The Marine Stewardship Council, which issues fishery certification programs, said that if the fishing continued at this rate, mackerel would start to fall below sustainable levels by 2012.
Let's imagine governments, along with the alphabet soup of NGOs the Time article mentioned (SFM, WWF, MSC, FIFVO, CFP, and PEG), are capable of optimally exploiting a common-pool resource. We can further imagine there are two possible stochastic states of the world: high price for mackerel and low price for mackerel. Additionally, there is a stock of fish that can be consumed, and after consumption, the remaining stock multiplies.
In this case, we might have a dynamic programming problem summarized by a value function with fish stock S, consumption C, price-state $$\theta$$, and price $$P(C,\theta)$$ and growth rate r:
With the law of motion for fish stock:
In this case, with appropriate parameters and functional forms, solving the value function above we may have a consumption pattern that has a non-singlet ergodic set for fish stock that is above zero. That is, in "low" price moments, we "under-fish." In "high" price moments, we "over-fish." Below, we graph out next period's stock against this period's stock under the high price and low price states (click to enlarge).
As one can see, the ergodic set stretches from where the 45 degree line (stock today is the same as stock tomorrow) intersects with a low shock (a stock cannot possibly go below this point, even with an infinite series of low shocks), to where the 45 degree line intersects with a high shock (a stock cannot possibly go above this point, even with an infinite series of high shocks).
The important stylized fact to understand from this is that it's possible not to have a "steady state" of fish but instead have an ergodic set of possible levels of fish, a similar notion that optimally exploits the resource, dipping into it in "bad" times and saving in "good."
I like this framework for thinking about sustainable fishing and it is worth noting that some biologists have thought about it this way (except without using the word ergodic). For example, some biologists have argued that its stupid to try to maintain a steady stock of a fish (say mackerel) because on top of the price shocks there are environmental shocks that affect the stock. Their definition of what is "sustainable" is to say how many shocks are required to get the stock back to its previous level (in other words how large is the ergodic set). You could draw an ergodic set where if you got the "worst" shock, it would take 10 "best" shocks to get you back to the original stock. If this example is the case then you might want to try to limit the worst shock because it is costly to have a low stock of mackerel for potentially years while the good shocks build the supply up (is the argument of the biologists).
ReplyDeleteAll that was just to illustrate the usefulness of your framework.