Dan 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 10 to Dan.

His utility function if he works hard is () = 5 10, where C is his consumption.

Dan lives in a village with many other farmers that have the same utility function as Dan 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.

Lets 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 provides the maximum possible level of consumption smoothing, ideally it completely eliminates risk to consumption at the expected value of his 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).

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For questions (e) through (i), lets 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 lets 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 10 summarizes these probabilities and income levels if the farmer relaxes.

Finally, if a farmer relaxes, he does not incur the 10-unit utility cost. Thus his utility if he chooses to relax is () = 5. (Recall that his utility if he instead works hard is () = 5 10.)

(e) What is Dans expected utility if he relaxes? Report in TWO decimal places if necessary.

(f) In the absence of the IRSA arrangement, would Dan 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 Dan choose to work hard or relax? i.e., if Dan 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.

(i) How would the informal insurance arrangement from question (d) 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.

Table 9. Income and Probabilities if a farmer works hard Pest Infestation Low Medium High Probability 2/5 2/5 1/5 Dan's Income 200 100 0 Table 10. Income and Probabilities if a farmer relaxes Pest Infestation Low Medium High Probability 1/3 1/3 1/3 Income 150 50 0 Table 9. Income and Probabilities if a farmer works hard Pest Infestation Low Medium High Probability 2/5 2/5 1/5 Dan's Income 200 100 0 Table 10. Income and Probabilities if a farmer relaxes Pest Infestation Low Medium High Probability 1/3 1/3 1/3 Income 150 50 0