BIOECONOMICS

Oscar CachoSchool of Economics

University of New England

AARES Pre-Conference WorkshopQueenstown, New Zealand

13 February 2007

2

• Definitions

• General models

• Solution techniques

• Incorporating risk

• Examples

• Extensions

• Useful literature

Outline

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Bioeconomics - DefinitionsIn its original coinage, "bioeconomics" referred to the study of how organisms of all kinds earn their living in "nature's economy," with particular emphasis on co-operative interactions and the progressive elaboration of the division of labor (see Hermann Reinheimer, Evolution by Co-operation: A Study in Bioeconomics, 1913). Today the term is used in various ways, from Georgescu-Roegen's thermodynamic analyses to the work in ecological economics on the problems of fisheries management.

Corning (1996)Institute for the Study of Complex Systems

Palo Alto, CA

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Bioeconomics - Definitions

Bioeconomics is what bioeconomists do.

Bioeconomics aims at the integration or ‘consilience’ (Wilson 1998) of two disciplines ...for the purpose of enriching both disciplines by substantially enlarging the theoretical and empirical bases which ultimately contribute to building of new hypotheses, theorems, theories and paradigms.

Landa (1999)Department of Economics, York University

Editor of the Journal of Bioeconomics

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Bioeconomics - Definitions

The use of mathematical models to relate the biological performance of a production system to its economic and technical constraints.

Allen et al (1984)

The idea of maximizing net economic yield while maintaining sustainable yield.

van der Ploeg et al. (1987)

The interrelations between the economic forces affecting the fishing industry and the biological factors that determine the production and supply of fish in the sea.

Clark (1985)

(more in line with how AARES members apply the term)

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Journal of Bioeconomics

• The Bioeconomics of Cooperation (new institutional economics)

• The Ecology of Trade (sustainability)

• Surrender Value of Capital Assets: The Economics of Strategic Virginity Loss (love)

• Making Good Decisions with Minimal Information: Simultaneous and Sequential Choice (ecological rationality)

• Altruism and Spite in a Selfish Gene Model of Endogenous Preferences (evolution)

• Evolutionary Theory and Economic Policy with Reference to Sustainability (behavioral economics)

A sample of titles (with keywords)

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• The Bioeconomics of Marine Sanctuaries

• The Bioeconomics of the Spatial Distribution of an Endangered Species: The Case of the Swedish Wolf Population

• Implementing a Stochastic Bioeconomic Model for the North-East Arctic Cod Fishery

• Optimization of Harvesting Return from Age-Structured Population

• Selective versus Random Moose Harvesting: Does it Pay to be a Prudent Predator?

• Using Genetic Algorithms to Estimate and Validate Bioeconomic Models: The Case of the Ibero-atlantic Sardine Fishery

Journal of BioeconomicsA sample of titles in resource management

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• Populations of natural organisms can be viewed as stocks of capital assets which provide potential flows of services.

• The critical characteristics of capital are:

• Durability: makes it necessary to apply intertemporal planning.

• Adjustment costs: force the decision maker to consider the future in order to spread out the cost of altering the capital stock.

• Types of decisions:

• Timing problem: ie. when to harvest a stand of trees.

• Harvest problem: i.e. how much of a resource to harvest each year.

• In both cases the flow of profits per time period depends upon the stock level (biomass) and the control variable (harvest).

Bioeconomics of renewable resources

Wilen (1985)

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• In the simplest case the value derived from natural resources is related to consumptive use by harvesting.

• The flows are usually measured in terms of number of organisms or biomass (weight).

• In the more complex cases the size of the stock may also have intrinsic value (i.e. number of birds available for birdwatchers).

• Models can be extended to include externalities.

Bioeconomics of renewable resources

Wilen, J.E.(1985).Bioeconomics of renewable resource use. In Kneese, A.V and Sweeney, J.L. (ed.), Handbook of natural resource and energy economics, Vol 1. North-Holland, Amsterdam 61-124.

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subject to:

General model in continuous time

T

tuTxFdtttutxRJ

0)(

)(),(),(Max

)(),( tutxfdt

dxx

ax )0(

)(),()(),(),( tutxftttutxRH

The Hamiltonian is:

x(t) =state variable (resource stock)u(t)= control variable (harvest rate)

equation of motion

initial state

reward final value

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FOC in continuous time

0)(

)(

tu

H

)(

)(

tx

H

)(

)(

t

Hx

ax )0(

))((')( txFT transversality condition

maximum condition

adjoint equation

equation of motion

initial state

This system is used to solve for the optimal trajectories u*(t), x*(t), *(t)

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subject to:

General model in discrete time

xt =state variable (resource stock)ut= control variable (harvest rate)

state transition

initial state

tttt uxfxx ,1

ax 0

The Hamiltonian is:

ttttt uxftuxRH ,,, 1

1

0

,,MaxT

tTtt

uxFtuxRJ

t

reward final value

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The Hamiltonian is the total rate of increase in the value of the asset (resource):

Interpretation

Value of net

returns at time t Shadow price of the

state variable (x) at

time t(user cost)

)(),,(),,( 11 ttttttt xxtuxRtuxH

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FOC in discrete time

This system has 3T+1 equations and 3T+1 unknowns:

ut for t = 0,...T1xt for t = 0,...T

t for t = 1,...T

0)(

tu

H

ttt x

H

)(

1

11

)(

ttt

Hxx

ax 0

)(' TT xF

t = 0,...,T1

t = 1,...,T1

t = 0,...,T1

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subject to:

Infinite horizon with discounting

The current-value Hamiltonian is:

state transition

initial state

tttt uxfxx ,1

ax 0

0

,Maxt

ttt

uuxRJ

t

r

1

1discount factor:

ttttt uxfuxRH ,, 1

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FOC of infinite horizon problem

ax 0

0)(

tu

H

ttt x

H

)(

1

11

)(

ttt

Hxx

In steady state:

ut+1 = ut = u

xt+1 = xt = x

t+1 = t = (1)

(2)

(4)

(3)rf

R

Rf u

u

xx

Solving (1) for steady state, substituting into (2) and rearranging yields the optimal condition:

This can be used to solve for (x*,u*) given the steady state condition from (3): 0),( uxf

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Summary of general model

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Typical problems in bioeconomics Problem tackled Model component Fishery Agriculture Invasives State variable (xt) biomass soil quality;

salinity number of individuals; area invaded

Control variable (ut) harvest stocking rate; fertilizer

pesticide; IPM

Adjoint variable (t) marginal value of biomass

marginal value of soil

marginal cost of organism

Reward tuxR tt ,, value of fish harvest

value of farm outputs

damage avoided

State transition equation ),,( tuxf tt

biomass growth soil degradation; salinity emergence

growth rate; rate of spread

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Solution techniques

• Numerical solution of optimal control model

• Nonlinear programming

• Dynamic programming

Here I will consider only optimal control and dynamic programming

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Numerical optimal control

simulation model

ut

H(xt,ut,t)

max?

t=T+1?

T =F’(xT)?

1

estimate t

yes

no

t=t+1

start

adjust

yesno

adjust

no

yes

initialguess

end

simulation model

ut

H(xt,ut,t)

max?

max?

t=T+1?

t=T+1?

T =F’(xT)?

T =F’(xT)?

1

estimate t

yes

no

t=t+1

startstart

adjust

yesno

adjust

no

yes

initialguess

endend

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Dynamic Programming (DP) 11,)( tttt

utt xVuxRMaxxV

t

subject to:

tttt uxfxx ,1

recursive equation

state transition

1,...1, TTt

4. Use this decision rule to derive the optimal path for any initial

state x0

2. Solve by backward recursion for a finite set of values Xxt

tt xu*

*tx

3. Obtain the optimal (state-contingent) decision rule

1. Set terminal value )( TT xFV

To solve:

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Dynamic programming algorithm

simulation model

recursive equation

max?

no

yes

adjustt=T

xt=xi ut=uj

xt+1

Save ut*(xi), Vt(xi)i=i+1

i=N+1?

no

yes

t=0?

t=t-1

no

yes

start

end

Set VT (xT)

simulation model

recursive equation

max?

max?

no

yes

adjustt=T

xt=xi ut=uj

xt+1

Save ut*(xi), Vt(xi)i=i+1

i=N+1?

i=N+1?

no

yes

t=0?

t=0?

t=t-1

no

yes

startstart

endend

Set VT (xT)

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Alternative DP solution techniques

Finite horizon models Backward recursion

Infinite horizon models

Function iteration

Policy iterationpractical only for infinite-horizon

problems (DP is converted into a root-finding problem)

(essentially the same as backward recursion but stop when V < tolerance)

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Introducing riskTwo general approaches exist for stochastic optimisation of bioeconomic models:

• Stochastic differential equations (Ito calculus) for continuous models

• Stochastic dynamic programming (SDP) for discrete models

Here I will only deal with SDP

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SDP BasicsAs before: the state variable xt can be observed before

selecting a value for the control ut which results in a known

reward R(xt,ut)

But now future returns are uncertain because the system is subject to stochastic influences:

),,(1 tttt uxfx

where t is a random variable with known probability, assumed

to be iid and therefore the stochastic process { t} is stationary

There is a fixed state set X and a fixed control set U, with n and m elements respectively

The time horizon may be fixed at T or

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Let the Markovian probability matrix:

The transition probability matrix (TPM)xnuP )(

uuxxxxuP titjtij ,|pr)( 1

denote the n × n state transition probabilities when policy u is followed:

(probability of jumping from state i to state j, given that action

u is taken)

To solve the problem first create an array P of dimensions

n × n × m

(P contains m transition probability matrices P(u))

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4. Recursion step; solve:

for all xX

SDP Algorithm

2. Run Monte Carlo simulation to create transition probability matrix

P(n,n,m) and reward matrix R(n,m)

1. Set dimensions (n,m) and initialise state set X and control set U

3. initialise terminal value vector VT(x) and set t=T-1

6. Decrease time counter and return to 4 until t=0 or convergence is achieved

jitijij

uit xVuPRMaxxV 1)()(

5. Save optimal decision rule Xut*

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7. For infinite horizon problem, create optimal transition probability

matrix P*(n,n) by selecting the elements of the Markovian probability matrices that satisfy the optimal decision rule for the given state statea:

SDP Algorithm (2)

8. Simulate the optimal state path by performing Monte Carlo

simulation for any initial state x0

))(*(*iijij xuPP

a Pij* is the probability of jumping from state i to state j in the

following period given that the optimal policy u*(xi) is

followed

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Example: Weed Control• A weed can be viewed as a renewable resource with the seed

bank representing the stock of this resource (x).

• The size of x changes through time due to depletion by weed management and new organisms being created via seed production.

• The change in the seed bank from one period to the next is

represented by the state transition equation xt+1-xt=f(xt,ut).

• The seed bank can be regulated through control u by targeting reproduction and seed mortality.

• The objective is to determine the level of control (u) in each

season that maximises profit over a period of T years.

Jones and Cacho (2000) A dynamic optimisation model of weed control http://www.une.edu.au/economics/publications/gsare/index.php

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Weed control model t

T

ttt uxRJ

0,max

tttt uxfxx ,1

subject to:

simulation model

The simulation model consists of a system of equations that represent the weed population dynamics, the effect of weed density on crop yields and the effect of herbicide on weed survival

The reward is net revenue: ytutty cupuxypR ,

xt=seedbank (seeds/m2)

ut=herbicide (l/ha)

y=crop yield (t/ha)

pi=price of i ($/unit)

cy=cropping cost ($/ha)

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Numerical optimal control (NOC)

is the net profit obtained from the existing level of xt and ut plus the

value of a unit change in xt valued at price t+1

The Hamiltonian:

t+1 , the costate variable, represents the shadow price of a unit of

the stock of the seed bank; its value is 0 because the state variable is bad for profits

tttytutty uxcupuxypH ,, 1

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0

50

100

150

200

250

300

350

400

450

500

0 1 2 3 4

NOC results The Hamiltonian and its components

u

H(x,u,)$/

ha

R(x,u)

f(x,u)

x=50, =-2

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0

100

200

300

400

500

600

0 2 4 6 8 10

NOC results Optimal paths

0

1

2

3

4

0 2 4 6 8 10

ut* xt*

t t

Control State

34

-3

-2

-1

0

0 2 4 6 8 10

NOC results The costate variable

u

t*

Shadow price of seedbank

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1. Solve simulation model: xj= xi + f(xi,u,), for = 1,…, k. where k is the number of Monte Carlo iterations, with each drawn from a lognormal distribution.

2. Use results from 1 to estimate Pij(u) given that

3. Calculate the reward Ri(u).

4. Repeat steps 1-3 for xi=x1,…xn to fill up the rows of P and R.

5. Repeat step 4 with u=u1,…um to fill up the columns of R and

the 3rd dimension of P.

6. Perform Backward recursion to solve SDP.

SDP model

j

ij uP 1)(

Use simulation model to generate transition probability matrix (P)

and Reward matrix (R):

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SDP model: TPM matrix

0 5 25 45 65

0 1 0 0 0 0

5 0.99 0.01 0 0 0

25 0.012 0.978 0.01 0 0

45 0.002 0.072 0.794 0.13 0.002

65 0.001 0.011 0.136 0.433 0.347

u = 1.0

fro

m s

tate

(x t)

to state (xt+1)

probabilities

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0 5 25 45 65

0 1 0 0 0 0

5 0.99 0.01 0 0 0

25 0.012 0.978 0.01 0 0

45 0.002 0.072 0.794 0.13 0.002

65 0.001 0.011 0.136 0.433 0.347

u = 3.0

0 5 25 45 65

0 1 0 0 0 0

5 0.99 0.01 0 0 0

25 0.012 0.978 0.01 0 0

45 0.002 0.072 0.794 0.13 0.002

65 0.001 0.011 0.136 0.433 0.347

u = 2.0

0 5 25 45 65

0 1 0 0 0 0

5 0.99 0.01 0 0 0

25 0.012 0.978 0.01 0 0

45 0.002 0.072 0.794 0.13 0.002

65 0.001 0.011 0.136 0.433 0.347

u = 1.0

TPM array

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0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 200 400 600

SDP results

x

Optimal decision rule

u*

39

0

100

200

300

400

500

0 2 4 6 8 10

SDP results

t

Optimal state path

x*

40

0 5 10 150

100

200

300

400

500

600

700

800

900

1000

SDP results

t

Optimal state path (Monte Carlo)

x*

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The optimal transition probability matrix is created by selecting the elements of the Markovian probability matrices that satisfy the optimal decision rule for the given state.

Optimal probability maps for any initial condition can then be

generated for any future time period t by applying (P*)t

0 5 25 45 65

0 1 0 0 0 0

5 0.894 0.106 0 0 0

25 0.004 0.89 0.106 0 0

45 0 0.21 0.75 0.04 0

65 0 0.017 0.627 0.324 0.028

P*=

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Example with multiple outputs

Recreation

0

2,000

4,000

6,000

8,000

10,000

0.0 0.2 0.4 0.6 0.8 1.0

Weed density

An

nu

al v

isit

ors

Biodiversity

0

20

40

60

80

100

0.0 0.2 0.4 0.6 0.8 1.0

Weed density

Pe

rce

nt

Agriculture

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Weed density

Yie

ld in

de

x

Odom et al (2002). Ecol Econ 44: 119-135

Optimal control of Scotch broom (Cytisus scoparius) in the Barrington Tops National Park

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Integrated weed management

1 2 3 4 5 Pd Ps fh wt

1 0 0 0 0 0 1.00 1.00 1.00 1.002 1 0 0 0 0 0.20 0.33 1.23 1.003 0 1 0 0 0 1.40 0.33 0.55 0.804 0 0 1 0 0 1.60 0.30 0.27 0.605 0 0 0 1 0 0.20 0.67 1.23 1.006 0 0 0 0 1 0.20 0.07 0.04 0.60... ... ... ... ... ... ... ... ... ...

29 1 0 1 1 1 1.30 0.32 0.46 0.4030 1 1 0 1 1 1.10 0.31 0.32 0.5031 1 1 1 0 1 0.70 0.02 0.80 0.3032 1 1 1 1 1 0.60 0.16 0.19 0.30

ui

parameterscontrol “package”

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0

500

1000

1500

2000

2500

3000

0 500 1000 1500 2000 2500 3000

Optimal state transition

xt

x t+1

45o

*x

45

Optimal paths

Barrington Tops National Park Scotch Broom Invasion

0

2

4

6

8

10

12

14

16

18

0 5 10 15 20 25 30 35 40 45 50

t

x*

Site

s in

vade

d (%

)

w / budget constraint

no constraint

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Optimal paths and policy

10

15

20

25

30

35

0 50 100 150

10

15

20

25

30

35

0 50 100 150t

x*

Opt

imal

soi

l C (

t/ha

)

t

With C creditsWith no C credits

Soil carbon content of an agoforestry system under optimal control

Wise and Cacho (2005). Dynamic optimisation of land-use systems in the presence of carbon payments: http://www.une.edu.au/carbon/wpapers.php

47

Extensions• Multiple state variables

• Multiple control variables

• Multiple outputs

• Spatially-explicit models

• Fish sanctuary models

• Multiple species / interactions

• Matrix population models

• Metamodelling

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Literature and LinksConrad, JM (1999). Resource Economics. Cambridge University

Press.

Conrad, JM and Clark, CW (1987). Natural Resource Economics: notes and problems. Cambridge University Press.

Fryer, MJ and Greenman, JV (1987). Optimisation theory: applications in OR and economics. Macmillan.

Judge, KL (1998). Numerical methods in Economics. The MIT Press.

Miranda, MJ and Fackler PL (2002). Applied computational economics and finance. The MIT Press.

NEOS Optimization Tree: http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/

Optimization Software Guide: http://www-fp.mcs.anl.gov/otc/Guide/SoftwareGuide/index.html

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Matlab optimal control modelWeedOC

optimisation 1 (0)

[Lx0]=TransvCond(x0,tmax,ubound,delta)

Lx0 = fminbnd(@ObjFn)

[xstar,ustar,Lxstar]=SolveWOC(x0,y,tmax,ubound,delta)

g = ObjFn(y)

[xstar,ustar,Lxstar]=SolveWOC(x0,Lx0,tmax,ubound,delta)

ustar = fminbnd(@ObjFn)

uopt=MaxHam(x,Lx,delta,ubound)

[x1,density]=seedbank(x,u)[H]=HWeed(x,u,Lx,df)

g = ObjFn(u)

profit=gm(density,u);

dHdx=dHweed(x,uopt,Lx,delta)

optimisation 2 (u*)

set x0

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Matlab SDP modelCreateMatrix

for t=nt:-1:1; for i=1:nx vopt=-inf; for k=1:nu fval = YM(i,:,k) * v(:,t+1); vnow = R(i,k) + delta*fval; if (vnow > vopt) vopt=vnow; uopt=k; end; end; v(i,t)=vopt; ustar(i,t)=uopt; end; end;

[x1,weeds]=seedbank(x,u)

profit=gm(density,u);

Generates TPM matrix (YM) and reward matrix (R)

SDP:

keep best control

value function

stage loopstate loop

control loopexpected vt+1