3. Consider a general equilibrium model where there are two persons, A and B, and two goods, 1 and 2 (good 2 is the numeraire," i.e., its price can be normalized to unit once you find the equilibrium price ratio). Person A has 6 units of good 1 and 3 units of good 2 while person B has 4 units of good 1 and 5 units of good 2. Person A's utility is represented by UA = 3 In (x1 ) +2In (x2 ) and person B's utility is given by UB= 2 In (x1 ) +3 In (x2 ). The corresponding Marshallian demands are given byx A = (3/5) (ma /P1 ), x2 = (2/5) (ma/p2), x1 = (2/5) (m/P1), and x2 = (3/5) (m /P2), where m' is the income of personj = A, B. (a) Find the marginal rates of substitution for both persons. Examine if the following three allocations are feasible and (for feasible allocations only) Pareto efficient: . x4 = (5.4, 6.0) and xB = (4.6, 3.2) . xA= (6.0, 6.0) and xB= (4.0, 2.0) . x4= (6.0, 3.2) and x= (4.0, 4.8) (b) Identify the income of each of the consumers and derive their gross demand functions. Derive their individual excess demand functions for both goods. (c) Derive the aggregate excess demands both goods. State Walras Law and use your findings to verify it. (d) Find the general equilibrium (i.e., the equilibrium price ratio). (e) Given the price ratio you found in part, find the final allocation of goods Hint: you might want to normalize the "numeraire" price before you proceed). Round your numbers to the third decimal place. Confirm that this allocation is Pareto efficient. Part 2: True or False. Each question is worth 5 points. In total 25 points for Part 2. Please provide a complete and relevant explanation. 1. The only Pareto efficient allocation of an apple between Adam and Eve (assuming that both like apples) is "half-half". 2. Consider Brian with utility given by u = 0.5 In W. Brian owns a car of value W = $15, 000. Driving a car is a risky activity and the probability of Brian being involved in a minor car accident is 10%. In that case the cost of repairs will be $2000. Brian is risk-averse and his willingness to pay for full coverage insurance is $35. 3. In a linear demand, constant cost industry with two identical firms, there are two possibilities for timing: Cournot where they produce simultaneously and Stackelberg where one produces before the other. The Stackelberg leader will necessarily earn a higher profit than it does under Cournot competition. 4. Consider two firms producing an identical product with marginal costsMC1 = $0.98 and MC2 = $1.00 correspondingly. The two firms set their prices simultaneously (Bertant compe- tition) and the lowest price gets the entire market. If they set the same price they split the market evenly. A set of equilibrium prices isp1 = $1.00 and p2 = $1.01. 5. The core in the Edgeworth box is the set of all allocations that are Pareto efficient
