1. Suppose that your roommate becomes ill and asks you to go to the store to pick up some food for them. They ask you to pick up some frozen pizzas (P), boxes of cookies (C), and gallons of milk (M). A bundle of goods is (P,M,C), the number of pizzas, boxes of cookies, and gallons of milk, respectively.

Your roommate gives you $30 to do the shopping with and gives you the following directions about what combinations of goods they prefer:

(4, 1, 1) (2, 2, 2)

(1, 3, 3) (0, 3, 4)

(3, 2, 2) (4, 1, 1)

(2, 2, 2) (1, 3, 3)

1. 1. If your roommates preferences are transitive, what can you say about their preference between bundles (4, 1, 1) and (0, 3, 4)? What about (1, 3, 3) and (3, 2, 2)? Explain.
   2. If your roommates preferences are rational and satisfy non-satiation, (i.e., more is better), which of the bundles listed above are at least as good as (2, 1, 1)? Explain.
   3. If you get to the store and find that the price of frozen pizzas is $8, the price of boxes of cookies is $4, and the price of gallons of milk is $3, which bundle of goods will you purchase for your roommate? Explain.
2. Suppose that a consumer with utility function given by U(x, y) = x^12 y^3/2 has wealth w = 40 and faces prices P_x = 5 and P_y = 6.
   1. Write the consumers Utility Maximization Problem (UMP) and the Lagrangian Function for that problem.
   2. Write the FOCs for the consumers UMP.
   3. Derive the consumers Equimarginal Condition.
   4. Use your answers to (b) and (c) to find x^* and y^* , the utility maximizing bundle of goods.
3. For each off the following utility functions, does it represent the same preferences as U(x, y) = x^12 y^3/2 or not?
   1. U¹ (x, y) = 5ln(x) + 10ln(y)
   2. U² (x, y) = x²y⁶ + 44
   3. U³ (x, y) = -1/(x^1/2 y^3/2)
   4. U⁴ (x, y) = (x+5)^1/2 (y+3)^3/2
   5. U⁵ (x, y) = e^(x25 y^6/5)

4. A consumer with utility given by U(x, y) = 2x + 4y + xy has wealth w and faces prices P_x and P_y.
   1. Write the consumers Utility Maximization Problem (UMP) and the Lagrangian Function for that problem.
   2. Write the FOCs for the consumers UMP.
   3. Derive the consumers Equimarginal Condition.
   4. Use your answers to (b) and (c) to find x^* (P_x, P_y, w) and y^* (P_x, P_y, w), the Marshallian Demand for goods x and y.
   5. Verify the Law of Demand for both goods.
   6. Using x^* (P_x, P_y, w), are the two goods complements or substitutes? Explain.
   7. Derive the Marginal Utility of Wealth as a function of P_x, P_y, and w.
   8. Is the Marginal Utility of Wealth increasing, decreasing, or constant in P_y? Give some intuition for your answer.