[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.]

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

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

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 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?

1. (6 points) Suppose a consumer has a utility functionU(x 1​ ,x 2​ )=x 1100​ x 2100​ and faces a budget constraintp 1​ x 1​ +p 2​ x 2​ =M. Derive her demand functions for good 1 and good 2 . Verify that both functions are homogeneous of degree 0 in(p 1​ ,p 2​ ,M). 2. (6 points) Suppose a consumer has a utility functionU(x 1​ ,x 2​ )=(x 1​ +x 2​ ) 2 and faces a budget constraintp 1​ x 1​ +p 2​ x 2​ =M. Derive her demand functions for good 1 and good 2 .