a. Again, this question will be answered on the basis of subjective judgment. Here are some possible types :

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1 If the two agree on the bet, then for both EU(et) > U(). a. If P(Rain tomorrow) =.1, and they agree on this probability, For,.1 U(4).9 U(-1) > U(), and E(Payoff) = -5. For,.1 U(-4).9 U(1) > U(), and E(Payoff) = 5. For each individual, the CE for the bet must be greater than U(). With an upward-sloping utility function, this implies CE >. Since E(Payoff) = -5 for, the only way for CE > is for the utility function to be convex. In fact, it must be convex enough for the inequality to hold. must be a risk seeker: EU(et)= U(CE) U() EU(et)= U(CE) U() -5 CE CE 5 For, the utility function might be concave, but not so much that CE <. could be risk-averse. b. If they agree that P(Rain tomorrow) =., then E(Payoff for ) = 5 and E(Payoff for ) = -5. Exactly the same arguments hold now as before, except that and have traded places. Now must be risk-seeking and could be risk-averse, risk-neutral, or risk-seeking. c. If we know nothing about their probabilities, it is possible that their utility functions are identical. For example, they could both be risk seeking. lso, they could have different assessments of the probability of rain such that EU(bet) for each would be greater than U(), given the same (possibly concave) utility function for each. For example, could think that P(Rain tomorrow) =.999 while assesses P(Rain tomorrow) =.1. In this case, each one could be very risk-averse yet still accept the bet. d. If they agree that P(Rain tomorrow) =., then the expected payoff for each is. However, if they have agreed to bet, then their CEs must each be greater than. Thus, in this case each one must be risk-seeking; only for a convex (risk-seeking) utility function is the CE for a bet greater than its expected value a. gain, this question will be answered on the basis of subjective judgment. Here are some possible types : The black-coffee drinker The coffee hater Proportion of coffee 1 Proportion of coffee 1

2 The coffee-with-a-little-milk lover Proportion of coffee 1 It is important to note that there may indeed be cases where the utility function is not monotonic; that is, that the high point on the utility curve is somewhere in the middle, like the coffee-with-a-little-milk lover. b. Take the coffee-with-a-little-milk lover, and let the peak be at c*. The decision tree, with c = c*, would be: Cup with c* coffee and (1 - c*) milk c* lack coffee (c = 1) (1-c*) Milk (c = ) Clearly, the expected proportions of coffee are the same: E(c ) = c* E(c ) = c*(1) (1-c*) = c* However, the cup in is preferred to either of the outcomes in because of the shape of the utility function. Thus, comparing the cups of coffee on the basis of the expected proportion of coffee is not a good preference model in this case a. The attributes are cash flows xi, and the weights are discount factors. Note that the weights (1 r) i and attributes are combined linearly. b. ssuming that the cash flows occur at the end of the period, we have the following: Riskless alternative: NPV = -$, For the risky alternative: NPV1 = -$, NPV = -$, $15, $5, $15, $5, = $51. $15, = $17,969 $5, = -$744 Thus, E(NPV) =.5 ($17,969).5 (-$744) = $51, which is the same as for the riskless alternative. Thus, NPV appears to be a risk-neutral decision criterion.

3 c. We might assess a utility function for wealth and use it instead of cash flow. It would also be reasonable to use a higher interest rate for the risky project (a risk-adjusted discount rate), which would decrease the NPV for the risky alternative. However, assessing the risk-adjusted discount rate is not a simple matter. Even though it is theoretically possible to match the risky project up with specific securities in a market and deduce from those securities an appropriate interest rate, actually doing so may be quite complicated. Moreover, identifying a matching security or portfolio requires substantial subjective judgments regarding the cash flows and uncertainty about both the risky project and the matching security a. The answer requires a subjective assessment. Most students will prefer on the grounds that it is less risky. It is certain that the total payoff will be $1,1, but unclear whether the will come in the first year or the second. b. ssuming end-of-period cash flows, E(NPV) is equal to $ for both and : Project : NPV1 = = $17, NPV = $1 $1 = $ E(NPV) =.5 ($17,591.11).5 ($175.91) = $ Project : NPV1 = NPV = $1 $1 = $ = $ E(NPV) =.5 ($958.48).5 ($858.54) = $ ecause they both have the same E(NPV), a decision maker using E(NPV) as a decision criterion would be indifferent between them. NPV Cash Flows i = 9% Year 1 Year alternative alternative alternative 4 5.% $17, % $ $1 $1 5.% $9,58.48 $1 5.% $8,58.54 $1 The decision tree is a linked tree where the outcome values are linked to the NPV calculations in the spreadsheet.

4 c. Using the logarithmic utility function, we have U() = ln(1,) = 9.1 U($1) = ln(1) = 4.61 gain assuming end-of-period cash flows, Project : NPU1 = = 16. NPU = = 8.11 E(NPU) =.5 (16.).5 (8.11) = 1.16 Project : NPU1 = = 1. NPU = = E(NPU) =.5 (1.).5 (11.98) = 1.16 gain, both projects have the same net present utility, so the decision maker using this decision criterion would be indifferent. alternative alternative alternative 4 NPU LN(Cash Flows) i = 9% Year 1 Year 5.% $16. $9 $9 5.% $8.1 $5 $5 5.% $1. $9 $5 5.% $11.98 $5 $9 The tree is a linked tree where the outcome values are linked to the NPU calculations in the spreadsheet. d. No, these calculations are not consistent with preferences in which either or is preferred. The utility function is not capturing the interactions among the annual cash flows. In particular, it is not recognizing that the cash flows in different periods can in some way act as substitutes for each other.

5 15.4. lternative : Year 1 Year NPV U EU() $17, $1 $1 $176. lternative : Year 1 Year NPV U EU() $1 $9, $1 $8,59.8 Now it is clear that lternative is preferred to. Thus, by using the exponential utility function, we have been able to incorporate a risk attitude. However, this model assumes that all we care about is the NPV (and the riskiness of NPV) of any given project, and not the actual pattern of the cash flows. alternative alternative alternative 4 NPV Cash Flows i = 9% Year 1 Year 5.% $17, % $ $1 $1 5.% $9,58.48 $1 5.% $8,58.54 $1 The tree is a linked tree where the outcome values are linked to the NPV calculations in the spreadsheet, and the calculations are then based on the expected utility defined by an exponential utility function with an R- value of 5.