Question

Bob's risk preference is represented by the following expected utility formula: U(π,c₁ ;1−π,c₂ )=πlog(c₁ )+(1−π)log(c₂) Bob can invest $100 into two stocks. Stock 1 returns $1 when the economy is good and returns 0 when the economy is bad while Stock 2 returns $0 when the economy is good and returns $1 when the economy is bad. The price of stock 1 is denoted by p₁ and the price of stock 2 is p₂ =0.2. Suppose the economy is good with probability 0.65 and is bad with probability 0.35. i) Write down Bob's utility maximization problem.

Answer 1

Bob's utility maximization problem can be expressed as follows:

`\[ \max_(c_1, c_2) \pi \log(c_1) + (1-\pi) \log(c_2) \]`

subject to the budget constraint:

`\[ p_1c_1 + p_2c_2 = 100 \]`

Step-by-step explanation

In this utility maximization problem, Bob aims to maximize his expected utility given the uncertainty in the economy. The utility function
`\( U(\pi, c_1; 1-\pi, c_2) \)` represents Bob's preferences, where
`\( \pi \)` is the probability of the economy being good. The logarithmic form reflects his risk aversion; logarithmic utility functions are common in modeling risk preferences.

The objective function
`\( \pi \log(c_1) + (1-\pi) \log(c_2) \)` represents the expected utility, considering the probabilities of the economy being good or bad. The budget constraint
`\( p_1c_1 + p_2c_2 = 100 \)` ensures that Bob allocates his $100 budget across the two stocks, where p1 and p2 are the prices of stocks 1 and 2, respectively. Here, p2 is given as 0.2.

The optimization problem involves finding the values of c1 and c2 that maximize Bob's utility while satisfying the budget constraint. Solving this problem will yield the optimal investment strategy for Bob, considering the probabilities and payoffs associated with the two stocks.