Harvesting stochastic, Harvesting stochastic, spatially connected spatially connected

fisheriesfisheries

Christopher Costello (UCSB)Christopher Costello (UCSB)Stephen Polasky (UMN)Stephen Polasky (UMN)

F3 All-Hands-OnF3 All-Hands-OnJuly 12, 2006July 12, 2006

Santa Barbara, CASanta Barbara, CA

Fisheries are in declineFisheries are in decline

Marine resources overexploited Marine resources overexploited 90% reduction in large predators 90% reduction in large predators

(Myers & Worm)(Myers & Worm) Long history of over-exploitation in Long history of over-exploitation in

near-shore (Jackson et al.)near-shore (Jackson et al.) ExploitationExploitation as the primary human as the primary human

disturbance of marine systemsdisturbance of marine systems Biological & Economic Biological & Economic

explanations, consequencesexplanations, consequences

Addressing over-Addressing over-exploitationexploitation

EconomistsEconomists: : Institutional failureInstitutional failure Solution is to regulate/privatize the commons Solution is to regulate/privatize the commons Example: individually transferable quotas (ITQs)Example: individually transferable quotas (ITQs)

BiologistsBiologists: : Over-exploitation (tragedy of commons, greed, or Over-exploitation (tragedy of commons, greed, or

myopia)myopia) Recent solution: Recent solution: “Marine Protected Areas”“Marine Protected Areas” Recommendations for 30% of oceans to be closedRecommendations for 30% of oceans to be closed Good for the fish, good for the fishermen Good for the fish, good for the fishermen

(Hastings & Botsford)(Hastings & Botsford)

Challenge to EconomistsChallenge to Economists

Biological sciences have Biological sciences have emphasizedemphasized Importance of spatial patternsImportance of spatial patterns Importance of variability and Importance of variability and

uncertaintyuncertainty Economics emphasizes incentives; Economics emphasizes incentives;

but much of the existing work is but much of the existing work is a-a-spatialspatial and and deterministicdeterministic (though (though there are notable exceptions)there are notable exceptions)

The worldThe world

Marine reserves in the Marine reserves in the worldworld

11% of land, 1% of ocean is set aside in reserve

Worldwide area in Worldwide area in reservesreserves

0

200000000

400000000

600000000

800000000

1000000000

1200000000

1400000000

1600000000

Year

Are

a o

f R

es

erv

es

(H

a)

Marine

Terrestrial

Reserves are being implemented at an increasing rate—Without guidance from economic analysis

2x size of Europe or Australia

20x size of Great Lakes

Rationale for reservesRationale for reserves

Two arguments are used for reservesTwo arguments are used for reserves1.1. Protect unexploited assemblages of Protect unexploited assemblages of

species – biodiversityspecies – biodiversity

2.2. Close an area, lose some harvest. But Close an area, lose some harvest. But production increases in closed area. production increases in closed area. Larvae “spill-over” into exploited area. Larvae “spill-over” into exploited area. Could this Could this increaseincrease overall harvestoverall harvest??

Second claim is largely Second claim is largely unsubstantiatedunsubstantiated. . We will evaluate this claim.We will evaluate this claim.

Inconsistent with economic Inconsistent with economic intuition?intuition?

““Unless we somewhat artificially Unless we somewhat artificially introduce an introduce an intrinsic value for intrinsic value for

biomass in the sanctuarybiomass in the sanctuary, there would , there would be no rationale for a marine sanctuary be no rationale for a marine sanctuary in a deterministic world with perfect in a deterministic world with perfect

management”management”

-J. Conrad (1999)-J. Conrad (1999)

Biologists as policy Biologists as policy makersmakers

Lack of data and formal analysis has not deterred biologists from giving policy advice:

“It is time to trust the insights of ecologists for once, press for the

establishment of marine reserves and place fisheries management and marine conservation on a sound basis at last.”

-Roberts (1997)

Economists, on biologistsEconomists, on biologists

““Owing to the lack of theoretical Owing to the lack of theoretical economic research, biologists have been economic research, biologists have been forced to forced to extend the scope of their own extend the scope of their own

thought into the economic spherethought into the economic sphere and in and in some cases have penetrated quite some cases have penetrated quite

deeply, despite the lack of the analytical deeply, despite the lack of the analytical tools of economic theory.”tools of economic theory.”

-- H. Scott Gordon (1954)-- H. Scott Gordon (1954)

A thought experiment…A thought experiment…

What would a fisherman do if he What would a fisherman do if he “owned” the fishery?“owned” the fishery? Secure “right” for exclusive accessSecure “right” for exclusive access Could harvest different amounts spatiallyCould harvest different amounts spatially Could account for dynamic and spatial Could account for dynamic and spatial

externalitiesexternalities Real-world institutions (e.g.):Real-world institutions (e.g.):

Harvest cooperativesHarvest cooperatives TURFsTURFs Area-based quotasArea-based quotas

Spatial bioeconomics under Spatial bioeconomics under uncertaintyuncertainty

Spatial issuesSpatial issues: : Most larvae disperse across space: 90% of Most larvae disperse across space: 90% of

near-shore marine harvested species have near-shore marine harvested species have sessile adults, larval dispersalsessile adults, larval dispersal

Economic payoffs may vary across space (e.g., Economic payoffs may vary across space (e.g., costs of harvest)costs of harvest)

Spatial externalities exist when an action in Spatial externalities exist when an action in one location affects other locationsone location affects other locations

StochasticityStochasticity:: Environmental variability affects life-history Environmental variability affects life-history

stages, state transitions, impact of policy, and stages, state transitions, impact of policy, and info about futureinfo about future

Dispersal itself may be variableDispersal itself may be variable

Significance for policySignificance for policy

Under a spatial/stochastic system, Under a spatial/stochastic system, even making predictions is difficult – even making predictions is difficult – much less designing optimal much less designing optimal spatial/temporal harvestspatial/temporal harvest

How should policy be designed to How should policy be designed to manage spatial/stochastic resources?manage spatial/stochastic resources?

Do harvest closures (MPAs) Do harvest closures (MPAs) emergeemerge as an economically optimal as an economically optimal management instrument?management instrument?

Research questionsResearch questions

Optimal spatial harvest under Optimal spatial harvest under uncertainty?uncertainty?

Role of spatial connections?Role of spatial connections? Permanent harvest closures ever Permanent harvest closures ever

optimal? How should they be designed?optimal? How should they be designed? Temporary harvest closures ever Temporary harvest closures ever

optimal? optimal? Optimal Optimal management outside a reserve?management outside a reserve?

Some of the relevant Some of the relevant literatureliterature

MPAs and fishery yield (MPAs and fishery yield (Hastings & BotsfordHastings & Botsford)) MPAs and open access or regulated open MPAs and open access or regulated open

access outside (access outside (Sanchirico & WilenSanchirico & Wilen)) Spatial harvesting, no reserves (Spatial harvesting, no reserves (Sanchirico & Sanchirico &

WilenWilen)) Optimal spatial harvesting in deterministic Optimal spatial harvesting in deterministic

environment (environment (Neubert, ClarkNeubert, Clark)) Game-theoretic spatial harvesting in Game-theoretic spatial harvesting in

deterministic environment (deterministic environment (Bjorndal et alBjorndal et al., ., MunroMunro, , Naito and PolaskyNaito and Polasky))

A-spatial optimal management under A-spatial optimal management under uncertainty (uncertainty (Reed, Costello et al.Reed, Costello et al.))

Flow, Fish, and FishingFlow, Fish, and Fishing

FlowFlow – how are resources connected – how are resources connected across space? [across space? [Resource ConnectivityResource Connectivity]]

FishFish – spatial heterogeneity of biological – spatial heterogeneity of biological growth [growth [Dynamic Externality and Spatial Dynamic Externality and Spatial HeterogeneityHeterogeneity]]

FishingFishing – harvesting incentives across – harvesting incentives across space, economic objectives, distributional space, economic objectives, distributional impacts over time [impacts over time [Economic OptimalityEconomic Optimality]]

““Patchy” dispersal vs. Patchy” dispersal vs. diffusiondiffusion

Simulation Result Diffusion Model

This relationship is highly variable – not a smooth dispersal kernelWe think of dispersal kernel as a probability distribution

A motivating example (2 A motivating example (2 patches)patches)

Current tends to flow towards B:Current tends to flow towards B:

State equation in A: State equation in A: XXt+1t+1=(1-=(1-)F(X)F(Xtt-H-Htt)) If profit is linear in harvest, want If profit is linear in harvest, want F’ F’

=1/=1/in both patchesin both patches If we close A:If we close A:

What is XWhat is Xssss? What is rate of return?? What is rate of return? Is this > or < 1/Is this > or < 1/??

A

B

x1K

(1-0)F(xt)

F(xt)

x0K

45o

(1-1)F(xt)

xt

xt+1

F’(x0K)<1/

F’(x1K)>1/

Dynamics in the closed patch (“A”)

(low spillover)

(high spillover)

x*

F’(x*)=1/

Generalizing the modelGeneralizing the model

Economics:Economics: Heterogeneous harvest cost, stock-effect Heterogeneous harvest cost, stock-effect

on MCon MC Constant priceConstant price

BiologyBiology Sessile adultsSessile adults Larval driftLarval drift

Variability & UncertaintyVariability & Uncertainty Production and survivalProduction and survival Where larvae driftWhere larvae drift

TimingTiming

Adult populationin a location

Settlement andsurvival to adulthood

Larval production

Spawning population(Escapement)

Harvest

Dispersal“Dij”

(Note here that harvestis location-specific)

Adding environmental Adding environmental variabilityvariability

Adult populationin a location

Settlement andsurvival to adulthood

Larval production

Spawning population(Escapement)

Harvest

zf

Random dispersal“Dij”

zS

z

(Note here that harvestis location-specific)

shock to adultsurvival

shocks to settlementand larval survival

shock to fecundity

The marginal economic The marginal economic conditionsconditions

p

c(s)

Biomass, s

The cost of catchingone more fish

The benefit of catchingone more fish

The net benefit of catchingone more fish

xe

Total profit from harvestingH fish (from initial population of x to an escapement of e)

H

MR, MC x

e

dsscexpex )()(),(

Stochastic dynamic Stochastic dynamic programprogram

ObjectiveObjective: Identify optimal : Identify optimal feedback control rulefeedback control rule (specifies harvest in each patch) to maximize (specifies harvest in each patch) to maximize expected present value from harvest over expected present value from harvest over TT periods periods

State vector: current stock in every patch (State vector: current stock in every patch (xx11, …, x, …, xII) ) Control vector: harvest (alternatively, escapement) Control vector: harvest (alternatively, escapement)

in every patch (in every patch (ee11, …, e, …, eII)) In each time period In each time period tt we have we have II state variables, and state variables, and II

control variables, control variables, t=1,….,Tt=1,….,T.. Typically optimal escapement will be a function of Typically optimal escapement will be a function of

the vector of current stocks: the vector of current stocks: eeitit* (x* (x1t1t, …, x, …, xItIt) ) for allfor all i=1,…,Ii=1,…,I

Problem setupProblem setup

Maximize E{NPV} of profits from Maximize E{NPV} of profits from harvest. Find optimal harvest harvest. Find optimal harvest strategy: strategy:

Equation of motion:Equation of motion:

Dynamic Programming Equation Dynamic Programming Equation (vector notation):(vector notation):

))(()(1

1,

I

jjijtj

fjti

Sititiitti DefzzezX

I

ittititi

ett XEVexxV

t 111 )(),(max)(

Solution procedureSolution procedure

Discrete-time stochastic dynamic Discrete-time stochastic dynamic programmingprogramming

If an interior solution exists, this If an interior solution exists, this problem is problem is state separable --state separable -- the the necessary conditions are necessary conditions are independent of the state vectorindependent of the state vector

This makes finding a solution This makes finding a solution tractabletractable

Interior solutionInterior solution

What is an What is an interior solutioninterior solution?? Optimal to have positive harvest in every Optimal to have positive harvest in every

patch in every time periodpatch in every time period Proposition 1Proposition 1: If an interior solution to : If an interior solution to

the DPE exists, optimal escapement will the DPE exists, optimal escapement will vary across space, but will be both time, vary across space, but will be both time, and state independentand state independent

Constant patch-specific escapement Constant patch-specific escapement levellevel

Corner solutions: harvest Corner solutions: harvest closures (MPAs)closures (MPAs)

Corner solution = harvest closureCorner solution = harvest closure Proposition 3Proposition 3: Patch i should be closed to : Patch i should be closed to

harvesting in period t if and only if xharvesting in period t if and only if xitit<e<eitit*, *, where ewhere eitit* satisfies equating marginal value * satisfies equating marginal value of current harvest with expected marginal of current harvest with expected marginal value of an additional unit of stockvalue of an additional unit of stock

If stock size in a patch is lower than a pre-If stock size in a patch is lower than a pre-determined escapement target, then it is determined escapement target, then it is optimal to close the patchoptimal to close the patch

Harvest closures (MPAs)Harvest closures (MPAs)

What bioeconomic conditions could lead to What bioeconomic conditions could lead to harvest closure? Three basic reasons:harvest closure? Three basic reasons:

1.1. Dispersal to a site is low relative to productivity of Dispersal to a site is low relative to productivity of the sitethe site

Ocean currents lead to poor dispersal to patch Ocean currents lead to poor dispersal to patch ii

2.2. High marginal harvest costsHigh marginal harvest costs

3.3. ““Bad” realizations of random variables lead to low Bad” realizations of random variables lead to low recruitment in a patchrecruitment in a patch

1&2 suggest permanent closure, 3 suggests 1&2 suggest permanent closure, 3 suggests temporary closuretemporary closure

Close patch to take advantage of high marginal Close patch to take advantage of high marginal productivity of site (relative to current returns)productivity of site (relative to current returns)

Optimal management Optimal management outside outside optimally designedoptimally designed

reservesreserves Suppose a patch (Suppose a patch (kk) is optimally ) is optimally

closed, what to do outside that patch?closed, what to do outside that patch?

Proposition 5: eProposition 5: eitit*(I*(Ikt+1kt+1=0)<e=0)<eitit*(I*(Ikt+1kt+1>0)>0)

Escape more fish outside (an optimally Escape more fish outside (an optimally designed) reserve to take advantage of designed) reserve to take advantage of high marginal returns in the reservehigh marginal returns in the reserve

Optimal management Optimal management outside outside sub-optimally sub-optimally

designeddesigned reserves reserves Suppose a patch is sub-optimally Suppose a patch is sub-optimally

closed to harvest. What to do outside closed to harvest. What to do outside that patch?that patch?

Proposition 6: Under A3 & A4, Proposition 6: Under A3 & A4, eeitit*(R*(Rkt+1kt+1=1)<e=1)<eitit*(R*(Rkt+1kt+1=0)=0)

Escape fewer fish outside (a sub-Escape fewer fish outside (a sub-optimally designed) reserve because of optimally designed) reserve because of low marginal returns in the reservelow marginal returns in the reserve

SummarySummary

1.1. Constant “patch-specific escapement” Constant “patch-specific escapement” (interior)(interior)

2.2. Reserves Reserves emergeemerge as an optimal as an optimal economiceconomic solutionsolution

3.3. Reserves can be optimal with & without Reserves can be optimal with & without stochastic environmentstochastic environment

4.4. Optimal harvest outside reservesOptimal harvest outside reserves If reserves optimal – harvest outside If reserves optimal – harvest outside decreasesdecreases If reserves sub-optimal – harvest outside If reserves sub-optimal – harvest outside increasesincreases

5.5. Shortcomings includeShortcomings include: known stock, constant : known stock, constant price, harvest cost, no fully optimal corner price, harvest cost, no fully optimal corner sol’n.sol’n.

Extensions to considerExtensions to consider

InstitutionsInstitutions How big should cooperatives be?How big should cooperatives be? How should TURFs be defined?How should TURFs be defined? How to solve the coordination problem?How to solve the coordination problem?

E.g. pay one guy not to fish through self-E.g. pay one guy not to fish through self-taxationtaxation

How hard is this framework to How hard is this framework to implement with real data? implement with real data? [tomorrow][tomorrow]