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 (x1) Jones will pay Smith $1

– tails (x2) Smith will pay Jones $1

• From Smith’s point of view,

2211)( xxXE π+π=

012

11

2

1=−+= )$()($)(XE

• 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 $2n

x1 = $2, x2 = $4, x3 = $8,

…,xn = $2n

• The probability of getting of getting a head on the ith trial is (½)i

π1=½, π2= ¼,…, πn= 1/2n

i

i i

iii xXE ∑ ∑

∞

=

∞

=⎟⎠

⎞⎜⎝

⎛=π=1 1 2

12)(

?)( =XE

• 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

)()(1

∑=

π=n

iii xUXE

• 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 x2 with probability q and x3 with probability (1-

q) expected utility (1) = q · U(x2) + (1-q) · U(x3)

– second gamble offers x5 with probability t and x6 with probability

(1-t) expected utility (2) = t · U(x5) + (1-t) · U(x6)

• The individual will prefer gamble 1 to gamble 2 if and only if q · π2 + (1-q) · π3 > t · π5 + (1-t) · π6

• 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) is a von Neumann-Morgensternutility index that reflects how the individualfeels about each value of wealth

The curve is concave to reflect theassumption that marginal utilitydiminishes as wealth increases

Risk Aversion

Utility (U)

Wealth (W)

U(W)

Suppose that W* is the individual’s currentlevel of income

W*

U(W*) is the individual’scurrent 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)

W*

U(W*)

W* + 2hW* - 2h

U(W*) > Uh(W*) > U2h(W*)

U2h(W*)

Uh(W*)

W* - h W* + 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

Return2% 8%

A

U

100

E(r)=5%

300Z

• 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

Return2% 8%

A

U

100

E(r)=5%

300Z

Utility Functions & Attitudes toward Risk

• Risk-Neutrality TU has positive, constant slope constant marginal utility

• Investor is indifferent between thecertain 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%

300Z

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

Income0 100K

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) = I½ 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?

• Choice Dollars Probability

A $1,000,000 1

B 5,000,000 .10

0 .01

1,000,000 .89

Kahneman-Tversky Experiments• Choice Dollars Probability

C $1,000,000 .11

0 .89

D 5,000,000 .10

0 .90

Kahneman-Tversky Experiments

• Choice Dollars Probability

E $30 1

F 45 .80

0 .20

Kahneman-Tversky Experiments• Flip a coin and if 2 heads in a row you can choose below:

• Choice Dollars Probability

G $30 1

H 45 .80

0 .20

• 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 50-50 win $11/lose $10 bet.

What is the largest value of Y for which Johnny will turn down a 50-50 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. 219-232.

• Another viewpoint: Loss Aversion & Mental Accounting