Question 1: Risk Preferences Anderson and Steve live the village of Los Reyes in
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Question 1: Risk Preferences
Anderson and Steve live 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 60% probability of having a GOOD harvest and a 40% chance of having a BAD harvest. If the harvest is GOOD, the farmer earns an income of $110. If the harvest is BAD, the farmer earns an income of only $35.
Activity 2: Full time construction work. This activity has no risk. An individual who decides to work full time in construction earns $77 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 $110 drops to 35%, and the probability of having a BAD harvest and earning only $35 increases to 65%. The individual also earns $17 with certainty as a construction worker (the person earns this $17 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 Anderson and Steve view risk differently, and that is why they have different utility functions (listed below).
Anderson: Steve:
?(?)=0.5? ?(?) = √?
(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
Anderson
1: Full time farming
Anderson
2: Full time construction work
Anderson
3: Part time farming
Steve
1: Full time farming
Steve
2: Full time construction work
Steve
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
(d) Which type of risk preferences describe each individual? (Risk Neutral, Risk Averse, or Risk Loving?)
Table 4
Individual Risk Preferences
Anderson
Steve
Anderson
Steve
Question 2: Conventional Insurance
Continue to use the same context from question 1, but now consider conventional insurance. Kelly 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 $30. At the end
of the season, Kelly pays farmers an indemnity payment of $72 if the farmer had a BAD harvest. If the farmer had a GOOD harvest, Kelly doesn’t pay the farmer anything. For parts a – d, assume the world is described by symmetric information. In other words, Kelly can write and enforce a contract that requires the farmer to choose full time farming.
(a) What is Kelly’s expected profit from this contract? (Kelly’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 Kelly’s insurance contract (Activity 4)?
(c) What is the expected utility associated with full-time farming with an insurance contract (Activity 4) for Anderson and Steve? Report TWO decimal places.
Table 5 Individual Expected Utility from Activity 4
Anderson Steve
(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 Kelly’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 Anderson
Steve
Now let’s make a more realistic assumption about information. Assume that Kelly 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 Kelly face?
(f) What is the expected utility associated with part-time farming with Kelly’s insurance contract (Activity 5)? Report TWO decimal places.
Table 7 Individual Expected Utility from Activity 5
Anderson
Steve
(g) Now assume that Anderson and Steve 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 Kelly’s insurance contract (Activity 5). If an individual is indifferent between two activities (say A or B), write “Activity A OR Activity B
Table 8
Individual Choice of Activity
Anderson Steve
(h) Do any of the individuals choose an activity with insurance? If no, explain why. If yes, what is Kelly’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 2/6, 3/6, and 1/6 respectively, and his farm income under Low, Medium and High infestation levels is 100, 81, and 1 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 ?(?) = √? − 1, where C is his consumption.
Table 9. Income and Probabilities if a farmer works hard
Pest Infestation Low Medium High
Probability Raghav’s Income 2/6 100
3/6 81
1/6 1
(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 1-unit utility cost. Thus his utility if he chooses to relax is ?(?) = √C
Table 10. Income and Probabilities if a farmer relaxes
Pest Infestation Low Medium High
Probability Income
1/3 81 1/3 64 1/3 0
(e) What is Raghav’s expected utility if he relaxes? Report TWO decimal places.
(f) IntheabsenceoftheIRSAarrangement,wouldRaghavprefertofarmandworkhardorinstead 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.
