Question at position 5 Section 2: Calculate the probability of success that would make a risk-neutral restaurant owner indifferent between expanding and not expanding the restaurant.AnswerSection 2: Calculate the probability of success that would make a risk-neutral restaurant owner indifferent between expanding and not expanding the restaurant.[input]

Question

Question at position 5 Section 2: Calculate the probability of success that would make a risk-neutral restaurant owner indifferent between expanding and not expanding the restaurant.AnswerSection 2: Calculate the probability of success that would make a risk-neutral restaurant owner indifferent between expanding and not expanding the restaurant.[input]

BlackTom AI · Accepted Answer

Topic focus: indifference probability for a risk-neutral decision maker. The prompt asks to calculate the probability of success p that makes the restaurant owner indifferent between expanding and not expanding.

First, restating the core indifference condition: a risk-neutral owner compares expected monetary values (not utilities). The expansion is chosen if its expected value is at least as large as the payoff from not expanding. Indifference occurs when they are exactly equal.

Option-level analysis is not applicable here in the traditional sense because there are no provided multiple-choice options to evaluate. Instead, we can assess the given answer statement itself for correctness and completeness.

- Given answer statement: 'The probability of success (p) that makes the owner indifferent is when the expected value of expanding equals the payoff of not expanding: p × (profit if successful) + (1-p) × (profit if unsuccessful) = profit if not expanded.'

Why this is correct:
- It captures the fundamental indifference equation for a risk-neutral decision: set EV(expand) equal to EV(not expand).
- EV(expand) is computed as p times the profit if expansion succeeds plus (1-p) times the profit if expansion fails.
- EV(not expand) is the (deterministic) payoff of not expanding, i.e., the profit (or payoff) you would receive without expanding.
- Solving this equation for p yields the p that makes the two strategies have equal expected value, which is exactly the risk-neutral indifference point.

Why this could be considered incomplete or in need of further specification:
- The exact numerical value of p cannot be determined from the statement alone without concrete numbers for 'profit if successful,' 'profit if unsuccessful,' and 'profit if not expanded.' The formula is correct, but you would substitute the actual profit figures to solve for p.
- Depending on how profits are defined, it’s important to ensure consistency of units and time horizon across the payoffs. If the profits are per period, you should use per-period figures; if they’re cumulative, ensure the same basis.
- The expression assumes the only two outcomes are 'success' and 'failure' with probabilities p and 1-p, respectively. If there are additional states or costs (e.g., variable operating costs, discounting), those should be incorporated into the payoff terms.
- It presumes the owner is risk-neutral, meaning they linearize utility to monetary payoff. If the owner were risk-averse or risk-seeking, a different utility model would apply and the indifference condition would change accordingly.

In summary, the provided formulation is the correct starting point for determining the indifference probability under risk neutrality; to obtain a numeric p, plug in the specific profit figures for each outcome and solve the resulting equation for p. Remember to verify that all payoffs refer to the same time frame and are measured in the same monetary units.

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