4. Calibration of Utility Functions. You are a decision analysis consultant, and you have a hydropower company as a client. You have asked some certainty equivalence-type questions of the CEO. You are going to use those answers to create a utility function for the amount of fish survival. (Adult fish have to migrate upstream from the ocean past their dam to the spawning grounds, while juvenile fish have to migrate downstream from where their eggs hatched down to the ocean. Passing through power turbines or through bypasses results in mortality of both adults and juvenile fish.) The hydropower company is considering removing the dam or investments in fish passage facilities that would lower mortality. Imagine that you know the following: x (least desirable alternative) = 10% reduction in fish mortality x* (most desirable alternative) = 80% reduction in fish mortality x (certainty equivalent) = 32% reduction in fish mortality, where this is a certainty equivalent for a lottery with a 0.5 chance of x and a 0.5 chance of x*. a. What is the expected utility of x? b. Fit an exponential U(X) to those 3 points (x,x ,x*), where U(X)= a-b*exp(-cX), where exp(y) = 2.7128. (L.e., find a,b,c that pass through those three points.) (Hint: three points define three equations, which can be used to solve for the three coefficients. However, if you use two equations to eliminate a and b, the remaining equation that involves just c might have to be solved through trial and error or numerically. Use a spreadsheet. We did this in class.) c. Repeat (b), but instead fit a quadratic function U(X) = a+bX+cX^2. Plot the two utility functions on the same graph. Do they differ? Is there any reason you might prefer to use the quadratic function to the exponential function, or vice versa? d. Using the exponential utility function: d. Using the exponential utility function: i. Determine the risk premium associated with a lottery with a 0.5 chance of x and a 0.5 chance of x*. ii. Determine the certainty equivalent and risk premium associated with a lottery with a 0.6 chance of a 30% reduction and a 0.4 chance of a 60% reduction e. Using the quadratic utility function, repeat d(i) and d(ii). Any surprises? f. Using the exponential utility function, which is the better alternative: i. A lottery with a 0.5 chance of x and a 0.5 chance of x* ii. A lottery with a 0.35 chance of x, a 0.5 chance of 35% reduction, and a 0.15 chance of x*. iii. Now calculate the certainty equivalent of (a) and (b); does the alternative with the better certainty equivalent have the higher expected utility? In words, what does the difference in their certainty equivalents tell you, and how does it help you interpret the difference in expected utility? g. Using the exponential utility function, answer the following question. Imagine that the utility executive instead answered a probability equivalence question in order to calibrate her utility function. In this probability equivalence question, she compared a lottery with chance (1-p) of x (least desirable alternative) and chance p of x* (most desirable alternative) with a certain alternative x = 45% reduction in fish mortality. Assume that the CEO answers the probability equivalence question in a way consistent with the exponential utility function you calibrated in (b) above; what answer p would she give? (I.e., it would result in the expected utility of both alternatives being equal?) 4. Calibration of Utility Functions. You are a decision analysis consultant, and you have a hydropower company as a client. You have asked some certainty equivalence-type questions of the CEO. You are going to use those answers to create a utility function for the amount of fish survival. (Adult fish have to migrate upstream from the ocean past their dam to the spawning grounds, while juvenile fish have to migrate downstream from where their eggs hatched down to the ocean. Passing through power turbines or through bypasses results in mortality of both adults and juvenile fish.) The hydropower company is considering removing the dam or investments in fish passage facilities that would lower mortality. Imagine that you know the following: x (least desirable alternative) = 10% reduction in fish mortality x* (most desirable alternative) = 80% reduction in fish mortality x (certainty equivalent) = 32% reduction in fish mortality, where this is a certainty equivalent for a lottery with a 0.5 chance of x and a 0.5 chance of x*. a. What is the expected utility of x? b. Fit an exponential U(X) to those 3 points (x,x ,x*), where U(X)= a-b*exp(-cX), where exp(y) = 2.7128. (L.e., find a,b,c that pass through those three points.) (Hint: three points define three equations, which can be used to solve for the three coefficients. However, if you use two equations to eliminate a and b, the remaining equation that involves just c might have to be solved through trial and error or numerically. Use a spreadsheet. We did this in class.) c. Repeat (b), but instead fit a quadratic function U(X) = a+bX+cX^2. Plot the two utility functions on the same graph. Do they differ? Is there any reason you might prefer to use the quadratic function to the exponential function, or vice versa? d. Using the exponential utility function: d. Using the exponential utility function: i. Determine the risk premium associated with a lottery with a 0.5 chance of x and a 0.5 chance of x*. ii. Determine the certainty equivalent and risk premium associated with a lottery with a 0.6 chance of a 30% reduction and a 0.4 chance of a 60% reduction e. Using the quadratic utility function, repeat d(i) and d(ii). Any surprises? f. Using the exponential utility function, which is the better alternative: i. A lottery with a 0.5 chance of x and a 0.5 chance of x* ii. A lottery with a 0.35 chance of x, a 0.5 chance of 35% reduction, and a 0.15 chance of x*. iii. Now calculate the certainty equivalent of (a) and (b); does the alternative with the better certainty equivalent have the higher expected utility? In words, what does the difference in their certainty equivalents tell you, and how does it help you interpret the difference in expected utility? g. Using the exponential utility function, answer the following question. Imagine that the utility executive instead answered a probability equivalence question in order to calibrate her utility function. In this probability equivalence question, she compared a lottery with chance (1-p) of x (least desirable alternative) and chance p of x* (most desirable alternative) with a certain alternative x = 45% reduction in fish mortality. Assume that the CEO answers the probability equivalence question in a way consistent with the exponential utility function you calibrated in (b) above; what answer p would she give? (I.e., it would result in the expected utility of both alternatives being equal?)