Economics 202: Intermediate Microeconomic Theory 1.HW #5 on website. Due Tuesday.

Expected Value Suppose that Smith and Jones decide to flip a coin –heads (x 1 )  Jones will pay Smith $1 –tails (x 2 )  Smith will pay Jones $1 From Smith’s point of view, Games which have an expected value of zero (or cost their expected values) are called actuarially fair games –a common observation is that people often refuse to participate in actuarially fair games –except if the stakes are small or they gain utility from playing the game (like state lottery or slot machines). We focus on risk aspect not consumption aspect of gambling.

St. Petersburg Paradox A coin is flipped until a head appears If a head appears on the nth flip, the player is paid $2 n x 1 = $2, x 2 = $4, x 3 = $8,…,x n = $2 n The probability of getting of getting a head on the ith trial is (½) i  1 =½,  2 = ¼,…,  n = 1/2 n Would you play?

Expected Utility Individuals do not care directly about the dollar values of the prizes –they care about the utility that the dollars provide If we assume diminishing marginal utility of wealth, the St. Petersburg game may converge to a finite expected utility value –this would measure how much the game is worth to the individual

Expected Utility Expected utility can be calculated in the same manner as expected value Because utility may rise less rapidly than the dollar value of the prizes, it is possible that expected utility will be less than the monetary expected value

Expected Utility Maximization Consider two gambles: –first gamble offers x 2 with probability q and x 3 with probability (1- q) expected utility (1) = q · U(x 2 ) + (1-q) · U(x 3 ) –second gamble offers x 5 with probability t and x 6 with probability (1-t) expected utility (2) = t · U(x 5 ) + (1-t) · U(x 6 ) The individual will prefer gamble 1 to gamble 2 if and only if q ·  2 + (1-q) ·  3 > t ·  5 + (1-t) ·  6 A rational individual will choose among gambles based on their expected utilities (the expected values of the von Neumann-Morgenstern utility index)

Risk Aversion Two lotteries may have the same expected value but differ in their riskiness –flip a coin for $1 versus $1,000 Risk refers to the variability of the outcomes of some uncertain activity When faced with two gambles with the same expected value, individuals will usually choose the one with lower risk

Risk Aversion Utility (U) Wealth (W) U(W)U(W) U(W) is a von Neumann-Morgenstern utility index that reflects how the individual feels about each value of wealth The curve is concave to reflect the assumption that marginal utility diminishes as wealth increases

Risk Aversion Utility (U) Wealth (W) U(W)U(W) Suppose that W* is the individual’s current level of income W*W* U(W*) is the individual’s current level of utility U(W*)

Risk Aversion The person will prefer current wealth to that wealth via a fair gamble The person will also prefer a small gamble over a large one

Risk Aversion Utility (U) Wealth (W) U(W)U(W) W*W* U(W*) W* + 2hW* - 2h U(W*) > U h (W*) > U 2h (W*) U 2h (W*) U h (W*) W* - hW* + h

Utility Functions & Attitudes toward Risk Total Utility curve plots the utility of different return levels The return is unknown beforehand and  uncertain Prob[Return = 2%] = 0.5 Prob[Return = 8%] = 0.5 Expected return = 2(.5) + 8(.5) = 5% E[U(r)]=200 U[E(r)]=225 Total Utility Return 2%8% A U 100 E(r)=5% 300 Z Expected Utility is the average utility you can expect to get from different possible returns E[U(r)] = 100(.5) + 300(.5) = 200 Utility from a certain return = U[E(r)] Risk-aversion is when a person prefers a certain return to an uncertain return giving the same expected return: U[E(r)] > E[U(r)]

Risk-aversion  TU has positive, diminishing slope  Diminishing Marginal Utility Risk-aversion is common –Fire-Insurance –U[E(r)] = utility of $1,000 premium –E[U(r)] = average utility of (-$100K*.01) + ($0*.99) = -$1,000 Depends on the probabilities and amount of loss E[U(r)]=200 U[E(r)]=225 Total Utility Return 2%8% A U 100 E(r)=5% 300 Z Utility Functions & Attitudes toward Risk

Risk-Neutrality  TU has positive, constant slope  constant marginal utility Investor is indifferent between the certain return and the risk of 2% or 8% return U[E(r)] = E[U(r)] E[U(r)] U[E(r)]= Return TU 2%8% A U E(r)=5% 300 Z Return TU 2%8% A U E(r)=5% Z U[E(r)] E[U(r)] > Risk-loving  TU has positive, increasing slope  increasing marginal utility Investor prefers the risk of 2% or 8% return U[E(r)] < E[U(r)] Example could be 10 lottery tickets Utility from $0 for sure < -50% return Decreases as stakes increase: Vegas! Utility Functions & Attitudes toward Risk

Methods to Minimize Risk (1) Insurance (fire, health, car, etc.) What is the max price a risk-averse person will pay? For an expected loss of $1,000, they will pay > $1,000 Why? They value $98,300 for sure the same as $99,000 expected income  $1,700 max Green line is expected utility line Will the insurance co. provide a policy? Pr(fire) =.01 so 1 out of 100 will burn Minimum ins. co. will accept = $1,000 per homeowner since 1 will burn. Room for mutually beneficial exchange (2) Diversification (variety of risks) Stocks in different industries, start-ups Total Utility Income 0100K A U 99K U* Z 98,300

Example: Risk Preference & Insurance In Japan, golfers who get a “hole-in-one” are expected to give gifts to relatives, fellow workers, and friends. These gifts can cost them the equivalent of thousands of dollars. The cost is so great that there actually exists a market for “hole-in-one” insurance. A Japanese golfer, who is an expected utility maximizer, has the following utility function, U(I) = I ½ where I is monthly income. Our golfer, who has a monthly income of $10,000, is quite accomplished and has a probability of making a hole-in-one equal to 1 in 100. If our golfer does indeed ace a hole, the typical gift expenditure is $3600. (a) Is the golfer risk-averse, risk-neutral, or risk-loving? Support your answer mathematically and explain. (b) Calculate the expected income and expected utility (indicate your units).

Example: Risk Preference & Insurance (c) Compute the maximum premium that would be paid to fully insure against the expected costs of getting a hole-in-one. (d) If the golfer’s monthly income were only $8,000, what would be the maximum he or she is willing to pay? Explain the relationship between income and the risk premium.

Segue In uncertain situations, economists typically assume individuals are concerned with the expected utility associated with various outcomes –if they obey the von Neumann-Morgenstern axioms, they will make choices in a way that maximizes expected utility

Kahneman-Tversky Experiments Is expected utility a reasonable assumption? ChoiceDollars Probability A$1,000,000 1 B 5,000, ,000,000.89

Kahneman-Tversky Experiments ChoiceDollars Probability C$1,000, D 5,000,

Kahneman-Tversky Experiments ChoiceDollars Probability E $30 1 F

Kahneman-Tversky Experiments Flip a coin and if 2 heads in a row you can choose below: Choice Dollars Probability G $30 1 H No coin flip Choice Dollars Probability I $30.25 J 45.20

Kahneman-Tversky Experiments People don’t simply care about the final rewards and their associated probabilities. They seem to care about how these are achieved which is inconsistent with expected utility hypothesis. Choose the correct answer: Assume Johnny is a risk-averse expected utility-maximizer who always turns down win $11/lose $10 bet. What is the largest value of Y for which Johnny will turn down a lottery in which the outcomes are lose $100/win $Y? (a) $110(b) $221(c) $2,000(d) $20,242 (e) $1.1 million(f) $2.5 billion (g) He will turn it down no matter what Y is (h) Need more information

Kahneman-Tversky Experiments Answer = Logic is that, under the expected utility framework, turning down a moderate- stakes gamble means that the MU of money must diminish very rapidly. “Expected utility is an ex-hypothesis!” Rabin and Thaler (2001), “Anomalies: Risk- Aversion,” Journal of Economic Perspectives, v. 15, no. 1, pp Another viewpoint: Loss Aversion & Mental Accounting