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##### Chapter 17 Choice Making Under Uncertainty

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**Calculating Expected Monetary Value**• The expected monetary value is simply the weighted average of the payoffs (the possible outcomes), where the weights are the probabilities of occurrence assigned to each outcome.**Expected Value**• Given: Two possible outcomes having payoffs X1 and X2 and probabilities of each outcome given by Pr1 & Pr2. • The expected value (EV) can be expressed as: EV(X) = Pr1X1+ Pr2X2**Expected Utility Hypothesis**• Expected utility is calculated the same way as expected monetary value, except that the utility associated with a payoff is substituted for its monetary value. • With two outcomes for wealth ($200 and $0) and with each outcome occurring ½ the time, the expected utility can be written: E(u) = (1/2)U($200) + (1/2)U($0)**Expected Utility Hypothesis**If a person prefers the gamble previously described, over an amount of money $M with certainty then: (1/2)U($200) + (1/2)U($0) > U(M)**Defining a Prospect**• The remainder of the chapter will cover lotteries, which will be referred to as prospects which offer three different outcomes. • The term prospect will refer to any set of probabilities (q1, q2, q3: and their assigned outcomes ($10 000, $6000 and $1000). • Note that the probabilities must sum to 1.**Defining a Prospect**• Such a prospect will be denoted as: (q1, q2, q3: 10 000, 6000, 1000) or simply: (q1, q2, q3)**Continuity assumption:**For any individual, there is a unique number e*, (0<e*<1), such that he/she is indifferent between the two prospects (0, 1, 0) and (e*, 0, 1-e*). This assumption guarantees that persons are willing to make tradeoffs between risk and assured prospects. Note - e* will vary across individuals. Deriving Expected Utility Functions**von Neuman-Morgenstern Utility Function**• Given any two numbers a and b with a>b, we could let U(10 000)=a and U(1 000)=b. We would then have to assign a utility number to $6000 as follows: U(6000) =ae*+b(1-e*)**von Neuman-Morgenstern Utility Function**• With the continuity assumption (and others) satisfied and the utility function constructed as shown, these important results are applicable: • If an individual prefers one prospect to another, then the preferred prospect will have a larger utility. • If an individual is indifferent between two prospects, the two prospects must have the same expected utility.**Subjective Probabilities**• The expected utility theory is often applied in risky situations in which the probability of any outcome is not objectively known or there exists incomplete information. • The ability to apply expected-utility theory in such scenarios is to use subjective probabilities.**The Expected Utility Function**• Assume there are 2 states of wealth (w1 and w2) which could exist tomorrow and they occur with probabilities (q and 1-q) respectively. • The expected utility function for tomorrow: U(q,1-q:w1w2) = qU(w1)+(1-q)U(w2)**The Expected Utility Function**• Two key features of this utility functions: • The U functions are cardinal, meaning that the utility values have specific meaning in relation to one another. • This expected utility function is linear in its probabilities (which simplifies MRS).**From Figure 17.1**• Figure 17.1 shows an indifference curve for utility level u. Wealth in state 1(today) and state 2 (tomorrow) are on each axis. • q and (1-q) are fixed. • The MRS (slope of u0) shows the rate at which an individual trades wealth in state 1 for wealth in state 2, before either of these states occur.**From Figure 17.1**• The slope of the indifference curve is equal to the ratio of the probabilities times the ratio of the marginal utilities. • Each marginal utility, however, is function of wealth in only one state, since the utility functions are the same in each state. • Therefore, the MRS equals the ratio of the probabilities.**From Figure 17.1**• Hence, along the 45 degree line, where wealth in the two states are equal, the slope of u0 is q/(1-q). • If q is large relative to (1-q) then u0 is relatively steep and vice versa. • In other words, if you believe state 1 is very likely (q is high) then you will prefer your wealth in state one rather than state two.**Optimal Risk Bearing**• Now that different attitudes toward risk have been defined, it is necessary to illustrate how attitudes toward risk affect choices over risky prospects. • An expected value line shows prospects with the same expected value. Note however that along this line, the risk of each prospect varies.**From Figure 17.3**• At point A there is no risk and that risk increases as the prospects move away from the 45 degree line. • The slope of the expected value line equals the ratios of the probabilities (relative prices) • Utility will be maximized when the individual’s MRS equals the ratios of the probabilities.**Optimal Risk Bearing**• The optimal amount of risk that a person bears in life depends on his/her aversion to risk. • The choices of risk averse persons tend toward the 45 degree line where wealth is the same no matter what state arises. • Risk inclined persons move away from the 45 degree line and are willing to take the chance that they will be better off in one state compared to the other.**Pooling Risk**• Risk Pooling is a form of insurance aimed at reducing an individual’s exposure to risk by spreading that risk over a larger number of persons. • Suppose the probability of either Abe or Martha having a fire is 1-q, the loss from such a fire is L dollars and wealth in period t denoted as wt.**Pooling Risk**• Abe’s expected utility is: u(q, L,w0) = qU(w0)+(1-q)U(w0-L). • If Abe’s house burns, his wealth is w0-L, and his utility U(w0-L). If it does not burn, his wealth is w0 and utility is U(w0).**Pooling Risk**• If Abe and Martha pool their risk (share any loss from a fire), there are now three relevant events: 1. One house burns. Probability = 2q(1-q), Abe’s Loss=L/2 2. Both houses burn. Probability = (1-q)2 , Abe’s Loss=L 3. Neither house burns. Probability = q2 , Abe’s loss = 0**Risk Pooling**• Abe’s expected utility with risk pooling: (1-q)2U(wo-L)+2q(1-q)U(w0L/2)+q2U(w0) • Rearranging and factoring Abe’s individual and risk pooling utility function shows he is better off if he is risk averse as: U(w0-L/2)>(1/2)U(w0-L)+(1/2)U(w0)**Risk Pooling**• When individuals are risk averse, they have clear incentives to create institutions that allow them to share (pool) their risks.**The Market for Insurance**• What is Abe’s reservationdemand price for insurance (the maximum he is willing to pay rather than go without)? • Set his expected utility without insurance equal to the certainty equivalent (assured prospect wce in Figure 17.6).**The Market for Insurance**• On the assumption that insurance companies are risk neutral, what is the lowest price they will offer full coverage? • This is the reservation supply price, denoted by Is in Figure 17.6**The Market for Insurance**• Ignoring any administrative costs, the expected costs are (1-q)L and the firm will write a policy if revenues (I) exceed costs.**The Market for Insurance**• As shown in Figure 17.6, there is a viable insurance market because the reservation supply price Is =(1-q)L is less than the reservation demand price (distance w0-wce). • Abe trades his risky prospect for the assured prospect and reaches indifference curve u*. • If no resources are required to write and administer insurance policies and if individuals are risk-averse, there is a viable market for insurance.