Spatial Bioeconomics Spatial Bioeconomics under Uncertainty (with under Uncertainty (with

Application)Application)Christopher Costello*Christopher Costello*

September, 2007September, 2007American Fisheries Society Annual MeetingAmerican Fisheries Society Annual Meeting

San Francisco, CASan Francisco, CA

with: with: D. Kaffine, S. Mitarai, S. Polasky, D. Siegel, J. D. Kaffine, S. Mitarai, S. Polasky, D. Siegel, J.

WatsonWatsonC. White, W. White C. White, W. White

* Bren School and Dept. Economics, UCSB. Costello@bren.ucsb.edu

Are marine reserves Are marine reserves consistent with economic consistent 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)

Research questionsResearch questions

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

Role of spatial connections?Role of spatial connections? Harvest closures ever optimal? How Harvest closures ever optimal? How

should they be designed?should they be designed? Effects of stochasticity on spatial Effects of stochasticity on spatial

management?management? How implement empirically?How implement empirically?

Flow, Fish, and FishingFlow, Fish, and Fishing

FlowFlow – how are – how are resources resources connected across connected across space?space?

FishFish – spatial – spatial heterogeneity of heterogeneity of biological growthbiological growth

FishingFishing – harvesting – harvesting incentives across incentives across space, economic space, economic objectivesobjectives

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 < Is this > or < ??

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)

Problem setupProblem setup

Maximize E{NPV} of profits from Maximize E{NPV} of profits from harvest. Find optimal patch-specific harvest. Find optimal patch-specific harvest strategy: harvest 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, special If an interior solution exists, special

structure allows us to break this into a less-structure allows us to break this into a less-complicated two period problem.complicated two period problem.

This makes finding analytical solutions This makes finding analytical solutions tractabletractable

Numerical approaches (e.g. VFI) are Numerical approaches (e.g. VFI) are intractableintractable

Corner solutions (reserves) difficult to Corner solutions (reserves) difficult to analyze explicitlyanalyze explicitly

Theoretical ResultsTheoretical Results

1.1. With sufficient heterogeneity, reserves With sufficient heterogeneity, reserves emerge as optimal solutionemerge as optimal solution

2.2. Design features: spatial siting, harvest Design features: spatial siting, harvest outsideoutside

3.3. If interior solution, constant patch-If interior solution, constant patch-specific escapement, differs by patch, specific escapement, differs by patch, protect “bioeconomic sources”protect “bioeconomic sources”

4.4. Stochasticity is sufficient, not necessary Stochasticity is sufficient, not necessary for reserves to be profit maximizingfor reserves to be profit maximizing

Effects of StochasticityEffects of Stochasticity

1.1. [Harvest] Higher variability causes increased [Harvest] Higher variability causes increased harvest in open patch that contributes larvae harvest in open patch that contributes larvae to closed patch.to closed patch.

2.2. [Reserves] Sufficiently high variability always [Reserves] Sufficiently high variability always gives rise to optimal (temporary) closures, gives rise to optimal (temporary) closures, typically relegates permanent reserves. typically relegates permanent reserves.

3.3. [Profits] Increasing variability tends to [Profits] Increasing variability tends to increase expected profits (system variability increase expected profits (system variability increases variability in stock, payoff is convex increases variability in stock, payoff is convex in stock)in stock)

The F3 Model: Simulation The F3 Model: Simulation to Optimizationto Optimization

Dynamic, discrete-time, discrete-patchDynamic, discrete-time, discrete-patch Requires: (a) connectivity matrix Requires: (a) connectivity matrix

(dispersal kernels), (b) spatial (dispersal kernels), (b) spatial production function, (c) spatial production function, (c) spatial economics. economics.

Delivers: Dynamics of stocks, harvest, Delivers: Dynamics of stocks, harvest, profits, etc. by patch for profits, etc. by patch for anyany spatial spatial managementmanagement

Plan: (1) Parameterize (2) Optimize Plan: (1) Parameterize (2) Optimize spatial harvest, including reserves, (3) spatial harvest, including reserves, (3) Analyze relative to alternative designsAnalyze relative to alternative designs

An application to An application to California’s South-Central California’s South-Central

CoastCoast Initial test species: kelp bassInitial test species: kelp bass Adults relatively sedentaryAdults relatively sedentary Larval dispersal via ocean currentsLarval dispersal via ocean currents

PLD=26-36 daysPLD=26-36 days Oceanographic model of currentsOceanographic model of currents

Settlement success and recruitmentSettlement success and recruitment Beverton Holt, associated with kelp Beverton Holt, associated with kelp

abundance in patchabundance in patch Constant price per unit harvest, stock-Constant price per unit harvest, stock-

effect on harvest cost functioneffect on harvest cost function

Heterogeneous Productivity Heterogeneous Productivity & Larval Survival& Larval Survival

-121.5 -121 -120.5 -120 -119.5 -119 -118.5

33.5

34

34.5

35

35.5

36

Un-harvested biomass by patch

0.5

1

1.5

2

2.5

3

x 106

Must look at all F3 components simultaneously:Flow, Fish, Fishing

Evaluating Spatial Harvest Evaluating Spatial Harvest ProfilesProfiles

1.1. Economic performanceEconomic performance Discounted profits over in infinite Discounted profits over in infinite

(discounted) horizon(discounted) horizon

2.2. Biological performanceBiological performance Overall system stock size in equilibriumOverall system stock size in equilibrium

Compare:Compare: Optimal spatial management (max profit)Optimal spatial management (max profit) Current reserves in regionCurrent reserves in region Randomly sited reserves (but same number)Randomly sited reserves (but same number)

All with optimal management outsideAll with optimal management outside

Current vs. OptimalCurrent vs. Optimal

-121.5 -121 -120.5 -120 -119.5 -119 -118.5

33.5

34

34.5

35

35.5

36

-121.5 -121 -120.5 -120 -119.5 -119 -118.5

33.5

34

34.5

35

35.5

36

Base Case PLDCurrent Optimal

Optimally sited reserves actually increase profits. Some overlap with existing reserves, but important differences.

1.7 1.75 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2

x 107

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55x 10

8

Stock

Pro

fitBase Case PLD

Current vs. Economically Current vs. Economically OptimalOptimal

Profit Maximizing

Current Reserves

Current Reserves vs. Profit Maximizing ReservesAbout 14% difference in profitsAbout 13% difference in stock

Relative to a null model…Relative to a null model…

1.7 1.75 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2

x 107

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55x 10

8

Stock

Pro

fitBase Case PLD

Profit and stock for 5000 simulated reserves with (roughly) equal total area

Profit Maximizing(1th percentile stock

100th percentile profit)

Current Reserves(90th percentile stock6th percential profit)

Could have increasedprofits and/or stocksat no cost

Recall Conrad…Recall Conrad…

What if we add an intrinsic value of What if we add an intrinsic value of stock biomass?stock biomass?

Multiple objectives:Multiple objectives: Infinite horizon discounted profitInfinite horizon discounted profit Stock size in equilibrium – constant Stock size in equilibrium – constant

value per fishvalue per fish

Max {Max {Stock + Profit}, for different Stock + Profit}, for different weights, weights,

Weighted biological and Weighted biological and economic objectiveeconomic objective

1.5 2 2.5 3 3.5 4 4.5 5

x 107

0

2

4

6

8

10

12

14

16x 10

7

Stock

Pro

fit

Profit Maximizing

Current Reserves

Efficiency Frontier

Close the OceanNote: it makes nosense to design a network that fallsinside the frontier.

ConclusionConclusion Under the F3 model with full stochasticity:Under the F3 model with full stochasticity:

Completely characterized optimal spatial harvest (for interior Completely characterized optimal spatial harvest (for interior solution)solution)

Closures “typically” emerge as an optimal solution, Closures “typically” emerge as an optimal solution, stochasticity sufficient not necessarystochasticity sufficient not necessary

General insights, but little practical design guidanceGeneral insights, but little practical design guidance Implementing the optimized F3 modelImplementing the optimized F3 model

Spatial optimization for deterministic system, kelp bass SB Spatial optimization for deterministic system, kelp bass SB ChannelChannel

Reserves emerge (about 26% of total area), maximizes profits, Reserves emerge (about 26% of total area), maximizes profits, does poorly for stockdoes poorly for stock

Joint objective - trace out efficiency frontier.Joint objective - trace out efficiency frontier. Next steps for optimization framework:Next steps for optimization framework:

Biological extensions (age or size structure, adult movement, Biological extensions (age or size structure, adult movement, multi-species interactions)multi-species interactions)

Economic extensions (TURFs, concessions, spatial ITQ, Economic extensions (TURFs, concessions, spatial ITQ, coordination mechanisms)coordination mechanisms)