## 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 partnerâs 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 borrowerâs 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 borrowerâs 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 otherâs 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 bankâs 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 borrowerâs 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 borrowerâs 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,â¦,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 â 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 â 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 â 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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