Sabotage in a Fishery
Ngo Van Long
Department of Economics, McGill University, Montreal H3A 2T7, Canada
and
Stephanie F. McWhinnie
School of Economics, University of Adelaide, SA 5005, Australia
Preliminary and Incomplete - Comments Welcome
1
SABOTAGE IN A FISHERY
Abstract: This paper presents a simple model of a common access fishery where
fishermen engage in sabotage activities in addition to productive fishing activity.
We model sabotage as reducing the catchability coefficient of rival fishermen. We
show that the steady-state fish stock is higher, the greater is the effectiveness of
the sabotage technology: it seems that sabotage can mitigate the over-exploitation
associated with the tragedy of the commons. We also consider the effect of sabotage
on steady-state profit and find some interesting results: in the presence of sabotage,
industry profit at first increases as the number of fishermen increases but later decreases, this non-monotonicity cannot happen in the standard model. Increasing the
effectiveness of sabotage reduces sabotage effort but has a non-monotonic effect on
levels of fishing effort.
2
1
Introduction
Sabotage is a fact of life. Reports of sabotage have been made in a variety
of fisheries including gear tampering, stealing and intimidation. A fisherman may
want to sabotage the fishing effort of his rivals for various reasons. One possible
motive for sabotage in an oligopolistic market is to raise rivals’ costs, thus reducing
their outputs. Alternatively, in a perfectly competitive market (where the fishermen
are price-takers), there is an incentive for a fisherman to sabotage if he shares the
fishing ground with a fixed number of rivals, because his sabotage would reduce their
catches, thus increasing next period’s stock (relative to what it would be without
his sabotage) and hence increases his future catch at any given effort level. A third
motive for sabotage is envy: an individual may care about his relative income, in
addition to the usual concern for absolute income.
In this paper we investigate whether sabotage may be helpful to a fishery: by
reducing catches it may mitigate the tragedy of the commons. We present a simple
model of a common access fishery where, in addition to productive fishing activity,
price-taking fishermen also engage in sabotage. We investigate how the steady-state
fish stock is dependent on the effectiveness of the sabotage technology. In particular,
we ask if sabotage can result in a higher steady-state stock than the socially optimal
stock and evaluate whether other model parameters such as the price/cost ratio lead
to higher or lower steady-state levels of sabotage. In addition, we consider the effect
of sabotage on steady-state profits relative to the non-sabotage profit level.
In our model, agents act in their own best interests when harvesting from a
common-pool resource. What differs from the standard Clark-Munro model is that,
instead of simply choosing effort to maximise profits, they can also choose to engage in sabotage activities to reduce the ‘catchability coefficient’ in the production
function of their rivals. We show that the greater the effectiveness of the sabotage
technology, the higher the steady-state fish stock: it seems that sabotage can mitigate
the over-exploitation associated with the tragedy of the commons. We also show that
lower sabotage effort does not necessarily imply higher catchability if the sabotage
3
technology has improved and that any change in the price/cost ratio that results
in a higher steady-state stock will result in a lower steady-state level of sabotage
effort. Our numerical analysis on the effect of sabotage on profits gives some interesting results: the response to an increase in sabotage effectivess is non-monotone:
industry profit at first increases as the number of fishermen increases but then it
reaches a peak and afterward further increases in the number of fishermen lead to
lower industry profit, this non-monotonicity cannot happen in the standard model
without sabotage.
There is a large literature on sabotage in the contexts of rent-seeking and tournament, see, for example, Gonzalez (2007), Konrad (2000, 2009), Amegashie and
Runkel (2007). In contrast, to our knowledge, there has been no formal analysis
of sabotage in a fishery. By considering sabotage we aim to highlight the effect of
non-profit-maximising behaviour on fisheries and, renewable resources more generally. In addition, by extending the analysis to allow for cooperative production and
the ability to engage in protection (as is currently in progress) we hope to shed light
on the effect of cooperative behaviour amongst fishermen.
2
The Model
The basis for our model is Clark and Munro’s (1975) dynamic, single species model.
That is, we consider a fish stock exploited by n cymmetric fishermen who live in
the same community. They do not coordinate their harvesting decision. We assume
that their outputs are sold in a large market, so that aggregate quantity of fish they
catch does not influence the market price, which we assume to be a constant p. Let
xt denote the stock size and Lit the effort level of fisherman i at time t. Following
Schaefer (1957) we assume that the amount harvested by each agent is hit = qit xt Lit ,
where qit is the catchability coefficient specific to agent i, and let the stock grow at
a natural rate minus the total harvest. The cost per unit of effort is c > 0.
We now diverge from the standard model and suppose that, in addition to productive fishing activity, fishermen may undertake sabotage activities against their
4
rivals. We model sabotage by rivals as affecting the catchability coefficient of each
agent. Let sijt denote the sabotage effort of agent j directed against agent i with cost
per unit of sabitage effort c > 0 (the same as the cost of fishing effort). Let Sti be
the sum of sabotage efforts (by all other fishermen) directed against i:
X
Sti =
sijt
j6=i
Then catchability, qit , depends on sabotage efforts by all rivals directed at i:
qit =
qb
≡ qi (Sti ) where 0 < σ ≤ 1
1 + (θ/σ)(Sti )σ
(1)
where qb is a positive constant, and θ > 0 is the effectiveness of the sabotage technology. The restriction 0 < σ ≤ 1 ensures that qi0 (S i ) < 0 and qi00 (S i ) > 0.
Incorporating sabotage in this way means that each agent’s harvest is directly
affected by the sabotage efforts of his rivals, and this affects both individual profits
and the growth function of the stock. In addition, each unit of effort that i puts into
sabotage reduces his own profits. That is, agent i’s profit, net of his fishing effort
cost and the cost of his own sabotage effort directed against his rivals, is:
X j
sit
πit = (pqit (Sti )xt − c)Lit − c
(2)
j6=i
and the net rate of growth of the stock is
n
xt X
ẋt = rxt 1 −
−
qit (Sti )xt Lit
K
i=1
where r > 0 is the intrinsic growth rate and K > 0 is the carrying capacity.
We assume that in the absence of sabotage, the industry is potentially profitable.
More precisely, we make the following assumption:
Assumption 1: There exists a stock level xt < K such that pb
q xt > c for xt > xt .
That is, pb
q xt > c iff xt > xt , where xt < K.
We restrict the sum of agent i’s fishing effort and sabotage effort directed against
other agents to not exceed Lit :
Lit +
X
sjit ≤ Lit
j6=i
5
(3)
In what follows, we will focus on the symmetric equilibrium where each agent acts
identically and sabotages all rivals equally. Then we can define:
sit ≡ sjit for all j 6= i
and agent i thus faces n − 1 identical agents, each allocating Ljt units of effort to
fishing, and sjt units of effort to sabotaging each rival. The effort constraint (3)
therefore reduces to:
Lit + (n − 1)sit ≤ Lit
We assume that each agent’s goal is to maximize the integral of discounted utility of
net profit, where utility is:
uit (πit ) =
πit1−α
1−α
where α ≥ 0.1
Player i takes the natural growth rate of the fish stock (xt ) and others’ fishing
and sabotage effort levels (Ljt and sjt , j 6= i) as given. His problem is to choose
his own fishing and sabotage effort levels (Lit and sit ) to maximise the discounted
stream of utility:
ˆ∞
e−δt
1−α
1 (pqit (Sti )xt − c)Lit − (n − 1)csit
dt
1−α
0
Taking into account the non-negativity and time constraints on fishing and sabotage effort and denoting agent i’s shadow price of the fish stock at time t as ψt we
get the Hamiltonian for agent i as:
H =
1−α
e−δt (pqit (Sti )xt − c)Lit − (n − 1)csit
1−α h i
xt +e−δt ψt rxt 1 −
− (n − 1)qjt (Stj )xt Ljt − qit (Sti )xt Lit
K
+e−δt ηt Lit + e−δt µt sit + e−δt νt (L¯it − Lit − (n − 1)sit )
1
(4)
In the steady-state, the parameter α plays no role but we retain this formulation to aid future
consideration of the motives for sabotage, that is, purely profit driven or behavioural motives such
as envy.
6
Assuming an interior solution, the first order conditions with respect to Lit , sit , xt
and ψt are:2
πit−α (pqit (Sti )xt − c) − ψt qit (Sti )xt = 0
−πit−α (n − 1)c − ψt (n −
πit−α pqit (Sti )Lit
∂qjt ∂Stj
1)xt Ljt j
∂St ∂sit
(5)
=0
(6)
2xt
j
i
− ψt (n − 1)qjt (St )Ljt − ψt qit (St )Lit = − ψ̇t − δψt
+ ψt r 1 −
K
(7)
xt rxt 1 −
− (n − 1)qjt (Stj )xt Ljt − qit (Sti )xt Lit = 0
(8)
K
Condition (5) says that at the agent’s optimum, today’s marginal utility obtained
by exercising fishing effort is equated to the imputed cost of having a lower stock
tomorrow. Provided that n − 1 6= 0, condition (6) says that today’s marginal cost of
sabotage effort against n − 1 rivals is equated to the benefit of reducing their catches
(in terms of inducing a higher stock tomorrow). If n − 1 = 0 then of course there
will be no sabotage; in what follows, we assume n − 1 > 0.
At the symmetric steady state, Lit = Ljt = L∗ , sjt = sit = s∗ , and qit = qjt =
q(S ∗ ) where S ∗ = (n − 1)s∗ . This means (5) and (6) can be rewritten as:
ψ∗ =
π ∗−α (pq(S ∗ )x∗ − c)
q(S ∗ )x∗
(9)
and
ψ∗ = −
π ∗−α c
∂q ∂S
x∗ L∗ ∂S
∂s
2
(10)
We have ignored corner solutions for simplicity. To take into account corner solutions, we note
that (i) fishing effort would be zero whenever the stock level is so low that the value marginal
product of fishing effort is lower than the marginal effort cost c, and that (ii) sabotage effort si
would be zero if the marginal benefit of sabotage (evaluated at s = 0) is lower than the marginal
effort cost, c. Clearly, if 0 < σ < 1 then sabotage is always positive when fishing effort is positive.
If σ = 1 then for sabotage to be positive, θ must be sufficiently large. In what follows, we focus on
the interior solution, and assume that θ is sufficiently large.
7
Equating (9) and (10) and finding
∂q ∂S
∂S ∂s
from (1) gives:
2
c 1 + σθ S ∗σ
pq(S ∗ )x∗ − c
=
q(S ∗ )
L∗ q̂θS ∗σ−1
and substituting in for q̂ from (1) gives:
θ ∗σ
S
c
1
+
σ
pq(S ∗ )x∗ − c =
(11)
L∗ θS ∗σ−1
which is the arbitrage condition that the marginal value from an extra unit of effort
used for fishing must equal the marginal value of effort used for sabotage. Substituting (9) into (7) and setting ψ̇ = 0 gives the steady-state relationship:
q(S ∗ )x∗
2x∗
∗
∗
+ pq(S )L
− nq(S ∗ )L∗
δ =r 1−
K
pq(S ∗ )x∗ − c
(12)
Further, (8) in steady-state can be rearranged to give:
r
L =
nq(S∗)
∗
x∗
1−
K
(13)
which can be substituted into (12) to give:
2x∗
r
x∗
pq(S ∗ )x∗
δ =r 1−
−
1−
n−
K
n
K
pq(S ∗ )x∗ − c
(14)
Equation (14) is the usual fisheries modified golden rule except that the catchability coefficient is a function of sabotage.3 However, as there are two endogenous
variables in (14), the equilibrium requires an additional equation. The other steadystate relationship we can use is the the intratemporal arbitrage condition (11) with
(13) substituted in, that is:
c 1 + σθ S ∗σ nq(S ∗ )
pq(S )x − c =
∗
θS ∗σ−1 r 1 − xK
∗
∗
3
(15)
The intuition of (14) is as usual: at the margin, the return from harvesting another fish today
must equal the return from leaving it in the ocean to grow for tomorrow, net of the change that a
rival will take it first.
8
This means that the steady-state equilibrium of this model is defined by both the
intertemporal arbitrage equation (14) and the intratemporal arbitrage equation (15).
In subsequent sections we use these conditions to evaluate the effect of sabotage on
equilibrium stock levels and the effect of sabotage on steady-state profits.
Lemma 1: Equilibrium steady state profit per fisherman is:
1
c
∗
∗
−1
π =
+ cS
θS ∗σ−1
σ
(16)
where S ∗ is the Nash equilibrium level of sabotage.4
Proof: In a symmetric equilibrium, we can see from (2) and (11) that:
π ∗ = (pq(S ∗ )x∗ − c)L∗ − cS ∗
and
c 1 + σθ S ∗σ
(pq(S )x − c)L =
θS σ−1
Substituting (18) into (17) gives (16).
∗
3
∗
(17)
∗
(18)
The case where 0 < σ < 1
In the case where 0 < σ < 1, we have a system of two equations in two unknowns,
x∗ and S ∗ , equations (14) and (15) which can be rewritten as:
r
x∗
c
2x∗
−
1−
(n − 1) −
δ =r 1−
K
n
K
pq(S ∗ )x∗ − c
r
x∗ θS ∗σ−1
c
=
1−
∗
n
K
qb
pq(S )x∗ − c
(19)
(20)
It turns out that these two equations can be solved in two steps. In the first
step, S ∗ can be expressed as an explicit function of x∗ and of other parameter values,
S ∗ = φ(x∗ ; qb, θ, σ, δ, n). In the second step, this function is substituted into eq (19)
to obtain an equation in x∗ . We state these results as Lemma 2 and Lemma 3.
In the special case where σ = 1, we get π ∗ = c/θ, which is rather surprising: it depends only
on c and θ.
4
9
Lemma 2: Given that n − 1 > 0, the steady-state stock x∗ uniquely determines
the steady-state sabotage effort by the following equation:
1
1−σ
θ
∗
S =
q̂
1
! 1−σ
∗ 2
r2 1 − xK
∗
n nδ − r + (n + 1) rx
K
(21)
Proof:
Equation (19) can be re-written as:
∗
δ − r 1 − 2x
c
K
+ (n − 1) =
x∗
r
∗
pq(S )x∗ − c
1− K
n
(22)
which gives the same right-hand-side as (20). Therefore, equating (22) and (20)
gives:
∗
δ − r 1 − 2x
K
r
x∗
1
−
n
K
r
+ (n − 1) =
n
x∗ θS ∗σ−1
1−
K
qb
and rearranging:
S ∗σ−1
"
#
∗
q̂ δ − r 1 − 2x
n
−
1
K
=
∗
2 + r
r
x∗
θ
1 − xK
1
−
n
n
K
q̂
rx∗
=
(n + 1)
nδ − r +
∗ K
nθ nr 1 − xK 2
(23)
From (23) we can see that S ∗σ−1 will be positive if and only if
r − nδ
x∗
>
K
(n + 1)r
(24)
We have already assumed that profits are positive which tells us that the right-handside of (22) must be positive. This implies the left-hand-side must also be positive,
that is:
∗
δ − r 1 − 2x
K
n−1+ r
x∗
1
−
n
K
10
>0
which can be rearranged to show that
x∗
r − nδ
>
K
(n + 1)r
Therefore, as profits are positive, S ∗σ−1 must be positive so we can denote S ∗ = φ(x∗ )
such that:5
1
1−σ
θ
∗
S =
q̂
1
! 1−σ
∗ 2
r2 1 − xK
= φ(x∗ ; δ, r, K, n, θ, qb)
∗
n nδ − r + (n + 1) rx
K
(25)
It can be seen from equation (21) that the derivative of S ∗ with respect to x∗
is negative for all x ≤ K, provided that condition (24) holds. Thus we obtain the
following proposition:
Proposition: Given that 0 < σ < 1, any change in a parameter (other than
n) that results in a higher steady-state stock x∗ will result in a lower steady-state
sabotage effort, S ∗ .
Note, however, that a lower steady-state sabotage effort does not necessarily
imply a higher steady-state catchability coefficient. For example, increasing the
sabotage effectiveness parameter θ may lead to less sabotage effort being necessary
to achieve the same or a lower catchability coefficient for rivals and hence higher
stocks.
Lemma 3: Assume that n − 1 > 0. Then there exists a steady-state stock of fish
x∗ < K. It is a solution of the following equation: 6
#σ/(1−σ)
σ/(1−σ) "
∗ 2
r2 1 − xK
θ
=
(θ/σ)
∗
qb
n nδ − r + (n + 1)r xK
(nδ − r + (n + 1)r(x∗ /K)) (p/c)b
q x∗
K(r − nδ)
− 1, where
< x∗ < K
∗
nδ + nr(x /K)
r(n + 1)
5
(26)
Note that (25) does not contain the parameter p/c, we will need another equation to determine
the effect of changing this ratio later.
6
Equation (26) determines the steady-state stock x∗ in the feasible interval (0, K). It does not
seem possible to prove uniqueness analytically; it is conceivable that there are several solutions, i.e.
multiple steady states).
11
Proof of Lemma 3:
Substitute into the rearranged modified golden rule equation (22) the equation
for q(S ∗ ) (1) and subsequently substitute S ∗ = φ(x∗ ; ...)
∗ δ − r 1 − 2x
c
K
+ (n − 1) =
p
∗
x∗
r
q
b
x
c
1− K
−c
n
1+ θ S ∗σ
σ
1
=
p
qbx∗
c
θ
1+ σ [φ(x∗ ;...)]σ
−1
which can be rearranged to:
1+
p
qbx∗
c
θ
[φ(x∗ ; ...)]σ
σ
⇒
1+
−1=
p
qbx∗
c
θ
[φ(x∗ ; ...)]σ
σ
δ−r 1−
r
n
2x∗
K
∗
1 − xK
+ (n − 1) nr 1 −
x∗
K
∗
r 1 − xK
=1+
∗ > 1
nδ − r + (n + 1) rx
K
(27)
The right-hand-side of (27) must be greater than one when x∗ < K and profits
are positive, that is, from condition (24). Further rearranging gives:
1+
Thus
p
qbx∗
c
θ
[φ(x∗ ; ...)]σ
σ
∗
nδ + n rx
K
=
∗
nδ − r + (n + 1) rx
K
"
#
rx∗
nδ
−
r
+
(n
+
1)
θ
p
K
[φ(x∗ ; ...)]σ = qbx∗
−1
∗
σ
c
nδ + n rx
K
(28)
Substituting (21) for φ(x∗ ; ...) in (28) gives us a condition for x∗ :
σ
!# 1−σ
#
"
σ "
∗ 2
∗
r2 1 − xK
θ θ 1−σ
p ∗ nδ − r + (n + 1) rx
K
−1
= qbx
rx∗
rx∗
σ qb
c
nδ + n K
n nδ − r + (n + 1) K
(29)
where from (24)
K(r−nδ)
r(n+1)
< x∗ < K.
The graph of the left-hand side of eq (29) against x in the interval
K(r−nδ)
r(n+1)
<
x∗ < K is a continuous curve, which approaches infinity when x is arbitrarily close
12
to
K(r−nδ)
,
r(n+1)
and which approach zero as x tends to K. The graph of the right-
hand side of eq (29) against x in the interval
K(r−nδ)
r(n+1)
< x∗ < K is a continuous
curve, which approaches −1 when x∗ is arbitrarily close to
K(r−nδ)
,
r(n+1)
and approaches
∗
(p/c)b
q K − 1 > 0 as x tends to K. Therefore the two curves must intersect at least
once. This completes the proof of Lemma 3.7
Remark: In the special case where σ =
1
,
2
we obtain from (29) the following
cubic equation:
2 x∗
x∗
θ2 2
δ+r
=
2 r 1−
q̂
K
K
x∗
pq̂x∗
x∗
x∗
nδ − r + (n + 1)r
nδ − r + (n + 1)r
−n δ+r
K
c
K
K
Defining z =
x∗
,
K
(30)
this cubic equation is:
2θ2 r2
pq̂K
2
(1−z) (δ+rz)−(nδ − r + (n + 1)rz) (nδ − r + (n + 1)rz)
z − n(δ + rz) = 0
qb
c
(31)
Numerical analysis for the case of σ = 21 .
4
In this section we explore the effect of changing various parameters on the level of
sabotage, the fish stock, and individual and industry profit under the assumption
that σ = 0.5. For our base parameters we use: r = 1, δ = 0.05, n = 10, σ = 0.5, qb =
1, K = 1, p = 222.2, c = 1, θ = 100
Substituting these initial parameter values into (31) gives a unique positive root:
z = 0.5. Thus the steady-state stock is x∗ = 0.5, which is at the maximum sustainable
yield level. We can now use equations (21), (1), (13) and (16) to calculate the
equilibrium sabotage (S ∗ = 0.25), catchability coefficient (q(S ∗ ) = 0.0099), fishing
effort (L∗ = 5.05), and individual (π ∗ = 0.255) and industry profit (nπ ∗ = 2.55).
Equation (29) determines the steady-state stock x∗ in the feasible interval (0, K). It does not
seem possible to prove uniqueness analytically. It is conceivable that there are multiple steady
states.
7
13
4.1
Changes in theta
x*
π*
S*
0.58
0.56
0.28
0.28
0.26
0.26
0.24
0.24
0.22
0.22
0.2
0.2
0.54
0.52
0.5
100
110
120
130
0.18
100
θ
nπ*
110
−3
x 10
2.6
120
130
0.18
100
110
θ
θ
q*
L*
120
130
120
130
5.1
10
5.09
2.4
9.5
5.08
2.2
5.07
9
2
1.8
100
5.06
110
120
130
8.5
100
110
θ
120
θ
130
5.05
100
110
θ
If we raise the effectiveness of sabotage, we can see that the steady-state stock
rises while effort devoted to sabotage falls. Both individual and industry profits fall.
Interestingly, effort devoted to fishing has a non-monotonic response.
14
4.2
Changes in the number of fishermen
x*
π*
S*
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
10
20
30
0
10
n
nπ*
3
0.025
2.5
0.02
20
30
10
n
n
q*
L*
20
30
20
30
15
10
2
0.015
1.5
0.01
5
1
0.005
10
20
30
0
10
n
20
n
30
10
n
As we raise the number of fishermen, we can see that, as usual, the steady-state
stock and individual profits fall. The effort devoted to sabotage and to fishing falls.
The non-monotonic response of industry profits that occurs here is not possible in
the standard model without sabotage: usually industry profits are strictly decreasing
in the number of fishermen.
15
4.3
Changes in the price
π*
S*
x*
0.3
0.52
0.35
0.28
0.51
0.3
0.26
0.24
0.5
0.25
0.22
0.49
200
220
240
0.2
200
220
p
nπ*
3
240
220
p
p
q*
L*
0.011
2.8
0.2
200
240
5.5
0.0105
2.6
0.01
5
2.4
0.0095
2.2
2
200
220
240
0.009
200
220
p
p
240
4.5
200
220
240
p
As the price rises (or equivalently costs fall), the steady-state stock falls while
fishing effort and individual and industry profits rise. Sabotage efforts rise.
16
5
Preliminary Conclusions
This paper models the effect of sabotage on stock status and profits in a dynamic,
renewable framework. In contrast to standard fisheries models, we allow fishermen
to undertake sabotage activities against their rivals as well as conduct productive
fishing. The effect of sabotage mitigates the usual tragedy of the commons result
of overexploitation of stocks. We show that increasing the sabotage effectiveness
parameter has a non-monotonic effect on fishing effort. In addition, in the presence
of sabotage, industry profits are non-monotonic in the number of fishermen, which
is in contrast to the model without sabotage.
While sabotage protects against the tragedy of the commons, we do not propose
that encouraging sabotage is an optimal policy response. Our next step is to consider
the effects on stocks and profits of cooperative production with profit sharing, as in
Sen (1966), and allowing fishermen to engage protection.
17
APPENDIX
Consider the special case where σ = 1.
Rewrite the modified golden rule (14) slightly:
2x∗
r
x∗
c
δ =r 1−
−
1−
(n − 1) −
K
n
K
pq(S ∗ )x∗ − c
(32)
and also rewrite the (11) with (1) and σ = 1:
c
θr
=
∗
∗
pq(S )x − c
q̂ n
x∗
1−
K
(33)
Substituting (33) into (32) we get a single equation that determines the steady-state
stock:
2
2x∗
r
x∗
r
x∗
θ
δ = r 1−
−
1−
(n − 1) +
1−
K
n
K
n
K
qb
2
x∗
r
(n + 1)r x∗ θ r2
=
−
+
1
−
n
n
K
q̂ n2
K
(34)
If this equation has a unique solution x∗ , it will be the unique candidate interior
steady-state stock.
Notice that (34) does not contain terms such as p and c. We must also use eq
(33), which (with (1)) can be written in the following form :
S ∗ = −1 +
qbpx∗
c+
(35)
cnb
q
∗
θr(1− xK )
and verify that S ∗ is indeed positive at the value x∗ that satisfies (34).
Proof of equation (35). Let us find S ∗ as a function of θ in the case where
σ = 1. From (33), with σ = 1,
pq(S ∗ )x∗ − c
nb
q
=
c
θr 1 −
we get
pq(S ∗ )x∗
nb
q
=
c
θr 1 −
x∗
K
∗
nb
q + θr 1 − xK
+
1
=
∗
x∗
θr 1 − xK
K
18
i.e.
∗
cnb
q + cθr 1 − xK
1
=
∗
1 + θS ∗
qbpx∗ θr 1 − xK
(36)
i.e.
∗
∗
q − cθr 1 −
qbpx∗ θr 1 − xK
qbpx∗ θr 1 − xK − cnb
−1=
θS =
∗
x∗
cnb
q + cθr 1 − K
cnb
q + cθr 1 − xK
∗
q px∗ − c] − (cnb
q /θ)
r 1 − xK [b
∗
S =
x∗
(cnb
q /θ) + cr 1 − K
∗
x∗
K
(37)
(So for S ∗ to be positive, it is necessary that qbpx∗ − c > 0.) Simplify
x∗
r
1
−
qbpx∗
K
S ∗ = −1 +
∗
r 1 − xK c + (cnb
q /θ)
= −1 +
qbpx∗
c+
cnb
q
∗
θr(1− xK )
Then, for S ∗ to be positive, we need
qbpx∗ > c +
cnb
q
∗
θr 1 − xK
This condition is satisfied if θ is sufficiently great.
We now state the following result:
Lemma A1: Assume σ = 1. Then there exists a unique positive candidate
steady-state stock x∗ < K if and only if the sabotage technology is sufficiently effective, so that the following inequality is satisfied:
θr2
r
>δ−
2
qbn
n
(38)
This candidate steady-state stock is admissible iff S ∗ as given by (35) is also positive.
Proof:
Let us consider the graph of the left-hand side of (34) where x is measured along
the horizontal axis. It is a straight line with positive slope, cutting the vertical axis
at δ − (r/n) and cutting the vertical line x = K at the value δ + r. The graph
19
of the right-hand side of (34) is downward sloping curve, cutting the verical axis
at θr2 /(b
q n2 ) and cutting the horizontal axis at x = K. Therefore the two curves
have a unique intersection at some x∗ < K if and only if the inequality (38) holds.
(Note: if the inequality (38) is reversed, then there are two roots, both of which are
non-admissible: one root is negative, and the other is greater than K.)
Remark: having found the steady-state stock x∗ , we can find the steady-state
sabotage S ∗ from
c
=
pq(S ∗ )x∗ − c
G(x∗ )
nx∗
θ
qb
and verify that S ∗ > 0.
Proposition A1: An increase in the effectiveness of the sabotage technology will
increase the steady-state fish stock.
Proof: An increase in θ shifts the graph of the right-hand side of (34) upwards,
and does not affect graph of the left-hand side of (34). Hence we obtain a higher
x∗ From (16), in the case σ = 1, steady-state profit is
c
+ cS ∗
π = (pq(S )x − c)L − cS =
∗
θ(S )σ−1
∗
∗
∗
∗
∗
1
−1
σ
c
θ
∗
It is rather surprising that π is independent of all other parameter values.
π∗ =
(39)
Let us calculate the effect of an increase in θ on the steady-state stock. Define
z≡
x∗
K
D≡
qbn2
θr2
and
Then (34) becomes
z 2 + βz + γ = 0
where
β ≡ −2 −
n+1
rD
n
20
(40)
r
D
n
The roots of the quadratic equation are real, because
√
−β ± ∆
z1,2 =
2
γ ≡ 1 − δD +
where
2
2
∆ ≡ β − 4γ = r D
2
n+1
n
2
+ 4D(δ + r) > 0
Assumption (38) implies that γ > 0, ie. both roots are positive, with the smaller
one in the interval (0, 1). The smaller positive root is
√
−β − ∆
z=
2
(It is not clear that an increase in θ will lead to an increase or a decrease in S ∗ :
if x∗ is kept unchanged, then from (35) the direct effect of an increase in θ is an
increase in S ∗ ; however, from Proposition 1, x∗ increases when θ increases.)
Steady-state fishing effort is
x∗
r
x∗
r
∗
∗
(1 + θS ) 1 −
L =
1−
=
nq(S ∗ )
K
nb
q
K
Upon substitution, using (36),
r
L∗ =
nb
q
!
∗
qbpx∗ θr 1 − xK
x∗
1−
∗
K
cnb
q + cθr 1 − xK
We can compare the steady-state profit under sabotage, π ∗ = c/θ, with the
steady-state profit when sabotage is not present because of a low θ. In the formercase,
profit is a constant:
π = c/θ
In the latter case, the steady-stock is given by the usual equation
G(x∗∗ )
c
0 ∗∗
δ = G (x ) −
(n − 1) −
nx∗∗
pb
q x∗∗ − c
21
and the steady-state profit is
π ∗∗ = (pb
q x∗∗ − c)
G(x∗∗ )
≡ (pb
q x∗∗ − c) L∗∗
x∗∗ nb
q
22
REFERENCES:
Amegashie, A., Runkel, M., 2007. Sabotaging Potential Rivals. Social Choice
and Welfare 28, 1343-1362.
Clark, C.W., Munro, G.R., 1975. The economics of fishing and modern capital
theory. Journal of Environmental Economics and Management 2, 92–106.
Gonzalez, F.M., 2007. Effective property rights, conflict and growth. Journal of
Economic Theory 137, 127-139.
Konrad, K.A., 2000. Sabotage in Rent-seeking Contests. Journal of Law, Economics, and Organization 16, 155-165.
Konrad, K.A., 2009. Strategy and Dynamics in Contests. Oxford University
Press.
Sen, A., 1966. Labor Allocation in a Cooperative Enterprise, Review of Economic
Studies, 33, 361-371.
Schaefer, M.B., 1957. Some Considerations of Population Dynamics and Economics in Relation to the Management of Marine Fisheries. The Fisheries Research
Board of Canada 14, 669-681.
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