Consider the following 3 × 3 pure exchange economy. The goods are; x, y andz. Individual utility functions are: u1 = 3x1 + 2y1 + z1; u2 = 2x2 + y2 + 3z2 andu3 = x1 + 3y1 + 2z1. The initial endowments are e1 = e2 = e3 = (1, 1, 1). Doinitial endowments(a) constitute an ‘equal division’ allocation ?(b) constitute a non-envious allocation?(c) constitute a Pareto Efficient allocation?

Consider consumers A and B with uilitiesuAxA1 ; xA2
= ln xA1 + (1 ) ln xA2uBxB1 ; xB2
= ln xB1 + (1 ) ln xB2 :Suppose the total endowment of good 1 in this economy is !1 and of good 2it is !2: Suppose !A1 ; !A2
is owned by consumer A and !B1 ; !B2
is owned byconsumer B: We have !A1 + !B1 = !1 and !A2 + !B2 = !2 by deÖnition.(a) What is the exchange economy equilibrium here.(b) What is a Pareto e¢ cient allocation? The social welfare is maximizedby allocating the goods to consumers A and B; and the allocation has torespect the resourse constraints. Formulate the search for such allocationsas an optimization program.(c) What can be said about the equilibrium prices at that allocation?

Consider a two person two goods pure exchange economy. The goods are; xand y. The utility functions are u1(.) = x21.y1 and u2(.) = x2.y2, and theinitial endowments are e1(.) = (25, 75) and e2(.) = (25, 75). Assuming p1 = 1,compute competitive equilibria for this economy

Suppose that in addition to the consumer described inExercise 1 we also have an another consumer, Consumer 2, whose pref-erences are given by the utility functionu2(x1, x2) = x1x2where x1 is the amount of good 1 the consumer consumes and x2 theamount of good 2. Let ωℓ2 be Consumer 2’s initial endowment of goodℓ. [This is the utility function you analysed in Homework 1.] Similarlylet u1(x1, x2) = min{x1, x2} be the utility function for Consumer 1 thatyou discussed in the previous exercise and ωℓ1 be Consumer 1’s initialendowment of good ℓ.(1) Repeat the analysis of the previous exercise for Consumer 2,finding the demand functions x12(p1, p2, ω12, ω22) andx22(p1, p2, ω12, ω22)

[20 points] Consider an economy with 3 goods and 30 agents. There are 10 agents each with the utility functionu(x 1​ ,x 2​ ,x 3​ )=lnx 1​ +2lnx 2​ +3lnx 3​ and endowmentse 1​ =(3,2,1). Also, the other 20 agents each have the utility functionu(z 1​ ,z 2​ ,z 3​ )=3lnz 1​ +2lnz 2​ +lnz 3​ and endowmentse 2​ =(1,2,3). Normalizep 3​ =1. (a) Derive all 3 market-clearing conditions. (b) Calculate the Walrasian equilibrium pricesp 1∗​ andp 2∗​ . [Hint: One may use any 2 of the 3 market-clearing conditions to solve for these 2 prices.]

Consider a contingent claim economy with two consumers (A and B), two states tomor-row and one good in each state. Agents have the following utility functions:uA(x0A, x1A, x2A) = ln x0A + ln x1A + ln x2Aand uB (x0B , x1B , x2B ) = ln x0B + ln x1B + ln x2BAs for endowments,ωA = (1, 1, 2) and ωB = (1, 2, 1).(a) (1 point) Write down the state-by-state budgent constraints for A and B. Use bsAand bsB to denote the units of contingent claims consumers buy for each state s.(b) (1 point) State the utility maximization problem for A and B without the use ofbsA and bsB mentioned above.(c) (3 points) Derive consumers’ demand functions with the Lagrangian method.(d) (3 points) Find the market equilibrium for this contingent claim economy.(e) (1 point) Verify that the equilibrium allocation is Pareto efficient.