Q.  Suppose a monopolist produces a good in two plants, with quantity q1 produced in Plant 1 and q2 produced in Plant 2. The demand and cost functions function of the good is given by:  

a)     Derive the FOC and SOC conditions of profit maximization for this firm. Show that SOC is satisfied (impose necessary conditions).

b)    Suppose that there is a shift in demand (change in α). Show that output will increase in both plants as a result.

c)     Which plant will have a greater increase in output? Please explain why.

Q. The demand and cost function of a firm is given respectively by: P = 100 – 4q and C = 50 + 4q. The firm chooses quantity q to maximize revenue subject to the constraint that profit = $334.

a.     Find the revenue maximizing q and the price at which the firm would sell this output. Show that SOC is satisfied.

b.     Find the output q and price if the firm was a profit maximizing firm.

c.      Compare the results in a and b. Please explain the difference. 

Q..  Suppose an individual has initial wealth W. There is a chance of losing X with a probability ∏. He can purchase insurance at an actuarily fair premium of ∏y for y dollars of coverage. The individual chooses the amount of coverage to maximize Expected Utility = ∏U(A) + (1-∏)U(B) where:

A= Individual’s wealth if he loses X but collects Y from the insurance company.

B = Individual does not lose X.

Note that in each case the individual has to pay the insurance premium ∏y.

a)     Derive the first order condition of utility maximization. Show that if U is concave, the SOC condition of utility maximization will be satisfied.

b)    How much insurance will this person buy to protect against the loss of X? Does this depend on risk aversion of the individual? (Hint: what does concavity suggest about risk aversion)