Spatial Bioeconomics under Uncertainty (with Application) Christopher Costello* September, 2007 American Fisheries Society Annual Meeting San Francisco, CA with: D. Kaffine, S. Mitarai, S. Polasky, D. Siegel, J. Watson C. White, W. White * Bren School and Dept. Economics, UCSB.

Are marine reserves consistent with economic intuition? “Unless we somewhat artificially introduce an intrinsic value for biomass in the sanctuary, there would be no rationale for a marine sanctuary in a deterministic world with perfect management” -J. Conrad (1999)

Research questions Optimal spatial harvest under uncertainty? Optimal spatial harvest under uncertainty? Role of spatial connections? Role of spatial connections? Harvest closures ever optimal? How should they be designed? Harvest closures ever optimal? How should they be designed? Effects of stochasticity on spatial management? Effects of stochasticity on spatial management? How implement empirically? How implement empirically?

Flow, Fish, and Fishing Flow – how are resources connected across space? Flow – how are resources connected across space? Fish – spatial heterogeneity of biological growth Fish – spatial heterogeneity of biological growth Fishing – harvesting incentives across space, economic objectives Fishing – harvesting incentives across space, economic objectives

A motivating example (2 patches) Current tends to flow towards B: Current tends to flow towards B: State equation in A: X t+1 =(1-  )F(X t -H t ) State equation in A: X t+1 =(1-  )F(X t -H t ) If profit is linear in harvest, want F’-1=  in both patches If profit is linear in harvest, want F’-1=  in both patches If we close A: If we close A: What is X ss ? What is rate of return? What is X ss ? What is rate of return? Is this > or or <  ? A B  

x1Kx1K (1-  0 )F(x t ) F(x t ) x0Kx0K 45 o (1-  1 )F(x t ) xtxt x t+1 F’(x 0 K )-1<  F’(x 1 K )-1>  Dynamics in the closed patch (“A”) (low spillover) (high spillover) x*x* F’(x*)-1= 

Generalizing the model Economics: Economics: Heterogeneous harvest cost, stock-effect on MC Heterogeneous harvest cost, stock-effect on MC Constant price Constant price Biology Biology Sessile adults Sessile adults Larval drift Larval drift Variability & Uncertainty Variability & Uncertainty Production and survival Production and survival Where larvae drift Where larvae drift

Timing Adult population in a location Settlement and survival to adulthood Larval production Spawning population (Escapement) Harvest Dispersal “D ij ” (Note here that harvest is location-specific)

Problem setup Maximize E{NPV} of profits from harvest. Find optimal patch-specific harvest strategy: Maximize E{NPV} of profits from harvest. Find optimal patch-specific harvest strategy: Equation of motion: Equation of motion: Dynamic Programming Equation (vector notation): Dynamic Programming Equation (vector notation):

Solution procedure Discrete-time stochastic dynamic programming Discrete-time stochastic dynamic programming If an interior solution exists, special structure allows us to break this into a less-complicated two period problem. If an interior solution exists, special structure allows us to break this into a less-complicated two period problem. This makes finding analytical solutions tractable This makes finding analytical solutions tractable Numerical approaches (e.g. VFI) are intractable Numerical approaches (e.g. VFI) are intractable Corner solutions (reserves) difficult to analyze explicitly Corner solutions (reserves) difficult to analyze explicitly

Theoretical Results 1. With sufficient heterogeneity, reserves emerge as optimal solution 2. Design features: spatial siting, harvest outside 3. If interior solution, constant patch-specific escapement, differs by patch, protect “bioeconomic sources” 4. Stochasticity is sufficient, not necessary for reserves to be profit maximizing

Effects of Stochasticity 1. [Harvest] Higher variability causes increased harvest in open patch that contributes larvae to closed patch. 2. [Reserves] Sufficiently high variability always gives rise to optimal (temporary) closures, typically relegates permanent reserves. 3. [Profits] Increasing variability tends to increase expected profits (system variability increases variability in stock, payoff is convex in stock)

The F3 Model: Simulation to Optimization Dynamic, discrete-time, discrete-patch Dynamic, discrete-time, discrete-patch Requires: (a) connectivity matrix (dispersal kernels), (b) spatial production function, (c) spatial economics. Requires: (a) connectivity matrix (dispersal kernels), (b) spatial production function, (c) spatial economics. Delivers: Dynamics of stocks, harvest, profits, etc. by patch for any spatial management Delivers: Dynamics of stocks, harvest, profits, etc. by patch for any spatial management Plan: (1) Parameterize (2) Optimize spatial harvest, including reserves, (3) Analyze relative to alternative designs Plan: (1) Parameterize (2) Optimize spatial harvest, including reserves, (3) Analyze relative to alternative designs

An application to California’s South- Central Coast Initial test species: kelp bass Initial test species: kelp bass Adults relatively sedentary Adults relatively sedentary Larval dispersal via ocean currents Larval dispersal via ocean currents PLD=26-36 days PLD=26-36 days Oceanographic model of currents Oceanographic model of currents Settlement success and recruitment Settlement success and recruitment Beverton Holt, associated with kelp abundance in patch Beverton Holt, associated with kelp abundance in patch Constant price per unit harvest, stock-effect on harvest cost function Constant price per unit harvest, stock-effect on harvest cost function

Heterogeneous Productivity & Larval Survival Must look at all F3 components simultaneously: Flow, Fish, Fishing

Evaluating Spatial Harvest Profiles 1. Economic performance Discounted profits over in infinite (discounted) horizon Discounted profits over in infinite (discounted) horizon 2. Biological performance Overall system stock size in equilibrium Overall system stock size in equilibrium Compare: Compare: Optimal spatial management (max profit) Optimal spatial management (max profit) Current reserves in region Current reserves in region Randomly sited reserves (but same number) Randomly sited reserves (but same number) All with optimal management outside All with optimal management outside

Current vs. Optimal CurrentOptimal Optimally sited reserves actually increase profits. Some overlap with existing reserves, but important differences.

Current vs. Economically Optimal Profit Maximizing Current Reserves Current Reserves vs. Profit Maximizing Reserves About 14% difference in profits About 13% difference in stock

Relative to a null model… Profit and stock for 5000 simulated reserves with (roughly) equal total area Profit Maximizing (1 th percentile stock 100 th percentile profit) Current Reserves (90 th percentile stock 6 th percential profit) Could have increased profits and/or stocks at no cost

Recall Conrad… What if we add an intrinsic value of stock biomass? What if we add an intrinsic value of stock biomass? Multiple objectives: Multiple objectives: Infinite horizon discounted profit Infinite horizon discounted profit Stock size in equilibrium – constant value per fish Stock size in equilibrium – constant value per fish Max {  Stock + Profit}, for different weights, 

Weighted biological and economic objective Profit Maximizing Current Reserves Efficiency Frontier Close the Ocean Note: it makes no sense to design a network that falls inside the frontier.

Conclusion Under the F3 model with full stochasticity: Under the F3 model with full stochasticity: Completely characterized optimal spatial harvest (for interior solution) Completely characterized optimal spatial harvest (for interior solution) Closures “typically” emerge as an optimal solution, stochasticity sufficient not necessary Closures “typically” emerge as an optimal solution, stochasticity sufficient not necessary General insights, but little practical design guidance General insights, but little practical design guidance Implementing the optimized F3 model Implementing the optimized F3 model Spatial optimization for deterministic system, kelp bass SB Channel Spatial optimization for deterministic system, kelp bass SB Channel Reserves emerge (about 26% of total area), maximizes profits, does poorly for stock Reserves emerge (about 26% of total area), maximizes profits, does 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, multi-species interactions) Biological extensions (age or size structure, adult movement, multi-species interactions) Economic extensions (TURFs, concessions, spatial ITQ, coordination mechanisms) Economic extensions (TURFs, concessions, spatial ITQ, coordination mechanisms)