ch04solution manual

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Chapter 4Consumer Choice

Solutions to Review Questions

1. If the consumer has a positive marginal utility for each of two goods, why will the consumer always choose a basket on the budget line?

Relative to any point on the budget line, when the consumer has a positive marginal utility for all goods she could increase her utility by consuming some basket outside the budget line. However, baskets outside the budget line are unaffordable to her, so she is constrained (as in “constrained optimization”) to choosing the most preferred basket that lies along the budget line.

2. How will a change in income affect the location of the budget line?

An increase in income will shift the budget line away from the origin in a parallel fashion expanding the set of possible baskets from which a consumer may choose. A decrease in income will shift the budget line in toward the origin in a parallel fashion, reducing the set of possible baskets from which a consumer may choose.

3. How will an increase in the price of one of the goods purchased by a consumer affect the location of the budget line?

If the price of one of the goods increases, the budget line will rotate inward on the axis for the good with the price increase. The budget line will continue to have the same intercept on the other axis. For example, suppose someone buys two goods, cups of coffee and doughnuts, and suppose the price of a cup of coffee increases. Then the budget line will rotate as in the following diagram:

Copyright © 2014 John Wiley & Sons, Inc. Chapter 4 - 1


4. What is the difference between an interior optimum and a corner point optimum in the theory of consumer choice?

With an interior optimum the consumer is choosing a basket that contains positive quantities of all goods, while with a corner point optimum the consumer is choosing a basket with a zero quantity for one of the goods. The tangency condition usually does not apply at corner optima.

5. At an optimal interior basket, why must the slope of the budget line be equal to the slope of the indifference curve?

If the optimum is an interior solution, the slope of the budget line must equal the slope of the indifference curve. If these slopes are not equal at the chosen interior basket then the “bang for the buck” condition will not hold. This condition states that at the optimum the extra utility gained per dollar spent on good must be equal to the extra utility gained per dollar spent on good . If this condition does not hold at the chosen basket, then the consumer could reallocate his income to purchase more of the good with the higher “bang for the buck” and increase his total utility while remaining within the given budget. Thus, if these slopes are not equal the basket cannot be optimal assuming an interior solution.

6. At an optimal interior basket, why must the marginal utility per dollar spent on all goods be the same?

At an interior optimum, the slope of the budget line must equal the slope of the indifference curve. This implies

This can be rewritten as

which is known as the “bang for the buck” condition. If this condition does not hold at the chosen interior basket, then the consumer can increase total utility by reallocating his spending to purchase more of the good with the higher “bang for the buck” and less of the other good.

7. Why will the marginal utility per dollar spent not necessarily be equal for all goods at a corner point?

The “bang for the buck” condition will not necessarily hold at a corner solution optimum. The consumer could theoretically increase total utility by reallocating his spending to purchase more of the good with the higher “bang for the buck” and less of the other good. Since the basket is a corner point, however, he is already purchasing zero of one of the goods. This implies that he cannot purchase less of the good with a zero quantity (since negative quantities make no sense) and therefore cannot reallocate spending.



8. Suppose that a consumer with an income of $1,000 finds that basket A maximizes utility subject to his budget constraint and realizes a level of utility U1. Why will this basket also minimize the consumer’s expenditures necessary to realize a level of utility U1?

In the utility maximization problem, the consumer maximizes utility subject to a fixed budget constraint. At the optimum the slope of the budget line will equal the slope of the indifference curve. If we now hold that indifference curve fixed, we can solve an expenditure minimization problem in which we ask what is the minimum expenditure necessary to achieve that fixed level of utility. Since the slope of the budget line and indifference curve have not changed, when the expenditure is minimized the budget line and indifference curve will be tangent at the same point as in the utility maximization problem. The same basket is optimal in both problems.

9. What is a composite good?

First, consumers typically allocate income to more than two goods. Second, economists often want to focus on the consumer’s response to purchases of a single good or service. In this case it is useful to present the consumer choice problem using a two-dimensional graph. Since there are more than two goods the consumer is purchasing, however, an economist would need more than two dimensions to show the problem graphically. To reduce the problem to two dimensions, economists often group the expenditures on all other goods besides the one in question into a single good termed a “composite good.” When the problem is shown graphically, one axis represents the composite good while the other axis represents the single good in question. By creating this composite good, the problem can be illustrated using a two-dimensional graph.

10. How can revealed preference analysis help us learn about a consumer’s preferences without knowing the consumer’s utility function?

By employing revealed preference analysis one can make inferences regarding a consumer’s preferences without knowing what the consumer’s indifference map looks like. For example, if a consumer chooses basket A over basket B when basket B costs at least as much as basket A, we know that basket A is at least as preferred as basket B. If the consumer chooses basket C, which is more expensive than basket D, then we know the consumer strictly prefers basket C to basket D. By observing enough of these choices, one can determine how the consumer ranks baskets even without knowing the exact shape of the consumer’s indifference map.



Solutions to Problems

4.1 Pedro is a college student who receives a monthly stipend from his parents of $1,000. He uses this stipend to pay rent for housing and to go to the movies (you can assume that all of Pedro’s other expenses, such as food and clothing have already been paid for). In the town where Pedro goes to college, each square foot of rental housing costs $2 per month. The price of a movie ticket is $10 per ticket. Let x denote the square feet of housing, and let y denote the number of movie tickets he purchases per month. a) What is the expression for Pedro’s budget constraint?b) Draw a graph of Pedro’s budget line.c) What is the maximum number of square feet of housing he can purchase given his monthly stipend?d) What is the maximum number of movie tickets he can purchase given his monthly stipend?e) Suppose Pedro’s parents increase his stipend by 10 percent. At the same time, suppose that in the college town he lives in, all prices, including housing rental rates and movie ticket prices, increase by 10 percent. What happens to the graph of Pedro’s budget line?

a) 2x + 10y ≤ 1000b)

c) The maximum amount of housing Pedro can purchase is his budget divided by the price of housing: $1,000/$2 per square feet = 500 square feet.d) The maximum number of movie tickets Pedro can purchase is his budget divided by the price of a movie ticket: $1,000/$10 per tickets = 100 tickets.e) His budget line does not change at all.

Initially, the budget line (with x on the horizontal axis and y on the vertical axis) has a horizontal intercept equal to 1000/2 = 500 and a vertical intercept equal to 1000/10 = 100. The slope of the budget line is -2/10 = - 0.20 (the price of housing divided by the price of movie tickets).



With the increase in Pedro’s stipend and the increases in prices we have: Horizontal intercept of budget line: 1000(1.10)/(2(1.10)) = 500 Vertical intercept of budget line: 1000(1.10)/(10(1.10)) = 100 Slope of budget line: -2(1.10)/(10(1.10)) = - 0.20.

These are the same as before and thus the budget line does not change.

4.2 Sarah consumes apples and oranges (these are the only fruits she eats). She has decides that her monthly budget for fruit will be $50. Suppose that one apple costs $0.25, while one orange costs $0.50. Let x denote the quantity of apples and y denote the quantity of oranges that Sarah purchases. a. What is the expression for Sarah’s budget constraint?b. Draw a graph of Sarah’s budget line. c. Show graphically how Sarah’s budget line changes if the price of apples increases to

$0.50.d. Show graphically how Sarah’s budget line changes if the price of oranges decreases to

$0.25.e. Suppose Sarah decides to cut her monthly budget for fruit in half. Coincidentally, the

next time she goes to the grocery store, she learns that oranges and apples are on sale for half price, will remain so for the next month, i.e., the price of apples falls from $0.25 per apple to $0.125 per apple and the price of oranges falls from $0.50 per orange to $0.25 per orange. What happens to the graph of Sarah’s budget line?

a) 0.25x + 0.50y ≤ 50.

b)

c)



d)

e) Sarah’s budget line would not change. Horizontal intercept of the budget line: (0.5)$50/((0.5)(0.25) = 200 Vertical intercept of the budget line: (0.5)$50/((0.5)(0.50) = 100 Slope of the budget line = -(0.5)(0.25)/((0.5)(0.50)) = 0.50

These are the same as before, and thus the budget line does not change.

4.3 In Problem 3.7 of Chapter 3, we considered Julie’s preferences for food F and clothing C. Her utility function was U(F, C) = FC. Her marginal utilities were MUF = C and MUC = F. You were asked to draw the indifference curves U = 12, U = 18, and U = 24, and to show that she had a diminishing marginal rate of substitution of food for clothing.



Suppose that food costs $1 a unit and that clothing costs $2 a unit. Julie has $12 to spend on food and clothing.a) Using a graph (and no algebra), find the optimal (utility-maximizing) choice of food and clothing. Let the amount of food be on the horizontal axis and the amount of clothing be on the vertical axis.b) Using algebra (the tangency condition and the budget line), find the optimal choice of food and clothing.c) What is the marginal rate of substitution of food for clothing at her optimal basket? Show this graphically and algebraically.d) Suppose Julie decides to buy 4 units of food and 4 units of clothing with her $12 budget (instead of the optimal basket). Would her marginal utility per dollar spent on food be greater than or less than her marginal utility per dollar spent on clothing? What does this tell you about how she should substitute food for clothing if she wanted to increase her utility without spending any more money?

a)

b) The tangency condition implies that

Plugging in the known information results in

Substituting this result into the budget line, , yields



Finally, plugging this result back into the tangency condition implies . At the optimum the consumer choose 6 units of food and 3 units of clothing.

c) At the optimum, . Note that this is equal to the ratio of the price of food to the price of clothing. The equality of the price ration and MRSF,C is seen in the graph above as the tangency between the budget line and the indifference curve for .

d) If the consumer purchases 4 units of food and 4 units of clothing, then

This implies that the consumer could reallocate spending by purchasing more food and less clothing to increase total utility. In fact, at the basket (4, 4) total utility is 16 and the consumer spent $12. By giving up one unit of clothing the consumer saves $2 which can than be used to purchase two units of food (they each cost $1). This will result in a new basket (6,3), total utility of 18, and spending of $12. By reallocating spending toward the good with the higher “bang for the buck” the consumer increased total utility while remaining within the budget constraint.

4.4 The utility that Ann receives by consuming food F and clothing C is given by U(F, C) = FC + F. The marginal utilities of food and clothing are MUF = C + 1 and MUC = F. Food costs $1 a unit, and clothing costs $2 a unit. Ann’s income is $22.a) Ann is currently spending all of her income. She is buying 8 units of food. How many units of clothing is she consuming?b) Graph her budget line. Place the number of units of clothing on the vertical axis and the number of units of food on the horizontal axis. Plot her current consumption basket.c) Draw the indifference curve associated with a utility level of 36 and the indifference curve associated with a utility level of 72. Are the indifference curves bowed in toward the origin?d) Using a graph (and no algebra), find the utility maximizing choice of food and clothing.e) Using algebra, find the utility-maximizing choice of food and clothing.f ) What is the marginal rate of substitution of food for clothing when utility is maximized? Show this graphically and algebraically.g) Does Ann have a diminishing marginal rate of substitution of food for clothing? Show this graphically and algebraically.

a) If Ann is spending all of her income then



b)

c) Yes, the indifference curves are convex, i.e., bowed in toward the origin. Also, note that they intersect the F-axis.

0

10

20

30

40

50

60

70

80

0 5 10 15 20 25 30 35

Food

Clothing

U=36

U=72

d)



e) The tangency condition requires that

Plugging in the known information yields

Substituting this result into the budget line, results in

Finally, plugging this result back into the tangency condition implies that . At the optimum the consumer chooses 5 units of clothing and 12 units of food.

f) The marginal rate of substitution is equal to the price ratio.

g) Yes, the indifference curves do exhibit diminishing . We can see this in the graph in part c) because the indifference curves are bowed in toward the origin. Algebraically,

. As increases and decreases along an isoquant, diminishes.



4.5 Consider a consumer with the utility function U(x, y) = min(3x, 5y), that is, the two goods are perfect complements in the ratio 3:5. The prices of the two goods are Px = $5 and Py = $10, and the consumer’s income is $220. Determine the optimum consumption basket.

This question cannot be solved using the usual tangency condition. However, you can see from the graph below that the optimum basket will necessarily lie on the “elbow” of some indifference curve, such as (5, 3), (10, 6) etc. If the consumer were at some other point, he could always move to such a point, keeping utility constant and decreasing his expenditure. The equation of all these “elbow” points is 3x = 5y, or y = 0.6x. Therefore the optimum point must be such that 3x = 5y.The usual budget constraint must hold of course. That is, . Combining these two conditions, we get (x, y) = (20, 12).

4.6 Jane likes hamburgers (H) and milkshakes (M). Her indifference curves are bowed in toward the origin and do not intersect the axes. The price of a milkshake is $1 and the price of a hamburger is $3. She is spending all her income at the basket she is currently consuming, and her marginal rate of substitution of hamburgers for milkshakes is 2. Is she at an optimum? If so, show why. If not, should she buy fewer hamburgers and more milkshakes, or the reverse?

From the given information we know that , , and Comparing the MRSH,M to the price ratio,

Since these are not equal Jane is not currently at an optimum. In addition, we can say that

which is equivalent to


(5,3)

(10,6)

y

x

(20,12)


That is, the “bang for the buck” from milkshakes is greater than the “bang for the buck” from hamburgers. So Jane can increase her total utility by reallocating her spending to purchase fewer hamburgers and more milkshakes.

4.7 Ray buys only hamburgers and bottles of root beer out of a weekly income of $100. He currently consumes 20 bottles of root beer per week, and his marginal utility of root beer is 6. The price of root beer is $2 per bottle. Currently, he also consumes 15 hamburgers per week, and his marginal utility of a hamburger is 8. Is Ray maximizing utility at his current consumption basket? If not, should he buy more hamburgers each week, or fewer?

Compare MUH/PH with MUR/PR, where the subscripts “H” and “R” refer respectively to hamburgers and root beer. We have all the information to make this comparison except for the price of a hamburger. But we can determine the price of a hamburger from Sam’s budget constraint:

PHH + PRR = Income, or PH(15) + 2(20) = 100.

So PH = $4 per hamburger.Now we can see that MUH/PH = 8/4 = 2 and MUR/PR = 6/2 = 3. Since the “bang for the buck” is higher for root beer than for hamburgers, he should buy fewer hamburgers (and more root beer).

4.8 Dave currently consumes 10 hot dogs and 6 sodas each week. At his current consumption basket, his marginal utility for hot dogs is 5 and his marginal utility for sodas is 3. If the price of one hot dog is $1 and the price of one soda is $0.50, is Dave currently maximizing his utility? If not, how should he reallocate his spending in order to increase his utility?

To determine if this situation is optimal, determine if the tangency condition holds.

Since the tangency condition does not hold, Dave is not currently maximizing his utility. To increase his utility he should purchase more soda and fewer hot dogs (since the ‘bang for the buck’ for sodas is higher).

4.9 Helen’s preferences over CDs (C) and sandwiches (S) are given by U(S, C) = SC + 10(S + C), with MUC = S + 10 and MUS = C + 10. If the price of a CD is $9 and the price of a sandwich is $3, and Helen can spend a combined total of $30 each day on these goods, find Helen’s optimal consumption basket.



See the graph below. The fact that Helen’s indifference curves touch the axes should immediately make you want to check for a corner point solution.

To see the corner point optimum algebraically, notice if there was an interior solution, the tangency condition implies (S + 10)/(C +10) = 3, or S = 3C + 20. Combining this with the budget constraint, 9C + 3S = 30, we find that the optimal number of CDs would be given by which implies a negative number of CDs. Since it’s impossible to purchase a negative amount of something, our assumption that there was an interior solution must be false. Instead, the optimum will consist of C = 0 and Helen spending all her income on sandwiches: S = 10.

Graphically, the corner optimum is reflected in the fact that the slope of the budget line is steeper than that of the indifference curve, even when C = 0. Specifically, note that at (C, S) = (0, 10) we have PC / PS = 3 > MRSC,S = 2. Thus, even at the corner point, the marginal utility per dollar spent on CDs is lower than on sandwiches. However, since she is already at a corner point with C = 0, she cannot give up any more CDs. Therefore the best Helen can do is to spend all her income on sandwiches: (C, S) = (0, 10). [Note: At the other corner with S = 0 and C = 3.3, PC / PS = 3 > MRSC,S = 0.75. Thus, Helen would prefer to buy more sandwiches and less CDs, which is of course entirely feasible at this corner point. Thus the S = 0 corner cannot be an optimum.]

4.10 The utility that Corey obtains by consuming hamburgers (H) and hot dogs (S) is

given by . The marginal utility of hamburgers is and the

marginal utility of steaks is equal to

a) Sketch the indifference curve corresponding to the utility level U = 12.b) Suppose that the price of hamburgers is $1 per hamburger, and the price of steak is $8 per steak. Moreover, suppose that Corey can spend $100 per month on these two foods. Sketch Corey’s budget line for hamburgers and steak given this budget.



c) Based on your answer to parts (a) and (b), what is Corey’s optimal consumption basket given his budget?

a) Some points on the U = 12 indifference curve include

S H U0 100 125 81` 1212 64 1221 49 1232 36 1245 25 1260 16 12

Connecting these points gives us the U = 12 indifference curve:

b) The equation of the budget line is H + 8S = 100

Graphing this on the same axes as the U = 12 indifference curve gives us:



c) The optimal consumption basket is S = 0, H = 100, i.e., point R in the figure below. There are several ways to see this. One way is to sketch a few more indifference curves (each corresponding to a different level of utility). This picture strongly suggests that the point of maximum utility occurs at point R.

Another way is to compare the marginal utility per dollar of spent on hamburger and the marginal utility per dollar spent on steak at point R. From the information given in the statement

of the problem, and and so at point R



Thus, at point R, the marginal utility per dollar spent on hamburger is greater than the marginal utility per dollar spent on steak, and so the consumer would like to purchase more hamburger and less steak. However, at point R, no further reduction in the quantity of steak is possible, and thus R is the optimal consumption basket.

4.11 This problem will help you understand what happens if the marginal rate of substitution is not diminishing. Dr. Strangetaste buys only French fries (F) and hot dogs (H) out of his income. He has positive marginal utilities for both goods, and his MRSH,F is increasing. The price of hot dogs is PH, and the price of French fries is PF.a) Draw several of Dr. Strangetaste’s indifference curves, including one that is tangent to his budget line.b) Show that the point of tangency does not represent a basket at which utility is maximized, given the budget constraint. Using the indifference curves you have drawn, indicate on your graph where the optimal basket is located.

a)

b) At point A, Dr. Strangetaste’s indifference curve, which is bowed out from the origin, is tangent to his budget line. This point is not an optimum because, for example, Dr. Strangetaste could move to point B on his budget line and achieve a higher level of total utility. Point B, though, is not an optimum either because Dr. Strangetaste could move to point C, a corner point, to achieve an even higher level of total utility. When the MRS is increasing, a corner point optimum will occur (with F = 0 in this picture, though it could equivalently be with H = 0 for another set of indifference curves).



4.12 Julie consumes two goods, food and clothing, and always has a positive marginal utility for each good. Her income is 24. Initially, the price of food is 2 and the price of clothing is 2. After new government policies are implemented, the price of food falls to 1 and the price of clothing rises to 4. Suppose, under the initial budget constraint, her optimal choice is 10 units of food and 2 units of clothing.a) After the prices change, can you predict whether her utility will be higher, lower, or the same as under the initial prices?b) Does your answer require that there be a diminishing marginal rate of substitution of food for clothing? Explain.

As given, Julie consumes F = 10 and C = 2 with an income of 24.Initially (with PF = PC = 2) she spends all her income: PFF + PCC = 2(10) + 2(2) = 24.To buy her initial basket at the new prices, she would only need to spendPFF + PCC = 1(10) + 4(2) = 18. Thus, her initial basket lies inside her new budget constraint (assuming her income stays at 24). With her new budget line she would be able to choose a new basket to the “northeast” of (i.e., a basket involving more food and clothing than) her initial basket, making her better off.

4.13 Toni likes to purchase round trips between the cities of Pulmonia and Castoria and other goods out of her income of $10,000. Fortunately, Pulmonian Airways provides air service and has a frequent-flyer program. Around trip between the two cities normally costs $500, but any customer who makes more than 10 trips a year gets to make additional trips during the year for only $200 per round trip.a) On a graph with round trips on the horizontal axis and “other goods” on the vertical axis, draw Toni’s budget line. (Hint: This problem demonstrates that a budget line need not always be a straight line.)b) On the graph you drew in part (a), draw a set of indifference curves that illustrates why Toni may be better off with the frequent-flyer program.c) On a new graph draw the same budget line you found in part (a). Now draw a set of indifference curves that illustrates why Toni might not be better off with the frequent-flyer program.

a) The budget line will have a kink where round trips = 10 and other goods = 5,000. Northwest of the kink, the budget line’s slope will be –500 . Southeast of the kink, the slope will be –200.



b) With the indifference curves drawn on the above graph, Toni is better off with the frequent flyer program (at point B) than she would be without it (at point C). Without the frequent flyer program the best she could achieve is point C, which lies on the hypothetical budget line where the price of round trips is always $500.

c) With the indifference curves drawn on graph below, Toni is no better off with the frequent flyer program than she would be without it (at point A). At this point, her indifference curve is tangent to a portion of the budget line where the frequent flyer program does not apply (less than 10 round trips).

4.14 A consumer has preferences between two goods, hamburgers (measured by H) and milkshakes (measured by M). His preferences over the two goods are represented by the utility function U = √H + √M. For this utility function MUH = 1/(2√H) and MUM = 1/(2√M).a) Determine if there is a diminishing MRSH,M for this utility function.b) Draw a graph to illustrate the shape of a typical indifference curve. Label the curve U1. Does the indifference curve intersect either axis? On the same graph, draw a second indifference curve U2, with U2 > U1.c) The consumer has an income of $24 per week. The price of a hamburger is $2 and the price of a milkshake is $1. How many milkshakes and hamburgers will he buy each week if he maximizes utility? Illustrate your answer on a graph.



a)

This utility function has a diminishing marginal rate of substitution since declines as H increases and M decreases.

b)

Since it is possible to have U > 0 if either H = 0 (and M > 0) or M = 0 (and H > 0), the indifference curves will intersect both axes.

c) We know from the tangency condition that

Substituting this into the budget line, , yields

Finally, plugging this back into the tangency condition implies . At the optimum the consumer will choose 4 hamburgers and 16 milkshakes. This can be seen in the graph above.

4.15 Justin has the utility function U = xy, with the marginal utilities MUx = y and MUy = x. The price of x is 2, the price of y is py, and his income is 40. When he maximizes utility subject to his budget constraint, he purchases 5 units of y. What must be the price of y and the amount of x consumed?



When Justin maximizes utility, his optimal consumption basket will be on the budget constraint and satisfy the tangency condition.Any basket on the budget line will satisfy pxx + pyy = I, or 2x + 5py = 40.The tangency condition requires that MUx / px = MUy / py, or that 5 / 2 = x / py. This implies that 5py = 2x.Putting these two equations together reveals that 5py + 5py = 40; thus py = 4.

4.16 A student consumes root beer and a composite good whose price is $1. Currently, the government imposes an excise tax of $0.50 per six-pack of root beer. The student now purchases 20 six-packs of root beer per month. (Think of the excise tax as increasing the price of root beer by $0.50 per six-pack over what the price would be without the tax.) The government is considering eliminating the excise tax on root beer and, instead, requiring consumers to pay $10.00 per month as a lump sum tax (i.e., the student pays a tax of $10.00 per month, regardless of how much root beer is consumed). If the new proposal is adopted, how will the student’s consumption pattern (in particular, the amount of root beer consumed) and welfare be affected? (Assume that the student’s marginal rate of substitution of root beer for other goods is diminishing.)

Assume the student is initially at an interior optimum, point A. Denote the initial price of root beer by P and the student’s income as M. Point A then consists of RA = 20 units of root beer and YA = M – 20P units of the composite good. The effect of the proposal is to rotate the budget line outward (the price change) and then shift it inward (the lump sum tax), for a total movement from BL1 to BL2. Notice that BL2 intersects BL1 exactly at point A: under the proposal, (RA, YA) costs the student 20(P – 0.5) + M – 20P = M – 10, which is equal to her income under the proposal.

Because A was initially optimal, MRSR,Y = P at point A. Yet the price ratio along BL2 is (P – 0.5). Hence MRSR,Y > PR / PY, so the student can increase her utility by purchasing more root beer and less of the composite good, at a point such as B depicted in the graph above. Thus, the proposal will make the student better off.



4.17 When the price of gasoline is $2.00 per gallon, Joe consumes 1,000 gallons per year. The price increases to $2.50, and to offset the harm to Joe, the government gives him a cash transfer of $500 per year. Will Joe be better off or worse off after the price increase and cash transfer than he was before? What will happen to his gasoline consumption? (Assume that Joe’s marginal rate of substitution of gasoline for other goods is diminishing.)

Assume Joe is initially at an interior optimum, point A, and that the price of other goods is $1. Let Joe’s income be M. Point A then consists of GA = 1000 units of root beer and YA = M – 2000 units of other goods. The effect of the proposal is to rotate the budget line inward (the price change) and then shift it outward (the cash transfer), for a total movement from BL1 to BL2. Notice that BL2 intersects BL1 exactly at point A: after the price increase, (GA, YA) costs Joe 1000*2.50 + M – 2000 = M + 500, which is equal to his income after the cash transfer.

Because A was initially optimal, MRSG,Y = 2 at point A. Yet the price ratio along BL2 is 2.5. Hence MRSG,Y < PG / PY, so Joe can increase his utility by purchasing less gas and more of the composite good, at a point such as B depicted in the graph above. Thus, the proposal will make Joe better off.

4.18 Paul consumes only two goods, pizza (P) and hamburgers (H) and considers them to be perfect substitutes, as shown by his utility function: U(P, H) = P + 4H. The price of pizza is $3 and the price of hamburgers is $6, and Paul’s monthly income is $300. Knowing that he likes pizza, Paul’s grandmother gives him a birthday gift certificate of $60 redeemable only at Pizza Hut. Though Paul is happy to get this gift, his grandmother did not realize that she could have made him exactly as happy by spending far less than she did. How much would she have needed to give him in cash to make him just as well off as with the gift certificate?

Paul’s initial budget constraint is the line AC, allowing him to purchase at most 50 hamburgers or at most 100 pizzas. The $60 cash certificate shifts out his budget constraint without changing the maximum number of hamburgers that he can buy. The new budget constraint is ABD and he can now buy a maximum of 120 pizzas.



Initially, Paul’s optimal basket contains all hamburgers and no pizza, at point A where (P, H) = (0, 50), because MUH /PH = 4/6 > MUP / PP = 1/3. His utility level at point A is U(0, 50) = 200. When he gets the gift certificate, Paul’s optimal basket is at point B, spending all of his regular income on hamburgers and the $60 gift certificate on pizza. So point B is where (P, H) = (20, 50) with a utility of U(20, 50) = 220. However, Paul could also achieve a utility of 220 by consuming 220/4 = 55 hamburgers. To buy the extra 5 hamburgers he would require 5*6 = $30. So, if he had received a cash gift of $30 it would have made Paul exactly as well off as the $60 gift certificate for pizzas.

4.19 Jack makes his consumption and saving decisions two months at a time. His income this month is $1,000, and he knows that he will get a raise next month making his income $1,050. The current interest rate (at which he is free to borrow or lend) is 5 percent. Denoting this month’s consumption by x and next month’s by y, for each of the following utility functions state whether Jack would choose to borrow, lend, or do neither in the first month. (Hint: In each case, start by assuming that Jack would simply spend his income in each month without borrowing or lending money. Would doing so be optimal?)a) U(x, y) = xy2, MUx = y2, MUy = 2xyb) U(x, y) = x2 y, MUx = 2xy, MUy = x2c) U(x, y) = xy, MUx = y, MUy = x

a) In this case, . If Jack neither borrows nor lends, then MRSx,y =

1050/(2*1000) = 0.525. Recall that if the interest rate is r, the slope of the budget line is –(1+r) = –1.05. Thus, if he neither borrows nor lends it will be the case that MRSx,y < 1 + r. That is, the “bang for the buck” for spending this month (good x) is less than that for spending next month (good y). Thus, Jack should lend some of his income this month (so x < 1000) in order to earn interest and have higher spending next month (y > 1050).b) Now MRSx,y = 2y/x. If Jack neither borrows nor lends, MRSx,y = 2.1 > (1 + r). Thus, Jack could increase his utility by borrowing in the first month (so that x > 1000 and y < 1050). c) Now MRSx,y = y/x. If Jack neither borrows nor lends, MRSx,y = 1.05 = (1 + r). Thus, Jack’s utility is maximized when he neither borrows nor lends, simply spending all of his income in each month: (x, y) = (1000, 1050).



4.20 The figure below shows a budget set for a consumer over two time periods, with a borrowing rate rB and a lending rate rL, with rL < rB. The consumer purchases C1 units of a composite good in period 1 and C2 units in period 2. The following is a general fact about consumers making consumption decisions over two time periods: Let A denote the basket at which a consumer spends exactly his income each period (the point at the kink of the budget line). Then a consumer with a diminishing MRSC1,C2 will choose to borrow in the first period if at basket A MRSC1,C2 > 1 + rB and will choose to lend if at basket A MRSC1,C2 < 1 + rL. If the MRS lies between these two values, then he will neither borrow nor lend. (You can try to prove this if you like. Keep in mind that diminishing MRS plays an important role in the proof.) Using this rule, consider the decision of Meg, who earns $2,000 this month and $2,200 the next with a utility function given by U(C1, C2) = C1C2, where the C’s denote the value of consumption in each month. Suppose rL=0.05 (the lending rate is 5 percent) and rB = 0.12 (the borrowing rate is 12 percent). Would Meg borrow, lend, or do neither this month? What if the borrowing rate fell to 8 percent?

The utility function implies that MRSC1,C2 = C2 / C1. At point A, MRSC1,C2 = 1.10, which lies between (1 + rL) = 1.05 and (1 + rB) = 1.12. Therefore Meg will neither borrow nor lend and will simply spend her entire income each month. If the borrowing rate falls to 8%, then the lower part of the budget line pivots outward, as depicted in the graph below. Then at point A, MRSC1,C2 > (1+ rB) > (1 + rL) since 1.10 > 1.08 > 1.05. So Meg can increase her utility by moving away from point A to a point like B, where she borrows money, spending more than her income this month (C1 > 2000) and less than her income next month (C2 < 2200).


BA


4.21 Sally consumes housing (denote the number of units of housing by h) and other goods (a composite good whose units are measured by y), both of which she likes. Initially she has an income of $100, and the price of a unit of housing (Ph) is $10. At her initial basket she consumes 2 units of housing. A few months later her income rises to $120; unfortunately, the price of housing in her city also rises, to $15. The price of the composite good does not change. At her later basket she consumes 1 unit of housing. Using revealed preference analysis (without drawing indifference curves), what can you say about how she ranks her initial and later baskets?

With the initial budget line, BL1, Sally chooses point A, where (xA, yA) = (2, 80). When her income and the price of housing increase, the budget line becomes BL2 and she chooses point B, where (xB, yB) = (1, 105). Importantly, because the equation for BL1 is 10x+ y = 100 and that for BL2 is 15x + y = 120, we can solve to see that these lines intersect at x = 4, to the right of point A on BL1.

Consider a hypothetical basket C on BL2 but northeast of A. We can then deduce that , because (i) B is at least as preferred as C (since B was chosen when C was affordable), and (ii) C


Slope= –1.05

Slope = –1.08

2000 2000+2200/1.08

2200+2000(1.05)

2200

C1

C2


is strictly preferred to A (since C lies to the northeast of A). By transitivity, B must be strictly preferred to A.

4.22 Samantha purchases food (F) and other goods (Y ) with the utility function U = FY. Her income is 12. The price of a food is 2 and the price of other goods 1.a) How many units of food does she consume when she maximizes utility?b) The government has recently completed a study suggesting that, for a healthy diet, every consumer should consume at least F = 8 units of food. The government is considering giving a consumer like Samantha a cash subsidy that would induce her to buy F = 8. How large would the cash subsidy need to be? Show her optimal basket with the cash subsidy on an optimal choice diagram with F on the horizontal axis and Y on the vertical axis. c) As an alternative to the cash subsidy in part (b), the government is also considering giving consumers like Samantha food stamps, that is, vouchers with a cash value that can only be redeemed to purchase food. Verify that if the government gives her vouchers worth $16, she will choose F = 8. Illustrate her optimal choice on an optimal choice diagram. (You may use the same graph you drew in part (b).)

a) MUY = F and MUF = Y, so MRSF,Y = Y/F, which diminishes as F increases along an indifference curve. Since the indifference curves do not intersect either axis, an optimal basket will be interior. At such an optimum two conditions must be satisfied: (1) tangency: MRSF,Y = PF / PY, or Y = 2F, and (2) budget line (“BL No subsidy” in the graph): 2F + Y = 12. This F = 3 and Y = 6. This optimum is depicted as point A in the graph.

b) We need to find an interior optimum with F = 8. As income increases, the consumer chooses a basket along the Income Consumption Curve, which consists of the tangency points Y = 2F. So Y = 2(8) = 16. Total expenditure will then be 2F + Y = 2(8) + 16 = 32. So the consumer needs an income of 32 (“BL Cash” in the graph). Since the consumer has an income of 12, she needs an additional income of 20 (=32 – 12). So the subsidy needed is 20. This optimum is shown as point B in the graph.

c) From part (b) we see that, with no restrictions on how the government subsidy can be spent, the consumer would like to buy 16 units of Y, more than her own budget (without subsidy) would permit. So we expect that with food stamps, she will use the voucher to purchase



the required 8 units of food and spend all of her own unrestricted income (12) on Y. In other words, this consumer will be at point C on the graph, at the kink on the budget constraint RCS (labeled “BL Food Stamps”).

We can verify that (F = 8, Y = 12) is her optimal choice by looking at the “bang for the buck” condition at C. MUF/price of food = Y/2 = 12/2 = 6. MUY/price of Y = F/1 = 8. So the consumer would like to substitute more Y for F, but cannot do so because at basket C she has purchased all the other goods she can given her budget constraint.

4.23 As shown in the following figure, a consumer buys two goods, food and housing, and likes both goods. When she has budget line BL1, her optimal choice is basket A. Given budget line BL2, she chooses basket B, and with BL3, she chooses basket C.

a) What can you infer about how the consumer ranks baskets A, B, and C? If you can infer a ranking, explain how. If you cannot infer a ranking, explain why not.b) On the graph, shade in (and clearly label) the areas that are revealed to be less preferred to basket B, and explain why you indicated these areas.c) On the graph, shade in (and clearly label) the areas that are revealed to be (more) preferred to basket B, and explain why you indicated these areas.

a) From figure 4.21 we can infer that , that , and therefore by transitivity that .



First, A must be strictly preferred to B since A is at least as preferred as D and D must be strictly preferred to B. Second, B must be strictly preferred to C since B is at least as preferred as E and E is strictly preferred to C. So by transitivity A must be strictly preferred to C.

b)

B will be strictly preferred to everything inside BL2. In addition, since B is strictly preferred to C, B will be strictly preferred to everything below BL3, including all the points along BL3 itself (since C is weakly preferred to everything on BL3).

c) See the graph in part (b). Everything to the northeast of B is strictly preferred to B. In addition, since A is strictly preferred to B [see part (a)], everything to the northeast of A must also be strictly preferred to B. Notice, however, that there are points on BL1 (both northwest and southeast of A) about which we cannot infer anything.

4.24 The following graph shows the consumption decisions of a consumer over bundles of x and y, both of which he likes. When faced with budget line BL1, he chose basket A, and when faced with budget line BL2, he chose basket B. If he were to face budget line BL3, what possible set of baskets could he choose in order for his behavior to be consistent with utility maximization?

Let point E denote the intersection of BL1 and BL3, and point F denote the intersection of BL2 and BL3 (see the figure below).



First, any point to the northwest of E, including E, is in the consumer’s budget set when he faces budget constraint BL1. The fact that he chose A over these points implies that A is at least as preferred to E and strictly preferred to the points northwest of E. However, point G lies on BL3 and is northeast of A, so . Therefore, by transitivity G is strictly preferred to E and all the points on BL3 northwest of E.Similarly, A is northeast of B so . Since B is at least as preferred as any point on BL2, including F, by transitivity we know that A is strictly preferred to F and all points on BL3 to the southeast of F. And since , we know that G is strictly preferred to these points as well. Therefore the consumer could choose any point between E and F on BL3, but neither E nor F themselves.

4.25 Darrell has a monthly income of $60. He spends this money making telephone calls home (measured in minutes of calls) and on other goods. His mobile phone company offers him two plans:• Plan A: pay no monthly fee and make calls for $0.50 per minute.• Plan B: Pay a $20 monthly fee and make calls for $0.20 per minute. Graph Darrell’s budget constraint under each of the two plans. If Plan A is better for him, what is the set of baskets he may purchase if his behavior is consistent with utility maximization? What baskets might he purchase if Plan B is better for him?

Let x denote the number of phone calls, and y denote spending on other goods. Under Plan A, Darrell’s budget line is JLM. Under Plan B, it is JKLN. These budget lines intersect at point L, or about x = 67.



If we know that Darrell chooses Plan A, his optimal bundle must lie on the line segment JL. No point between L and M would be optimal under this plan because then Darrell could have chosen a point under Plan B, between L and N, that would have given him more minutes, and left him with more money to buy other goods. However, we cannot exclude point L itself (Darrell could, for instance, have perfect complements preferences with an “elbow” at point L). Thus, if Darrell chooses Plan A his optimal basket could be anywhere between J and L, including either of these points.Similarly, if he chose Plan B then his optimal basket must lie between L and N. Any point between L and K (but not including point L) would be dominated by a point under Plan A between L and J. Thus, if Darrell chooses Plan B his optimal basket could be anywhere between L and N, including either of these points.

4.26 Figure 4.17 illustrates the case in which a consumer is better off with a quantity discount. Can you draw an indifference map for a consumer who would not be better off with the quantity discount?

With this set of indifference curves, the tangency with the budget line occurs on the portion of the budget line where the quantity discount has not taken effect. Therefore, the consumer does not receive any benefit from the quantity discount.



4.27 Angela has a monthly income of $120, which she spends on MP3s and a composite good (whose price you may assume is $1 throughout this problem). Currently, she does not belong to an MP3 club, so she pays the retail price of an MP3 of $2; her optimal basket includes 20 MP3s monthly. For the past several months Asteroid, a media company, has offered her the chance to join their “Premium Club”; to join the club she would need to pay a membership fee of $60 per month, but then she could buy all the MP3s she wants at a price of $0.50. She has decided not to join the club. Asteroid has now introduced an “Economy Club”; to join, Angela would need to pay a membership fee of $30 per month, but then she could buy all the MP3s she wants at a price of $1. Draw a graph illustrating (1) Angela’s budget line and optimal basket when she joins no club, (2) the budget line she would have faced had she joined the Premium Club, and (3) her budget line if she joins the Economy Club. Will Angela surely want to join the Economy Club? If she were to join the club, how many MP3s per month might she buy? Show how you arrive at your answers using a revealed preference argument.

The budget lines for the “no club,” “Economy Club,” and “Premium Club” opportunities available to Angela are drawn below.

She chooses A with “no club,” and chooses not to join the “Premium Club” when given the chance. Thus, A is weakly preferred to any basket on WZ, and strongly preferred basket on the segment CS (except for C). So, if she does join the Economy Club, she would not choose a basket on CS (except possibly for C).

With “no club” she chooses A when she could have afforded B; thus A > B. Since the rest of the segment RB lies inside the “no club” BL, these baskets are strongly inferior to A. So, if she does join the Economy Club, she would not choose a basket on RB (except possibly for B).

To summarize, if she joins the Economy Club, she might consume any basket on the segment BC, including B and C. So her consumption in that case would be 30 < MP3s < 60.

But she might not join the Economy Club at all! Although we have established that A > B and A > C, we cannot say how she ranks A with the baskets between B and C.


Y

MP3s

BL – no club

BL1 – Premium Club

BL – Economy ClubA(20,80)

120

90

B(30,60)

C(60,30)

60

120 90 60

R

S Z

W


4.28 Alex buys two goods, food (F) and clothing (C). He likes both goods. His preferences for the goods do not change from month to month. The following table shows his income, the baskets he selected, and the prices of the goods over a two-month period.

Month PF PC Income Basket Chosen

1 3 2 48 F=16, C=02 2 4 48 F=14, C=5

a) On the graph with F on the horizontal axis and C on the vertical axis, plot and clearly label the budget lines and consumption baskets during these two weeks. Label the consumption bundle in week 1 by point A on the graph and the consumption basket in week 2 by point B. Using revealed preference analysis, what can you say about Alex’s preferences for baskets A and B (i.e., how does he rank them)?b) In month 3 Alex’s income rises to 57. The prices of food and clothing are both 3. Assuming his preferences do not change, describe the set of baskets he might consume in month 3 if he continues to maximize utility. Show this set of baskets in the graph.

a)

B is weakly preferred to C, which is strongly preferred to A. By transitivity, B is strongly preferred to A.

b) We know that he will choose a basket on BL3.He will not choose any basket on or inside BL1. Why? He could choose B, and B>A (by part A).Given BL1, he chose A instead of other baskets on BL1, all of which were affordable.He will not choose a basket on BL3 to the southeast of B. Why? Given BL2 he chose B, and B is strictly preferred to any basket inside BL3, including those on BL2 to the southeast of B.Therefore, he might choose any basket on BL3 between B and E, not including E. This set of baskets is illustrated with the heavy line in the graph.



4.29 Brian consumes units of electricity (E) and a composite good (Y), whose price is always $1. He likes both goods. In period 1 the power company sets the price of electricity at $7 per unit, for all units of electricity consumed. Brian consumes his optimal basket, 20 units of electricity and 70 units of the composite good. In period 2 the power company then revises its pricing plan, charging $10 per unit for the first 5 units and $4 per unit for each additional unit. Brian’s income is unchanged. Brian’s optimal basket with this plan includes 30 units of electricity and 60 units of the composite good. In period 3 the power company allows the consumer to choose either the pricing plan in period 1 or the plan in period 2. Brian’s income is unchanged. Which pricing plan will he choose? Illustrate your answer with a clearly labeled graph.

Brian’s income is I = (7)(20) + 1(70) = $210.His initial budget line is the solid curve RJ.His initial optimal basket is A.

His new budget line is the dotted, piecewise linear curve RST. He chooses basket B, which costs 10(5) + 4(30 – 5) + 1(60) = 210. In period 2 Basket A costs (10)(5) + (4)(20 – 5) + 1(70) = $180, so Basket A lies inside the new budget line.

In period 3 Brian will choose the plan from the second period. B is weakly preferred to a basket like C, which, in turn is strictly preferred to A (because C lies to the northeast of A). Thus he strictly prefers B to A, and he can only reach B by choosing the plan from period 2.

4.30 Carina consumes two goods, X and Y, both of which she likes. In month 1 she chooses basket A given budget line BL1. In month 2 she chooses B given budget line BL2, and in month 3 she chooses C given budget lineBL3. Assume her indifference map is unchanged over the three months. Use the theory of revealed preference to show whether her choices are consistent with utility maximizing behavior. If so, show how she ranks the three baskets. If it is not possible to infer how she ranks the baskets, explain why not.



From BL1 we would infer that A is weakly preferred to S, and that S is strongly preferred to C; by transitivity we conclude that A is strongly preferred to C.

From BL3 we would infer that C is weakly preferred to R, which is strongly preferred to A; by transitivity we conclude that C is strongly preferred to A.

It cannot be simultaneously true that A is strongly preferred to C and C is strongly preferred to A, for this would imply that A is strongly preferred to C, which is strongly preferred to A, or that A is strongly preferred to itself. The preferences are either intransitive, or else the consumer is failing to maximize utility in each of the three time periods.
