Economics 1: Spring 2012: Problem Set 5:
J. Bradford DeLong: U.C. Berkeley: Due at first section after March 12, 2012 lecture
Headnote: Remember that the utility—that is, the happiness—of the consumer is higher when the function:
U = (x1)θ(x2)(1-θ)
is higher. For this function the marginal rate of substitution is—that is, the utility stays the same as you move from (x1, x2) to (x1+Δx1, x2-Δx1) for small changes Δx1 and Δx2 if:
Δx1 = ((1-θ)/θ)(x1/x2)Δx2
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?
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?
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?
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.)
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.
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.
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 is actually: Uj = (Vj)θ(Rj/Rc)(1-θ), where “j” as a subscript means that this applies to Jonah, and “c” as a subscript means this applies to Channing. And Uc = (Vcc)θ(Rc/Rj)(1-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...)
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.