Economics 202: Intermediate Microeconomic Theory

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Economics 202 Intermediate Microeconomic Theory

• HW 5 on website. Due Tuesday.

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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 Smiths 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.

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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
• ?1½, ?2 ¼,, ?n 1/2n

• Would you play?

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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

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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

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Expected Utility Maximization

• A rational individual will choose among gambles
  based on their expected utilities (the expected
  values of the von Neumann-Morgenstern utility
  index)

• 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 ?2 (1-q) ?3
  t ?5 (1-t) ?6

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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

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Risk Aversion
Utility (U)
The curve is concave to reflect the assumption
that marginal utility diminishes as wealth
increases
Wealth (W)
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Risk Aversion
Utility (U)
U(W)
Wealth (W)
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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

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Risk Aversion
U(W) Uh(W) U2h(W)
Utility (U)
U(W)
U(W)
Uh(W)
U2h(W)
Wealth (W)
W
W 2h
W - 2h
W - h
W h
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Utility Functions Attitudes toward Risk

• Total Utility curve plots the utility of
  different return levels
• The return is unknown beforehand and ? uncertain
• ProbReturn 2 0.5
• ProbReturn 8 0.5
• Expected return 2(.5) 8(.5) 5

Total Utility
U
300
Z
UE(r)225
EU(r)200

• Expected Utility is the average utility you can
  expect to get from different possible returns
• EU(r) 100(.5) 300(.5) 200

100
A
Return
2
8
E(r)5

• Utility from a certain return UE(r)
• Risk-aversion is when a person prefers a certain
  return to an uncertain return giving the same
  expected return
• UE(r) EU(r)

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Utility Functions Attitudes toward Risk

• Risk-aversion ? TU has positive, diminishing
  slope ? Diminishing Marginal Utility
• Risk-aversion is common
• Fire-Insurance
• UE(r) utility of 1,000 premium
• EU(r) average utility of
  (-100K.01) (0.99) -1,000
• Depends on the probabilities and amount of loss

UE(r)225
EU(r)200
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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
• UE(r) EU(r)

UE(r)
EU(r)

• Risk-loving ? TU has positive, increasing slope ?
  increasing marginal utility
• Investor prefers the risk of 2 or 8 return
• UE(r)
• Example could be 10 lottery tickets
• Utility from 0 for sure
• Decreases as stakes increase Vegas!

U
TU
Z
EU(r)
UE(r)
A
Return
2
8
E(r)5
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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

U
Total Utility
Z
U
Income
A
0
100K
98,300
99K
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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).

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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 golfers 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.

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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

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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

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Kahneman-Tversky Experiments

• Choice Dollars Probability
• C 1,000,000 .11
• 0 .89
• D 5,000,000 .10
• 0 .90

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Kahneman-Tversky Experiments

• Choice Dollars Probability
• E 30 1
• F 45 .80
• 0 .20

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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

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Kahneman-Tversky Experiments

• People dont 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

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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