Assignment:
Question 1. [Risk Aversion and Insurance]
Kevin owns a house worth $500,000, and the utility from the house value (H) follows the function given below:
U(H) = In(H)
a) If there is a 20 percent probability that Kevin's house catches on fire and incurs a 20 percent loss of its current value, what is the house's expected utility?
b) Suppose he can buy insurance against fire at an actuarially fair premium of $20,000. If he chooses to buy the insurance, what is the expected utility? Which one would he prefer - being insured or not being insured?
c) What is the maximum amount that he would be willing to pay to insure the possible loss?
Question 2. [Marginal Productivity and Cost Minimization]
Consider the following production function for a microeconomic textbook 'A':
q = 0.2k0.3∫0.7,
where q is the number of textbooks printed, k is the number of hours of printing machines used, and 1 is the number of labor hours employed. Suppose both k and 1 cost the same amount ($20 per hour), and 20 books are to be printed given a budget of $4,000.
a) If two inputs (again, k and 1) are hired in equal $ amounts, how much of each input will be hired and how much will be the total cost?
b) If instead, the production follows the equal marginal productivity rule (i.e., input quantities need to be chosen such that their marginal productivities are equal), how much of each input will be hired and how much will be the total cost?
c) Assuming that textbooks are produced under the equal marginal productivity rule, let's suppose that all budgets ($4000) are spent for printing books. How many more extra books can be printed? How is your answer related to the returns to scale of the given production function?
Question 3. [Costs - Short Run vs Long Runj]
Following is a firm's production function for soccer balls:
q = √k1l
Where k1 is the fixed amount of k(capital equipment) in the short run, and I is the amount of labor.
a) Calculate the firm's short-run total costs as a function of q, v. w and k1 (where v = unit capital price; w = unit labor price)
b) Given q, v, and w, how should the capital stock be chosen to minimize short-run total cost?
c) Using your answer from b) above, calculate the long-run total cost of soccer ball production.
d) For v=$1, w=$4, and k1=100, at what level of q is short-run cost function tangent to the long-run cost function?
Question 4. (Contingent Input Demand and Production Function]
Suppose the total-cost function for a firm is given by C = q v2/3w1/3,
where q is the production output quantity, v is the capital rental rate and w is the wage rate.
a) Use Shephard's lemma to compute the contingent input demand for capital (k) and labor (l).
b) Using your answer from a) above, calculate the underlying production function for q.