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 \times Value\ if\ Win) + (Probability\ of\ Loss \times Value\ if\ Loss)
For the example bet:
- Expected\ 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 \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) = \sqrt{C}
Initial consumption: C0 = 100, Initial utility: U0 = 10
Expected utility of the gamble:
- If win: U(225) = \sqrt{225} = 15
- If lose: U(0) = \sqrt{0} = 0
- Expected\ 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 \times C), the individual is risk neutral.
In this case, the expected utility of the initial gamble is 0.5(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) = \frac{C^2}{1000}), the individual is risk-loving.
The expected utility of the gamble is 0.5 \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 \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) = \sqrt{C}
Expected utility without insurance:
- 0.01 \times \sqrt{40000 - 30000} + 0.99 \times \sqrt{40000} = 199
Expected utility with insurance:
- \sqrt{40000 - x}, where x is insurance premium.
Setting the expected utilities equal to find the price you'd be willing to pay: \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

People are risk-loving:
- Disproved by the fact that Americans spend heavily on insurance.
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.
Entertainment:
- People gamble because they find it entertaining; the thrill is in the utility function.
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.