Question

Homework 6: Present Value

We sought out a soothsayer, who did sayeth some sooth. She stirred her cauldron and foresaw that terrible things would happen to Evanston. 100 years from this very day, the crimes of John Evans will come back to punish the residents of this town, causing $300 million dollars of damages. However, we can avert this terrible fate at the low, low cost of just $6 million dollars today (paid to descendants of those Evans wronged). That’s right, for just $6 million dollars now, we can avert $300 million dollars of damage to future Evanston residents! You can’t beat this deal!

1. What is the most we would be willing to pay to avert this future harm if our discount rate is 1.4% per year?

2. What is the most we would be willing to pay to avert this future harm if our discount rate is 4% per year?

3. What is the most we would be willing to pay to avert this future harm if our discount rate is 10% per year?

Suppose that we could buy a bit of Evanston lakefront for $130 million and build a lovely public beach that would deliver social benefits of $5 million dollars per year forever, starting one year from now.

4. What is the most we would be willing to pay to build this park if our discount rate is 1.4% per year?

5. What is the most we would be willing to pay to build this park if our discount rate is 4% per year?

6. What is the most we would be willing to pay to build this park if our discount rate is 10% per year?

7. Think of the basic Pigouvian Externality situation.

Private Marginal Benefit = 600 - 2*Q

Private Marginal Cost = 30 + Q

Marginal Damage = 90

Private market equilibrium quantity = Q_P = (600-30)/(2+1) = 190

What is the optimal Pigouvian tax and socially optimal quantity?

8. Same setup as in the previous problem, except that the Marginal Damage doesn’t occur now, but will actually happen in 10 years. Let the discount rate be 3%.

What is the optimal Pigouvian tax and socially optimal quantity today?

9. Same setup as in the previous problem, except we just had an election, and so now the discount rate is 7%. What is the optimal Pigouvian tax now? What is the optimal social quantity today?

Answer 1

Question 1

Amount of future harm (F) = $300 million

Time period (n) = 100 years

Interest rate (i) = 1.4% or 0.014

We have to calculate the present value of furure harm (F) at the given interest rate to determine the maximum amount we will be willing to pay to avert this future harm.

Calculate the Present Value -

PV = F/(1+i)ⁿ

PV = $300 million/(1+0.014)¹⁰⁰

PV = $300 million/(1.014)¹⁰⁰

PV = $300 million/4.0160

PV = $74.70 million

The most we would be willing to pay to avert this future harm if our discount rate is 1.4% per year is $74.70 million.

Question 2

Amount of future harm (F) = $300 million

Time period (n) = 100 years

Interest rate (i) = 4% or 0.04

We have to calculate the present value of furure harm (F) at the given interest rate to determine the maximum amount we will be willing to pay to avert this future harm.

Calculate the Present Value -

PV = F/(1+i)ⁿ

PV = $300 million/(1+0.04)¹⁰⁰

PV = $300 million/(1.04)¹⁰⁰

PV = $300 million/50.5049

PV = $5.94 million

The most we would be willing to pay to avert this future harm if our discount rate is 4% per year is $5.94 million.