3. Assume an economy with two goods, X and Y. Three individuals derive satisfaction from consuming these goods, as represented by the following utility functions: Peter's utility function is U(x, y) = xy; Bridget's utility function is U(x, y) = 2x + 4y; Sarah's utility function is U(x, y) = min{3x, y}, where U(x, y) is the amount of satisfaction, expressed in utils.

a. For each of these individuals graph the indifference curve showing bundles that produce 12 utils of satisfaction.

b. Define the MRS and find it for Peter's and Bridget's utility functions (not graded: try to find it for Sarah).

c. What is the actual value of the MRS at bundle A (X = 10, Y = 20) for Peter and Bridget (not graded: try to find it for Sarah).

4. Chris gets satisfaction from both pizza and beer. Suppose he has a budget of I = $120 a month to spend on these goods, where pizza costs $4 (per slice) and a bottle of beer cost $2.

a. Graph Chris's budget line, with Pizza on the x-axis. What is the price of Pizza in terms of beer?

b. Show what happens to the budget line if his budget increases to I = $160.

c. Show what happens to the budget line if the price of Pizza increases to $6 (assume that I is equal to $120 again).

d. Assume that Chris's favorite pub is giving away five bottles of beer each month (per customer) at no charge, in order to promote the business. Use Chris's original income and prices and show what happens to his budget line following this new promotional campaign.