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:
•
SAFE farmers have a good harvest all the time,
they earn revenues of $700 with 100% probability.
•
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 “Figure 1”
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 “Figure 1”
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 Brian’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 Brian’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, Peter’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. Let’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,
let’s think about Peter’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
Peter’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 Peter’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:
.gif”>
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 Quagmire’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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