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ARE/ECN 115A Fall 2023
Problem Set 4. RISK, RISK TAKING AND INSURANCE
Due: Tuesday, December 5th at 11:59 PM
Instructions
During this problem set, you will apply concepts of risk and risk preferences to explore how risk
affects peoples’ choice of economic activities and their income. You will also explore how the
presence of insurance – both formal insurance and informal risk sharing arrangements – can
change peoples’ choice of activities and welfare. Finally, you will explore how asymmetric
information can adversely affect the performance of insurance markets.
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Question 1: Risk Preferences
Alvaro and Estela live in the village of Los Reyes in the state of Michoacan, Mexico1.
They each have zero wealth, so their consumption is equal to the income they earn from
their economic activity. Each of them must choose one (and only one) of the following
three activities:
• Activity 1: Full time farming. Strawberry farming is risky because of a combination of weather
and pests. Under full time farming, the farmer works 7 days per week on their farm. There is
a 65% probability of having a GOOD harvest and a 35% chance of having a BAD harvest. If
the harvest is GOOD, the farmer earns an income of $200. If the harvest is BAD, the farmer
earns an income of only $50.
• Activity 2: Full time construction work. This activity has no risk. An individual who decides
to work full time in construction earns $144 with certainty.
• Activity 3: Part-time farming. In this third activity, the farmer works during the week as a
strawberry farmer and works in construction during the weekend. Since he is not able to work
full time on the farm, the probability of having a GOOD harvest and earning $200 drops to
40%, and the probability of having a BAD harvest and earning only $50 increases to 60%. The
individual also earns $25 with certainty as a construction worker (the person earns this $25
from construction in addition to his farm income under both a GOOD and BAD harvest).
(a) What is the expected value of consumption for each activity? Report your answers in Table
1 below.
Table 1
Activity Expected Value of Consumption: E(C)
1: Full time farming
2: Full time construction work
3: Part time farming
1Mexico has become one of the leading exporters of strawberries in the past 15 years. Michoacan is
the most important strawberry producing region in Mexico.
We also know that Alavaro and Estela view risk differently, and that is why they have different
utility functions (listed below).
Alvaro:
Estela:
() = 2
() = √
(b) Using those utility functions, compute the certainty equivalent (CE), the risk premium (RP)
and expected utility (EU) associated with each of the three activities for each individual.
Report your answers in Table 2 below. Report precise final results, which means that you
should use all decimals of your intermediate results to get your final answer. Round your
final answers to TWO decimal places.
Table 2
Individual Activity EU CE RP
Alvaro 1: Full time farming
Alvaro 2: Full time
construction work
Alvaro 3: Part time farming
Estela 1: Full time farming
Estela 2: Full time
construction work
Estela 3: Part time farming
(c) Which activity will be chosen by each individual? If an individual is indifferent between two
activities (say A or B), write "Activity A OR Activity B"
Table 3
Individual Choice of Activity
Alvaro
Estela
(d) Which type of risk preferences describe each individual? (Risk Neutral, Risk Averse, or
Risk Loving?)
Table 4
Individual Risk Preferences
Alvaro
Estela
Question 2: Conventional Insurance
Continue to use the same context from question 1, but now consider conventional insurance. Juana
is an insurance agent who offers conventional crop insurance contracts only to full time farmers.
She is not interested in offering insurance to part time farmers. The contracts are
straightforward. At the beginning of the season, farmers pay a premium of $52.5. At the end
of the season, Juana pays farmers an indemnity payment of $150 if the farmer had a
BAD harvest. If the farmer had a GOOD harvest, Juana doesn’t pay the farmer anything. For
parts a - d, assume the world is described by symmetric information. In other words, Juana
can write and enforce a contract that requires the farmer to choose full time farming.
(a) What is Juana’s expected profit from this contract? (Juana’s profit is just the premium she
collects from the farmer minus the indemnity payment she makes to the farmer).
(b) What is the expected consumption for an individual who chooses full-time farming with
Juana’s insurance contract (Activity 4)?
(c) What is the expected utility associated with full-time farming with an insurance contract
(Activity 4) for Alvaro and Estela? Report TWO decimal places.
Table 5
(d) Now assume that each individual can choose between the four available activities: Full
Time Farming without Insurance (Activity 1 above), Full time construction work (Activity
2 above), Part Time Farming without insurance (Activity 3 above) and Full Time Farming
with Juana’s insurance contract (Activity 4). Which activity will each individual choose?
If an individual is indifferent between two activities (say A or B), write "Activity A OR Activity B
Table 6
Individual Choice of Activity
Alvaro
Estela
Individual Expected Utility from Activity 4
Alvaro
Estela
Now let’s make a more realistic assumption about information. Assume that Juana cannot
observe and enforce the amount of time that individuals work on their farm. She can only
observe if the individual does any farming. This means that an individual may purchase the
insurance contract and then choose to either farm full time or farm part time. An individual who
chooses full time construction work cannot purchase an insurance contract.
(e) What type of asymmetric information problem does Juana face?
(f) What is the expected utility associated with part-time farming with Juana’s insurance
contract (Activity 5)? Report TWO decimal places.
Table 7
(g) Now assume that Alvaro and Estela can choose between the five available activities: Full
Time Farming without Insurance (Activity 1 above), Full time construction work (Activity
2 above), Part Time Farming without insurance (Activity 3 above), Full Time Farming with
Kelly’s insurance contract (Activity 4) and Part Time Farming with Juana’s insurance
contract (Activity 5). If an individual is indifferent between two activities (say A or B),
write "Activity A OR Activity B
Individual Expected Utility from Activity 5
Alvaro
Estela
Table 8
Individual Choice of Activity
Alvaro
Estela
(h) Do any of the individuals choose an activity with insurance? If no, explain why. If yes,
what is Juana’s expected profit from these insurance contracts? Will she be willing to offer
the insurance contract? Why or why not?
Question 3: Informal Risk Sharing Arrangements
Raghav is a farmer with zero wealth (so his consumption will equal his income). His farm
income, y, is subject to risk from pests. Pest infestation can take three possible values: Low,
Medium and High. If he works hard (which we will assume he does for parts (a) – (d)) then the
probabilities of getting Low, Medium and High infestation levels are 2/5, 2/5, and 1/5
respectively, and his farm income under Low, Medium and High infestation levels is 200, 100,
and 0 respectively. Table 9 summarizes these probabilities and incomes. Working hard imposes a
utility cost of 8 to Raghav. His utility function if he works hard is () = 5√ − 8, where C is
his consumption.
Table 9. Income and Probabilities if a farmer works hard
Pest Infestation Probability Raghav’s Income
Low 2/5 200
Medium 2/5 100
High 1/5 0
(a) What is the expected value of income from farming and working hard?
(b) What is Raghav’s expected utility if he farms and works hard? Report TWO decimal places.
(c) Is Raghav risk averse, risk neutral, or risk loving? Explain.
Raghav lives in a village with many other farmers that have the same utility function as Raghav
and face the same risks associated with pests given by Table 9. The village decides to implement
an informal risk sharing arrangement (IRSA). The arrangement works as follows. _, _,
and _ are the amount of money a farmer must transfer into the village insurance fund when
that farmer has Low, Medium and High levels of pest infestation. A negative transfer means the
farmer gets to withdraw money from the village insurance fund.
Let’s start by assuming that all the villagers have farms very near to each other, so they can observe
how hard everybody works. We will thus assume for question (d) that villagers have symmetric
information and that everybody will work hard.
(d) Find the values of _, _, and _ in an optimal informal risk sharing arrangement
(IRSA). An optimal IRSA satisfies the following two characteristics: 1) It allows
farmers to perfectly smooth consumption and guarantees that the value of consumption is
equal to the expected value of their income and; 2) The expected value of transfers is zero
(this means that, on average, the same amount of money is going into the village pot as
out of the village pot).
For questions (e) through (g) let's change our assumption about the information environment.
Let's now assume that villagers live and work on farms that are far away from each other."This
means that villagers face asymmetric information because they cannot observe if other farmers
are working hard or not. Now let’s allow farmers to choose how hard they work. They can either
work hard (as above) or they can relax. Compared to working hard, the probabilities of getting
the different pest infestation levels change (it becomes more likely to get high infestation
levels). However, income levels under the different infestation levels do not change.
Table 10 summarizes these probabilities and income levels if the farmer relaxes.
Finally, if a farmer relaxes, he does not incur the 8-unit utility cost. Thus his utility if he chooses
to relax is () = 5√.
Table 10. Income and Probabilities if a farmer relaxes
Pest Infestation Probability Income
Low 1/4 200
Medium 2/5 100
High 7/20 0
_ =
_ =
_ =
(e) What is Raghav’s expected utility if he relaxes? Report TWO decimal places.
(f) In the absence of the IRSA arrangement, would Raghav prefer to farm and work hard or instead
farm and relax? (Assume he cannot work for wage labor off-farm.) Explain.
(g) If the ideal insurance arrangement that you found in question (d) were available, would
Raghav choose to work hard or relax? i.e., if Raghav receives the transfers _, _,
and _ that you found in question (d) when he has Low, Medium and High infestation
even if he chooses to relax, would he choose to Work Hard or Relax? Explain.
(h) Based on your answer to question (g), would the ideal insurance arrangement from question
(d) be feasible for the village? Explain.