ch05Solution manual

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Chapter 5 The Theory of Demand Solutions to Review Questions 1.

What is a price consumption curve for a good?

The price consumption curve plots the set of optimal bundles for two goods, say X and Y, by changing the price of one good while holding the price of the other good and income constant. 2.

How does a price consumption curve differ from an income consumption curve?

The price consumption curve plots the set of optimal bundles for two goods as the price of one good changes while the price of the other good and income remain constant. The income consumption curve, on the other hand, plots the set of optimal bundles for two goods as the consumer’s income changes while holding the prices of both goods constant. 3. What can you say about the income elasticity of demand of a normal good? of an inferior good? With a normal good, when income increases, consumption of the good will increase. This implies the income elasticity for a normal good will be positive. With an inferior good, when income increases consumption of the good will decrease. This implies the income elasticity for an inferior good will be negative. 4. If indifference curves are bowed in toward the origin and the price of a good drops, can the substitution effect ever lead to less consumption of the good? If indifference curves are bowed in toward the origin and the price of, say, good X falls, consumption of X will always increase; so the substitution effect will always be positive. A decrease in the price of X implies that the slope of the budget line becomes flatter. When indifference curves are bowed in, a direct consequence of this change in relative prices is that any tangency will occur “southeast” of the original bundle along the initial indifference curve. The only way for consumption to fall when price falls is for the income effect to be negative (an inferior good) and for its magnitude to more than offset the substitution effect. In this rare situation, the good is known as a Giffen good. 5. Suppose a consumer purchases only three goods, food, clothing, and shelter. Could all three goods be normal? Could all three goods be inferior? Explain. If the consumer purchases only three goods and income increases, it is possible that consumption of all three goods will increase. For example, the consumer might allocate one-third of the increase to each of the three goods. Thus, it is possible for all three goods to be normal. If the consumer purchases only three goods and income increases, it is not possible that consumption of all three goods will decrease. Recall that if consumption falls when income increases the Copyright © 2014 John Wiley & Sons, Inc.

Chapter 5 - 1

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

good is inferior. If this were to occur, the consumer would be spending less income than he did prior to the income increase. Thus, it is not possible for all three goods to be inferior. 6. Does economic theory require that a demand curve always be downward sloping? If not, under what circumstances might the demand curve have an upward slope over some region of prices? Generally speaking demand curves are downward sloping. Economic theory, however, suggests a special case of an inferior good whose negative income effect is greater than its positive substitution effect. In this event, consumption of the good falls when the price of the good falls. This type of good is known as a Giffen good. While economic theory suggests that such a good could exist, in practice no such good has been confirmed for humans (although the text suggests an experiment on rats where a good was determined to be a Giffen good). 7.

What is consumer surplus?

Consumer surplus is the difference between the maximum amount a consumer is willing to pay for a good and what he must actually pay when he purchases it in the marketplace. For example, if Joe is willing to pay $20 for a cap but purchases it at the store for only $5, Joe will receive $15 in consumer surplus. This measure indicates how much better off the consumer is after purchasing the good. 8. Two different ways of measuring the monetary value that a consumer would assign to the change in price of the good are (1) the compensating variation and (2) the equivalent variation. What is the difference between the two measures, and when would these measures be equal? Compensating variation answers the question, “How much would the consumer be willing to give up after a price reduction to achieve the same level of satisfaction as she had before the price change?” Equivalent variation, on the other hand, answers the question, “How much money would we have to give the consumer before a price reduction to leave her level of satisfaction the same as it would be after the price reduction?” In essence, both of these are measures of the “distance” between the initial and final indifference curves after a price change. Typically the compensating and equivalent variation measures will not be the same. In the case of quasi-linear utility functions, however, the compensating and equivalent variation measures will always be the same (they will be equal to the change in consumer surplus). In general, these two measures will be identical when there is no income effect associated with a price change. 9. Consider the following four statements. Which might be an example of a positive network externality? Which might be an example of a negative network externality? (i) People eat hot dogs because they like the taste, and hot dogs are filling. (ii) As soon as Zack discovered that everybody else was eating hot dogs, he stopped buying them. (iii) Sally wouldn’t think of buying hot dogs until she realized that all her friends were eating them. (iv) When personal income grew by 10 percent, hot dog sales fell.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

(i) No network externality (ii) Negative network externality (iii) Positive network externality (iv) Since sales fall when income increases, this might be a negative network externality if some consumers stopped buying hot dogs not only because of a lower income, but also because other consumers bought fewer hot dogs. 10. Why might an individual supply less labor (demand more leisure) as the wage rate rises? When the wage rate rises, the substitution effect will induce a worker to supply more hours of labor. The income effect, on the other hand, may induce the worker to increase the amount of leisure and decrease the amount of labor. If the income effect reduces the amount of labor supplied more than the substitution effect increases it, the worker will ultimately supply less hours of labor.

Solutions to Problems 5.1 Figure 5.2(a) shows a consumer’s optimal choices of food and clothing for three values of weekly income: I1 = $40, I2 = $68, and I3 = $92. Figure 5.2(b) illustrates how the consumer’s demand curve for food shifts as income changes. Draw three demand curves for clothing (one for each level of income) to illustrate how changes in income affect the consumer’s purchases of clothing. Py

D2 (I=68)

4

D3 (I=92) D1 (I=40) y 5

8

11

5.2 Use the income consumption curve in Figure 5.2(a) to draw the Engel curve for clothing, assuming the price of food is $2 and the price of clothing is $4.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

5.3 Show that the following statements are true: a) An inferior good has a negative income elasticity of demand. b) A good whose income elasticity of demand is negative will be an inferior good. %Q Q / Q  Q  I       %I I / I  I   Q I and Q must be greater than zero. In addition, assume income increases, i.e., I  0 . If the good is inferior, then Q  0 . Thus, the first term (Q / I )  0 and the second term ( I / Q)  0 . Multiplying these two terms together implies  Q , I  0 . Inferior goods have a negative income elasticity of demand. a)

 Q,I 

b)

If income elasticity of demand is negative then  Q  I   Q,I      0.  I   Q

Since I and Q must be greater than zero, for  Q , I to be negative, we must have Q 0. I This can only happen if either a) Q  0 and I  0 or b) Q  0 and I  0 . In both instances, the change in quantity demanded moves in the opposite direction as the change in income implying the good is inferior. 5.4 If the demand for a product is perfectly price inelastic, what does the corresponding price consumption curve look like? Draw a graph to show the price consumption curve.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

If demand for good X is perfectly price inelastic then the demand curve is a vertical line and quantity remains constant as price changes. Graphing the price consumption curve for good X on an optimal choice diagram would appear as Y

Price consumption curve

X The price consumption curve is a straight line because the level of consumption of X is constant. 5.5 Ann consumes five goods. The prices of all goods are fixed. The price of good x is px. She spends 25 percent of her income on good x, regardless of the size of her income. a) Show that her income elasticity of demand of good x is the same for any level of income, and determine its value. b) Would the value of the income elasticity of demand for x be different if Ann always spends 60 percent of her income on good x? a) Since she spends 25% of her income on x, it must be true that pxx/I = 0.25. Thus x/I = 0.25/px. This means that x/I is a constant. If I increases by 1%, x must also increase by 1%. Since the percentage increase in x is the same as the percentage increase in I, the income elasticity must be 1. b) The income elasticity of demand would still be 1. Now x/I = 0.6/px. This means that x/I is a constant. If I increases by 1%, x must also increase by 1%. 5.6 Suzie purchases two goods, food and clothing. She has the utility function U(x, y) = xy, where x denotes the amount of food consumed and y the amount of clothing. The marginal utilities for this utility function are MUx = y and MUy = x. a) Show that the equation for her demand curve for clothing is y = I/(2Py). b) Is clothing a normal good? Draw her demand curve for clothing when the level of income is I = 200. Label this demand curve D1. Draw the demand curve when I = 300 and label this demand curve D2. c) What can be said about the cross-price elasticity of demand of food with respect to the price of clothing? a)

At the consumer’s optimum we must have

Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

MU x MU y  Px Py y x  Px Py Substituting into the budget line, Px x  Py y  I , gives   P  Px  y  y   Py y  I   Px  2 Py y  I y b)

I 2 Py

Yes, clothing is a normal good. Holding Py constant, if I increases y will also increase.

c) The cross-price elasticity of demand of food with respect to the price of clothing must be zero. Note in part a) that with this utility function the demand for y does not depend on the price of x. Similarly, you can show that the demand for food is x = I / (2Px), which does not depend on the price of y. In fact, the consumer divides her income equally between the two goods regardless of the price of either. Since the demands do not depend on the prices of the other goods, the cross-price elasticities must be zero. 5.7 Karl’s preferences over hamburgers (H) and beer (B) are described by the utility function: U(H, B) = min(2H, 3B). His monthly income is I dollars, and he only buys these two goods out of his income. Denote the price of hamburgers by PH and of beer by PB. a) Derive Karl’s demand curve for beer as a function of the exogenous variables. b) Which affects Karl’s consumption of beer more: a one dollar increase in PH or a one dollar increase in PB?

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

a) Karl’s optimal bundle will always be such that 2H = 3B. If this were not true then he could decrease the consumption of one of the two goods, staying at the same level of utility and reducing expenditure. Also, at the optimal bundle, it must be true that PH H  PB B  I . Substituting the first condition into the second we get B (1.5PH  PB )  I which implies that the demand curve for beer is given by, B 

I (1.5 PH  PB )

b) You can answer this just by looking at the demand curve. Because it has a larger coefficient, the price of hamburgers affects the demand for beer more than the price of beer. A one dollar increase in PH decreases demand for beer more than a one dollar increase in PB . 5.8 David has a quasi-linear utility function of the form U(x, y) = √x + y, with associated marginal utility functions MUx = 1/(2√x) and MUy = 1. a) Derive David’s demand curve for x as a function of the prices, Px and Py. Verify that the demand for x is independent of the level of income at an interior optimum. b) Derive David’s demand curve for y. Is y a normal good? What happens to the demand for y as Px increases? a)

Denoting the level of income by I, the budget constraint implies that

the tangency condition is

1 2 x



px x  p y y  I

and

px py2 x , which means that . The demand for x does not py 4 px2

depend on the level of income. b)

From the budget constraint, the demand curve for y is, y 

py I  px x I   . py p y 4 px

You can see that the demand for y increases with an increase in the level of income, indicating that y is a normal good. Moreover, when the price of x goes up, the demand for y increases as well. 5.9 Rick purchases two goods, food and clothing. He has a diminishing marginal rate of substitution of food for clothing. Let x denote the amount of food consumed and y the amount of clothing. Suppose the price of food increases from Px1 to Px2. On a clearly labeled graph, illustrate the income and substitution effects of the price change on the consumption of food. Do so for each of the following cases: a) Case 1: Food is a normal good. b) Case 2: The income elasticity of demand for food is zero. c) Case 3: Food is an inferior good, but not a Giffen good. d) Case 4: Food is a Giffen good. a)

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Y

B C

A

X Substitution Effect Income Effect b) Y

B A C X Substitution Effect Income Effect = 0 c)

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Y

B A C X

Substitution Effect Income Effect d) Y

B A C X

Substitution Effect Income Effect 5.10 Reggie consumes only two goods: food and shelter. On a graph with shelter on the horizontal axis and food on the vertical axis, his price consumption curve for shelter is a vertical line. Draw a pair of budget lines and indifference curves that are consistent with

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

this description of his preferences. What must always be true about Reggie’s income and substitution effects as the result of a change in the price of shelter?

Food

Price Consumption Curve

Shelter A pair of possible indifference curves and budget lines are shown above. For the Price consumption curve to be a vertical line, it must be that Reggie’s demand for shelter does not change even when the price of shelter changes and the budget line rotates. The fact that his optimal bundle stays the same, despite a price change, means that Reggie’s income and substitution effects as a result of a change in the price of shelter must cancel each other out so as to leave a net zero effect. For example, if the price of shelter were to decrease, the substitution effect would be positive and this would imply a negative income effect, just large enough to cancel out the substitution effect. In other words, the two effects have the same magnitude but opposite signs. This also implies that Reggie views shelter as an inferior good. 5.11 Ginger’s utility function is U(x, y) = x2y, with associated marginal utility functions MUx = 2xy and MUy = x2. She has income I = 240 and faces prices Px = $8 and Py = $2. a) Determine Ginger’s optimal basket given these prices and her income. b) If the price of y increases to $8 and Ginger’s income is unchanged, what must the price of x fall to in order for her to be exactly as well off as before the change in Py?

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

a)

The budget constraint is 8 x  2 y  240 and the tangency condition is

2y 8   4. x 2

Solving, the optimal bundle is (x, y)=(20, 40) with a utility of 202(40)=16,000. b) Now py=8. We need to calculate px such that, with the new prices, Ginger reaches exactly the same indifference curve as before. The new optimal bundle (x,y) must be such that: 2 y px  p x x  8 y  240, and x 2 y  16000 . The tangency condition now implies that that is, x 8 p x x  16 y. Substituting this into the budget constraint we find that y=10. Using the condition x 2 y  16000 , we find that x = 40. Finally, substituting the values of x and y back into the budget constraint, we can see that p x 40  8(10)  240 , or px=4. Therefore, if the price of y were to increase to $8, Ginger would need the price of x to decrease to $4 in order to be exactly as well off as before. 5.12 Ann’s utility function is U(x, y) = x + y, with associated marginal utility functions MUx = 1 and MUy = 1. Ann has income I = 4. a) Determine all optimal baskets given that she faces prices Px = 1 and Py = 1. b) Determine all optimal baskets given that she faces prices Px = 1 and Py = 2. c) What is demand for y when Px = 1 and Py = 1? What is demand for y when Px = 1 and Py > 1? What is demand for y when Px = 1 and Py < 1? Plot Ann’s demand for y as a function of Py. d) Repeat the exercises in a), b) and c) for U(x, y) = 2x + y, with associated marginal utility functions MUx = 2 and MUy = 1, and with the same level of income. a) Notice that MUx / MUy = 1 for all x and y. In this case indifference curves are straight lines with slope 1. Therefore, when Px = 1 and Py = 1 all pairs of x and y such that x + y = 4 are optimal baskets. b) Optimal consumption in this case is at a corner point. Since the price of x is smaller than the price of y and marginal utility of each good is the same, consumer is better off purchasing only x. (Another way to see this is to note that MU X/PX = 1/1 > MUY/PY = ½.) Hence, the optimal basket consists of 4 units of x and zero units of y. c) When price of y is lower than 1 there are zero units of x in the optimal basket. Hence, for Px = 1 and Py < 1 the demand for y equals to I / Py.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Py

1

4

y

d) By the same argument Ann purchases only x when Px = 1 and Py = 1. Marginal utility per dollar from consumption of x is higher than marginal utility per dollar of y. When Px = 1 and Py = 2 marginal utilities per dollar are the same for both goods. Hence, all baskets such that 2x + y = 4 are optimal. Construction of the demand curve is similar as in part c). 5.13 Some texts define a “luxury good” as a good for which the income elasticity of demand is greater than 1. Suppose that a consumer purchases only two goods. Can both goods be luxury goods? Explain. Consider any change in income I . For the budget constraint to hold, it must be true that I  Px x  Py y .

(For example, if income increases then some of it may be spent on x and some on y, but the total new expenditures must be equal to the change in income.) Since we are interested in income elasticities, it helps to rewrite the previous equation as 1  Px

x y  Py I I

Since  x , I   x I  I / x  and  y , I   y I  I / y  , we can write this as 1  Px

x y  x , I  Py  y , I I I

Or I   Px x   x , I   Py y  y , I

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

But if both goods are luxury goods, then  x , I  1 and  y , I  1 so that the previous equation implies I   Px x 1   Py y 1

Thus, if both x and y are luxury goods then I > I, which obviously is untrue! Therefore, both goods cannot simultaneously be luxury goods. 5.14 Scott consumes only two goods, steak and ale. When the price of steak falls, he buys more steak and more ale. On an optimal choice diagram (with budget lines and indifference curves), illustrate this pattern of consumption. When the price of steak falls, the budget line rotates from BL1 to BL2. The consumer now maximizes utility on U2 at point B on BL2. The amounts of steak and ale consumed at point B are greater than the initial amounts consumed at point A. This is shown in the following figure. Ale

B A

U2 U1 BL1

BL2 Steak

5.15 Dave consumes only two goods, coffee and doughnuts. When the price of coffee falls, he buys the same amount of coffee and more doughnuts. a) On an optimal choice diagram (with budget lines and indifference curves), illustrate this pattern of consumption. b) Is this purchasing behavior consistent with a quasi-linear utility function? Explain. a)

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Doughnuts

B BL1 A

BL2 Coffee

In the diagram above, the consumer purchases the same amount of coffee and more doughnuts after the price of coffee falls. b) No, this behavior is not consistent with a quasi-linear utility function. While it is true that there is no income effect with a quasi-linear utility function, the substitution effect would still induce the consumer to purchase more coffee when the price of coffee falls. 5.16 (This problem shows that an optimal consumption choice need not be interior and may be at a corner point.) Suppose that a consumer’s utility function is U(x, y) = xy + 10y. The marginal utilities for this utility function are MUx = y and MUy = x + 10. The price of x is Px and the price of y is Py, with both prices positive. The consumer has income I. a) Assume first that we are at an interior optimum. Show that the demand schedule for x can be written as x = I/(2Px) − 5. b) Suppose now that I = 100. Since x must never be negative, what is the maximum value of Px for which this consumer would ever purchase any x? c) Suppose Py = 20 and Px = 20. On a graph illustrating the optimal consumption bundle of x and y, show that since Px exceeds the value you calculated in part (b), this corresponds to a corner point at which the consumer purchases only y. (In fact, the consumer would purchase y = I/Py = 5 units of y and no units of x.) d) Compare the marginal rate of substitution of x for y with the ratio (Px/Py) at the optimum in part (c). Does this verify that the consumer would reduce utility if she purchased a positive amount of x? e) Assuming income remains at 100, draw the demand schedule for x for all values of Px. Does its location depend on the value of Py? a)

If we are at an interior optimum the tangency condition must hold: P y  x x  10 Py Py y  Px ( x  10)

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Substituting into the budget line, Px x  Py y  I , yields Px x  Px ( x  10)  I 2 Px x  10 Px  I 2 Px x  I  10 Px x b)

I 5 2 Px

If I  100 , then x

100 5 2 Px

x

50 5 Px

Since we must have x  0 , we must have 50 5  0 Px 50 5 Px 50  5 Px Px  10 So the consumer would only purchase x for prices less than 10. c)

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Given Px  Py  20 , the slope of the budget line is –1. At the corner point optimum, the slope of the indifference curve is 

MU x y 5 1    MU y x  10 10 2

Because the indifference curve has a flatter slope than the budget line, the consumer would like to substitute more y for x , but has no more x to give up at the corner point. MU x 5 10 MU y    . If the consumer were to purchase any x , since the “bang for Px 20 20 Py the buck” for x is less than the “bang for the buck” for y , the consumer would reduce total utility by increasing x above zero. d)

e)

As shown in part a), the demand for x depends only on I and Px . Therefore, the location of the demand curve does not depend on Py .

5.17 The figure below illustrates the change in consumer surplus, given by Area ABEC, when the price decreases from P1 to P2. This area can be divided into the rectangle ABDC and the triangle BDE. Briefly describe what each area represents, separately, keeping in mind the fact that consumer surplus is a measure of how well off consumers are (therefore the change in consumer surplus represents how much better off consumers are). (Hint: Note that a price decrease also induces an increase in the quantity consumed.)

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

P1 P2

As the figure shows, a decrease in the price from p1 to p2 induces an increase in quantity from q1 to q2. The resulting change in consumer surplus is due to two things: First, the consumer is paying a lower price, per unit, on all the units of the good that he was consuming before the price change. That is, for the q1 units he was earlier consuming, he now pays a lower price and therefore enjoys a higher consumer surplus, denoted by the area of the rectangle ABCD. Another way of putting this is that if he continued to consume q1 even after the price change his consumer surplus would increase by only area ABCD. Second, the lower price induces him to consume more of the good in question. In fact he consumes (q2 – q1) more units. The additional benefit he gets from this is the area of triangle BDE. 5.18 The demand function for widgets is given by D(P) = 16 − 2P. Compute the change in consumer surplus when price of a widget increases for $1 to $3. Illustrate your result graphically. For price of a widget equal to $1 consumer surplus is CS$1 = ½ ∙ (8 – 1) ∙ D(1) = ½ ∙ 7 ∙ 14 = 49. When price is equal to $3 consumer surplus is CS$3 = ½ ∙ (8 – 3) ∙ D(3) = ½ ∙ 5 ∙ 10 = 25.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

P

$8

Area of ABE triangle CS when P = $3 is 25

A

D(P) = 16 – 2P

$3

$1

E

B

D

Area of ACD triangle CS when P = $1 is 49

C

D(P)

5.19 Jim’s preferences over cookies (x) and other goods (y) are given by U(x, y) = xy with associated marginal utility functions MUx = y. and MUy = x. His income is $20. a) Find Jim’s demand schedule for x when price of y is Py = $1. b) Illustrate graphically the change in consumer surplus when the price of x increases from $1 to $2. a) Jim’s optimal basket is a solution to equations MUx / MUy = Px / Py and Px x + Py y = I. Hence, we have y / x = Px and Px x + y = 20 with solution x = 10 / Px and y = 10. Demand schedule for x is D(Px) = 10 / Px. b) Px

$2 $1

A

B C

D 5

10

x

The change in consumer surplus is area of region ABCD under the demand curve. The are of this region can be computed by simple integration: –∫[1,2] 10/p dp = – 10 ln(2).

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

5.20 Lou’s preferences over pizza (x) and other goods (y) are given by U(x, y) = xy, with associated marginal utilities MUx = y and MUy = x. His income is $120. a) Calculate his optimal basket when Px = 4 and Py = 1. b) Calculate his income and substitution effects of a decrease in the price of food to $3. c) Calculate the compensating variation of the price change. d) Calculate the equivalent variation of the price change. a)

Using the tangency condition,

y  4 , and the budget constraint, 4 x  y  120 , Lou’s x

initial optimum is the basket (x, y) = (15, 60) with a utility of 900. b)

First we need the decomposition basket. This would satisfy the new tangency condition,

y  3 and would give him as much utility as before, i.e. xy  900 . This gives x

or approximately (17.3,51.9). Now we need the final basket, which satisfies the same tangency condition as the decomposition basket and also the new budget constraint: 3 x  y  120. Together, these conditions imply that (x, y) = (20, 60). The substitution effect is therefore 17.3 – 15 = 2.3, and the income effect is 20 – 17.3 = 2.7. ( x, y )  (10 3 ,30 3 )

c) The compensating variation is the amount of income Lou would be willing to give up after the price change to maintain the level of utility he had before the price change. This equals the difference between the consumer’s actual income, $120, and the income needed to buy the decomposition basket at the new prices. This latter income equals: 3*17.3 + 1*51.9 = 103.8. The compensating variation thus equals 120 – 103.8 = $16.2. d) The equivalent variation is the amount of income that Lou would need to be given before the price change in order to leave him as well off as he would be after the price change. After the price change his utility level is 20(60)=1200. Therefore the additional income should be such that it allows Lou to purchase a bundle (x, y) satisfying the initial tangency condition,

y  4, x

and also such that xy  1200. This implies that ( x, y )  (10 3 ,40 3 ) or approximately (17.3, 69.2). How much income would Lou need to purchase this bundle under the original prices? He would need 4(17.3) + 69.2 = 138.4. That is he would need to increase his income by (138.4 – 120) dollars in order to be as well off as if the price of pizza were to decrease instead. Therefore his equivalent variation is $18.4. 5.21 Carina buys two goods, food F and clothing C, with the utility function U = FC + F. Her marginal utility of food is MUF = C + 1 and her marginal utility of clothing is MUC = F. She has an income of 20. The price of clothing is 4. a) Derive the equation representing Carina’s demand for food, and draw this demand curve for prices of food ranging between 1 and 6. b) Calculate the income and substitution effects on Carina’s consumption of food when the price of food rises from 1 to 4, and draw a graph illustrating these effects. Your graph need not be exactly to scale, but it should be consistent with the data.

Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

c) Determine the numerical size of the compensating variation (in monetary terms) associated with the increase in the price of food from 1 to 4. a) MUF = C + 1 MUC = F Tangency: MUF/MUC = PF / PC. (C + 1)/ F = PF/4 => 4C + 4 = PFF. (Eq 1) Budget Line: PFF + PCC = I . PFF + 4C = 20. (Eq 2) Substituting (Eq 1) into (Eq 2): 4C + 4 + 4C = 20. Thus C = 2, independent of PF. From the budget line, we see that PFF + 4(2) = 20, so the demand for F is F = 12/PF . 6 Demand for food

   PF5 4

 

3 2 1 2

3 4

6

   F

12 

b) Initial Basket: From the demand for food in (a), F = 12/1 = 12, and C = 2. Also, the initial level of utility is U = FC + F = 12(2) + 12 = 36. Final Basket: From the demand for food in (a), we know that F = 12/4 = 3, and C = 2. (Also, U = 3(2) + 3 = 9.) Decomposition Basket: Must be on initial indifference curve, with U = FC + F = 36 (Eq 5) Tangency condition satisfied with final price: MUF/MUC = PF / PC. (C + 1)/ F = 4/4 => C + 1 = F. (Eq 3) Eq 5 can be written as F(C + 1) = 36. Using Eq 3, (C + 1)2 = 36, and thus, C = 5. Also, by Eq 3, F = 6. So the decomposition basket is F = 6, C = 5. Income effect on F: Ffinal basket – Fdecomposition basket = 3 – 6 = -3. Substitution effect on F: F decomposition basket – Finitial basket = 6 – 12 = -6. c) PFF + PCC = 4(6) + 4(5) = 44. So she would need an additional income of 24 (plus her actual income of 20). The compensating variation associated with the increase in the price of food is -24.

Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Clothing 11 10 9 Initial U

8 7

Final U

6

Decomp Basket

5 4 Final BL 3

1 0

Initial Basket

Final Basket

2

Decomp BL Initial BL

0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 Food

Inc Effect = -3

Subst Effect = -6

5.22 Suppose the market for rental cars has two segments, business travelers and vacation travelers. The demand curve for rental cars by business travelers is Qb = 35 − 0.25P, where Qb is the quantity demanded by business travelers (in thousands of cars) when the rental price is P dollars per day. No business customers will rent cars if the price exceeds $140 per day. The demand curve for rental cars by vacation travelers is Qv = 120 − 1.5P, where Qv is the quantity demanded by vacation travelers (in thousands of cars) when the rental price is P dollars per day. No vacation customers will rent cars if the price exceeds $80 per day. a) Fill in the table to find the quantities demanded in the market at each price.

b) Graph the demand curves for each segment, and draw the market demand curve for rental cars. c) Describe the market demand curve algebraically. In other words, show how the quantity demanded in the market Qm depends on P. Make sure that your algebraic equation for the market demand is consistent with your answers to parts (a) and (b). Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

d) If the price of a rental car is $60, what is the consumer surplus in each market segment? a) Price ($/day) 100 90 80 70 60 50

Business (000 cars/Week) 10.0 12.5 15.0 17.5 20.0 22.5

Vacation (000 cars/Week) 15.0 30.0 45.0

Market Demand (000 cars/Week) 10.0 12.5 15.0 32.5 50.0 67.5

Price

b) 160 140 120 100 80 60 40 20 0

Vacation

Business

Market

-

50.0

100.0

150.0

200.0

1000 Cars/Week

c) For price greater than $80, vacation traveler’s demand will be zero. So above P  80 , market demand is Qb  35  0.25P . For price between $0 and $80, market demand is the sum of the vacation and business demand, Qm  Qb  Qv , or Qm  35  0.25 P  120  1.5 P Qm  155  1.75P Above a price of $140, no purchases will be made so market demand is zero. In summary, when P  140  0,  Qm   35  0.25 P, when 80  P  140  155  1.75P, when P  80  d)

Copyright © 2014 John Wiley & Sons, Inc.

Chapter 5 - 22

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Business Demand

Price

150

CS=0.5(20)(80) CS=800

100 50 0 -

10.0

20.0

30.0

40.0

1000 Cars/Week

Price

Vacation Demand 100 80 60 40 20 0

CS=0.5(30)(20) CS=300

-

20.0

40.0

60.0

80.0

100.0

120.0

140.0

1000 Cars/Week

5.23 There are two types of customers in a market for sheet metal. Let P represent the market price. The total quantity demanded by Type I consumers is Q1 = 100 - 2P, for 0< P < 50. The total quantity demanded by Type II consumers is Q2 = 40 - P, for 0< P < 40. Draw the total market demand on a clearly labeled graph. P 

DII 

    

50  40 

Market D  (heavy line)  DI 

20  40 

100 

Copyright © 2014 John Wiley & Sons, Inc.

  140 

   Q 

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

5.24 There are two consumers on the market: Jim and Donna. Jim’s utility function is U(x, y) = xy, with associated marginal utility functions MUx = y and MUy = x. Donna’s utility function is U(x,y) = x2y, with associated marginal utility functions MUx = 2xy and MUy = x2. Income of Jim is IJ = 100 and income of Donna is ID = 150. a) Find optimal baskets of Jim and Donna when price of y is Py = 1 and price of x is P. b) On separate graphs plot Jim’s and Donna’s demand schedule for x for all values of P. c) Compute and plot aggregate demand when Jim and Donna are the only consumers. d) Plot aggregate demand when there is one more consumer that has identical utility function and income as Donna. a) Jim’s optimal basket is a solution to equations MUx / MUy = P / Py and P x + Py y = IJ. Hence, we have 2xy / x2 = P and P x + y = 100 with solution x = 200 / (3P) and y = 100 / 3. Analogous system of equations for Donna is y / x = P and P x + y = 150 with solution x = 75 / P and y = 75. b)

Approximate shape of the demand curve for Jim and Donna is depicted below.

Px

x

c)

Aggregate demand is Dx(P) = 200 / (3P) + 75 / P = 445 / (3P).

d) When there is one more consumer that has preferences identical to Donna’s then her demand is also 75 / P and hence aggregate demand is Dx(P) = 200 / (3P) + 75 / P + 75 / P = 650 / (3P).

Copyright © 2014 John Wiley & Sons, Inc.

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Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Shape of the demand curve in this case is the same as in part b). 5.25 One million consumers like to rent movie videos in Pulmonia. Each has an identical demand curve for movies. The price of a rental is $P. At a given price, will the market demand be more elastic or less elastic than the demand curve for any individual. (Assume there are no network externalities.) The market demand and individual demand will have the same price elasticity given any particular price. Denote an individual’s demand curve by Qi(P). With 1,000,000 identical individuals the market demand curve will be Qm(P) = 1,000,000Qi(P). At a given price P, an individual’s demand curve will have elasticity  Qi , P   Qi P  P / Qi  . Since Qm(P) = 1,000,000Qi(P), it must also be true that Qm Qi  1,000,000 P P The elasticity for the market demand curve will be Q

m ,P



Qm P Qi P Qi P  1,000,000    Qi , P P Qm P 1,000,000Qi P Qi

In other words, with identical consumers the elasticity of the market demand curve will equal the elasticity of the individual demand curve at any price P. 5.26 Suppose that Bart and Homer are the only people in Springfield who drink 7-UP. Moreover their inverse demand curves for 7-UP are, respectively, P = 10 − 4QB and P = 25 − 2QH, and, of course, neither one can consume a negative amount. Write down the market demand curve for 7-UP in Springfield, as a function of all possible prices. Bart will only consume when the price is less than 10. Therefore his demand curve for 7-UP is

10  P , when P