Fisheries Enforcement TheoryContributions of WP-3

A summary

Ragnar Arnason

COBECOS COBECOS Project meeting 2Project meeting 2

London September 5-7 2007

Introduction

• Task of WP-3 : Develop fisheries management theory

– To understand the process better– To support the empirical work– To support the programming work

I. Basic model

Social benefits of fishing: B(q,x)-·q

Shadow value of biomass

Enforcement sector:Enforcement effort: e

Cost of enforcement: C(e)

Penalty: f

Announced target: q*

Private benefits of fishing: B(q,x)

Exogenous

Model (cont.)

Probability of penalty function (if violate): (e)

(e)

e

1

Model (cont.)

q

(q;e,f,q*)

q*

(e)f

Private costs of violations: (q;e,f,q*)=(e)f(q-q*), if qq*

(q;e,f,q*) = 0 , if q<q*

Model (cont.)

Private benefits under enforcement

Social benefits with costly enforcement:

B(q,x)-(e)f(q-q*), q q*

B(q,x), otherwise

B(q,x)-q-C(e)

Private behaviour

Maximization problem: Max B(q,x)-(e)f(q-q*)

Enforcement response function: q=Q(e,f,x)

Necessary condition:Bq(q,x)-(e)f=0

Can show: Q1, Q2<0, Q3>0

q

e

q*

[lower f][higher f]

Free access

q

Enforcement response function

Optimal enforcement

Social optimality problem

eMax B(q,x)-q-C(e).

subject to: q=Q(e,f,x), e0, f fixed.

Necessary conditions

( ( ( , , ), ) ) ( , , ) ( )q e eB Q e f x x Q e f x C e , if q=Q(e,f,x)>q*

Q(e*,f,x)=q*, otherwise

Social optimality: Illustration

e

$

e*

( )q eB Q eC

eC

e°

Some observations

1. Costless enforcement traditional case (Bq=)

2. Costly enforcement i. The real target harvest has to be modified

(....upwards, Bq<)ii. Optimal enforcement becomes crucial iii. The control variable is enforcement not “harvest”!iv. The announced target harvest is for show only

3. Ignoring enforcement costs can be very costlyi. Wrong target “harvest”ii. Inefficient enforcement

Practical guidance

• Seek to determine e* (and q*)

• For that(i) Set q* low enough

(ii) Find e* that solves

• Need to know B(q,x), C(e), π(e) and f

( ( ( , , ), ) ) ( , , ) ( )q e eB Q e f x x Q e f x C e

Empirical data needs

• B(q,x): bioeconomic model

• C(e): Enforcement cost function. Need data on enforcement costs and enforcement effort. Standard econometrics

• π(e): Probabilty of paying a penalty function. Estimate somehow! Non-standard

• f: The penalty structure (expressed in monetary terms)

Extension ISeveral management measures and

enforcement tools

• Vector of fishing actions; s• Vector of management measures s*

– s≤s*, quite unrestrictive!– If s(i) unrestricted, just set s*(i) very high– If s(i)≥s*(i), just redefine s’(i)=-s(i), s’*(i)=-s*(i).

Harvesting function: q=Q(s,x)

• Vector of enforcement tools; e

Probability function: (e)

Fishers:

1

( , ) ( ) ( *) ( ) ( ( , ) *)I

i i i q qi

Max B x f s s f Q x q

s

s e e s

( , , *, *)x s q S e; f

Enforcers

( ( , , ), ) ( ( , , ), ) ( )Max B x x Q x x CE e

S e f S e f e

1 1

( ) 0, 0, ( ) =0, i j i j

I Ii i

s e j s e ji ij j

s sp B CE e p B CE e

e e

j=1,2…J

1

( ) 0,i j

Ii

s ei j

sp B CE

e

all ej>0.

Basically the same theory applies!

Conclusion

Extension IIUncertain fishers’ response function

Why?

1. Many fishers with different risk attitudes

2. Fishers seeking ways to bypass enforcement

3. Erratic enforcement personnel

( , , ) ( )q Q e x f g u

2( ) , (0, )uug u e u N

Distribution of actual harvest: An example

(Given e,f and x; u=0.2; 1000 replications)

0 20 40 600

20

40

Harvest

Freq

uenc

y

Optimal stochastic enforcement

( ( ( , , ), ) ) ( , , ) ( )q e eB Q e f x x Q e f x C e

Compare to the non-stochastic optimum condition:

Necessary condition:

[ ( ( , , ) ( ), ) ) ( , , ) ( )] ( )q e eE B Q e f x g u x Q e f x g u C e

Complicated function of the random varaible, u !

Two important results

Result 1

If and only if will optimal enforcement be characterized by the non-stochastic condition.

(( ) , ( )) 0q eCov B Q g u

Result 2

If then e*>e° and vice versa.

(( ) , ( )) 0q eCov B Q g u

ee1*

$

e2*

Ce

2( ) ( )q eB Q g u

1( ) ( )q eB Q g u

The effect of a high random term

A numerical example

2

( , )q

B q x p q cx

Private fishing benefits:

( ) ( )C e eCost of enforcement:

( )e

eb e

Probability of penalty:

Shadow value of biomass: (assumed known) (can calculate on the basis of bioeconomic model)

Numerical assumptions

Parameters Values p 1 c 1 f 1 0.4 a 0.05 b 2 x 100

u 0.2

Example (cont.)

( ( ) )

2

p e f xq

c

Enforcement response function:

f=2p

f=p

f=0.5p

0 5 100

20

40

60

Enforcement effort

Act

ual h

arve

st

0 5 100

5

10

Socialbenefits

Enforcement effort

Nonstochastic benefits, u=0

Expected benefit

function

Expected stochastic and non-stochastic benefit functions

Optimal and sub-optimal enforcement effort

•Table 2Optimal and suboptimal enforcement effort(1000 replications) Enforcement effortLevelExpected harvestExpected social benefitsVariance of social

benefitse*1.4629.68.533.6%e°1.2631.58.496.3%

Enforcement effort

Level

Expected harvest

Expected social benefits

Variance of social benefits

e* 1.46 29.6 8.53 3.6% e° 1.26 31.5 8.49 6.3%

7.5 8 8.50

100

200

e° policy

e* policy

Histograms for benefits under the optimal and sub-optimal (non-stochastic) policies

Extension IIIFully dynamic context

• In the basic enforcement theory, is taken to be exogenous

• At a given point of time (and in continuous time) it is

• However, for the optimal dynamic enforcement policy we need to include

Essential model

0{ }

( ( , ; ), ) ( ) r t

eMax V B Q e x f x C e e dt

( ) ( , ; )Subject to x G x Q e x f

( , ; ) ( ( , ) ( ) )Q e x f Max B q x e f q

Q(e,x,f) is fishers’ behaviour

Maximization

( ( , ; ), ) ( ) ( ( ) ( , ; ))H B Q e x f x C e G x Q e x f

(1) ( ) , all q e eB Q C t

(2) ( ), all q x x x xr B Q B G Q t

Hamiltonian

Some necessary conditions

(1) is the basic social enforcement rule!!(2) describes the optimal evolution of

How to calculate ?

• Generally not easy to obtain the path of • Jointly determined with e and x

• With a bioeconomic-enforcement model can work out (t), all t, (in principle)

For optimal enforcement need to solve the dynamic model at each point of time.

Optimal equilibrium

x q x

x x

B B Q

r G Q

e x x ex

q e e

C Q B QG r

B Q C

x q x

x x

B B Q

r G Q

Costly enforcement No or costless enforcement

xx

q

BG r

B

x

x

B

r G

So, enforcement modifies the marginal stock effect

• In traditional fisheries models, marginal stock effect, >0

• Under costly enforcement it may be of any sign

• Likely that

• Thus likely that (enforcement)<(costless enforcement)

0eC

Optimal approach paths (conjectures)

0x

Ce>0

Ce=0

Biomass, x

Harvest, q

Numerical example

1.1

( )q

p q c f e qx

10( ( ))

( , , )1.1

p f e xQ e x f

c

( )e

eb e

21t t t t tx x x x q

2( )C e a e

0.67 0.0004 p 20 f 100 a 1000 b 4 c 5363 r 0.05

Parameters

Approximately optimal paths

-100

0

100

200

300

400

500

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Biomass

Harvest

Growth Costly enforcement Costless enforcement

Approximately optimal pathsof biomass, harvest and enforcement

0.0

200.0

400.0

600.0

800.0

1000.0

1200.0

1400.0

2008 2012 2016 2020 2024 2028

0.000

0.200

0.400

0.600

0.800

1.000

1.200

Harvest (left axis) Biomass (left axis) Enforcement effort (right axis)

Extension IVAvoidance activities

• Avoidance possible• Another control for the fishers (e,u) probability becomes endogenous• Behaviour:

Q(e,f,x)

U(e,f,x)

• Qe and Qf may be positive!

• The theory becomes substantially more complicated

Empirical considerationsApplication to case studies

• Data and estimation:

• Dynamics and the shadow value of biomass

• Deal with uncertainty

Data & estimation

• Observations (cross-section, time-series) ons: Management controlse: Enforcement effortsC: Enforcement costs : Probablity of penalty (if violate)

• Estimate the probability and cost functions

– Best procedures available

Dynamics

• Basically should solve the dynamic maximization problem for enforcement controls

• This is generally a major undertaking Short-cuts are desirable

Approximating the shadow value of biomass

({ }, ( ))( )

( )

V e x tt

x t

( ) ( )q x x

x x x x

B Q B

r Q G r Q G

ˆ( )

q x x

x x

B Q B

r Q G

0, if *ˆ 0, if *

( )0, if *x x

x x

x xr Q G

x x

Theory:

Theory:

Approximation:

Error:

Biomass, x(0)

Economically minimum biomass

Optimalequilibrium

Theoretical

Biomass, x(0)

Optimalequilibrium

Approximation

Dealing with uncertainty

– Guess or estimate the uncertainty:

– Solve by simulations

2( ) , (0, )uug u e u N

END