The Value We Place on the Earth

We can find an estimate that economics sets on the value of Earth we leave behind in the following way.
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 of
this 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  
  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  
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
\[S_{FUTURE} = \frac{A(1/(1+r/100))^{75}}{1-1/(1+r/100)}\]
\[S_{FUTURE} \simeq 0.026 £A\]
  which seems a very low value to put on the Earth, or the future.

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