Suppose we can make use of the Earth's resources at the rate of
\[£ A\]
per year. If we assume long term interest rates are \[r %\]
then the present value ofthis years resources is
\[£ A\]
next years resources is
\[£ \frac{ A}{1-1/{(1+ \frac{r}{100}})}\]
the following years resources is
\[£ \frac{ A}{1-1/(1+ \frac{r}{100})^2}\]
and so on. This is a geometric series with first term
\[£A\]
and common ratio \[\frac{1}{1+ \frac{r}{100}}\]
.If we add up the net present value of 75 years use of the Earths resources we obtain the expression
\[S_{75}= \frac{A(1- (1/(1+r/100))^{75})}{1-1/(1+r/100)}\]
.The net present value of all the Earths resources used to eternity is
\[S=\frac{A}{1-1/(1+r/100)}\]
.IN a sense the value we face on the Earth or the value we place on the future is the difference between these two. It is
\[S_{FUTURE} = \frac{A(1/(1+r/100))^{75}}{1-1/(1+r/100)}\]
.If
\[r=5%\]
then \[S_{FUTURE} \simeq 0.026 £A\]
which seems a very low value to put on the Earth, or the future.