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Optimization over time

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Optimization over time

A tricky question,Today versus tomorrow.Use present value.

- Partly because of inflation, a general increase in prices over time.
- But also because of preferences for sooner rather than later.
- In this lecture, assume no inflation.

- Assume that banks or other financial institutions are available to trade money today for money tomorrow at interest rate r through loans and savings accounts.
- This assumption is key because it allows us to push money back and forth over time. (Money is no longer like an ice cream cone that you have to eat right away.)

- Put $100 in the bank at 5% interest and in one year you’ll have $100(1.05) = $105, in two years you’ll have $100(1.05)2 = $110.25, and in n years you’ll have $100(1.05)n.
- The future value of $x invested in the bank for n years at interest rate r is FV=x(1+r)n.
- Example: The FV of $100 in 14 years at 5% interest is 100(1+0.05)14 = $197.99.
- Rule of 70/72: money doubles in ≈72/5 years.

- The present value of receiving a lump sum payment of $x at the end of n years at interest rate r (e.g., r = 0.05 for 5%) is:
- More generally, present value tells us how much we need to put in the bank today to finance one or more payments in the future.

- Example: The present value of receiving $200 at the end of 14 years with r = 0.05 is:
- The rule of 70/72 says that money in the bank at r % interest doubles in ≈ 70/r or 72/r years.
- Equivalently, every ≈ 70/r or 72/r years you go into the future cuts your present value in half.

- An annuity is a constant payment every year (or every month, etc.) for a fixed number of years, e.g., a home mortgage or lottery payout.
- To calculate the present value of a stream of payments you can always just break it into a series of lump sum payments… but sometimes (as with annuities) there’s an easier way.

- The present value of an annuity paying $x at the end of each year for n years at interest rate r (e.g., r = 0.05 for 5%) is:
- Example: $100 at the end of each year for 3 years has a present value of $272.32 if r = .05.

- Example: $100 at the end of each year for 3 years has a present value of $272.32 if r = 0.05.
- Put $272.32 in the bank today at 5% interest.
- After year 1 you’ll have (1.05)(272.32)=$285.94.
- After taking out 1st $100 you’ll have $185.94.
- After year 2 you’ll have (1.05)(185.94)=$195.24.
- After taking out 2nd $100 you’ll have $95.24.
- After year 3 you’ll have (1.05)(95.24)=$100.00.
- After taking out 3rd $100 you’ll have $0 left.

- The present value of an annuity paying $x at the end of each year for n years at interest rate r (e.g., r = 0.05 for 5%) is:
- Put this amount in the bank today at interest rate r and then make annual payments of $x and you’ll have a zero balance after n years.

- Proof: For an annuity paying $x at the end of each year for n years at interest rate r we want
- Multiply both sides by (1+r):
- Now rewrite, putting the 2nd equation first:

- Subtract; note that almost all terms cancel!

- A perpetuity is a perpetual annuity, i.e., a constant payment every year (or every month, etc.) forever.
- The present value of a perpetuity paying $x at the end of each year forever at interest rate r (e.g., r = 0.05 for 5%) is:
- Example: A $100 perpetuity at 5% interest has a present value of 100/0.05 = $2000.

- Proof #1: Similar to annuity proof:

- Proof #2: Take the limit of the annuity formula as the number of years n goes to infinity:
- Proof #3 (and intuition!): Living off the interest: Put $2000 in the bank at 5% interest and you can take out $100 every year forever!

- Example: $100 at the end of each year forever has a present value of $2000 if r = 0.05.
- Put $2000 in the bank today at 5% interest.
- After year 1 you’ll have (2000)(1.05) = $2100.
- “Live off the interest” by taking out $100, leaving you with $2000.
- Rinse and repeat.

- Example: $100 at the end of each year forever has a present value of $2000 if r = 0.05.
- Q: Is PV > $2000 or < $2000 if r = 0.10 instead?
- A: PV =$1000, which is < $2000. Intuition?
- Intuition: You have to put less in the bank today in order to finance future payments.
- In general, higher interest rates make the future less important.

- Example: $100 at the end of each year forever has a present value of $2000 if r = 0.05.
- Q: What is PV of $100 annuity for 100 years?
- A: Use annuity formula to get PV = $1984.79.
- Difference is $15.21. (Not much!) Intuition?
- Q: Put $15.21 in the bank at 5% interest for 100 years and what do you get?
- A: Use FV formula to get FV ≈ $2000. (!)

- Maximizing present value of catch is not the same thing as maximum sustainable yield!

- Q: Since oil doesn’t grow like fish or trees, why would resource owners hold on to it instead of selling it and putting the proceeds in the bank? (Same question for minerals/stock/etc.)
- A: The price is expected to go up (or the costs of extraction are expected to go down).
- So (returning to the Big Question) optimizing individuals and companies have strong incentives to think hard about Peak Oil.