Uncertainty

Question 1

Expected value

Answer 1

A random variable X can take the values x1,x2,…xkand each value occurs with probability p1,p2,…pk. Then the expected value of X is E[X] = x1∗p1 + x2∗p2 + ….xkpk

In other words, the expected value is the sum

Question 2

Fair gamble

Answer 2

Zero expected value

Question 3

Expected utility

Answer 3

– Probability-weighted average of utility
EU[X] = u(x1) ∗ p1 + u(x2) ∗ p2 + ….u(xk) ∗ pk

– EU = Pr(Lose) U(Lose) + Pr(Win) U(Win)
– Different than the utility of expected value, since utility functions usually concave (diminishing MU of income)! Diminishing MU of income means that the next dollar is worth less to you than the last one was in terms of happiness you gain

Question 4

Expected value

Answer 4

EV = probably win * amount you win + probability lose * amount you lose

Question 5

Individuals maximize…

Answer 5

expected utility (not expected value)

Question 6

Risk-averse

Answer 6

Most individuals are risk-averse, won’t take gambles even with small expected value

Question 7

How does diminishing marginal utility relate to expected utility?

Answer 7

Because of diminishing marginal utility, you value winning money less than you value losing money (i.e. you really don’t want to lose money, and you only care a little about winning money)

Diminishing MU of income means that the next dollar is worth less to you than the last one was in terms of happiness you gain

Question 8

Why would someone not take a “more than fair” gamble?

Answer 8

If someone is risk averse, then money will have diminishing marginal utility to them. This means that winning makes them less happy than losing makes them sad. Even if a gamble has positive expected value (is “more than fair”), they might still not want to take it because they weigh the losses greater than they weigh the gains. Risk loving or risk neutral individuals would take the gamble, because of its positive expected value (risk loving individuals would get even more utility from the uncertainty). A linear utility function indicates that someone is risk neutral, so they would also take a “more than fair” gamble.

Question 9

Risk neutral

Answer 9

Linear expected utility, equal to expected value

You’re indifferent to winning and losing so you’ll take the gamble

Question 10

Risk loving

Answer 10

Convex expected utility
Increasing marginal utility
You love risk

Question 11

How do the different shapes of utility functions relate to an individual’s risk behavior?

Answer 11

Concave utility functions indicate diminishing marginal utility, which creates risk aversion. Linear utility functions indicate constant marginal utility, which creates risk neutrality. Convex utility functions indicate increasing marginal utility, which creates risk-loving behavior.

Question 12

As the risk gets smaller, you become…

Answer 12

less risk-averse, more risk-neutral

Locally linear and therefore risk-neutral (linear utility is constant marginal utility; indifferent to winning/losing)

Larger gambles make you more risk-averse. With smaller gambles, your utility function will be closer to locally linear, which makes you closer to risk-neutral. As the gambles get larger, your utility function will appear more and more concave. Intuitively, it makes sense that risk-averse individuals would be more worried about the increased risk from larger gambles.

Question 13

As you become wealthier, you become ______ with respect to any given gamble if your utility function is U=sqrt(C)

Answer 13

More risk-neutral.

Becoming wealthier makes you more risk-neutral. Any given gamble will now appear smaller relative to your underlying resources, so you will be more willing to take the gamble. Graphically, these relatively smaller gambles will appear more locally linear, which makes your utility function appear more risk-neutral. You will not become risk-loving, because your utility function will never look convex.

Question 14

Risk premium

Answer 14

The risk premium is amount you are willing to pay for insurance above the expected cost of the risk you are insuring against. The risk premium will increase as one acts more risk averse and will decrease as one acts less risk averse, because risk aversion will increase your desire to insure against a given risk. Since higher initial income makes you less risk averse, it will decrease the risk premium. Since a larger gamble makes you more risk averse, it will increase the risk premium.

Question 15

What will be the price of insurance in a perfectly competitive/non-cooperative Bertrand oligopoly?

Answer 15

In a perfectly competitive market or a non-cooperative Bertrand oligopoly, the price of insurance will be the marginal cost to the insurance company. For the risk-neutral insurance company, the marginal cost of taking on a client is the expected cost of the risk that they are insuring against for that client

Question 16

What will be the price of insurance in a monopoly/cooperative oligopoly?

Answer 16

In a monopoly market or a cooperative oligopoly (a cartel), the insurance company will have complete market power and be able to price insurance at one’s willingness to pay, which will equal the expected cost of the risk plus one’s risk premium.

Question 17

Which of the following are plausible explanations for the popularity of the lottery?

Answer 17

As a form of entertainment & people are misinformed about lottery payouts.

We know people are not risk-loving because of the popularity of insurance. We know that “People are risk averse over small gambles and risk loving over large gambles” is not the explanation because scratch-offs (small gambles) are more popular than megabucks (big gambles). “People play the lottery as a form of entertainment” is a plausible explanation, because if people got separate utility from the act of playing the lottery, then this might override their risk aversion. “People are misinformed about lottery payouts” would also be a plausible explanation, because people may not realize they are taking a less than fair gamble. Information about the fairness of the lottery is also not very well publicized.

Question 18

Is the lottery good or bad for society?

Answer 18

If people play the lottery as a form of entertainment, then the government is simply providing a beneficial service and using the revenue from selling that service to finance important public goods, which would generally be good for society. If people are misinformed about lottery payouts, then the government is tricking people (predominantly poor people) into wasting their money.