Determine the slutzky equation with endowments

Question

Assignment:

Question 1 A person is deciding whether to invest 1,000 Dollars into a project that has the following payoffs:

1. With probability 0.5, the project's value is 200;

2. With probability 0.25, the project's value is 1,000;

3. With probability 0.15, the project's value is 2,000;

4. With probability 0.1, the project's value is 10,000.

(a) What is the project's expected return?

(b) Suppose the person has a Bernoulli utility function u(x) = log(x). The person's income is m. Specify the person's expected utility of investing into the project (as a function of m).

(c) Determine numerically the value of m at which the person is indifferent between investing and not investing. Does the person invest if the person's actual income is higher or lower than this cutoff value?

(d) Now suppose that utility is u(x) = -1/x. Determine numerically the value of m at which the person is indifferent between investing and not investing, and explain why this new cutoff is higher.

Question 2 A person has a wealth of $20,000 but faces an accident that results in a loss of $12,000 with probability, p. Suppose that Bernoulli utility is given by u(x) = -1/x.

1. Determine the maximum amount of money the person is willing to pay for complete coverage (as a function of p).

2. Now suppose that an insurance company offers an insurance contract with a deductible of $2,000. Again, determine the maximum amount of money a person is willing to pay for this insurance contract.

Question 3 Suppose a person's Bernoulli utility function is given by u(x) = x².

(a) Is the utility function concave, i.e., is the person risk averse?

(b) Suppose the person has an income of 100 Dollars, and considers whether or not to invest all of the money in a stock that pays either 0 Dollars with probability 0.6 or 200 Dollars with probability 0.4. Will the person make the investment? Would any risk averse or risk neutral person make the investment?

Question 4 Consider the Slutzky equation with endowments.

1. Suppose that ∂h_i(p, u)/∂p_j = 2. The person's endowment of good j is 10 units, and the current consumption is 4 units. Finally, suppose that an increase in the price of good i increases demand by one unit. Determine ∂x_i(p, I)/∂I.

2. Suppose that the person's demand for a good k equals the person's endowment. Suppose that ∂x_k(p, pe)/∂p_k = -2. What does this imply about the change of Hicksean demand for good k when the p_k changes?

3. Suppose that a good is inferior and that the person is a net-seller of the good. Does demand for the good increase or decrease when the price is increased?

4. Can an increase of the interest rate result in a decrease of saving? Explain your answer.

Question 5 Suppose that a person's endowment is e = (10, 0), prices are p₁ = p₂ = 1 and utility is u(x₁, x₂) = x₁ x₂. Suppose that the price of good 1 increases to p₁ = 2.

In class we defined the Slutsky equation with endowment for infinitesimally small changes. The same concept can be used for discrete changes, and for the Slutsky version of the Slutsky equation (i.e., using the Slutsky substitution effect).

Determine the following: (a) The endowment income effect, (b) the (standard) income effect, (c) the (Slutsky) substitution effect.

Question 6 Suppose there are two periods i = 1, 2. Let x_i denote consumption in each period. Utility is given by u(x₁, x₂) = min{x₁, x₂}. Suppose the person is endowed with 10 units of the good in period 1 and 0 in period 2. Let r be the interest rate.

Determine the person's consumption in period 1 as a function of r.

Do savings increase or decrease when the interest rate is increased?

How do your results change when you replace the utility function by u(x₁, x₂) = log(x₁) + β log(x₂), where 0 < β < 1?