1. Exercise 1 (20 pts)

There are two goods in this economy, Cigarettes and "all other goods," sold at aper unit price of pc and po = $1. Consider a representative consumer with CobbDouglaspreferences u(xc, xo) = x1/3c x2/3o , where xc is the quantity of cigarettes and xois the quantity of "all other goods" (assume both amounts can be any positive realnumber). This consumer has a $9 income. Suppose congress enacts a $0.75 quantitytax on cigarettes (a quantity tax is a per unit tax). Show the original and new budgetconstraint for a representative consumer.

(1) Compute the pre- and post-tax optimal bundle. Then decompose the change incigarette consumption into income and substitution effects.(2) What is the maximum amount the smoker would have been willing to pay thegovernment not to impose the quantity tax? What is the name of this concept?

(3) What amount of income would have to be given to the smoker after the tax to restore his or her utility to the pre-tax level? What is the name of this concept?

(4) What is amount of taxes the government collects from the consumer? Show that the government could levy a lump sum tax on the consumer without changing the price of cigarettes, collect the same amount of revenue, and leave the consumer better off.

Exercise 2 (21pts)

Two consumers, A and B, consume books and wine. Consumer A's initial endowmentis (60, 10); that is, A has has 60 books and 10 bottles of wine. Consumer B's initialendowment is (20, 30). They have no other assets, and make no trades with anyoneother than each other. For A, a book and a bottle of wine are perfect substitutes; herutility function is uA(xb, xw) = xb + xw, where xb is the number of books she consumes,and xw is the number of bottle of wines she consumes. Consumer B's preferences aregiven by the utility function uB(xb, xw) = xbxw.(1) Draw an Edgeworth box with B's consumption measured from the upper righthandcorner of the box. On this diagram, mark the initial endowment, and labelit e. Use red ink to draw A's indifference curve that passes through her initialendowment. use blue ink to draw B's indifference curve that passes through herinitial endowment.

(2) At any Pareto optimal allocation, where both people consume some of each good,it must be that their marginal rates of substitution are equal. Why? What are Aand B's marginal rates of substitution? What is the equation of the contract curve?Use black ink to draw the contract curve on the diagram

(3) In a competitive equilibrium, A will consume some books and wine. But inorder for her to do so, what must the ratio of prices be? What is the competitive equilibrium of this exchange economy?

Exercise 3 (20t pts)

Martina has the utility function u(c1, c2) = c1/21 +13c1/22, where ci is the consumption in period i = 1, 2. Her income in period 1 is $100, and her income in period 2 will be$125. The price of the good is $1 in period 1, and 75 cents in period 2. She can borrowor lend at an interest rate of 10%.

(1) Write an equation that describes Martina's budget constraint.(2) What is Martina's optimal consumption stream?

Exercise 4 (20 pts)

Consider an exchange economy with two consumers and two goods. The consumptionsets of the consumers are R2+. Each consumer has the same utility function u(x1, x2) =min{x1, x2}. Provide a complete characterization of all the allocations which are in thecore of this economy. (Recall that an allocation is in the core if it is i) Pareto optimal and ii) Individually rational (weakly preferred to the endowment))

Exercise 5 (36 pts)

Emilie loves profiteroles. She makes a profiterole by combining one unit of chocolatesauce (say 8 fl oz) and two puffs (stuffed with paste). She receives one unit of utilityper profiterole. Bertrand just likes to eat. He receives one unit of utility per puff orper unit of chocolate sauce. Bertrand and Emilie are on a diet that assigns points tofood: pp is the number of points for a puff, and pc is the number of points for a unit ofchocolate sauce. They would simply like to maintain their current weight, so the diet rules are as follows: They can consume any combination of chocolate sauce and puffs as long as the total number of points is less than (or equal to) the point-value of their endowment.

(1) Let xci R+ be the amount of chocolate sauce and xpi R+ be the quantity of puffs consumed by consumer i (i is Emilie or Bertrand). Write down both agents'utility functions. (Note: Suppose any amount of profiteroles in R+ is possible.)

(2) What kind of goods are chocolate sauce and puffs for Bertrand? Show his indifference curves on a graph.

(3) What kind of goods are chocolate sauce and puffs for Emilie? Show her indifference curves on a graph.

(4) Suppose that Emilie and Bertrand decide to have a private party. Bertrandbrings one bottle of chocolate sauce (8 units) and Emilie brings one box of puffs (16puffs). Represent this two-consumer, two-good exchange economy in an Edgeworthbox, remembering to indicate the endowment point.

(5) Find the contract curve in this economy.(6) What is the equilibrium point in this Edgeworth box? What is the equilibrium point ratio pp/pc?

Exercise 6 (20 pts)

Consider an exchange economy with two consumers, A and B, and two goods, 1 and2. Consumer A's initial endowment is (1, 2), and B's initial endowment is (2, 1). Theirpreferences are represented by the utility functions uA(xA1, xA2) = (xA1)0.25(xA2)0.75 anduB(xB1, xB2) = 2(xB1)0.1(xB2)0.9.

(1) Represent the initial situation in an Edgeworth box (draw the indifference curves through the endowment).

(2) What is the equation of the contract curve? Draw it on the diagram.

(3) Compute the Walrasian equilibrium of this economy.