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Unformatted text preview: Rice University Professor Diamond PROBLEM SET 3: UTILITY AND DEMAND INCOME AND SUBSTITUTION EFFECTS Due September 18, 2008 Economics 370 Microeconomic Theory I. Suppose that an individual's indifference curves are described by straight lines with slopes equal to b, and that, as usual, prices and income, p1 , p2 , m are determined exogenously. 1. What is the demand function for x1 ? 2. What is the demand function for x 2 ? II. Suppose an individual's utility function (defined over two goods) is expressed u( x1 , x2 ) = x1 x22 . 1. Derive the demand function for x1 . 2. Derive the demand function for x 2 . III. Suppose now that the utility function is u( x1 , x 2 ) = ln( x1 ) + 2 ln( x 2 ) . 1. Again, derive the demand functions for both x1 and x 2 . 2. Are your answers for II and III the same? Why or why not? 3. For demand functions derived in II and III, what is the share of income spent on each of the two consumption goods? 4. Now consider an individual utility function that can be expressed as u ( x1 , x2 ) = x 2 x3 . 1 2
What is the share of income spent on each good in this case? (Hint: You should not have to calculate the demand functions to solve this problem.) IV. Consider income and price offer curves. 1. What is the difference between income offer curve and a price offer curve? 2. Draw an example of both. V. Suppose x1 is an inferior good when income is higher than a certain level m. 1. Describe intuitively what is meant by an inferior good. 2. Graph the change in x1 when income increases from m to m'. 3. Draw the corresponding income offer curve and Engel curve. VI. Suppose that a consumer has the utility function U(x1 ,x2 )=x1 x2 and an income of $24. Prices are p1 =1 and p2 =2. 1. Find the demand functions for x1 and x2 and, using those functions, find the optimal consumption of both goods. 2. Suppose that the price of good 2 rises to $3, while the price of good 1 stays at $1. How much would income have to change in order for the consumer to afford the original consumption bundle? 3. With the purchasing power held constant after the price change, what is the new demand for good 2? What is the Slutsky substitution effect (in units of good 2)? 4. However, in reality income does not change. It stays at $24. (We only calculate a new level of income to isolate the substitution effect.) Calculate the demand for good 2 with the new price, but the original income level. What is the Slutsky income effect (in units of good 2)? 5. What is the total effect of the change in the price of good 2 on the demand for good 2? 6. Graphically show the price change and its corresponding Slutsky substitution and income effects. VII. Graphically show the Slutsky substitution and income effects on good 1 that result from a decrease in the price of good 1. Assume that good 1 is a normal good. VIII. Graphically show the Hicks substitution and income effects on good 1 that result from a decrease in the price of good 1. Assume that good 1 is a normal good. ...
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