# Problem Set 3 Due July 23 (in sections) ARE/ECN 115A Summer Session 1, 2015 Last Name: ____________.

Problem Set 3

Due July 23 (in sections)

ARE/ECN 115A Summer

Session 1, 2015

Last Name: ____________ First Name:

____________ Student

ID:____________

Part A. Credit Markets

In part A of this problem set, we will

think systematically about how the equilibrium interest rate is determined in

the credit market. In particular, we will explore how the interest rate, the

size of the economic surplus and the distribution of the surplus are affected

by asymmetric information in the context of limited liability loans.

In the village of Quahog, people can

choose to either farm or work in the factory. If a person works in the factory,

she makes $100 with certainty. People in Quahog are either born a SAFE farmer

or a RISKY farmer. Half of the people are SAFE farmers and half are RISKY

farmers. If either type of person chooses to farm, she will need $300

investment in order to farm. Thus, the opportunity cost of farming is the $100

one could have had earned if she instead worked in the factory. The only

difference between SAFE and RISKY farmers is as follows:

â€¢

SAFE farmers have a good harvest all the time,

they earn revenues of $700 with 100% probability.

â€¢

RISKY farmers have a good harvest with 60%

probability, in which they earn revenues of $1100 with 60% probability; and

they have a bad harvest with 40% probability, in which they earn revenues of

$0.

1 Perfect Information

Brian is a moneylender who lives in Quahog. His opportunity

cost of money is 0.20 (i.e., he would earn 20% if he invested the money in a

business instead of lending it to farmers). Brian offers limited liability

loans, so a farmer does not have to repay the loan if she has a bad harvest.

Since Brian lives in Quahog, he has perfect information about farmers.

Specifically, he knows who is a SAFE farmer and who is a RISKY farmer.

1. Let

Ysbe

the farming income of a SAFE farmer and YRbe the farming income of

a RISKY farmer. The farming income for any famer Yis equal to revenues minus

all costs (including opportunity cost). Derive expressions for the expected

value of farming income E(YS)and E(YR)

as functions of the interest rate i. Your expressions should take the

form of E(Y) = A+ Bi,

where you have to find Aand B.

(a) What

are the functions E(YS)

and E(YR)

(b) Graph

your functions E(YS)

and E(YR);

place ion

the horizontal-axis; title the graph â€œFigure 1â€

2. Let

?be

the profit of Brian the moneylender. Derive expressions for the expected value

of profit for a loan to a safe farmer E(?S)and a loan to a

risky farmer E(?R)

as functions of the interest rate i. Your expressions should take the

same form of E(?) = A+ Bi,

where you have to find Aand B.

(a) What

are the functions E(?S)

and E(?R)

(b) Graph

your functions E(?S)

and E(?R)

in the same â€œFigure 1â€

3. Using

your questions and graph, answer the following questions:

(a) What

is the highest interest rate a SAFE farmer would be willing to payfor a loan from Brian?

(b) What

is the highest interest rate a RISKY farmer would be willing to payfor a loan from Brian?

(c) What

is the lowest interest rate Brian would be willing

to chargeon a loan to a SAFE farmer?

(d) What

is the lowest interest rate Brian would be willing

to chargeon a loan to a RISKY farmer?

4. First,

assume that the loan market is perfectly

competitive. There are many other lenders who would charge a lower interest

than Brian, if Brian is making a profit.

(a) What

is the equilibirum interest rate Brian would charge a SAFE farmer?

(b) What

is the equilibrium interest rate Brian would charge a RISKY farmer?

(c) What

is Brianâ€™s total expected profit E(?) = E(?S)+ E(?R)?

(d) What

is total expected income across all types of famers E(Y) = E(YS)+ E(YR)?

5. Now,

assume that all the other lenders left Quahog, and now Brian is a monopolistmoneylender.

(a) What

is the equilibirum interest rate Brian would charge a SAFE farmer?

(b) What

is the equilibrium interest rate Brian would charge a RISKY farmer?

(c) What

is Brianâ€™s total expected profit E(?) = E(?S)+ E(?R)?

(d) What

is total expected income across all types of famers E(Y) = E(YS)+ E(YR)?

2 Asymmetric Information

Brian has decided to leave Quahog in search of a better life.

Peter has come from a far away town, and has decided to live in Quahog. He is

considering offering limited liability loans. Like Brian, Peterâ€™s opportunity

cost of money is 0.20 (20%), and he is a monopolist

since all the other moneylenders in Quahog have left. Peter, however, does

not know the people in Quahog, so he cannot tell who was born a SAFE farmer and

who was born a RISKY farmer. All he knows is that half of the people are SAFE

and half are RISKY farmers. Thus, Peter can charge only one interest rate. In

contrast to Brian, Peter suffers from asymmetric information. So when Peter

thinks about the single interest rate he will charge, he must think about who

will want the loan.

1. Letâ€™s

think carefully about who will want a loan depending on the interest rate.

(a) What

is the maximum interest rate Peter can charge so that both types of farmers

would want to borrow?

(b) What

is the maximum interest rate Peter can charge so that at least one type of

farmer would want to borrow?

2. Now,

letâ€™s think about Peterâ€™s expected profit function E(?):

(a) Derive

an expression for E(?)as

a function of the interest rate i, for all interest rates below the iyou

identified in (2.1.a).

(b) Derive

an expression for E(?)as

a function of the interest rate i, for all interest rates above the iyou

identified in (2.1.a) and below the iyou identified in (2.1.b)

(c) Derive

an expression for E(?)as

a function of the interest rate i, for all interest rates above the iyou

identified in (2.1.b).

3. Graph

Peterâ€™s expected profit function E(?)on a range of i= 0to i= 3(i.e.

0 to 300%)

4. Recall

that Peter is a monopolist:

(a) What

is the equilibrium interest rate that Peter sets?

(b) What

is Peterâ€™s total expected profit E(?) = E(?S)+ E(?R)?

(c) What

is total expected income across all types of famers E(Y) = E(YS)+ E(YR)?

Part B. Risk and Informal Insurance

In part B of this problem set, we introduce risk preferences

and study an alternative to formal insurance contracts. We evaluate how

asymmetric information may also affect informal insurance arrangements.

3 Risk Preferences

There are three farmers in a village: Chris with utility

function U= C0.5?2, Meg with utility

function U= C2,

and Stewie with utility function U= 15+2C. All three farmers have certain

wealth equal to $100, and they earn random farm income Ywhich depends on the unknown

level of pest infestation:

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1. Find

the following for Chris:

(a) Expected

utility

(b) Certainty

equivalent

(c) Risk

premium

(d) Risk

preference (risk loving, risk neutral or risk averse)?

2. Find

the following for Meg:

(a) Expected

utility

(b) Certainty

equivalent

(c) Risk

premium

(d) Risk

preference (risk loving, risk neutral or risk averse)?

3. Find

the following for Stewie:

(a) Expected

utility

(b) Certainty

equivalent

(c) Risk

premium

(d) Risk

preference (risk loving, risk neutral or risk averse)?

4. Quagmire

is offering an insurance contract to farmers with premium equal to $90. This

insurance scheme pays out $0 if there is low infestation, $125 if there is

medium infestation, and $200 if there is high infestation. Quagmire knows

perfectly well who Chris, Meg and Stewie are (i.e. there is no asymmetric

information).

(a) What

is Quagmireâ€™s expected profit?

(b) Will

Chris want to purchase this insurance contract?

(c) Will

Meg want purchase this insurance contract?

(d) Will

Stewie want to purchase this insurance contract?

4 Informal Risk Sharing Arrangements

Quagmire, Meg and Stewie decide to leave the village. Chris

stayed, and many other people migrated into the village. These new people are exactlylike Chris: they have the same utility

function U= C0.5?2, have the same certain

wealth of $100 and face the same random income Y(same amounts and

probabilities).

We know that all these Chris-types would

prefer some insurance, but no formal insurance is offered since Quagmire is

gone. Thus, they discuss amongst each other and decided to implement an informal risk sharing arrangement(IRSA).

All of the pest infestation risk in the village is idiosyncratic (that is, the

risk is uncorrelated across all the people in the village).

Let TL,TM,THdenote

the transfer made by a farmer into the village insurance fund when that farmer

has Low, Medium and High levels of pest infestation (a negative transfer means

the farmer receives a payment). Assume that the transfers are out of income,

not wealth. An optimal IRSAis a set of

transfers TL,TM,TH

that satisfies the following two criteria: (1) first-best: it

provides the maximum possible level of consumption smoothing (ideally it

completely eliminates risk to consumption) and; (2) affordable: the expected

value of transfers is zero for an individual (this means that, on average, the

same amount of money is going into the village pot as is coming out of the

village pot).

1. Find

the values of TL,TM

and THin an optimal IRSA.

2. Show

that the values you specified in 4.1 are (a) first-best and (b) affordable

(i.e. show that offering a higher value of consumption than your answer in 4.1

is not affordable).

3. What

is the expected utility for Chris for this optimal IRSA?

5 Asymmetric Information

Now, assume that each Chris can now choose to relax by taking

multiple breaks and not working hard on the farm. None of the people in the

village can see whether a person is working hard or relaxing. If he relaxes,

then three things change: his utility function, the random income amounts, and

the probabilities. If he relaxes, his utility function becomes U(C) = C0.5since he does not incur the disutility of 2 units

from working hard, and his random income is now given by the following:

.gif”>

1. First,

assume that no IRSA is available.

(a) What

is the expected utility of relaxing on the farm?

(b) Will

Chris choose to work hard on the farm or relax on the farm?

2. Now,

assume that the IRSA you determined in (4.1) was available.

(a) What

is the expected utility of relaxing on the farm?

(b) Will

Chris choose to work hard on the farm or relax on the farm?

3. Is

the IRSA that you determined in (4.1) affordable if each Chris can choose to

relax on the farm?

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