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.

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?

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)

Assume that the economy consists of two infinitely lived consumers namedi=1, 2. There is one nonstorable consumption good. Consumericonsumesc ti​ at time t. Consumer i's preferences over consumption streams are represented by \[ \mathrm{E}_{0} \sum_{t=0}^{\infty} \beta^{t} \log \left(c_{t}^{i}\right) \quad \beta \in(0,1) \] The endowments are decided each period by a toss of a fair coin. If the outcome of the coin toss is heads(H), consumer 1 receives one unit of the consumption good, and consumer 2 receives nothing. If the outcome is tails(T), consumer 2 receives one unit of the consumption good, and consumer 1 receives nothing. (i) Lets t​ ={H,T}denote the state of the economy at datet.q t0​ (s t​ )denotes the time 0 price of an Arrow-Debreu security that pays one unit of consumption in datetconditional on states t​ , andc t′​ (s t​ )denotes consumer is consumption at datetin states t​ . Define an Arrow-Debreu competitive equilibrium. (ii) Compute the equilibrium under the assumption that Arrow-Debreu securities are traded in time 0 after the realizations 0​ =H. Explain your answer.

Exercise 1. Consider an economy with two consumers (named Con-sumer 1 and Consumer 2) and two goods (called good 1 and good 2).Suppose that Consumer 1’s preferences are given by the Cobb-Douglasutility functionu1(x11, x21) = x11x21where x11 is the amount of good 1 that Consumer 1 consumes andx21 the amount of good 2 that Consumer 1 consumes. Suppose thatConsumer 1 is endowed with 9 units of good 1 and 2 units of good 2.Suppose that Consumer 2’s preferences are given by the Leontiefutility functionu2(x12, x22) = min{x12, x22}where x12 is the amount of good 1 that Consumer 2 consumes andx22 the amount of good 2 that Consumer 2 consumes. Suppose thatConsumer 2 is endowed with 1 units of good 1 and 8 units of good 2.Normalise the price vector so that p2 = 1.(1) The contract curve is defined as the set of Pareto optimal alloca-tions that make each consumer at least as well off as they are atthe initial allocation. Find the contract curve. [Hint: Think beforeyou start calculating! It may help to start with a draft answer topart (3).](2) Find the equilibrium price and the equilibrium allocation.(3) Represent this economy in an Edgeworth Box diagram. Clearlymark the initial endowment, the equilibrium allocation, and theequilibrium price vector. Draw the indifference curves through theinitial endowment for the two consumers, labelling clearly whichis which. Also draw the contract curve and clearly label it. Drawyour Edgeworth Box as neatly and accurately as you can.Date: First Semester, 2024.1

[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.]