Discounting, Public Goods, Etc.

  1. Calculate the future value five years from today of a $1,000 savings deposit (made today) yielding a nominal 5% annual rate of return, if the compounding period is (a) annual, (b) monthly, (c) weekly, (d) daily, (e) hourly and (f) each minute.  Calculate the future value if interest is compounded instantaneously:  FV = e(rT)PV.

  2. In 1626 the Canarsee Indians sold the island of Manhattan to Peter Minuit's New Amsterdam colony for trade goods valued at 60 guilders, or approximately $24. How much would their $24 be worth today if the Canarsee had invested this money at 6.5% interest, compounded annually?   At 7%?   At 7.5%?
    (Historical background: the Canarsee were from Brooklyn, and Manhattan wasn't actually theirs; it belonged to the Wappani.   If there had been a Brooklyn Bridge, the Canarsee would have been happy to sell that too.)

  3. What is the mathematical relationship between interest rate r and the number of years t required to double your money?  Use the discrete discounting formula $FV/$PV = (1+r)t = 2, to calculate the doubling times t for values of r ranging from 0.02 to 0.12. For each interest rate, multiply r times doubling time t; you should get approximately the same doubling constant D = rt every time; what is it?
    Use the same procedure to calculate the equivalent doubling constant when discounting is continuous:
    $FV/$PV = ert = 2.

  4. Assets A, B and C have current prices of $1,000, $2,000 and $3,000 respectively.  In 3 years A will be worth $1,150.  In 6 years B will be worth $2,600.  In 9 years C will be worth $4,500.  Which asset yields the highest implicit rate of return (compounded annually)?

  5. Corporate bonds A, B and C each pay $10,000 at the end of their terms, which are 5, 10 and 15 years respectively (there are no interim interest payments, aka “coupons”).  Their Moody’s bond ratings indicate 2%, 1% and 3% annual risks of default respectively.  If you assume a defaulted bond is a total loss, you are otherwise risk-neutral.  How much would you bid for each bond so it yields a 5% expected rate of return (compounded annually)?

  6. Suppose you are doing a benefit-cost analysis of three alternative proposals for a vacant site.  Proposal A (petting zoo) would yield net benefits of $30, $35, $40 and $15 in the first four years, then zero net benefits after that.  Proposal B (condos) would yield an up-front net benefit of $110, and zero net benefits in subsequent years.  Proposal C (nature preserve) would yield a net benefit stream of $5 per year forever. 
    Calculate the net present value (NPV) of each proposal when r = 0.04, r = 0.06 and r = 0.08.  Which proposal has the highest NPV under each discount rate?  Determine the threshold value of r (to 4 decimal places) at which A and B yield the same NPV.  Determine the threshold value of r at which A and C yield the same NPV.

  7. Suppose you own three oil fields, A, B and C, each containing one million barrels, with constant extraction costs of $20, $25 and $30 per barrel respectively.  You have the capacity to extract up to 200,000 barrels per year from each field.  The price of oil is currently $50/barrel, and is assumed to increase 3% annually over the long run.
    1. For each oil field, over a 20-year time horizon, calculate the percent increases in marginal rent for each year over the previous year.
    2. Assume your objective is a minimum 5% annual rate of return on each oil field, so ideally you would have each field about halfway depleted when its annual rate of rent growth has declined to 5%.  Over what 5-year periods would you extract and sell off the oil in each field?
    3. Suppose you didn’t own these fields, but were considering buying them.  What is the most you would be willing to pay today for each field, for a target annual return of at least 5% on each?

  8. A good is "non-rival" if multiple people can use or enjoy it, and the degree to which one person uses or enjoys it does not affect another user's enjoyment of  it.  A good is "non-excludable" if you can't prevent people from enjoying it.  Give two examples (not the ones below!) of each, with brief explanations:
    1. a "club" good that is excludable but non-rival
    2. a "common-property" good that is non-excludable but rival
    3. a pure "public" good that is both non-excludable and non-rival

  9. A town has 62 voters with diverse environmental attitudes, and they are trying to decide the quantity of land Q to purchase for a nature preserve to protect the endangered Pickled Strumpet.  Protection of this species is a pure public good. There are 2 Activists, each with marginal WTPA=16-Q; 4 Birdwatchers, each with marginal WTPB=12-Q; 8 Concerned citizens, each with marginal WTPC=8-Q; 16 Distracted citizens, each with marginal WTPD = 4-Q; and 32 Exasperated citizens with zero WTP.  The unit cost of Q is $14.
    1. With pure free-ridership and no cooperation, what is the maximum amount of Q that would be purchased?
    2. If the town held a referendum for a public purchase, with the purchase price to be split equally among all 62 voters, what is the maximum purchase of Q that would be approved via majority vote?
    3. If the two Activists cooperate and purchase Q based on their combined WTP, how much will they purchase?
    4. Graph the aggregate WTP schedule for the entire town (assuming individual WTP's are positive or zero, not negative). Identify the level of Q at which aggregate WTP = $14. Which categories of voter have positive WTP for this quantity?
    5. Knowing the WTP's of these people, how would you split the cost of providing this optimal level of Q between the Activists, Birdwatchers, Concerned citizens and Distracted citizens?
    6. Calculate the aggregate potential consumer surplus from this socially optimal solution.

  10. Here's a problem taken from J.A. Paulos' Beyond Numeracy (Alfred A. Knopf, NY, 1991), which Paulos borrowed from W.F. Lucas, who formalized it from a problem originally discussed by Condorcet.  Suppose there are 5 candidates for a public office, A, B, C, D and E.  There are 55 voters with the following preferences:
    18 voters prefer A > D > E > C > B
    12 voters prefer B > E > D > C > A
    10 voters prefer C > B > E > D > A
     9 voters prefer D > C > E > B > A
     4 voters prefer E > B > D > C > A
     2 voters prefer E > C > D > B > A
    Since no candidate is the first choice of a majority of voters, you would have to choose some voting rule to determine a winner:
    1. If the winner is the candidate with the most first-place votes, who is the plurality winner?
    2. If you compare each candidate pairwise versus each other candidate, who is the pairwise winner?
    3. If you hold a runoff between the two candidates getting the most first-place votes, who is the top-two runoff winner?
    4. If the candidate with the fewest first-place votes is eliminated, and there is a runoff among the remaining four, and the runoff candidate with the fewest first-place votes is eliminated, and there is another runoff among the remaining three, and the candidate with the fewest first-place votes in that runoff is eliminated, who wins the final two-candidate runoff?
    5. If you assign a Borda count to preferences, so that most-preferred gets 5 points, second most-preferred gets 4 points, ... and least-preferred gets 1 point, who is the winner on points?

  11. In March 1964, a young woman named Kitty Genovese was knifed to death by a stranger over a period of about 10 minutes outside her apartment building in Queens, NY. At least a dozen neighbors were aware the assault was happening, but nobody helped her or even called the police until it was too late. Explain the economic rationale for the neighbors' inaction.