✏️ L4 Multiple Choice Practice

Both questions and answers are AI generated, so they may contain errors. However, based on the way I generated them, they should hew very closely to the material we covered.

✏️ A lottery offers three prizes:

• $200 with probability 0.3
• $50 with probability 0.5
• $0 with probability 0.2

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

a. $75
b. $80
c. $85
d. $90

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Answer: c. $85

Explanation:
Calculate EV by multiplying each outcome by its probability and summing:
$EV = (0.3 \times 200) + (0.5 \times 50) + (0.2 \times 0) = 60 + 25 + 0 = \$85. \quad\text{✅}$

✏️ A candidate has two summer job options:

• Option A: A guaranteed salary of $3,000.
• Option B: A 40% chance of earning $5,000 and a 60% chance of earning $1,500.

Which option has the higher expected value?

a. Option A
b. Option B
c. Both have the same EV
d. Cannot be determined

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Answer: a. Option A

Explanation:
For Option B:
$EV = 0.4 \times 5{,}000 + 0.6 \times 1{,}500 = 2{,}000 + 900 = \$2{,}900.$ Since $3,000 > $2,900, the safe Option A has the higher EV.  ✅

✏️ An investor’s utility function yields $U(\$50) = 20 \text{ utils}$ and $U(\$150) = 50 \text{ utils}$. The investor faces a gamble paying $50 with probability 0.7 and $150 with probability 0.3. What is the expected utility of the gamble?

a. 29 utils
b. 30 utils
c. 35 utils
d. 40 utils

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Answer: a. 29 utils

Explanation:
$EU = 0.7 \times 20 + 0.3 \times 50 = 14 + 15 = 29 \text{ utils.} \quad\text{✅}$

✏️ Which statement best explains why expected utility theory is considered superior to expected value theory in decision-making under uncertainty?

a. It always leads to choosing the option with the highest monetary outcome.
b. It accounts for risk preferences by incorporating diminishing marginal utility of wealth.
c. It ignores probabilities in favor of outcomes.
d. It is simpler to compute than expected value.

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Answer: b. It accounts for risk preferences by incorporating diminishing marginal utility of wealth.

Explanation:
Expected utility theory transforms monetary outcomes into utilities, reflecting how additional dollars often provide less extra satisfaction (diminishing marginal utility), thus capturing risk aversion or risk seeking.  ✅

✏️ A individual faces two prospects: a guaranteed gain of $100 versus a 50–50 gamble between $50 and $200. If the individual’s certainty equivalent for the gamble is $90, what does this imply about their risk attitude?

a. Risk-loving
b. Risk-neutral
c. Risk-averse
d. Cannot be determined

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Answer: c. Risk-averse

Explanation:
The gamble’s expected monetary value is $\$125$ [$(0.5 \times 50) + (0.5 \times 200)$], yet the individual’s certainty equivalent is only $90. Since $CE < EV$, the person is risk-averse.  ✅

✏️ If a utility function for income exhibits increasing marginal utility (i.e., the slope gets steeper as income increases), what risk attitude does this indicate?

a. Risk-averse
b. Risk-neutral
c. Risk-loving
d. Indifferent to risk

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Answer: c. Risk-loving

Explanation:
Increasing marginal utility implies that additional income yields progressively more utility, a hallmark of risk-loving behavior.  ✅

✏️ A decision maker’s utility of income function is linear. What risk attitude does this represent?

a. Risk-averse
b. Risk-neutral
c. Risk-loving
d. Indeterminate

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Answer: b. Risk-neutral

Explanation:
A linear utility function implies constant marginal utility of income, so the decision-maker is risk-neutral—they evaluate options solely by their expected monetary value.  ✅

✏️ An individual with $40,000 income faces a 20% chance of incurring a $15,000 loss. What is the expected loss (expected cost) to the insurance company computed using expected value?

a. $1,500
b. $3,000
c. $6,000
d. $15,000

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Answer: b. $3,000

Explanation:
$\text{Expected loss} = 20\% \times 15{,}000 + 80\% \times 0 = \$3{,}000. \quad\text{✅}$

✏️ A risk-averse consumer faces a lottery with a best outcome of $60,000, a worst outcome of $40,000, and an EV of $55,000. They have a certainty equivalent of $52,000. What is the maximum premium the consumer would be willing to pay to guarantee the best outcome?

a. $2,000
b. $3,000
c. $4,000
d. $8,000

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Answer: d. $8,000

Explanation:
The insurance contract guarantees them $60,000. After a premium of $8,000, they will have $52,000 left over. Because this is their CE, $8,000 is the maximum they would ever possibly pay for insurance.  ✅

✏️ An insurance policy covers a potential loss of $100,000 that occurs with a 10% probability. What is the expected cost per policy (ignoring administrative costs)?

a. $1,000
b. $5,000
c. $10,000
d. $90,000

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Answer: c. $10,000

Explanation:
$\text{Expected cost} = 0.1 \times 100{,}000 = \$10{,}000. \quad\text{✅}$

✏️ A consumer currently earns $72,000 for sure. They are offered a new job with a 60% chance of paying $80,000 and a 40% chance of paying $60,000. If the consumer’s utility function is concave (reflecting risk aversion), which is most likely true?

a. They will definitely choose the new job because its EV is higher.
b. They will reject the new job because their certainty equivalent is likely below $72,000.
c. They are risk-neutral and will choose the option with the higher EV.
d. They will be indifferent between the options.

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Answer: b. They will reject the new job because their certainty equivalent is likely below $72,000.

Explanation:
You can decide which option you will choose by looking at the CE of every option and choosing the option with the higher CE. In this case, your current job with a guaranteed income of $72,000 has a CE of $72,000. If the new job has a CE less than that, you will reject it. Likewise, if the new job has a CE that is higher than $72,000, you will accept it.

The risky job’s EV ($0.6 \times 80{,}000 + 0.4 \times 60{,}000 = \$72{,}000$) is equal to their current income. Risk aversion implies the certainty equivalent will be lower than the EV (ie $CE < EV$ when someone is risk averse).

If the EV is $72,000 and $CE < EV$, then $CE < \$72{,}000$ — making the current guaranteed income more attractive.

Old job: you get $72k for sure, so $CE = \$72\text{K}$
New Job: EV is $72K, and $CE < EV$, so $CE < \$72\text{K} \quad\text{✅}$

✏️ Jeff’s income and utility are as follows:

• $30K → 20 utils
• $50K → 40 utils
• $70K → 55 utils
• $90K → 65 utils

He can choose between a secure job paying $70K and a risky job that pays $90K with probability 0.3 and $30K with probability 0.7. Which job should he choose based on expected utility?

a. Secure job
b. Risky job
c. Both yield the same utility
d. Cannot be determined

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Answer: a. Secure job

Explanation:
Secure job utility $= U(\$70\text{K}) = 55 \text{ utils.}$
Risky job $EU = 0.3 \times U(\$90\text{K}) + 0.7 \times U(\$30\text{K}) = 0.3 \times 65 + 0.7 \times 20 = 19.5 + 14 = 33.5 \text{ utils.}$
Since $55 > 33.5$, the secure job is preferable.  ✅

✏️ An individual faces a gamble with an expected income of $80,000 and a certainty equivalent of $70,000. If the best possible outcome of the gamble is $100,000, what is the maximum premium they would be willing to pay to secure a guaranteed income of $100,000?

a. $10,000
b. $20,000
c. $30,000
d. $40,000

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Answer: c. $30,000

Explanation:
The maximum premium is the difference between the best outcome and the certainty equivalent: $\$100{,}000 - \$70{,}000 = \$30{,}000. \quad\text{✅}$

✏️ A salesperson must choose between a fixed salary of $40,000 and a commission-based job that pays $60,000 with probability 0.5 and $20,000 with probability 0.5. Given a concave utility function (risk aversion), which option will she most likely choose?

a. Commission-based job
b. Fixed salary
c. She will be indifferent
d. It depends on additional benefits

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Answer: b. Fixed salary

Explanation:
The EV of the commission job equals $40,000, but a risk-averse individual will have a certainty equivalent lower than the EV. Hence, she prefers the fixed, guaranteed income.  ✅

✏️ An insurance company covers a risk where each client faces a 5% chance of incurring a $90,000 loss. What is the expected cost per policy? If the insurer charges a premium of $5,200 (with no administrative costs), what is the expected profit per policy?

a. Expected cost $4,500; expected profit $700.
b. Expected cost $4,500; expected profit –$700.
c. Expected cost $5,000; expected profit $200.
d. Expected cost $4,000; expected profit $1,200.

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Answer: a. Expected cost $4,500; expected profit $700.

Explanation:
$\text{Expected cost} = 0.05 \times 90{,}000 = \$4{,}500.$
$\text{Expected profit} = \text{Premium} - \text{Expected cost} = 5{,}200 - 4{,}500 = \$700. \quad\text{✅}$

Certainty Equivalent and Risk Premium

Question:

✏️ A firm is considering a project with a 70% chance of returning $120,000 and a 30% chance of returning $80,000. If the firm is risk-averse, what is most likely true about its evaluation of the project?

a. Its certainty equivalent equals $108,000 (the EV).
b. It will evaluate the project solely by its expected monetary return.
c. Its certainty equivalent will be lower than $108,000.
d. It will prefer this project over any risk-free alternative offering $108,000.

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Answer: c. Its certainty equivalent will be lower than $108,000.

Explanation:
A risk-averse firm discounts risky outcomes; even though the EV is $108,000, the certainty equivalent is lower because of the risk premium demanded to accept uncertainty.  ✅

✏️ What is the primary difference between expected value (EV) and expected utility (EU) in decision-making under uncertainty?

a. EV accounts for risk preferences, while EU does not.
b. EU incorporates the decision-maker’s risk preferences via the utility function, whereas EV does not.
c. EV is always numerically higher than EU.
d. EV is used only in insurance contexts, while EU is used only in investments.

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Answer: b. EU incorporates the decision-maker’s risk preferences via the utility function, whereas EV does not.

Explanation:
The main problem with EV is that it can’t account for risk aversion or risk-loving behavior. EU accounts for these via the utility function.  ✅

✏️ Diminishing marginal utility of income implies that as income increases, each additional dollar provides:

a. A larger increase in utility than the previous dollar.
b. The same increase in utility as the previous dollar.
c. A smaller increase in utility than the previous dollar.
d. An unpredictable change in utility.

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Answer: c. A smaller increase in utility than the previous dollar.

Explanation:
By definition, diminishing marginal utility means each extra unit of income adds less to overall satisfaction than the previous unit.  ✅

✏️ Under what condition might a risk-averse individual accept a gamble?

a. When the gamble’s expected value is significantly lower than their current wealth.
b. When the gamble’s expected utility exceeds the utility of their current, certain wealth.
c. Only when the gamble offers a guaranteed win.
d. When the gamble’s risk is completely eliminated.

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Answer: b. When the gamble’s expected utility exceeds the utility of their current, certain wealth.

Explanation:
Even risk-averse individuals will take on risk if the expected utility (after adjusting for risk) is higher than that of their current guaranteed situation.  ✅

✏️ In an insurance context, the “certainty equivalent” is best defined as:

a. The guaranteed amount that provides the same utility as a risky prospect.
b. The expected monetary value of the risky outcome.
c. The maximum possible loss.
d. The difference between the best and worst outcomes.

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Answer: a. The guaranteed amount that provides the same utility as a risky prospect.

Explanation:
The certainty equivalent is the sure amount a decision maker views as equally desirable to the uncertain prospect.  ✅