Expected utility function-indifference curve-marginal cost

Question

Question 1: Joe has the expected utility function u (c₁,c₂) = p₁c₁^0.5 + p₂c₂^0.5 where c₁ and c₂ are consumption levels in states 1 and 2, which occur with probabilities p₁ = 3/8 and p₂ = 5/8 

a) Calculate Joe’s expected  utility if c₁ = 0 and c₂ = 100. Calculate the expected value of this lottery, and the utility of receiving this expected value with certainty, i.e., of receiving this expected value in both states of nature. Which is larger: the expected utility of the lottery, or the utility of the expected value of the lottery? What does this tell you about  Joe’s attitude toward risk?

b) The certainty equivalent of the lottery is a payoff which, if received with certainty, would make the person indifferent between the lottery and receiving the certainty equivalent in each state. Calculate the certainty equivalent of the lottery in (a). How does it compare to the expected value?

c) Show that, in general, if one is risk averse, then the certainty equivalent of a lottery falls short of its expected value. The difference between these two is often called the risk premium for the lottery.

d) Find the formula for an indifference curve, giving c₂ as a function of c₁, identifying combinations of c₂ and c₁ that give the same expected utility.

e) Find the slope of this indifference curve when c₂ = c₁. Explain why it has this slope.

Question 2: A monopolist has the production function y = z₁^1/2z₂^1/3. Demand is given by p = a - by.

a) Find the conditional input demand for input 1.

b) Find the long-run cost function of the firm.

Question 3: All consumers in a community have the expected utility function

u (c₁,c₂) = p₁c₁^0.5+ p₂c₂^0.5. Now c₁ = 36 and c₂ = 64, which occur with probabilities p₁ =1/2 and p₂ =1/2

a) Find the expected utility of this lottery.

b) Suppose now there are two agents. Each faces this lottery, but the outcomes are independent. Hence, with probability 1/4 each receives 36, with probability 1/4  each receives 64, and with probability 1/2 , one receives 36 and the other 64. The two agents make an agreement that if one receives 36 and the other 64, the fortunate agent transfers 14 to the unfortunate agent. Calculate their expected utility under this arrangement. Such an arrangement is the essence of an insurance policy: people facing uncorrelated risks can use the fact that they are unlikely to both experience losses to eliminate some of the risk they face.

c) Suppose again that there are two agents, each facing this lottery, but with perfectly correlated lotteries. Hence, either both receive 36 or both receive 64. Can an arrangement like that of [b] give both agents a higher expected utility than that calculated in [a]?

d) In light of your answers to [a] - [c], explain why it may be easier to insure houses against fires than against floods. What you have just discovered is that if insurance is to be effective, it must involve uncorrelated rather than correlated risks. An insurance policy cannot help with risks that tend to either impose a loss on everyone, or on no one. What risks are likely to fall into this category?

Question 4: A monopolist has a cost function c (y) =100 + ay + by²and faces demand p₁ = 100 – y₁ and p₂ = 150 – 2y². Define y = y₁ +y₂.

a) Find the marginal cost function of the firm.
b) Find the profit-maximizing prices in the two markets.
c) Does market 2 affect market 1’s price, and if so, why?
 
Question 5:

a) John has a von-Neumann-Morgenstern utility function over Bernoulli utility u(c) = c^1/2. John has $1000. If John is offered a riskless bond that will pay him $1000 in four years’ time, and the interest rate is r, what price should John be willing to pay for the bond today?

b) Now suppose that the bond is not so riskless. There is a 10% chance that the bond will be worthless when it comes due. What is the most John should be willing to pay?
 
Question 6: A monopolist has a cost function c (y) =100 + ay. Demand is given by y = 100 – bp(y).

a) The monopolist can perfectly price discriminate (i.e. first-degree). Find the monopoly profits as a function of a and b.

b) Now suppose that there are 10 consumers who each have the demand function y = 100 – bp(y). The monopolist cannot perfectly price discriminate, but can use two-part pricing, setting a fixed fee for access to the good, and then a per-unit price.
Find the optimal fixed fee and the optimal per-unit price.  Find the monopoly profits per consumer as a function of a and b. 

c) What is the consumer surplus per consumer in case (b)?