Spatial management of renewable resources under

uncertainty

Preliminary results on the economics of coupled Flow, Fish and Fishing

Christopher Costello* and Steve Polasky

October 22, 2004

Specific research questions

1. How to optimally manage a fishery subject to spatial heterogeneity in economics and biology?

2. How does optimal spatial management change under uncertainty and variability?

More research questions

3. Are permanent reserves ever optimal? Under what conditions? How should they be designed?

4. Are temporary reserves ever optimal? Under what conditions?

5. If a reserve is implemented, what is the optimal spatial management outside the reserve?

(a) If reserve is optimally placed?

(b) If reserve is arbitrarily placed?

More research questions

6. What is the economic value of an optimally designed reserve? What is the economic cost of an arbitrarily placed reserve?

7. What information is required to implement the recommended management? How should management proceed with less information?

Timing of production & harvest

Adult populationin a location

Settlement andsurvival to adulthood

Larval production

Spawning population(Escapement)

Harvest

zf

Random dispersal“Kij”

zS

zm

(Note here that harvestis location-specific)

shock to adultsurvival

shocks to settlementand larval survival

shock to fecundity

Uncertainty and variability

Adult populationin a location

Settlement andsurvival to adulthood

Larval production

Spawning population(Escapement)

Harvest

zf

Random dispersal“Kij”

zS

zm

(Note here that harvestis location-specific)

shock to adultsurvival

shocks to settlementand larval survival

shock to fecundity

Conceptual approach

• Instead of asking: What are the economic implications of implementing a reserve?

• We ask: What is the optimal spatial management of this resource over time?– Patches without harvest are “reserves” (permanent vs.

temporary)

– Determine optimal management outside any given reserve

– Derive “cost” of reserve.

General assumptions

• Finite set of discrete “patches”– Adult population in each patch is known

• Harvest is perfectly implemented• 4 random variables each period

– Random shocks independent of time, distributions known, may be correlated across space.

• Fishery manager seeks to: choose patch-specific harvest each period to maximize expected discounted profit over T year planning horizon.

Economic assumptions

• Price exogenously given (may be random)

• Cost of harvesting in patch i depends on:– How much is harvested and– The population in that patch (stock effect)

• But MC of harvest depends only on stock size (in that patch) at time of harvest.

The marginal conditions

p

c(x)

Population, x

The cost of catchingone more fish

The benefit of catchingone more fish

The net benefit of catchingone more fish

AY

Total profit from harvestingH fish (from initial population of A to an escapement of Y)

H

MR, MC A

Y

dxxcYApYA )()(),(

Uncertainty & Non-linearities

• Idea is to derive optimal escapement as a function of adult population in every patch: Yi*(A1,A2,…,AI).

• How Y* is affected by uncertainty depends on non-linearities in the problem– If objective is linear in uncertain parameters,

uncertainty has no effect on optimal sol’n.– If non-linear, shape will determine whether

more uncertainty leads to higher or lower Y*

What are the non-linearities?

• Biology– Larval production = f(escapement)– Adult survival = M(escapement)– Larval survival to adulthood = S(successful

settlement)

• Economics– Harvest cost = c(harvest, adult population)

• TODAY: focus on deterministic environment

Problem setup

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

• Equation of motion:

• Dynamic Programming Equation:

))(()(1

1,

I

jjijtj

fjti

Sititi

mitti KyfzSzymzA

I

ittititi

ytt AEVAyAV

t 111 )(),(max)(

Solution procedure

• Discrete-time stochastic dynamic programming.– Start at period T (end of planning horizon)– Work backwards analytically

• State variables: A1t,A2t,…,Ait

• Control variables: Y1t,Y2t,…,Yit

• Each control is a function of the state vector.• From this, can find

– Value of the resource (when used optimally)– Optimal spatial harvest policy as function of the state.

Solution

• Can we get any traction analytically on this problem?

• Turns out to have a simple solution:

• “Constant Patch-Specific Escapement”– There is a Y* for each patch. Every year, harvest

down to Y*.

Answers to the research questions

1. [optimal management] Constant patch-specific escapement.

• If interior everywhere then can disregard larval transport – extra production in highly productive patches, extra harvest where not highly productive.

• If some closures, may be more difficult

Answers to the research questions

2. [under uncertainty and variability] Same general approach, different y* level.

• Under reasonable conditions, spikiness doesn’t matter.

• If high variability, might affect answer because may want closure in future

Answers to the research questions

3. [permanent reserves] Yes. Since base escapement on productivity, close areas with adult pops that are low relative to productivity.

• Suggests closure of high productivity areas that have low adult populations.

Answers to the research questions

4. [temporary reserves] Yes. Bad shock (e.g. low larval dispersal to a site) can cause stock to be lower than that path’s y*. Optimal rule is to close that area until it rebounds above y*.

Answers to the research questions

5. [optimal management outside reserve]• If reserve is optimal, then harvest less outside

reserve (because growth is high in reserve, so want to send extra larvae there)

• If reserve is not optimal, then harvest more outside reserve (because growth in reserve is low)

Answers to the research questions

6. [Economic value of reserves, economic cost of reserves]…??

7. [Information]…??