A college town in the West consists of three types of individuals: undergraduate

Question

A college town in the West consists of three types of individuals: undergraduate students, graduate students, and professors. There are 100 of each type in the town of 300 residents. The three types differ in their preferences for the number of exercise machines at the public athletic facilities provided by the university. Undergraduates, who are more active, get the highest benefits from exercise machines. They each get benefits according to: Bu = 10M - .5 M2, where M is the number of exercise machines. Graduate students are too busy to exercise daily so they get less total benefits from exercise machines: Bg = 8M - M2. Professors do not actually work out and simply get a fixed benefit of 10 from the mere existence of exercise machines—just from knowing that other people are working out. Each machine costs $600.

a. How many exercise machines should the university buy? (Show how you get your answer.)

b. The University would like to collect payments from individuals in order to buy the socially optimal number of exercise machines (from part a)). Describe the Lindahl or benefit-tax scheme the university could implement if it had perfect information about preferences. How much would it charge each type per machine?

c. Suppose instead that the university has very little information about preferences and decides just to charge all individuals the same price per exercise machine that it buys. Assume that the university charges just enough to cover costs. The college town residents vote to decide on the number of exercise machines that should be bought. Which number of machines will always have the majority of votes when compared to any other alternative (in pairwise voting)? Is this outcome efficient? Why of why not?

Explanation / Answer

a. University should buy he number of machines which maximizes the social benefit.

So, University should buy till

Marginal cost or price of Machine = Marginal Social Benefit

MBu = dBu/dM = 10 - M

MBg = 8 - 2M

MBp = 0

Marginal Social benefit = 100*MBu + 100*MBg + MBp

= 100(10 - M) + 100(8 - 2M) + 100*0

= 1000 - 100M + 800 - 200M

= 1800 - 300M

Now, equating MC = MSB

600 = 1800 - 300M

M = 1200/300 = 4

b. Amount charged by each group should be equal to their Benefits

Price charged to undergraduates =  Bu = 10M - .5 M2 = 10*4 - 0.5*(4)^2 = 40 - 8 = 32

  Price charged to graduates =  Bg = 8M - M2 = 8*4 - (4)^2 = 32 - 16 = 16

Price charged to Professors = Bp = 10

c.

Number of Machines Undergraduates prefer ,

  MC = 100*MBg

600 = 800 - 100M

   M = 400/100 = 4

Number of Machines graduates prefer ,

  MC = 100*MBg

600 = 800 - 200M

   M = 200/200 = 1

Amount paid by everyone = 600/300 = 2

This outcome is n't efficient as group of graduates and professors can be made better off by decreasing the M, as they are paying $2 wheras getting benefit of 0 and 1 .hence it's n't efficient.

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