Question 3. Bruce and Sheila both consume the same goods (shrimp and Fosters) in a pure exchange econ- omy. Bruce is originally endowed with 4 units of good 1 (shrimp) and 6 units of good 2 (Fosters). (As usual, think of both shrimp and Fosters as being infinitely divisible.) Sheila is originally en- dowed with 8 shrimp and 4 Fosters. They both have the utility function U(x₁, x₂) = x₁x₂. Let shrimp be the numeraire, so that p₁ = 1. (a) What is the equation describing the Pareto efficient allocations? Which of these allocations are individually rational? (b) How many shrimp does Bruce wish to buy (or sell) in terms of the price of Fosters? At what price of good 2 is Bruce satisfied with his endowment (i.e., he does not want to buy or sell any shrimp)? (c) How many shrimp does Sheila wish to buy (or sell) in terms of the price of Fosters? (d) Determine the equilibrium price of Fosters. (e) Draw an Edgeworth box to illustrate the your answers above. Question 1. We reconsider the exchange economy from question 3 in Homework 5 to explore the possibility of price manipulation. Bruce and Sheila both consume the same goods (shrimp and Fosters) in a pure exchange economy. Bruce is originally endowed with 4 units of good 1 (shrimp) and 6 units of good 2 (Fosters). (As usual, think of both shrimp and Fosters as being infinitely divisible.) Sheila is originally endowed with 8 shrimp and 4 Fosters. They both have the utility function U (x₁, x₂) = x₁x₂. Let shrimp be the numeraire, so that p₁ = 1. (a) Suppose the Walrasian auctioneer knows the utility functions of Bruce and Sheila, and cal- culates the equilibrium price on the basis of reported endowments. Suppose Bruce and Sheila honestly report their endowments to the Walrasian auctioneer. What price p2 will the auctioneer announce? Which consumer buys shrimp and which consumer sells shrimp (and how much)? [Hint: This is easy, and is just to get us started. If it is relevant, you may use the information from the solution to Homework 5.] (b) Suppose Sheila contemplates under-reporting her endowment of shrimp. What would be the resulting announcement of p2 by the auctioneer if Sheila were to report an endowment of 6 shrimp and 4 Fosters? (c) Suppose that Sheila must buy and sell the quantities implied by her endowment report. What will be her trades and final consumption of shrimp and Fosters when she reports an endowment of 6 shrimp and 4 Fosters? Which does Sheila prefer and why: reporting truth- fully or under-reporting shrimp? (d) Suppose (just for this part) that Sheila does not need to buy and sell the quantities implied by her endowment report. Is it consistent with market clearing for Sheila to still believe that she can buy and sell as much as she would like at the announced prices? (e) Now consider the n-fold replica economy. In this economy, there are n clones of Bruce (with identical endowments and preferences), and n clones of Sheila (with identical endowments and preferences). Suppose all clones of Bruce and all clones of Sheila report their endow- ments truthfully. What price p₂ does the auctioneer announce? (f) Continuing with our analysis of the n-fold replica economy, suppose now that one clone of Sheila reports an endowment of 6 shrimp and 4 Fosters, while all other consumers an- nounce truthfully. What is the resulting announcement of p₂ by the auctioneer? What hap- pens to this announcement as n gets arbitrarily large?