discount rates, how to make calculation

Here is a passage from Environmental Economics 8th edition, by Barry C Field, pg 118

"The logic is even more compelling if we consider a future cost. One of the reasons that environmentalists have looked askance at discounting is that it can have the effect of downgrading future damages that result from today’s economic activity. Suppose today’s generation is considering a course of action that has certain short-run benefits of \$10,000 per year for 50 years, but that, starting 50 years from now, will cost 1 million a year forever. This may not be too unlike the choice faced by current generations on nuclear power or on global warming. To people alive page, the present value of that perpetual stream of future cost discounted at 10 percent is only \$85,000. These costs may not weigh particularly heavily on decisions made by the current generation. The present value of the benefits (\$10,000 a year for 50 years at 10 percent, or \$99,148) exceeds the present value of the future costs. From the standpoint of today, therefore, this might look like a good choice, despite the perpetual cost burden placed on all future generations."

How does the author calculate \$85,000 and \$99,148?

As the author said it is calculated through discounting. Author just uses the standard time value of money rules for calculating present value.

Present value of 1 mil perpetuity that is starting 50 years from now will be:

$$PV= \frac{\frac{V}{r}}{(1+r)^t} =\frac{\frac{1 mil}{0.1}}{(1+0.10)^{50}} \approx 85185.51 \approx 85k$$

The value in the numerator is the present value of perpetuity at year 50 and then dividing it by the denominator further discounts it as a lump sum 50 years.

The present value of 10k benefit can be calculated through present value of annuity formula (more general version of the present value of perpetuity formula - you get the perpetuity formula when number of time periods goes to infinity):

$$PV= \frac{V}{r} \left(1-\frac{1}{(1+r)^t} \right) = \frac{10k}{0.1} \left(1-\frac{1}{(1+0.1)^{50}} \right) \approx 99148.14 \approx 99148$$

• I'm familiar with the standard time value of money formula that the author presented earlier in the text but I'm not familiar with the 'r' under the 'V'. I wonder why this is. Is anyone able to explain why 'V' is divided by 'r'? Commented Mar 27 at 19:31
• @KevinPreston $r$ is the standard letter for rate of interest or discount rate, sometimes people use $i$; $V$ is the value. As my answer explains $V/r$ is what you get when $t \to \infty$ in the annuity formula. Take a limit of the last formula as $t \to \infty$ and you are end up with $V/r$
– 1muflon1
Commented Mar 27 at 20:05