Question

PART 1:

Bob lives two periods: today and tomorrow. His preference is represented by the following Cobb-Douglas utility function:

U(c1,c2) = c1c20.9

where c1 is today’s consumption level and c2 is tomorrow’s consumption level. Suppose Bob’s income today is y1 = $100 and his income tomorrow is y2 = $200. Interest rate is denoted by r.

Suppose today Bob can borrow at most $50 (i.e., c1 ≤ 150). Then determine Bob’s optimal consumption bundle (c1∗,c2∗) as a function of r.

PART 2:

Carol’s risk preference is represented by the following expected utility formula:

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A)Suppose Carol is indifferent between the following two options: the first option A returns $64 with probability 1/2 and $36 with probability 1/2, and the second option B returns X for sure. Determine X. Is X smaller than the expected return of A? Explain why.

B)Consider the following three lotteries: L1 = (0.8,$100; 0.2,$25), L2 = (0.6,$100; 0.4,$36), and L3 = (0.5,$225; 0.5,$1). What is the ranking of these lotteries for Carol? Calculate the risk premiums of these lotteries for Carol.

Answer 1

Bob's optimal consumption allocation between today and tomorrow is a utility maximization problem determined by his incomes, borrowing limit, and interest rate (r). Carol's financial decision-making is based on her expected utility, reflecting her risk preferences and leading to calculations of risk premiums.

Step-by-step explanation:

The optimal consumption bundle for Bob, who is faced with the decision of how to allocate his limited resources across two periods with consumption levels c1 and c2, can be found through utility maximization given his incomes y1 and y2 and the constraint of the interest rate r. Based on the provided utility function U(c1,c2) = c1c20.9, and considering he can only borrow up to $50, he must choose a combination of c1 and c2 that falls along his budget constraint, which will in turn depend on the rate r. Bob, like the character Quentin in the reference examples, will compare different scenarios to maximize his utility, considering his preferences for consuming today versus saving for tomorrow.

Similarly, Carol's decision-making process involving assessing the expected utility of different financial options can be analyzed to determine her risk preferences. For example, given her indifference between a certain outcome of receiving X dollars and a lottery with different probabilities, calculated figures can reveal her value of X and her attitude towards risk. The questions also involve calculating risk premiums for different lotteries, which further reflect how Carol valuates sure outcomes compared to risky ones.