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Individual transferable fishery quotas under uncertainty Kusuda, Hisafumi 1993

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INDIVIDUAL TRANSFERABLE FISHERY QUOTAS UNDERUNCERTAINTYByHisafumi KusudaBachelor of Law, The University of Tokyo, 1978A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESECONOMICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJune 1993© Hisafumi Kusuda, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(SignatureDepartment of  EconomicsThe University of British ColumbiaVancouver, CanadaDate^June 30, 1993DE-6 (2/88)AbstractA model of a fishery with an uncertain fish stock is proposed to compare alternativemanagement systems with individual transferable quotas (ITQs). Assumptions of themodel include: (1) the fish stock fluctuates randomly year by year; (2) in-season stockdepletion is small; (3) the total allowable catch (TAC) set by the quota authorities has adefinite relation with the fish stock level; (4) the true value of the stock level is revealedonly at the middle of each season, when the authorities revise the TAC; (5) fishers formrational expectations on future quota prices. The principal results are: (a) If fishers arerisk-neutral, the share quota (SQ) system and the quantity quota (QQ) system generatethe same amount of fishery rent, although the division of the rent between fishers andthe authorities under one system is different from the other. If the TAC is proportionalto the stock level, the more price-inelastic the demand for fish is, the more likely it isthat fishers are better off under the QQ system at the expense of the authorities. (b) Aquota tax and a harvest tax that collect the same amount of revenue for the authoritiesresult in the same division of the fishery rent among heterogeneous fishers. The quotatax and the profit tax differ in this respect. Which fishers will prefer a quota tax overa profit tax will depend on fishers' shares of the initial quota endowment and in totalinframarginal profits afterward. (c) If fishers are risk-averse, the SQ system and the QQsystem are not equivalent in their allocative efficiency. An example shows that the SQsystem is potentially better than the QQ system when fishers prefer the latter and theauthorities prefer the former. This conclusion has to be modified if risk-neutral tradersparticipate in the quota marketiiTable of ContentsAbstract^ iiList of Tables^ viList of Figures^ viiAcknowledgements viii1 Introduction^ 12 The Basic Model^ 112.1 Assumptions  ^112.2 Notation  ^172.3 Fishers' problem and rational expectations equilibrium  ^182.4 Interpretation of the model: spacial fish stock distribution  ^202.5 Ideas behind results  ^233 Comparison of Alternative ITQ Systems^ 273.1 Equilibrium quantities  ^273.2 Allocative equivalence of alternative ITQ systems ^  363.3 Distributive effect  ^373.4 Adjustment phase ^  484 Tax Effects^ 514.1 Equilibrium under taxation  ^52iii4.2 Comparison of tax effects  ^564.3 Adjustment phase ^  625 The Case of Risk-averse Fishers^ 655.1 No risk-neutral traders case  ^665.1.1 New assumptions and notation ^  665.1.2 Rational expectations equilibrium  ^685.1.3 Fishers' behaviour at equilibrium  ^695.1.4 Equilibrium quota prices  ^715.1.5 Example: piecewise linear utility function  ^745.1.6 Equilibrium quota prices  ^765.1.7 Comparison of the two ITQ systems ^  825.1.8 Numerical examples ^  835.2 Risk-averse fishers with risk-neutral traders  ^896 Conclusion^ 95Bibliography 100Appendices^ 103A Proof of Proposition 1^ 103B Propositions 2, 3, and 4 116B.1 Proof of Proposition 2: profit tax ^  116B.2 Proof of Proposition 3: quota tax  117B.3 Proof of Proposition 4: harvest tax ^  117ivC Propositions 5 and 6^ 118C.1 Proof of Proposition 5: the SQ system ^  118C.2 Proof of Proposition 6: the QQ system  119D Proposotions 7 and 8^ 122D.1 Proof of Proposition 7  122D.2 Proof of Proposition 8 ^  122E VQQ and GQQ^ 127F Table 5.4 131G Equilibrium conditions for the open SQ system^ 133VList of Tables1.1 The quota authorities' revenues and expenditures in New Zealand^. . 61.2 Basic statistics of Hoki fishery in New Zealand ^ 75.3 Critical values of the degree of risk-aversion 835.4 The quota authorities' deficits under the two ITQ systems ^ 865.5 Quota price movements under the QQ system ^ 87viList of Figures2.1 Basic model ^ 142.2 TAC adjustments 142.3 TAC under alternative ITQ systems ^ 153.4 Inframarginal profit under proportional TAC ^ 413.5 Fishers' expected profits under the two ITQ systems ^ 474.6 Lorentz curves for profit share (aj) and quota share (71j) ^ 615.7 Piecewise linear utility function ^ 755.8 Risk-aversion, government's revenue, and fishers' utility ^ 84viiAcknowledgementsI would like to express my deep gratitude to Professors Gordon Munro and Philip Neherwho guided my study in natural resource economics and supported this research fromthe beginning. The constant encouragement and valuable suggestions given to me byProfessor Colin Clark (Mathematics) are gratefully acknowledged. I appreciate the help-ful comments of Professors John Cragg, Mukesh Eswaran, William Schworm. and Mar-garet Slade. I also thank Professors Anthony Charles (Saint Mary's University), MichaelHealey (Westwater Research Centre), and Carl Walters (Zoology/Fisheries Centre) fortheir thoughtful comments and good suggestions at the final oral examination. Theadvice of Professor Quentin Grafton (University of Ottawa) has substantially improvedboth the content and the style of this thesis. Mr. Philip Major (Director of FisheriesPolicy, the Ministry of Agriculture and Fisheries, New Zealand) kindly provided severalkey data. Finally, I would like to give my special thanks to Professor Keizo Nagataniwho has encouraged me ever since I came to UBC five years ago. I was supported by aGovernment of Canada Scholarship for Foreign Nationals and A. D. Scott Fellowship inEconomics during the preparation of this thesis.viiiChapter 1IntroductionIndividual quota management systems are used in several countries of the world as ameans of making fisheries more efficient. Under an individual quota system fishery quotasare allocated to each fisher or vessel and the sum of the quotas which is the total allowablecatch (TAC) is controlled by the quota authorities. The most comprehensive scheme hasbeen adopted in New Zealand since 1986. Australia, Canada, Iceland, and several othercountries have individual quota systems in particular fisheries.' New Zealand's systemcovers most of the inshore and deepwater fisheries. Their system is outstanding notonly in its comprehensiveness but also in the fact that their quotas are permanent andfully transferable. The systems in the other countries are more limited in their coverage,quota tenure, and quota transferability. In Australia an individual quota system hasbeen used for Southern bluefin tuna fishery since 1984. The quotas are fully transferable,although the coverage is limited only to that fishery. Canada has had an individualquota system, known as Enterprise Allocations, since 1984 in several of its Atlanticfisheries. These fisheries include the offshore ground fishery and the inshore cod fisheryalong the west coast of Newfoundland (1984 the offshore lobster fishery (1985the offshore scallop fishery (1986 and the northern shrimp fishery (1987 UnderEnterprise Allocations, however, companies cannot freely buy and sell their allocations.The government has discretionary powers over such actions. In Iceland all demersalfisheries have been managed by an individual quota system since 1984. Annual catch'See Ian Clark et al.[13] for New Zealand, Geen and Nayar [15] for Australia, Fraser and Jones [14]for Canada, and Amason [5] for Iceland.1Chapter 1. Introduction^ 2quotas can be transferred only between licensed vessels. Quotas are allocated to licensedvessels based on their recent average catch and the duration of quotas is a few years at atime. In Norway individual catch quotas are not transferable and are allocated each yearat the discretion of the government. Although the characteristics of individual quotasystems vary from country to country, it is generally recognized that wider coverage,fuller transferability, and longer tenure of quotas are desirable long-term managementgoals.The common property nature of fish resources poses two main problems which haveto be handled by management authorities: (1) if there is no regulation, excessive entry tothe fishery results in the dissipation of resource rent and the depletion of the resource; (2)if the authorities put restrictions on the amount of seasonal harvest through some meansbut fail to curb the intensity of fishing activities, excessive fishing effort and excessiveinvestment in vessels competing for the limited harvests still result in the dissipationof resource rent. Munro and Scott [24] refer to these as Class 1 and Class 2 commonproperty problems, respectively.Many fishery management schemes, such as vessel licensing or shortening of fishingseasons, have failed because they are inappropriate in resolving the Class 2 commonpropety problem. For example, shortening of fishing seasons has led to excessive fishingeffort within seasons and more idle capacity between seasons; Vessel licensing programshave stimulated the installation of ever more powerful fishing equipment. The appealof the individual quota system is that it has the potential to solve problems (1) and(2) simultaneouly. Under the individual quota system, the total allowable catch (TAC)set by the authorities prevents the excessive depletion of the resource. The system alsoprevents the dissipation of resource rent because the overexpansion of fishing activitiesis curbed by individual quota limits. Under this system each fisher will try to harvest upto their quota limit with minimal costs.Chapter 1. Introduction^ 3However, there is no gurantee that the aggregate cost in the fishery is minimized toharvest the aggregate quota limit (= TAC) unless quotas are fully transferable betweenfishers. If quotas are transferable, quota prices formed in the quota market constituteopportunity costs of fishing which fishers have to take into account when they decide thescale of their operation. Then, in principle, quota transferability forces the fishery tooperate in the optimal manner by allowing the shift of fishing effort from less efficientfishers to more efficient ones. We refer to such a system as the individual transferablequota (ITQ) system.Incidentally, the authorities can utilize the quota system for yet another purpose.Those fisheries for which the introduction of quota management is contemplated bythe authorities are most often plagued with excessive numbers of fishers and vessels.In such fisheries the authorities can reduce those numbers to their target levels rathersmoothly without much opposition from fishers by buying back a part of the quotasinitially allocated gratis to fishers. This initial buyback scheme constitutes an importantaspect of quota management systems in real situations.As Clark [10] points out, a correctly calculated tax on catch has, in principle, thesame allocational effect as the ITQ system. The advantage of the ITQ system lies inthe fact that under the system the authorities have direct control over the fish stock andcan attain the (approximately) optimal stock level more easily. This is because they donot need to have detailed information about each fisher, particularly about his harvestfunction and cost function, which is necessary to design the optimal tax scheme. Quotamarket prices are determined in the quota market in a decentralized manner reflecting allthe private information about each fisher.' This is the reason why, despite many practicaldifficulties, ITQ's are considered to be the long-term management goal in many fisheries.'2 Clark [10] gives a comprehensive theoretical analysis of the regulatory measures of fisheries. Clark[11] (Chapter 4) and Clark [12] (Chapter 8) discuss the same topic in sinipler forms.'See Scott and Neher [29], Canada [9], Fraser and Jones [14], and Geen and Nayar [15].Chapter I. Introduction^ 4In view of the important role the ITQ management system will play in the future, inthis thesis I investigate the economic performance of the ITQ system. In particular, westudy ITQ systems which offer permanent tenure and full transferability of quotas.There are basically two variants of the ITQ system depending on how the quotaauthorities adjust the level of total allowable catch (TAC) within fishing seasons andfrom season to season. One is a share quota (SQ) system and the other is a quantityquota (QQ) system. In the SQ system, quotas are allocated as a percentage of the TACof each season and sum to 100 percent. For example, if a fisher possesses 1 percentshare quota and the TAC is 1,000 ton per season, then the fisher is allowed to catch 10ton of the fish in each season, as long as the TAC remains the same. If the authoritiesannounce a change of TAC from 1.000 to 1,500 ton, the fisher's seasonal allowable catchis automatically changed from 10 to 15 ton from that season onwards. The authoritiesneed not sell or buy quotas in the market'.On the other hand, in the QQ system, quotas are denominated in absolute tonnageunit.' If a fisher has 10 ton of quantity quotas, the fisher can harvest up to 10 ton in eachseason regardless of the current TAC. as long as the fisher holds that amount of quota atthe end of each season. Suppose that the current TAC is 1,000 ton and the authoritiesdecide to change it to 1,500 ton. Mere announcement of the change in the TAC has noeffect in the QQ system. In order to put the change into effect, the authorities have toissue and sell 500 ton of quotas in the quota market. In other words, the authoritiesmust trade quota to adjust the TAC. 6 In either system, fishers can sell or buy quotas in'Except possibly at the beginning of the initial year. If the total initial quota endowments in per-centage unit are different from 100 percent, the quota authorities sell or buy the difference in the quotamarket at the beginning of the initial year to adjust the total to 100 percent.'Even if quotas are denominated in tonnage unit, they are equivalent to share quotas, if each fisher'squota holdings are adjusted on a pro-rata basis when the level of TAC is changed.'Throughout this thesis we assume that the quantity quotas obtained by the authorities from fishersare automatically cancelled. This means that we do not count the quantity quotas retained or obtainedby the authorities as a part of the TAC.Chapter 1. Introduction^ 5the market so as to maximize their profit.If the abundance of fish is uncertain but if information increases as fishing activitiesproceed in a season, it will be better for the authorities to adjust the TAC based onnewly acquired information than to stick to an old TAC. When the fish stock turns outto be less (more) abundant, a decrease (increase) in TAC will be desirable. As describedabove, the SQ system and the QQ system are different in the way the TAC is changed. Itis natural to expect, therefore, that the two systems may have different economic effects,even if they have an identical TAC path over time.The recent experience in New Zealand provides a good example of this issue. NewZealand introduced an ITQ system in 1986. It was originally a QQ system', but over timethe authorities incurred a deficit in quota operations when adjusting the TAC's. (SeeTables 1.1 (p.6) and 1.2 (p.7).) In spite of strong opposition from the fishing industry,the New Zealand government decided to change the system from a QQ system to a SQsystem in 1989. 8 The reason for the fishing industry's opposition may be explained byfishers' aversion to income fluctuations under the SQ system in which fishers get nocompensation for their harvest loss when the authorities decrease TAC. But there maybe other economic reasons for the opposition that do not depend on fishers' attitudetoward risk. A related issue is why authorities should lose money through quota marketoperations and if this is a general phenomenon or not. Suppose that authorities tend tolose money with a QQ system but fishers are better off under that system. Which ITQsystem is then better for the society as a whole? Careful theoretical arguments seemnecessary to answer these questions.Few theoretical analyses have been done so far about these problems. Anderson [2]gives a diagrammatic analysis of the economics of the ITQ system in general.' But7See Ian N. Clark et al[13] about the ITQ sytem in New Zealand.8 See Macgillivray [22], p.7.9 Libecap [20] gives supplementary comments on [2].Chapter I. Introduction^ 6Table 1.1: The quota authorities' revenues and expenditures in New Zealand(All values expressed in current $ NZ)Ministry of Agriculture and FisheriesREVENUE1984/85($000)Actual1985/86($000)Actual1986/87($000)Actual1987/88($000)Actual1988/89($000)Budget Quota Leases 0 0 0 3,373 2,000Quota Sales 0 0 60,738 22,835 0Resource Rentals 0 0 0 12,500 20,027Foreign Access Fees 15,129 13,613 19,410 22,974 9,000Deepwater Royalty 2,264 5,720 8,656 0 0Other 745 3,185 2,306 3,204 3,686Total Revenue 18,138 22,518 91,110 64,886 34,713EXPENDITUREPersonnel 9,951 11,406 14,647 16,623 18,557Operating 5,928 7,085 8,995 10,701 13,592Capital 662 782 2,486 538 2,166MAFFish operating 16,541 19,273 26,128 27,862 34,315Buybacks 0 0 44,630 1,364 3,000Other Assistance 2,509 715 486 1,567 74MAFFish Grants 2,509 715 45,116 2,931 3,074Total Expenditure 19,050 19,988 71,244 30,793 37,389SURPLUS/(DEFICIT) (912) 2,530 19,866 34,093 (2,676)(Source: Macgillivray [22], p.11.)Chapter 1. Introduction^ 7Table 1.2: Basic statistics of Hoki fishery in New ZealandYear b TACc[t]Catch[t]Fish price'[S/t]Quota prices[$/t]Volume traded9[t]86/87 250,029 158,171 430 998h (370-4,167) 33,58687/88 250,062 216,240 281 639 (352-2,200) 6,40988/89 251,036 182,310 350 979 (350-2,000) 589/90 251,036 208,546 520 662 (200-2,500) 2,44590/91 201,897 218,172 1 500 846 (400-1,283) 15,721a. Hoki is one of the major fisheries in New Zealand which accounts for about20% of the total catch value in 1991. Cf. The source for e. f, g (1991) p.51.b. 86/87, 87/88, 88/89 — QQ system; 89/90, 90/91 — SQ system.c. TAC's (Total Allowable Commercial Catches) are at year end dates(i.e., 30 September). Here TAC under the QQ system seems to be defined byTAC = total quantity quota held by fishers and the authorities.If this is the case, a reduction of TAC under the QQ system occurs only whenthe authorities cancel any quota held by them. Notice that this is differentfrom our definition of TAG which does not count the quotas retained by theauthorities as a part of the TAC. Cf. New Zealand [27], sections 6, 8 of pp.16-8.e. Indicative port prices.f. Weighted average (low - high).g. Volume of quota traded in perpetuity.h. Numerical average.i. Quota holders were able to fish against up to 10% of uncaught quota carriedover from 1989-90. See Annala [4], p.100.(Source. c,d: Atlas of Area Codes and TACCs 1991/1992 published by Clementand Associates; e,f,g: New Zealand Fishing Industry Review 1986-87, 1987-88,1988-89, 1990, 1991 published by the NZ Fishing Industry Board.)Chapter 1. Introduction^ 8the effect of uncertainty in fish abundance is not discussed there. Arnason [6] analysesthe relative advantage of the two ITQ systems with regard to the optimization of adeterministic and non-seasonal fishery model. As such, his model is not appropriateto address the issue of seasonal TAC adjustment under uncertainty. Hannesson [17]analyses the economic performance of the QQ system under stock uncertainty in a mannersimilar to mine but with a much simpler aggregated model. His model, however, tacitlyassumes that TAC adjustments are carried out through seasonal quota lease betweenfishers and the authorities. Hence his results, although very suggestive, do not directlyapply to the QQ systems in which the level of TAC's is adjusted through permanentquota trades between fishers and the authorities as assumed in my analysis.' One ofthe purposes of this thesis is to bridge the gap in the existing literature by providing ananalytical framework suited for the study of ITQ systems with permanent quota understock uncertainty and to find the answers to the above questions.With ITQ's, the rents generated in the fishery accrue to the original quota holders.Those fishers who are originally endowed with large amounts of quota may, therefore, reapsubstantial capital gain. It can be argued that this kind of "windfall gain" for priviledgedfishers is not desirable from the perspective of social equity. Those who consider equitabledistribution very important may argue that all (or most) of the fishery rent should revertto the state (the general public) who owns the fishing ground and the fishery resource.It has been shown that this problem can be solved by using both ITQ's and taxes onfishers at the same time. In principle, fishery rents can be split up between fishers andthe authorities at any desired ratio with this method.' There are many different taxforms that can serve this purpose. They include profit taxes, quota taxes, harvest taxes,etc. The problem is that they may have different effects on the distribution of fishery'Andersen [1] gives some comments on [1 7] .11 See Clark [12] Chapter 8.Chapter 1. Introduction^ 9rents.The New Zealand case again provides an example. In the ITQ system of New Zealand,a resource rental is levied on fishers. The resource rental is set according to the quotatrading values. The reason is that under this system "... the value at which quota istraded is the best guide to economic rent, and therefore should be the determinant of theresource rental." 12 However, "The New Zealand fishing industry has consistently arguedthat quota trading data are inappropriate for setting resource rentals. This has led tolong and vigorous debate between the industry and government." 13 So far, there are notmany theoretical works in the literature about advantages and disadvantages of varioustaxes in the fishery under an ITQ management system. Grafton [16] gives some analysisabout this matter, although his model does not take into account fish stock fluctuationswhich can play a major role in ITQ management systems. This thesis will show that thetheoretical framework developed here can be applied to the analysis of tax effects in ITQmanagement systems under stock uncertainty.The real situations surrounding ITQ systems are very complicated both economicallyand biologically. I have to make many simplifying assumptions in my model in order toget clear analytical results. In chapter 2 I explain the basic structure and assumptionsof the model. In chapter 3 I derive equilibrium quantities for the SQ system and the QQsystem with risk-neutral fishers. A comparison of the two systems is then made withregard to the division of the fishery rents between fishers and the authorities. In chapter4 I examine tax effects in a fishery managed under an ITQ system. Using equilibriumquota prices under taxation, I compare the effects of taxes on the division of the fisheryrents among fishers with different levels of efficiency. In chapter 5 the comparison of theSQ system and the QQ system is extended to the case of risk-averse fishers. I discuss12Ian N. Clark et al. [13] pp.334-337.13 [13] p.337.Chapter 1. Introduction^ 10the limitations and the possible extentions of our model in chapter 6.Chapter 2The Basic Model2.1 AssumptionsAny economic model of a fishery incorporates many simplifying assumptions reflectingthe complexity of the subject. The complexity arises from both the biological and eco-nomic sides of the problem. Uncertainty and dynamics are the main ingredients of thiscomplexity. The difficulty surrounding the estimation of stock-recruitment relationshipsis well-known. There are too many unknown factors which affect the relationship. Notonly the fish stock dynamics but also the fish stock size at any point of time are veryhard to estimate. On the economic side, it is difficult to model the behaviour of fisherswho face uncertain fish stock dynamics and uncertain fish stock size. If fishery quotasare traded in the quota market, the movement of future quota prices is also uncertain.Fishers' behaviour depends on their expectations of these highly uncertain factors. Theauthorities' behaviour concerning the determination and adjustment of the level of TACcan also be very complicated.The model in this thesis is based on the standard single-species, single-cohort, seasonalfisery model which is commonly used in the literatures. The uncertainty of the fish stocklevel is crucial for the TAC management under the individual transferable quota (ITQ)system, whereas the inclusion of uncertainty into the model makes the analysis muchharder. We make the following assumptions as a compromise.1 See, eg., Clark [10]. [11].IAChapter 2. The Basic Model^ 121. Under the TAC management, the level of the fish stock. X t , fluctuates randomlyfrom year to year according to a known probability distribution with expectedvalue x over a range [b, , where b > 0. This implies that we can ignore thestock-recruitment relationship in the fish population dynamics.2. The fish stock does not deplete significantly in each season even under the pressureof harvesting activities because the TAC is assumed to be small compared withthe total fish stock. The fish stock level, Xt , therefore, is approximately constantthroughout each season.3. One fishing season occupies a part of one year. Each fishing season consists of twoperiods with equal length, period A and period B (see Figure 2.1, p.14). Duringperiod A the exact level of the fish stock in the year is not known. It is revealedonly at the end of period A, presumably the result of catch-effort data analysisconducted by the authorities.'4. The quota authorities try to set the level of TAC equal to a variable proportionK(Xt ) of the total fish stock X t . 3 When the exact level of fish stock is unknown,they set the TAC equal to n.(x)x based on the average stock level x. 4 (The overalloptimization problem of how the authorities should relate TAC to the total stocklevel is beyond the scope of this thesis. 5 )2 Hilborn and Walters [19] give detailed accounts about the techniques used in the actual stockassessment.3The actual practice of TAC setting in New Zealand's ITQ system is well documented in Annala [4]about its scientific bases and in New Zealand [27] about its legal aspects.'Assumption 1 (stable probability distribution) and Assumption 4 (TAC adjustments) are not inde-pendent of each other in our model. We can think that the stable probability distribution of fish stocksize around the average level x (Assumption 1) is maintained year by year because of the authorities'TAC adjustments accommodating fish stock fluctuations (Assumption 4). This point becomes clear ifwe imagine the situation in which the authorities do not impose the TAC limit. Fishers will harvest asmuch as they like and the stability of fish stock level will be lost in overfishing.5 Hannesson [18] gives some analysis about this matter.Chapter 2. The Basic Model^ 135. The quota market opens twice a year at the beginning of each period.(a) Just before the market opens in period A, the authorities change the level ofTAC from last year's K(X t _ i )Xt _ i to the provisional level K(x)x. At this pointthe true stock level Xt of the year is unknown. (See Figure 2.2, p.14.)(b) At the end of period A, knowing the true stock level X t , the authorities changethe level of TAC from the provisional K(x)x to K(Xt )Xt .6. We consider a very general hybrid ITQ system that incorporates the genuine QQsystem and the genuine SQ system as two special cases. In this hybrid system theauthorities divide the overall TAC into two parts, one for share quotas and theother for quantity quotas (see Figure 2.3, p.15): 6TAC for share quotas [ton] = f(z)TAC for quantity quotas [ton] = K(z)z — f(z)where z = x for period A, z = X t for period B, and0 < f(z) K(z)z Vz E [b,Each share quota is denominated as a fixed proportion of the TAC for share quotas.The authorities announce the TAC for share quotas at the beginning of each period.If the total amount of quantity quotas owned by fishers is different from the TACfor quantity quotas, the authorities have to either sell or buy back the difference inthe market to implement the TAC.7. We denote by N the number of fishers who remain in the fishery from year 1onwards. N may change over years and may be larger or smaller than the originalnumber of fishers before year 1.6 f(z) E. 0 case corresponds to the genuine QQ system and f(z)^K.(z)z case corresponds to thegenuine SQ system.randomfluctuationknown(X2 )N^unknownChapter 2. The Basic Modelfish stockXtknown.(X 1)\J kunknownrandomfluctuationFigure 2.1: Basic modelA^A14^ tfishing season^fishing seasonyear 1 year 2Figure 2.2: TAC adjustmentsTAC [ton]>7KX2A Afishing season fishing seasonyear 1^year 2QQ (SQ)> xt0TAC [ton]Genuine QQ (SQ) systemN./b15Chapter 2. The Basic ModelFigure 2.3: TAC under alternative ITQ systemsTAC [ton]General hybrid system^Ic(Xt)Xtf(Xt)SQ>^ XtChapter 2. The Basic Model^ 16(a) Fisher i's variable effort cost function ci(.) is the same for all periods. A crucialassumption is that this function is strictly convex. In addition to variable costs,fisher i incurs annual setup costs^> 0 if they participate in the fishery inthat season. Fisher i's total cost in year t is thenci( eAt + c, (e tBi) +where etli and eri are the fisher's effort levels in period A and period B,respectively. We assume that effort level eit i (j = A, B) is constant duringperiod j. Effort levels measure the volume of sea water screened by vessels ina unit time. A typical unit for effort levels is [m 3 /period].(b) Fisher i's harvest function in period A and B of any year t has the standardform'yA ytBi - qxtetBi= ate tAi andwhereq catchability coefficient which is assumed to be constant. 8(c) Fishers are all risk-neutral. Namely, fisher i maximizes the expected presentvalue of his profits E 6'141, where 'et is the fisher's net annual profitand 6 is the discount factor. (1/6) —1 is assumed to coincide with the marketrate of interest. Fishers form rational expectations on future quota prices.This implies that their anticipated distribution of future quota prices is con-sistent with realized prices.8. The market demand for the fish depends on the price of landed fish, pt , throughthe demand function D(pt ). p i is determined so that D(pt ) equals the supply of7See Section 2.4 for more about this.8 For a detailed account of the definition and the meaning of catchability coefficient, see Clark [11],Sections 2.1 and 2.2.eiChapter 2. The Basic Model^ 17the fish St , i.e.,pt = D -1 (Si).This demand function is assumed to be known to fishers and the authorities.2.2 NotationWe use the following notation throughout this thesis. Fisher i's annual net income inyear t is7rt =^rpiqXte —^wt • (11P^) + wi • ( - 6 )] -j=A,Bwhere 9Pt = the price of landed fish (unit: $/ton)q = catchability coefficient (unit: /m3 )Xt^the level of the fish stock (unit: ton)et^fisher i's effort level in period j (unit: m3 /period)quantity quota price in period j (unit: $/ton)fisher i's before-trading qauantity quota holdings in period j (unit: ton)at'^fisher i's after-trading quantity quota holdings in period j (unit: ton)share quota price in period j (unit: $/TAC for share quotas)fisher i's before-trading share quota holdings in period j (unit: %)fisher i's after-trading share quota holdings in period j (unit: %)= seasonal setup costs (unit: S)'As a matter of fact, the share quota -price"^A, B) is the total monetary value of sharequotas in the market.Chapter 2. The Basic Model^ 18andat-1^ h t ^att^t(tBil = otAi, C;1i = jBitbecause after-trading quota holdings in each period are carried over to the next period.2.3 Fishers' problem and rational expectations equilibriumOne of the key points of the ITQ management is that fishers are not allowed to sell morefish than their final quota holdings each season. At the end of each season they have topossess enough quotas to cover their harvest of the season, if they want to sell all theharvested fish. We assume that there is no way for fishers to carry over their harvest ofthis year to next year. Hence, each fisher has to satisfy the following quota constraint:qxtetAi qxtetBi < aiBi^f(Xt)where the LHS is fisher i's total harvest in year t and the RHS is the fisher's final quotaholdings in tonnage unit in the same year. aP i (Sri • f (Xt )) [ton] is his season-end quantity(share) quota holdings.Given the initial quota endowments and based on his expectations about future quotaprices {4, WtA; wr,,^fisher i chooses a contingency plan {e tAi , a tAi , Cim ; e tB, aPi CBilrxt t=1 7 J=1about his effort level and quota holdings of each period so as to maximize the expecteddiscounted sum of his profits{E E.- 6 t_i..r.,ittt=isubject to his quota constraintqxt etAi qxt eiBi < atBi ) for t = 1, 2, 3, ...Chapter 2. The Basic Model^ 19It is assumed that fishers form rational expectations on the quota price sequencef A^A^B^.131cx)iwt " t ; wt^7t^The sequence is, in general, a stochastic process which de-pends on the stochastic movements of Xt. Then fisher is optimal contingency plan for{ eAt aAt eSt aBt cri}toci is also a stochastic process that depends on the stochasticpath of Xi . The rational expectations hypothesis implies that fishers perceive the correctrelationships between the fish stock level and quota prices.Our fundamental hypothesis about the working of the quota market is that the currentquota price in any period is determined so as to clear the market. Formally, we assumethat the quota prices wt > 0 and 14/1 > 0 (j = A, B) are adjusted in the market so thatthe following market equilibrium conditions hold in the quantity quota market and inthe share quota market, respectively:1 Ei ate < K(z)z — f (z), = if wi > 0,^Ei a' < 1 ,^= if WI > 0,where z = x for j = A, and z = X i for j = B. The LHS's are total quota demands andthe RHS's are total quota supplies.A rational expectations equilibrium' for this model is defined by stochastic processesw2t4 ,Willi4,wtB and , (i 1,...,N) that satisfy thefollowing two equilibrium conditions':n1. Given fishers' optimal contingency plans for setting {^-tAi^-t -t^1 t=1tA(i = 1,^N), the stochastic process for quota prices^w wtB,^i-wtBr  clearsthe quota market in each period, i.e.,1 E i ajt i < k(z)z — f(z), =Ei Ciii < 1, = if 'ail' > 0,if II/ti > 0, "Newberg and Stiglitz [26] (Ch. 10) succinctly explain and defend the concept of rational expectationsequilibrium in the context of agricultural production."For the definition of a rational expectations equilibrium in this form, see Sargent [28], p.403.Chapter 2. The Basic Model^ 20where z = x for j = A, and z = Xt for j = B.2. When fishers face the stochastic process {4, w-4 ; wB , 1413 } ctl), as price takers, the{ 6At atAi ,^eSti,aBti,(tBiltocistochastic process^ maximizes expected present valueE {E,"=1 6' 1 70 for each fisher i = 1,^, N.2.4 Interpretation of the model: spacial fish stock distributionThere are two possible interpretations about the spacial distribution of the fish stock inmy model: (1) uniform distribution, and (2) patchy distribution.1. Uniform distributionLike many standard fishery models, we can interpret our model assuming that thefish population Xt is uniformly distributed over the fishing ground. Consequently, theproduction function yi = qXt e't (j = A, B) is easily understood. However, there isa difficulty about this interpretation. Suppose it is literally the case that the spacialdistribution of fish is (approximately) uniform. Then fishers will realize the true stockdensity at a very early stage of period A from their harvest records. Then it is naturalto expect that they will adjust their effort levels and quota holdings long before thebeginning of period B according to this new information about the true fish stock level.This leads to a model inconsistency, because one of our basic assumptions is that fishers donot adjust their effort levels and quota holdings until the beginning of period B. Therefore,if fishers do not adjust their effort levels and quota holdings until the beginning of periodB as assumed in our model, it has to be the case that the spacial distribution of fish isnonuniform (patchy) and fishers cannot form an accurate estimate of the fish stock leveluntil after having been at sea for a substantial time. Therefore, we are led to the secondinterpretation with patchy distribution.Chapter 2. The Basic Model^ 212. Patchy distribution'Suppose that the fish stock in the fishing ground consists of many schools which areall of the same size. We denote the total number of schools in year t by Xt . Let us denoteby At [school/hour] the expected rate of encounter with fish schools when the effort levelis el IT [m3 /hour] (j = A, B), where T hours = 1 period. Then^X t [school]^1^•= — [m3 /hour] x V^ [m3] = VT e't Xt [school/hour],Twhere V [m3] is the total volume of water in the fishing ground.We assume that"1. The probability of encountering one school in time ( s, s ds) = Vt ds+o(ds), whereo(ds) limas– o ds = 0;2. The probability of encountering two or more schools in time (s, s ds) = o(ds);3. The number of encounters in nonoverlapping time intervals are independent.Then the probability of encountering n schools in a time interval s is given by the followingPoisson distribution 14 :Pr (np4) = (A sr^S^(j = A, B)This distribution has mean Vi s and variance .1t s. Therefore the coefficient of variation0 if s is large."-VVt.Since the spacial distribution is patchy and the encounter with schools is random,fishers cannot know the true average rate of encounter )it immediately. As a fishing12 See Clark [11], Chapter 2.4.13 See Mangel and Clark [23], pp.19-25.14e in the expression is the base of natural logarithm.15 1f s is large, the central limit theorem guarantees that the above Poisson distribution approximatelyfollows the normal distribution with mean V, s and variance Vt s.Chapter 2. The Basic Model^ 22season proceeds, they update their estimation based on their harvest records. Then ourbasic assumption about fishers' behaviour amounts to an assumption that fishers changetheir effort levels and quota holdings only at the beginning of period B when they haveaccumulated enough evidence about the true stock level (i.e., the existence of a threshold).This seems to be a reasonable assumption. However, we have another problem. Sinceencounters with fish schools are random, the total catch yi 4 [school] in period A andyP [school] in period B (so yid + y1-3 in year t) are also random. Let us assume thatfishers can catch all the schools they encounter. We interpret the catchability coefficientq as q = 1/V [1/m3 ]. Then )4 = qXt eUT [school/hour] holds and the average number ofschools caught in period j when the effort level is et will be given by )471 = qXt e't [school].Namely, y tA and y tB follow the Poisson distribution with means qX t e tA [school] and qXte tB[school], respectively. Therefore if the spacial distribution of fish schools is patchy, theharvest functions yt = ateit (j = A, B) in our basic model cannot be exact relationships.However, as noted above, the coefficient of variation of yi is 1/04 (j = A, B) which issmall when the length T of one fishing period is large enough with the level of At given.Therefore, the magnitude of variation of yi relative to the mean values qX4 is smallif each fishing period is sufficiently long. Hence, we can interpret our harvest functionyti = qXt elt = A, B) as a first approximation.If the authorities monitor all the fishers' harvest records, the authorities' estimate ofthe fish stock level is more accurate than each individual fisher's estimate by the law oflarge numbers. This can be another reason that fishers wait to make decisions abouttheir behaviour until the beginning of period B when the authorities announce a newTAC level based on their latest stock estimate. Our basic assumption that after periodA the authorities' stock estimate is sufficiently close to the true value Xt is also justifiedon this ground.Chapter 2. The Basic Model^ 232.5 Ideas behind resultsUsing the basic model described above, we examine the economic characteristics of ITQmanagement systems. Before going into detailed arguments, we explain the basic ideasbehind them. They will clarify the meaning of the results derived below. Those ideasare simple and intuitive.Imagine a fishery managed under the genuine QQ system defined above. Let epi bethe profit maximizing effort level in period B for fisher i. Let mBt be the quota rentalprice (for the season) in period B. Then1not = Pt — qXtc(e,Bi)has to hold at equilibrium. On the RHS, is the marginal cost of harvestingone more unit of fish, where qXt is the harvest from a unit effort. The RHS, therefore,expresses the maximum amount of money fisher i is ready to pay for renting one moreunit of quota for the season. In a quota market equilibrium this value should be equalto the quota rental price m tB. In other words the value of 'e tBi is adjusted so as to satisfythe above equality, where the values of X i and pi are known to fishers with certainty inperiod B.The true fish stock level Xi is not known in period A. Suppose that Xi turns outto be larger than the average x in period B. Then the quota authorities have to sellkXt — kx(> 0) of quantity quotas.' If Xi < x, then the authorities have to buy backkx — kXt (> 0). Now the capital price of quota, wr, will be determined reflecting therental price, 772P, and they will move together in the same direction for different Xi values.Therefore we can infer the following: 17'Here we assume for simplicity that the TAC is a fixed proportion k of the fish stock level.17The same idea is used in Hannesson [17](pp.461-462), although his model implicitly assumes an ITQsystem with one - year quantity quotas for which zr2f3 43.Chapter 2. The Basic Model^ 24• If m tP moves in the same direction as the fish stock level X t , then in period Bthe authorities{sell quotas when X t and 4 are largebuy quotas when X t and t1,13 are smallAs a result, the authorities gain money through quota market operations under theQQ system.• If n-43 moves in the direction opposite to Xt , then in period Bthe authorities I sell quotas when quota price tvP is lowbuy quotas when quota price wr is high,which will cause loss for the authorities.Thus the relative movement of X t and m-P is the key to understand the outcome of thequota market operations under the QQ system.The formula1int = Pt ^eVri)qXt(i = 1, ... ,N),suggests that there are three elements which may affect the movement of mr relative toXt : (1) Pt , (2) 1/qXt , and (3) c(ePi).1. If we assume a downward sloping demand curve for fish, the price of fish Pt movesin the opposite direction of fish supply kXt^TAC). In this case, pt moves in theopposite direction of Xt.2. 1/qXt is the amount of effort to harvest one unit of fish when the fish stock levelis Xt . This moves in the opposite direction of Xt .Since pt and l/qXt move in the same direction for different X t values, we are notsure whether mr and Xt move in the same direction or not.Chapter 2. The Basic Model^ 253. ci ( '6ei) is the marginal effort cost at equilibrium. Without further information, wecannot determine in which direction this term moves relative to Xi .The rest of this thesis is devoted mainly to the investigation of the conditions underwhich we can determine the relative movement of Xi and me (or we).Remark: We have not mentioned the market operations of the authorities in periodA. The reason is as follows. Suppose that quota price in period A, 4, is constantover time. 18 Since the total quota holdings in the previous year (= kXi_ i ) fluctuateevenly around the provisional level of TAC of this year (= kx), the effect of quotamarket operations on the authorities' revenue cancel out over time. Therefore when weconsider the authorities' revenue in the long run, we can ignore the effect of quota marketoperations in period A. 19How about the genuine SQ system? How can we compare the SQ system and theQQ system? Here we follow the argument stated in Newbery and Stiglitz [26] (Ch.15).Although our model is dynamic whereas theirs is not, their results are general and canbe used as a useful heuristic principle.The principle takes the following form when applied to our fishery model. Supposethe following conditions are satisfied:1. Fishers are risk-neutral.2. Fishers form rational expectations on future prices.3. Fishers behave as price takers.4. Free entry into and exit from the fishery.18 We will show in Chapter 3 that this is indeed the case under certain conditions."This is true for fixed proportion TAC rules, but not for general TAC rules.Chapter 2. The Basic Model^ 26Then under a given TAC policy of the authorities, an ITQ management system, whetherit is a QQ system or a SQ system, should bring about the efficient outcome that maximizesthe total monetary benefit from the fishery. Thus a QQ system and a SQ system underthe same TAC policy should be equivalent to each other in their allocative efficiency.They should give rise to the same amount of total fishery rent.This means that fishers and the authorities play a "constant-sum game" under a givenoverall TAC adjustment rule. For example, consider a QQ system and a SQ system withinitial quota endowments equivalent to each other. Then whether fishers are betteroff under the QQ system or not is equivalent to whether under the QQ system theauthorities' expected revenue through quota market operations (after the initial quotasales) is negative or not. Therefore the above argument about the relative movement ofm tB (or wP) and X t that affects the authorities' revenue under the QQ system have adirect implication on the relative advantage of the two ITQ systems.Chapter 3Comparison of Alternative ITQ Systems3.1 Equilibrium quantitiesAs mentioned in the Introduction, there is a dispute concerning the relative advantageof the SQ system and the QQ system when the fish stock size fluctuates randomly fromseason to season. We analyze this problem by using our basic model with a generalhybrid system which incorporates these two systems.We make the following assumptions about equilibrium quota prices (w^A; wr , wtB ):I. (4, I/1/A; wr, I/VtB) are described by stationary (i.e., fixed) functionswi4 =wB =^OA, Xi ),WiA (1)(-W,WtB = (Hi, 0^),where HA (0-ii) is the aggregate quantity (share) quota holdings of all fishers atthe beginning of year t.2. Implicit in the above is that (4, WtA; wr, WtB ) may depend on the level of theaggregate initial quota holdings HA and (V but not on the distribution of HA and(4- among (heterogeneous) fishers.In Proposition 1 we focus on the equilibrium in which quota prices take the aboveforms. Rational expectations hypothesis then implies that fishers know (or correctlyguess) these functions. Proposition 1 gives various quantities at equilibrium. The propo-sition is followed by detailed remarks (pp.29 - 36) which explain the meaning and the27Chapter 3. Comparison of Alternative ITQ Systems^ 28intuition behind it.Proposition 1 Suppose that under the hybrid ITQ system the authorities endow fisheri with hP ton of quantity quotas and OP • 100 percent of share quotas before year 1(i = 1,2, ...). Suppose also that entry to and exit from the fishery are free. Let us denotethe expectation operator with respect to Xt by Et {•} .^Then the equilibrium number offishers N and the equilibrium effort levels e: it i (i = 1, .^, N; j = A, B) satisfiy^^Ai^Aie t^=^e^=^constant,^ (3.1)Et feaefli )} , (3.2)^c ti(eri ■^=^cAt _(eBN)^g(Xt),)^= (3.3)Et {g(Xt) • ( oN^eSt N)}^=^Et {cN(oN) eN (.13t N ) 67 } (3.4)at(eAl^eri)^K(Xt)Xti=1on the assumption that1(3.5)0^VXt^d and (3.6)Pt^g(Xt)^> E [b,^,qXtABt^>^0^Vi=1,...,N;et VXt E [b,ar (3.7)Here the numbering of fishers is such that for i < NEt {g(Xt)^(eAi^eBi ) }^>^Et f ei(oi ) i)i } (3.8)and fishers 1, ... , N include all the fishers for whom (3.8) holds.Fisher i's quota demands ajt i and^(i = 1, .^N; j = A, B) are indeterminate onthe condition that they satisfy the quota market equilibrium conditionsNE^= ri(z)z — f(z)^ (3.9)i=1E c/a^1 (3.10)Chapter 3. Comparison of Alternative ITQ Systems^ 29where z = x for j = A and z = X t for j = B.The equilibrium quantity quota prices [$1 ton] areAWt _ 1 — SEt {Pt — qXtg(Xi1wB = Pi — qXt g(Xt)+ Sz-vAand the equilibrium share quota prices [$1 TAC for share quota] are==: (3.11 )(3.12)1 1 6 Et{[Pt q^Xtg(X0] • f(X t )} =: WA (3.13)wtB = 1Pt ^qXt g(Xt)] • f (xt) + SWA . (3.14)The expected present value of fisher i's net profit Vi(•,•) is17 i (joi/ eiti ■)^-/1)AhAi _L^_L 1^bEt {g(xt).(e .ki^e i )^,i11^1 (3.15)where hl i (OP) is fisher^initial quantity (share) quota endowments before year 1.The authorities' expected revenue G(•,.) through initial quota sales (or buybacks) andsubsequent quota market operations isG(I-V ,^=^HiA — WA + 1 1^Et {[pi — x1 i g(Xt )] • K(Xt )Xt }q^ 3.16)where HA (Of) is the sum of initial quantity (share) quota endowments of all fishers.Appendix A gives the proof of this proposition.Remark1. (3.1) and (3.3) say that the equilibrium effort level in period A, e-m , is constant overtime, while the equilibrium effort level in period B, q3i, may depend on the fish stocksize Xt of the year. The condition (3.2), = Et {cz( ipi ) }, says that e fti is someChapter 3. Comparison of Alternative ITQ Systems^ 30average level of EP i . The assumption (3.7), etBi > 0 (i = 1, . . . , N; VXt E [b, d]),implies that regardless of the fish stock size X t of the year, all the fishers (includingrelatively inefficient ones) who are active in period A can continue to harvest prof-itably in period B. This facilitates the derivation of equilibrium quantities. Withoutthis assumption, some of the less efficient fishers' optimal effort levels Jpi may be-come zero in period B (i.e., corner solutions) when the true fish stock level X t isrevealed and the level of TAC is changed accordingly in period B. This possibility ofcorner solutions will make the calculation of equilibrium quantities more difficult.2. As explained below (see page 33),Et {g(Xt) • (oi etBi ) _ ci(oi) -^_expresses the expected inframarginal profit for fisher i in year t. Therefore thecondition (3.4)Et {g(Xt) (eAN e tB N )1 = Et feN(esAN ) etv(eBN ) &).means that the expected inframarginal profit of the marginal fisher N is equal tozero. The condition (3.8)Et {g(Xt) • (eA '^)}^Et f ci (0 ' )^ci( e't5 '^( i^N )says that only those fishers (i = 1, . . , N) who can expect nonnegative infra-marginal profits remain in the fishery from year 1 onward.3. The condition (3.5)i qxt (e-4i +^= K(x,)-( ti .1says that total harvest equals TAC each year.Chapter 3. Comparison of Alternative ITQ Systems^ 314. The meaning of the assumption (3.6), pt — --)1-(ty g(Xt ) > 0 VXt , is the following.g(Xt ) = cii (e}i3i) (i — 1,  ^V) is the marginal effort cost of fisher i in period B.qXt is the marginal harvest from a unit effort. ThenqXtSince he can sell one unit of fish for pt [$], pt — ---k g(Xt ) expresses the maximumamount fisher i is willing to pay in period B for renting a unit quota for the seasonto harvest one more unit of fish. In a competitive equilibrium, this is equal tothe market rental price of a unit quota in period B. Therefore the assumptionpt — tq g X t > 0 VX-t guarantees that the rental price is always positive. Thisimplies that fishers harvest up to their quota holdings limits, because if the rentalprice is positive, there is no point for fishers to leave unused quotas without rentingthem out.5. Fisher i's after-trading quota holdings in period A (kli and &ti) are indeterminate.This is because the equilibrium quota prices (3.11) - (3.14) satisfywA =Et _B1lLut j and WA = Et {1,17tB } .Since the fisher forms rational expectations on wia and WtB by assumption, the RI-ISof the above equalities coincide with the fisher's expectation on the quota pricesin period B. Therefore, how much quota fisher i holds in period A does not affecthis expected profit perceived by him in the year. For example, even if fisher i doesnot buy enough quota in period A, he can always buy more quotas in period B atthe price whose expected value is the same as the price in period A. Although eachfisher's quota demands in period A (41i and ( 11 ) are indeterminate, the conditions((3.9) - (3.10) for / = A)1g(Xt ) = fisher i's marginal cost of harvesting one more unit of fish in period B.A Aitotal quantity quota demand (E a t )^total quantity quota supply (k(x)x — f(x))Chapter 3. Comparison of Alternative ITQ Systems^ 32total share quota demand (E ";'1i ) = total share quota supply (1)i=imust hold at the quota market equilibrium.6. As we saw above, the assumption (3.6), pt — —k g(Xt ) > 0, gurantees that fisher i'squota constraint binds at equillibrium, i.e.,qXt e -4i qXteSti^aBti^f(Xi).Fisher i's quota holdings in period B, a (quantity quotas) and^(share quotas),satisfy this equation. However, the division between quantity quotas and sharequotas is indeterminate in period B. In other words, how fisher i divides his quotaholdings in period B into quantity quotas and share quotas does not affect hisexpected profits. The reason is clear. At equilibrium, quota prices are adjusted sothat both quotas are equally profitable for fishers. As in the case of quota demandin period A, however, the conditions ((3.9) - (3.10) for j = B)^total quantity quota demand^total quantity quota supplyN(i.e., Ear K(xi )xt — f(Xt ))^total share quota demand^total share quota supply(i.e., E^= 1)i = 1have to be satisfied at equilibrium.7. We can interpret the expressions of the equilibrium quantity quota prices1^1fpt —^g(Xt )}1 — qXt1= Pt qXtg(Xt) 61-DAandas^Chapter 3. Comparison of Alternative ITQ Systems^ 33the cost of acquiring^expected profit fromone more ton of quota^harvesting one more ton of fishin period A^in all subsequent years,andthe cost of acquiring^profit from harvesting^discounted sales value ofone more ton of quota = one more ton of fish^+ one more ton of quotain period B^in period B^in period A of next yearLet us denote the equilibrium share quota prices in quantity unit by WA [Slton]and VMS Iton] instead of WA [$/TAC for share quota] and 1/1/73 [5/TAC for sharequota]. Because, by definition, f(z) [ton] = T4C for share quota (z = x in periodA and z Xt in period B), we have WA = WA I f(x) and WiB = WiB i,f(Xt)• Thus1^(^ 11 _ 6 Et [Pt qXt^f(x)9(Xt)] (Xt) }Pt^1^g(Xt)+ (5VVA  f(x) qXt f(Xt).The interpretation of these expressions is similar to the one given above.8. In the expression (3.15)vi(12.1", op)^ti)A^WolAif(x)1 1 6 Et{g(xt)•(0 , -Feri)—q(eAi)—c(en-0the first two terms on the RHS express the monetary value of fisher i's initialquantity and share quota endowments, respectively. The last term expresses thediscounted sum of fisher i's inframarginal profit. This interpretation of the lastterm is derived as follows. From the expression of wB, we obtaing(Xt)= [Pt — (wr — 6'17/1)]qXtWAWB =Chapter 3. Comparison of Alternative ITQ Systems^ 34wherept — (wr — 6fOA) = revenue (net of opportunity cost) from a unit harvestqXt = harvest from a unit effort.Thereforeg(Xt ) = revenue (net of opportunity cost) from a unit effort.Hence^Et {g(Xt ) • (E li^eSt i)^ci (eAi)^i)^i}expected inframarginal profit for fisher i in year t.9. In the expression (3.16)G(1-4,14) =^—^1 1^6 Et {[Pt ^g(Xt)] • n(Xt)Xt}— qX tthe first two terms on the RHS express (the negative of) the total monetary valueof the initial quantity and share quota endowments, respectively, that are givengratis to fishers initially. The last term corresponds to the expected present valueof the so-called "management rent" from the fishery when the authorities adjustthe level of TAC according to TAC = K(z)z (z = x in period A and z = Xt inperiod B).10. Although fishers are indifferent about the division of their quota holdings intothe two kinds of quotas from year 1 onward, they are not indifferent about thecomposition of their initial quota endowments. The expected present value of fisheri's profit isvi(hAi i . 0Ai^3;vIftoiAif(x)^1 —Et {9(-Xt) • (eAl + ‘r i ) — ci(e^—Chapter 3. Comparison of Alternative ITQ Systems^ 35hAi +OP f (x) [ton] is fisher i's total quota holdings in tonnage unit at the beginningof period A of year 1. Suppose that initially fisher i is given this value and is givenan opportunity to choose the ratio of quantity quotas and share quotas. Then thefollowing is clear from the expression of Vi(hP• If 2.0 > WA [$/ton], fisher i prefers initial endowments before year 1 with alarger quantity quota ratio.• If ft) A < WA [$/ton], fisher i prefers initial endowments before year 1 with alarger share quota ratio.The reason is the following. In the case of t-vA > WA , a unit quantity quotais priced higher than a corresponding amount of share quota. Since we assumerational expectations, this must be the reflection of fishers' expectation that havingquantity quotas will be more profitable than having share quotas. This profitabilitydifference between the two kinds of quotas is the result of the market operationsof quantity quotas. If we remember the argument in. Section 2.5 (Ideas behindresults), we can infer that the case of tOA > WA corresponds to the situation inwhichthe authorities buy quantity quotas when the quantity quota price is highsell quantity quotas when the quantity quota price is low.This mechanism serves as an implicit subsidy to fishers who hold quantity quotasinstead of share quotas. ZliA > WA is the reflection of this situation.However, only those fishers with initial endowments of quantity quotas who getquantity quotas for nothing can benefit from this "subsidy". For from year 1onward equilibrium quantity quota prices are determined fully incorporating thebenefit of this "subsidy" under the assumption of rational expectations. HenceChapter 3. Comparison of Alternative ITQ Systems^ 36fishers cannot expect extra gain from purchasing quantity quotas. The high priceexactly offsets the benefit of the "subsidy". This is why from year 1 onward fishersare indifferent about the division of their quota holdings between quantity quotasand share quotas.3.2 Allocative equivalence of alternative ITQ systemsNow we can make a comparison of alternative ITQ systems using the results in Proposi-tion 1.First, let us look at the total expected rent generated by the fishery. The total rent isdefined to be the sum of fishers' profits and the authorities' revenue. Under our hybridITQ system, this isNvi(hi4i,^) , -A^W A Ael + G(H14 ,0N.,,where HA (Op) is the sum of the initial quantity (share) quota endowments to the fisherswho do not participate in the fishery from year 1 onward, i.e.,NE hAi^=iNi= 16.11. z__ 0A1Substituting the expressions for V i (hP, eh and G(/-41. , Oil ) in Proposition 1, and using;;Y_ gXt (eiti^K(Xt)Xt, we can easily verify thattotal expected rent = 1 6^ Et fpt • s(Xt)Xt —— This clearly shows that under our overall TAC adjustment rule (TAC = k(z)z, wherez = x in period A and z = Xt in period B),• the amount of initial quota endowments• the division of initial quota endowments into share quota and quantity quotaa)=1Chapter 3. Comparison of Alternative ITQ Systems^ 37• the division rule of overall TAC into share quota and quantity quota (f(z)TAC for share quota, K(z)z — f(z) = TAC for quantity quota)do not affect the total expected rent generated by the fishery. In other words, alternativeITQ management systems with a common overall TAC adjustment rule are equivalentin their allocative efficiency. For instance, the genuine QQ system (f(z)^0) and thegenuine SQ system (f(z)^k(z)z) have the same allocative efficiency, as long as theyhave the same overall TAC adjustment rule, TAC = K(z)z.3.3 Distributive effectAlthough alternative ITQ systems with a common overall TAC adjustment rule generatethe same amount of rent from the fishery, they are different in their effects on the distri-bution of the fishery rent. We verify this in the following concentrating on the differencebetween the genuine QQ system and the genuine SQ system.Remember that under the two systems with a common TAC adjustment rule (TAC = K(z)z)fisher i's expected profits areLL QQu'h Ai Hi andlfi,,), 15Q OP ii:(x)x^Itwhere: =^1 6Et {g (xt) ( eAi eB^ki(eAi) ci (e tB iis the expected present value of fisher i's inframarginal profit, andWQ Q [8/ton]WSQ [S/ton]:=:= WA1^11 — 6 Et {pt ^g(qX t -Xt) }1^1 ^ K(Xt)Xt11 — 6Et {[Pt^qA-t g(-N t k(x)xChapter 3. Comparison of Alternative ITQ Systems^ 38Here, for comparison's sake, we express the share quota price tbsAQ [$/ton] in the sameunit as for the quantity quota price ti4Q [$/ton]. (Cf. page 33.)Let us consider the situation in which= 0 i ii(x)x [ton],^i.e.,the initial quota allocations to fisher i are equivalent to each other in the two systems.In this case, it is clear thatVsQ >V ' Q "<-> ft4Q^if hP = ON(x)x > 0,if VP =^= O.Therefore, in general, fisher i's expected profit in one system is different from in theother if he has positive initial endowments (i.e., h -j4i = 011i K(x)x > 0). In the rest of thissection, we assume that fisher i has positive initial quota endowments unless otherwisestated.If we look at the expressions for zi4, c) and ii4Q , we notice that there are four factorswhich may affect the relative magnitude of ii)4Q and Cv:9Q :• the form of TAC adjustment rule, K(z)z;• the form of effort cost functions, ci(.) (cf.^=:.g(-V0);• the form of demand function, D(pi );• the form of the probability distribution of fish stock X t .Corresponding to whether K(Xi ) is constant, increasing, or decreasing in X i , thereare three cases about TAC adjustment rule.Chapter 3. Comparison of Alternative ITQ Systems^ 391. Proportional TACThis is the case in which ti(z) = k = constant. It is easy to see from (3.1) — (3.5),(3.8), and (3.12) in Proposition 1 (page 28) that equilibrium quantities in this case satisfythe following':^6"Ai = e^ Bi^^i = constant =^cc(e i ) =^,N(eN) =: rNcN(eN ) N^/2^ci(e i) rN^( = 1,...,N)e^Nk2q^= ^1 1,1 — 6Et {pt qXt TN}1 = Pt — qXt TN + 64Q under the QQ system3,,-A""SQ^= ^ {1 — 6 Et [pt qXt rN, x1 1^ Xt^ —1tt) = pt qXtrN + 6w SQ —x under the SQ systemQ Xton the assumption that1Pt — qXt rN > 0 VXt .Therefore, whether 4 under the QQ system increases or decreases when X t increasesdepends on the relative movement of pt and .-fctq rN .We can interpretej • [rN ci (e')+ W21e i1 (3.3), cc(e 131 ) = • • • = cily (ei9 N)^g(Xt ), shows that q31,^, i.p2v move all in the same directionfor different Xt values. (3.5) is nowq.X t (eli + epi)^kXt , which becomes E(Oi + epi)^k/qfor all X. It follows that ep ,^N) cannot change at all for different Xt values, i.e.. eSt =constant (i = 1, ..., N). Then, by the strict convexity of ci(.), eAa =^= 1,^, N) must holdas a result of cost minimization by fishers.NE e-AwQQwtChapter 3. Comparison of Alternative ITQ Systems^ 40Aas the inframarginal profit of fisher^ [pt — (wp — AzI,in each period, where TN =^ qXtis the revenue (net of opportunity cost) from a unit effort (cf. p. 33). This inframarginalprofit is zero for the marginal fisher N. See Figure 3.4, p. 41.2. More responsive TACSuppose K(z) = TAC/z is increasing in z. TAC in this case is more responsive to Xtthan in the above proportional TAC case. From (3.3), (3.5), and (3.12) in Proposition 1,the relationship between the stock level Xt and the equilibrium quota price 43 in periodB under the genuine QQ system isXtT^K(xt)^(i = 1, • • • , N)^g(Xt)the decrease of qA.  g(Xt ) is smallerwP is lower than otherwise,where qXt g(Xt ) is the marginal cost of harvesting one more unit of fish. This means thatunder the QQ system with a more responsive TAC adjustment rule, the authorities sell(buy) quotas at lower (higher) prices in period B. If the effect of wP is stronger thanthe effect of TAC = k(Xt )Xt T , the authorities' expected surplus (deficit) under the QQsystem through quota market operations will be smaller (larger).3. Less responsive TACThis is the case in which K(z) = TAC/z is decreasing in z. The relationship betweenXt and zup is thenk(Xt)^eBz ( = 1 , ,^ N) =,' g(Xt )the decrease of -1-g(Xt ) is largerqxtwP is higher than otherwise.Ci (ei) Cii(eN)Ci(ei ) -1-ti/2eir1rNr•0Chapter 3. Comparison of Alternative ITQ Systems^ 41Figure 3.4: Inframarginal profit under proportional TACChapter 3. Comparison of Alternative ITQ Systems^ 42Therefore, under the QQ system with a less responsive TAC adjustment rule, the author-ities sell (buy) quotas at higher (lower) prices. 2 If the effect of wP is stronger than theeffect of K(X t ) J., the authorities' expected surplus (deficit) under the QQ system will belarger (smaller).Thus if TAC is not proportionally responsive to X t , whether the authorities' surplus(or deficit) is larger or smaller than in the proportionally responsive TAC case is deter-mined by two factors (w iB and ( X t)) whose effects are opposite to each other. Whicheffect dominates the other depends on the functional forms of (1) TAC rule (i(z)z), (2)cost functions (ci(ci)), (3) demand function (D(pt )), and (4) the probability distributionof Xt . We can determine the effects of the responsiveness of TAC on the authorities'revenue under the QQ system (and the relative advantage of the two ITQ systems) afterspecifying the forms of these four functions. However, there are too many combinationsof possible functional forms to derive general results about the effects. Therefore, herewe only point out that it is possible to calculate the effects case by case.Example: Proportional TAC and demand with constant price elasticityIn the rest of this section, we concentrate on the case of proportional TAC rule whichreduces the possible combinations to a manageable range. As we saw above, undera proportional TAC adjustment rule TAC = ti:(z)z kz (k : constant), we have thefollowing:Vi ^>SQ^QQ ^ t) sQ < WQQ-p — —rN p —qx^qx 'Andersen [1] (p.481) points out this possibility.Chapter 3. Comparison of Alternative //Y2 Systems^ 43where x = Et {Xt } and= Et {ptt}^p = Et{Pi}, 7 = Ei{Vt}.Notice that if Xt fluctuates year by year,holds because of the Jensen's inequality.Since it is difficult to compare the two systems in general settings, we confine ouranalysis to the case in which the fish demand function takes the simple formpt = vD t- P (p > 0, v = constant)which has a constant price elasticity 1/p. D i is the demand for fish in year t. Forsimplicity, we take the unit of fish stock appropriately so thatthe average stock level x =Then Xt fluctuates around 1. Equilibrium fish price in year t is given by the conditionD t = St , where St is the supply of the fish in year t. Since St = kXt by (3.5) inProposition 1,Dt = kXt .This leads topt = uXt nat equilibrium, where u := vk - fi = constant.1. p = 0 (pt = u = constant) case:This is the case where elasticity = oc. It is easy to see that> 1/:(i2(2Chapter 3. Comparison of Alternative ITQ Systems^ 44in this case (because /3 = /5 = u and -y > 1, so that fv:45Q > WQQ ). Namely, if the priceof fish is constant and the fish stock fluctuates year by year, fishers with positive quotaendowments are better off under the SQ system regardless of the probability distribution ofXt . Under the QQ system, however, the difference V:4Q — 176 Q accrue to the authoritiesthrough their quota market operations. The reason for this is transparent. When Xtturns out to be large under the QQ system, quota priceWt= Pt^rN , —QQqXtbecomes high and the authorities sell quotas according to their proportional TAC adjust-ment rule at this high quota price. On the other hand, when Xt turns out to be small,quota price becomes low and the authorities buy back quotas at this low price.'2. p =1 (pt = nXt-1 ) case:This is the case of elasticity = 1. pt = uXt-1 yields/3 = Et I /3' X'^(x 1)xij = Et{pil = wy.Therefore tq c2 -yi-14Q , so thatUVQ < Vr' C?if there is uncertainty about the fish stock level year by year. In short, when the priceelasticity of the demand for fish is unity, fishers are better off under the QQ system,regardless of the probability distribution of the fish stock level X t . On the other hand,the authorities' quota operations account under the QQ system will have a deficit on theaverage in this case.3 Hannesson [17] obtains a result similar to this from a much simpler model. See [17], pp.461-2. His"quota price" (or quota rent), however. means quota rental price which corresponds to pt —^rN inour context.Chapter 3. Comparison of Alternative ITQ Systems^ 453. p^1 (13t = Dt P ) case:Here we further assume that the fish stock X t follows the uniform distribution over therange [1 - a, 1 + a], where a (0 < a < 1) expresses the degree of fish stock fluctuation.Then1 fl+a^1^1^r2au Xt-PdXt =   [(1 a) l-P - (12a 1 - p1ra^1  1uX1- dVt =2a 2 - p [(1 + a)2- P2a 1—a^t1 fl+a 1^+ a 2a A-a Xt dXj^2ain1- ap-^Et {pt}= Et 13' )(t1x I= Et {k}Hence—a) 11 u,— ( 1 — a) 21 u,VsQ< VQ Q^ (.1 - 1) -1 rN PM > R( a, p),whereR(a, p)M1 ^1-^[1(1 +^+a)1-P - (1 - a) 1- P^(1 o) 2--P - (1 - a) 2- Play -1  1 - p^2 - p1-TN .qM is the marginal cost of harvesting one more unit of fish when Xt = 1. n is theequilibrium price of fish when Xt = 1. HenceM—u = the marginal cost/price ratio at the average stock level.Since our assumption1^ r,v > 0 VXtq-Xtimplies marginal cost < price for all Xt , we have 111/tt < 1 under this assumption. Nowit is easy to see the following:Chapter 3. Comparison of Alternative ITQ Systems^ 46• If a = 0 (no stock fluctuation), then clearly V,4. Q = 1.7ci2Q .• If a^0, then R(a, p)^p.^(by L'HOpital's rule)• If p = 0, then I/12 > KiN as seen above.• If p^1, then 1,7,iQ < 1/'", Q for a > 0.From these and numerical calculations, we can draw diagrams like the following forparticular values of M/u < 1. (See Figure 3.5.) The dividing curve in the figure representsthe combinations of a and p values that satisfy M/u = R(a, p).From the above considerations, we can conclude that when TAC is set equal to a fixedproportion of the fish stock level,• fishers with positive quota endowments are better off under the QQ system (at theexpense of the authorities) if the demand for fish is inelastic;• fishers with positive quota endowments are better off under the SQ system if thedemand for fish is sufficiently elastic;• fishers with no initial quota endowments are indifferent to which of the two ITQsystems is chosen.These are valid for the case in which the demand for fish has a constant price elasticityand the fish stock level follows the uniform distribution. The case with demand functionsand probability distributions different from these require more complicated calculations.It will not be worthwhile to carry out these calculations at this level of abstraction. Wecan expect, however, that the basic qualitative nature of our result will be true for thosemore general cases.Finally, we note that if the TAC adjustment rule is non-proportional and the devi-ation from proportional TAC rules is large, the above result may have to be modified11-- Q < vemvi > viSQ VQQChapter 3. Comparison of Alternative ITQ Systems^ 47Figure 3.5: Fishers' expected profits under the two ITQ systemsinelasticdemandP"1M u elasticdemand ---Chapter 3. Comparison of Alternative ITQ Systems^ 48considerably. However, it will involve complicated calculations to determine the neces-sary modifications. In order to carry out such calculations, we need detailed informationabout the functional forms of (1) TAC adjustment rule, (2) cost functions, (3) demandfunction, and (4) the probability distribution of Xt . It will be very unlikely to obtainsimple relationships between the nature of non-proportionality and that of the necessarymodifications, because all the above functions interact with each other in determiningthe effect of non-proportionality of TAC adjustment rule. Therefore, we do not pursuethis line of investigation any further in this thesis.3.4 Adjustment phaseWe have assumed in the above analysis that the equilibrium quantities are attainedwith no adjustment period. For example, the number of fishers are adjusted rapidlybecause of free entry/exit assumption. Effort levels and quota demands are optimallychosen by fishers from the beginning based on equilibrium quota prices. Quota pricesattain stationarity from the outset. Fishers quickly form rational expectations on quotaprices realized under a new ITQ management system in which they may have no priorexperience. There are many difficulties about these assumptions. Suppose that the ITQsystem introduced to the fishery is new to the fishers. For instance,1. The levels of TAC after the introduction of the ITQ system may be quite differentfrom prior harvest history. Then fishers' past experiences may not give them muchguidance in figuring out the resulting probability distribution of the fish stock levelafter year 1.2. Fishers may not have any experience in quota trading. As a result, it is implausiblefor them to predict future quota prices (or price functions) correctly, at least at theoutset.Chapter 3. Comparison of Alternative ITQ Systems^ 49In short, we have assumed away all the complexities accompanying the initial adjustmentprocess. In reality, however, the initial adjustment period can be rather long. Some mayeven argue that most of important real economic issues are related to short or mediumterm adjustment problems. Nevertheless, this does not necessarily mean that the resultsobtained above have no relevance to real issues. On the contrary, the results presented inthe above propositions provide us with useful information about the comparison of oneequilibrium with another, as we shall see below.Let us consider the situation in which fishers are better off under the QQ system thanunder the SQ system (at the expense of the authorities) if the adjustment to the newITQ system is instantaneous as assumed in the above propositions. This is the case ofVQ Q > V. Then as we saw in Section 3.3,> qc, -;=:> wQ Q > WQ .Although 4, and 't-i4Q may not be attained immediately if the adjustment to the cor-responding ITQ system is not instantaneous, eventually those equilibria will be reachedafter the initial adjustment period is over. Now suppose that the authorities introducethe QQ system initially but some time later they decide to change it to the SQ system.Quotas held by fishers at that point which are not their initial endowments must havebeen bought by them at some prior time in the market. Even if we admit the differencebetween the initial adjustment phase and the subsequent equilibrium state, the prices atwhich fishers bought those quotas must have, to some extent, reflected the quota pricelevel irti4Q in the subsequent equilibrium. Then it will be reasonable to assume that thoseprices are more or less higher than fiygcd which is realized in equilibrium under the SQsystem. After the authorities' announcement of the change from the QQ system to theSQ system, quota price will start declining toward this new equilibrium level ti4Q . Thiswill result in the loss of fishers, because the authorities' "subsidy' (through quota marketChapter 3. Comparison of Alternative ITQ Systems^ 50operations) which fishers "counted on" when they bought quotas at relatively high pricesunder the original QQ system will disappear under the new SQ system. Fishers cannotfully avoid this loss by selling their quotas upon the authorities' announcement, since thequota price fall will start as soon as the system change is announced if there are fisherswho are aware of the effect of the "subsidy cut."This scenario seems to fit the case experienced in New Zealand. Originally introducedas a QQ system in 1986, their ITQ system was switched to an SQ system in 1989 in spiteof the opposition of the fishing industry. Deficit in quota market operations account wasone of the reasons the quota authorities in New Zealand decided to relinquish the originalQQ system.The argument in this section is unavoidably qualitative and less precise compared withthe clear-cut results stated in the previous propositions. However, it is almost impossibleto establish definite quantitative results about what will happen during a transitionalphase from one equilibrium state to another. There are many uncertain factors that willaffect the course of adjustment (e.g., the response of fishers' expectations to the systemchange). We have to content ourselves with the above somewhat vague analysis basedon our clear results for equilibrium states.Chapter 4Tax EffectsIn the previous chapter, we compared alternative ITQ management systems and showedthat they may have different effects on the distribution of the rent generated in the fishery.We derived this conclusion on the assumption that the quota authorities intervene in thefishery only through TAC adjustments. In reality, however, various taxes are proposedto capture the rent in the fishery. The distribution of the rent between fishers and theauthorities will surely be affected by these taxes. It is also natural to expect that thesetaxes will affect the rent distribution among fishers. We will examine tax effects on therent distribution among fishers with different cost functions using our basic model for theQQ system. The same analysis applies to any hybrid system. In this chapter, we continueto assume that the authorities use a fixed proportion TAC adjustment rule: TAC = kz(k = constant, z = x in period A, z = Xt in period B).We consider three types of taxes: profit tax, quota tax, and harvest tax.• A profit tax is levied as a fixed proportion 7.2, of each fisher's profit.• A quota tax is levied as a fixed proportion Tq of the market value of each fisher'squota holdings.'• A harvest tax Th [S/ton] is levied on each ton of harvested fish.1 We consider a quota tax levied on the quota holdings in period B of each season. The analysis iseasily extended to a quota tax which is levied on the quota holdings of each period.51Chapter 4. Tax Effects^ 52Fisher i's after- tax net profit in year t,^under each tax system is as follows:1= (1 - To E [mate, - ci(eati)] + 4 . (h/tli — db ) + wr • (a't4i — ajt3i ) — i3 .A,BEEptateii - ei(e)]^wt . (hAi^•^ —t^tl^(1 + Tg )ajt3i ] — ei,j=A,B^[(pt - roateiti - ci(ejt i)] + wA • (hAi^aAt i ) tvi•B . (aA, - atBi) -tj=A,B4.1 Equilibrium under taxationIn the following propositions, we state equilibrium quantities under the above three taxes.We give only a part of the proofs in Appendix B, since the procedure is the same as forProposition 1. For brevity, we define H i and G byH i := {2rN et^2ca)1 _1 6 — —G :=1= 1-1 b[Pqxrivikx,Et1^{[pt^rN]kXt}— 6^q-Xtwhere /3 = Et {pt Xt /x} and rN = c'# i ) (i = 1,..., N). fl i is the expected present valueof fisher i's inframarginal profit (cf. Figure 3.4, page 41). G corresponds to the expectedpresent value of the so-called "management rent" (cf. page 34).No taxFor comparison's sake, we first state the equilibrium quantities under the QQ systemwith a fixed proportion TAC rule and no tax. Proposition 1, when applied to this case,gives:1^1^11 — Et {pt atNT} 1 — S P— —qx rNWt^Pt qxt^ rN 614Q,I/4 (hi4i )GQQ i4), 7,A 7,^Fri—QQ -1 "A,Au'QQ,Chapter 4. Tax Effects^ 53where rN :=^= • • • = CAI EN and y = Ei lx/Xt l. (See Proportional TAC case,page 39.)Profit taxProposition 2 Suppose that the authorities introduce the QQ system with a profit taxwith tax rate Tv into the fishery from. year 1. Then at equilib•im, the number of fishersN, the effort levels in each period e i (i 1, , N), and quota demands A, B)are determined as in the no tax case. The equilibrium quota prices are also the same asin the no tax case, i.e.,^wt^1 —A 1^1^1^- 7^Et {pi qXt rN} 1 — (p —qx )1wt = N 64Q .qAtThe expected present value of fisher i's after-tax net profit with initial quota endowmentshiti is17cQ(h1.1-i) = (1 —^+ H i ]^(i = 1,^, N).The authorities' expected revenue from the profit tax and their quota market operationsisGQQ^= —(1 — T7,)^G^Ewhere 1114 is the sum of the initial quota endowments to all the fishers.Remark:The only difference for fishers between the profit tax case and the no tax case is thatthe expected present values of fishers' future profits are reduced according to the profittax rate rp . In this case it is theoretically possible to drive the expected present valueof after-tax net profit down to zero by setting rp = 1 (i.e., 10(1 Vc profit tax). Then the„„-AluCKPChapter 4. Tax Effects^ 54authorities appropriate the whole fishery rent G E1=i 1F including the inframarginalprofits E .1\1:= 1 fij .Quota taxWe have the following proposition for the QQ system with a quota tax.Proposition 3 At equilibrium under the QQ system with quota tax rate Tg , the numberof fishers N, effort levels es i (i 1,... , N), and quota demands edi A, B) are thesame as in the no tax case. The equilibrium quota prices under the quota tax areA ^1^1^1Wt = 1 _ 6 + Tq Et { pt^rN}   p --qXt^1 — 6 + 7.9, (qxwB = ^ (pt 1 + Tq^qXt TN -4- 64Q ) •„„--Au'QQ'The expected present value of fisher i's after-tax net profit is-1/ Q (hAi i ) = WQQh1 +^(i = 1, ... N).The authorities' expected revenue isGQ(2(H . 1 ) =^+ G.Remark:The above equilibrium quota prices dIAQQ and wtB for the quota tax case are lower thanthose in the no tax and the profit tax cases by the factors determined by the quota taxrate Tq . Notice also that /74 Q = Et {4}.Here, unlike in the profit tax case, the authorities cannot appropriate the infra-marginal profit TP (j = 1... , N) of fishers however high quota tax rate Tq is imposed.A high quota tax rate simply drives down the price of quota, which in turn reduces themarket value of quotas and the authorities' tax revenue. The maximum amount of moneyChapter 4. Tax Effects^ 55the authorities can expect to raise is the management rent G =^Et {[pt — Xt }qXteither by setting the total initial quota endowments Hill = 0 or by setting a very highquota tax rate TqHarvest taxThe following proposition holds for the fishery under the QQ system with the harvesttax Th [$/ton].Proposition 4 At equilibrium under the QQ system with a harvest tax Th , the numberof fishers N, effort levels ei (i = 1, , N), and quota demands etIt i (j = A, B) are thesame as in no tax case. The equilibrium quota prices under the harvest tax areWt1 1 6Et{pt 7h ^rN} =qA t1— Th — r N) :qxWQQ,1 — 61Wt = Pt — Th qXrNtassuming pt — Th - -±rN > 0 V.,Ki [b,d].The expected present value of fisher i's after-tax net profit isVQQ(hi ti ) = z4Q h 1li^(i = 1,^N).The authorities' expected revenue isGQQ(Hi4 ) = -4QH1 + G.Remark:As in the quota tax case, the authorities cannot appropriate the inframarginal profit11 i (i = 1, ... , N) as long as the harvest tax rate Th is low and the condition pt — Th -±q N > 0 VXt E [b, d] is satisfied. If the tax rate is so high that this condition is violated,some of marginal fishers may be driven out of the fishery and we have to recalculate theresulting equilibrium based on this fewer number of fishers. We do not go into thispossible complication any further.Chapter 4. Tax Effects^ 564.2 Comparison of tax effectsIt is clear from the above propositions that the three types of taxes do not affect theallocative efficiency of the ITQ management system.' This is because the number offishers and effort levels at equilibrium do not change under these taxes. However, theyhave different implications on the distribution of the fishery rent among fishers andthe authorities. The issue of rent distribution between the authorities and fishers wasdiscussed in the previous chapter. In the following we will concentrate on the problemof rent distribution among heterogeneous fishers. Namely, we compare the three typesof taxes under the circumstance in which rp , Tq , and Th are so fixed that the authorities'expected revenues are the same under these taxes.First, notice that the quota tax and the harvest tax give rise to the same formulae forfishers' expected net profit 1722Q (h -;1 ') (i =1,... , N) and the authorities' expected revenueGQQ(W) as in no tax case:li,rciN (hP) = fv- QAQ hAi  Z -I- H i^,...,N),GQQ(-1/i)^—u-/QAQHP + G.The only difference lies in the value of quota price Ct4 c, (and we), i.e.,1—QQ^1— 6 + Tq gx rNfor quota tax1  (_ ^)—QQ 1— 6^—^--qxrN^for harvest taxTherefore if rg, and Th are so set that1 - 6+T9 (p qx rN) 1 1 6 (p -Th 1sr.N)2 This is on the condition that the profit tax rate rp and the harvest tax rate rh are not too high.Chapter 4. Tax Effects^ 57holds, then both taxes result in the same values of 1/6 Q (1-41 i) (i = 1. . , N) and GQQ(Ilii, 4:).In other words, the quota tax and the harvest tax which have the same initial quotaallocation and give the same amount of expected revenue to the authorities are equivalentnot only in allocative but also in distributive effects. Whatever can be said about thecomparison of the profit tax and the quota tax, therefore, is also true for the comparisonof the profit tax and the harvest tax.'We consider the QQ system with the profit tax and the QQ system with the quotatax. We assume that the initial quota allocation is identical for these two systems. Let usdenote the values of the variables under the profit tax by a subscript p and those underthe quota tax by a subscript q. Then from the previous propositions{^L ^(__^-ylip' (hili ) = (1 — Tp) 1 - 6 p - -qx r N hl ^iliVqi (hP) = ^1^-y1 - 6 + r q (jj - --qxr N) h i i + Il l1for i = 1, . . . , N andGp (HiA ) = — (=— 6^ ( q—rN Hl + G+Tp E— 1-rN)^+G.qx- Tp11 — + Tq3=1(I)As mentioned above, we consider the situation where 7-7, and 7, are so chosen thatGp (1-P14, - ) = G q (Hi4 ).This condition is transformed into- Tp^I ^ETAT^Hi_ 6^1 — 6 + rg )^--qxrN) 111(4.17) 'This may depend on our assumption p t — -,4 ;g 7.N > 0 VATt which guarantees that fishers alwaysharvest up to their quota limits.rprli^113 = (1 - rpP^1 - 6^— 6 + rq)1 N U11j= 1Chapter 4. Tax Effects^ 58If^= Gq (HiA ), then rp and 'rq have to satisfy (4.17).The difference in fisher i's expected profit between the two taxes is(17:( h 1 i ) Vqi (hP ) = li _ 7;^1 — °, , r ) /5 — -2-r N hP — Tp ll i .^(4.18)-t" q^qx JJJLetrii = fisher i's share in the initial quota endowments 10hiP1Then by multiplying both sides of (4.17) by 7/„ we getSubstituting this into the RHS of (4.18) yieldsNV i (hAi ) - Vgi (h iP) = •p^1 )^ (4.19)Hencewhere. >^.^.^>^HiV i (hAi)^V"(hill) ^ qi CriP 1 q 1 ElY fli3 =1(4.20)o-j = fisher i's share in the total inframarginal profits.ThusV ip (hA )^Vqi (hAi i ) -<==>- fisher i's share qi in1 fisher i's share ai in>quota endowments < inframarginal profitsTherefore the relative magnitude of Vpi (12.11i ) and Ki (h'ili ) depends on both• the profitability of the fisher reflected in his inframarginal profit share, and• the initial quota allocation to the fisher.Chapter 4. Tax Effects^ 59If we specify the initial quota endowments, we can get more information about who ben-efits from which tax. Let us consider the following three cases with different initial quotaallocations. For simplicity, we assume that the fishers who get initial quota endowmentscoincide with those N fishers who remain in the fishery afterward.1. qi = 1/N (i = 1,^, N) case:In this case, the initial quota endowments are equally allocated among N fishers. Forfisher i who earns more than average inframarginal profit, 1/N < of holds. Hence forsuch a fisherlii < Qi ,^ i.e.,Vpi (hP) < Vqi (hAi i ) .Thus under equal initial quota allocation, those fishers who earn above average infra-marginal profits are better off under the quota tax. The opposite is true for the fisherswhose inframarginal profits are below average.2. ii = o-i (i = 1, . . . , N) case:In this case, probably by accident, initial quotas are allocated in proportion to fishers'inframarginal profit shares ai (i = 1, . , N). ThenVpi(q i ) = ii:(hiP) for all i = 1, . . . , N.Hence under this "proportional" quota allocation, the profit tax and the quota tax areequivalent to all fishers in the fishery.3. The case of "progressive" allocation:This is the case in which more profitable fishers get initial quota endowments morethan proportional to their inframarginal profit shares cri . (Here profitability is measuredby the share cri in the total inframarginal profits). In this case, the initial quota allocationChapter 4. Tax Effects^ 60exaggerates the difference in profitability among fishers. Then the more profitable fisheri is, the more likely it is that he is better off under the profit tax than under the quotatax.The following examples illustrate the above three cases. Suppose that after year 1four fishers have inframarginal profit shares a1 0.4, a2 = 0.3, cr3 = 0.2, and a4 = 0.1,respectively.'• In Case 1, each fisher is equally endowed with 25 percent of total initial quotas.• In Case 2, initial quota allocation (gi) is proportional to inframarginal profit shares(ai).• In Case 3, initial quota allocation (?)i) exaggerates the difference in profitability.These cases are schematically presented in Figure 4.6 (p.61) by using "Lorentz curves".In each case, fisher i whose part of the "Lorentz curve" has a slope (a, Irii) greater than1 (i.e., rii < o-i ) is better off under the quota tax. In Case 3, for instance, fisher 1 withthe largest inframarginal profit share of = 0.4 is better off under the profit tax, becausethe slope (0.4/0.6) is smaller than 1 and g i > a1.Remark: From a little different model Grafton [16] (p.503) derives a similar result: "Afisher who earns a higher average profit per unit of quota owned will pay proportionatelyless rent to a regulator with a quota tax than with an equivalent profit tax ...." Tocompare our results with his, rii should be understood as fisher i's share in the initialquota holdings at equilibrium. Then under the situation of no stock fluctuations in [16],o-i fisher is inframarginal profits total quota holdingsrii fisher is quota holdings total inframarginal profits[average profit per unit of quota owned] • constant.This ordering of fishers may be different from the ordering used in Proposition 1. Namely. fisher 4need not be a marginal fisher.CaselFigure 4.6: Lorentz curves for profit share^and quota share (7/)Ea-Jr,1.00.25 0.5^0.750.60.3Case277- ai(1) 0.4 = 0.4(2) 0,3 0.3(3) 0.2 = 0.2(4) 0.1 =^0.1^0.3Ea-)hi crj(1) 0.25 < 0.4(2) 0.25 < 0.3(3) 0.25 > 0,2(4) 0.25 > 0.11 ,^0.1 0.4^1,01 .00.60.3Chapter 4. Tax Effects^ 61(1)0.6 > 0.4(2) 0.3 = 0.3(3) 0.1 < 0.2(4) 0.0 < 0.1Chapter 4. Tax Effects^ 62We can say that in [16] fisher i's profitability is measured by ai /r/ i not byThe main lesson of the above analysis is that we cannot compare the effects of theprofit tax and the quota tax on fishers without taking into account the distribution ofinitial quota endowments. In most cases of actual ITQ management systems, initialquotas are allocated to fishers based on their past harvest records. Although thereseems to be no simple relation between harvested amount in the past and profitability'afterward, it is likely that profitable fishers harvest more fish than less profitable fishersbefore initial quota allocation. If this is true, Case 1 above (equal initial allocation) isnot so plausible. Furthermore, if the past harvest record exaggerates the difference inprofitability among fishers, it is possible that the profit tax is more advantageous forprofitable fishers as illustrated above. Of course, it is not easy to determine which is thecase for a particular fishery. More detailed study will be necessary to draw conclusionsabout tax effects.4.3 Adjustment phaseThe results obtained above were based on the assumption that the fishery reaches anew equilibrium immediately after the introduction of an ITQ management system withtaxation. We assumed away the existence of an adjustment period before the new equi-librium. This assumption is doubtless unrealistic. However, the results based on theunrealistic assumption can give us insight when we compare one equilibrium state withanother under different taxes. For example, consider the following situation. In year tthe fishery is in equilibrium under an ITQ system with a quota tax, when the authoritiesannounce a change from the quota tax to a profit tax. We assume that the profit tax isso designed that the authorities' expected revenue is the same as before under the new5As stated above, we measure fisher is profitability by his inframarginal profit share ui.Chapter 4. Tax Effects^ 63profit tax. Then what kind of effects does this policy change have to different fishers?Suppose that a new equilibrium under the profit tax is reached in year T(> t). Theinitial quota holdings in year T, hTi (i = 1, . . . , N) will be the same as in the case withno tax change. (Remember that the equilibrium effort levels are the same under differenttax schemes.) Let us denote the expected present value of fisher i's profit under the newprofit tax by 17; : (4i). We denote by 17:(itl) the expected present value of fisher i's profitat the beginning of year T for the case with no tax change in year t. Then, as we sawabove,17;1, 01^Vq (1^ •=:>. 71i <where7/i = fisher i's share in the total quota holdings at the beginning of year To-i = fisher i's share in the total inframarginal profitsFor fisher i with 71 i > o-i,ViP (11 Ai > V (P i )T^T 'If the transition from the original equilibrium to the new one is smooth, fisher i's after-taxnet profit will also change smoothly fromE [psqxs eJs. i — q (eis.i)} +^( ios i _^w sB^— (1 + Tq )aBs i ls +j=A,Bfor s < t toir's = ( 1 — 7-p)^E [psqxs is — c ij=A,Bfor s > T. Then, for fisher i with 7/ > o-i , the average of ;k si will start increasing in yeart after the change from the quota tax to the profit tax. In year T it will reach a newequilibrium level higher than before. Thus the tax change is beneficial for fisher i withq > o-i• On the other hand, the tax change will cause a loss to fisher j with 7ij < o-j.16, jsi)] + wAs . (iltsli — iiiis i) + wBs . (Os i — anChapter 4. Tax Effects^ 64Now suppose that for the particular fishery the distribution of quota holdings in equi-librium exaggerates the difference in "profitability" among fishers. 6 Then fisher i withrelatively large share in total inframarginal profits has quota holdings more than propor-tional to his inframarginal profits (i.e., qi > ai). For such a fisher,> v:( 14i).Thus the fishers who harvest relatively large amount of fish and possess large amountof quotas will benefit from the change from the quota tax to the profit tax under thiscircumstance. For fisher j whose harvests (i.e., quota holdings) are relatively small,< o j holds in this fishery where the distribution of quota holdings in equilibriumexaggerates the difference in profitability among fishers. Hence the tax change will notbe beneficial for relatively small scale fishers in such a fishery. In other words, fisherswith relatively large shares in total quota holdings (= total harvests) will benefit fromthe tax change at the cost of those fishers who have smaller shares in total quota holdings(= total harvests). If fishers with larger harvests have more power in the fishing industrythan small scale fishers, it is understandable that those large scale fishers put pressureon the government as the representative of the industry to change the tax system in theirfavour.The point is that there can be a conflict of interest among fishers concerning thechoice of tax, whether the existence of the conflict is recognized by fishers or not.'This corresponds to Case 3 (progressive allocation) in the previous section. As before, profitabilityis measured as the fisher's share in total inframarginal profits.Chapter 5The Case of Risk-averse FishersIn Chapter 3 we compared the SQ system and the QQ system on the assumption thatfishers are all risk-neutral. The conclusion was that if we take into account the possibilityof fish price changes caused by fish stock fluctuations, the two ITQ systems can havedifferent effects on the distribution of the fishery rent between the authorities and fishers,even if fishers are risk-neutral. As mentioned in the Introduction, the fishing industry inNew Zealand opposed the transition to the SQ system from the QQ system. The analysisin Chapter 3 provides one possible explanation. Namely, if the price elasticity of the fishdemand is not large, risk-neutral fishers can expect more profits under the QQ systemat the cost of the balance in the authorities' quota operations account.Another possible explanation is that fishers are not risk-neutral and they prefer incomestability under the QQ system. It sounds plausible that the QQ system contributes tostabilizing the income fluctuations of fishers, because under this system the authoritiesbuy back quotas in bad harvest years and sell quotas in good harvest years, wherebythose operations seem to damp fishers' income fluctuations. In this chapter we examinethe validity of this claim. We use the same basic model as before for the genuine QQsystem and the genuine SQ system.We consider two cases in the following:1. The only participants in the quota market are the authorities and fishers. Thereare no other risk-neutral traders in the market.65Chapter 5. The Case of Risk-averse Fishers^ 662. There are risk-neutral traders in the quota market and they can freely buy/sell/leasequotas in the market.The existence of risk-neutral traders in the quota market may change the resulting marketequilibrium considerably. The answer to the above question can critically depend on thispoint, as we shall see below.5.1 No risk-neutral traders case5.1.1 New assumptions and notationFor simplicity, we add the following assumptions:1. Proportional TAC rule: TAC = kz (k : constant), where z = x in period A andz = Xt in period B.2. The price of landed fish, p, is constant.3. There are N identical fishers. They have the same initial quota endowments inthe fishery, the same cost function c(c.[I) c(eP) and the same utility function47_0 defined over their annual net profit 7r t . u(.) is a concave function reflectingthe risk-aversion of fishers.4. There is no credit market and insurance market available to fishers (except forcostless in-season credit). Hence fishers do not have means to average out year-to-year income fluctuations over time except for selling and buying quotas in themarket.5. Fishers' net profits of each year are not carried over to next year. When fishers buyquotas and other goods at the beginning of a year, the payments are made whenChapter 5. The Case of Risk-averse Fishers^ 67they sell their harvest of the season.'The representative fisher's annual net income in year t under a genuine (QQ or SQ)ITQ management system in our model is7rt^pgxtetA _ c(0) wtit (h iA atA) •pqxtetB _ c ( eiB) w13 (hr atB) 4 ,wherep = the price of landed fish^(unit: $/ton)q^catchability coefficient^(unit: /m3 )Xt = the level of the fish stock^(unit: ton)the fisher's effort level in period j^(unit: m3 /period)wt^quota price in period j^(unit: $/ton)frit^the fisher's before-trading quota holdings in period j^(unit: ton)ait^the fisher's after-trading quota holdings in period j^(unit: ton)seasonal setup costs^(unit: $).Under the QQ system it is natural to express quota prices and quota holdings in theabove units. Since it is always possible to translate the amount of share quotas frompercentage unit to tonnage unit based on the current TAC level, we express quota pricesand quota holdings under the SQ system also in the same units as above. For example,if the TAC is 1,000 ton and the price of a 1 percent share quota is 100 dollars, the quotaprice is equivalent to 10 [$/ton] = 100 [$] -4- (1,000 [ton] x 0.01). 2'I.e., they can defer the payments costlessly within the season. This is an assumption of convenienceto highlight the effect of fishers' aversion to income fluctuations year by year.`Notice that the unit for share quota in this chapter (8/ton) is the same as WA (WP) [8/ton] (page 33)and ft;'1Q (4) [8/ton] (page 37) but is different from ' ,VA (i7',B) [8/TAC for share quota] in Proposition1 (page 28). We make this choice for expositional convenience.Chapter 5. The Case of Risk-averse Fishers^ 68Under the QQ system, fishers' quota holdings change only when they buy or sell quotain the quota market. Therefore, after-trading quota holdings in each period are carriedover to the next period:a B = hA and aA — hBt-i — t ^  t (5.21)Under the SQ system fishers' quota holdings in percentage unit change only whenthey buy or sell quotas in the market. After-trading quota holdings in percentage unitare carried over to the next period under the SQ system. In our model the authoritiesset TAC equal to kXt-1 before quota trading in period B of year t — 1. The next TACchange occurs when the authorities set TAC equal to the provisional level kx beforeperiod A of year t. At the end of period A of year t they change TAC again from kx tokXt . Therefore, if we express each fisher's quota holdings in percentage unit, we havethe following relations:a t- 1^ ht-4^and^ -4 ^h t t  3kx kx kXtHenceX B^ t Aat-i =^and —xa t = h tB^(5.22)where aB 1 , hA , 4, and hr are defined as before in tonnage unit.Formally, (5.21) and (5.22) are the only difference between the SQ sytem and the QQsystem in our model and play an important role in our comparison of these two systems.5.1.2 Rational expectations equilibriumGiven the initial quota endowments and based on his expectations about future quotaprices {wA wBr . . the representative fisher chooses a contingency plan { e4, ^loot^t t=P th^t i^fi ^h^ti^l^f t "t aBt t=13These conditions are the same as (i132 1 = OA and^= op on page 18.Chapter 5. The Case of Risk-averse Fishers^ 69for his effort level and quota holdings of each period so as to maximize his expected dis-counted sum of utiltyE {E 6t-1u(71-t)}subject to his quota constraintqXt e l qXt eP < aP for t = I, 2, 3, . . .As before, a rational expectations equilibrium is defined by stochastic processesetA , atA ,^atB}too_itiltB l tp° 1 and^that satisfy the following two equilibrium conditions:• Given the representative fisher's optimal contingency plan (stochastic process)for setting {0, 41-, 44, an t"_ 1 , the stochastic process for quota prices {tvi 4 , wr} tc°_ 1clears the quota market in each period, i.e.,I N4 < kx, = if mil > 0 for period A,NaP < k Xt , = if wiB > 0 for period B.• When fishers face the stochastic process {w4, w iB } t°L 1 as price-takers, the stochasticprocess {0, a tA, e tB , an t'_ 1 maximizes the expected present value of the representa-tive fisher's utility, E{Ei"_1 6t-1u(7rt)}.5.1.3 Fishers' behaviour at equilibriumIf fishers are identical, we can easily determine the effort levels and the quota holdingsB , ap) ,of each fisher at equilibrium in our model. Let us denote these values by (eAt atA et = 1, 2, 3, .... Suppose that at equilibrium, quota prices satisfy 4 > 0 and wr > 0 andthat the quota constraint binds, i.e., VW qX t eB = 41 . Then we have( ^A A ee at^tThe reason is as^(2 k ^kx k 1:7Xiq AT ^ N )'Chapter 5. The Case of Risk-averse Fishers^ 701. From the assumption w tA > 0 and te > 0, the quota market equilibrium conditionsare Na tA^kx and NaP = kXt , because by assumption there are no tradersother than N fishers in the market. Then these conditions yield^kx/N andaB = kXt /N.2. Since qXt ei4 qXt eP = ap by assumption and ait3 kxt/N by the above, we haveaipt3 kXtN •Namely, at equilibrium, each fisher harvests kXt /N. Since the equilibrium effortlevels should be cost minimizing, we know thate" At and eB minimize c(e tA) c(eSt )subject to qXt e:/t 1 qXt e.B =t^NThen e"-t`4- = eB = k/2Nq follows from the strict convexity of c(.).We define e by := ^B— e tkThe condition2Nq •p — 1 ci (e) > 0^VXt E [b, crguarantees that our assumptionsA > 0 B^ Aw t^, W t > 0, and ate t TXt 6B = aBare all satisfied at equilibrium.kXtChapter 5. The Case of Risk-averse Fishers^ 715.1.4 Equilibrium quota pricesWe can use the above equilibrium quantities (0, 'etit, ep,ap) = (e,kx/N,6,kXt/N) toderive the conditions that quota prices have to satisfy at equilibrium.` Let us denote thenet profit at equilibrium by irt . Thenp Xtlrt =^2c(e) + w;-A. otA — wtB (h tB a .:3 ) _Suppose that under the SQ system the authorities endow each fisher with 100/Npercent of share quotas before year 1. The authorities set TAC equal to (kx, kX1 ) inyear 1, (kx, kX2 ) in year 2, and so on. Hence under this SQ system the representativefisher's before-trading quota holdings at the beginning of each period are (hp, 143 ) =(kx /N, kXt/N) = 43) for all t = 1, 2, 3, ... Therefore at equilibrium the net profitt isp Xtart = ^ 246)—Notice that in depends only on Xt under the SQ system.Next, suppose that under the QQ system the authorities give each fisher initial quotaendowments a,139 = hi = kX0/N before year 1. Xo is set by the authorities and neednot be the stock level in year 0. Under the QQ system the representative fisher's before-trading quota holdings at the beginning of each period are (hI hp ) = (43 1 , at) for allt = 1, 2, 3, ... Then, at equilibrium,(ht ht)hf3 ) = (a? a ,t,^kx) _ (kxt_i •^)N^N ' t = 1. 2. 3Hence the net profit is given bypkXt^kx^B kx kXtt = ^ 2c(e)^("A't,-1^N )+wt ( N  )• 'This method is similar to the one in Lucas [21]. In a sense, our model is a special case of his AssetPricing Model. See also Blanchard and Fischer [8] pp.510-512 and Stokey et al. [30] pp.300-304.Chapter 5. The Case of Risk-averse Fishers^ 72Notice that in depends on not only X t but also X t _ i at equilibrium under the QQ system.^Now we state the conditions that quota prices 41 and 7.1)13 (t^1,2,3, ...) have tosatisfy at equilibrium.Proposition 5 Suppose that under the SQ system with no risk-neutral traders the au-thorities endow each fisher with 100/N percent of share quotas before year 1. Supposealso that p — qXt c'(e) > 0 VXt E [b, d] is satisfied. Then, at equilibrium, quota prices zui4and w:r (t = 1, 2, 3, ...) satisfyA^1 Et { uv.t) .2c1 [p al t el ( e ) ] 1=: WAWt 1 - 6^Et {uVrt)}W.,B = p — 1 c'(e) + 6 Et+i-tu'(Irt-1-1)} _A XW —qXt^Weirt)^Xtwhere Et is the expectation operator with respect to X t and 1r 1 = pk_Kt IN — 2c(e) —Since in = pk Xt IN — 2c(e) — depends on neither 14 nor 43 , we know that 4 tofor some constant CvA for all t = 1, 2, 3, ... Then if the functional forms of u(.), c(•), andthe probability distribution of Xt are given, we can calculate wA and wB explicitly from(5.23) and (5.24). Notice that zet9 depends on the stock level X t .Similar quota pricing formulae for the QQ system are given in the following proposi-tion.Proposition 6 Suppose that under the QQ system with no risk-neutral traders the au-thorities endow each fisher with k Xo l N ton of quantity quotas before year 1. Supposealso that p — qX t c'(e) > 0 VXt E [b, d] is satisfied. Then, at equilibrium, quota prices wit land wfi (t = 1, 2, 3, .. .) satisfyEt {4% r t)[P^601 Wt^ + 1 _6 6 E-E"-i{z/(;:+flu)/(Pit }qxt+leV)]} (5.25)AEt {u (,, t )}Et+1 {u l0-1-1-1)}  wA^(5.26)1^,=_- ^C (6) + 6q Xt(5.23)(5.24)Chapter 5. The Case of Risk-averse Fishers^ 73where Et is the expectation operator with respect to Xt and^Pk—Vt^kx^B kx kXt2C(e)^N ) wt • (^N^ )N Since in contains a Xt _ 1 term, we know from (5.25) and (5.26) that wi A- dependson Xt _ 1 and that wB depends on X t _ 1 and Xt . Therefore the above equations areactually two simultaneous functional equations for two unknown functions 4I (Xt _ 1 ) andw tB(Xt_ i , Xt ). In general, it is hard to get solution functions for these equations. Onlyby specifying simple forms for u(.), we can expect to obtain explicit solutions.The proofs of these propositions are found in Appendix C.Although the above formulae look complicated, they have very simple interpretations.Let us consider the SQ system case.The quota price formulae (5.23) and (5.24) can be rewritten as follows:Et {u'(.1i-t)1 Et f u ,cfr, t) kXt ^ c,(01}^iv 4 ikoxo }^(5.27)1 —^100^atkXt .^1 kxti ' 043 kX t =100^u'(Ir') 100 P qXt c'(e)1+ Et+1 {ut+1)4+1 1001 (5.28)It is easy to see that one more ton of harvest brings him an income increase of p— —1- Tq c' (e),where _,*q ci(e) is the marginal cost of harvesting one more ton of fish at equilibrium. Then(5.27) says that the cost and benefit of purchasing one more percent of share quotas inperiod A have to balance with each other at equilibrium. The condition (5.28) says thatthe cost and benefit of possessing one more percent of share quotas in period B have tobalance with each other at equilibrium.The interpretation of the equations (5.25) and (5.26) for the QQ system is similar.Chapter 5. The Case of Risk-averse Fishers^ 745.1.5 Example: piecewise linear utility functionPropositions 5 and 6 state the conditions quota prices have to satisfy at equilibrium.As we mentioned there, the conditions for the QQ system are expressed in simultane-ous functional equations with two unknown functions w,A (Xt _ i ) and wBt (Xt _ i ,Xt ). Tocompare the relative advantage of the two systems, we have to calculate the solutionfunctions explicitly. However, if fishers' utility functions are not linear, it is, in gen-eral, very difficult to solve these functional equations. To proceed further, we need somesimplifications to the model.We make the following additional assumptions.1. There are only two possible fish stock levels with equal probability:(1 + a)x with probability 1/2 (good year)Xt =(1 — a)x, with probability 1/2 (bad year),where a = the degree of fish stock fluctuations, 0 < a < 1.2. Fishers have a piecewise linear utilty function u(ir t ) with the following form (seeFigure 5.7, p.75):u(7t ) = { (1 — )) 7rt + Ov, if 7rt >(1 + 3)ir t — /3v, if ar t < v,Vwhere = the degree of risk-aversion, 0 <13 < 1, and'pkxN^2c(e)the average annual income under the SQ systemat equilibrium: Ecrirt l. (cf. page 71)Since fishers' income will fluctuate around this v, we can expect that this fuction5 1n the following, we omit the seasonal setup costs for brevity. This omission does not change theresults derived below.Chapter 5. The Case of Risk-averse Fishers^ 75Figure 5.7: Piecewise linear utility function—VChapter 5. The Case of Risk-averse Fishers^ 76will capture the effect of their risk-aversion.In the following, right-superscripts, + or - indicate the harvest of this year andleft-subscripts, + or -, indicate the harvest of last year where + means a good year and- means a bad year. For example,_7rt+^net income in year t when year t is a good year andyear t - 1 is a bad year.= quota price in period B of year t when year t is a bad year andyear t - 1 is a good year.rri = net income in year 1 when year 1 is a good year.5.1.6 Equilibrium quota pricesWe know that the net profit at equilibrium in year t ispkXt= ^ 2c(e)Nfor the SQ system andk t-i pkXt^X^kx^zoB . ( kx kXt ,N^ 2c(e) +^.t N N^ )for the QQ system.When the stock level is (1- a)x and the effort level is e, the marginal cost of harvestingone more unit of fish is q(11 a)sle). We denote this by it, i.e.,it := q(1 - 1- a)x ci(e).i is the marginal harvesting cost at equilibrium in bad years. Then1 -^1^c'(e)1 + a II = q( 1 + a)xChapter 5. The Case of Risk-averse Fishers^ 77is the marginal harvesting cost at equilibrium in good years.Now, by using the general results in the previous section, we can derive the equilibrimquota prices for the SQ and QQ systems.Proposition 7 Suppose that under the SQ system the authorities endow each fisher with100/N percent of share quotas before year 1. Suppose also that p > ,a is satisfied. Then< V <^(5.29)and the equilibrium quota prices are given bywtA1 — 6 [(1 — a3)p — (1 — a)p] = 20 (constant)^(5.30)B+^1 - a^1^ 60  1wt = p ^^1 + a 11+ 1 — 13^1 + a^ (5.31)wiB- = P — P + 1 6 ,170- A  1  . (5.32)1 + /3^1 —a.In order to determine equilibrium quota prices for the QQ system unambiguously, wehave to impose a condition on p and it a little stronger than p >Proposition 8 Suppose that under the QQ system the authorities endow each fisher withkx IN ton of quantity quotas before year 1. Suppose also that2(1 — 6)1p > + 62is satisfied. Then for all 0 < 3 < 1,'7;1 < v <^(5.33)__"17T^<v < (5.34)and the equilibrium quota prices for year t (t > 2) are as follows:When year t — 1 is a good year,11 ^6 ^1)(P 1 + a/..t) =1 — /3 1 —VVA (constant) (5.35)period A :^= (1 +A^— a)^6^1= P 1^+a + 1 — 6 (P 1+ a l2)period B : wB+ = _wB+tB-WB1^= + Wt •period A : (5.41)(5.42)(5.43)Chapter 5. The Case of Risk-averse Fishers^ 78period B : + 4++ w t -p 1^ — a it + 1 — ) 6+u_,A=^1 a^1 — )1 + /3 6_ t-v-A= p — .1— ,3(5.36)(5.37)When year t —1 is a bad year,period A :^_wtA = 1(1 + 6^1 (5.38)1 + )1 — 6 )(P^1+ a it) = _w-A^(constant)period B :^_4+ = 1 — ap 1 — ) A1 + a (5.39)+ to+ w1+ ,3B-1+ ,a 4—_Wt = (5.40)p, + 1+ )^•The equilibrium prices for year 1 areRemark:1. The condition p > [1 + 2(1 — 6)162]ii is not much stronger than p > 1.1 for ordinary6 values. For example,1+ 2(1 — 6)^1.111 if 6 = 0.956. 21.247 if 6 = 0.902. Notice that the expression Et+iftt i (irt+i )llui(irt ) in the general formulawt = p 1  cv)^6 -Ei+ 1 { u'( 7r^ i-1-1)}  wAq-Kt^u'(7rt)^t-Fiin Proposition 6 (page 72) appears here as1 ^1+)^1 -3^1+31 — /3'^1 —^1+).^1+3Chapter 5. The Case of Risk-averse Fishers^ 79in (5.36), (5.37), (5.39), and (5.40). The quota price differences due to the differencesbetween these marginal utility ratios have important effects in the following on the resultof quota market operations under the QQ system.The proof of these propositions are given in Appendix D.We use these equilibrium quota prices to calculate the expected discounted sum ofutility of the representative fisher at the beginning of year 1. The calculation is easy forthe SQ system. Under the SQ system the net profit of year t at equilibrium is•fr t+^pk(1 + a)x 2c(e)^if year t is a good yearpk(1 — a)x2c(e)^if year t is a had yearTherefore in < u <^and the expected utility of year t at equilibrium isEt -0.4/1701 = 2 [(1 — 0)irrtl- +13v]^2[(1 +13)r t  —13v}= (1 — a,(3) P7,v--„kx — 2c(e).This holds for all t = 1,2,3, ... Hence the expected discounted sum of utility ispkx^2c(e)]. (5.44)s^= 1 —^a/3) ATThe calculation of / Q is more involved (see Appendix E). The final result is1^pkx^a^1^kaxr^2c(e)] (5.45)+^_ 6 ) 1 + a P N •= 1 — 6 NUnder the QQ system the expected present value of the authorities' revenue is1G =a 2^6^1+ )^1 (5.46)QQ^1 — 6p,kax1 + 1 — 6^(1 — 6) 2 j 1 —^1 +^p)kaxSee Appendix E for the derivation.Chapter 5. The Case of Risk-averse Fishers^ 80We can interpret these formulae for VSQ, VQQ, and GQQ as follows. In (5.44), VsQdecreases when a and/or increases. This is an understandable result, for if the degreeof fish stock fluctuation (a) and/or the degree of risk-aversion (0) become larger, therisk-averse fisher's expected utility will decrease.'The results in (5.45) and (5.46) need more careful explanation. In (5.46) the authori-ties' revenue GQQ decreases when the degree of risk-aversion /3 increases. The reason forthis is basically the following:buy quotas when quota price is highIf 13 is large, the authoritiessell quotas when quota price is low.Let us look at the situation more closely. When ,a (0 < h3 < 1) is large, we know thefollowing from the equilibrium quota price formulae (5.37) and (5.39) in Proposition 8:Period Bcase 1. +43^= p^I + 1 ,36_a-) A • • • high1 1 — a^1 — . A^low.B   + 1 + (3 6+ th^• • •case 2. -w+t^p 1+ ay ^In case 1, year t turns out to be a bad harvest year and the authorities reduce the TACof the year accordingly. Hence they have to buy back quotas at the beginning of period Bat a high price. In case 2, year t turns out to be a good harvest year and they have to sellquotas at the beginning of period B at a /ow price to increase TAC. A similar argumentapplies for the TAC adjustment at the beginning of period A:Period Acase 1. + w-A = (1 + ^ )(P ^it) • • • high1 — /31 — 6^1-+ a-case 2. _ u ,A = (1 + 1 + /3 1 — 6 )(P^1 + a it) • • • low.'Under the SQ system with identical initial quota endowments to fishers, no actual quota trades occurat equilibrium. Hence we can ignore the effect which quota price change due to risk-aversion might haveon fishers' profits.1^S 11^6 1Chapter 5. The Case of Risk-averse Fishers^ 81The authorities lose money because of these quota trades. Since fishers trade quotas inthe reverse direction, they will gain from these quota trades. We can regard this as animplicit "subsidy" to fishers through quota market operations.Incidentally, in the expression^2 ^ 1^itkax 6[1-^6 + (1-6 6) 2 ] 1 —13 02(P 1 + 11)kax,GQQ ^1^a— 51  +the term[ ^13^1 6 (1 — 6) 2 1 —^(p 1+ ap)kaxcorresponds to the authorities' deficit due to the TAC adjustments in period A, and theterm1-  2^f3^1 6Li _ 6]^/32 (p 1+ a ii)kaxcorresponds to the deficit due to the TAC adjustments in period B. (Cf. the last partof Appendix E.) As is easily seen, the effect of period A adjustments which involvesthe coefficient 6/(1 — 6) 2 is larger than that of period B adjustments which involves thecoefficient 2/(1-6). For 6 = 0.9, for example, we have 6/(1-6) 2 = 90 and 2/(1-6) = 20.Now let us examine the effect of the risk-aversion coefficient /3 on the fisher's expectedutility VQQ:1 pkx^a^1^kaxVQQ =^6[ N^2c(e)]+ (0 1— S)1+ a N^ .Given the degree of fish stock fluctuation a, the direct effect of a larger /3 value is thedecrease of expected utility of fishers as we can see from the shape of the utility function.However, a larger 0 value also has an indirect effect through quota price change. Forwhen /3 is large, the authorities' quota market operations serve as a "subsidy" mechanismto fishers as we saw above. The larger the value of 0. the larger is the "subsidy".When value increases in our specific model, this indirect effect of "subsidy increase"outweighs the direct negative effect on utility. As a result, -17QQ becomes larger as 0Chapter 5. The Case of Risk-averse Fishers^ 82becomes larger. But this specific movement of VQQ value relative to is not crucial forthe subsequent argument. What is important is that when we compare VSQ and VQQ ,the former decreases faster than the latter as /3 value becomes larger and eventuallyVSQ < VQQ holds for sufficiently large ,3 values.5.1.7 Comparison of the two ITQ systemsNow we can make a comparison of the two ITQ systems. The difference of the repre-sentative fisher's expected discounted sum of utility between the two systems is given by(5.44) and (5.45):1 „pkx a a pkxVQQ — VSQ = ^ N (43 6^)1 + NHence> 1-1-a VQQ <VsQ^03 2 _4_ 1-8' 1-1-0For the authorities' revenue under the QQ system, we have by (5.46)(5.47)1 ^a6 ^1 (^1 G Q Q 0 '4=' 1 — 1 + apo_ ,(92^2) [ 1 — 6 + (1 — 6) 2^1 + ap)) < 0^ 3 2 + A) — 1 0,where1-1-aA:=6(2+ 1 _ 6 ) a [ p 1^ 1 + al and111 = q(1 — a)xeV).Hence>^< —A+ VA 2 +4G Q Q < 0 'ti==> ,39(5.48)Chapter 5. The Case of Risk-averse Fishers^ 83GQQ is positive if 3 is small and GQQ is negative if 3 is large. As we saw above, thereason for this is:If /3 is large, the authorities5.1.8 Numerical examplesI buy quotas when quota price is highsell quotas when quota price is low.We illustrate the above analysis by numerical examples. We specify four parameter val-ues, /3, a, p/p , 6, i.e., the degree of risk-aversion, the degree of fish stock fluctuation, theprice-marginal cost ratio, and the discount factor. For the sake of simplicity, the discountfactor is fixed at 6 = 0.9 in all the examples. The result does not change substantially,even if we choose other plausible values for 6. For pl y we choose two values, pl y = 1.25and pl p = 1.5. For a we choose also two values, a = 0.1 and a = 0.5. Thus there arefour combinations of pl p and a, i.e., (p/ p, a) = (1.5, 0.1), (1.25. 0.1), (1.5, 0.5), (1.25, 0.5).Table 5.3 shows the values of 3F and /3G for each case. Figure 5.8 (p.84) illustrates theTable 5.3: Critical values of the degree of risk-aversion6 = 0.9 /3G 13FCase 1 p = 1.5p 0.016 0.057a = 0.1Case 2 p = 1.25p 0.030 0.068a = 0.1Case 3 p = 1.5p 0.040 0.213a = 0.5Case 4 p = 1.25p 0.058 0.253a = 0.5situation. We notice that there is a range of /3 value (3G < 3 < ,3F) where both fishersand the authorities are better off under the SQ system. This case poses no problem. Ifthere is a conflict of interest between fishers and the authorities, it must be the case thatChapter 5. The Case of Risk-averse Fishers^ 84Figure 5.8: Risk-aversion, government's revenue, and fishers' utility(1) GQQ>0^GQQ<oI- --1--0fl 0.016 0.057I- I AV <VQQ SQ(2) GQQ>0^GQQ<0R^io^I^I 11^110.030 0.068^F I 4V f, V<Qw SQ^VQQ>VSQ(3) GQQ >o GQQ<0^1-^-i  -Io 1f3^1^10.040^III0.213^11-1F  ^ I- AV <VQQ SQ VQQ >VSQ(4) GQQ >0^0QQ <01- 4 ^ A0^ 113 1 I^II II----10.058 0.253V > VQQ SQI- ^ I-V <\TQQ SQ 4VQ Q>VsQChapter 5. The Case of Risk-averse Fishers^ 85either 3 < OG or 8 > /3F.Let us consider the situation in which fishers' degree of risk-aversion ,3 is large enoughthat 3 > OF and > /3G both hold. Then by (5.47) and (5.48)• Fishers are better off under the QQ system (VQQ > VsQ).• However, under the QQ system the authorities' expected revenue is negative(GQQ < ).In this situation a conflict of interest occurs between the authorities and fishers concerningthe choice of the quota system. Fishers will support the QQ system which brings themhigher expected utility. On the other hand, the authorities will try to adopt the SQ systemin order to avoid deficit in their quota operations account. This seems to correspond tothe situation which was observed in New Zealand. Now the problem is, which is betterfor the society as a whole, the SQ system or the QQ system ?Table 5.4 summarizes the calculations of• the expected present value of the authorities' deficit under the QQ system, and• the present value of the authorities' constant annual expenditure necessary to com-pensate fishers for their utility loss under the SQ systemfor the above four cases (see Appendix F for further details).^values are so chosenthat both /3 > OF and > OG are satisfied. Compare the figures in the table with theexpected present value of total fish sales11 — Spk.x = l0pkx.In all these cases in which GQ Q < 0 and VQQ > tisQ hold at the same time, the SQsystem is potentially better than the QQ system for both fishers and the authorities.Chapter 5. The Case of Risk-averse Fishers^ 86Table 5.4: The quota authorities' deficits under the two ITQ systems6 = 0.9authorities' deficitin quota tradingsunder QQ(discounted sum)authorities' compensationto fishersunder SQ(discounted sum)(1) p = 1.5p,^a = 0.1 3 = 0.1 0.333pkx > 0.045pkx3 = 0.5 2.535pkx > 0.470pkx(2) p = 1.25p, a = 0.1 3 = 0.1 0.200pkx > 0.034pkx,3 = 0.5 1.727pkx > 0.464pkx(3) p = 1.5/L,^a = 0.5 3 = 0.3 7.955pkx > 0.455pkx3 = 0.5 17.222pkx > 1.5pkx(4) p = 1.25p, a = 0.5 = 0.3 6.282pkx > 0.246pkx= 0.5 14.066pkx > 1.3pkxThere is another observation about Table 5.4. It is clear that when a is large, theauthorities' deficit can become very large under the QQ system. Case 3 and Case 4 giveexamples. In these cases the degree of fish stock fluctuation a is 0.5. This meansXt = 1.5x if year t is a good year0.5x if year t is a bad yearThe harvest in good years is three times as high as in bad years. The fish stock levelfluctuates wildly from year to year. Let us look at case 3. If the degree of risk-aversion offishers, 3, is 0.3, the authorities' expected present value of deficit under the QQ systemis 7.955pkx. If we compare this value with the expected present value of total fish sales,l0pkx, we can easily grasp the magnitude of the deficit. If, instead, the authorities adoptthe SQ system, the present value of the necessary compensation is 0.455pkx, i.e., 4.55percent of total fish sales. Of course, the authorities will not make even this amount ofpayment to fishers. Then fishers will be worse off under the SQ system. They surely willoppose the SQ system. Therefore if the fish stock fluctuation is very large and fishersare sufficiently risk-averse, there seems to be no way to implement ITQ managementChapter 5. The Case of Risk-averse Fishers^ 87of the fishery to the satisfaction of all the parties concerned. We might say that thisprovides a theoretical ground for a report on Canadian fisheries (Commission on PacificFisheries Policy [9]) which refused to recommend a quota system for the salmon fisheryon the basis of the wide and unpredictable fluctuations to which harvests in the fisheryare subject. ?All these seem reasonable except that the authorities' deficit are too large in somecases. We need to understand the reason for these huge numbers. This helps us to checkthe plausibility of the above scenario which seems to support government decisions suchas to adopt the SQ system instead of the QQ system. The point lies in the equilibriumquota prices under the QQ system with risk-averse fishers. The following table showsequilibrium quota prices under the parameter values given above (see Table 5.5).Table 5.5: Quota price movements under the QQ systemS = 0.9p = 1.5pa = 0.10 = 0.1p = 1.25pa = 0.10 = 0.1p = 1.5pa = 0.50 = 0.3p = 1.25pa = 0.5I3 = 0.3+ z-vA 4.333p 3.000p 7.689p 6.467p+ 4+ 4.355p 3.045p 7.706p 6.553p+ wr- 4.312p 2.955p 7.690p 6.380p_CvA 3.617p 2.504p 4.402p 3.697p_4+ 3.645p 2.554p 4.509p 3.867p_wr- 3.589p 2.454p 4.295p 3.528pIn all cases,a bad harvest year (t - 1) is followed by low quota prices (_wit B+^B--■Wt^Wt ), anda good harvest year (t - is followed by high quota prices ( +0, +4 + , + wr - ).7See Canada [9], Chapters 8,9 and 10. Particularly page 105. See also Munro and Scott [24],pp.661-664 and Anderson [3],pp.258-266.Chapter 5. The Case of Risk-averse Fishers^ 88This can be understood as follows. Suppose that last year t — 1 was a had harvest year.After last year's bad harvest, fishers start this year t with a smaller amount of initial quotaholdings. This causes lower expected net income 7rt this year (so 7r t < v, v : constant),since fishers are likely to end up spending more money for the acquisition of additionalquotas. With this low net income, fishers' marginal utility of income is high (= 1 + 0).(See Figure 5.7, p. 75.) If last year t — 1 is a good harvest year, the opposite is true andfishers' marginal utility of income is low (= 1— /3) this year. Hence, in short, this year t's1 dollar has more value to fishers after a bad harvest year t — 1 than after a good harvestyear t — 1. Then, with low net income in year t following a bad harvest in year t — 1,fishers are reluctant to increase their quota acquisition by giving up money. This has theeffect of depressing fishers' quota demand this year and results in low quota prices (_20,B+ B-\-Wt -Wt 1.Let us look at Case 1 and suppose that last year t — 1 was a bad year and thatthis year t turns out to be a good year in period B. The equilibrium quota price is_ze -F 3.645p. In this case, quota prices in next year will be + 71;A = 4.333p in periodA and _F tutB+1 = 4.355p or +4471 = 4.312p in period B. Quota price rises more than 20 %from this year to next year. (This would not happen if there are risk-neutral traders inthe market.) At the beginning of year t 1, the authorities have to buy back quotas atthis very high price to reduce the supply of quota down to the average level kx. Similarly,if last year t — 1 was a good year and this year t is a bad year ( + wt13- = 4.312p), theauthorities have to sell quotas at a very low price (_Cv A = 3.617p) at the beginning ofnext year. This large fluctuation of quota price is responsible for the authorities' deficitand fishers' gain in our model under the QQ system with no risk-neutral traders.The existence of such large fluctuations of quota price becomes possible in our examplebecause ofChapter 5. The Case of Risk-averse Fishers^ 89I. relatively large risk aversion (/3) of fishers, and2. the nonexistence of risk-neutral traders in the market.If these conditions are not met, quota price fluctuations cannot be so large. Then sucha situation as above in which fishers are better off under the QQ system will be unlikelyto occur (at least in our example). 8In the case of New Zealand's ITQ management system, there is no institutional re-striction on the participation of risk-neutral traders in the quota market, because "Mem-bership in the [quota trading] exhange is through subscription and is available to any-one."(Ian N. Clark et al.[13] p.333.) If this means that there are many risk-neutral tradersin the quota market other than fishers and the authorities, the validity of the above resultfor this case may be limited. The existence of risk-neutral traders will surely affect thenature of the resulting equilibrium. In the next section we explore the consequence ofthe existence of risk-neutral traders in the quota market.5.2 Risk -averse fishers with risk-neutral tradersAs quoted above, in the ITQ management system of New Zealand there is almost norestriction on the participants of the quota market exchange. Ian N. Clark et al. [13]point out one of the benefits of this as follows:The lack of conditions on the identity of quota owners also has the advantage ofallowing nonfishers to influence the value and distribution of quota. Theoretically,fish retailers could purchase quota and lease to fishers with conditions guaranteeing8 However, by assuming a constant fish price p, we are neglecting in this chapter the effect of the fishprice fluctuations due to inelastic demand for fish. If we take this effect into consideration, quota pricefluctuations can be large even when there are risk-neutral traders in the market. Then it may still bepossible that fishers are better off under the QQ system by the same mechanism as in the risk-neutralfishers' case in Chapter 3.Chapter 5. The Case of Risk-averse Fishers^ 90supplies of catch, financial institutions could acquire quota as security, recreation-ists or tourist guides could purchase quota away from commercial fishers, or privateor government conservationists could purchase quota to reduce catches. Some ofthese activities have begun taking place. ([13] p.332.)In the previous section we assumed away the existence of these outsiders (except the quotaauthorities) in the quota market. In this section we introduce risk-neutral outsiders intoour model and examine its effect. We assume that there are many potential participantsin the market (other than fishers and the authorities) who are all risk-neutral. Wealso allow quota leasing between outsiders and fishers. The following assumptions areretained:• identical fishers with a concave utility function (not necessarily piecewise linear);• constant fish price;• no credit market and no insurance market available to fishers.Let us consider the case of the SQ system. The representative fisher's net profit inyear t is7rt^pateAt _ c(eAt^wtA (hAt^aAt) nrtilitAmxtet13 c ( GtB) wtB ( Xt aA a .p) _ m B/B _^t ^t^(5,wherem ' ^price of a unit quota for one season in period jquantity of quotas leased to the fisher in period jThe existence of risk-neutral traders imposes strong restrictions on equilibrium quotaprices. Namely, quota prices have to satisfy no - arbitrage conditions. In our model theseChapter 5. The Case of Risk-averse Fishers^ 91conditions are expressed as follows:A^Awt Mt =^1 wt^f,c Awt mt = uwt+i xtFor example, the second condition has the following meaning for outsiders:(5.49)(5.50)wBrntAwA ^t+i xtcost of purchasing one ton of quotas in period Brevenue from leasing one ton of quotas in period Bdiscounted sales of x/Xt ton of quotas in period A of next yearRisk-neutral traders can get benefit from arbitrage transactions whenever these condi-tions are not satisfied. Such transactions will rapidly restore the quota prices that satisfythe above conditions.Given a quota price sequence {4, tvPM 1 and a quota rental price sequence{ m tA , mr} t"_ i that satisfy (5.49) and (5.50), the representative fisher tries to maximizehis expected present value of utilityE f Eblu(rt)}t.isubject to his quota constraintqxteAt^qxtetB < atB^1.r.Then the maximized value function Vs (hi4 ) will satisfy the Bellman's equation:7,21\VsQkut^max Et max [u(7)d- WsQ(h At+i)]eP,aP,1Pwhere the second maximization is subject to the above quota constraint andhA x aBxt t •Chapter 5. The Case of Risk-averse Fishers^ 92Let us define e bye = 2Nq.Then we know as before that equilibrium quantities satisfy(on aAt^lts) kx, kXtN' e ' N )(5.51)on the condition thatp — , (e) > 0^V Xt E [b, dqAtThe same procedure as before yields the first order conditions that have to be satisfied atequilibrium (see Appendix G). These conditions together with (5.49), (5.50), and (5.51)determine, in principle, all the equilibrium quantities and equlibrium prices. In practice,however, it is not easy to solve this problem even if we specify a simple functional formfor u(.).Therefore, we do not try to solve the problem here. Instead, we make the followingobservation. Namely, the existence of risk-neutral traders in the quota market under theSQ system makes fishers better off. This is because it is always possible for fishers tochoose quantitieskx kXt )(6;1 ,43 ,^, cri8 ) = (‘ N" Nby setting tifri = lB = 0 and these quantities give exactly the same annual net profitpkXt7rt =^, ^2c(e) —as in the no risk-neutral traders case. Hence fishers cannot be worse off under theSQ system due to the existence of risk-neutral traders. Rather fishers will utilize theopportunity of quota leasing to reduce their income fluctuations. For instance, fishers inbad years will sell their quotas to outsiders and rent back those quotas from outsiders inorder to make up for their income loss. In good years they will buy back quotas fromChapter 5. The Case of Risk-averse Fishers^ 93outsiders. As a result, fishers' income fluctuations will be smoothed out over time andrisk-averse fishers will be better off due to the existence of risk-neutral traders and quotaleasing opportunities.We can use the same method as above to derive the conditions that equilibriumquantities and equilibrium prices satisfy under the QQ system. However, it is not possibleto ascertain that fishers are better off with the existence of risk-neutral traders underthe QQ system. They may be worse off because of risk-neutral traders, since the sourceof fishers' gain under the QQ system (i.e., large fluctuations of quota price) will beeliminated by risk-neutral traders.As we said before, it is difficult to obtain explicit solutions for equilibrium quantitiesand equilibrium prices for the open SQ and QQ systems. Without these values, we cannotcalculate fishers expected utility under these systems (and the authorities' expectedrevenue under the QQ system). Therefore we do not pursue any further the comparisonof the two ITQ systems with risk-neutral traders. The following two questions remainopen when risk-neutral traders exist in the quota market:• Under which system are fishers better off?• Which of the two ITQ systems is better for the society as a whole?Our conjecture is that the existence of risk-neutral traders will contribute to reduce thedifference between the two systems with risk-averse fishers by1. reducing quota price fluctuations due to risk-aversion and2. allowing fishers to smooth out their income fluctuations.Then factors which we did not take into consideration in this chapter such as• the price elasticity of the demand for fish orChapter 5. The Case of Risk-averse Fishers^ 94• administrative costsmight become important in determining the relative advantage of the two alternativeITQ systems.Chapter 6ConclusionAmong various regulatory measures of fishery management, individual transferable quo-tas (ITQ's) have attractive features which other measures do not share. In principle,an ITQ based management system can overcome both the biological overexploitationof fishery resources and the economic overexpansion of fishing capacity in a simple andunified way. Overexploitation is effectively prevented by setting an appropriate level oftotal allowable catch. This is implemented by adjusting quota volume to that level. Theproblem of overcapacity in a profitable fishery under stock protection is resolved in adecentralized manner through voluntary restraint of fishers based on their individuallyrational economic calculations. Quota prices which are formed in a competitive quotamarket guide those calculations automatically and efficiently. The management authori-ties do not have to engage in other detailed and complicated regulatory measures exceptfor monitoring fishers' harvests.These advantages of ITQ management systems over other measures is the reasonthat now growing number of countries are using ITQ's in their fishery management. Butwe have to admit that ITQ management systems are not free from problems. One ofthe problems is the uncertainty of fish stock level. Fluctuations in the stock level areoften unpredictable and sometimes very large. When the fish stock level fluctuates, themanagement authorities have to adjust the TAC level correspondingly to attain theirbiological and economic goals. These TAC adjustments clearly affect fishers' profits. Ifthe authorities participate in the quota market transactions. their revenue is also affected95Chapter 6. Conclusion^ 96by fish stock fluctuations and resulting TAC adjustments. This thesis focused on thisissue.The purpose of this thesis was• to provide a simple model of ITQ management systems under fish stock uncertainty;• to compare the performance of alternative ITQ systems by using the model.We obtained the following results from our simplified model:• The share quota (SQ) system and the quantity quota (QQ) system with a commonTAC adustment rule are equivalent for the society as a whole. The two systemsgenerate the same amount of rent from the fishery. Under the SQ system the to-tal rent minus the initial quota sales (or plus the initial quota buybacks) of theauthorities accrues to fishers. Under the QQ system with a proportional TAC ad-justment rule, fishers earn more than this amount at the expense of the authoritiesif the demand for fish is inelastic. If the demand is sufficiently elastic, a part ofthis amount is transferred from fishers to the authorities. Under the hybrid systemwith a definite rule of TAC division into quantity quotas and share quotas, fishersare indifferent to the difference of quotas after the initial endowments are allocatedto them.• Taxes on fishers can affect the distribution of fishery rent among fishers. In thecomparison of a profit tax and a quota tax, we found that the former favours thosefishers who have more share in total quota endowments than in total inframarginalprofits. Depending on the initial quota allocation, fishers with larger inframarginalprofit shares may be better or worse off under a profit tax.• Risk-aversion of fishers may explain their preference for the QQ system withoutassuming a downward sloping demand curve for fish, if the effect of risk-neutralChapter 6. Conclusion^ 97traders in the quota market is negligible. In this case, it is possible that theSQ system is better than the QQ system for both fishers and authorities, if theauthorities make necessary compensation to fishers under the SQ system. If thequota market is open to outsiders, risk-aversion may not be a decisive factor ofthe fishers' preference for the QQ system. The SQ system open to risk-neutraloutsiders is better than the closed SQ system for the society as a whole.The policy implications of our results are:• If the authorities decide to use the QQ system, they should be prepared for aresulting long-term deficit in their quota market operations account. This deficitis more likely to occur under smaller price elasticity of the demand for fish.• The authorities should be prepared for the opposition of fishers to the change fromthe QQ system to the SQ system. With a TAC rule given, fishers and the authoritiesplay a "constant-sum game" if fishers are risk-neutral.• If the authorities have to raise certain amount of money through taxation on fishers,the authorities should be aware of the fact that different tax schemes favour differentgroups of fishers depending on their profitability and scales of operation.• If we take into account the risk-aversion of fishers, neither the QQ system nor theSQ system may bring about satisfactory result, especially in fisheries with largeyear-to-year stock fluctuations. The QQ system is particularly vulnerable becauseof the authorities' deficit. The SQ system will also be difficult to implement unlessthe authorities are ready to compensate fishers for their loss caused by large year-to-year income fluctuations.Chapter 6. Conclusion^ 98These findings depend on the assumptions of our model. The most significant dif-ferences of our model from the standard one are the following about fish populationdynamics:1. The fish stock level fluctuates from year to year independently.2. The fish stock level remains constant throughout each fishing season.Firstly, in the standard seasonal model, the fish stock level at the beginning of each yeardepends on the escapement of the year before through the stock-recruitment function.'Assumption 1 ignores this dependence. Secondly, the standard model takes into accountthe fish stock depletion during each fishing season. Assumption 2 ignores the effects ofthis in-season depletion. Because of these assumptions, our model lacks the aspect ofthe intertemporal externality that plays the central role in the standard fishery model.On the other hand, owing to these simplifications, we were able to derive equilibriumquota price formulae rather easily. Apart from this technical convenience, we can justifyAssumption 2 if the TAC level is much smaller than the stock level. It is very difficultto formulate and solve a model without Assumption 2 (small in-season depletion) whenwe take into account, as we have done in our model, the possibility of the informationalchange about the fish stock level in fishing seasons. It is not difficult, in principle,to remove Assumption 1 (uncorrelated fluctuations) in our model as long as we retainAssumption 2. Suppose that a good (had) year tends to be followed by a good (bad)year, i.e., the fish stock level fluctuates with some persistence from year to year. Wecan model this situation by assuming some joint probability distribution of this year'sstock level Xt and next year's stock level X t+1 . (If the stock level is discretized, the jointdistribution is expressed by a matrix of transition probabilities.) Although the derivation'See Clark [11] Chapter 6.Chapter 6. Conclusion^ 99of equilibrium quantities may become more complicated, there is no intrinsic difficultyin this generalization. Most of the qualitative aspects of our results will be retained.Another limitation of our results which has practical importance is that the findingsabout the distributional effects of alternative ITQ systems are obtained for the propor-tional TAC adjustment rule. If an actual TAC rule is different from the proportionalrule, we have to modify our results. But the precise nature of the necessary modificationdepends on the interaction of the following four factors: (1) TAC adjustment rule, (2)cost functions, (3) fish demand function, and (4) the probability distribution of fish stocklevel. They interact with each other in complicated ways and it is hard to obtain generalresults that hold for wide variety of cases. In dealing with non-proportional TAC rules,we will have to examine the outcome of such rules case by case by specifying the detailsof the above four factors.An issue related to both of the above two points, 1 and 2, is the problem of the optimalTAC adjustment rule. If we could literally ignore in-season depletion regardless of thelevel of fishing activities, the "optimal" policy would be to set TAC sufficiently large andlet fishers harvest as much as they wish. The fact that this is not the case reflects theimportance of in-season depletion and the stock-recruitment relationship. Therefore, inorder to determine the optimal TAC rule, we will have to consider these two aspects offish population dynamics explicitly. Since this is a tough problem under the fish stockuncertainty of our model, we ignored the dynamics and concentrated on the positiveanalysis of the problem, i.e., the comparison of alternative ITQ systems with a givenTAC adjustment rule. The normative analysis of various TAC adjustment rules underITQ management systems will be a topic of future research.Bibliography[1] Andersen, P. 1989. Comments on Hannesson [17] and [18]. In Neher et al. [25],pp.481-483.[2] Anderson, L. G. 1989. Conceptual constructs for practical 1TQ management policies.In Neher et al. [25], pp.191-209.[3] Anderson, F. J. 1985. Natural Resources in Canada. Toronto: Methuen.[4] Annala, J. H. (Comp.) 1992. Report from the Fishery Assessment Plenary, May1992: stock assessments and yield estimates. 222 p. (Unpublished report held inMAF Fisheries Greta Point library, Wellington.) The Ministry of Agriculture andFisheries, New Zealand.[5] Arnason, R. 1986. Management of the Icelandic demersal fisheries. In N. Mollett(ed.) Fishery access control programs worldwide. Alaska Sea Grant Report No.86-4,University of Alaska.[6] Arnason, R. 1990. Minimum information management in fisheries. Canadian Journalof Economics 23:630-653.[7] Birchenhall, C., and P. Grout. 1984. Mathematics for Modern Economics. Oxford:Philip Allan.[8] Blanchard, 0. J., and S. Fischer. 1989. Lectures on Macroeconomics. Cambridge,Mass.: The MIT Press.[9] Canada, Commission on Pacific Fisheries Policy. 1982. Turning the Tide: A NewPolicy for Canada's Pacific Fisheries. Ottawa.[10] Clark, C. W. 1980. Towards a predictive model for the economic regulation of com-mercial fisheries. Canadian Journal of Fisheries and Aquatic Sciences 37:1111-1129.[11] Clark, C. W. 1985. Bioeconomic Modelling and Fisheries Management. New York:Wiley-Interscience.[12] Clark, C. W. 1990. Mathematical Bioeconomics: The Optimal Management of Re-newable Resources, Second Edition. New York: Wiley-Interscience.100Bibliography^ 101[13] Clark, I., P. Major, and N. Mollet. 1989. The development and implementation ofNew Zealand's ITQ management system. Marine Resource Economics 5(4): 325-350.Also printed in Neher et al. [25], pp.117-145.[14] Fraser, C. A., and J. B. Jones. 1989. Enterprise allocations: the Atlantic Canadianexperience. In Neher et al. [25], pp.267-288.[15] Geen, G., and M. Nayar. 1989. Individual transferable quotas in Southern bluefintuna fishery: an economic appraisal. Marine Resource Economics 5(4): 365-388.Also printed in Neher et al. [25], pp.355-387.[16] Grafton, R. Q. 1992. Rent capture in an individual trasnsferable quota fishery. Cana-dian Journal of Fisheries and Aquatic Sciences 49:497- 503.[17] Hannesson, R. 1989. Catch quotas and the variability of allowable catch. In Neheret al. [25], pp.467-480.[18] Hannesson, R. 1989. Fixed or variable catch quotas? The importance of populationdynamics and stock dependent costs. In Neher et al. [25], pp.459-465.[19] Hilborn, R., and C. Walters. 1992. Quantitative Fisheries Stock Assessment: Choice,Dynamics and Uncertainty. New York: Chapman and Hall.[20] Libecap, G. D. 1989. Comments on Anderson [2]. In Neher et al. [25], pp.210-214.[21] Lucas, R. E., Jr. 1978. Asset prices in an exchange economy. Econometrica 46:1429-1445.[22] Macgillivray, P. B. 1990. Assessment of New Zealand's Individual Transferable QuotaFisheries Management. Economic and Commercial Analysis Report No.75, Depart-ment of Fisheries and Oceans, Canada.[23] Mangel, M. and C. W. Clark. 1988. Dynamic Modeling in Behavioral Ecology. Prince-ton, N.J.: Princeton University Press.[24] Munro, G. R. and A. D. Scott. 1985. The economics of fisheries management. InA. V. Kneese and J. L. Sweeny (eds.), Handbook of Natural Resource and EnergyEconomics, vol. 2, pp.623-676. Amsterdam: North-Holland.[25] Neher, P. A., R. Arnason, and N. Mollet (eds.) 1989. Rights Based Fishing. Dor-drecht, The Netherlands: Kluwer Academic Publishers.[26] Newbery, D. M. G. and J. E. Stiglitz. 1981. The Theory of Commodity Price Stabi-lization. Oxford: Oxford University Press.Bibliography^ 102[27] New Zealand. 1992. TACC and management review for the 1992/93 fishing year, 11August 1992. Unpublished. The Ministry of Agriculture and Fisheries, New Zealand.[28] Sargent, T. J. 1987. Macroeconomic Theory, Second Edition. Orlando, Fla.: Aca-demic Press.[29] Scott, A. D., and P. A. Neher (eds.). 1981. The Public Regulation of CommercialFisheries in Canada. Ottawa: Supply and Service Canada.[30] Stokey. N. L. and R. E. Lucas, Jr. with E. C. Prescott. 1989. Recursive Methods inEconomic Dynamics. Cambridge, Mass.: Harvard University Press.Appendix AProof of Proposition 1The proof consists of five steps.Step 1Under the hybrid ITQ system fisher i's net profit in year t isE^— ci (eij i)] wtA . (hAi — +^ B^Ai^Bi\^t ^at )^tot • a t — a)t j=A,BwtA otAi (tilt) wt.8 ((tAi (pi)where hitli ar_i1 and OtAi = (P ip By assumption, equilibriumquota prices w iA , WtA , wtB ,and WtB satisfyWtA = (HA , OA), WA = 4) (11 , OM,wt =^,(t1  _Kt ),^WtB =^(HtA ,(-V , Xt)•The quota prices perceived by fisher i coincide with these price functions by our ratio-nal expectations hypothesis. Through the dependence of wt and WI on Hia and (V,the maximized present value of profits expected by fisher i at the beginning of year tdepends on the aggregate quota holdings Hill and 1;-.V as well as his own quota hold-ings hAi and O -tAi. We denote the maximized expected value by V i (qi, OtAi ) suppressingthe possible dependece on HA and OA. Hereafter V i (hifrii, OM should be understood as, HA,74103Appendix A. Proof of Proposition 1^ 104The maximized value function 1/ i (hA i OA ') has to satisfy the follwing dynamict^t pro-gramming equation:^Vi( hAi °Ai ) = max Et^max [ 74 4_ 6 -vi^oit^t 't+1 , t+1,]}where the second maximization is subject to+ q-KteSt < aBt + ('t6i • f(Xt),and hAi 1 =— aP i and OtAi!' i =The second maximization problem  is:max^zi +^Ai Aiv h^g^e it3i, c1 P , CP^tsubject to qXteAt i + ate tBi < aBt^(tBi f (Xi)given^Ai otAi;^ht;(wit4 , WA) = (6(11, eh , 4) (k , e it1 ))( 143 ,WtB) = ((HA 24', Xt),k I1 ( 11-t A , OA, Xi))Let the Lagrangian be£ = 7r^i • [at^CtBi •^^( —t )^gxte .4i^qxterijt' + At^+ ./ Ais the Lagrangian multiplier of this problem. Let us denote the maximizers by e thand We assume that Epi > 0, aBi > 0, and 13i > 0 are satisfied. The twoequalities are assumed to occur not as corner solutions but as "just touching" cases.These assumptions are justified in Step 5. Then the first order necessary conditions areet Bi^ptqXt — c'.(eB )^t q- t = 0^(A.52)a tBi :^j_ 6  a^i pidAi^nt+1^hA +11 vt+" I^= "^(A.53)(Bi^+ 6 00-4ia^ vi( hAi 8t+i1) A t  • f (x = 0^(A.54)Appendix A. Proof of Proposition _I^We denote the maximized value by^'ON1), i.e.,+^i (it 't1 1 ,^=max^+ (w i t' A A?: aAiki6t+1,vt-Fi)e .13i Bi ".13i, atsubject to qXte tAi qXter i < atBi + e i • f(xi),whereti piqXietAiPtate i —^ci ( eAi) + 4 (hAi cet ti )^otAi _ CiA.1 )+ct(er i) wB . Ai^- Bit ^( a at )^14/tB • (Cili^-t21 )Ai^A-^h i /^.13i' t+1 = aBi^t t+1 = St •By using the envelope theorem s , we obtain^a^sTritlAiaeAi[ t -v kut+1, ,t+i)] = ptqxt — cact^101 !I tA^17" kAdi- °tA+i ) .11^— w^w ta r^acAi^[ 7-r; +^ötA+ii)]^_WA + 1-'17,, tB •— A it qXtThe first maximization problem is:max . Et {mot 6 V i Ch tA+1 1 , a tAdi 1)}e tlz ,cq4 t ,CiAtgiven 7, Ai plAi"t( D; W71 ) = (0(N 131;1 ), 431 ( -H;^1))(4^= (OW, OA, X^tA et 4 Xt)) •'See, e.g. Birchenhall and Grout [7] p.247.105Appendix A. Proof of Proposition 1^ 106We denote the maximizers by E .P.i , a ^and Ci4i . We also denote the values of EP^7and^at^ci4i) by Eit3i , aRi and Cp i , respectively.Since fishers are assumed to be risk-neutral and^is linear in cr4i and (i4i , it is clearthat^Et {— w tA wtB} = 0^(A.55)E t { — WtA WtB} = 0 (A.56)have to hold in order for the quota market to be in equilibrium in period A. The maxi-mizers Cett i and ( 4. i are inderterminate under these conditions.We assume that Oi > 0. This assumption is justified in Step 5. The first ordercondition with respect to eAZ isEt {ptqXt — eAt i^AqXt} = 0.^(A.57)The maximized value isV' (41 , Ai = Et { .-fr + 6V i (hAi 19Ait+17whereptqxte tAi _ ei (E.41) totA (htAi — atAi) WAptqXter — c i (e i )^• (a j4i — d'6i) WB • (Vi(T i )—ht +iAi^6Ai -t+1^t •Finally, applying the envelope theorem to the maximized value function 1/"(hil i , Oh,we obtain^   vi(hAi,^) = Er {wA} ,.^•^•aolu vi(hAi,^0 4 =^{wtAl(A.58)(A.59)Appendix A. Proof of Proposition 1^ 107Step 2Let (0i,a:At i, i,4i ; e‘pi,a-pi,j) be the quantities at equilibrium. In this step we showthat"Aietci (eB 1 )cif ( 6-Ai)NE(eAl e tBi)i=1e^= constant= cN(etEiN )Et { ci'(eSt i )}1—K(Xt )•(i = 1,^, N)1, ... ,N)(A.52) — (A.59) must hold at equilibrium. Since 0 i , wtZ , and WtA i do not depend onthe fish stock level Xt in year t which is still unknown in period A, we have the following:Et {gi ('eAi )} = c(eAi).^A, E t ( Wt = w„ and Et {WA} WA .Then the following conditions corresponding to (A.52) — (A.59) have to be satisfiedsimultaneously (t = 1,2,3, . ..):PtqXt — caer) — A it qXt = 0^—wr + •^ V i (letli+1 +a—wtB + ^V i (11;Itli Ojt 14..i i ) + At • f (xt)^DeNi^+ ,Et {pt qXt — A it qXt} — cae't4i )+ Et {4} = 0_wtA + Et {KT}a^pahAi , ,, ^oi,,vtaaBAi VAi aAi\000= 0w t A(A.60)(A.61)(A.62)(A.63)(A.64)(A.65)(A.66)(A.67)Appendix A. Proof of Proposition 1^ 108whereAi = hAi^//Ai —t-1-1 —8Ai^arti1^vl^t+1 — '3t •From (A.66)pjAi^wA"t+11 v HA*By substituting this into (A.61),+ 64+1 + = 0, i.e.,= —14 qXt (54+0Xt .Substituting this into (A.60) gives1Wt = Pt — Bi) 6'4 1•+qXtLet us assume that 2total fish supply = E( qXt0i qXt er i) = K(Xt)Xt.(A.68)(A.69)Under this assumption, total fish supply depends only on X t . Then fish price pi isindependent of i/ tAi and OtAi , since Xt does not depend on hA i and 8Ai. Equilibrium quotaprices wr and 4+1 are also independent of ie and OtAi by assumption. Therefore, qiwhich satisfies (A.68) is independent of hA i and BAZ.(A.68) must hold for all i = 1,2, ... , N. Henceefi(eri ) = c2(eB2 ) =^=^(eBN) (A.70)Since (A.60) holds for any Xt E [b, d we haveEt {pi qXt — c(er i ) - A 2t qXt } = 0.2 This assumption is justified in Step 5.aAppendix A. Proof of Proposition 1^ 109From this and (A.63), we obtainoi ) , Et fr( essi ) }^ (A . 71 )From this we know that both e -iti and eBi are independent of 11, 1 i and 8t i , since êpi isindependent of h't4i and /ki as seen above.From (A.69)N^e tBt ) = —1 K (Xt ).Now we show that eAi = eAi = constant and that epi depends only on Xt.that (A.70) — (A.72) hold for any values of H -:1. (s = 1,...,t), e (s = 1,...,t),(s = 1,...,t — 1). Let Z stand for any one of these variables.Differentiating (A.70) with respect to Z giveseBlAp-Bi^h,^BN)^ —^(wit (.r )N--t k^az azHence, aer i/OZ (i =^N) all have the same sign. AlsoEt {c7(eiBl) ) az} = • = Et c'k (et ) az f .^BN a6PN1Differentiating (A.71) with respect to Z yieldscneA-i )  t  =Et {cNBt az'^ },azFrom (A.73) and (A.74), we haveN Al^ NAN_7 1 nAl   = — civfl ., 7t4N) et ) az az(A.72)Noticeand X,(A.73)(i = 1, 9 ,^, (A.74)Hence ae-;ivaz (i = 1,...,N) all have the same sign.Differentiating (A.72) with respect to Z yieldsaetAi^N aeBiaz + E azt = 0 (A.75)Appendix A. Proof of Proposition 1^ 110Therefore, (aeiti vaz,...,aer !az) and (aeriaz,...,&B,Niaz) have opposite signsor are all zero. Then, (A.74) implies thatA o iDe i^a, ^t^0^(i^1,^, N).az azThus, in particular, e't4i and epi are independent of HA and (V.To sum up, ei4i and ejt3i are independent of ivit li , i4-': , _W 1- , and O. Since equilibriumquantities 41i and eri are the maximizers of the dynamic programming equationV i (1^ {4", e i , W O, M = max Et^max [74 + 60 (hN 01 , tA4i. 1 , HA (+1 , V )+1 ]at ,airli , (;4i^el3i ,api ,cpwe can conclude from the recursive structure that^Ai^= constant for t = 1,2, ... , andet-Bi depends only on Xtetas asserted. Let us denote^Ai^^Aiet^= (i = 1,2, ... , N)^4(eB1) =^= 4(er)^g(Xt).Then by (A.71),(A.76)(A.77)Ai^Ci e t Et{g(Xt)} = 1,2,...,N).Step 3In Step 3 we derive equilibrium quota prices. From (A.68) and (A.77)wB v 6 A= Pi —^Wt+1qXtSubstituting this into (A.64) gives1^, 4W tA = Et {Pt —^AXt^6Et twi+i(IA t(A.78)(A.79)Appendix A. Proof of Proposition 1^ 111t > 2 case:For t > 2, HA^— f (Xt-i) and OA = 1 at equilibrium. Thenwt = 0(W,^= 46 ( K(Xt-i)Xt-i — f(xt_i),i)HenceEt-1 {w } Et-1 {0(11;4) et)} = Et {46 (H4_1, (4+1)} = Et {wi4+1} •Now taking the expectation of both sides of (A.79) with respect to Xt-1 yieldsEt -1 {wtA l Et {Pt qXtg(Xt)} + 6 Et {4+1 1 , i.e.,Et^ A .1 ^Et {Pt — —1 c1 (e i)} (t 2 ).1 – 6^qXtBy substituting this into (A.79), we get^wtA = 1 1 6,Et {pt qxtg(Xi)} =: fvA (I> 2)^(A.80)By (A.78) and (A.80),?Di^Ptq^Xtg(Xt)d- 6 ^(t > 2)^(A.81)If we substitute (A.80) (t = 2) into (A.78) and (A.79) (t = 1), we can easily see that(A.80) and (A.81) hold also for t^1.We can use the same argument to derive equilibrium share quota prices, WA andwiBTo sum up, the equilibrium quota prices are given by the following:1^1 – 6 Et {pi^1  g(Xt)}q-Kt1^.W.tB = Pt at g(Xt) + 60WtA = ^11 – 6 Et {{Pt – 1—qXt g(Xt)} • f(Xt)}1 WtB = [Pt qXt g(Xt)] • f (Xi) + 6WAAwt = W-- 4 (A.82)(A.83)wA^(A.84)(A.85)Appendix A. Proof of Proposition 1^ 112wherec'i(eV)^• = c'N(eB lv) =: g(Xt),c'i (e2' i )^caeAi) = Et {g(Xt )}^(i = 1 , 2,^,^•1E(eAi el3i) = -K(Xt).i=1Step 4We use the above quota prices to calculate the expected present values of fisher i'sprofits, Vi(hP, OP) (i = 1,2, ..., N), and the authorities' net revenue, G(.11i/", ON. Wecontinue to assume that the quota constraint for each fisher binds at equilibrium:qXteAi qXt eSt --^i c;tBi • f (Xt ).^(A.86)Fisher i's net profit in year t at equilibrium is= poxt e‘Ai ci (eki) ,thA^aim) + wt.BPtateB i ci(e i3i ) + WA '^+ WtB •(PA — ei) -where iriu = hi i and BAi = 61i when t = 1. If we take the expectation oft with respectto Xt , the terms involving h4 i and cancel out on the RHS, because CvA = Et {4}and WA = Et {W 3 } by (A.64) and (A.65). By using (A.83), (A.85), and (A.86) inthe above expression of fr ti, it is straightfoward to derive the following expression for themaximized value function of fisher i:vi(hAi i, eA1 i) "7'6=^+ 6 1^ 6 E,_1Et{ .A Ai^- 4 Ai1 6W^+ IV"^+ ^Et {9(Xt) • (e Ai + t )— i(e.A zIn the course of calculation, we notice that terms involving aBi and -t13 'i cancel out alltogether, as long as ar and c;tBi satisfy (A.86). Therefore, how fisher i divides his quotaAppendix A. Proof of Proposition 1^ 113holdings qXt e i qXtet i in period B into quantity quotas aBt i and share quotas '.tBi • f (Xt )has no influence on the maximized value Vi(hP, Oh, as long as (A.86) is satisfied.From the assumption on the numbering of fishers in Proposition 1, it is clear thatv i ( hp op) > ti-) A^WA0:4i for i < N,v i^01.4i) <^w-Aop for i > N.Hence, under the assumption of free entry and exit, only fisher i (< N) operate in thefishery regardless of their initial quota endowments. Fisher j (> N) are better of byselling their initial quota endowments at the beginning of year 1 and exiting the fishery.The authorities' revenue in year 1 and in year t (t > 2) are:G1 = ti/ • [K(x)x — f (x) — HiA ]wf3 • [K(Xi )Xi - f(X1) - (K(x)x - f(x))] + WA • (1 —Gt = wA • [K(x)x - f(x) - (k(Xt-i)Xt_i - f(Xt-i))]+ 4 • [k(Xt)Xt - f(Xt)- (K(x)x - f (x))]Using &I = Et {4} and (A.83), it is straigtforward to show thatG(Hji ,^) = E1 {C-1} + 6 1 1 6^ Et_iEt {Gd= — zi)A^— WA 0'14. + 1 1 6^. -E't {[Pt^gx1 t^ g(-3(t)]K(Xt)Xi}Step 5As the last step of our proof, we verify that the assumptions made so far are indeedsatisfied by the equilibrium quota prices and effort levels derived above if the conditions1Pt — ---g(Xt) > 0 V.Xt E [b, d andqXtAppendix A. Proof of Proposition 1^ 114^BiCt^> 0^(i = 1, . . . , N)are met.We derived (A.52) — (A.54) and (A.57) (i.e., (A.60) — (A.63)) on the assumptionthat at equilibrium,"Bi^0, etkit > 0,^> 0 , e tAi > 0.Also in Step 2 we derivedeAi = e"Ai and^"^g(Xt) (i = 1, . . . , N)on the assumption thattotal fish supply = TAC = ti(X2)Xt•First, the assumption epi > 0 together with caeh = E t { c',; (eP i )} yieldeitti > 0 (i = 1 ,..• N).Therefore the derivations of (A.52) and (A.57) are justified.Secondly, since (A.52) and caer i) = g(Xt) hold at equilibrium,ptqXt — g(Xt ) — A;qX t = 0.Then pt — —k g(Xt) > 0 eXt E [b, d implies A > 0 at equilibrium. Hence the quotaconstraint binds at equilibrium, i.e.,qXteitt i^q )(teSt —^13i^4.131: a t + t . .fFrom our assumption p t — 9 ,g(Xt ) > 0 VXt E [1), d , we have= ^1 11— 6Et {pi — qXtg(Xt)} > 0, so1^iy\= Pt —g ( -At) vW >q-XtAppendix A. Proof of Proposition 1^ 115Hence^= K(Xt )Xt — f(Xt ) at equilibrium. If f (Xi ) = 0 for all X t E [b, d], then(xt) = f (Kt) is trivially satisfied. If f (Xt ) > 0 for some interval of Xt E [b, d ,thenti/A1 -1^E^atgt {[Pt —^(Xt^>)] • f(Xt) }^0^soPt 1—^g(Xt)1 • f(Xt) SWA > 0,qXtwhich implies^(,131: = 1 at equilibrium, so that E liv_ i^f (Xt) = f (Xt) is satisfiedin this case, too. ThereforewtBN^ Ntotal fish supply = E(qxtet^ te+q^t^tx 3i ) — E(e i + ( tBi f(Xt)) K (Xt)X, = TAC,i.1 j=1which justifies our assumption.Finally, as we saw in Step 4, the division of the quota holdings in period B, qXt eAiqXt e-P i , into api and r"i15 i • f (Xt) is immaterial for fisher i at equilibrium. How fisheri divides his quota holdings in period B into quantity quotas and share quotas doesnot affect his expected profit. Hence the derivation of (A.53) and (A.54) based on theassumptions eir > 0 and c; Pi > 0 where the equalities do not arise as corner solutions isjustified.This completes the proof of Proposition 1.Appendix BPropositions 2, 3, and 4B.1 Proof of Proposition 2: profit taxLet V ic,' Q (1-th be the maximized value function for fisher i with quota holdings hitu atthe beginning of period A of year t. Then V i(: , Q (h tAi ) satisfies the following Bellman'sequation:V6Q(ftAi) = max Et max kffi SVQi Q (h tli+i iAi ,a t^et ,Bit^t twhere the second maximization is subject to qX4i qXt er < aP i and(1 - 7.p) E [mate - ci(e )1 + wtA (h tAi atAi) wtB (atAi atBi)7rtj=A,BL Ai^Bint+1 — at •Proceeding as in the proof of Proposition 1, we obtain the following conditions whichhave to be satisfied at equilibrium:(1 — rp) {That — eVr i )] Aq,Kt = 0,(1 — rp )(-4)+ 6176Q t (it/tlii i ) = 0,Et {(1 — rp ){pt qXt — c(e 4')] — A it qXt } = 0,Et {(1 — Tp)^W-;4 + WI 3 )1 = 0,V6 Q '(7 b) = Et{( 1— Tp)w }.wherejt AiBi=t+1^at •I}116Appendix B. Propositions 2, 3, and 4^ 117The rest of the proof is basically the same as for Proposition 1.B.2 Proof of Proposition 3: quota taxFrom the Bellman's equation, we obtain the following conditions at equilibrium:pt qXt — caer) — A it qXt = 0,_(1 +7.9)4 _4_ SVQQ ' (h Ati 0,Et {pt qXt — caeAi ) — A itgXi}Et {_ wtil^wr}VQQ" (ilAi )0,0,EtWe omit the rest of the proof.B.3 Proof of Proposition 4: harvest taxThe Bellman's equation for this case yields the following conditions at equilibrium:(Pt - Th)at -^AitqXt = 0,+ 1143 + 61'&2' ( 11 't4-1!.1) +^= 0 ,Et {(pt - Th)ai - caeAi) - /1q.Xt} = 0,Et { wAt + ton = 0,1‘7:2Q /(iiitti) = Et { wtAThe rest of the proof is omitted.Appendix CPropositions 5 and 6C.1 Proof of Proposition 5: the SQ systemLet 4 and wr be the equilibrium quota prices in each period of year t. Let VsQ OM bethe maximized value function of a fisher with initial quota holdings 41 in tonnage unitat the beginning of period A. 1 Then VsQ(141 ) has to satisfy the Bellman's equation:VsQ (hAt = max Et max[u(7t) 617sQ(hAt-Fi )1 •e t ,a t^etl3 ,a.13where the second maximization is subject to qXtq- qXt eP < ar , andE [7,axte .ti_ c(4) , +wtA . (htA_ atA )+ w tB ( X t7rt^ utatXhAt+1 = A, at •From the first order necessary conditions for the maximization problem on the RHSof the Bellman's equation and from the equilibrium quantitieskx kXt(et^,eN" Nwhere e = k/2Nq, we obtain the following conditions that have to be satisfied at equi-librium:tl(*‘ t) • [PqXt — V)] — Atq -Vt = 0uVrt ) • (—wr)-f- 61/ Q (it A ) x--,,T + A t = 0(C.87)(C.88)'For the general asset pricing model, see Lucas[21]. See also Stokey et. al.[30] pp.300-304. 118Et { u / ( irt )[mXt - c'( e )] - )tat =XtEt {u'( 71-t) (^+ trt —)qc) (11 A )0^ (C.89)= 0 (C.90)= Et {z1(7r t )wit l } ,^(C.91)Appendix C. Propositions 5 and 6^ 119wherepkX t^N^2c(e) —kx—a t =XtBy substituting (C.91) into (C.88), we get—A t qXt = uVrt)( — wrat) SEt { u 1 (7"rt)4 qx.Substituting this into (C.87) gives1^Etlui('frt)} A XW ^ (C.92)w:t8 = p — qXteVe) + (5uVir t )^1 Xt'where 4 does not depend on Xt , because Xt is not known in period A. Substitutingthis into (C.90) leads to^1 ^Xt SEt fEt {uVrt)(-4) u'01.0[13^^ )], ^ C (e—^uVrt)}4 = O.}co tThereforeA 1 Et {nV*t)[P ^Wt = 1^Et{nTirt)}(C.93)C.2 Proof of Proposition 6: the QQ systemThe proof of Proposition 6 is basically the same as the previous one. Let VQQ(q) be themaximized value function of the representative fisher with the initial quota holdings hitiin tonnage unit at the beginning of period A of year t. Then VQQ (hi4 ) has to satisfy theBellman's equation:Aht ) = max Et rnax[u(^(51/Q 9 (h -t4+1 )]ci4 ,crit^eD,aPAppendix C. Propositions 5 and 6^ 120where the second maximization is subject to qXt eAt + qXt eP < aB , and7rt =^ix x tejt c(e;h1 + _A 7, ^A \)1- tut ^(ht^atj=A,BhA^att+1 t •The conditions corresponding to (C.87)—(C.91) aren'(it t)pqXt — c'(e)] — AtqXt = 0u f (irt)(-4) + (N(2041) + At = 0Et {u'(rt) pqxt — c' (E)] — AtqXt} = 0Et {u'(it)(-4 + 4)1 = 01 6Q( 14) = EctuVrt)41.a t2[pqXt e — 40] wtA (kit a tA) wtA (41 a tB)pkXt kx^B kx kXt^ 2c(e)+ w t kXt-1 ( N )+' N N- (  k  kx  k  kXt2Nq' N 2Nq' N ).From (C.98)) = Et + i { u' ( irt+i )4+0.By substituting this into (C.95),li f (li't)( — WtB ) + Et+1 {?- 110- )wi4+11 + At = 0 , i.e.,A— AtqXt = (%s t)( — wtB at) + 6Et-Fifut ( ,‘rt+i)tt' t lqX+ 1^t•Hereand(C.94)(C.95)(C.96)(C.97)(C.98)1 ^et( , )-11qxt+i Et {u'(fr i )}(C.102){ Vrt+i)[PAppendix C. Propositions 5 and 6^ 121Substituting this into (C.94) givesOwt  1u / (7rt ) • [(Pi — wt )0it — c (e)] + 6 Et+1 {z/(irt+ +1 1 qXt = 0, i.e.,1^tE -Filu'(irt+i)w -t4+1 1B'I./i = P ^cV) + 6 q-X-t til(i-t)Substituting (C.99) into (C.97) yields(C.99){Et-FittiV-t+OwtA-Fill}u'eirt)^q--will + (p — 1 c' (e)) + 6^ = 0, i.e.,Et^Xt^tiVrt),,— Etfu'(irt)w tA l + Et { uVrt)[P q-gtc, (e)]} + 6 EL Et+i {i/ (ii-t-1-04-1-1} = 0. (C.100)This must hold for any Xt _ 1 . Therefore, by taking the expectation Et _ 1 {•} and noticingthat the last term of the LHS does not depend on X t _ 1 , we get{—Et_i_EtfuVrt)41 + Et_i_Et u/(i-t)Sincep — qA, cV)] + bEtEt+ ituVrt+ i)wi4+1 1 = 0.tEt_1Etfu'(71-t)tviA l = EtEt+1lu /(irt+1)wiA+11, and1 { Et_ i Et u'(7"rt)[P^c'(e)]^= EtEt+i uVirt-Fibi ^qXt qXt-Fi c' (0}we obtain1ci (e)] .^(C.101)EtEt+ifu'Frt+i)wi4+1 1 = 1 _ 6 -r-t -Lt -I- 1 u'(it -1-14 ^1qXt-1-Substituting (C.101) into (C.100) givesA Et ttiVrt){1) — -j—cV)]}^6 EtEt+iwt —^Et {uVrt)}^1 — 6Appendix DProposotions 7 and 8D.1 Proof of Proposition 7The net profit at equilibrium under the SQ system is i-+7rtpk(1 a)x N^2c(e) > v if year t is a good year,pk(1 — a)x^c ,^ 2^< v if year t is a bad year.Hence we haveuVrt) 1 1 — ,3 if year t is a good year1 + /3 if year t is a bad yearEt {11(7rt )} = 1Et {uVrt4^X t l— qxt cV)]^= (1— cEO)p— (1— a)It•Substituting these into the general formulae in Proposition 5 and using ft —we get the desired results.1 ^(q(1-a)x."" \ ) 1D.2 Proof of Proposition 8 Let us suppose that at equlibrium, profit levels under the QQ system satisfy_7rt 7r" t+ < v < + 7‘rt-,^(t^2 )+7ri^<^< 7ri •(D.103)(D.104)1 As in the text, we omit the seasonal setup costs122u'(7rt)^+ 1 — b^uVrt)(5^1 ,r 1 ^l= (1 +1 —61—,3 ) LP 1+a itEt {u / r t)(P —^(0)}^6 EtEt+i {72' r*ii-i)[P qx,+ , i (0]} Appendix D. Proposotions 7 and 8^ 123First we derive equilibrium quota prices based on this supposition. Then we verifythat the above inequalities are indeed satisfied under these quota prices. (Rememberthat profit levels depend on quota prices.) Finally, it is easy to show that under theassumption p > [1 + 2(1 — (5)/6 2 ] fit , there is no equilibrium at which the above inequalitiesare violated.From (D.103), the marginal utility in year t at equilibrium isuVir t) = 1 — 0 if year t — 1 is a good year1 + 0 if year t — 1 is a bad yearregardless of the harvest in year t (t > 2). ThenEt+1 tit'Ctd-i)[P^1  c, ^( 1 — ))[P —qXt-1-1^(1 + OIPiLii] if year t is a good year1+a pi if year t is a bad year11EtEt+ifuVrt+i)[1) qx^ (e)]} = P^+ it •By using the general formulae in Proposition 6, we get the following: If year t —1 (t > 2)is a good year, the equilibrium quota price in period A of year t isFrom (D.104),1 — i3 if year 1 is a bad year1 + ) if year 1 is a good yearE1 {u'( 71-1)} = 11^1 — a/3El {u' eitd[p qxi^ cV)]} = p^1 + a ttAppendix D. P•oposotions 7 and 8^ 124Hence by (5.25) in Proposition 6p ^1+ a 1-1 + 1 6— (5 (^1+A^1 - a3All the other formulae in Proposition 8 are derived similarly.Now we have to verify that our suppositions (D.103) and (D.104) are true under thesequota prices and the assumption p > [1 + 2(1 — 6)16 2 ] p. We know that at equilibriumthe net profit ispkXj _ i^A kXt _ i kx \^B kx k)(i\2c(e)+ w • (^N^N)+wt .\^N^andtpkXi kx A kx^ 2c(e) + w t  • ( ^N ) (since h 1^N )We show(a) jr; < v <^, (b)^< v <^, and (c) _art^, irT < 4-7 .(a), (b) and (c) imply that the suppositions (D.103) and (D.104) are true.(a) We have v pkx/N — 2c(e) by definition, and+7"r^pk(1 + ^2c ( e)^+210^+( k(1 + a)x kx\^( kx k(1 + a)x +N^N 4-wP N^Npkx^+N L11l +— aft + (1 6) 46,A ]k A.aSimilarlypNkx^ kax^ 2c(e) — + (1 —Therefore^< v <(b) SincepkX1 .^kx kX 1^ ),71= N^^2c(e) + wr ( N^N1=Appendix D. Proposotions 7 and 8^ 125we havepk(1 — a)x^1 +^kx k(1 — a)x*i-- = ^N^2c(e) + [p it + 3^1 — 3 6 -01( N N )pkx kax 1 + /3 ,. (^1 ^6  )1-^1  j kax  ^2c(e)^ii^N N ^1 _ ,13 ° 1 + 1 + /3 1 _ s LP 1+ a l  N= 4-pkx^„^1— a kax 1— /3 (5 ( i +  1 ^S  ) rp ^1 ^ikax 7r— = ^1^N^ c(e)+ 1^ + ce P N^1 + 13^1 — 31 — 6 ) l^1+ aitNThen, by using+ 3^1^6 \ ,, xt 1^6 \^6  6 6(1 +1— 3^1+ 31— 6 1 '--- `-' \ ' -i- 16 1?- 2 1 ^6^and1 — /36(1 + 1^6 1^6  S1 + /3^1 — /31 — 6 1 ?" 21 — 6'it is straightforward to verify that our assumption p — [1 + 2(1 — 6)/6 2] ,u > 0 implies7rT — v > 0 and v — irl . > 0.(c) By definition we have+ii-' t = pk(1 — a)x^ 2c(e) + .01,Akax + +Lvr_kaxN N^N—pk(1 + a)x^_IpAkax^B+ kax_7r^t+ . ^N2c(e)N^_Lot Nir - = ^1^pk(1 — a)x2c(e)+ wB-kaxN 1 N= pk(1 N+ a)x2c(e) — w19+ kax^1 ^'Since +4- wf3- and _4+ wf3 + in Proposition 6, we getkax^,+^_Akax—^= +20N > 0, and 711 _7r t =   > O.Thus we have verified that the quota price formulae in Proposition 8 which are derivedbased on suppositions (D.103) and (D.104) are consistent with them.Suppose that inequalities different from (D.103) and (D.104) hold at equilibriumin some way or other. Proceeding as before, we can derive "equilibrium" quota pricesAppendix D. Proposotions 7 and 8^ 126corresponding to these inequalities instead of (D.103) and (D.101). Under the assumptionp > [1 + 2(1 — 6)/6 2 ] Ec, however, it can be shown that all these "equilibrium" prices leadto contradictions to the assumed inequalities. We omit the proof which is straightforwardbut needs tedious calculations.2c(e) — 32'1 — a^1 —^ A) kaxp +(p^1 + a^  /3 6 + w^7,2c(e) +4_ 3v.1 + /3 A) kaxlp^_ 0 6 _w Nw+ and wT: in Proposition 8, we geta)xa)Appendix E1Q Q and GQQWe show that at equilibriumkx 1 I pkx a^1^pkaxVQQ( 7v )(a) =6 N2c(e)1 + (0 — 1 — 6 ) 1 + a^N(b) GQQ (kx) = 1^a pkax^6 2 + 6 11 1 —— 6 1 + (1 — 6) 2 1-32 (p 1 + a p)k a x(a) It is easy to see that VQQ(kx/N) satisfieskx^1 1  6VQQ( — ) =^+^+ 4 _ dt(-1- 1r7)+u(1-i-iF )+In the following we calculate the RHS of (E.105) explicitly.We know thatu(_ir7)+ u(_Irt+)]. (E.105)+ pk(1 + a)x= ^ 2c(e) wB+kaxN^ 1 Npk(1 — a)x kaxir1 =Since^< v <^(see Appendix D),= (1 + 3)/ ^3v,= (1 —^+2c(e) + WB-^.1 NSubstituting the formulae for= (1 +3) [Pk(1;u(2r1) = (1 — „3)[Pk(1 —N127Appendix E. VQQ and GQQ^ 128Therefore1^,2-[u(ril") + u(irT)]^pkx^1 + a N 23 - kax 1^ kax2c(e)+     +^[(1 /3) -CDA -^)) d-dJA N • (E.106)Next we derive the expression for the second term of the RHS of (E.105). Since^< v <^t- ,^, we haveu(+mo7) = (1-)+9vu( = - + ievu( _irT) = (1 + 0) - 1rT - Ovu(_iriE) = - - -Henceu( + fr t+)u(_frt )u(_irtf )Therefore=^0)[pk(1 N- a)x= (i 0) 1pk(1+ a)x N[a.= ( 1 + 0) pk(1 -N a) (1 + 0) [pk(1+ a)xN=2c(e) +2c(e)2c(e)2c(e) -^_ A Lax^B _ kax+ w —N +wt —N OvA kax^B+ kax^ w —^- + w t^/3vA^_k x^B _ kax_WA^ -Wt N - 13v^_wkax^B+ kax-wt^0v•A^4 1 - (5[u( + Ir')^u( + irrt)^u(_fr)]- /3) +olkaxf pkx^ , ^[(1^ A^1 - 6 1 N^`'c6) 7)" I)) -wp. kax4l+- [(1 - ,3) + wr - + (1 +^- (1 - 0)+4+ - ( 1 + 0) - wt +]L •Appendix E. VQQ and GQQ^ 129Since^( 1- ))^+ (1 + 0)^= 2(p - it) + 26(1 + /3) _20, and1 --(1 - 0) +4+ - (1 + 0)_4+ = -2(p ^+ a it) 26 ( 1 - /3 ) +thA ,we obain1  641 - 6[n( + ii-T)^ u(._ir'7)]= 1 6it6  ipkx 2c(e„) 1 +a a Nkeel 12^kax6 [(1 + 0) _WA — (1 — 13) -Al+ ^N . (E.107)— [ NSubstituting (E.106) and (E.107) into the RHS of (E.105) yields the desired expressionfor VQQ(kx IN).(b) Next, we derive the expression for the authorities' expected revenue, GQQ(kx), wherekx is the initial quota endowments. Using superscripts and subscripts in the same wayas before, the authorities' revenue in a particular year is expressed as follows:+ G -tf = (— + WA + + wB +)kax,^_Gt+- = ( A + _wr+)kax,_F G7^( — 4.26A - +43- )kax,^_GT = ( _WA — _wr - )kax,G1^wBi+kcxx = _wB+kax,= —w B-^B -kax — — + w t kax.At the beginning of year 1, the expected present value of the authorities' revenue isccGQQ(kx) = E {E 6 iGtt=1=^{Gi} + 6E1{E2(G2Ix1)} + 62E1{ E2[E3(G3IX2)IXI]} + • • •^=^1 6+ ^Et_i{Et(CtIXt-i)}, (E.108)since, by assumption, X t (t = 1, 2, 3....) are independently and identically distributed.Using the formulae in Proposition 8 for _4 + , +4 - , + WA, and _wB, we obtainE1 {G i } = -1 Gt + -1 GT2^2Appendix E. VQQ and GQ Q^ 130and—2( -- .11)13+^-1-Wr)kaX^iukax6(2 + 1 _)6 61 3 0(2 P 1 +1 + a 1 a ii)kax (E.109)14-( + G+ + + GT +^+ _GT)t11 . + Alkax^ +4+ +^12^th _74+)kax - -( +4- + _4 - )kax4 4a 1^,3^1(E.110)By substituting (E.109) and (E.110) into the RHS of (E.108), we get the desired expres-sion for GQQ(kx):1^a^2^6 ^1 GQC? (kx) = 1— 61 + a^ ykax 6 [ i _ 6 + (1 _ 6)2 1 _ /32 (p 1 + a p)kax.From the above calculations, it is easy to verify that the term concerning the quotaadjustments in period A is 1 6— .7 2(— + CO + _z-vA )kax, i.e.,1 [(1 6^)2j 1 .±3 (P02^1 4. a p)kaxand the term concerning the quota adjustments in period B is1 6 1 + apkax 6 {1 -2 6 1 -I-3^11 - ^ap)kax.The (absolute) effect of the former concerning period A adjustments is much larger thanthat of the latter for period B adjustments as we can see easily by substituting 6 = 0.9into the expressions 0 202 and 127.1 + a pkax 6(1 + 1 — 6 ) 1 — /32(p 1 + a p)kax.Appendix FTable 5.4Since the procedure of the calculation is the same for all cases, we illustrate it with Case1 in which 6 = 0.9, pl = 1.5, and a = 0.1. By substituting these parameter values intothe expressions for GQQ, VQQ, and VsQ . we get^GQQ^(0.091 - 5.8411 !3,32)fikx^VQQ — VsQ^ . 17 .513 — 1)11^NIf 13 = 0.1,^GQQ^—0.499pkx —0.333pkx [8] andVQQ — VsQ = 0.068 Pk = 0.045 Pkx [utility unit].Remember that VQQ — 17sQ is measured in utility, not monetary, unit. Under the SQsystem the net profit at equilibrium is in = pkXt IN — 2c(e). When a = 0.1,+^pkx^7r t = 1.1 -N^2c(e),^and_^_pkx^2c(e).irt = 0.9 N^^2By definition we have v = pkx/N — 2c(e).Suppose that the authorities pay= (1 — 0.9)0.045pkx dollars131Appendix F. Table 5.4^ 132annually to each fisher under the SQ system. (In the following we show that this isjust enough to compensate for each fisher's loss under the SQ system.) Then after-compensation net profit levels are given simply by 'Ir t- + a and "* .t+ cr, 1 and we have+ a < v < irt + a.Before-compensation expected utility in year t is1-2[(1 - 0)7^r't+ + )3v] + -2 [(1^)31 )/i-7 - Ov]while after-compensation expected utility in year t is1-2 [( 1 0)(irt + a) + /9v] + -2 [( 1 0)( 71-7 + a) - dv]= 2 [(1 — /3)iriE Ov] + -2 [(1^0)11-.7 - Ov]+ o-.Thus the compensation •[$] increases the expected utility in year t by the amount a[utility unit]. Therefore the compensation increases the fisher's expected discounted sumof utility under the SQ system by the amount11 6a- = 0.045 pN [utiliy unit],which is just enough to compensate for the fisher's utility loss under the SQ systemVQQ - VSQ). Hence the constant annual compensation that makes fishers as well-off asunder the QQ system is indeeda = 0.0045 pkx [$].Then the authorities' total expenditure (discounted sum) for the compensation isN a = 0.045pkx [s].1 — 611 Remember that there is no actual quota trade taking place at equilibrium under the SQ system withidentical fishers with identical initial quota endowments^(1/N) • 100[%]).Appendix GEquilibrium conditions for the open SQ systemThe Bellman's equation isVsQ (h it = max Et { max {u(7rt ) -I-- 6V. sQ (hAt+1 )] ,e . 1 ,4,/f1^eP,ap,ipwhere the second maximization is subject toatetA cixter < + + randar t = Pq-Kte tA — c(et) + A (htA a )^A I t — ^+XtpqXt e tB — c(e tB) wtB • (—xa tA — a tB ) — rn tB lr —h t + 1 X= ^aLet us denote equilibrium quantities by a hat. Then the following conditions have to besatisfied at equilibrium.uTirt)[pqXt — g(et)] — AtqXt = 0^ (G.111)u'^+ 6VsQ(iti4 1 )—x + At = 0 (0 < 4< +00)^(G.112)+(—oc < lB < +0c)^(G.113)(G.114)(0 < a tA < +co)^(G.115)(-00 < It < +0c)^(G.116)t)(—mTh + A t^= 0Et laVrt)[PqXt — (1 (0)] — A t qXt } = 0Et {z1 (irt)( — 4 +^B, X t ) = 0Et {11' (irt)( — T4 )^At} = 0133qQ^) = Et {u'(7t)wtA }B — rnB = 6wA xro t+1wt — m^E {wB —XtEt^t x(0.117)(G.118)(G.119)Appendix G. Equilibrium conditions for the open SQ system^ 134whereA X ;, Bt''t+1AtWe assumed 0 < ai9 < -boo etc. (G.111)— (G.113) are from the second maximizationproblem, (G.114)—(G.116) are from the first maximization problem, and (G.117) is fromthe envelope theorem applied to the Bellman's equation. (G.118) and (G.119) are theno-arbitrage conditions in the quota market.From (G.111)1At = WO-t) {p — ',AqXtBc et Susbstituting this into (G.113) yields^CI (0).771t = p qXtThe quota rental price in period B is equal to the marginal profit of harvesting one moreunit of fish.


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