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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.