Econ. 101
Spring 2008
Tauchen
Practice Problems and Homework on Consumer Theory
1. Describe the effect on the budget line of each of the changes listed below. You should be able to show each change on a graph.
a. The price of good X decreases.
b. The price of good Y increases.
c. Income increases.
d. Both prices decrease proportionately, (for example, both decrease by 50%):
e. Income and both prices increase proportionately:
f. Income and the price of good X increase proportionately
2. John’s budget line for goods X and Y has intercepts of 20 units of good X and 5 units of good Y. John’s income is $200 per time period. What are the prices of goods X and Y? What is the slope of the budget line? Provide the equation for the budget line.
3. Karen’s MRSXY at the bundle (12,2) is .6 .
a. Provide a verbal interpretation of MRSXY.
b. How many units of good Y is Karen willing to give up for an additional unit of good X?
c. How many units of good Y must Karen be given to compensate for the loss of a unit of good X?
4. Roy’s utility function is U(x,y) = a x.3 + ln y which is defined for x≥ 0 and y>0.
a. Provide an expression for the marginal utility of good X and for the marginal utility of good Y.
b. Provide an expression for MRSxy(x,y).
c. Determine MRSxy(1,1) for a=5.
5. Sylvia has utility U(x,y) = x3 y2 .
a. Provide an expression for the marginal utility of good X and for the marginal utility of good Y.
b. Provide an expression for MRSxy(x,y).
c. Determine the utility for the bundle (5,1). Suppose that the individual has 1 unit of good X. Determine the amount of Y required for the individual to obtain the same utility as at the bundle (5,1). Compute the MRSxy for the two bundles. Are the values are consistent with diminishing MRSxy?
6. Answer the above question for Butch with utility function U(x,y)= x3 y2 + x.
7. Eli consumes two goods and has utility U(x,y) = x y . Complete the following table and then graph the indifference curves for utility 1, 4, and 9.
Amount of good X / Amount of Y required to obtain utility 1 / Amount of Y required to obtain utility 4 / Amount of Y required to obtain utility 92
4
6
8
10
Eli’s brother Peyton has utility U(x,y) = x2y2 . Complete the following table and then graph the indifference curves for utility 1, 16, and 81.
Amount of good X / Amount of Y required to obtain utility 1 / Amount of Y required to obtain utility 16 / Amount of Y required to obtain utility 812
4
6
8
10
Eli’s friend Tom has utility U(x,y) = x.5y.5. Complete the following table and then graph the indifference curves for utility 1, 2, and 3
Amount of good X / Amount of Y required to obtain utility 1 / Amount of Y required to obtain utility 2 / Amount of Y required to obtain utility 32
4
6
8
10
Compare Eli’s, Peyton’s, and Tom’s indifference maps.
8. Provide expressions for Eli’s, Peyton’s, and Tom’s marginal utility for good X. For each individual determine whether the marginal utility for X is increasing in the amount of good X, decreasing in the amount of good X, or independent of the amount of good X.
9. The “more-preferred-to-less” assumption discussed in class is as follows.
Assumption 2: Suppose that A contains at least as much of each good as bundle B and that A contains more of at least one good. Then A is preferred to B.
An alternative form of the “more-preferred-to-less” assumption is:
Assumption 2′: Suppose that A contains at least as much of each good as bundle B and that A contains more of at least one good. Then A is liked at least as well as B.
a. Let A be the bundle (1,2,3) and B be the bundle (1,2,2).
i. Does Assumption 2 imply that A is necessarily preferred to B?
ii. Does Assumption 2′ imply that A is necessarily preferred to B?
iii. Suppose that Jan prefers A to B. Are Jan’s preferences consistent with Assumption 2′?
iv. Suppose that Jan prefers B to A. Are Jan’s preferences consistent with Assumption 2′?
b. Are thick indifference curves consistent with Assumption 2′? Explain.
10. In class, we mentioned that the indifference map for left and right shoes is Leontief, which is the formal name for an indifference map with “L-shaped” indifference curves. The justification for representing the preferences for left and right shoes in this fashion is that a left-shoe is valuable only if matched with a right shoe. Thus, the bundle (5,10) provides the same level of well-being as (5,5) where the first number listed is the number of left shoes and the second is the number of right shoes. In the past, however, some students have suggested that the Leontief representation of the preferences is too simple.
Explanation 1 for why the Leontief representation of the preferences for left and right shoes is too simple.
If an individual lives in a small dorm room, then extra shoes require valuable space. Thus the bundle (5,10) is worse than the bundle (5,5). Similarly (10,5) is worse than (5,5). More pairs of shoes are preferred to fewer, and (5,5) is preferred to (3,3). But unmatched shoes are a burden.
Explanation 2 for why the Leontief representation of the preferences for left and right shoes is too simple.
Having an extra left (or right) shoe could be an advantage. Suppose that the family dog chews only a right shoe or that the wearer steps in a puddle with only the right shoe. Then, having an extra right shoe provides additional well-being.
Which of the preference maps below is most consistent with Explanation 1? with Explanation 2? Explain.
11. Hillary consumes two goods and has utility U(x,y) = max{y,3x}. Show a graph with Hillary’s indifference map. [Hint: Identify bundles that provide 3 units of utility and show these on a graph. These bundles are on the indifference curve for u=3. Do the same for u=6. By observing the pattern for these indifference curves, you can figure out the general properties of the indifference map.] Also show the indifference map for the utility function U(x,y) = min{y,3x}.
12. Laura consumes two goods (X and Y) and the market prices of the goods are px = $4 and py =$8. Given her income, Laura finds it optimal to consume positive amounts of both goods. What is Laura’s marginal rate of substitution at her optimal bundle? Explain.
13. Sangeeta consumes two goods -- food and entertainment. Show an example of a preference map for which Sangeeta consumes only food at a low income level but positive amounts of both goods at a high income level. [Hint: Construct a graph on which you show an indifference map for which it is optimal for Sangeeta to purchase only food for budget lines with low income. But, as the budget line shifts out, it is optimal for Sangeeta to purchase both food and entertainment. There are many indifference maps consistent with the description of Sangeeta’s behavior and your indifference maps will not all be identical. In constructing your graph, be sure to label the axes.]
14. Howard’s income is $1800. At the initial price for goods X and Y of $40 and $20 respectively, Howard selects the bundle (30,30). The prices of goods X and Y then change to $20 and $40 respectively.
a. Assume that Howard has Leontief preferences. Does Howard continue to consume the bundle (30,30)? Construct a graph to support your argument.
b. Now, assume instead that Howard’s preferences satisfy the usual assumptions. His indifference curves are downward sloping and bowed in towards the origin. Does Howard continue to consume the bundle (30,30)? Construct a graph to support your argument. [Hint: Make your graph large enough and construct your graph carefully enough that you can show Howard’s indifference map in detail near the optimal bundle.]
15. The indifference curves shown to the right are parallel to one another (relative to the X axis) which means that the vertical distance between any two of them is the same at x=2, x=4, x=6, x=8, and any other value of x. Also, the slope at x=2 is the same for each indifference curve. Indeed, select any value of x and all of the indifference curves have the same MRSxy at that value of x, (although the MRSxy differ at x=2 and other x values.)
Construct a budget line consistent with the individual selecting the bundle (2,4). Now suppose that the individual’s income changes with no change in the prices. Explain the effect of the change in income on the individual’s optimal choice.
16. Rudy’s utility function is U(x,y) = 2x.5 + y.
a. By definition, his indifference curve for utility level 9 consists of bundles (x,y) for which 2x.5 + y = 9. Solve the equality for y and graph the indifference curve.
b. The indifference curve for utility level u consists of bundles for which 2x.5 + y = u. Solve for y. Compare your expression for y with the expression that you obtained from part a. How do they differ? What does your answer imply about the characteristic of the indifference map for this quasilinear utility function (with y entering linearly)? [Hint: You might look at the graph for the problem above and compare the characteristics of the two indifference maps.]
17. Judy regards 7-up and Sprite as perfect substitutes. Hence, her grandmother recommends that she purchase whichever sells at the lower price. We want to verify that our consumer theory model is consistent with grandmother’s common sense.
Construct an indifference map for 7-up and Sprite. Construct Judy’s budget line assuming that the price of 7-up is less than the price of Sprite. Use the graph to explain why Judy consumes only 7-up. (Note that for this problem, you must decide upon the prices and income.)
18. Show an indifference map for which Y is a “good” and X is a product in which the individual eventually becomes satiated.