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Economics

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Question 1 A utility function is given by u(x1, x2) = x 0.6 1 + x 0.3 2 . Prices are p1 = 3, p2 = 3. Determine the equation of the income offer curve. Plot this curve using the attached grid. A good is defined as luxury good if in response to an income increase, a person spends a larger fraction of income on the good. A necessary good is the opposite of a luxury good. Is one of the goods a necessary good or a luxury good? Are goods normal or inferior? Question 2 Suppose utility is given by u(x1, x2) = 20 log(x1 + x2) + x2. Price are p1 = 1, p2 = 2. Determine the equation of the income offer curve, and plot it using the attached grid. Is one of the goods a necessary good or a luxury good? Are goods normal or inferior? Question 3 Consider the utility function u(x1, x2) = x 2 1 x2. Prices are p1, p2, and income is I. (a) Determine the (Walrasian) demand functions for goods 1 and 2, x1(p1, p2, I) and x2(p1, p2, I). Are goods 1 and 2 gross substitutes, gross complements, or neither. (b) Determine the Hicksean demand functions h1(p1, p2, u), h2(p1, p2, u). (c) Determine the expenditure function, e(p1, p2, u). Question 4 Consider the utility function u(x1, x2) = √ x1 + √ x2. Prices are p1, p2, and income is I. (a) Determine the (Walrasian) demand functions for goods 1 and 2, x1(p1, p2, I) and x2(p1, p2, I). Are goods 1 and 2 gross substitutes, gross complements, or neither. (b) Determine the Hicksean demand functions h1(p1, p2, u), h2(p1, p2, u). (c) Determine the expenditure function, e(p1, p2, u). Question 5 Suppose utility is again u(x1, x2) = √ x1 + √ x2. Use the results from question 3 to answer the following questions. Suppose that prices are p1 = 1, p2 = 4 and income is I = 64. Then due to a tax of 3 Dollars per unit, the price of good 1 increases to p1 = 4. 1 Determine the government’s tax revenue from the consumer. Then determine the amount of money ˆI the person would need at the before-tax prices to get the aftertax utility. The difference, I − ˆI measures the loss to the consumer from the tax. Determine the deadweight loss of the tax, and compare this to the result we derived in class for Cobb-Douglas utility u(x1, x2) = x1 x2. Is the deadweight loss larger or smaller compared to the Cobb-Douglas case? Provide intuition. Question 6 Utility is given by u(x1, x2) = min{x1, 4×2}. (a) Determine the (Walrasian) demand functions for goods 1 and 2. (b) Determine the Hicksean demand functions h1(p1, p2, u), h2(p1, p2, u). (c) Determine the expenditure function, e(p1, p2, u). Question 7 Utility is given by u(x1, x2) = x1 + 2×2. (a) Determine the (Walrasian) demand functions for goods 1 and 2. (b) Determine the Hicksean demand functions h1(p1, p2, u), h2(p1, p2, u). (c) Determine the expenditure function, e(p1, p2, u).

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