Economics 410

Spring 2008

Tauchen

Practice Problems on Consumer Theory -- Answers

  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.
  1. The price of good X decreases: Answer: The Y intercept is unchanged. The X intercept is larger.
  2. The price of good Y increases. Answers: The X intercept is unchanged. The Y intercept is smaller.
  3. Income increases: Answer: The BL shifts out; the slope is unchanged.
  4. Both prices decrease proportionately, (for example, both decrease by 50%): Answer: The budget line shifts out. The slope is unchanged. The way in which the graph changes is the same as for an increase in income.
  5. Income and both prices increase proportionately: Answer: No change in the BL.
  6. Income and the price of good X increase proportionately. Answer: The X intercept is unchanged. The Y intercept is larger.
  1. 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? Give the equation for the budget line.

Answer: The prices of goods X and Y are $10 and $40 respectively. Consider a graph with X measured on the horizontal and Y on the vertical axis. The slope of the budget line is –px/py = -1/4. The equation for the budget line is y=5-x/4.

  1. Karen’s MRSXY at the bundle (12,2) is .6 .
  1. Provide a verbal interpretation of MRSXY. Answer: The number of units of good Y that Karen is willing to give up for a unit of good X or the number of units of good Y required to compensate her for the loss of a unit of good X.
  2. How many units of good Y is Karen willing to give up for an additional unit of good X? .6
  3. How many units of good Y must Karen be given to compensate for the loss of a unit of good X? .6
  1. Roy’s utility function is U(x,y) = a x.3 + ln y which is defined for x≥ 0 and y>0.
  1. Provide an expression for the marginal utility of good X and for the marginal utility of good Y.
  2. Provide an expression for MRSxy(x,y).
  3. Determine MRSxy(1,1) for a=5.

Answer: The marginal utility of good X is .3ax-.7 . The marginal utility of good Y is 1/y.

MRSxy(x,y)= .3ax-.7 y = .3ay/x.7. MRSxy(1,1)=1.5

  1. Sylvia has utility U(x,y) = x3 y2 .
  1. Provide an expression for the marginal utility of good X and for the marginal utility of good Y.
  2. Provide an expression for MRSxy(x,y).

Answer: The marginal utility of good X is 3x2y2 and the marginal utility of good Y is 2x3 y.

MRSxy(x,y)=1.5 y/x.

  1. 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?

Answer: U(5,1) = 125. Determine the value of Y for which U(1,y) =125 or y2=125 or y=11.18

MRSxy(5,1) = 1.5/5 and MRSxy(1,11.18) =1.5*11.18 . The bundles (5,1) and (1,11.18) are on the same IC. The MRSxy is smaller at the larger value of X, which is consistent with decreasing MRSxy.

  1. Answer the above question for Butch with utility function U(x,y)=x3 y2+x.

Answer: The marginal utility of good X is is 3x2y2 +1 and the marginal utility of good Y is 2x3 y.

MRSxy(x,y)=1.5 y/x + 1/(2x3 y).

U(5,1) = 130. Determine the value of Y for which U(1,y) =130 or y2+1=130 or y=11.36.

MRSxy(5,1) = 1.5/5 + 1/(2*125) and MRSxy(1,11.36) =1.5*11.36 + 1/(2*11.36) . The bundles (5,1) and (1,11.36) are on the same IC. The MRSxy is smaller at the larger value of X, which is consistent with decreasing MRSxy.

  1. 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 9
2 / 0.50 / 2.00 / 4.50
4 / 0.25 / 1.00 / 2.25
6 / 0.17 / 0.67 / 1.50
8 / 0.13 / 0.50 / 1.13
10 / 0.10 / 0.40 / 0.90

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 81
2 / 0.50 / 2.00 / 4.50
4 / 0.25 / 1.00 / 2.25
6 / 0.17 / 0.67 / 1.50
8 / 0.13 / 0.50 / 1.13
10 / 0.10 / 0.40 / 0.90

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 3
2 / 0.50 / 2.00 / 4.50
4 / 0.25 / 1.00 / 2.25
6 / 0.17 / 0.67 / 1.50
8 / 0.13 / 0.50 / 1.13
10 / 0.10 / 0.40 / 0.90

Compare Eli’s, Peyton’s, and Tom’s indifference maps.

Answer: The graph shows Eli’s indifference curves for utility levels 1, 4, and 9. The graph also shows Peyton’s ICs for utility levels 1, 16, and 81 and Tom’s ICs for utility levels 1, 2, and 3. The numbering of the indifferences curves differs for the three individuals but the three have the same preferences. In saying that they have the same preferences, we mean that for any two bundles A and B all three individuals have the same ranking. Either all three prefer A to B, all three prefer B to A, or all three are indifferent between A and B – although the individuals attach different utility numbers to bundles except for the bundles with utility 1.

Marginal Utility of X
Eli / MUx(x,y) = y
Peyton / MUx(x,y) = 2xy2
Tom / MUx(x,y) = .5 x-.5 y.5
  1. 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.

Answer: Peyton’s marginal utility for good X is an increasing function of good X; the larger the value of X, the larger Peyton’s marginal utility for good X. Eli’s marginal utility for good X does not depend upon the amount of good X. Tom’s marginal utility for good X is a decreasing function of good X; the larger the value of good X, the smaller Tom’s marginal utility for good X.

  1. The “more-preferred-to-less” assumption discussed in class is listed below.

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.

  1. Let A=(1,2,3) and B=(1,2,2).
  1. Does Assumption 2 imply that A is necessarily preferred to B?Answer: Yes.
  2. Does Assumption 2′imply that A is necessarily preferred to B? Answer: No
  3. Suppose that Jan prefers A to B. Are Jan’s preferences consistent with Assumption 2′?

Answer:Yes.

  1. Suppose that Jan prefers B to A. Are Jan’s preferences consistent with Assumption 2′?

Answer:No. Bundle A contains the same amount of the first two goods as does bundle B and A contains more of the third good. Assumption 2′ implies that A is liked at least as well as B.

  1. Are thick indifference curves consistent with Assumption 2′? Explain.

Answer: Thick indifference curves are consistent with Assumption 2′ (but, as shown in class are not consistent with Assumption 2).

In the graph to the left, all of the points on the lowest line and all the points below the lowest line are an IC. All of the points on the second line and all of the points between the lowest and the second lines are an IC. All of the points on the highest line and all of the points between the highest and the middle lines are an IC. With these thick ICs, there are many points such as A and B with A containing more of both goods than B and with both being in the same IC. These preferences are consistent with Assumption 2′ since 2′ requires that for such bundles A be liked at least as well as B.

  1. In class, we mentioned that the indifference map for left shoes and right shoes was 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 it is matched with a right shoe. Thus the bundle (5,10) provides the same level of well-being as the bundle (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, some students have suggested that the Leontief representation of the preferences is too simple.

Explanation 1 for why the Leontief representation of the preferences 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 still preferred to fewer, and thus (5,5) is preferred to (3,3). But unmatched shoes are a burden – assuming that the individual must keep any unmatched shoes and cannot simply toss them out.

Explanation 2 for why the Leontief representation of the preferences is too simple.

Having an extra left (or right) shoe could be an advantage. Suppose that the family dog chews only the left shoe or that the wearer steps in a puddle with only the left shoe. Then, having an extra left shoe provides additional well-being.

Which of the preference maps below is most consistent with Explanation 1? with Explanation 2? Explain.

Answer: The graph on the right is consistent with Explanation 1. Consider the bundle A which involves three left shoes and nine right shoes. This bundle is below the indifference curve for the bundle (3,3) for the indifference map to the right. Having extra right shoes makes the individual worse off. (And extra left shoes also makes the individual worse off.) In describing the set-up, we have assumed that the individual cannot simply through the shoes out. Saying that the individual has the bundle (9,3) means that the individual must deal with 12 shoes. An alternative set-up would be to assume free disposal. With free disposal, an individual who had the bundle (9,3) could costlessly dispose of six of the shoes. With free disposal, the preferences would be Leontief.

The graph on the left is consistent with Explanation 2. Having an extra left (or right) shoe could be an advantage in case the family dog chews only one shoe. For the indifference map shown to the left, the bundle (3,9) is on a higher IC than is the bundle (3,3). The extra right shoe makes the individual better off.

  1. (correction) Hillary consumes two goods and has utility U(x,y) = max {y,3x}. Show a graph with Hillary’s indifference map.

Answer: The bundles (x=1,y=3), (x=1,y=0), and (x=0,y=3) are all on the IC for 3 utiles.

Question that was not asked on the HW but is possibly interesting: 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.]

Answer: Bundles that provide three units of utility include (x=1,y=3), (x=3,y=3), (x=1,y=9). Bundles that provide 6 units of utility include (x=2,y=6), (x=4,y=6) and (x=2,y=8).

These might be the preferences for an individual who likes coffee only with cream and likes cream only in coffee. Specifically, the individual drinks coffee only if each 4 ounces of coffee consists of 1 ounce of cream and 3 ounces of coffee. Good X is cream (measured in ounces per time period) and good Y is coffee (measured in ounces per time period). We also assume that the individual can costlessly discard any unused cream or coffee. Further, we have shown an individual for whom more cups of coffee are preferred to fewer – no satiation.

  1. Laura consumes two goods (X and Y) and purchases positive amounts of both. The market prices of the goods are px = $4 and py =$8. What is Laura’s marginal rate of substitution at her optimal bundle? Explain. Answer: If Laura’s preferences satisfy the usual assumptions, then the IC for the optimal bundle is tangent to the BL at the optimal bundle. Thus, at the optimal bundle MRSxy equals the price ratio, 4/8.
  1. 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.]

Answer: The dark lines are the budget lines. In this example, the budget line for a higher income has X and Y intercepts of 10. The budget line for the lower income has X and Y intercepts of 3. With the higher budget line, Sangeeta consumes positive amounts of both goods. With the lower income, she consumes only good Y. [Many other indifference curve and budget line graphs would satisfy the description of Sangeeta’s preferences.]

  1. 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.
  1. Assume that Howard has Leontief preferences. Does Howard continue to consume the bundle (30,30)? Construct a graph to support your argument.

Answer: The Y intercept of the initial BL is 90, and the Y intercept for the new budget line is 45. The bundle (30,3) lies on both budget lines. The initial budget line is solid, and the new budget line is dotted. The Leontief indifference curve through the bundle (30,30) is also shown on the graph. The point (30,30) is the optimal bundle for the initial budget line and for the new budget line.

Let’s also consider Howard’s budget line if goods X and Y could be purchased only as paired bundles, as for a pair of shoes. If the price of a pair is $60, then the only bundle on the budget line is the bundle (30,30). Assuming that Al could throw away shoes, the budget set consists of all bundles below and to the right of (30,30). His optimal choice is the bundle (30,30).

  1. 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.]

Answer: The two straight lines are the budget lines. The Y intercept for the initial budget line is 90 and for the new budget line is 45. The new budget line is the darker line. With the initial BL, the optimal bundle is (30,30). With the new budget line, Howard can afford bundles that he likes better. Bundles on the new budget line to the right of (30,30) are preferred to (30,30).

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.

Answer: At a BL for higher income, the optimal bundle contains two units of good X. With these preferences, an increase in income has no effect on the consumption of good X. Increases in income are used to purchase more of good Y.

There are not many “interesting” examples of such product. As mentioned in class previously, tooth paste might be such a product, at least for middle-class income ranges. For most individuals, overall dental care would depend upon income even in the range of middle-class incomes.

  1. Rudy’s utility function is U(x,y) = 2x.5 + y.
  1. By definition, his indifference curve for utility level 9 consists of bundles (x,y) for which 2x.5 + y = 10. Solve the equality for y and graph the indifference curve.

Answer: y= 10 – 2 x.5

  1. 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.]

Answer: y= u – 2 x.5 The utility level enters the expression for the IC linearly. Thus, let’s consider two different utility levels, say utility 10 and 15. The IC for utility 15 is 5 units higher than the IC for utility 10. In other words, the indifference curves are parallel.