PS444

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University of California, Davis *

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115A

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Economics

Date

Feb 20, 2024

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Question 1: Risk Preferences A lvaro and Estela live in the village of Los Reyes in the state of Michoacan, Mexico 1 . 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 % chan c e 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 $1 44 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 4 0%, and the probability of having a BAD harvest and earning only $50 increases to 6 0%. 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 1 Mexico has become one of the leading exporters of strawberries in the past 15 years. Michoacan is the most important strawberry producing region in Mexico. (0.65×200)+(0.35×50) = 147.5 144 (0.40×200+25) + (0.60×50+25)= 135

We also know that A lavaro and Estela view risk differently, and that is why they have different utility functions (listed below). A lvaro : Estela : ?(𝐶) = 2𝐶 ?(𝐶) = √𝐶 (b) Using those utility function s , 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 A lvaro 1: Full time farming A lvaro 2: Full time construction work A lvaro 3: Part time farming Estela 1: Full time farming Estela 2: Full time construction work Estela 3: Part time farming 295 147.5 147.5 288 144 144 270 135 135 11.66 136.25 68.13 12 144 72 11.20 125.44 62.75

(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 A lvaro Estela (d) Which type of risk preferences describe each individual? (Risk Neutral, Risk Averse, or Risk Loving?) Table 4 Individual Risk Preferences A lvaro 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 wri te 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). full time construction risk loving full time farming risk averse

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(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 A lvaro 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 A lvaro Estela Individual Expected Utility from Activity 4 A lvaro Estela 0 295 activity 1 OR activity 4 147.5 12.15 Activity 4

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 A lvaro 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 Juana ’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 A lvaro Estela There are 2 main risks. Moral Hazard, Adverse Risk. The first being that she cannot observe the work that they are completing. there is a chance that they are not working or they. The second problem she faces is that the worker may choose to jepordize the crop in order to recieve the insurace pay since there is incetive for a bad crop. Since there is no way for her to observe the actual farming there is no way for her to accurately price the risk at hand. 345 13.13

Table 8 Individual Choice of Activity A lvaro 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 Ragh av 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 20 0, 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 ?(𝐶) = 5 √𝐶 − 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 Activity 5 Yes, they both choose activity 5, which is part time Farming with insurance. Juana's expected under the farmer Activity 5 is 37.5, which is lower than the previously calculated because the part time farming leads to a higher probability of bad harvests and therefore her making indemnity payouts. Because her expected profit is negative, she will not offer this insurance contract, assuming she is aware that Activity 5 is optimal for both farmers. Activity 5

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(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. 120 -4.24 Raghav exhibits risk averse behavior because you can tell he is more concerned with income loss then income gain. He is more sensitve to it. He values the certenty of income. You can see it by observing the concavity of the utility function.

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 ?_𝐻 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 ?_𝐻 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 ?(𝐶) = 5 √𝐶 . Table 10. Income and Probabilities if a farmer relaxes Pest Infestation Probability Income Low 1/4 20 0 Medium 2/5 10 0 High 7/20 0 ?_? = ?_? = ?_𝐻 = -80 20 120

(e) What is Raghav ’s expected utility if he relax es? 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 ?_𝐻 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. 1.72 In the absence of IRSA it would be in his best interest to relax because the utility is higher. It is 1.72 instead of -4.24 due to the fact he does not have to pay the utility cost of 8. Farming and relaxing allows him to maximize his utility and income. He would choose to relax because, as reitered above, the utility is signiciantly higher. In the sinario where insurance comes into play, it does not effect his descion because regardless of the insurance, his utility is still higher when he works harder. There is more incentive to not do better due to the transfers, he will remain relaxed at within any infestation because the utility is still higher.

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