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Rather than culling males at random, however, the optimal economic decision would

be to collect ivory from individuals that have died naturally or harvest only senes-

cent males (Basson et al. 1991). Elephant tusks increase in size exponentially with

age and large tusks are worth more, gram for gram, than small tusks, because large

tusks are much more valuable to carvers. As a result, the optimal economic solution

should reinforce conservation needs.

There are two quite distinct phases to a cropping operation: first, the population must

be reduced below its unharvested density (capital reduction), and then it must be

harvested at precisely the rate it seeks to bounce back (sustained-yield harvesting).

Biologists tend not to think too much about the capital-reduction phase because they

look forward to the prospect of a yield sustainable into the indefinite future.

If you were offered $1000 now as against $1000 in 10 years’ time you would take

the money now. However, if you were offered $400 now as against $1000 in 10 years’

time, the decision is no longer clear cut. Against money in the hand you are offered

a guarantee of sure future benefit, but the monetary value of that future benefit is

unclear. How much is $1000 in 10 years actually worth? A simple answer is that it

is worth a present sum which, when prudently invested, yields $1000 ten years hence.

If we assume that capital expands at about 10% per year, then $1000 in 10 years is

worth $385 now, or even less if the currency is inflating. With this knowledge the

answer to $400 now or $1000 in 10 years’ time is clear. Take the $400 now; it is

worth more. By the same reasoning a game animal harvested in 10 years’ time is worth

nothing like an animal harvested now. All future earnings must be discounted by the

time it takes to get the money, and the economics of the harvesting operation can

be dictated by the ratio of present to future earnings.

Discounting can be represented fairly simply, by rearranging the terms of the expon-

ential growth model. Instead of growing exponentially, however, the present value

of future harvests (PV) declines exponentially at the discount rate δ:

PV(t) =profit e−δt

What this means, of course, is that future profits are not valued as highly as current

profits. The higher the discount rate, the less the future is valued. A concrete exam-

ple may prove illuminating. A mature Bolivian mahogany tree currently brings in around

$396, for the lumber it can supply (Gullison 1998), yet it takes roughly a century

for such a tree to mature. At the 17% inflation rate in Bolivia in the early 1990s

(δ=0.17), the present value of a mahogany tree worth $396 a century from now is

around 1 penny, almost 10 times more expensive than required for replanting. Small

wonder that replanting is a low priority for most logging firms!

The present value of all future harvests can be calculated by integration:

If the per unit price p=0.75, cost per unit effort =0.5, rmax=1, and K=100, then

for a discount rate of 5% the total present value of all future harvests would be $246

at the optimal effort level (E=25). Change in the discount rate to 10% (still very

PV = 
 d

0

∞

et−δtΠ

352 Chapter 19

19.9 Game cropping and the discount rate