]
Which types will borrow (comparing the payoff to borrowing to the
outside payoff)? Justify your answer.
i.
In words, briefly explain how group lending can
overcome the adverse selection problem.

2. Project A
pays $75,000 (i.e. succeeds) with probability 9/10 and pays 0 (i.e. fails) with
probability 1/10. Project B pays
$100,000 (i.e. succeeds) with probability 3/5 and pays 0 (i.e. fails) with
probability 2/5.
[Note: the expected value for any random amount X,
that takes on values {x1, x2, …, xN} with
probabilities {p1, p2, …, pN} respectively,
equals .gif”>. For instance, if profits equal $40 with
probability 1/3, $60 with probability
1/2, and 0 with probability 1/6,
then the expected value of profits (i.e. expected profits) equals 1/3 * $40 + 1/2 * $60 + 1/6 * 0 = $43.33.]
a.
What is the expected gross payoff, i.e. expected gross
value, of project A? What is the
expected gross payoff, i.e. expected gross value, of project B?
b.
Assume each project requires an investment of $55,000
to undertake. Consider a person with
$55,000 to invest who just cares about maximizing expected payoffs. Would this
person prefer to invest in project A or B?

Now consider a person who is borrowing the $55,000 in order to
invest in project A or B. Assume the
interest rate charged is r = 8%, whichever project is chosen. Assume the borrower puts up collateral C and
faces no liability other than that if the project fails. Thus, if the project fails, the borrower
loses C (this is equivalent to paying C); if the project succeeds, the borrower
repays (1+r) $55,000.
c.
What is the borrower’s expected payment to the bank
(counting collateral seized as making a payment) under project A? under project B? Leave your answers in terms of C.
d.
What is the borrower’s expected net payoff under
project A? under project B? Leave your answers in terms of C.
e.
If C = 0 and the borrower maximizes expected payoffs,
which project will the borrower choose?
f.
If C = (1+r)$55,000 and the borrower maximizes expected
payoffs, which project will the borrower choose?
g.
What is the minimum collateral level necessary for the
borrower to make the same choice as if his own money were at stake (i.e. the
answer of part b.)?
h.
In words, what is causing the moral hazard problem
here?

Consider now joint liability. There is now a group of two, each of which is
liable for the other’s loan.
Specifically, if one succeeds and the other fails, the one
who succeeds owes an extra $16,500 (i.e. 30% of the loan principal,
c=30%). The interest rate r is now
reduced to compensate for the extra liability: r = 5%. [This keeps the bank’s expected profits the
same as before provided borrowers choose the safe project, as you could
check.] To review, now if the borrower
succeeds and his partner succeeds, he repays $57,750 (= [1+5%] * $55,000); if
he succeeds and his partner fails, he repays $74,250; and if he fails, he loses
collateral C.
i.
What is a borrower’s expected payment to the bank
(counting collateral seized as making a payment) if both he and his partner
choose project A? if they both choose
project B? Leave your answers in terms
of C. [Hint: Remember that the expected
repayment is the amount paid in each state of the world multiplied by the
probability of being in that state of the world. For both succeeding, the probability is just
p2; for one succeeding and the other failing, it is p*(1?p); and for failing, it is (1?p).]
j.
What is the borrower’s expected net payoff if both
group members choose project A? if both
group members choose project B? Leave
your answers in terms of C.
k.
What is the minimum collateral level necessary for a
pair of borrowers that choose the same project to make the same choice as if
their own money were at stake (i.e. project A)?
l.
In words, how does joint liability affect the moral
hazard problem here?

3. Here we
explore how dynamic lending can reduce limited enforcement. In particular, consider the model introduced
in class, where the borrower repays when and only when the financial benefit of
the future
relationship, B(rt+1,Lt+1,…,rT,LT),
is at least as great as the financial cost of repaying the current loan, (1+rt)Lt. Let t=0 and T=3 – that is, the borrower
anticipates 3 more loans after repaying the current one. Assume that the borrower has a project that
always doubles the value of the capital input, without risk – e.g. a $100 loan
can be turned into a certain $200 gross return.
If the borrower repays, the net return would then subtract off the cost
of the capital, principal and interest – e.g. if r=20% and L=$100, the cost of
capital would be $120 and the net return would be $200-$120 = $80. Thus the value of one loan would be $80. In calculating benefits of future loans,
assume the borrower does not discount the future (i.e. values current and
future income equally) and that he plans to repay all future loans when
deciding whether to repay the current loan.
a.
Assume all loans sizes and interest rates are the same:
L0=L1=L2=L3=$100 and r0=r1=r2=r3=r,
say. At what interest rates r would the
borrower choose to repay the current (0th) loan?
b.
Assume r0=r1=r2=r3=60%,
and compare repayment behavior when L0=L1=L2=L3=$100
to when L0=$25, L1=$75, L2=$125, L3=$175. In each case, which loan will the borrower
default on?
c.
Now add the expectation of a 4th loan: i.e.
T=4. Consider r0=r1=r2=r3=r4=60%
and
L0=L1=L2=L3=L4=$100. Will the borrower repay the current (0th)
loan? How does this compare to repayment
behavior under these loan sizes and interest rates when T=3?
d.
Return to the assumptions of part a. However, assume that capital is worth less to
borrowers, so that their projects only increase the value of the capital by 50%
rather than doubling it. (E.g. a $100
loan grosses $150 with certainty, rather than $200.) At what interest rates r would the borrower
choose to repay the current (0th) loan?

 

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