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# Write down the consumer\'s budget constraint and make a picture of it in (5, 5 )

INSTRUCTIONS TO CANDIDATES

1. (Hand in) A consumer has utility function

u(5, 5 ) = 5A51—A,

where 5 represents hours of leisure, 5 is food, and A is a parameter that satisfies 0 < A < 1. The consumer has 24 hours that she can divide between leisure and work. For each hour that she works, she earns a salary s > 0, and each unit of food costs p. She has initial wealth u (i.e., this wealth is independent of how much she works).

(a) Write down the consumer's budget constraint and make a picture of it in (5, 5 )

space.

(b) Solve the consumer's utility maximization problem and obtain the demand for leisure 5(p, s, u) and food 5 (p, s, u).

(c) Find a condition on the parameters such that the consumer does not work (Hint: examine the inequality 5(p, s, u) ≥ 24).

(d) How does the consumer's demand for leisure respond to a change in initial wealth u? How does the consumer's demand for leisure respond to a change in the salary s? Use partial derivatives to answer both questions.

2. (Hand in) Consider a consumer with utility function u(x, y) = x + ln y.

(a) Solve for the consumer's demand functions (ignore corner solutions. i.e., where

x = 0 or y = 0).

(b) Calculate the consumer's indirect utility function and verify that it is homoge- neous of degree zero.

(c) State a condition on px, py, and I such that the consumer's utility maximizing choice is x = 0. What is the consumer's demand for good 2 under this condition?

What is the consumer's indirect utility function under this condition?

(d) Draw an income-consumption curve and a price-consumption curve. For the price- consumption curve, use px as the moving part.

(e) Is y a normal good or an inferior good?

(f) Obtain the expenditure function e(px, py, u¯) by inverting the indirect utility func- tion and then obtain the compensated demand functions xC(px, py, u¯) and yC(px, py, u¯) by applying Shephards' lemma.

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