University of California, Berkeley
Economics 1
Spring 2012
Prof. DeLong
Problem Set 5 Solutions 1. Suppose we have students going to Railroad Monopoly University who spend their money on only two things all semester: vacations in Cabo San Lucas (V) and renting BMWs for the weekend (R). And suppose that their utility function is the Cobb-­‐Douglas function with θ =1/3, and suppose that a student named Jonah takes vacations in Cabo on three weekends and rents a BMW for the other 15 weekends of the semester. What, for that consumption pattern, is Jonah’s marginal rate of substitution between Cabo vacations and renting BMWs? That is, if he takes an additional vacation, by how many BMW rentals could he cut back his BMW renting and still be as happy, still be on the same indifference curve? 1 − θ x1
given Δx1 =
Δx θ x2 2
For all problems, let x1 = vacations (V) and x2 = rentals (R). Right now, Jonah is taking 3 vacations and renting 15 BMWs. Since θ = 1/3, he spends 1/3 of his income on vacations and 2/3 of his income on BMW rentals. If he wants increase vacations by 1 (ΔV = 1), what is the necessary ΔR? Rearranging the above equation, we get θ R
ΔR =
ΔV
1−θ V
1 / 3 15
=
(1) 2/3 3
= 2.5
Jonah needs to reduce his car rentals by 2.5 to enjoy 1 additional vacation. 2. Suppose that Channing is also a student at Railroad Monopoly University, with the same utility function as Jonah. But suppose that Channing takes vacations in Cabo on six weekends and rents a BMW for six weekends of the semester. What, for that consumption pattern, is Channing’s marginal rate of substitution between Cabo vacations and renting BMWs—that is, if he takes an additional vacation, by how many BMW rentals per average semester could he cut back his BMW-­‐renting and still be as happy, still be on the same indifference curve? Channing currently consumes V = 6, R = 6; to get ΔV = 1, what is the necessary ΔR? 1
θ R
ΔV
1−θ V
1/ 3 6
=
(1) 2/3 6
= 0.5
ΔR =
He needs to cut back by 0.5 rentals to enjoy additional 1 vacation. 3. Suppose that renting a BMW costs $50 a weekend and taking a vacation in Cabo costs $500, and that Jonah has $2250 to spend and Channing $3300. Is either Channing or Jonah making a mistake in choosing their consumption pattern? If only one is, which one is making a mistake? Why are they making a mistake? At the optimal point, the slope of the utility curve should be the same as the slope of the budget constraint, which is just the ratio of the prices: ΔV PR
=
ΔR PV
1 − θ V PR
=
θ R PV
We can rewrite this to show that the ratio of total expenditure on the two goods is the same as the ratio of the budget shares, (1-­‐θ) and θ: 1 − θ V PR
=
θ R PV
V × PV
θ
=
R × PR 1 − θ
Jonah has income of $2250, and consumes V = 3 and R = 15. Is he consuming optimally? V × PV
θ
= (?)
R × PR
1−θ
3 × 500
= (?)1 / 2 15 × 50
1500 / 750 > 0.5
2 > 0.5
Jonah is spending ⅔ of his income on vacations, but he should be spending only 1/3 according to his utility function. He is taking too many vacations and renting too few BMWs. 2
University of California, Berkeley
Economics 1
Spring 2012
Prof. DeLong
For Channing, he consumes 6 of each good with an income of $3300. V × PV
θ
= (?)
R × PR
1−θ
6 × 500
= (?)1 / 2 6 × 50
3000 / 300 > 0.5
10 > 0.5
Channing is consuming way too many vacations and should consume more car rentals. 4. Explain to either Channing or Jonah—whichever one you think is making a mistake, or both—
how they could make themselves happier (or at least more dissipated) if they changed their consumption pattern. In what direction do you think they should change their consumption pattern(s)? How far do you think they should change their consumption pattern(s)? (Or, if you think neither is making a mistake, explain why you think both are doing what they ought to do. As mentioned previously, both guys should consume fewer vacations and more car rentals. They should spend twice as much income on rentals as they do on vacations, but they are spending far more on vacations right now because a) they are taking a lot of vacations and b) vacations are very expensive relative to car rentals. 5. Brie, with only $1100 per semester to spend, has different tastes and preferences. Her utility function has θ=5/6. If Cabo vacations cost $500 and BMW rentals cost $50, is she happiest buying 0, 1, or 2 vacations and spending the rest of her money on BMW rentals? Explain why her optimal ratio of vacations to rentals is different than the optimal ratio for Channing and Jonah. V × 500 5 / 6
=
R × 50 1 / 6
V
(Brie’s optimal vacation: rental ratio) ×10 = 5
R
V / R = 0.5
Subject to her budget constraint, Brie prefers to rent twice as many cars as the number of vacations that she takes. With $1100, she can purchase 1 vacation and 2 car rentals, which costs $600, or she can exhaust her income and consume 2 vacations and 2 car rentals For Jonah and Channing, they want to spend a smaller share of their income on vacations than Brie (θ = 1/3 vs. θ = 5/6), so they seem to value vacations less than Brie. Moreover, vacations 3
are much more expensive than car rentals, so their optimal vacation to rentals ratio (V/R) is much lower than Brie’s: V × 500 1 / 3
=
R × 50 2 / 3
V
×10 = 1 / 2 (Jonah and Channing’s optimal vacation: rental ratio) R
V / R = 0.05
6. Suppose that there is a BMW shortage. BMWs now rent for not $50 a weekend but $500 a weekend. And suppose that Jonah, Channing, and Brie have $2500, $3500, and $1000 to spend, respectively. How should each of the three spend his or her money? Explain your reasoning V × PV
θ
V × PV
θ
=
=
R × PR 1 − θ
R × PR 1 − θ
V × 500
V × 500
= 1 / 2 (Jonah and Channing) = 5 (Brie) R × 500
R × 500
V / R = 0.5
V /R=5
Jonah and Channing should consume twice as many rentals as vacations. Given this, Jonah will spend his $2500 and Channing his $3500 at a ratio of R = 2V: PV × V + PR × R ≤ 2500
PV × V + PR × R ≤ 3500
500V + 500(2V ) ≤ 2500
500V + 500(2V ) ≤ 3500
(Jonah) (Channing) 1500V ≤ 2500
1500V ≤ 3500
V ≤ 1.67
V ≤ 2.33
Jonah will purchase 2 vacations and spend the remaining money on 3 rentals. Channing will purchase 2 vacations and 5 rentals. Brie will consume 5 times as many vacations as rentals: V = 5R  R = 0.2V PV × V + PR × R ≤ 1000
500V + 500(0.2V ) ≤ 1000
600V ≤ 1000
V ≤ 1.67
Brie would like to consume a lot more vacations than rentals, but cannot afford more than one of each or two of only one good. However, her utility is not separable across goods, so she must 4
University of California, Berkeley
Economics 1
Spring 2012
Prof. DeLong
consume a positive amount of both in order to have any utility. Thus, Brie will consume 1 vacation and 1 rental. 7. Suppose Phil and Chris notice that neither Channing nor Jonah actually likes riding around in BMWs. What they like, instead, is impressing each other by renting more BMWs than their co-­‐
star—and they feel unhappy when their co-­‐star rents more BMWs than they do. That is, the utility function for Jonah and Channing are actually: 𝑢𝑗=𝑉𝑗𝜃𝑅𝑗𝑅𝑐1−𝜃 𝑢𝑐=𝑉𝑐𝜃𝑅𝑐𝑅𝑗1−𝜃 Phil and Chris calculate how many vacations and BMW rentals, if BMW rentals cost $50 and Cabo vacations cost $500, Channing and Jonah should spend their money on to collectively make them the happiest. What do they conclude? Explain your reasoning. (Hint: suppose Phil and Chris decide to calculate the geometric mean of Channing’s and Jonah’s utility, and then to try to make that product as large as possible...) (
Using the hint, the geometric mean of u j , uc = VjVc
)
θ /2
So, Phil and Chris decide that instead of trying to outdo one another with BMWs, Jonah an Channing should maximize their expenditures on vacations. 8. Suppose that Phil and Chris are right, that you are in charge of Railroad Monopoly PDC, and that you try to make both Channing and Jonah happier by imposing a tax on BMW rentals. How high a tax do you think you should impose? Explain your reasoning. From #7, the combined utility of Channing and Jonah depends solely on expenditure on Vacations. Thus, in order to keep both guys happy, we need to impose a high tax rate on car rentals, so that they are only inclined to spend on vacations. (thereby increasing their happiness) 5