1. Preferences and Utility PDF

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Preferences and Utility Review Questions 1.

What is a basket (or a bundle) of goods?

A basket is a collection of goods and services that an individual might consume. 2. What does the assumption that preferences are complete mean about the consumer’s ability to rank any two baskets? By requiring preferences to be complete, economists are ensuring that consumers will not respond indecisively when asked to compare two baskets. A consumer will always be able to state that either A is preferred B, B is preferred to A, or that she is indifferent between A and B. 3. Consider Figure 3.1. If the more is better assumption is satisfied, is it possible to say which of the seven baskets is least preferred by the consumer? Unfortunately, it is impossible to say definitively whether D, H, or J is the least preferred basket. Since more is better, baskets to the northeast are more preferred and baskets to the southwest are less preferred. In this case, H has more clothing but less food than D, while J has more food but less clothing than D. Without more information regarding how the consumer feels about clothing relative to food, we cannot state which of these baskets is the least preferred. 4. Give an example of preferences (i.e., a ranking of baskets) that do not satisfy the assumption that preferences are transitive. If a consumer states that A is preferred to B and that B is preferred to C, but then states that C is preferred to A, she will be violating the assumption of transitivity. The third statement is inconsistent with the first two. 5. What does the assumption that more is better imply about the marginal utility of a good? If more is better, then the marginal utility of a good must be positive. That is, total utility must increase if the consumer consumes more of the good. 6.

What is the difference between an ordinal ranking and a cardinal ranking?

An ordinal ranking simply orders the baskets, but does not give any indication as to how much better one basket is when compared with another; only that one is better. A cardinal ranking not only orders the baskets, but also provides information regarding the intensity of the preferences. For example, a cardinal ranking might indicate that one basket is twice as good as another basket.

7. Suppose Debbie purchases only hamburgers. Assume that her marginal utility is always positive and diminishing. Draw a graph with total utility on the vertical axis and the number of hamburgers on the horizontal axis. Explain how you would determine marginal utility at any given point on your graph.

Utility

Slope of this line measures marginal utility at this level of consumption, H’ Total Utility

H’

Hamburgers

Marginal utility would be measured as the slope of a line tangent to the total utility curve in the graph above. 8.

Why can’t you plot the total utility and marginal utility curves on the same graph?

The two cannot be plotted on the same graph because utility and marginal utility are not measured in the same dimensions. Total utility has the dimension U , while marginal utility has the dimension of utility per unit, or DU / Dy where y is the number of units purchased. 9. Adam consumes two goods: housing and food. a) Suppose we are given Adam’s marginal utility of housing and his marginal utility of food at the basket he currently consumes. Can we determine his marginal rate of substitution of housing for food at that basket? b) Suppose we are given Adam’s marginal rate of substitution of housing for food at the basket he currently consumes. Can we determine his marginal utility of housing and his marginal utility of food at that basket? a)

Yes, we can determine the MRS as

MU h MU f b) No, when we know the MRS, all we know is the ratio of the marginal utilities. We cannot “undo” that ratio to determine the individual marginal utilities. For example, if we know that MRSh,f = 5, it could be the case that MUh = 5 and MUf = 1, but it could equivalently be the case that MUh = 10 and MUf = 2. Clearly, there are countless combinations of MUh and MUf that could lead to some particular value of MRSh,f, and we have no way of inferring which is the right one. MRSh , f =

10. Suppose Michael purchases only two goods, hamburgers (H) and Cokes (C). a) What is the relationship between MRSH,C and the marginal utilities MUH and MUC ? b) Draw a typical indifference curve for the case in which the marginal utilities of both goods are positive and the marginal rate of substitution of hamburgers for Cokes is diminishing. Using your graph, explain the relationship between the indifference curve and the marginal rate of substitution of hamburgers for Cokes. c) Suppose the marginal rate of substitution of hamburgers for Cokes is constant. In this case, are hamburgers and Cokes perfect substitutes or perfect complements? d) Suppose that Michael always wants two hamburgers along with every Coke. Draw a typical indifference curve. In this case, are hamburgers and Cokes perfect substitutes or perfect complements? a)

MRSH ,C =

MU H MU C

b)

C

H The indifference curve in this case will be convex toward the origin. The marginal rate of substitution is measured as the absolute value of the slope of a line tangent to the indifference curve. As can be seen in the graph above, this slope becomes less negative as we move down the indifference curve, implying a diminishing MRS. c) If the MRS was constant, this would imply that at any consumption level the consumer would be willing to trade a fixed amount of one good for a fixed amount of the other. This occurs with perfect substitutes. d) If the consumer wishes to always consume goods in a fixed ratio, then the goods are perfect complements. In this case, the indifference curves will be L-shaped.

C

1 2

H

11. Suppose a consumer is currently purchasing 47 different goods, one of which is housing. The quantity of housing is measured by H. Explain why, if you wanted to measure the consumer’s marginal utility of housing (MUH) at the current basket, the levels of the other 46 goods consumed would be held fixed. Marginal utility is defined as the change in total utility relative to a change in consumption for a particular good. In order to accurately measure the change in total utility, the levels of the other goods would need to be held constant. If they were not, the change in total utility would occur as a result of multiple goods changing and it would be impossible to determine what portion of the change in total utility should be assigned to each good.

Problems 3.1 Bill has a utility function over food and gasoline with the equation U = x2y, where x measures the quantity of food consumed and y measures the quantity of gasoline. Show that a consumer with this utility function believes that more is better for each good. By plugging in ever higher numerical values of x and ever higher numerical values of y, it can be verified that U increases whenever x or y increases. 3.2 Consider the single-good utility function U(x) = 3x2, with a marginal utility given by MUx = 6x. Plot the utility and marginal utility functions on two separate graphs. Does this utility function satisfy the principle of diminishing marginal utility? Explain. The two graphs are shown below. It can be seen from both graphs that this function does not satisfy the law of diminishing marginal utility. The first figure shows that utility increases with x, and moreover, that it increases at an increasing rate. For example, an increase in x from 2 to 3, increases utility from 12 to 27 (an increase of 15), while an increase in x from 3 to 4 induces an increase in utility from 27 to 48 (an increase of 21).

This fact is easier to see in the second figure. The marginal utility is an increasing function of x. Higher values of x imply a greater marginal utility. Therefore this function exhibits increasing marginal utility. U(x) = 3x2

MUx = 6x

3.3 Jimmy has the following utility function for hot dogs: U(H) = 10H − H2, with MUH = 10 − 2H a) Plot the utility and marginal utility functions on two separate graphs. b) Suppose that Jimmy is allowed to consume as many hot dogs as he likes and that hot dogs cost him nothing. Show, both algebraically and graphically, the value of H at which he would stop consuming hot dogs. The first figure below shows Jimmy’s utility function for hotdogs. You can see that the point at which H = 5 corresponds to the flat portion of the utility function, i.e. the point at which the marginal utility of hotdogs is zero, and beyond which the marginal utility is negative. Alternatively using the second graph it is clear that the point H = 5 is when the marginal utility intersects the x-axis, and beyond which it is negative. Both graphs tell you that to maximize his utility Jimmy should only consume 5 hotdogs and not more. To answer this question algebraically, you should first recognize from the marginal utility function that Jimmy has a diminishing marginal utility of hotdogs. Therefore the point at which he should stop consuming hotdogs is the point at which MU H = 0, or 10 - 2 H = 0. This gives H = 5.

U(H) = 10H – H2

MUH = 10 – 2H

3.4 Consider the utility function U(x, y) = y√x with the marginal utilities MUx = y/(2√x) and MUy = √x. a) Does the consumer believe that more is better for each good? b) Do the consumer’s preferences exhibit a diminishing marginal utility of x? Is the marginal utility of y diminishing? 3.4 a) Since U increases whenever x or y increases, more of each good is better. This is also confirmed by noting that MUx and MUy are both positive for any positive values of x and y. b) Since MUx = y 2 x , as x increases (holding y constant), MUx falls. Therefore the marginal utility of x is diminishing. However, MUy = x . As y increases, MUy does not change. Therefore the preferences exhibit a constant, not diminishing, marginal utility of y. 3.5 Carlos has a utility function that depends on the number of musicals and the number of operas seen each month. His utility function is given by U = xy2, where x is the

number of movies seen per month and y is the number of operas seen per month. The corresponding marginal utilities are given by: MUx = y2 and MUy = 2xy. a) Does Carlos believe that more is better for each good? b) Does Carlos have a diminishing marginal utility for each good? a) By plugging in ever higher numerical values of x and ever higher numerical values of y, it can be verified that Carlos’ utility goes up whenever x or y increases. b) First consider the marginal utility of x, MUx. Since x does not appear anywhere in the formula for MUx, MUx is independent of x. Hence, the marginal utility of movies is independent of the number of movies seen, and so the marginal utility of movies does not decrease as the number of movies increases. Next consider the marginal utility of y, MUy. Notice that MUy is an increasing function of y. Hence, the marginal utility of operas does not decrease in the number of operas seen. In this case, neither good, movies or operas, exhibits diminishing marginal utility.

3.6 For the following sets of goods draw two indifference curves, U1 and U2, with U2 > U1. Draw each graph placing the amount of the first good on the horizontal axis. a) Hot dogs and chili (the consumer likes both and has a diminishing marginal rate of substitution of hot dogs for chili) b) Sugar and Sweet’N Low (the consumer likes both and will accept an ounce of Sweet’N Low or an ounce of sugar with equal satisfaction) c) Peanut butter and jelly (the consumer likes exactly 2 ounces of peanut butter for every ounce of jelly) d) Nuts (which the consumer neither likes nor dislikes) and ice cream (which the consumer likes) e) Apples (which the consumer likes) and liver (which the consumer dislikes) a)

Chili

U2 U1 Hot Dogs b)

Sweet’N Low

Slopes = –1

U1

U2 Sugar

c)

Jelly

U2

2 U1

1 2

Peanut Butter

4

d)

Ice Cream

U2

U1 Nuts e)

Liver U1 U2

Apples

3.7 Alexa likes ice cream, but dislikes yogurt. If you make her eat another gram of yogurt, she always requires two extra grams of ice cream to maintain a constant level of satisfaction. On a graph with grams of yogurt on the vertical axis and grams of ice cream on the horizontal axis, graph some typical indifference curves and show the directions of increasing utility. Grams of Yogurt 3

U3>U2>U1

U2

U1

U3 Directions of Increased Satisfaction

2

1

1

2

3

4

5

6

Grams of Ice Cream

3.8 Joe has a utility function over hamburgers and hot dogs given by U = x + √y , where x is the quantity of hamburgers and y is the quantity of hot dogs. The marginal utilities for this utility function are MUx = 1 and MUy = 1/(2√y ). Does this utility function have the property that MRSx,y is diminishing? This utility function does have the property of diminishing MRSx,y. One way to verify this is to graph several indifference curves. Another way to tell is to use algebra. Recall that

MRSx ,y =

MU x . Applying that general formula to this case gives us MRS x, y = 2 y . As we MU y

move “down” the indifference curve, x increases and y decreases. As y decreases, 2 y decreases. Thus, MRSx,y decreases.

3.9 Julie and Toni consume two goods with the following utility functions: UJulie = (x + y)2, MUJuliex = 2(x + y), MUJuliey = 2(x + y) UToni = x + y, MUTonix = 1, MUToniy = 1 a) Graph an indifference curve for each of these utility functions. b) Julie and Toni will have the same ordinal ranking of different baskets if, when basket A is preferred to basket B by one of the functions, it is also preferred by the other. Do Julie and Toni have the same ordinal ranking of different baskets of x and y? Explain. Indifference curves corresponding to U = 2 are shown for both Julie and Toni in the graph below. Notice that the indifference curves are parallel everywhere – indeed, MRSx,y = 1 for both Julie and Toni, for all values of x and y. Toni’s indifference curve for the utility level UToni = 2 is the same as Julie’s indifference curve for the utility level UJulie = 4. So whenever Julie ranks bundle A higher than bundle B, Toni would have the same ranking, and vice-versa. So Julie and Toni will have the same ordinal ranking of bundles of x and y. (Julie will associate each bundle with a higher utility level than Toni will, but that is a cardinal ranking.)

UToni = 2 UJulie = 2

3.10 The utility that Julie receives by consuming food F and clothing C is given by U(F, C) = FC. For this utility function, the marginal utilities are MUF = C and MUC = F. a) On a graph with F on the horizontal axis and C on the vertical axis, draw indifference curves for U = 12, U = 18, and U = 24. b) Do the shapes of these indifference curves suggest that Julie has a diminishing marginal rate of substitution of food for clothing? Explain.

c) Using the marginal utilities, show that the MRSF,C = C/F. What is the slope of the indifference curve U = 12 at the basket with 2 units of food and 6 units of clothing? What is the slope at the basket with 4 units of food and 3 units of clothing? Do the slopes of the indifference curves indicate that Julie has a diminishing marginal rate of substitution of food for clothing? (Make sure your answers to parts (b) and (c) are consistent!) a) 14 12

Clothing

10 8 6 4 2 0 0

5

10

15

Food

b) Yes, since the indifference curves are bowed in toward the origin we know that MRSF,C declines as F increases and C decreases along an indifference curve. MU F C MRSF ,C = c) = MUC F When F = 2 and C = 6, MRSF,C = 3. The slope of the indifference curve is –3. When F = 4 and C = 3, MRSF,C = 0.75, so the slope of the indifference curve is –0.75. Since the marginal rate of substitution falls as F increases and C decreases, she has a diminishing marginal rate of substitution.

3.11 Sandy consumes only hamburgers (H) and milkshakes (M). At basket A, containing 2 hamburgers and 10 milkshakes, his MRSH,M is 8. At basket B, containing 6 hamburgers and 4 milkshakes, his MRSH,M is 1/2. Both baskets A and B are on the same indifference curve. Draw the indifference curve, using information about the MRSH,M to make sure that the curvature of the indifference curve is accurately depicted.

Milkshakes Slope = –8 A

10

Slope = –½ B

4

Hamburgers 2

6

3.12 Adam likes his café latte prepared to contain exactly 1/3 espresso and 2/3 steamed milk by volume. On a graph with the volume of steamed milk on the horizontal axis and the volume of espresso on the vertical axis, draw two of his indifference curves, U1 and U2, with U1 > U2. Volume of Espresso 3 U1 2 U2 1

1

2

3

4

5

6

Volume of Steamed Milk

3.13 Draw indifference curves to represent the following types of consumer preferences. a) I like both peanut butter and jelly, and always get the same additional satisfaction from an ounce of peanut butter as I do from 2 ounces of jelly. b) I like peanut butter, but neither like nor dislike jelly. c) I like peanut butter, but dislike jelly. d) I like peanut butter and jelly, but I only want 2 ounces of peanut butter for every ounce of jelly.

In the following pictures, U2 > U1. a) Jelly 4 2 U1

U2

1

2

Peanut Butter

b) Jelly U1

U2

Peanut Butter

c) U1

Jelly

U2

Peanut Butter

d) Jelly U1

U2

2 1 2

4

Peanut Butter

3.14 Dr. Strangetaste buys only food (F) and clothing (C) out of his income. He has positive marginal utilities for both goods, and his MRSF,C is increasing. Draw two of Dr. Strangetaste’s indifference curves, U1 and U2, with U2 > U1.

Clothing

U1

U2 Food

The following exercises will give you practice in working with a variety of utility functions and marginal utilities and will help you understand how to graph indifference curves. 3.15 Consider the utility function U(x, y) = 3x + y, with MUx = 3 and MUy = 1. a) Is the assumption that more is better satisfied for both goods? b) Does the marginal utility of x diminish, remain constant, or increase as the consumer buys more x? Explain. c) What is MRSx, y? d) Is MRSx, y diminishing, constant, or increasing as the consumer substitutes x for y along an indifference curve? e) On a graph with x on the horizontal axis and y on the vertical axis, draw a typical indifference curve (it need not be exactly to scale, but it needs to reflect accurately whether there is a diminishing MRSx, y). Also indicate on your graph whether the indifference curve will intersect either or both axes. Label the curve U1. f ) On the same graph draw a second indifference curve U2, with U2 > U1. a) Yes, the “more is better” assumption is satisfied for both goods since both marginal utilities are always positive. b) The marginal utility of x remains constant at 3 for all values of x. MRS x ,y = 3 c) d) The MRS x, y remains constant moving along the indifference curve. e & f) See figure below

Y

U1 U2

X

3.16 Answer all parts of Problem 3.15 for the utility function U(x, y) = √xy. The marginal utilities are MUx = √y/(2√x) and MUy = √x/(2√y). a) Yes, the “more is better” assumption is satisfied for both goods s...