ch05solution manual

Besanko & Braeutigam – Microeconomics, 5th editionSolutions Manual

Chapter 5The 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


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.



(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.

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.



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.

a)

and must be greater than zero. In addition, assume income increases, i.e., . If the good is inferior, then . Thus, the first term and the second term .

Multiplying these two terms together implies . Inferior goods have a negative income elasticity of demand.

b) If income elasticity of demand is negative then

.

Since and must be greater than zero, for to be negative, we must have

.

This can only happen if either a) and or b) and . 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.



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

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



Substituting into the budget line, , gives

b) Yes, clothing is a normal good. Holding constant, if increases 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?



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 . Substituting the first condition into the second we get which implies that the

demand curve for beer is given by,

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 decreases demand for beer more than a one dollar increase in .

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 and

the tangency condition is , which means that . The demand for x does not

depend on the level of income.

b) From the budget constraint, the demand curve for y is, .

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)



b)

c)



d)

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



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?

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?

a) The budget constraint is and the tangency condition is .

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:

. The tangency condition now implies that that is,

Substituting this into the budget constraint we find that y=10. Using the condition , we find that x = 40. Finally, substituting the values of x and y back into the budget


Shelter

Price Consumption Curve

Food


constraint, we can see that , 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 MUX/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.

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).


Py

y

1

4


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 . For the budget constraint to hold, it must be true that

.

(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

Since and , we can write this as

Or

But if both goods are luxury goods, then and so that the previous equation implies

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.



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)

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:

Substituting into the budget line, , yields

b) If , then

Since we must have , we must have



So the consumer would only purchase for prices less than 10.

c)

Given , the slope of the budget line is –1. At the corner point optimum, the slope of the indifference curve is

Because the indifference curve has a flatter slope than the budget line, the consumer would like to substitute more for , but has no more to give up at the corner point.

d) . If the consumer were to purchase any , since the “bang for

the buck” for is less than the “bang for the buck” for , the consumer would reduce total utility by increasing above zero.

e)



As shown in part a), the demand for depends only on and . Therefore, the location of the

demand curve does not depend on .

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.)

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


P2

P1


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.

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.


$8

$1

$3

D(P)

Area of ABE triangleCS when P = $3 is 25

D(P) = 16 – 2P

A

CD

BE

Area of ACD triangleCS when P = $1 is 49

P


b)

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).

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, , and the budget constraint, , Lou’s

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,

and would give him as much utility as before, i.e. . This gives

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: 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.

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.


Px

xxxx10

$1

$2

5

A B

C

D


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, , and

also such that This implies that 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.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 = FTangency: 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 .

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.

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.


Final Basket Decomp BL

Initial U

Final U

Final BL

Initial BL

20191817161514131211109876543210

11

10

9

8

7

6

5

4

3

2

1

0

Food

Clothing

Food

Clothing

Subst Effect = -6 Inc Effect= -3

Initial Basket

Decomp Basket


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).d) If the price of a rental car is $60, what is the consumer surplus in each market segment?

a)

Price ($/day)Business (000

cars/Week)Vacation (000

cars/Week)Market Demand (000 cars/Week)

100 10.0 - 10.0 90 12.5 - 12.5 80 15.0 - 15.0 70 17.5 15.0 32.5 60 20.0 30.0 50.0 50 22.5 45.0 67.5

b)



c) For price greater than $80, vacation traveler’s demand will be zero. So above , market demand is .For price between $0 and $80, market demand is the sum of the vacation and business demand,

, or

Above a price of $140, no purchases will be made so market demand is zero. In summary,

d)



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.

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.



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).

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 . Since Qm(P) = 1,000,000Qi(P), it must also be true that

The elasticity for the market demand curve will be


Px

x


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

when P<10 and zero otherwise. Homer will only consume if the price is less than

25 so his demand curve is when P < 25 and zero otherwise.

Therefore the market demand curve for 7-UP as a function of all possible values of price is:

5.27 Joe’s income consumption curve for tea is a vertical line on an optimal choice diagram, with tea on the horizontal axis and other goods on the vertical axis.a) Show that Joe’s demand curve for tea must be downward sloping.b) When the price of tea drops from $9 to $8 per pound, the change in Joe’s consumer surplus (i.e., the change in the area under the demand curve) is $30 per month. Would you expect the compensating variation and the equivalent variation resulting from the price decrease to be near $30? Explain.

a) If the income consumption curve is vertical the utility function has no income effect. This will occur, for example, with a quasi-linear utility function. This utility function will have the same marginal rate of substitution for any particular value of tea regardless of the level of total utility. If the price of tea falls, flattening the budget line, the consumer will reach a new optimum where the marginal rate of substitution is equal to the slope of the new budget line. Since the budget line has flattened, this cannot occur at the previous optimum amount of tea. The substitution effect implies that this new optimum level of tea will be greater than the previous level. Thus, when the price of tea falls, the quantity of tea demanded increases, implying a downward sloping demand curve. This can be seen in the following figure.



b) Yes, the values will be exactly $30. When the income consumption curve is vertical, the consumer’s utility function has no income effect. As stated in the text, when there is no income effect, compensating and equivalent variation will be identical and these will also equal the change in consumer surplus measured as the change in the area under the demand curve.

5.28 Consider the optimal choice of labor and leisure discussed in the text. Suppose a consumer works the first 8 hours of the day at a wage rate of $10 per hour, but receives an overtime wage rate of $20 for additional time worked.a) On an optimal choice diagram, draw the budget constraint. (Hint: It is not a straight line.)b) Draw a set of indifference curves that would make it optimal for him to work 4 hours of overtime each day.

a)

Because the wage rate changes for any hours worked over eight (leisure less than sixteen) the budget line has a kink at sixteen hours of leisure.



b)

With this set of indifference curves, the consumer reaches an optimum at 12 hours of leisure and 12 hours of labor, or $160 of income.

5.29 Terry’s utility function over leisure (L) and other goods (Y ) is U(L, Y ) = Y + LY. The associated marginal utilities are MUY = 1 + L and MUL = Y. He purchases other goods at a price of $1, out of the income he earns from working. Show that, no matter what Terry’s wage rate, the optimal number of hours of leisure that he consumes is always the same. What is the number of hours he would like to have for leisure?

If Terry’s wage rate is w, then the income he earns from working is (24 – L)w. Since PY = 1, the number of units of other goods he purchases is Y = (24 – L)w. Now at an optimal bundle, Terry’s must equal the price ratio w/PY = w. Therefore, the

tangency condition tells us that . The two conditions imply . This

means that the optimal amount of leisure is L = 11.5. You can see that this does not depend on the wage rate.

5.30 Consider Noah’s preferences for leisure (L) and other goods (Y ), U(L, Y) = √L + √Y. The associated marginal utilities are MUL = 1/(2√L) and MUY = 1/(2√Y). Suppose that PY = $1. Is Noah’s supply of labor backward bending?

If Noah’s wage rate is w, then the income he earns from working is (24 – L)w. Since PY = 1, the number of units of other goods he purchases is Y = (24 – L)w. Also, the tangency condition gives

us . Combining the two conditions, , or . Clearly, the amount

of leisure that Noah consumes decreases with an increase in the wage rate, and this is true no matter what the wage rate is. Since the amount of labor that Noah supplies equals (24 – L), we



see that his supply of labor always increases with an increase in the wage rate. So, his labor supply curve is always positively sloped – that is, it is not backward bending.

5.31 Raymond consumes leisure (L hours per day) and other goods (Y units per day), with preferences described by The associated marginal utilities are

The price of other goods is 1 euro per unit. The wage rate is

w Euros per hour.a) Show how the number of units of leisure Raymond chooses depends on the wage rate?b) How does Raymond’s daily income depend on the wage rate? c) Does Raymond work more when the wage rate rises?

a) For this utility function, it turns out that the amount of leisure can be determined from the

tangency condition alone. The tangency condition for an optimum is , or

. Thus w2 = 1/L, or L = 1/w2.

b) When Raymond consumes L units of leisure, he works (24 – L) hours, and receives an income of w(24 – L) Euros per day. His expenditure on other goods is Y Euros per day. His budget constraint will have income equal to expenditures, or w(24 – L) = Y. In (a) we learned from the tangency condition that L = 1/w2; substituting this into the budget equation reveals that w(24 – [1/w2]) = Y, which can be rewritten as Y = 24w – (1/w).

c) We can answer this in two ways. First, from part (a) we see that Raymond consumes less leisure as the wage rate rises. Thus he works more as the wage rate rises. Alternatively, Raymond works (24 – L) hours per day, i.e., (24 – [1/w2]) hours per day; this increases as w rises.

5.32 Julie buys food and other goods. She has an income of 400 per month. The price of food is initially $1.00 per unit. It then rises to $1.20 per unit. The prices of other goods do not change. To help Julie out, her mother offers to send her a check each month to supplement her income. Julie tells her mother, “Thanks, Mom. If you would send me a check for $50 per month, I would be exactly as happy paying $1.20 per unit as I would have been paying $1.00 per unit and not receiving the $50 from you.” Which of the following statements is true? Explain. The increased price of food has:a) an income effect of + $50 per month.b) an income effect of – $50 per month.c) a compensating variation of +$50 per monthd) a compensating variation of –$50 per monthe) an equivalent variation of +$50 per monthf) an equivalent variation of –$50 per month



The answer is (d). Let’s call Julie’s initial basket A; this is the basket she chooses when she faces a price of food of $1.00 per unit and has a monthly income of $400. Julie’s statement to her mother indicates that she is indifferent between her initial basket A and a different basket (call it basket B) which she would purchase if she had to pay $1.20 per unit of food, but received an extra monthly income of $50. (Basket B would be the decomposition basket if we were analyzing income and substitution effects associated with the increase in the price of food.) So her compensating variation is - $50 per month.

5.33 Gina lives in Chicago and very much enjoys traveling by air to see her mother in Italy. On the graph below, x denotes her number of round trips to Italy each year. The composite good y measures her annual consumption of other goods; the price of the composite good is py, which is constant in this problem. Several indifference curves from her preference map are drawn below, with levels of utility U1 < U2 < U3 < U4 < U5. If she spends all her income on the composite good, she can purchase y* units, as shown in the graph below. When the initial price of air travel is 1000, she could purchase as many as 18 round trips if she spends all her income on air travel to Italy.a) Make a copy of the graph below, and use it to determine the income and substitution effects on the number of round trips Gina makes as the price of a round trip increases from $1000 to $3000. Clearly label these effects on the graph.b) Using the graph, estimate the numerical size of the compensating variation associated with the price increase? You may refer to the graph to explain your answer.c) Will the consumer surplus measured using Gina’s demand for air travel to Italy provide an exact measure of the monetary value she associates with the price increase? In a sentence explain why or why not.


y

x0 1862 4 8 10 12 14 16

U1

U2

U3

U4

U5

y*


a)

Income Substitution. Effect Effect

With the initial budget line (labeled “Initial” on the graph), Gina’s income is $18,000 (she could make a maximum of 18 trips, each costing $1000). She chooses initial basket A, making 6 round trips. With the final budget line (labeled “Final” on the graph), she could use her $18,000 to purchase only 6 trips (the horizontal intercept of the final budget line). She chooses basket C, making 2 round trips.The decomposition budget line (labeled “Decomposition”) is parallel to the final budget line, but tangent to the initial indifference curve (U4) at basket B (with 4 trips).The income effect is XC – XB = (2 – 4) trips = - 2 trips.The substitution effect is XB – XA = (4 – 6) trips = -2 trips.

b) The compensating variation is the additional amount of money we would have to give Gina so that she can be as well off as she is initially (U4), but making purchases at the final price, in which case she would choose basket B. The decomposition budget line has a horizontal intercept of 9 trips, which represents an income of $27,000 at the final price of $3,000 per trip. Thus, we would have to compensate Gina by giving her and extra $9,000 (above her initial income of $18,000) to keep her as well off as she was initially. Because the price is rising, the compensating variation is negative. So the compensating variation is -$9,000.

c) No. From the graph we can see that the income effect is nonzero (for example, baskets B and C are associated with different numbers of round trips). So consumer surplus will not provide an exact measure of the monetary value of the price increase for Gina.


Decomposition BLFinal

BLInitial BL

C

B

A

y

x01862 4 8 10 12 14 16

U1

U2

U3

U4

U5

y*

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