The Limits To Growth - Exponential Reserve Index

Exponential Reserve Index

One key idea within the The Limits to Growth is the notion that if the rate of resource use is increasing, the amount of reserves cannot be calculated by simply taking the current known reserves and dividing by the current yearly usage, as is typically done to obtain a static index. For example, in 1972, the amount of chromium reserves was 775 million metric tons, of which 1.85 million metric tons were mined annually (see exponential growth). The static index is, but the rate of chromium consumption was growing at annually (Limits to Growth, pp 54–71). If instead of assuming a constant rate of usage, the assumption of a constant rate of growth of annually is made, the resource will instead last

(note that the book rounded off numbers).

In general, the formula for calculating the amount of time left for a resource with constant consumption growth is :

where:

y = years left;

r = 0.026, the continuous compounding growth rate (2.6%).

s = R/C or static reserve.

R = reserve;

C = (annual) consumption.

The authors list a number of similar exponential indices comparing current reserves to current reserves multiplied by a factor of five:

		Years
Resource	Consumption growth rate, annual	Static index	Exponential index	5 times reserves exponential index
Chromium	2.6%	420	95	154
Gold	4.1%	11	9	29
Iron	1.8%	240	93	173
Petroleum	3.9%	31	20	50

The static reserve numbers assume that the usage is constant, and the exponential reserve assumes that the growth rate is constant.

The exponential index has been interpreted as a prediction of the number of years until the world would "run out" of various resources, both by environmentalist groups calling for greater conservation and restrictions on use, and by skeptics criticizing the index when supplies failed to run out. What The Limits to Growth actually has is the above table, which has the current reserves (that is no new sources of oil are found) for oil running out in 1992 assuming constant exponential growth. In Limits to Growth: The Thirty Year Update there are several pages explaining that new resources are found over time and that the current reserves therefore change but that ultimately resources are finite. (Earlier editions did explain this as well, but not in as much detail.) The standard model includes a resource base of double that of what they have calculated, but the book includes model runs where the assumed resources are infinite, but those model runs still result in overshoot and collapse from other factors.