(24 points) Consider a quasi-linear utility function, U(X, Y) = X^(1/2) + Y, with P_x and P_y being the price of X and Y respectively.

4. (24 points) Consider a quasi-linear utility function, U(X, Y) = X^(1/2) + Y, with P_x and P_y being the price of X and Y respectively.

a. (6 pts) Find the demand functions for X* and Y* for both the interior and the corner solutions.

b. (4 pts) Ceteris paribus, how does X* change when the price of X increases? When income Increases? Show your work.  

c. (6 pts) Now, let P_y = $1, I = 24, and Px increases from $0.5 to $2. Decompose the effect of this price increase into the substitution effect and the income effect, and calculate their magnitude.

d. (8 pts) For the same price change (i.e., P_x increases from $0.5 to $2), find the Compensating Variation (CV) and the Equivalence Variation (EV). How does your answer to this part relate to parts (b) and (c)?

5. (14 pts) Prove the following statements.

a. (7 pts) For a univariate production function, the marginal product always intersects the average product at the maximum of average product.

b. In a two-commodity world, goods X and Y cannot be luxury goods at the same time.

6. (12 points) Paul consumes only two goods, pizza (P) and hamburgers (H), and his utility function is given by U(P, H) = P + 4H. The price of pizza is $3 and the price of hamburgers is $6, and Paul’s income is $300. Knowing that he likes pizza, Paul’s grandmother gives him a birthday gift certificate of $60 redeemable only at Pizza Hut. Though Paul is happy to get this gift, his grandmother could have made him exactly as happy by spending far less than she did. Find the amount of cash that the grandmother could give to make Paul just as well off as with the pizza certificate.     

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