4. (4x1 = 4 points) David gets $3 per week as an allowance to spend any wag.r he pleases. Because he likes only peanut butter (b) and jelly sandwiches 0'), he spends the entire amount on peanut butter (at $0.05 per ounce) and jelly (at: $0.10 per ounce}. Bread is provided he of charge by a concerned neighbor. David is a particular eater and makes his sandwiches with exactly 1 ounce of jelly and 2 ounces of peanut butter. He is set in his ways and will never change these proportions. a. Write down David's utility Jnction and his budget constraint. b. How much peanut butter and jelly will David buy with his $3 allowance in a week? Suppose the price of jelly were to increase to $0.15 an ounce. How much of each commodity would be bought? c. By how much should David's allowance be increased to compensate for the increase in the price of jelly i.e., if David wants to consume the bundle obtained from solving part (b) but at the higher price of jelly ($0. 1 5 an ounce), how much additional allowance would he need? d. Graph your results in parts (a) to (c). 5. (3x2 = 6 points) Assume that utility is given by U{x. y) = xu'ayu'? a. Use the uncompensated demand Jnctions to compute the indirect utility function and the expenditure function for this case. b. Use the expenditure Jnction calculated in part (a) together with Shephard's lemma to compute the compensated demand function for good x. c. Use the results 'om part (b) together with the uncompensated demand function for good I to show that the Slutsky equation holds in this case. 6. (2 points) For a three-good CES utility function U (x, y, z) = i i i, the demand function for good I is given by: I I p: + as}. + as: ' Use this demand function for x to determine whether 31: and y, or x and z are gross substitutes or gross complements. x