c) Call the allocation where Ace and Butch each have half of each good (50 units of Hostess cakes and 50 bottles of water) the equitable allocation. Would each be willing to give up their endowments to move to the equitable allocation? Is the equitable allocation Pareto optimal? (5 points) d) Assume that the price of Hostess cakes is pyy = 1. Calculate the competitive equilibrium price of water pw such that the markets for both Hostess cakes and Evian water clear (i.e. total amount of each good is exactly equal to the sum of quantities demanded by each individual). (10 points) 3. (25 points total) Samsung and LG produce liquid crystal display (LCD) panels for televisions sets. Suppose (for now) the panels they produce are identical and Samsung and LG compete in the quantities of display panels they produce. Samsung can produce a panel for a marginal cost of $50 per unit and LG can produce a panel for a marginal cost of $80 per unit. Demand for LCD panels is given by the following inverse demand function p=500 -0.01Q. a) If the two firms must produce their quantities simultaneously, what is the Cournot-Nash equilibrium in this market? In other words, what quantity does each firm produce and what is the market price? What is each firm's profit? (5 points) b) Suppose instead that Samsung chooses its quantity first, followed by LG. What is the Stackelberg equilibrium in this market? In other words, what quantity does each firm produce and what is the market price? What is each firm's profit? (5 points) Suppose now Samsung focuses on LCD panel while LC focuses on plasma panels. As a result, the two manufacturers produce imperfect substitutes. Each firm's costs are the same as above. Samsung's demand function is now 45 = 50000 - 100pg + 100p, and LC's demand function is now q, = 50000 - 100p, + 100pg. a) What is LC's optimal price for any given price Samsung charges py, i.e. LG's reaction function? (5 points) b) If Samsung and LG have to set prices simultaneously, what is the Bertrand-Nash equilibrium in this market? In other words, what price does each firm charge and what quantity does each firm produce? (5 points) c) Suppose instead that Samsung chooses its price first, followed by LG. What are the equilibrium prices and quantities for each firm in the sequential game? (5 points) 4. (25 points total) Ned has the utility function UN(W) = W, Arya has the utility function UA(W) = VW, and Sansa has the utility function ((W) = W2. For each of these individuals, answer the following questions. a) Is the individual risk averse, risk neutral, or risk seeking? (5 points) b) Suppose each chooses to invest & proportion of their wealth in a risky investment that returns $8 for every $1 in- vested with probability , and $0 with probability , and (1-x) proportion of their wealth in a riskless investment that pays $2 for every $1 invested with certainty. What is the optimal choice of a for each person? (10 points) (Hint: Remember second-order conditions. If the second derivative is negative, setting the first-derivative equal to zero provides a maximum. If the second derivative is positive, setting the first-derivative equal to zero provides a minimum.) c) Suppose each individual has $10,000. In addition, each believes that, over the course of the year, with probability 0.99 they will suffer no injuries, but with probability 0.01 they will break one of their legs. The loss for each individual of breaking a leg (in medical bills, lost wages, etc.) is $10,000. What is the maximum annual premium each individual will be willing to pay UPMC to insure themselves against the loss of breaking a leg? (10 points) (Hint: It is best to think of this problem as a 99% chance of keeping $10,000 and a 1% chance of ending up with nothing)
