Clark's ``Harvest to extinction'' theorem

One of the key strategies in sustainable development is the use of ``extractive reserves'', i.e., managed lands that preserve considerable biodiversity (or ecosystem service) value while allowing extraction of valuable resources for sale on the open market. So the value of extractive reserves is often expressed in terms of the dollar value of goods sold from the reserves.² Although extractive reserves are typically thought of in a terrestrial environment - rubber tappers and Brazil nut harvesters are the canonical example -, it's easier to see the paradox I'm about to describe if we focus on a fishery resource.³

Let's suppose that we're harvesting cod in the North Atlantic, and let's suppose that we can treat the particular stock we're harvesting as a single population that grows according to the standard logistic equation of population growth

$$\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right) \quad ,$$

where $r$ is the intrinsic rate of increase and $K$ is the carrying capacity. It's not too difficult to show that the maximum sustained yield, i.e., the maximum number of fish that can be removed every year withing depleting the stock is

$$MSY = r\left(\frac{K}{42}\right) \quad ,$$

where $r$ is the intrinsic rate of increase and $K$ is the carrying capacity (see Appendix A for details).

You know the old saying ``A bird in the hand is worth two in the bush''? Well, there's an analog here. ``Money in my pocket now is worth more than money in my pocket next year.'' That's a phenomenon that economists refer to as ``the time value of money.'' It arises, in part, because we can use money we have now to make more money - if we invest it wisely. Suppose I have $100 now, and I invest it for 10 years at 5% interest. How much money will I have in 10 years?

$$100e^{(0.05)10} = 164.87 \quad .$$

So suppose I gave you the choice of taking $100 from me now or waiting ten years and taking $150 from me then, knowing that you could earn 5% on the $100 for the next years. Which would you choose?⁴

Either you're concerned for my financial well-being⁵ or you chose to take the $100 from me now, because it's worth more than $150 ten years from now - assuming a 5% discount rate. Yes, I just changed the terminology on you. When thinking about events in the future we ``discount'' them at some rate, i.e., reduce their value and the percentage rate at which we discount the future is called the discount rate. We can also reverse the calculation and figure out how much $150 ten years from now is worth now:

$$\begin{eqnarray*} xe^{(0.05)10} &=& 150 \\ x &=& 150e^{-(0.05)10} \\ &=& 90.98 \quad . \end{eqnarray*}$$

So the net present value of $150 is $90.98, assuming a 5% discount rate. Rather than calculating how much $100 would be worth in 10 years, you can calculate how much $150 in ten years is worth to you now. The formula is

$$NPV = FVe^{-rt} \quad ,$$

where $NPV$ is the net present value, $FV$ is the future value, $\rho$ is the discount rate, and $t$ is the time.

If you have a constant stream of income, at a rate of $d$ dollars per year, and that stream extends indefinitely into the future, then then net present value of all of that future income is⁶

$$\begin{eqnarray*} NPV &=& \int_0^\infty de^{-\rho t} dt \\ &=& d \left[ -\frac... ...ft(-\frac{1}{\rho}\right)\right] \\ &=& \frac{d}{\rho} \quad , \end{eqnarray*}$$

i.e., the annual value of the income divided by the discount rate.

Let's apply this result to get the net present value of our maximum sustained yield, assuming that each fish bring $f$ dollars of profit when sold at market:

$$NPV(MSY) = \frac{fr\left(\frac{K}{4}\right)}{\rho} \quad .$$

On the other hand, suppose we caught every cod in this population right now and sold it with the same margin of profit per fish and that there are $N$ fish in the population. Then we get

$$NPV(extinction) = fN \quad.$$

For managing the fishery sustainably to make sense, in terms of the dollar value recovered from sale of fish, we require

$$\begin{eqnarray*} NPV(MSY) &>& NPV(extinction) \\ \frac{fr\left(\frac{K}{4}\right)}{\rho} &>& fN \\ \frac{rK}{4\rho} &>& N \quad . \end{eqnarray*}$$

If we imagine that the population is currently at the level needed for its maximum sustained yield, $K/2$, this becomes

$$\frac{r}{2\rho} \quad > 1$$

But remember, $r$ is the intrinsic growth rate of our population. Typical values for long-lived animal species are on the order of 2-3%. Let's be generous and say that it's 3%. $\rho$ is the financial discount rate. When making investment decisions, businesses typically use rates of at least 5% and often higher. That means that the left-hand side of the inequality is less than 1, i.e, $NPV(MSY) < NPV(extinct)$!

Our income will be higher if we harvest the cod to extinction now rather than harvesting them sustainably into the indefinite future. The same principle holds for any resource. If the intrinsic rate of increase associated with that resource is less than the financial discount rate, it makes more sense to harvest it to extinction immediately than to manage it sustainably.