1. INTRODUCTION

Johnson and Libecap [7] stressed the heterogeneity of the fishers, particularly

in fishing skills, in their study of the Texas shrimp fishery. They detail the prac-

tical difficulties that heterogeneity causes for fishery managers in trying to get a

fishery to operate in a more efficient manner.1 Merrifield [11] returns to this point

and stresses the need to examine potential management measures for their ability

to promote homogeneity among fishers. The present paper will use a theoretical

bioeconomic model in the Gordon - Shaefer tradition to highlight further some

differences in the operation of a single species fishery with heterogeneous fishing

firms as opposed to the homogeneous case.2 The emphasis will be on the manner

in which an open access fishery is converted to management by individual trans-

ferable quotas (ITQs). Of particular interest will be the question of who gains

from the introduction of ITQs.

The general economic assumptions for the Gordon - Shaefer model are as

follows. Fishing firms are assumed to choose input (effort) levels to maximize

current profits from fishing net of their opportunity costs of entering the fishery.

1Other empirical studies of variations in catch rates among fishers include Hilborn and Led-better [6], Palsson and Durrenberger [13] and Salvanes and Steen [14].

2An early theoretical discussion of variations in the opportunity cost of effort across a fishingfleet is in Copes [4].

Secondly, the supply of firms to the fishery is competitive in the sense that the

number of vessels active in the fishery adjusts rapidly so that the least profitable

vessel makes zero economic returns from fishing. Thirdly, if fishers have to acquire

quotas equal to their catch and these quotas are freely transferable in a market.

then this market is competitive and adjusts rapidly to establish an equilibrium

price for these quotas.3

The case of a fishery accessible to a large number of identical vessels will now

be described. The purpose is to introduce the issues that will later be examined in

the context of a heterogeneous fishery.4 The conditions above imply that vessels

choose the effort level which minimizes their average costs (total costs divided by

their catch). With a certain separability assumption, this effort level does not

depend on conditions in the fishery.

Suppose now a management authority imposes a total allowable catch (TAC)

on this fishery which previously operated under conditions of open access and that

the TAC is below the open access catch. Effort levels do not adjust to this change

so initially vessel catches will not change either. The only change is therefore a

reduction in the number of vessels active in the fishery. In addition, suppose the

3Morgan [12] discusses problems which might arise with the operation of such secondarymarkets.

4The comments made here can be verified as a special case of the mathematical modelpresented below.

2

TAC has to be acquired by fishers in the form of individual transferable quotas.

It does not matter at this stage how the quotas are supplied. At the moment

the TAC is first introduced the market quota price will be zero as all fishers are

making zero profits and have no cash flow which can be used to purchase quota.

Once the TAC is established, the surplus production of the fish stock will exceed

the fishing fleet’s catch and the biomass will begin to grow. Vessel catches will

also grow and there will be further retirement of vessels as the fleet catch remains

constant. Vessel revenues will also grow while vessel costs remain constant because

vessel effort does not change. The fishery will be generating an increasing resource

rent RR which here is the difference between fleet revenues which are constant

and fleet costs which are falling due to the reduction in the size of the active

fleet. Since the quota market is competitive, the market value of quotas will equal

RR and as the supply of quotas is constant, the market price of quotas must be

increasing.5

These early transitional effects are illustrated in Figure 1.6 The fishery is

5This paper is not about the socially optimal way of setting the TAC over tine but in theshort term is about a practical way of beginning the rationalization of the fishery.

6In Figure 1, TR = pQ where p is the price of fish and Q is the catch of the fleet. TC = cNwhere N is the number of active vessels. Assuming vessel catch is qM , where M is the biomass,Q = qMN and TC = (c/q)(Q/M). TRtac and TCtac are the case of a TAC of Q. In openaccess, it is asumed that Q =M(1−M) so TRoa = pM(1−M) and TCoa = (c/q)(1−M). Theexample is calibrated on the open access biomass t. Since pt(1− t) = (c/q)(1− t), (c/q) = pt.

3

initially at point A in full open access equilibrium.. The imposition of a TAC

shifts the fleet revenue and cost curves and moves the fishery to a position such as

point B. Subsequently, the biomass begins to grow and the fishery moves to where

point C and point D indicate total revenue and total cost respectively. In this

case, this situation maximizes sustainable resource rents. In other cases, further

adjustments upwards in the TAC will be required to reach this optimum.7

FIG. 1. here

The effects of changing the management of the fishery from open access to a

system which causes the fishery to operate in an optimal manner will be called

the long term effects of that system. Here the long term effects are no change in

vessel effort levels and no change in profits from fishing as they remain at zero.

However, RR becomes positive so if the system is chosen so that fishers receive

some of the RR, they will be made better off. An example is an ITQ system

where fishers are allocated a share of the TAC for free. The final long term effect

is that the number of active vessels will be reduced. This is because for the fishery

to be dynamically stable, it must be the case that if vessel numbers increase, then

7Figure 1 and all other figures in this paper were drawn using the mathematical softwareMaple 6. The programs can be downloaded from the URL http://www.sfu.ca/˜heaps/hetfishas Maple text files.

4

ceteris peribus fleet profitability must fall. The transition from open access to the

optimum can be accomplished in a way such that the fishery is always close to

biological equilibrium and fleet profits are increasing.

This paper will examine the effects mentioned above when the fishery is ex-

ploited by heterogeneous fishing enterprises. The model used is a slightly more

general version of a model analyzed by Terrebonne [15]. A key aspect of the model

is that in any circumstances enterprises earn quasi-rents (producer surpluses). The

impact of a management measure on the distribution of these quasi-rents then be-

comes a nontrivial issue. It should also be noted that the social surplus SS now

exceeds RR by the amount of aggregate quasi-rents. The objective of society will

be assumed to be to maximize the sustainable social surplus.8 The objective of

a sole owner of the fishery would be to maximize the present value of resource

rents. As Terrebonne points out, this means that a sole owner with a zero discount

rate would desire a biomass greater than the socially optimal biomass unlike the

case of homogeneous fishers.. This result is rederived here and illustrated with an

example.9

8Thus, the social discount rate is being set at zero. This is appropriate when the main policyconcern is the income of fishers rather than the economic benefits generated for society.

9Arnason [1] has proposed managing fisheries with ITQs where the TAC is set to maximizethe market value of the quotas. This is equivalent to maximizing RR and so the proposal onlyworks if the fishing fleet is homogeneous.

5

2. THE MODEL

The following is a brief review of Terrebonne’s [15] model.10 Fishing en-

trepreneurs are indicated by x and each has a catch function of the form q(x) =

f(k,M, x) when k is an index of inputs used in fishing, M is the biomass of the

fish stock and the dependance of the catch on x indicates the differing fishing

abilities of the entrepreneurs. The normal assumptions about the catch functions

are:

fk > 0, fkk ≤ 0, fM > 0 and fx < 0

Fishing profits for an entrepreneur are then

(p− v)f(k,M, x)− c(k, x)

where p is the price of fish, v is the price of the quota needed to catch the

fish, and c(k, x) is the cost of using the input including the opportunity cost of

10The model was originally developed by Clark [3] where the number of vessels was modelledas a discrete variable. Similar models may be found in Heaps and Helliwell [5] and Mattiasson[9]. In Terrebonne’s version of the model the number of vessels is treated as a continuousvariable for mathematical convenience. The same results can be obtained with Clark’s versionof the model. Boyce [2] and Matulich and Sever [11] also analyze fisheries with heterogeneousfishers. However, their fisheries are seasonal in nature and the issue of stock conservation is notaddressed.

6

the fisher. As fishermen maximize profits, the input level k∗(x) used by fisher x

who participates in the fishery will satisfy

(p− v)fk(k∗(x),M, x)− ck(k∗(x), x) = 0

Note that

∂k∗(x)∂v

=ck

(p− v)πkk∂k∗(x)∂M

= −(p− v)fkMπkk

where πkk = (p− v)fkk − ckk. is assumed to be negative.

The fishery is otherwise assumed to operate in an open access manner. The

equilibrium conditions are then that the marginal fisher N should make 0 fishing

profits and the catch of all fishers combined should equal the TAC Q which is

assumed to be set by a management agency. That is

(p− v)f(k∗(N),M,N)− c(k∗(N), N) = 0

Z N

0f(k∗(x),M, x)dx = Q

7

These conditions determine the quota price and the number of active vessels

in the fishery fromM and Q. Unlike Terrebonne [15], however, in the short term,

the stock is not assumed to be in biological equilibrium. The interest here is to

examine the immediate effects of the imposition of or a change in a TAC. The

intuitive comparative static results are that if Q falls, then v should increase and

N should fall. In addition, the effects of growth in the biomass over periods

where the TAC is held constant will be looked at. One would expect also that

during such periods v would increase and N would fall. It does not seem possible,

however, to get the results concerning N in the case of a general catch function.

Terrebonne introduced the common assumption of proportional catching tech-

nology to further his analysis of the model. Here something similar, but a little

more general will be done It will be assumed that for all fishers11

f(k,M, x) = h(k, x)φ(M)

Then fkM = fkφ0(M)/φ(M) and

∂k∗(x)∂M

= −φ0(M)φ(M)

ckπkk

11φ(M) is assumed to strictly increasing inM and it will also be assumed that ln(φ) is concavein M . Mβ where β > 0 is an example.

8

(p− v)h(k∗(N), N)φ(M)− c(k∗(N), N) = 0

φ(M)Z N

0h(k∗(x), x)dx = Q

With some manipulation and letting q(x) = h(k∗(x), x)φ(M) be the catch of

vessel x, the results of totally differentiating the last two equations can be written

as

−q(N) πx(k∗(N), N)

1(p−v)2

RN0 (c

2k/πkk)dx q(N)

∂v/∂M ∂v/∂Q

∂N/∂M ∂N/∂Q

=

−(p− v)q(N)φ0(M)/φ(M) 0

−φ0(M)φ(M)

Q+ φ0(M)(p−v)φ(M)

RN0 (c

2k/πkk)dx 1

The determinant of the coefficient matrix of these equations is

|A| = −q(N)2 − 1

(p− v)2πx(k∗(N), N)

Z N

0(c2k/πkk)dx < 0

9

and the comparative static results are

∂v

∂M=(p− v)φ0(M)

φ(M)+

πx(k∗(N), N)Qφ0(M)/φ(M)

|A| > 0

∂N

∂M=q(N)Qφ0(M)/φ(M)

|A| < 0

∂v

∂Q= −πx(k

∗(N), N)|A| < 0

∂N

∂Q= −q(N)|A| > 0

These equations confirm the results mentioned above and some further short

term effects are discussed in the next section.12

3. EARLY TRANSITIONAL EFFECTS

Using the formulas of the previous section, the short term effects on fishing

effort are

∂k∗(x,M)∂M

=∂k∗

∂v

∂v

∂M+

∂k∗

∂M

=ckΓQφ

0(M)(p− v)πkkφ(M) < 0

12The results for a homogeneous fishery can be confirmed by putting πx(k∗(N), N) = 0.

10

where Γ = πx(k∗(N), N)/ |A| > 0.

∂k∗(x,M)∂Q

=∂k∗

∂v

∂v

∂Q= − ckΓ

(p− v)πkk > 0

Thus during a period of a fixed TAC or at a time the TAC is adjusted down-

ward, fishing effort will be reduced.

At the same time, the remaining active vessels will have their profits from

fishing reduced. Letting these profits be

π(x) = (p− v)h(k∗(x), x)φ(M)− c(k∗(x), x)

then the same formulas give the results

∂π(x)

∂M= −q(x)ΓQφ

0(M)φ(M)

< 0

∂π(x)

∂Q= q(x)Γ > 0

Thus all fishers are made worse off if the management agency is able to collect

the full market value of the quotas that it issues. Also of interest are the aggregate

impacts on social welfare. The social surplus SS generated by the fishery is

11

the sum of the fishing profits earned by fishing firms and the resource rent RR

generated by the fishery. That is

SS =Z N

0((p− v)h(k∗)φ(M)− c(k∗))dx+ vQ

Using the equilibrium conditions

∂SS

∂M=Z N

0(− ∂v

∂Mh(k∗)φ(M) + (p− v)h(k∗)φ0(M))dx+ ∂v

∂MQ

= (p− v)Qφ0(M)/φ(M) > 0

∂SS

∂Q=Z N

0(− ∂v

∂Qh(k∗)φ(M))dx+

∂v

∂QQ+ v = v ≥ 0

Thus during a period where the TAC is held constant aggregate social welfare

will improve. However, if a TAC is first imposed or is at some time reduced the

opposite will occur. One implication is that even if the quotas are given out free

so that fishers get the whole of the social surplus, at least some of the fishers must

be made worse off in the short term. This situation can be further explored by

defining what Terrebonne calls full profits to be the income fishers would receive

if they were given for free an initial quota which is a share s0(x) of the TAC. This

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is

π∗(x) = π(x) + s0(x)vQ

for those fishers who remain active in the fishery. The effects of changes in M

or Q on these full profits are

∂π∗(x)∂M

=∂π(x)

∂M+ s0(x)

∂v

∂MQ

= (s0(x)(p− v) + (s0(x)− s(x))ΓQ)Qφ0(M)φ(M)

∂π∗(x)∂Q

=∂π(x)

∂Q+ s0(x)(

∂v

∂QQ+ v)

= s0(x)v + (s(x)− s0(x))ΓQ

where s(x) = q(x)/Q is the share of the fisher in the aggregate catch after

trades are made in the quota market. The effect on full profits depends on how

this share compares to the share of the initial allocation of quota. The imposition

of a TAC must result in higher shares of the fleet catch for at least some fishers.

These fishers will have their full income reduced in the short term if s0(x) is their

share of the catch in the open access fishery. Figure 2 shows, however, there

are cases where some still active fishers have their shares of the aggregate catch

13

reduced by the imposition of a TAC. This figure and all subsequent figures are

based on the example presented in Appendix A.13 Figure 3 shows the change in

full profits which occurs in case 2 of Figure 2 if s0(x) is based on catch history.

Immediately after the imposition of the TAC, all remaining active fishers are worse

off as their shares in the catch have increased.14 However, subsequent growth in

the biomass improves the full profitability of all. In case 1, not shown, all fishers

are better off than in open access after the 25% increase in the biomass.

FIG. 2. here

Those who choose to sell their whole allocation will get its market value as

their full income. These full incomes are also compared to open access profits in

Figure 3 to the right of the kinks in the curves. The least efficient in the open

access fishery will definitely gain from the imposition of the TAC as their open

access profits are near zero. The figure shows however that some of the exiters

may have their welfare reduced. The question of who gains in the longer term will

be addressed below.

13The example contains 3 parameters: t the open access biomass, m an index of heterogenityand w an economic parameter. In both cases t = 2/5 and m = 10. In case 1, w = 1/2 and incase 2, w = 2. The TAC is set at 75% of the open access catch.14In case 1, with the TAC increased to 90% of the open access catch, some of the less efficient

but still active fishers have their welfare improved.

14

FIG. 3. here

4. LONG TERM EFFECTS

In this section, the biologically sustainable equilibria given by the additional

condition Q = g(M) are compared. Here g(M) is the natural growth function

of the fish stock.15 The change in a sustainable variable Z with respect to M

can be computed as ∂Z/∂M = ∂Z/∂M + g0(M)∂Z/∂Q where the derivatives on

the right hand side are those computed in the previous section. Table I gives

the results of these calculations. The signs of the results depend on the sign

of D = g(M)φ0(M)/φ(M) − g0(M). It will be assumed that this expression is

positive for all M not less than the open access biomass.16 Then the long term

effects of an increase in the sustainable biomass on v, N , k∗(x) and π(x) are

qualitatively the same as the early transitional effects and the same comments

apply. The condition determining the socially optimal biomassM∗ (∂SS/∂M = 0)

can be interpreted as the marginal benefit to the fleet of increasing the biomass

((p − v) RN0 fMdx = (p − v)gφ0/φ) must equal the loss to society of sustainable15g(M) is assumed to be strictly concave in M .16This assumption definitely holds for an interval of M beginning below the biomass which

maximizes g(M). Another justification, given in Terrebonne [15], is that open access equilibriumshould be dynamically stable. Here suppose the configuration of the fishing fleet is fixed. The

dynamics of the biomass would be given by•M= g(M) − φ(M)R where R is fixed. At the

equilibrium given by g(M)/φ(M) = R, g0(M)−φ0(M)R = −D < 0 is required for local stability.

15

rent (−vg0(M)).17

Table I

The Sustainable Derivatives

Z ∂Z/∂M

v (p− v)φ0/φ+ ΓD > 0

N q(N)D/ |A| < 0

k∗(x) ckΓD/(p− v)/πkk < 0

π(x) −q(x)ΓD < 0

SS (p− v)gφ0/φ+ vg0

RR ∂SS/∂M + gΓD

π∗(x) s0(x)∂SS/∂M + (s0(x)− s(x))gΓD

Γ = πx(k∗(N), N)/ |A| > 0 : D = gφ0/φ− g0 > 0

WhenM =M∗, ∂RR/∂M > 0 so resource rent is maximized by a biomassMr

which is higher than the socially optimal biomass as was shown by Terrebonne[15].

Figure 4 provides an illustration of the difference between these 2 biomasses.18

17If a social discount rate δ is used to determine the optimal long run target for the biomass,then the condition which determines this target becomes ∂SS/∂M = δv where v is the financialcapital created by catching another fish and δv is the social rate of return on this capital.18The parameter values are t = 2/5, m = 10 and w is varied from 0 to 50.

16

FIG. 4. here

A major problem in establishing an individual transferable quota system for

a fishery that was previously open access is deciding how to supply quota to the

fishing entrepreneurs. As illustrated in the short term, the allocation method

determines how the benefits of rationalizing the fishery are distributed among

fishers and the regulators. The ITQs could be sold to fishers at a price which

balances the fleet demand for quotas with the TAC. However, such a scheme

would be very unpopular because as shown above it reduces the profits of all the

fishers. Kaufman and Geen [8] state that in practise quotas are initially allocated

at zero or very low cost according to some administrative formula. One proposal

would be to assign initial quotas to fishers equal to their socially optimal shares

in the TAC. This would mean that some fishers would have to be given a zero

allocation in order to reduce vessel numbers while the other fishers would have to

be given a different share of the total catch than they had historically. Such an

allocation rule would also be unlikely to be politically acceptable.

The allocation formulas discussed by Kaufman and Geen are based on some

combination of historical catches and the historical use of some inputs. One

group of quota receivers would sell their entire allocation and receive an income

that was proportional to the market value of the aggregate quota vg(M). These

17

entrepreneurs would be in favour of reducing the TAC below its socially optimal

level.

A second group would wish to remain active in the fishery. The effect of

increasing M on the full income π∗(x) of this group is shown in Table I. In the

case where these fishers received initial allocations of quota equal to their socially

optimal allocation, this derivative would be zero at M = M∗. Thus they are all

made as better off as possible by setting the TAC equal to g(M∗). An initial

allocation based on catch history will however produce different results, contrary

to the claim in Terrebonne [15]. It must be the case that s0(x) < s(x) for at least

some of the surviving entrepreneurs and such a fisher would prefer a stock size

Mfp(x) that was smaller than M∗.

Figure 5 shows 2 examples of the desired full profit maximizing biomasses

when the initial allocation share is the share sq0(x) of the fisher’s catch in the

open access fishery.19 These biomasses may be different for different fishers. They

are also below or aboveM∗ when the socially optimal share is greater than or less

than the initial allocation.20

19The cases for this and the next two figures are the same as the cases for Figure 2. A plot ofthe changes in the shares from open access to the socially optimal fishery for these two cases issimilar to Figure 2.20For a Cobb-Douglas production function f = h(x)kαMβ, s0(x)/s(x) does not depend on x.

All remaning active fishers then do agree on which stock size is most beneficial for them but itis below M∗. For details, see http://www.sfu.ca/˜heaps/hetfish.

18

FIG. 5. here.

5. WHO GAINS?

In a fishery where quotas are free, the original open access fishers receive the

entire increase in the social surplus so that at least some of them will be better

off than they were in the full open access fishery. Indeed, those most marginal

in the open access situation go from nearly zero returns to a positive share of

the resource rent and are thus clearly made better off. Matulich and Sever [11]

show that there are initial allocations of ITQs that make everybody better off.

This turns out also to be the case in the fishery where the initial allocation is

based on catch history and a proof is provided in Appendix B.21 Figure 6 shows

two examples of the gains received by fishers from moving to free ITQs. It is

interesting to see that in one of the cases marginal fishers actually benefit more

than do some of the highliners.

FIG. 6. here

The case where the initial allocation of shares sk0(x) is based on the fisher’s

application of effort in the open access fishery will also be examined. The ratio

21The result must also hold when the long run target biomass is set on the basis of a lowsocial discount rate.

19

of the two allocations (sq/sk) is a constant times h(k∗(x), x)/k∗(x) which is catch

per unit effort by fisher x in the open access fishery. An increase in x will di-

rectly reduce this CPUE but will also reduce k∗ which will cause h(k)/k to rise.

A comparison of the two allocations therefore seems to require some additional

assumptions about the technology. The following assumptions are satisfied by the

examples mentioned above. First h is separable in k and x. That is h = d(k)e(x)

where d is strictly concave and e0(x) < 0 for all x. Secondly kd0(k)/d(k) is non-

increasing in k and thirdly c(k, x) is independent of x and concave in k.22 Then,

making use of the condition determining k∗, the ratio sq0(x)/sk0(x) is then a

constant times

d(k∗)k∗d0(k∗)

c0(k∗)

The two terms in this expression are nondecreasing in k∗ by the assumptions

made and since k∗ is lower for higher x this means sq0(x)/sk0(x) is decreasing in

x. Thus the effort allocation favours lowliners over highliners as compared to the

catch allocation. Figure 7 illustrates this comparison for case 2 of the example. In

this particular case, all fishers are still made better than they were in the full open

access equilibrium. However, there are other cases in which highliners are actually

22For a concave d(k), both the average and marginal functions are decreasing in k. Theassumption says that the average falls no more quickly than the marginal.

20

made worse off when the free initial allocation of shares is based on effort.23

FIG. 7. here

A third allocation formula which is even more favourable towards the lowliners

is to give all the participants in the open access fishery the same share of the TAC.

This formula may also reduce the full profits of the highliners. The interesting

observation here is that there are cases of the example where the full profits of all

entrepreneurs are improved by the application of this equitable formula.24

6. CONCLUSIONS

The immediate effects of the imposition of a TAC on a heterogeneous fishery

include a nonzero price in the quota market and a reduction in the social surplus.

This means that even if fishers are given free quotas, some of the original fishers

will be made worse off. Other early transitional effects include falling effort levels,

falling profits from fishing and falling numbers of active vessels.

Another significant point is that before and after the introduction of ITQs,

different firms have different shares of the aggregate catch. These shares tend to

increase with the change in management because of the reduction in the number

23For example,t = 2/5, m = 25 and w = 0.24For example, t = 2/5, m = 1 and w = 2.

21

of active firms but this effect is offset by the reduced effort levels. An example was

given where the shares of some of the less efficient firms were actually reduced. Full

profits are fishing profits plus the market value of the initial allocation of quota to

the firm, that is the income the firm would receive if there was a free allocation of

quota. The early transitional effects on full profits were shown to depend on how

the share in the initial allocation compares to the share in the aggregate catch.

This reinforced the point that even with free quotas, some fishers will be made

worse off.

The long term effects also included reductions in fishing efforts, fishing profits

and vessel numbers. The issue of whether a free allocation of quota will make

all fishers better off was addressed. It was shown that if the TAC is set at its

socially optimal level, then this will happen if the initial allocation is based on

catch history but may not happen if this allocation is based on effort history or is

equitable. Finally, Terrebonne’s [15] claim that the socially optimal biomass will

also maximize the full profits of all individual entrepreneurs is shown to be true

only in special circumstances.

APPENDIX A

The example used to illustrate some of the points made in this paper uses the

22

production function

f = ln(1 + ak)M/(1 +mx)

The profit function is (p−v) ln(1+ak)M/(1+mx)−rk−y. Then the optimal

choice of k is given by

1 + ak∗(x) =a(p− v)Mr(1 +mx)

The output of and profits earned by the xth fisher with this choice of the input

are

q(x) =M

1 +mxln(a(p− v)Mr(1 +mx)

)

π(x) =(p− v)M1 +mx

ln(a(p− v)Mer(1 +mx)

) + r/a− y

Putting π(N) = 0, this condition for open access equilibrium can then be

written as

(p− v)M/(1 +mN) = r

aew

where w is the unique nonnegative solution of the equation (w − 1)ew =

(ya)/r − 1 and has values ranging from 0 to ∞. Using this condition q(x) =

M(w + ln((1 + mN)/(1 + mx)))/(1 + mx) and the other equilibrium condition

23

RN0 q(x)dx = Q has the explicit solution

ln(1 +mN) =qw2 + 2mQ/M − w

Note that as w increases. N decreases as the fishery becomes less profitable.

The full open access equilibrium will be denoted by t and for the purposes of

illustration will be treated as a basic parameter together with m and w. From

the first equilibrium condition (v = 0 ) and assuming that the natural growth

function of the biomass is g(M) =M(1−M),

pt =r

aexp(

qw2 + 2m(1− t)) = r

aew(1 +mNt)

where Nt is the number of vessels in the open access fishery. For the fishery to

be profitable at all, it must be the case that t < 1. As exp(qw2 + 2m(1− t))/t is

decreasing in t, this requires that pa/r > ew.

The first equilibrium condition can now be written as

p− v = pt(1 +mN)

M(1 +mNt)

24

This formula can be used to show that the fishing profits of vessel x are

π(x) =pt

(1 +mx)(1 +mNt)((w − 1)m(N − x) + (1 +mN) ln(1 +mM

1 +mx))

The socially optimal biomass can now be calculated using the condition given

in Table I which becomes

∂SS/∂M = (p− v)(1−M) + v(1− 2M) = 0

This can be rearranged as

2M − 11 +mN

=t

1 +mNt

The solution is increasing in w as the fishery becomes less profitable and is

approximately (1 + t)/2 for w large.

The sustainable rents earned in the fishery are

vM(1−M) = p(M − t(1 +mN)(1 +mNt)

) (1−M)

25

Using the formula ∂N/∂M = −(1 +mN)/(w + ln(1 +mN))

p−1∂RR/∂M = 1− 2M +t(1 +mN)

(1 +mNt)(1 +

m(1−M)w + ln(1 +mN)

)

so that rents are maximized by the biomass which solves

2M − 1(1 +mN)

=t

(1 +mNt)(1 +

m(1−M)w + ln(1 +mN)

)

and which is greater than M∗ provided 0 < t < 1, that is that positive returns

can be obtained from the fishery. The formula for ∂RR/∂M can be used to show

that the term gΓD in Table I is given by

gΓD =t(1 +mN)m(1−M)

(1 +mNt)(w + ln(1 +mN)

Then the formula there for the derivative of full profits can be used to show that

the sustainable full profit maximizing biomass satisfies

2M − 1(1 +mN)

=t

(1 +mNt)(1 +

m(1−M)w + ln(1 +mN)

(1− s(x)

s0(x)))

From the formula for q(x) above, the shares of the fishers in the total catch of

26

the fishery are

s(x) =q(x)

Q=

M

Q(1 +mx)(ln(

1 +mN

1 +mx) + w)

where Q =M(1−M) in the long term.

The initial allocation of shares in the TAC based on historical catches is

sq0(x) =1

(1− t)(1 +mx)(ln(1 +mNt1 +mx

) + w)

.Finally, using the first equilibrium condition, the historical optimal use of inputs

is given by

1 + ak∗(x) = ew(1 +mNt)/(1 +mx)

and the shares based on historical input use are

sk0(x) =m(ew(1 +mNt)/(1 +mx)− 1)ew(1 +mNt) ln(1 +mNt)−mNt

Appendix B

Suppose some fisher who remains active in the ITQ fishery is made worse off

27

than he was in the open access fishery. Then

(p− v∗)h(k∗)φ(M∗)− c(k∗) + so(x)v∗g(M∗) < ph(k∗t )φ(t)− c(k∗t )

where k∗, k∗t maximize profits when M =M∗ andM = t the open access level

of the biomass respectively. It is also then the case that

(p− v∗)h(k∗t )φ(M∗)− c(k∗t ) ≤ (p− v∗)h(k∗)φ(M∗)− c(k∗)

Combining the two inequalities leads to

(p− v∗)h(k∗t )φ(M∗) + so(x)v∗g(M∗) < ph(k∗t )φ(t)

Now if the initial allocation is based on historical catch shares, sq0(x) =

h(k∗t )φ(t)/g(t) and it must be the case that

(p− v∗)g(t)φ(M∗)/φ(t) + v∗g(M∗) < pg(t)

Since g/φ is decreasing in M by the assumption that D > 0 (see Table I), this

implies that g(M∗) < g(t).

28

Now the socially optimal biomass satisfies the condition

(p− v∗)g(M∗)φ0(M∗)/φ(M∗) + v∗g0(M∗) = 0

Thus

v∗ =pg(M∗)φ0(M∗)

g(M∗)φ0(M∗)− g0(M∗)φ(M∗)

p− v∗ = − pg0(M∗)φ(M∗)g(M∗)φ0(M∗)− g0(M∗)φ(M∗)

Substituting these expressions in the critical inequality above gives

−g0(M∗)g(t)φ(M∗)2/φ(t) + g(M∗)2φ0(M∗) < g(t)(g(M∗)φ0(M∗)− g0(M∗)φ(M∗))

This inequality can be reorganized to

g(t)g0(M∗)(1− φ(M∗)/φ(t)) < g(M∗)(g(t)− g(M∗))φ0(M∗)/φ(M∗)

Now if g(M) is concave

g(t)− g(M∗)M∗ − t ≤ −g0(M∗)

29

Furthermore if lnφ(M) is concave

φ0(M∗)φ(M∗)

(M∗ − t) ≤ lnφ(M∗)− lnφ(t)

Then using the inequality lnx ≤ x− 1 if x ≥ 1

φ0(M∗)φ(M∗)

(M∗ − t) ≤ φ(M∗)φ(t)

− 1

Multiplying this inequalitiy with the inequality involving g and then further

multiplying by the inequality g(M∗) < g(t) contradicts the critical inequality

above.

30

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