Add a Maximum Economic Yield (MEY) section to the MSY fishery lecture

[mmcky]

Follow-up to #763 (the new MSY lecture, lectures/msy_fishery.md).

The lecture stops at MSY and notes in its summary that MSY "ignores economic costs and prices." MEY is the natural next step and fits the intro level well — it's the same one-variable concave maximization, reusing the existing Gordon–Schaefer setup and parameters.

Proposal: add a short Maximum Economic Yield section (after Sustainable yield, before Randomness) covering:

• sustainable rent $\pi(e) = p,y^*(e) - c,e$, maximized at $e_{MEY}$
• the three reference points on one diagram: $e_{MEY} < e_{MSY} < e_{\text{open access}}$, where the open-access (zero-rent) point delivers Gordon's tragedy-of-the-commons result — currently gordon1954 is cited but this result isn't covered
• one line tying it back to the collapse analysis: since $x_{MEY} > x_{MSY} = K/2$, MEY also leaves a larger safety buffer (more profit and more robust)
• brief real-world case: Australia's Northern Prawn Fishery switching its target from MSY to MEY

Scope: keep it static (maximise sustainable rent). Defer dynamic/discounted MEY (Clark's modified golden rule) to the advanced series.

Refs: add grafton2007 (Grafton, Kompas & Hilborn, "Economics of Overexploitation Revisited," Science); clark1990 already in the bib.

[mmcky]

While scoping this, I checked whether the shock-driven collapse dynamics from #763 also occur under MEY. Short version: they don't, and the reason is itself a nice economics/policy point.

The collapse in the lecture is a property of the constant-quota policy pinned to MSY, not of "taking the yield" in general. MSY $= rK/4$ is the peak of the growth dome, so a fixed quota there merges the two steady states into a knife-edge tangent at $x=K/2$ with zero margin — any downward shock ratchets the stock to zero.

MEY sits strictly to the left of that peak, which removes the knife-edge. Using the lecture's parameters ($r=0.5$, $K=1000$, $q=0.01$) with $p=1$, $c=4$:

	effort $e$	stock $x^*$	sustainable catch
MEY	15	700	105
MSY	25	500 (=$K/2$)	125
open access	30	400	120

Because the MEY catch (105) is below the dome's peak (125), a constant quota at the MEY level has two distinct steady states — stable at $x_{MEY}=700$, unstable at $x_-=300$ — i.e. a 400-tonne basin of attraction vs zero at MSY.

Collapse probability within 100 years (4000 paths, same lognormal shocks as the lecture):

                       σ=0.15   σ=0.25   σ=0.35   σ=0.45
  MSY constant quota    67.7%    81.2%    89.3%    93.8%
  MEY constant quota     0.0%     0.0%     0.1%     2.6%
  (constant-effort policies never collapse at either target)

So even the literal "fixed quota every year" reading of MEY — the version that wrecks MSY — is robust right up to extreme noise.

The economics: $e_{MEY} = e_{MSY}(1 - c/pqK)$, so accounting for the cost of effort automatically pulls fishing below the MSY peak (here to 84% of MSY) — the same margin that msy_ex1 introduces by hand as a fraction $\alpha<1$, except the economic optimum supplies it for free. "More profit and more robust." One caveat worth stating: the buffer is proportional to the cost ratio — if effort were costless ($c\to0$) then $e_{MEY}\to e_{MSY}$ and MEY collapses back onto the knife-edge, so the robustness comes from costs, not from MEY being magic.

@jstac — do you think this is worth demonstrating in the MEY section, or does it risk over-loading an intro lecture? It's a one-paragraph addition (reusing collapse_fraction pointed at y_mey/x_mey) rather than a new dynamics section.

[jstac]

Thanks @mmcky !

I'm thinking of turning this into a sequence of lectures -- including a proper treatment of MEY and then leading into dynamic programming. This is very helpful.

I also want to play around with the biological dynamics -- I suspect that even some MEY type behaviors can lead to collapse, especially when the true state (biomass is not observable).

This will lead into a Bayesian discussion: we get noisy observations on the hidden state (stock/biomass) and have to try to back out that hidden state from the observations --- and then make appropriate decisions.

This injects another level of risk.

It will be fun to explore all of these ideas.

TL;DR: I think MEY will overload this lecture but the comments will be very useful for follow up lectures!

[mmcky]

Sounds amazing! I like where this is going -- what a great series. Thanks @jstac.

[jstac]

Thanks @mmcky ❤️

Public policy ultimately runs into the brick wall of politics, but it doesn't mean we shouldn't try!

So many problems are international now, so developing collaboration is crucial...