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Fisheries Enforcement TheoryContributions of WP-3
A summary
Ragnar Arnason
COBECOS COBECOS Project meeting 2Project meeting 2
London September 5-7 2007
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Introduction
• Task of WP-3 : Develop fisheries management theory
– To understand the process better– To support the empirical work– To support the programming work
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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
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Model (cont.)
Probability of penalty function (if violate): (e)
(e)
e
1
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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*
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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)
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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
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q
e
q*
[lower f][higher f]
Free access
q
Enforcement response function
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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
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Social optimality: Illustration
e
$
e*
( )q eB Q eC
eC
e°
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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
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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
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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)
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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)
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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
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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.
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Basically the same theory applies!
Conclusion
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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
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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
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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 !
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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
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ee1*
$
e2*
Ce
2( ) ( )q eB Q g u
1( ) ( )q eB Q g u
The effect of a high random term
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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)
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Numerical assumptions
Parameters Values p 1 c 1 f 1 0.4 a 0.05 b 2 x 100
u 0.2
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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
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0 5 100
5
10
Socialbenefits
Enforcement effort
Nonstochastic benefits, u=0
Expected benefit
function
Expected stochastic and non-stochastic benefit functions
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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%
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7.5 8 8.50
100
200
e° policy
e* policy
Histograms for benefits under the optimal and sub-optimal (non-stochastic) policies
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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
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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
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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
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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.
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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
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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
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Optimal approach paths (conjectures)
0x
Ce>0
Ce=0
Biomass, x
Harvest, q
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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
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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
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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)
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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
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Empirical considerationsApplication to case studies
• Data and estimation:
• Dynamics and the shadow value of biomass
• Deal with uncertainty
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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
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Dynamics
• Basically should solve the dynamic maximization problem for enforcement controls
• This is generally a major undertaking Short-cuts are desirable
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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:
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Biomass, x(0)
Economically minimum biomass
Optimalequilibrium
Theoretical
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Biomass, x(0)
Optimalequilibrium
Approximation
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Dealing with uncertainty
– Guess or estimate the uncertainty:
– Solve by simulations
2( ) , (0, )uug u e u N
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