Part 1: RISK PREFERENCES AND CONVENTIONAL INSURANCE

Kelly, Isaac and Daniel are sunflower farmers in the village of Girasol. 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:

ï‚·

Activity 1: Full time farming. Sunflower 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 50% probability of having a GOOD harvest

and a 50% change of having a BAD harvest. If the harvest is GOOD, the farmer earns an income of $150. If the

harvest is BAD, the farmer earns an income of only $50.

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Activity 2: Full time construction work. This activity has no risk. An individual who decides to work full time in

construction earns $95 with certainty.

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Activity 3: Part-time farming. In this third activity, the farmer works during the week as a sunflower farmer, and

works in construction during the weekend. Since she is not able to work full time on the farm, the probability of

having a GOOD harvest and earning $150 drops to 25%, and the probability of having a BAD harvest and earning

only $50 increases to 75%. The individual also earns $20 with certainty as a construction worker (the person earns

this $20 from construction

in addition to

her farm income under both a GOOD and BAD harvest).

1. What is the expected value of consumption for each activity?

2.

Kelly, Isaac, and Daniel view risk differently, and that is why they have different utility functions (listed below).

Using those utility function, 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 1 below.

You should

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 when it is necessary. [Suggestion: Use Excel to do this

part!]

ARE/ECN 115A

Winter 2020

Problem Set #4: RISK, RISK TAKING AND INSURANCE

Due: Tuesday, Mar 10

th

Expected Value of Consumption: E(C)

Activity 1: Full time farming

Activity 2: Full time construction work

Activity 3: Part time farming

2

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Kelly:

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(

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)

=

5í¶

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Isaac:

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(

í¶

)

=

í¶

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Daniel:

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(

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=

0.05

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2

3. Which activity will be chosen by each individual?

Table 1. Certainty Equivalent, Risk Premium and Expected Utility for 3 Activities

Individual

Activity

EU

CE

RP

Kelly

1. Full Time Farming

Kelly

2. Full Time Construction Work

Kelly

3. Part Time Farming

Isaac

1. Full Time Farming

Isaac

2. Full Time Construction Work

Isaac

3. Part Time Farming

Daniel

1. Full Time Farming

Daniel

2. Full Time Construction Work

Daniel

3. Part Time Farming

3.

Choice of Activity

Kelly

Isaac

Daniel

3

4. Which type of risk preferences describe each individual? (Risk Neutral, Risk Averse, or Risk Loving?)

Oscar is an insurance agent who offers conventional crop insurance contracts only to

full time

farmers. He 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 $50. At the end of the season, Oscar pays farmers an indemnity payment of $100 if the farmer

had a BAD harvest. If the farmer had a GOOD harvest, Oscar doesnâ€™t pay the farmer anything. For questions 5 – 8,

assume that Oscar has perfect information about the farmerâ€™s activity choice. In other words, he can write and enforce a

contract that requires the farmer to choose full time farming.

You need to show your work that how you get your final answer to earn full credits for Question 5 â€“ 7.

5. What is Oscarâ€™s expected profit from this contract? (Oscarâ€™s profit is just the premium he collects from the farmer

minus the indemnity payment he makes to the farmer).

6. What is the expected consumption for an individual who chooses full-time farming with Oscarâ€™s insurance contract?

4.

Risk Preferences

Kelly

Isaac

Daniel

4

7. What is the expected utility associated with full-time farming with an insurance contract(Activity 4) for Kelly, Isaac

and Daniel?

8. 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 Oscarâ€™s insurance contract (Activity 4). Which activity will each

individual choose?

Now letâ€™s make a more realistic assumption about information. Assume that Oscar

cannot

observe and enforce the

amount of time that individuals work on their farm. He 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.

9. What type of asymmetric information problem does Oscar face? [Moral Hazard or Adverse Selection]

Expected Utility from Activity 4

Kelly

Isaac

Daniel

8.

Choice of Activity

Kelly

Isaac

Daniel

5

10. What is the expected utility associated with part-time farming with an insurance contract (Activity 5) for Kelly, Isaac

and Daniel?

Report your final answer only.

11. Now assume that each individual 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 Oscarâ€™s insurance contract (Activity 4) and Part Time Farming with

Oscarâ€™s insurance contract(Activity 5). Which activity will each individual choose?

Expected Utility from Activity 5

Kelly

Isaac

Daniel

11.

Choice of Activity

Kelly

Isaac

Daniel

6

12. i). Do any of the individuals choose an activity with insurance?

ii). If so, what is Oscarâ€™s expected profit from these insurance contracts?

iii). Will he be willing to offer the insurance contract? Why or why not?

Part 2: INFORMAL RISK SHARNG ARRANGEMENTS

JosÃ© 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 13 â€“ 16) 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 121, 16, and 0 respectively. Table 2

summarizes these probabilities and incomes:

Table 2. Income and Probabilities if a farmer works hard

Pest Infestation

Probability

JosÃ©â€™s Income

Low

2/5

121

Medium

2/5

16

High

1/5

0

13. What is the expected value of income from farming and working hard?

Show your work of calculation.

12(i)

12(ii)

12(iii)

7

Working hard imposes a utility cost of 1 to JosÃ©. His utility function if he works hard is

, where

C

is his

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(

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)

=

í¶

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1

consumption.

14. Is JosÃ© risk averse, risk neutral, or risk loving?

Support your answer.

JosÃ© lives in a village with many other farmers that have the same utility function as JosÃ© and face the same risks

associated with pests given by Table 2. 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

í‘‡

í¿

í‘‡

í‘€

í‘‡

í»

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.

Letâ€™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 14 that villagers have

symmetric information

and that everybody will thus work

hard.

15. Find the values of ,

, and

in an optimal informal risk sharing arrangement (IRSA).

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í‘‡

í‘€

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An optimal IRSA satisfies the following two characteristics: 1) It provides the maximum possible level of

consumption smoothing, ideally it completely eliminates risk to consumption 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).

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í¿

=

í‘‡

í‘€

=

í‘‡

í»

=

8

For questions 16 through 19, letâ€™s assume that our villagers work on farms that are far away from each. This means that

villagers face

asymmetric

information because they cannot observe if other farmers are working hard or not. Now letâ€™s

allow farmers to choose how hard they work. They can either work hard (as above) or they can relax. Compared to

working hard, three things happen if a farmer relaxes. First, the probabilities of getting the different pest infestation levels

change (it becomes more likely to get high infestation levels). Second, income levels under the different infestation levels

decrease. Table 3 summarizes these probabilities and income levels if the farmer relaxes:

Table 3. Income and Probabilities if a farmer relaxes

Pest Infestation

Probability

JosÃ©â€™s Income

Low

1/3

100

Medium

1/3

4

High

1/3

0

Finally, if a farmer relaxes, he does not incur the 1-unit utility cost. Thus his utility if he chooses to relax is

.

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(

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)

=

í¶

(Recall that his utility if he instead works hard is

.)

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(

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)

=

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1

16. i) What is JosÃ©â€™s expected utility if he farms and works hard?

ii)

What is

JosÃ©â€™s expected utility if he farms and works relax?

Show your work of calculation.

iii) In the absence of the IRSA arrangement, would JosÃ© prefer to farm and work hard or instead farm and relax?

(Assume he cannot work for wage labor off-farm.)

16 (i)

16 (ii)

16(iii)

9

Now letâ€™s assume that it is NOT possible for farmers to observe and enforce each othersâ€™ effort levels. In this case, JosÃ©

(and all the other farmers) will choose the effort level (either work hard or relax) that maximizes their expected utility.

17. If the ideal insurance arrangement that you found in question 15 were available, would JosÃ© choose to work hard or

relax? i.e., if Jose receives the transfers ,

, and

that you found in question 15 if he has Low, Medium and

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í‘‡

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High infestation even if he chooses to relax, would he choose to Work Hard or Relax?

Support your answer.

18. Based on your answer to question 17, would the ideal insurance arrangement from question 15 be feasible for the

village?

19. How would the informal insurance arrangement from question 15 be modified so that it would still induce farmers to

work hard? (Just describe a basic strategy; you do not need to find the exact transfer levels.)