1. [MRS] Calculate the marginal rate of substitution (MRS) of the following utility functions.

(a) U(x, y) = 2x 2y 3

(b) U(x, y) = 4x 4y 6 + 10

(c) U(x, y) = √ x + 0.5y

(d) U(x, y) = 3x α + 4y 2.

[Cobb-Douglas Preferences] Asad lives in a world with two goods (N=2), apples and bananas. He has a Cobb-Douglas utility function over these goods U(a, b) = a 0.7 b 0.3 . Asad has an income I = 200, and faces prices pa = 2, pb = 1

(a) We know that if Asad is optimizing (and consumes a bundle that contains both apples and bananas), that the slope of the budget constraint should be equal to his MRS. Use this and his budget constraint to solve for his optimal bundle.

(b) Suppose the price of apples rises to 4, and the price of bananas rises to 3. What is his new optimal consumption bundle?

(c) Suppose Asad instead had a utility function given by U(a, b) = a 1.4 b 0.6 + 1. Given the same prices and income as in part (b), what is his optimal bundle?