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)

Mary (consumer 1) and Lucy (consumer 2) are the only two consumers in the economy.Each of them consumes only two goods, fish (good x) and chips (good y), which they also own.Consumer 1’s utility function is given byU 1(x, y) = x2y3She has 25 units of x and 5 units of y. Consumer 2’s utility function is given byU 2(x, y) = ln x + ln y.She has 6 units of x and 20 units of y. Let p be the price of x and normalise the price of y to 1.(a) Draw the Edgeworth Box of this economy, marking clearly the endowment point. For eachconsumer, sketch the indifference curve passing through the endowment point.(b) Calculate the marginal rate of substitution for each consumer at the endowment point.(c) Who is going to sell x and buy y? Why?(d) Find the demand of consumer 1 for x—denote it as x1—and the demand of consumer 1for y—denote it as y1.(e) Find the demand of consumer 2 for x—denote it as x2—and the demand of consumer 2for y—denote it as y2.(f) Find the equilibrium price of good x.(g) Find the equilibrium consumption bundle of each consumer.(h) Mark your answer to part (g) in the Edgeworth Box you have drawn for part (a). Drawthe budget line under the equilibrium price. For each consumer, draw the indifferencecurve passing through the equilibrium consumption bundle.1

There are two goods in the economy, wine and cheese. The price of wine is p1 dollars per litre of wine. The price of cheese is p2 per kilogram of cheese. The consumer has wealth M dollars to spend on wine and cheese. Let x1 denote the quantity of wine in litres she chooses to buy and let x2 denote the quantity of cheese in kilograms she chooses to buy. Suppose throughout that she can only purchase non-negative quantities of either good. Solve the following utility maximization problems when the preferences of a consumer are characterized by the following utility functions. For each part clearly specify what is the quantity of each good the consumer wishes to purchase given her budget constraint. (a) (10 points) U (x1; x2) = x1 x 4 2 [Hint: MU1 (x1; x2) = x 4 2 and MU2 (x1; x2) = 4x1x 3 2 .] (b) (15 points) U (x1; x2) =  x1 + 2x2 ￾ x 2 2 if x2  1 x1 + 1 if x2 > 1 [Hint: MU1 (x1; x2) = 1 and MU2 (x1; x2) = 2 (1 ￾ x2). First work out the solution for the case in which x 1 > 0 and x 2 > 0. Then show if p2 > 2p1 it is optimal for our consumer to spend her entire wealth on wine.] (c) (5 points) U (x1; x2) = maxfx1; x2g That is, U (x1; x2) =  x1 if x1  x2 x2 if x1 < x2 . Warning: In this case MU1 (x1; x2) and MU2 (x1; x2) do not exist! [Hint: Reason from Örst principles how the individual should allocate her wealth between the two goods if her aim is to maximize a utility function of this form. To do this, I recommend drawing a diagram to graph the budget set for di§erent price ratios. Can you Ögure out which 2 bundle or bundles maximise her utility for each particular budget set? To complete your answer you should now be able to work out for general prices and income levels which bundle or bundles maximize her utility.]

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?

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