1 Ghanaian farmers face a lot of risks and often have no insurance. This question is inspired by the reading reaction/podcast. Assume farmers face two states of the world; good rainfall" (with output of $10000), and "bad rainfall [with output of $625) with equal probability. Farmers have a Utility function U-output1/4 a) Assume the cash grant intervention provides farmers with $1000. How much better off are farmers with the cash grant (compared to no cash grant) in terms of expected utility and expected income? (recall you get the cash grant regardless of the weather) b) Karlan insurance company offers a rainfall insurance policy to farmers. However, the insurance will not fully cover the losses from bad rainfall. Find what the insurance payout needs to be so that a farmer would be indifferent between the cash grant and insurance? c) Calculate the actuarially fair price of the insurance policy that pays out the amount you calculated in question 3. (hint: the actuarially fair price just means that the price of insurance is equal to the expected payments the insurance company has to make) d) Calculate the risk premium for a farmer given the utility function and risks. e) Suppose that the government had 100 farmers who were spread out enough such that the probability of rainfall was independent across farmers (i.e. each farmer faces a 0.5 probability of good rainfall). Which public policy option would be more cost effective - insurance or cash grants? (hint which policy option would cost less for a given level of effect") 1) Briefly explain why insurance is more "valuable when a person's risks aversion is higher [you may want to use a graph to show what happens as people become more risk averse) I a 2. Christian's utility of income function is U = In(I + 1) and lan's utility of income function is U = 0.4 125 (where I = income) Based on these utility functions show and explain whether each individual is risk loving, risk averse, or risk neutral? b. Sketch each individual's utility vs income curves and sketch the risk premium for an arbitrarily chosen gamble. (Just sketch the utility vs income curves and show the risk premium for a gamble that you choose.) 3. Nora is a farmer. She receives a weather forecast that shows there is a 50-50 chance that next growing season will be abnormally rainy (e.g. floods). Her expected utility has the form U = 0.5 In (YNR) + 0.5ln (YVR) Where Ynr and Yvr represent her income in the states of "normal rain" and "very rainy' respectively. Suppose the farmer must choose between two crops that yield the following income prospects Yur Ynr Crop Wheat $28,000 $19,000 $10,000 $15,000 Com (a) Compute the Expected income and Expected utility for each crop. Which crops will she plant? (b) Suppose Nora can plant half her field with each crop. Would she choose to do so? Explain your answer (C) Suppose Slim-Shady Insurance company offers insurance to farmers who only grow wheat. The policy costs $4,000 but pays $8,000 in the event of a very rainy growing season. Would she be willing to purchase such a policy? Explain (don't forget she has to decide whether to grow only corn, only wheat, or a 50-50 mix) (d) In the absence of an insurance policy what is the mix of corn and wheat that would provide her with the greatest expected utility? I 4. Hotel Jasmine is a five-star hotel. The hotel faces a risk that it will suffer a fire causing a $200 million loss with a probability of 0.02. The owner of the hotel, Kristen, has a utility function of U = W0.5, where W is the owner's wealth (measured by the value of the hotel in millions of dollars). Suppose the initial value of the hotel is $225 million (W-225) a) what is Kristen's expected loss? b) what is Kristen's expected utility? c) What is Kristen's risk premium? d) What is the most Kristen is willing to pay for insurance ? 5. (Bonus) Khal and Daenerys are a newlywed couple. Since winter is approaching they must hunt and gather for supplies. There are two good hunting locations nearby. The table below shows the mean and variance of the supplies collected (per person). Mean Supplies Collected Variance of Supplies collected Northern Forest 250 75 Southern Forest 200 50 Assume that the mean and variance do not change if both people go to the same Forest. Using the insights from class briefly discuss why the couple might be better off sending one person to the Northern Forest and the other to the Southern Forest rather than both individuals going hunt and gather in the same forest? 6. Bonus 2: Why might a neighborhood group have a harder time self-insuring for flood damage versus fire damage? 1 Ghanaian farmers face a lot of risks and often have no insurance. This question is inspired by the reading reaction/podcast. Assume farmers face two states of the world; good rainfall" (with output of $10000), and "bad rainfall [with output of $625) with equal probability. Farmers have a Utility function U-output1/4 a) Assume the cash grant intervention provides farmers with $1000. How much better off are farmers with the cash grant (compared to no cash grant) in terms of expected utility and expected income? (recall you get the cash grant regardless of the weather) b) Karlan insurance company offers a rainfall insurance policy to farmers. However, the insurance will not fully cover the losses from bad rainfall. Find what the insurance payout needs to be so that a farmer would be indifferent between the cash grant and insurance? c) Calculate the actuarially fair price of the insurance policy that pays out the amount you calculated in question 3. (hint: the actuarially fair price just means that the price of insurance is equal to the expected payments the insurance company has to make) d) Calculate the risk premium for a farmer given the utility function and risks. e) Suppose that the government had 100 farmers who were spread out enough such that the probability of rainfall was independent across farmers (i.e. each farmer faces a 0.5 probability of good rainfall). Which public policy option would be more cost effective - insurance or cash grants? (hint which policy option would cost less for a given level of effect") 1) Briefly explain why insurance is more "valuable when a person's risks aversion is higher [you may want to use a graph to show what happens as people become more risk averse) I a 2. Christian's utility of income function is U = In(I + 1) and lan's utility of income function is U = 0.4 125 (where I = income) Based on these utility functions show and explain whether each individual is risk loving, risk averse, or risk neutral? b. Sketch each individual's utility vs income curves and sketch the risk premium for an arbitrarily chosen gamble. (Just sketch the utility vs income curves and show the risk premium for a gamble that you choose.) 3. Nora is a farmer. She receives a weather forecast that shows there is a 50-50 chance that next growing season will be abnormally rainy (e.g. floods). Her expected utility has the form U = 0.5 In (YNR) + 0.5ln (YVR) Where Ynr and Yvr represent her income in the states of "normal rain" and "very rainy' respectively. Suppose the farmer must choose between two crops that yield the following income prospects Yur Ynr Crop Wheat $28,000 $19,000 $10,000 $15,000 Com (a) Compute the Expected income and Expected utility for each crop. Which crops will she plant? (b) Suppose Nora can plant half her field with each crop. Would she choose to do so? Explain your answer (C) Suppose Slim-Shady Insurance company offers insurance to farmers who only grow wheat. The policy costs $4,000 but pays $8,000 in the event of a very rainy growing season. Would she be willing to purchase such a policy? Explain (don't forget she has to decide whether to grow only corn, only wheat, or a 50-50 mix) (d) In the absence of an insurance policy what is the mix of corn and wheat that would provide her with the greatest expected utility? I 4. Hotel Jasmine is a five-star hotel. The hotel faces a risk that it will suffer a fire causing a $200 million loss with a probability of 0.02. The owner of the hotel, Kristen, has a utility function of U = W0.5, where W is the owner's wealth (measured by the value of the hotel in millions of dollars). Suppose the initial value of the hotel is $225 million (W-225) a) what is Kristen's expected loss? b) what is Kristen's expected utility? c) What is Kristen's risk premium? d) What is the most Kristen is willing to pay for insurance ? 5. (Bonus) Khal and Daenerys are a newlywed couple. Since winter is approaching they must hunt and gather for supplies. There are two good hunting locations nearby. The table below shows the mean and variance of the supplies collected (per person). Mean Supplies Collected Variance of Supplies collected Northern Forest 250 75 Southern Forest 200 50 Assume that the mean and variance do not change if both people go to the same Forest. Using the insights from class briefly discuss why the couple might be better off sending one person to the Northern Forest and the other to the Southern Forest rather than both individuals going hunt and gather in the same forest? 6. Bonus 2: Why might a neighborhood group have a harder time self-insuring for flood damage versus fire damage