Exercise 1. Suppose consumer's preferences are given by U (X, Y ) = min{X, 2Y } that is goods X and Y are perfect complements. Please, make sure to label all your graphs accurately.

(a) Suppose the consumer has $9 to spend and the price of good Y is $3. Sketch the price-consumption curve for the prices of PX = $1.5, PX = $3 and PX = $4.5. To do this, carefully draw the budget constraints associated with each of the prices for good X, and indicate the bundles that the consumer chooses in each case. Also draw indifference curves that pass through the optimal bundles.

(b) Suppose the prices for good X and Y are PX = PY = $1. Sketch the income-consumption curve for the values of M = $3, M = $4.5 and M = $6. To do this, carefully draw the budget constraints associated with each of the value for income, and indicate the bundle that the consumer chooses in each case. Also draw indifference curves that pass through the optimal bundles.

(c) Based on your solution to part (b) conclude whether goods X and Y are normal or inferior. Explain.

(d) Draw Engel Curve for good Y (Hint: the quantity of good Y should be on the X axis).

Exercise 2. Suppose a consumer's utility function is given by U (X, Y ) = XY that is MUX = Y and MUY = X. Also, the consumer has $20 to spend, and the price of X, PX = $4 and the price of Y , PY = $1.

a) How much X and Y should the consumer purchase in order to maximize her utility?

b) How much total utility does the consumer receive?

c) Now suppose PX decreases to $1. What is the new bundle of X and Y that the consumer will demand?

d) Of the total change in the quantity demanded of X, how much is due to the substitution effect and how much is due to the income effect?

Exercise 3. Suppose there are two consumers, A and B. There are two goods, X and Y . There is a TOTAL of 8 units of X and a TOTAL of 8 units of Y . The consumers' utility functions are given by: UA(X, Y ) = 2X + Y , UB (X, Y ) = XY . You may find usefull to know that MUAX = 2, MUAY = 1, MUBX = Y and MUBY = X. Consider arbitrary allocation: (XA, YA) and (XB, YB ). Find the conditions this allocation should satisfy to be Pareto efficient (Hint: restrict your attention to interior point at which both goods are consumed in positive amount by each consumer). Draw contract curve using Edgeworth Box. Make sure to label your Edgeworth Box carefully and accurately.

Exercise 4. Suppose there are two consumers, A and B, and 2 goods, X and Y . The utility functions of each consumer are given by: UA(X, Y ) = min{X, 3Y }, UB (X, Y ) = X + 2Y . The initial endowments are: XA = 4, YA = 2, XB = 4 and YB = 2.

a) Using an Edgeworth Box, illustrate the initial endowments. Also, for each consumer, draw the indifference curve that runs through their bundle. Be sure to label your graph carefully and accurately.

b) Is the initial allocation Pareto Efficient? If your answer is no, find allocation that Pareto dominates given allocation.

c) Repeat parts a) and b) for UA(X, Y ) = XY and UB (X, Y ) = X +2Y
keeping everything else unchanged.

d) Repeat parts a) and b) for UA(X, Y ) = XY and UB (X, Y ) = min{X, 3Y }.