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Homework 4: Econ 500

The homework is due on Wedneday, October 3 at 4pm. Each question is worth 0.5 points

Question 1 A utility function is given by u(x1, x2) = x 0.6 1 + x0.3

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).

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).

Question 5 Suppose utility is again u(x1, x2) = √

x1 + √

x2. Use the results from ques-

tion 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.

Determine the government’s tax revenue from the consumer. Then determine the

amount of money Î the person would need at the before-tax prices to get the after-

tax utility. The difference, I − Î 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? Pro-

vide intuition.

Question 6 Utility is given by u(x1, x2) = min{x1,4x2}.

(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).

Question 7 Utility is given by u(x1, x2) = x1 + 2x2.

(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).

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