Uncertainty and Decision Making

Uncertainty and Decision Making

Introduction

  • Most decisions in life are made under uncertainty.
  • Examples include studying for exams, carrying an umbrella, buying a house, and getting insurance.
  • A realistic model of decision-making must account for uncertainty.

Expected Value

  • Consider a bet: heads, win $125; tails, lose $100.
  • Many people are hesitant to take this bet.
  • Expected value is the probability of winning times the payout, plus the probability of losing times the loss.
    • Expected Value=(Probability of Win×Value if Win)+(Probability of Loss×Value if Loss)Expected\ Value = (Probability\ of\ Win \times Value\ if\ Win) + (Probability\ of\ Loss \times Value\ if\ Loss)
  • For the example bet:
    • Expected Value=(0.5×125)+(0.5×−100)=12.5Expected\ Value = (0.5 \times 125) + (0.5 \times -100) = 12.5
  • A fair bet has an expected value of 0.
  • A bet with a positive expected value is a more than fair bet.

Risk Aversion

  • Risk aversion means individuals value each dollar of winning less than they devalue each dollar of losing.
  • This is related to consumer theory and can be analyzed using expected utility theory.

Expected Utility Theory

  • Expected utility theory rewrites the expected value equation in terms of utility rather than dollar values.
  • Expected Utility=(Probability of Win×Utility if Win)+(Probability of Loss×Utility if Loss)Expected\ Utility = (Probability\ of\ Win \times Utility\ if\ Win) + (Probability\ of\ Loss \times Utility\ if\ Loss)
  • If utility functions were linear, expected utility would be the same as expected value.
  • Utility functions are typically concave, reflecting diminishing marginal utility of consumption.
  • Diminishing marginal utility of consumption naturally leads to risk aversion.
  • With a concave utility function, each additional dollar makes you less happy than losing a dollar makes you sad.
Example
  • Utility function: U(C)=CU(C) = \sqrt{C}
  • Initial consumption: C<em>0=100C<em>0 = 100, Initial utility: U</em>0=10U</em>0 = 10
  • Expected utility of the gamble:
    • If win: U(225)=225=15U(225) = \sqrt{225} = 15
    • If lose: U(0)=0=0U(0) = \sqrt{0} = 0
    • Expected Utility=(0.5×15)+(0.5×0)=7.5Expected\ Utility = (0.5 \times 15) + (0.5 \times 0) = 7.5
  • The expected utility (7.5) is lower than the initial utility (10), so a risk-averse person would not take the bet.

Graphical Representation

  • Figure 20-1 shows wealth/consumption on the x-axis and utility on the y-axis.
  • Point A represents the initial state ($100, 10 utils).
  • The gamble has a 50% chance of ending up at 0 and a 50% chance of ending up at point B ($225).
  • The expected outcome of the gamble is point C, which is below point A due to the concavity of the utility function.
  • The concavity reflects diminishing marginal utility: losing hurts more than winning helps.
  • Even though the bet is more than fair, individuals are unwilling to take it due to risk aversion.

Willingness to Pay to Avoid a Gamble

  • A person would be willing to pay to avoid the gamble.
  • The gamble leaves you at the same level of happiness as having $56.25. So, you'd be willing to pay 100 - 56.25 = $43.75 to avoid the gamble.
  • Even with a more than fair gamble, people are willing to give up a significant amount of wealth to avoid it.
  • The winning payoff would have to be very high to induce someone to take the gamble (e.g., win $300, lose $100).
  • The core concept is that the distress of going to zero is greater than the happiness of gaining above the current level.

Risk Neutrality

  • If utility is a linear function of consumption (e.g., U(C)=0.1×CU(C) = 0.1 \times C), the individual is risk neutral.
  • In this case, the expected utility of the initial gamble is 0.5(0.1×225)+0.5(0.1×0)=11.250.5(0.1 \times 225) + 0.5(0.1 \times 0) = 11.25, which is greater than the initial utility of 10, so the bet would be accepted.
  • Risk-neutral individuals only care about expected value.

Risk Loving

  • If the utility function is convex (e.g., U(C)=C21000U(C) = \frac{C^2}{1000}), the individual is risk-loving.
  • The expected utility of the gamble is 0.5×22521000+0.5×021000=25.31250.5 \times \frac{225^2}{1000} + 0.5 \times \frac{0^2}{1000} = 25.3125, higher than the initial utility of 10, making the gamble attractive.
  • Risk-loving individuals have increasing marginal utility of consumption.
  • They will take even unfair gambles and may pay for the opportunity to gamble.
  • For example, with heads you win 75, tails you pay me 100. So it's a negative expected value of 12.5. The person would still take the bet because it yields an expected utility of 15.3.

Gamble Size Relative to Resources

  • When the gamble is small relative to resources, the utility function becomes locally linear, and individuals become more risk-neutral.
  • If the original bet (win $12.50, lose $10) is scaled down, more people are willing to take it.
  • The expected utility of this scaled gamble is 0.5×112.5+0.5×90=10.050.5 \times \sqrt{112.5} + 0.5 \times \sqrt{90} = 10.05, which is higher than the initial expected utility of 10.
  • Risk aversion is relative; what matters is the size of the gamble relative to initial wealth.
  • As the gamble gets smaller or initial wealth gets bigger, you become more risk neutral.

Cardinal vs. Relative Values

  • Utility values are only meaningful in relation to alternative choices.
  • Expected utility theory assumes a specific linear combination of utility in different states.
  • This assumption may not always hold, and more complex models may be needed to explain certain paradoxes.

Applications: Insurance

  • Insurance is a significant part of the US economy (10% of GDP).
  • People buy insurance to avoid risk.
Example
  • A 25-year-old single male in Cambridge, MA, with an income of $40,000.
  • 1% chance of being hit by a car, resulting in a $30,000 hospital bill.
  • Insurance pays the medical bill in exchange for a premium.
  • Utility function: U(C)=CU(C) = \sqrt{C}
  • Expected utility without insurance:
    • 0.01×40000−30000+0.99×40000=1990.01 \times \sqrt{40000 - 30000} + 0.99 \times \sqrt{40000} = 199
  • Expected utility with insurance:
    • 40000−x\sqrt{40000 - x}, where x is insurance premium.
  • Setting the expected utilities equal to find the price you'd be willing to pay: 40000−x=199  ⟹  x=399\sqrt{40000 - x} = 199\implies{x = 399}
  • Willingness to pay is $399 for insurance with an expected value of $300.
  • The extra $99 is the risk premium.
  • As the size of the loss rises, the risk premium rises.
  • As income rises, the risk premium falls.

Applications: Lottery

  • The lottery in the US is a very unfair bet (expected value of $0.50 per dollar spent).
  • However, it's popular and a major source of revenue for states.
Theories for Lottery Popularity
  1. People are risk-loving:
    • Disproved by the fact that Americans spend heavily on insurance.
  2. People are both risk-averse and risk-loving (Friedman-Savage preferences):
    • People are risk-averse for small gambles but risk-loving for large gambles.
    • Empirically false because most lottery money is spent on scratch tickets (small gambles), not Mega Millions.
  3. Entertainment:
    • People gamble because they find it entertaining; the thrill is in the utility function.
  4. People are uninformed or making mistakes:
    • People don't understand the odds or are not thinking it through properly.
  • The government's role (supporting vs. discouraging lotteries) depends on which theory is correct.
  • If it's entertainment, the government can make money through a voluntary tax.
  • If it's a mistake, the government should discourage it.
  • In some low-income communities, people spend up to 20% of their income on the lottery.

Information Asymmetry and Market Failure

  • Government also provides insurance (6-7% of GDP) because private people buy too little due to information asymmetry.
  • Information asymmetry is when some parties have information that others do not, which can cause market failure.
The Lemons Problem (Akerlof)
  • Information asymmetry can cause market failure.
  • Example: Used car market without Carfax.
  • John has a 10-year-old car worth $5,000.
  • Andrew values the car at $6,000.
  • Welfare-improving transaction should happen.
  • Andrew knows that on average, 10-year-old cars need $2,000 in repairs.
  • Andrew doesn't know John's car is pristine.
  • Andrew's value: $6,000 - $2,000 = $4,000.
  • The market fails; the transaction doesn't happen.
  • With perfect information (e.g., Carfax), the transaction would occur.
  • The information asymmetry is that John knows how good the car is, but Andrew doesn't.
  • This can cause an entire market to collapse, even with no other problems (monopoly, etc.).
Insurance and Adverse Selection
  • Applying this to insurance, Andrew (insurance company) worries about adverse selection.
  • Adverse selection: the people who want to buy insurance are only the ones who really need it.
  • The buyer knows more than the seller.
  • If Andrew knew that John is a clean-living, non-skydiving guy, he would happily sell him insurance.
  • But without that information, he's worried that John is someone who runs in the middle of the street and gets hit a lot.
  • Since Andrew is worried he's going to lose money if he gives John insurance, he won't insure John, even though society would benefit from it.
  • Partial information leads to a marketing failure.
  • This leads to people who could have insurance at a fair price not getting it.
  • The government addresses this market failure through things like mandates, taxes, subsidies, single-payer coverage.