Econ 2Z03: 4 - Uncertainty

What is the main focus of the study of uncertainty in economics?

To understand how economic agents make decisions when outcomes are uncertain.

What are examples of choices that have uncertain outcomes?

Buying a lottery ticket, going to college, starting a new job, purchasing a car or a house.

What is the first step an economic agent must take when making decisions in uncertain situations?

Consider all possible outcomes of the risky decision.

What are objective probabilities?

Probabilities that can be calculated statistically, such as the likelihood of flipping heads or tails on a fair coin.

What are subjective probabilities?

Probabilities assigned based on personal beliefs or opinions, which may differ among individuals.

What is a lottery in the context of economic decision-making?

A risky decision that consists of a list of possible outcomes and probabilities assigned to each outcome.

What must the probabilities assigned to outcomes in a lottery satisfy?

They must be nonnegative and sum to 1.

In Example 1, what are the outcomes and their associated wealth when tossing a fair coin?

Heads results in $30 (w1 = 30), tails results in $10 (w2 = 10), with each outcome having a probability of 1/2.

In Example 2, what are the outcomes and wealth associated with rolling a die?

Even outcomes (2, 4, 6) yield $30, odd outcomes (1, 3, 5) yield $10, with each outcome having a probability of 1/6.

What does Example 3 illustrate about certainty in lotteries?

It shows a situation with no uncertainty where the outcome is the same ($20) regardless of the event, leading to no risk.

What is expected value in the context of lotteries?

A probability-weighted average of the wealth outcomes from a given lottery.

How can the framing of a lottery affect decision-making?

The presentation of the lottery can influence how individuals perceive risk and make choices.

What is the significance of understanding risk and uncertainty in economic choices?

It helps to reveal the attitudes of agents towards risk and their decision-making processes.

What is the relationship between risk and utility in economic decisions?

Economic agents may experience different levels of utility based on the outcomes of their risky choices.

What is the role of probabilities in decision-making under uncertainty?

Probabilities help agents assess the likelihood of different outcomes and make informed choices.

What is a common challenge when assigning probabilities to economic outcomes?

Many outcomes cannot be assigned objective probabilities due to the complexity and variability of real-world factors.

What is the difference between monetary and non-monetary outcomes in risky decisions?

Monetary outcomes involve financial gains or losses, while non-monetary outcomes may include personal satisfaction or social implications.

How does the concept of risk aversion relate to economic decision-making?

Risk-averse individuals prefer certain outcomes over uncertain ones, even if the uncertain option has a higher expected value.

What can the comparison of risky and certain outcomes reveal about an agent's preferences?

It can indicate the agent's attitude towards risk and their willingness to accept uncertainty for potential rewards.

What is the importance of modeling risk in economics?

Modeling risk allows economists to predict behavior and outcomes in uncertain environments.

What is the significance of the sum of probabilities in a lottery?

The sum of probabilities must equal 1 to ensure that all possible outcomes are accounted for.

What is an example of a situation where subjective probabilities might be used?

Estimating the likelihood of getting a dream job after college based on personal beliefs and experiences.

How do economic agents typically respond to lotteries with identical outcomes and probabilities?

They treat them the same way, regardless of how the lottery is framed or presented.

What is the expected value (EV) of a lottery?

The probability-weighted average wealth the agent will obtain from the lottery.

How is expected value calculated?

EV = p1w1 + p2w2 + … + pnwn, where pi is the probability and wi is the wealth outcome.

What does expected value represent?

The average wealth of the agent after playing the lottery, considering the likelihood of each outcome.

Can different lotteries have the same expected value?

Yes, different lotteries can have the same expected value but may have different risks associated with them.

What is the concept of expected utility?

Expected utility is the average utility an agent expects to obtain from a lottery, calculated as EU = p1u(w1) + p2u(w2) + … + pn u(wn).

How does expected utility differ from expected value?

Expected utility takes into account the utility derived from wealth outcomes, while expected value focuses solely on wealth.

What is the utility function in the context of lotteries?

A function u that represents the utility an agent derives from different levels of wealth.

What does a risk-averse agent prefer?

A risk-averse agent prefers the expected value of a lottery to playing the lottery itself.

What characterizes a risk-loving agent?

A risk-loving agent prefers the lottery to its expected value, valuing the potential for higher outcomes.

What defines a risk-neutral agent?

A risk-neutral agent is indifferent between a lottery and its expected value, showing no preference for risk.

How can risk preferences be inferred from a utility function?

By examining the second derivative of the utility function: concave (risk-averse), convex (risk-loving), or linear (risk-neutral).

What does it mean if a utility function is concave?

It indicates that the agent is risk-averse, as they prefer certain outcomes over uncertain ones with the same expected value.

What does it mean if a utility function is convex?

It indicates that the agent is risk-loving, as they prefer uncertain outcomes that could yield higher returns.

What does it mean if a utility function is linear?

It indicates that the agent is risk-neutral, showing no preference for risk in their decision-making.

What is the expected utility for an agent with utility function u(w) = √w when facing a lottery with outcomes 10 and 30?

EU = 1/2(√30 + √10) ≈ 4.32.

How does an agent's utility function affect their decision-making under risk?

Different utility functions lead to different risk preferences, influencing whether they prefer lotteries or their expected values.

What is the relationship between expected value and expected utility for a risk-averse agent?

For a risk-averse agent, u(EV(L)) > EU(L), meaning they prefer the certain expected value to the uncertain lottery.

What is the expected utility for an agent with utility function u(w) = √w facing a lottery with guaranteed wealth of $20?

EU = u(20) = √20 ≈ 4.47.

What does the inequality u(EV(L)) > EU(L) signify?

It signifies that the agent prefers the certainty of the expected value over the uncertainty of the lottery outcomes.

What is the expected utility for a risk-neutral agent?

For a risk-neutral agent, u(EV(L)) = EU(L), indicating indifference between the lottery and its expected value.

What happens to an agent's risk preferences as their utility function changes?

As the utility function changes from concave to convex, the agent's risk preference shifts from risk-averse to risk-loving.

What is a key characteristic of risk-averse agents in decision-making?

They are primarily concerned with avoiding potential losses rather than seeking potential gains.

What does the expected utility represent in the context of decision-making?

It represents the average utility an agent expects to obtain from a lottery, factoring in the probabilities of different outcomes.

What does a negative second derivative of a utility function indicate about an agent's risk preference?

It indicates that the agent is risk-averse.

What is the utility function for a risk-averse agent that is commonly used?

u(w) = √w.

What are the derivatives of the utility function u(w) = √w?

u′(w) = 1/(2√w) and u′′(w) = -1/(4w^(3/2)).

What does the second derivative u′′(w) = -1 indicate for the utility function u(w) = log(w)?

It indicates that the agent is also risk-averse.

What is the utility function for a risk-loving agent?

u(w) = w².

What are the derivatives of the utility function u(w) = w²?

u′(w) = 2w and u′′(w) = 2.

What characterizes a risk-neutral agent's utility function?

It is linear, typically written as u(w) = a + bw.

What is the expected value (EV) of a lottery where an agent has wealth w0 = $50000, with a 60% chance of losing $20000?

EV = $38000.

How do you calculate the expected utility (EU) of a lottery for an agent with a utility function u(w) = √w?

EU = p1u(w1) + p2u(w2), where w1 and w2 are the possible wealth outcomes.

What is the certainty equivalent (wce) for an agent who is indifferent between a lottery and a certain amount of wealth?

It satisfies u(wce) = EU.

What is the certainty equivalent for an agent with utility function u(w) = √w and expected utility of 193?

wce = 37249.

What is the risk premium for a risk-averse agent?

It is the amount they are willing to pay above the expected value to avoid risk.

What is the risk premium for a risk-neutral agent?

It is zero, as they do not pay to avoid or incur risk.

How can you determine which of two risk-averse agents is more risk-averse?

By comparing their risk premiums or certainty equivalents.

What does a higher certainty equivalent indicate about an agent's risk preference?

It indicates that the agent is less risk-averse.

What does the point C represent in the graphical representation of risk preferences?

It represents the expected value of the lottery and the expected utility of the lottery.

What is the relationship between expected value (EV) and certainty equivalent (wce) for risk-averse agents?

EV > wce, indicating they would pay to avoid risk.

What is the expected utility for a risk-neutral agent with utility function u(w) = w?

EU = $38000, equal to the expected value.

What does it mean if an agent's utility function is concave?

It indicates that the agent is risk-averse.

What is the formula for the expected value of a lottery with two outcomes?

EV = p1w1 + p2w2.

What does the term 'risk premium' refer to?

The additional amount a risk-averse agent is willing to pay to avoid risk.

How does the utility function u(w) = w affect an agent's risk preference?

It indicates that the agent is risk-neutral.

What is the significance of the second derivative of a utility function being positive?

It indicates that the agent is risk-loving.

What does the notation p1 and p2 represent in the context of lotteries?

They represent the probabilities of each outcome occurring.

What is the wealth level w1 for an agent who faces a 60% chance of incurring $20000 in medical bills?

w1 = $30000.

What is the expected value (EV) in the context of insurance?

The expected value is the average outcome of a lottery or gamble, calculated as the sum of all possible outcomes weighted by their probabilities.

What does the certainty equivalent (wce) represent?

The certainty equivalent is the guaranteed amount of wealth that an individual considers equally desirable to a risky lottery.

How is the risk premium defined?

The risk premium is the difference between the expected value of a lottery and the certainty equivalent, reflecting the amount a risk-averse agent is willing to pay to avoid risk.

What is the relationship between expected utility (EU) and the utility of expected value (u(EV)) for a risk-averse agent?

For a risk-averse agent, the utility of the expected value (u(EV)) is greater than the expected utility (EU), indicating a preference for the expected value over the lottery itself.

What does the agent's willingness to pay for insurance indicate?

The agent's willingness to pay reflects their risk aversion, as they would pay up to the difference between their initial wealth and the certainty equivalent to avoid risk.

What is the expected payout for the insurance company when insuring an agent with a 60% chance of incurring a $20,000 cost?

The expected payout is calculated as 0.6 * $20,000 + 0.4 * $0 = $12,000.

What fee should the insurance company charge to ensure profitability?

The insurance company should charge a fee of at least $12,000 to cover the expected payout and ensure profitability.

What is the significance of the insurance company's risk attitude?

The insurance company's risk-neutral attitude means it is indifferent to risk and focuses on expected costs, allowing it to charge fees based on average outcomes.

What are the conditions under which insurance is welfare-improving?

Insurance is welfare-improving when the insured party is more risk-averse than the insurer, allowing for a mutually beneficial exchange.

What happens when both parties in an insurance scenario are risk-neutral?

If both parties are risk-neutral, insurance does not create welfare gains or losses, as both are indifferent to risk.

How does the risk premium of the agent compare to that of the insurance company?

In the example, the agent's risk premium was $751 while the insurance company's risk premium was $0, indicating the agent is more risk-averse.

What is the impact of market competition on insurance pricing?

In a competitive market, the price of insurance may decrease, benefiting the insured party more than in a monopolistic scenario.

What is full insurance?

Full insurance is when an insurance policy covers the entire loss incurred by the insured party.

What is an example of insurance with a deductible?

In insurance with a deductible, the insurer covers only part of the losses, requiring the insured to pay a specified amount before coverage kicks in.

What utility function is used in the example of an agent with a car accident risk?

The utility function used is u(w) = ln(w), indicating logarithmic utility based on wealth.

What are the probabilities of different outcomes for the agent in the car accident example?

The probabilities are: 0.9 for no accident, 0.09 for a small accident, and 0.01 for a large accident.

What are the damages associated with a small and large car accident in the example?

The damages from a small accident total $3,000, while the damages from a large accident total $50,000.

What is the role of insurance companies in the context of risk?

Insurance companies accept fees from insured parties and cover costs associated with risks, allowing individuals to transfer risk.

What is the expected loss for the agent in the insurance example?

The expected loss for the agent is the difference between their initial wealth and the expected value of the lottery.

What does it mean for an agent to be risk-averse?

A risk-averse agent prefers certain outcomes over uncertain ones and is willing to pay a premium to avoid risk.

What is the expected utility for the insurance company when insuring an agent?

The expected utility for the insurance company is calculated based on the probabilities of the agent's health outcomes and the associated costs.

What is the relationship between the agent's certainty equivalent and the insurance company's expected value?

The agent's certainty equivalent is lower than the insurance company's expected value, indicating greater risk aversion on the part of the agent.

What is the agent's wealth without insurance in the best outcome?

$100,000

What is the agent's wealth with a loss of $3,000?

$97,000

What is the agent's wealth with a loss of $50,000?

$50,000

How is the expected utility calculated without insurance?

EU = 0.9 ln(100000) + 0.09 ln(97000) + 0.01 ln(50000)

What is the expected utility without insurance?

11.5032

What does a deductible of $2,000 mean in the insurance policy?

The agent pays the first $2,000 of damages; the insurance covers the rest.

What is the agent's wealth with insurance in the best outcome?

$100,000 - f