INDIVIDUAL TRANSFERABLE FISHERY QUOTAS UNDER UNCERTAINTY By Hisafumi Kusuda Bachelor of Law, The University of Tokyo, 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES ECONOMICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1993 ©HisafumKd,193 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature Department of Economics The University of British Columbia Vancouver, Canada Date ^ DE-6 (2/88) June 30, 1993 Abstract A model of a fishery with an uncertain fish stock is proposed to compare alternative management systems with individual transferable quotas (ITQs). Assumptions of the model include: (1) the fish stock fluctuates randomly year by year; (2) in-season stock depletion is small; (3) the total allowable catch (TAC) set by the quota authorities has a definite relation with the fish stock level; (4) the true value of the stock level is revealed only at the middle of each season, when the authorities revise the TAC; (5) fishers form rational expectations on future quota prices. The principal results are: (a) If fishers are risk-neutral, the share quota (SQ) system and the quantity quota (QQ) system generate the same amount of fishery rent, although the division of the rent between fishers and the authorities under one system is different from the other. If the TAC is proportional to the stock level, the more price-inelastic the demand for fish is, the more likely it is that fishers are better off under the QQ system at the expense of the authorities. (b) A quota tax and a harvest tax that collect the same amount of revenue for the authorities result in the same division of the fishery rent among heterogeneous fishers. The quota tax and the profit tax differ in this respect. Which fishers will prefer a quota tax over a profit tax will depend on fishers' shares of the initial quota endowment and in total inframarginal profits afterward. (c) If fishers are risk-averse, the SQ system and the QQ system are not equivalent in their allocative efficiency. An example shows that the SQ system is potentially better than the QQ system when fishers prefer the latter and the authorities prefer the former. This conclusion has to be modified if risk-neutral traders participate in the quota market ii Table of Contents Abstract^ ii List of Tables^ vi List of Figures^ vii Acknowledgements^ viii 1 Introduction^ 1 2 The Basic Model^ 11 2.1 Assumptions ^11 2.2 Notation ^17 2.3 Fishers' problem and rational expectations equilibrium ^18 2.4 Interpretation of the model: spacial fish stock distribution ^20 2.5 Ideas behind results ^23 3 Comparison of Alternative ITQ Systems ^ 3.1 Equilibrium quantities 27 ^27 3.2 Allocative equivalence of alternative ITQ systems ^ 3.3 Distributive effect 36 ^37 3.4 Adjustment phase ^ 4 Tax Effects^ 48 51 4.1 Equilibrium under taxation ^52 iii ^56 4.2 Comparison of tax effects 4.3 Adjustment phase ^ 5 The Case of Risk-averse Fishers^ 62 65 ^66 5.1 No risk-neutral traders case 5.1.1 New assumptions and notation ^ 66 5.1.2 Rational expectations equilibrium ^68 5.1.3 Fishers' behaviour at equilibrium ^69 5.1.4 Equilibrium quota prices ^71 5.1.5 Example: piecewise linear utility function ^74 5.1.6 Equilibrium quota prices ^76 5.1.7 Comparison of the two ITQ systems ^ 82 5.1.8 Numerical examples ^ 83 5.2 Risk-averse fishers with risk-neutral traders ^89 6 Conclusion^ 95 Bibliography^ 100 Appendices^ 103 A Proof of Proposition 1^ 103 B Propositions 2, 3, and 4^ 116 B.1 Proof of Proposition 2: profit tax ^ 116 B.2 Proof of Proposition 3: quota tax ^ 117 B.3 Proof of Proposition 4: harvest tax ^ 117 iv C Propositions 5 and 6^ C.1 Proof of Proposition 5: the SQ system ^ 118 C.2 Proof of Proposition 6: the QQ system ^ 119 D Proposotions 7 and 8^ E 118 122 D.1 Proof of Proposition 7 ^ 122 D.2 Proof of Proposition 8 ^ 122 VQQ and GQQ^ F Table 5.4 127 ^ 131 G Equilibrium conditions for the open SQ system^ V 133 List of Tables 1.1 The quota authorities' revenues and expenditures in New Zealand ^. . 6 1.2 Basic statistics of Hoki fishery in New Zealand ^ 7 5.3 Critical values of the degree of risk-aversion 5.4 The quota authorities' deficits under the two ITQ systems 5.5 Quota price movements under the QQ system ^ vi ^ ^ 83 86 87 List of Figures 2.1 Basic model ^ 14 2.2 TAC adjustments ^ 14 2.3 TAC under alternative ITQ systems ^ 15 3.4 Inframarginal profit under proportional TAC ^ 41 3.5 Fishers' expected profits under the two ITQ systems 4.6 Lorentz curves for profit share (aj) and quota share (71j) 5.7 Piecewise linear utility function ^ 75 5.8 Risk-aversion, government's revenue, and fishers' utility ^ 84 vii ^ ^ 47 61 Acknowledgements I would like to express my deep gratitude to Professors Gordon Munro and Philip Neher who guided my study in natural resource economics and supported this research from the beginning. The constant encouragement and valuable suggestions given to me by Professor Colin Clark (Mathematics) are gratefully acknowledged. I appreciate the helpful comments of Professors John Cragg, Mukesh Eswaran, William Schworm. and Margaret Slade. I also thank Professors Anthony Charles (Saint Mary's University), Michael Healey (Westwater Research Centre), and Carl Walters (Zoology/Fisheries Centre) for their thoughtful comments and good suggestions at the final oral examination. The advice of Professor Quentin Grafton (University of Ottawa) has substantially improved both the content and the style of this thesis. Mr. Philip Major (Director of Fisheries Policy, the Ministry of Agriculture and Fisheries, New Zealand) kindly provided several key data. Finally, I would like to give my special thanks to Professor Keizo Nagatani who has encouraged me ever since I came to UBC five years ago. I was supported by a Government of Canada Scholarship for Foreign Nationals and A. D. Scott Fellowship in Economics during the preparation of this thesis. viii Chapter 1 Introduction Individual quota management systems are used in several countries of the world as a means of making fisheries more efficient. Under an individual quota system fishery quotas are allocated to each fisher or vessel and the sum of the quotas which is the total allowable catch (TAC) is controlled by the quota authorities. The most comprehensive scheme has been adopted in New Zealand since 1986. Australia, Canada, Iceland, and several other countries have individual quota systems in particular fisheries.' New Zealand's system covers most of the inshore and deepwater fisheries. Their system is outstanding not only in its comprehensiveness but also in the fact that their quotas are permanent and fully transferable. The systems in the other countries are more limited in their coverage, quota tenure, and quota transferability. In Australia an individual quota system has been 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 individual quota system, known as Enterprise Allocations, since 1984 in several of its Atlantic fisheries. These fisheries include the offshore ground fishery and the inshore cod fishery along the west coast of Newfoundland (1984 the offshore lobster fishery (1985 the offshore scallop fishery (1986 and the northern shrimp fishery (1987 Under Enterprise Allocations, however, companies cannot freely buy and sell their allocations. The government has discretionary powers over such actions. In Iceland all demersal fisheries 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. 1 Chapter 1. Introduction^ 2 quotas can be transferred only between licensed vessels. Quotas are allocated to licensed vessels based on their recent average catch and the duration of quotas is a few years at a time. In Norway individual catch quotas are not transferable and are allocated each year at the discretion of the government. Although the characteristics of individual quota systems vary from country to country, it is generally recognized that wider coverage, fuller transferability, and longer tenure of quotas are desirable long-term management goals. The common property nature of fish resources poses two main problems which have to be handled by management authorities: (1) if there is no regulation, excessive entry to the 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 means but fail to curb the intensity of fishing activities, excessive fishing effort and excessive investment in vessels competing for the limited harvests still result in the dissipation of resource rent. Munro and Scott [24] refer to these as Class 1 and Class 2 common property problems, respectively. Many fishery management schemes, such as vessel licensing or shortening of fishing seasons, have failed because they are inappropriate in resolving the Class 2 common propety problem. For example, shortening of fishing seasons has led to excessive fishing effort within seasons and more idle capacity between seasons; Vessel licensing programs have stimulated the installation of ever more powerful fishing equipment. The appeal of 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 also prevents the dissipation of resource rent because the overexpansion of fishing activities is curbed by individual quota limits. Under this system each fisher will try to harvest up to their quota limit with minimal costs. Chapter 1. Introduction^ 3 However, there is no gurantee that the aggregate cost in the fishery is minimized to harvest the aggregate quota limit (= TAC) unless quotas are fully transferable between fishers. If quotas are transferable, quota prices formed in the quota market constitute opportunity costs of fishing which fishers have to take into account when they decide the scale of their operation. Then, in principle, quota transferability forces the fishery to operate in the optimal manner by allowing the shift of fishing effort from less efficient fishers to more efficient ones. We refer to such a system as the individual transferable quota (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 by the 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 rather smoothly without much opposition from fishers by buying back a part of the quotas initially allocated gratis to fishers. This initial buyback scheme constitutes an important aspect of quota management systems in real situations. As Clark [10] points out, a correctly calculated tax on catch has, in principle, the same allocational effect as the ITQ system. The advantage of the ITQ system lies in the fact that under the system the authorities have direct control over the fish stock and can attain the (approximately) optimal stock level more easily. This is because they do not need to have detailed information about each fisher, particularly about his harvest function and cost function, which is necessary to design the optimal tax scheme. Quota market prices are determined in the quota market in a decentralized manner reflecting all the private information about each fisher.' This is the reason why, despite many practical difficulties, 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 ^ 4 In view of the important role the ITQ management system will play in the future, in this thesis I investigate the economic performance of the ITQ system. In particular, we study ITQ systems which offer permanent tenure and full transferability of quotas. There are basically two variants of the ITQ system depending on how the quota authorities adjust the level of total allowable catch (TAC) within fishing seasons and from season to season. One is a share quota (SQ) system and the other is a quantity quota (QQ) system. In the SQ system, quotas are allocated as a percentage of the TAC of each season and sum to 100 percent. For example, if a fisher possesses 1 percent share quota and the TAC is 1,000 ton per season, then the fisher is allowed to catch 10 ton of the fish in each season, as long as the TAC remains the same. If the authorities announce a change of TAC from 1.000 to 1,500 ton, the fisher's seasonal allowable catch is automatically changed from 10 to 15 ton from that season onwards. The authorities need not sell or buy quotas in the market'. On the other hand, in the QQ system, quotas are denominated in absolute tonnage unit.' If a fisher has 10 ton of quantity quotas, the fisher can harvest up to 10 ton in each season regardless of the current TAC. as long as the fisher holds that amount of quota at the end of each season. Suppose that the current TAC is 1,000 ton and the authorities decide to change it to 1,500 ton. Mere announcement of the change in the TAC has no effect in the QQ system. In order to put the change into effect, the authorities have to issue and sell 500 ton of quotas in the quota market. In other words, the authorities must 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 percentage unit are different from 100 percent, the quota authorities sell or buy the difference in the quota market 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's quota 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 fishers are automatically cancelled. This means that we do not count the quantity quotas retained or obtained by the authorities as a part of the TAC. Chapter 1. Introduction^ 5 the market so as to maximize their profit. If the abundance of fish is uncertain but if information increases as fishing activities proceed in a season, it will be better for the authorities to adjust the TAC based on newly acquired information than to stick to an old TAC. When the fish stock turns out to be less (more) abundant, a decrease (increase) in TAC will be desirable. As described above, the SQ system and the QQ system are different in the way the TAC is changed. It is 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. New Zealand introduced an ITQ system in 1986. It was originally a QQ system', but over time the authorities incurred a deficit in quota operations when adjusting the TAC's. (See Tables 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 SQ system in 1989. 8 The reason for the fishing industry's opposition may be explained by fishers' aversion to income fluctuations under the SQ system in which fishers get no compensation for their harvest loss when the authorities decrease TAC. But there may be other economic reasons for the opposition that do not depend on fishers' attitude toward risk. A related issue is why authorities should lose money through quota market operations and if this is a general phenomenon or not. Suppose that authorities tend to lose money with a QQ system but fishers are better off under that system. Which ITQ system is then better for the society as a whole? Careful theoretical arguments seem necessary 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.' But See Ian N. Clark et al[13] about the ITQ sytem in New Zealand. See Macgillivray [22], p.7. 9 Libecap [20] gives supplementary comments on [2]. 7 8 6 Chapter I. Introduction^ Table 1.1: The quota authorities' revenues and expenditures in New Zealand (All values expressed in current $ NZ) Ministry of Agriculture and Fisheries 1984/85 ($000) Actual 1985/86 ($000) Actual 1986/87 ($000) Actual 1987/88 ($000) Actual 1988/89 ($000) Budget REVENUE Quota Leases Quota Sales Resource Rentals Foreign Access Fees Deepwater Royalty Other 0 0 0 15,129 2,264 745 0 0 0 13,613 5,720 3,185 0 60,738 0 19,410 8,656 2,306 3,373 22,835 12,500 22,974 0 3,204 2,000 0 20,027 9,000 0 3,686 Total Revenue 18,138 22,518 91,110 64,886 34,713 EXPENDITURE Personnel Operating Capital 9,951 5,928 662 11,406 7,085 782 14,647 8,995 2,486 16,623 10,701 538 18,557 13,592 2,166 MAFFish operating 16,541 19,273 26,128 27,862 34,315 Buybacks Other Assistance 0 2,509 0 715 44,630 486 1,364 1,567 3,000 74 MAFFish Grants 2,509 715 45,116 2,931 3,074 Total Expenditure 19,050 19,988 71,244 30,793 37,389 (912) 2,530 19,866 34,093 (2,676) SURPLUS/(DEFICIT) (Source: Macgillivray [22], p.11.) Chapter 1. Introduction^ 7 Table 1.2: Basic statistics of Hoki fishery in New Zealand Volume traded 9 [t] Year b TACc [t] Catch [t] Fish price' [S/t] Quota prices [$/t] 86/87 250,029 158,171 430 998h (370-4,167) 33,586 87/88 250,062 216,240 281 639 (352-2,200) 6,409 88/89 251,036 182,310 350 979 (350-2,000) 5 89/90 251,036 208,546 520 662 (200-2,500) 2,445 90/91 201,897 218,172 1 500 846 (400-1,283) 15,721 a. Hoki is one of the major fisheries in New Zealand which accounts for about 20% 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 by TAC = total quantity quota held by fishers and the authorities. If this is the case, a reduction of TAC under the QQ system occurs only when the authorities cancel any quota held by them. Notice that this is different from our definition of TAG which does not count the quotas retained by the authorities 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 carried over from 1989-90. See Annala [4], p.100. (Source. c,d: Atlas of Area Codes and TACCs 1991/1992 published by Clement and 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^ 8 the effect of uncertainty in fish abundance is not discussed there. Arnason [6] analyses the relative advantage of the two ITQ systems with regard to the optimization of a deterministic and non-seasonal fishery model. As such, his model is not appropriate to address the issue of seasonal TAC adjustment under uncertainty. Hannesson [17] analyses the economic performance of the QQ system under stock uncertainty in a manner similar to mine but with a much simpler aggregated model. His model, however, tacitly assumes that TAC adjustments are carried out through seasonal quota lease between fishers and the authorities. Hence his results, although very suggestive, do not directly apply to the QQ systems in which the level of TAC's is adjusted through permanent quota trades between fishers and the authorities as assumed in my analysis.' One of the purposes of this thesis is to bridge the gap in the existing literature by providing an analytical framework suited for the study of ITQ systems with permanent quota under stock 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, reap substantial capital gain. It can be argued that this kind of "windfall gain" for priviledged fishers is not desirable from the perspective of social equity. Those who consider equitable distribution very important may argue that all (or most) of the fishery rent should revert to 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 on fishers at the same time. In principle, fishery rents can be split up between fishers and the authorities at any desired ratio with this method.' There are many different tax forms 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] . See Clark [12] Chapter 8. 11 Chapter 1. Introduction^ 9 rents. 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 quota trading values. The reason is that under this system "... the value at which quota is traded is the best guide to economic rent, and therefore should be the determinant of the resource rental." 12 However, "The New Zealand fishing industry has consistently argued that quota trading data are inappropriate for setting resource rentals. This has led to long and vigorous debate between the industry and government." 13 So far, there are not many theoretical works in the literature about advantages and disadvantages of various taxes in the fishery under an ITQ management system. Grafton [16] gives some analysis about this matter, although his model does not take into account fish stock fluctuations which can play a major role in ITQ management systems. This thesis will show that the theoretical framework developed here can be applied to the analysis of tax effects in ITQ management systems under stock uncertainty. The real situations surrounding ITQ systems are very complicated both economically and biologically. I have to make many simplifying assumptions in my model in order to get clear analytical results. In chapter 2 I explain the basic structure and assumptions of the model. In chapter 3 I derive equilibrium quantities for the SQ system and the QQ system with risk-neutral fishers. A comparison of the two systems is then made with regard to the division of the fishery rents between fishers and the authorities. In chapter 4 I examine tax effects in a fishery managed under an ITQ system. Using equilibrium quota prices under taxation, I compare the effects of taxes on the division of the fishery rents among fishers with different levels of efficiency. In chapter 5 the comparison of the SQ system and the QQ system is extended to the case of risk-averse fishers. I discuss 12 13 Ian N. Clark et al. [13] pp.334-337. [13] p.337. Chapter 1. Introduction^ the limitations and the possible extentions of our model in chapter 6. 10 Chapter 2 The Basic Model 2.1 Assumptions Any economic model of a fishery incorporates many simplifying assumptions reflecting the complexity of the subject. The complexity arises from both the biological and economic sides of the problem. Uncertainty and dynamics are the main ingredients of this complexity. The difficulty surrounding the estimation of stock-recruitment relationships is well-known. There are too many unknown factors which affect the relationship. Not only the fish stock dynamics but also the fish stock size at any point of time are very hard to estimate. On the economic side, it is difficult to model the behaviour of fishers who face uncertain fish stock dynamics and uncertain fish stock size. If fishery quotas are 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. The authorities' behaviour concerning the determination and adjustment of the level of TAC can also be very complicated. The model in this thesis is based on the standard single-species, single-cohort, seasonal fisery model which is commonly used in the literatures. The uncertainty of the fish stock level is crucial for the TAC management under the individual transferable quota (ITQ) system, whereas the inclusion of uncertainty into the model makes the analysis much harder. We make the following assumptions as a compromise. 1 See, eg., Clark [10]. [11]. IA Chapter 2. The Basic Model^ 12 1. Under the TAC management, the level of the fish stock. X t , fluctuates randomly from year to year according to a known probability distribution with expected value x over a range [b, , where b > 0. This implies that we can ignore the stock-recruitment relationship in the fish population dynamics. 2. The fish stock does not deplete significantly in each season even under the pressure of harvesting activities because the TAC is assumed to be small compared with the total fish stock. The fish stock level, Xt , therefore, is approximately constant throughout each season. 3. One fishing season occupies a part of one year. Each fishing season consists of two periods with equal length, period A and period B (see Figure 2.1, p.14). During period A the exact level of the fish stock in the year is not known. It is revealed only at the end of period A, presumably the result of catch-effort data analysis conducted by the authorities.' 4. The quota authorities try to set the level of TAC equal to a variable proportion K(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 overall optimization problem of how the authorities should relate TAC to the total stock level is beyond the scope of this thesis. 5 ) 2 Hilborn and Walters [19] give detailed accounts about the techniques used in the actual stock assessment. 3 The 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 independent of each other in our model. We can think that the stable probability distribution of fish stock size 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 if we imagine the situation in which the authorities do not impose the TAC limit. Fishers will harvest as much as they like and the stability of fish stock level will be lost in overfishing. 5 Hannesson [18] gives some analysis about this matter. 13 Chapter 2. The Basic Model^ 5. 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 of TAC from last year's K(X t _ i )Xt _ i to the provisional level K(x)x. At this point the true stock level X t 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 change the 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 QQ system and the genuine SQ system as two special cases. In this hybrid system the authorities divide the overall TAC into two parts, one for share quotas and the other for quantity quotas (see Figure 2.3, p.15): 6 TAC 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, and 0 < 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 TAC for quantity quotas, the authorities have to either sell or buy back the difference in the market to implement the TAC. 7. We denote by N the number of fishers who remain in the fishery from year 1 onwards. N may change over years and may be larger or smaller than the original number 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 the genuine SQ system. 14 Chapter 2. The Basic Model Figure 2.1: Basic model fish stock Xt known.(X 1) \J random fluctuation k unknown random fluctuation A known(X2 ) N^ unknown ^ >t ^ A fishing season^fishing season year 1^year 2 Figure 2.2: TAC adjustments TAC [ton] 7 KX2 A A fishing season fishing season year 1^year 2 15 Chapter 2. The Basic Model Figure 2.3: TAC under alternative ITQ systems TAC [ton] General hybrid system ^Ic(Xt)Xt f(Xt) SQ ^> Xt TAC [ton] Genuine QQ (SQ) system QQ (SQ) 0 b N./ > xt 16 Chapter 2. The Basic Model^ (a) Fisher i's variable effort cost function ci(.) is the same for all periods. A crucial assumption 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 in that season. Fisher i's total cost in year t is then ci( eA t where etli and eri + c, (e tBi) + are the fisher's effort levels in period A and period B, respectively. We assume that effort level e it i (j = A, B) is constant during period j. Effort levels measure the volume of sea water screened by vessels in a 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 standard form' yA = ate tAi and ytBi - q xtetBi where q catchability coefficient which is assumed to be constant. 8 (c) Fishers are all risk-neutral. Namely, fisher i maximizes the expected present value of his profits E 6'141, where 'et is the fisher's net annual profit and 6 is the discount factor. (1/6) —1 is assumed to coincide with the market rate of interest. Fishers form rational expectations on future quota prices. This implies that their anticipated distribution of future quota prices is consistent with realized prices. 8. The market demand for the fish depends on the price of landed fish, pt , through the demand function D(p t ). p i is determined so that D(p t ) equals the supply of See Section 2.4 for more about this. For a detailed account of the definition and the meaning of catchability coefficient, see Clark [11], Sections 2.1 and 2.2. 7 8 Chapter 2. The Basic Model^ 17 the fish St , i.e., pt = D -1 (Si). This demand function is assumed to be known to fishers and the authorities. 2.2 Notation We use the following notation throughout this thesis. Fisher i's annual net income in year t is 7rt =^rpiqXte —^wt • (11P ^ j=A,B ) + wi • ( - 6 )] - where 9 the price of landed fish (unit: $/ton) Pt = q = catchability coefficient (unit: /m 3 ) 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: %) ei = seasonal setup costs (unit: S) 'As a matter of fact, the share quota price"^A, B) is the total monetary value of share quotas in the market. - Chapter 2. The Basic Model^ 18 and t-1^h t ^at a t-1^t^t^t (tBil = otAi, Bi C;1i = j t because after-trading quota holdings in each period are carried over to the next period. 2.3 Fishers' problem and rational expectations equilibrium One of the key points of the ITQ management is that fishers are not allowed to sell more fish than their final quota holdings each season. At the end of each season they have to possess enough quotas to cover their harvest of the season, if they want to sell all the harvested fish. We assume that there is no way for fishers to carry over their harvest of this 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 quota holdings in tonnage unit in the same year. aP i (Sri • f (X t )) [ton] is his season-end quantity (share) quota holdings. Given the initial quota endowments and based on his expectations about future quota prices {4, WtA; wr, ,^fisher CBilJ=1 i chooses a contingency plan {e tAi , a tAi , Cim ; e tB, aPi rx t t=1 7 about his effort level and quota holdings of each period so as to maximize the expected discounted sum of his profits E {E. 6 t_i..r - ., itt t=i subject to his quota constraint q xt etAi q x t eiBi < atBi ) for t = 1, 2, 3, ... 19 Chapter 2. The Basic Model^ It is assumed that fishers form rational expectations on the quota price sequence f 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 { eA t a At eS t aB t cri}toci is also a stochastic process that depends on the stochastic path of Xi . The rational expectations hypothesis implies that fishers perceive the correct relationships between the fish stock level and quota prices. Our fundamental hypothesis about the working of the quota market is that the current quota price in any period is determined so as to clear the market. Formally, we assume that the quota prices wt > 0 and 14/1 > 0 (j = A, B) are adjusted in the market so that the following market equilibrium conditions hold in the quantity quota market and in the share quota market, respectively: ^Ei a' 1 E i ate < K(z)z — f < (z), = if wi > 0, 1 ,^ = if WI > 0, where z = x for j = A, and z = X i for j = B. The LHS's are total quota demands and the RHS's are total quota supplies. A rational expectations equilibrium' for this model is defined by stochastic processes w2t4 ,Willi4,wtB and , (i 1,...,N) that satisfy the following two equilibrium conditions': n Ai 1. Given fishers' optimal contingency plans for setting {^-t ^-t -t^1 t=1 wtBr tA wtB,^ (i = 1,^N), the stochastic process for quota prices^w i- clears the quota market in each period, i.e., 1 E i ajt i < k(z)z — f(z), = if 'ail' > 0, Ei Ciii < 1 , = if II/ti > 0, "Newberg and Stiglitz [26] (Ch. 10) succinctly explain and defend the concept of rational expectations equilibrium 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^ 20 where 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 , { 6A ti,aB ti,(tBiltoci t atAi ,^eS stochastic process^ maximizes expected present value E {E,"=1 6' 1 70 for each fisher i = 1,^, N. 2.4 Interpretation of the model: spacial fish stock distribution There are two possible interpretations about the spacial distribution of the fish stock in my model: (1) uniform distribution, and (2) patchy distribution. 1. Uniform distribution Like many standard fishery models, we can interpret our model assuming that the fish population Xt is uniformly distributed over the fishing ground. Consequently, the production function yi = qXt e't (j = A, B) is easily understood. However, there is a difficulty about this interpretation. Suppose it is literally the case that the spacial distribution of fish is (approximately) uniform. Then fishers will realize the true stock density at a very early stage of period A from their harvest records. Then it is natural to expect that they will adjust their effort levels and quota holdings long before the beginning 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 do not 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 period B as assumed in our model, it has to be the case that the spacial distribution of fish is nonuniform (patchy) and fishers cannot form an accurate estimate of the fish stock level until after having been at sea for a substantial time. Therefore, we are led to the second interpretation with patchy distribution. 21 Chapter 2. The Basic Model^ 2. Patchy distribution' Suppose that the fish stock in the fishing ground consists of many schools which are all of the same size. We denote the total number of schools in year t by Xt . Let us denote by At [school/hour] the expected rate of encounter with fish schools when the effort level is el IT [m3 /hour] (j = A, B), where T hours = 1 period. Then ^X t [school]^1^• = [m 3 /hour] x ^ = e't Xt [school/hour], — T V [m3] VT where V [m 3 ] 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), where o(ds) lima s – 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 following Poisson distribution 14 : Pr (np4) = (A sr^ S ^ (j = A, B) This distribution has mean Vi s and variance .1t s. Therefore the coefficient of variation 0 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 fishing See Clark [11], Chapter 2.4. 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 approximately follows the normal distribution with mean V, s and variance Vt s. 12 13 Chapter 2. The Basic Model^ 22 season proceeds, they update their estimation based on their harvest records. Then our basic assumption about fishers' behaviour amounts to an assumption that fishers change their effort levels and quota holdings only at the beginning of period B when they have accumulated 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. Since encounters with fish schools are random, the total catch yi 4 [school] in period A and yP [school] in period B (so y id + y1-3 in year t) are also random. Let us assume that fishers can catch all the schools they encounter. We interpret the catchability coefficient q as q = 1/V [1/m 3 ]. Then )4 = qXt eUT [school/hour] holds and the average number of schools caught in period j when the effort level is et will be given by )47 1 = qX t 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, the harvest 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 is small 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 small if each fishing period is sufficiently long. Hence, we can interpret our harvest function yti = qXt etl = A, B) as a first approximation. If the authorities monitor all the fishers' harvest records, the authorities' estimate of the fish stock level is more accurate than each individual fisher's estimate by the law of large numbers. This can be another reason that fishers wait to make decisions about their behaviour until the beginning of period B when the authorities announce a new TAC level based on their latest stock estimate. Our basic assumption that after period A the authorities' stock estimate is sufficiently close to the true value Xt is also justified on this ground. Chapter 2. The Basic Model^ 23 2.5 Ideas behind results Using the basic model described above, we examine the economic characteristics of ITQ management systems. Before going into detailed arguments, we explain the basic ideas behind them. They will clarify the meaning of the results derived below. Those ideas are simple and intuitive. Imagine a fishery managed under the genuine QQ system defined above. Let epi be the profit maximizing effort level in period B for fisher i. Let mBt be the quota rental price (for the season) in period B. Then not = Pt 1 — qXt c(e,Bi) has to hold at equilibrium. On the RHS, is the marginal cost of harvesting one 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 more unit of quota for the season. In a quota market equilibrium this value should be equal to the quota rental price m tB. In other words the value of 'e tBi is adjusted so as to satisfy the above equality, where the values of X i and p i are known to fishers with certainty in period B. The true fish stock level X i is not known in period A. Suppose that Xi turns out to be larger than the average x in period B. Then the quota authorities have to sell kXt — kx(> 0) of quantity quotas.' If Xi < x, then the authorities have to buy back kx — kXt (> 0). Now the capital price of quota, wr, will be determined reflecting the rental 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. same idea is used in Hannesson [17](pp.461-462), although his model implicitly assumes an ITQ system with one year quantity quotas for which zr2f3 43. 17 The - Chapter 2. The Basic Model^ 24 • If m tP moves in the same direction as the fish stock level X t , then in period B sell quotas when X t and the authorities { 4 are large buy quotas when X t and t1,13 are small As a result, the authorities gain money through quota market operations under the QQ system. • If n-43 moves in the direction opposite to X t , then in period B the authorities I sell quotas when quota price tvP is low buy 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 the quota market operations under the QQ system. The formula int = 1 Pt ^ eVri) qXt (i = 1, ... ,N), suggests that there are three elements which may affect the movement of mr relative to Xt : (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 moves in the opposite direction of fish supply kXt^TAC). In this case, p t moves in the opposite direction of Xt. 2. 1/qXt is the amount of effort to harvest one unit of fish when the fish stock level is Xt . This moves in the opposite direction of X t . Since pt and l/qX t move in the same direction for different X t values, we are not sure whether mr and Xt move in the same direction or not. Chapter 2. The Basic Model^ 25 3. ci ( '6ei) is the marginal effort cost at equilibrium. Without further information, we cannot determine in which direction this term moves relative to Xi . The rest of this thesis is devoted mainly to the investigation of the conditions under which we can determine the relative movement of X i and me (or we). Remark: We have not mentioned the market operations of the authorities in period A. The reason is as follows. Suppose that quota price in period A, 4, is constant over time. 18 Since the total quota holdings in the previous year (= kXi _ i ) fluctuate evenly around the provisional level of TAC of this year (= kx), the effect of quota market operations on the authorities' revenue cancel out over time. Therefore when we consider the authorities' revenue in the long run, we can ignore the effect of quota market operations in period A. 19 How about the genuine SQ system? How can we compare the SQ system and the QQ 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 can be used as a useful heuristic principle. The principle takes the following form when applied to our fishery model. Suppose the 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^ 26 Then under a given TAC policy of the authorities, an ITQ management system, whether it is a QQ system or a SQ system, should bring about the efficient outcome that maximizes the total monetary benefit from the fishery. Thus a QQ system and a SQ system under the 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 given overall TAC adjustment rule. For example, consider a QQ system and a SQ system with initial quota endowments equivalent to each other. Then whether fishers are better off under the QQ system or not is equivalent to whether under the QQ system the authorities' expected revenue through quota market operations (after the initial quota sales) is negative or not. Therefore the above argument about the relative movement of m tB (or wP) and X t that affects the authorities' revenue under the QQ system have a direct implication on the relative advantage of the two ITQ systems. Chapter 3 Comparison of Alternative ITQ Systems 3.1 Equilibrium quantities As mentioned in the Introduction, there is a dispute concerning the relative advantage of the SQ system and the QQ system when the fish stock size fluctuates randomly from season to season. We analyze this problem by using our basic model with a general hybrid 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) functions wi4 = WiA (1)(-W, wB =^OA, Xi ), WtB = (Hi, 0^), where HA (0-ii) is the aggregate quantity (share) quota holdings of all fishers at the beginning of year t. 2. Implicit in the above is that (4, WtA; wr, WtB ) may depend on the level of the aggregate 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 above forms. Rational expectations hypothesis then implies that fishers know (or correctly guess) these functions. Proposition 1 gives various quantities at equilibrium. The proposition is followed by detailed remarks (pp.29 - 36) which explain the meaning and the 27 Chapter 3. Comparison of Alternative ITQ Systems^ 28 intuition behind it. Proposition 1 Suppose that under the hybrid ITQ system the authorities endow fisher i 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 denote the expectation operator with respect to X t by Et {•} .^Then the equilibrium number of fishers N and the equilibrium effort levels e: ti i (i = 1, .^, N; j = A, B) satisfiy ^^Ai^Ai e t^=^e^=^constant,^ (3.1) Et feaefli )} ,^ ti(e ri ■)^=^ ^=^cAt _(eBN) ^g(Xt), ^c Et {g(Xt) • i=1 ( oN^eSt N)}^=^E t {cN(oN) at(eAl^ 67 } t N) eN (.13 (3.2) (3.3) (3.4) (3.5) K(Xt)Xt eri)^ on the assumption that 1 d 0^VXt^ E [b,^, Pt^g(Xt)^>^ qXt ABt^ et >^0^Vi=1,...,N; (3.6) and VXt E [b,ar (3.7) Here the numbering of fishers is such that for i < N Et { g(Xt)^(e Ai^e Bi ) }^>^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 on the condition that they satisfy the quota market equilibrium conditions N E^= ri(z)z — f(z)^ (3.9) i=1 E c/a^1^ (3.10) Chapter 3. Comparison of Alternative ITQ Systems^ 29 where z = x for j = A and z = X t for j = B. The equilibrium quantity quota prices [$1 ton] are A Wt _ wB ==: 1 — SEt {Pt qXtg(Xi 1 g(Xt)+ Sz-vA = Pi — qXt — ( 3.11 ) (3.12) and the equilibrium share quota prices [$1 TAC for share quota] are 1 wtB = 1 6 Et{[Pt ^ qXtg(X0] • f(Xt )} =: 1 Pt ^ g(Xt)] qXt •f (xt) + SW A . WA (3.13) (3.14) The expected present value of fisher i's net profit Vi(•,•) is 17i (joi iti ■ AhAi _L^_L e ^1^1 /1) / ) bEt { g(xt).(e.ki^ 1^ e i )^,i1 (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) and subsequent quota market operations is G(I-V ,^=^HiA — W A + 1 1^ Et {[pi —q^(3.16) x 1 i g(Xt )] • K(Xt )Xt } where HA (Of) is the sum of initial quantity (share) quota endowments of all fishers. Appendix A gives the proof of this proposition. Remark 1. (3.1) and (3.3) say that the equilibrium effort level in period A, e -m , is constant over time, while the equilibrium effort level in period B, q3i, may depend on the fish stock z( ) size Xt of the year. The condition (3.2), = Et {c ipi }, says that e fti is some Chapter 3. Comparison of Alternative ITQ Systems^ average level of EP i . The assumption (3.7), 30 e tBi > 0 (i = 1, . . . , N; VXt E [b, d]), implies that regardless of the fish stock size X t of the year, all the fishers (including relatively inefficient ones) who are active in period A can continue to harvest profitably in period B. This facilitates the derivation of equilibrium quantities. Without this assumption, some of the less efficient fishers' optimal effort levels Jpi may become zero in period B (i.e., corner solutions) when the true fish stock level X t is revealed and the level of TAC is changed accordingly in period B. This possibility of corner 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 the condition (3.4) Et {g(Xt) (e AN e tB N )1 = Et f eN(esAN ) etv(e BN ) &). means that the expected inframarginal profit of the marginal fisher N is equal to zero. The condition (3.8) Et {g(Xt) • (e A ' ^ f (0 ' )^ci( e't5 '^( i^N ) )}^Et ci says that only those fishers (i = 1, . . , N) who can expect nonnegative inframarginal profits remain in the fishery from year 1 onward. 3. The condition (3.5) ii .1 q xt (e-4i +^= K(x,)-( t says that total harvest equals TAC each year. Chapter 3. Comparison of Alternative ITQ Systems^ 31 t ) > 0 VX t , is the following. 4. The meaning of the assumption (3.6), pt — --t y )1(g(X g(X t ) = cii (e }i3 i) (i — 1, ^V) is the marginal effort cost of fisher i in period B. qXt is the marginal harvest from a unit effort. Then 1 g(Xt ) = fisher i's marginal cost of harvesting one more unit of fish in period B. qXt Since he can sell one unit of fish for p t [$], pt — ---k g(X t ) expresses the maximum amount fisher i is willing to pay in period B for renting a unit quota for the season to harvest one more unit of fish. In a competitive equilibrium, this is equal to the market rental price of a unit quota in period B. Therefore the assumption pt — qt g X t > 0 VX-t guarantees that the rental price is always positive. This implies that fishers harvest up to their quota holdings limits, because if the rental price is positive, there is no point for fishers to leave unused quotas without renting them 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) satisfy _ B1 w A =Et lLut j and W A = Et {1, 17tB } . Since the fisher forms rational expectations on wia and WtB by assumption, the RI-IS of the above equalities coincide with the fisher's expectation on the quota prices in period B. Therefore, how much quota fisher i holds in period A does not affect his expected profit perceived by him in the year. For example, even if fisher i does not buy enough quota in period A, he can always buy more quotas in period B at the price whose expected value is the same as the price in period A. Although each fisher's quota demands in period A (41i and ( 11 ) are indeterminate, the conditions ((3.9) - (3.10) for / = A) total quantity quota demand (E a Ai t )^total quantity quota supply (k(x)x — f(x)) A Chapter 3. Comparison of Alternative ITQ Systems^ 32 total share quota demand (E ";'1i ) = total share quota supply (1) i=i must hold at the quota market equilibrium. 6. As we saw above, the assumption (3.6), pt — —k g(Xt ) > 0, gurantees that fisher i's quota constraint binds at equillibrium, i.e., qXt e 4i - ti^ ti^aB qXteS 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 share quotas is indeterminate in period B. In other words, how fisher i divides his quota holdings in period B into quantity quotas and share quotas does not affect his expected profits. The reason is clear. At equilibrium, quota prices are adjusted so that both quotas are equally profitable for fishers. As in the case of quota demand in period A, however, the conditions ((3.9) - (3.10) for j = B) ^total quantity quota demand^total quantity quota supply (i.e., N Ear K(xi )xt — f(Xt )) ^total share quota demand^total share quota supply(i.e., E^= 1) i=1 have to be satisfied at equilibrium. 7. We can interpret the expressions of the equilibrium quantity quota prices 1^1 fpt —^g(Xt )} 1 —^qXt = Pt as 1 qXt g(Xt) 61-DA and ^ Chapter 3. Comparison of Alternative ITQ Systems ^ 33 the cost of acquiring^expected profit from one more ton of quota^harvesting one more ton of fish in period A^in all subsequent years, and the cost of acquiring^profit from harvesting^discounted sales value of one more ton of quota = one more ton of fish^+ one more ton of quota in period B^in period B^in period A of next year Let us denote the equilibrium share quota prices in quantity unit by W A [Slton] and VMS Iton] instead of W A [$/TAC for share quota] and 1/1/73 [5/TAC for share quota]. Because, by definition, f(z) [ton] = T4C for share quota (z = x in period A and z Xt in period B), we have W A = W A I f(x) and WiB = WiB i,f(Xt)• Thus WB 1 1 9(Xt)] (Xt) } _ 6 Et [Pt qXt^f(x) ^(^ WA 1 = Pt^1^ g(Xt)+ (5VV A 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-0 , the first two terms on the RHS express the monetary value of fisher i's initial quantity and share quota endowments, respectively. The last term expresses the discounted sum of fisher i's inframarginal profit. This interpretation of the last term is derived as follows. From the expression of wB, we obtain g(Xt)= [Pt — (wr — 6'17/1)]qXt Chapter 3. Comparison of Alternative ITQ Systems^ 34 where pt — (wr — 6fO A ) = revenue (net of opportunity cost) from a unit harvest qXt = harvest from a unit effort. Therefore g(Xt ) = revenue (net of opportunity cost) from a unit effort. Hence ^Et {g(Xt ) • (E t i) li^eS ^ci (eAi)^i)^i} expected inframarginal profit for fisher i in year t. 9. In the expression (3.16) 1 G(1-4,14) =^—^^6 Et {[Pt ^ g(Xt)] • n(Xt)Xt} 1 —^qX t the first two terms on the RHS express (the negative of) the total monetary value of the initial quantity and share quota endowments, respectively, that are given gratis to fishers initially. The last term corresponds to the expected present value of the so-called "management rent" from the fishery when the authorities adjust the level of TAC according to TAC = K(z)z (z = x in period A and z = Xt in period B). 10. Although fishers are indifferent about the division of their quota holdings into ^1 the two kinds of quotas from year 1 onward, they are not indifferent about the composition of their initial quota endowments. The expected present value of fisher i's profit is vi(hAi i . 0 Ai^3;vIftoiAif(x) — Et {9(-Xt) • (e Al + ‘r i ) — ci(e ^ — Chapter 3. Comparison of Alternative ITQ Systems^ 35 hAi +OP f (x) [ton] is fisher i's total quota holdings in tonnage unit at the beginning of period A of year 1. Suppose that initially fisher i is given this value and is given an opportunity to choose the ratio of quantity quotas and share quotas. Then the following is clear from the expression of Vi(hP • If 2.0 > W A [$/ton], fisher i prefers initial endowments before year 1 with a larger quantity quota ratio. • If ft) A < W A [$/ton], fisher i prefers initial endowments before year 1 with a larger share quota ratio. The reason is the following. In the case of t -v A > W A , a unit quantity quota is priced higher than a corresponding amount of share quota. Since we assume rational expectations, this must be the reflection of fishers' expectation that having quantity quotas will be more profitable than having share quotas. This profitability difference between the two kinds of quotas is the result of the market operations of quantity quotas. If we remember the argument in. Section 2.5 (Ideas behind results), we can infer that the case of tOA > W A corresponds to the situation in which the authorities buy quantity quotas when the quantity quota price is high sell quantity quotas when the quantity quota price is low. This mechanism serves as an implicit subsidy to fishers who hold quantity quotas instead of share quotas. Zli A > W A is the reflection of this situation. However, only those fishers with initial endowments of quantity quotas who get quantity quotas for nothing can benefit from this "subsidy". For from year 1 onward equilibrium quantity quota prices are determined fully incorporating the benefit of this "subsidy" under the assumption of rational expectations. Hence Chapter 3. Comparison of Alternative ITQ Systems^ 36 fishers cannot expect extra gain from purchasing quantity quotas. The high price exactly offsets the benefit of the "subsidy". This is why from year 1 onward fishers are indifferent about the division of their quota holdings between quantity quotas and share quotas. 3.2 Allocative equivalence of alternative ITQ systems Now we can make a comparison of alternative ITQ systems using the results in Proposition 1. First, let us look at the total expected rent generated by the fishery. The total rent is defined to be the sum of fishers' profits and the authorities' revenue. Under our hybrid ITQ system, this is N vi(hi4i,^) , -A^W A A el + G(H114 ,0N.,, where HA (Op) is the sum of the initial quantity (share) quota endowments to the fishers who do not participate in the fishery from year 1 onward, i.e., N N E h Ai^= i= 1 i Substituting the expressions for ;Y_ gXt (eiti^K(Xt)Xt, ; 6.11. z__ 0 A1 V i (hP, eh and G(/-41, Oil ) in Proposition 1, and using . we can easily verify that total expected rent = ^ Et fpt • s(Xt)Xt — 1— 6 a) =1 This clearly shows that under our overall TAC adjustment rule (TAC = k(z)z, where z = 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 quota Chapter 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, alternative ITQ management systems with a common overall TAC adjustment rule are equivalent in their allocative efficiency. For instance, the genuine QQ system (f(z)^0) and the genuine SQ system (f(z)^k(z)z) have the same allocative efficiency, as long as they have the same overall TAC adjustment rule, TAC = K(z)z. 3.3 Distributive effect Although alternative ITQ systems with a common overall TAC adjustment rule generate the same amount of rent from the fishery, they are different in their effects on the distribution of the fishery rent. We verify this in the following concentrating on the difference between 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 are h Ai Hi and LL QQu' l fi,,),5Q 1 OP ii:(x)x^ where := ^1 6 Et { g ( xt) It ( e Ai eB^ki(eAi) ci (e tB i is the expected present value of fisher i's inframarginal profit, and WQ Q [8/ton] := WSQ [S/ton] := WA 1^1 Et {pt ^ g( -Xt) } 1—6 qX t 1^1 ^ K(Xt)Xt1 Et {[Pt^qA-t g(-N t 1—6 k(x)x Chapter 3. Comparison of Alternative ITQ Systems^ 38 Here, for comparison's sake, we express the share quota price tb sAQ [$/ton] in the same unit as for the quantity quota price ti4 Q [$/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 that VsQ > 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 the other if he has positive initial endowments (i.e., h -j4i = 011i K(x)x > 0). In the rest of this section, we assume that fisher i has positive initial quota endowments unless otherwise stated. If we look at the expressions for zi4, c) and ii4Q , we notice that there are four factors which may affect the relative magnitude of ii)4 Q 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(p i ); • the form of the probability distribution of fish stock X t . Corresponding to whether K(Xi ) is constant, increasing, or decreasing in X i , there are three cases about TAC adjustment rule. ^ Chapter 3. Comparison of Alternative ITQ Systems^ 39 1. Proportional TAC This 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 satisfy the following': ^6"Ai ^cc(e i ) = e^Bi^^i = constant = =^,N(eN) =: rN N/2^ci(e i ) cN(e N )^ rN^(i e^N N = 1,...,N) k 2q Ee -A w QQ 3-,A wt 1^1 , = ^ E t {pt qXt TN} 1—6 1 = Pt — qXt TN + 64 Q under the QQ system ^1 ^1^ Xt ^=""SQ ^ Et {[pt ^ — 1—6 qXt rN, x tt) = pt 1 qXtrN x under the SQ system + 6w SQ — Q Xt on the assumption that Pt Therefore, whether 4 1 — qXt r N > 0 VX t . under the QQ system increases or decreases when X t increases depends on the relative movement of pt and-f q . cr t N. We can interpret ej • [rN ci (e')+ W21 e i 1 (3.3), cc(e 131 ) = • • • = cily (ei9 N)^g(Xt ), shows that q31,^, i.p2v move all in the same direction for different X t values. (3.5) is nowq.X t (eli + epi)^kX t , which becomes E(Oi + epi)^k/q for all X. It follows that ep ^N) cannot change at all for different X t values, i.e.. S t constant (i = 1, ..., N). Then, by the strict convexity of ci(.), eAa = ^= 1,^, N) must hold as a result of cost minimization by fishers. , e = Chapter 3. Comparison of Alternative ITQ Systems^ [pt — (wp — AzI, A as the inframarginal profit of fisher^ in each period, where T N =^ 40 q Xt is the revenue (net of opportunity cost) from a unit effort (cf. p. 33). This inframarginal profit is zero for the marginal fisher N. See Figure 3.4, p. 41. 2. More responsive TAC Suppose K(z) = TAC/z is increasing in z. TAC in this case is more responsive to Xt than 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 period B under the genuine QQ system is XtT^K(xt)^(i = 1, • • • , the decrease of qA. N)^g(Xt) g(Xt ) is smaller wP is lower than otherwise, where qXt g(Xt ) is the marginal cost of harvesting one more unit of fish. This means that under 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 than the effect of TAC = k(Xt )Xt T , the authorities' expected surplus (deficit) under the QQ system through quota market operations will be smaller (larger). 3. Less responsive TAC This is the case in which K(z) = TAC/z is decreasing in z. The relationship between Xt and zup is then k(Xt)^e B z (=1 , N) =,' g(X t ) ,^ the decrease of -qxt 1-g(Xt ) is larger wP is higher than otherwise. Chapter 3. Comparison of Alternative ITQ Systems^ Figure 3.4: Inframarginal profit under proportional TAC Ci(e i ) -1- ti/2 ei rN r• 0 Ci (ei) Cii(eN) r1 41 Chapter 3. Comparison of Alternative ITQ Systems ^ 42 Therefore, under the QQ system with a less responsive TAC adjustment rule, the authorities sell (buy) quotas at higher (lower) prices. 2 If the effect of wP is stronger than the effect of K(X t ) J., the authorities' expected surplus (deficit) under the QQ system will be larger (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 determined by two factors (w iB and ( X t)) whose effects are opposite to each other. Which effect dominates the other depends on the functional forms of (1) TAC rule (i(z)z), (2) cost functions (ci(ci)), (3) demand function (D(p t )), and (4) the probability distribution of 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) after specifying the forms of these four functions. However, there are too many combinations of possible functional forms to derive general results about the effects. Therefore, here we only point out that it is possible to calculate the effects case by case. Example: Proportional TAC and demand with constant price elasticity In the rest of this section, we concentrate on the case of proportional TAC rule which reduces the possible combinations to a manageable range. As we saw above, under a proportional TAC adjustment rule TAC = ti:(z)z kz (k : constant), we have the following: Vi ^ SQ^QQ ^> t) s Q < WQQ p — —r N p — qx^qx 'Andersen [1] (p.481) points out this possibility. Chapter 3. Comparison of Alternative //Y2 Systems^ 43 where x = Et {X t } 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 our analysis to the case in which the fish demand function takes the simple form pt = 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. For simplicity, we take the unit of fish stock appropriately so that the average stock level x = Then Xt fluctuates around 1. Equilibrium fish price in year t is given by the condition D t = S t , where St is the supply of the fish in year t. Since St = kXt by (3.5) in Proposition 1, Dt = kXt . This leads to p t = uX t n at equilibrium, where u := vk fi = constant. - 1. p = 0 (p t = u = constant) case: This is the case where elasticity = oc. It is easy to see that > 1/:(i2(2 44 Chapter 3. Comparison of Alternative ITQ Systems^ in this case (because /3 = /5 = u and -y > 1, so that fv:5Q 4 > WQ Q ). Namely, if the price of fish is constant and the fish stock fluctuates year by year, fishers with positive quota endowments are better off under the SQ system regardless of the probability distribution of Xt . Under the QQ system, however, the difference V:4 Q — 176 Q accrue to the authorities through their quota market operations. The reason for this is transparent. When X t turns out to be large under the QQ system, quota price Wt= Pt^rN , qXt — QQ becomes high and the authorities sell quotas according to their proportional TAC adjustment 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 = E t I /3' X'^(x 1) x ij = Et{pil = wy. Therefore tq c2 -yi14Q , so that - UVQ < V r' C? if there is uncertainty about the fish stock level year by year. In short, when the price elasticity 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 the average 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 in our context. —^ Chapter 3. Comparison of Alternative ITQ Systems ^ 45 3. p^1 (13t = Dt P ) case: Here we further assume that the fish stock X t follows the uniform distribution over the range [1 - a, 1 + a], where a (0 < a < 1) expresses the degree of fish stock fluctuation. Then p-^Et 1 fl+a^1^1^r u Xt-P dXt = [(1 a) l- P - (1 2a 2a 1 - p {pt} 1ra^1 1 uX 1- dV t = = Et 13' )(t1 xI = Et 2a 1—a^t 2a 2 - p [ 1 fl+a^1^+ a {k} 2a A-a Xt dXj^2ain1- (1 + a) 2- P —a) 1 — ( 1 u, 1 — a) 2 1 u, a Hence (.1 - 1) -1 rN P VsQ< VQ^ Q M > R( a, p), where 1 ^1 R(a, p) 1(1 +^+ a) 1- P - (1 - a) 1- P^(1 o) 2-- P - (1 - a) 2- P lay -^[ -11 1 - p^2 - p 1 -T N . M q M is the marginal cost of harvesting one more unit of fish when Xt = 1. n is the equilibrium price of fish when X t = 1. Hence M — = the marginal cost/price ratio at the average stock level. u Since our assumption 1 ^ r,v > 0 VXt q-Xt implies marginal cost < price for all Xt , we have 111/tt < 1 under this assumption. Now it 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 = 1.7c2i Q . • 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,i Q < 1/'", Q for a > 0. From these and numerical calculations, we can draw diagrams like the following for particular values of M/u < 1. (See Figure 3.5.) The dividing curve in the figure represents the 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 fixed proportion of the fish stock level, • fishers with positive quota endowments are better off under the QQ system (at the expense of the authorities) if the demand for fish is inelastic; • fishers with positive quota endowments are better off under the SQ system if the demand for fish is sufficiently elastic; • fishers with no initial quota endowments are indifferent to which of the two ITQ systems is chosen. These are valid for the case in which the demand for fish has a constant price elasticity and the fish stock level follows the uniform distribution. The case with demand functions and probability distributions different from these require more complicated calculations. It will not be worthwhile to carry out these calculations at this level of abstraction. We can expect, however, that the basic qualitative nature of our result will be true for those more general cases. Finally, we note that if the TAC adjustment rule is non-proportional and the deviation from proportional TAC rules is large, the above result may have to be modified Chapter 3. Comparison of Alternative ITQ Systems^ Figure 3.5: Fishers' expected profits under the two ITQ systems inelastic P" demand 11-- Q 1 e < vm M u elastic demand --- vSQ vi i >VQQ 47 Chapter 3. Comparison of Alternative ITQ Systems^ 48 considerably. However, it will involve complicated calculations to determine the necessary modifications. In order to carry out such calculations, we need detailed information about the functional forms of (1) TAC adjustment rule, (2) cost functions, (3) demand function, and (4) the probability distribution of Xt . It will be very unlikely to obtain simple relationships between the nature of non-proportionality and that of the necessary modifications, because all the above functions interact with each other in determining the effect of non-proportionality of TAC adjustment rule. Therefore, we do not pursue this line of investigation any further in this thesis. 3.4 Adjustment phase We have assumed in the above analysis that the equilibrium quantities are attained with no adjustment period. For example, the number of fishers are adjusted rapidly because of free entry/exit assumption. Effort levels and quota demands are optimally chosen by fishers from the beginning based on equilibrium quota prices. Quota prices attain stationarity from the outset. Fishers quickly form rational expectations on quota prices realized under a new ITQ management system in which they may have no prior experience. There are many difficulties about these assumptions. Suppose that the ITQ system 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 different from prior harvest history. Then fishers' past experiences may not give them much guidance in figuring out the resulting probability distribution of the fish stock level after year 1. 2. Fishers may not have any experience in quota trading. As a result, it is implausible for them to predict future quota prices (or price functions) correctly, at least at the outset. Chapter 3. Comparison of Alternative ITQ Systems^ 49 In short, we have assumed away all the complexities accompanying the initial adjustment process. In reality, however, the initial adjustment period can be rather long. Some may even argue that most of important real economic issues are related to short or medium term adjustment problems. Nevertheless, this does not necessarily mean that the results obtained above have no relevance to real issues. On the contrary, the results presented in the above propositions provide us with useful information about the comparison of one equilibrium with another, as we shall see below. Let us consider the situation in which fishers are better off under the QQ system than under the SQ system (at the expense of the authorities) if the adjustment to the new ITQ system is instantaneous as assumed in the above propositions. This is the case of VQ Q > V. Then as we saw in Section 3.3, > q c, -;=:> wQ Q > WQ . Although 4, and 'ti4Q may not be attained immediately if the adjustment to the cor- responding ITQ system is not instantaneous, eventually those equilibria will be reached after the initial adjustment period is over. Now suppose that the authorities introduce the 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 have been bought by them at some prior time in the market. Even if we admit the difference between the initial adjustment phase and the subsequent equilibrium state, the prices at which fishers bought those quotas must have, to some extent, reflected the quota price level irti4 Q in the subsequent equilibrium. Then it will be reasonable to assume that those prices are more or less higher than fiygcd which is realized in equilibrium under the SQ system. After the authorities' announcement of the change from the QQ system to the SQ system, quota price will start declining toward this new equilibrium level ti4Q . This will result in the loss of fishers, because the authorities' "subsidy' (through quota market Chapter 3. Comparison of Alternative ITQ Systems^ 50 operations) which fishers "counted on" when they bought quotas at relatively high prices under the original QQ system will disappear under the new SQ system. Fishers cannot fully avoid this loss by selling their quotas upon the authorities' announcement, since the quota price fall will start as soon as the system change is announced if there are fishers who are aware of the effect of the "subsidy cut." This scenario seems to fit the case experienced in New Zealand. Originally introduced as a QQ system in 1986, their ITQ system was switched to an SQ system in 1989 in spite of the opposition of the fishing industry. Deficit in quota market operations account was one of the reasons the quota authorities in New Zealand decided to relinquish the original QQ system. The argument in this section is unavoidably qualitative and less precise compared with the clear-cut results stated in the previous propositions. However, it is almost impossible to establish definite quantitative results about what will happen during a transitional phase from one equilibrium state to another. There are many uncertain factors that will affect the course of adjustment (e.g., the response of fishers' expectations to the system change). We have to content ourselves with the above somewhat vague analysis based on our clear results for equilibrium states. Chapter 4 Tax Effects In the previous chapter, we compared alternative ITQ management systems and showed that 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 the fishery only through TAC adjustments. In reality, however, various taxes are proposed to capture the rent in the fishery. The distribution of the rent between fishers and the authorities will surely be affected by these taxes. It is also natural to expect that these taxes will affect the rent distribution among fishers. We will examine tax effects on the rent distribution among fishers with different cost functions using our basic model for the QQ system. The same analysis applies to any hybrid system. In this chapter, we continue to 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's quota holdings.' • A harvest tax T h [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 is easily extended to a quota tax which is levied on the quota holdings of each period. 51 ^[(pt Chapter 4. Tax Effects^ 52 Fisher i's after tax net profit in year t,^under each tax system is as follows: - 1 = (1 - To E 3 .A,B [mate, ei(e)]^ Eptateiii E - ci(eati)] + - j=A,B - 4 . (h/tli — db ) + wr • (a't4i — a jt3i )—i . (hAi^ •^ — (1 + Tg )a jt3i ] — ei, t^tl^ wt A t i ) tvi•B . (a A, - atBi) roateiti - ci(ejt i)] + wAt • (hAi ^a j=A,B 4.1 Equilibrium under taxation In 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 for Proposition 1. For brevity, we define H i and G by H i := G := 1 1 _ 6 {2rN et^2ca) —^— 1 Et {[p t^rN]kXt} 1^ — 6^q-Xt 1 = 1- b [P qx rivikx, where /3 = Et {pt Xt /x} and r N = c'# i ) (i = 1,..., N). fl i is the expected present value of fisher i's inframarginal profit (cf. Figure 3.4, page 41). G corresponds to the expected present value of the so-called "management rent" (cf. page 34). No tax For comparison's sake, we first state the equilibrium quantities under the QQ system with a fixed proportion TAC rule and no tax. Proposition 1, when applied to this case, gives: 1 1 ^1^ T} Et {pt 1 S P— — 1— atN qx rN — Wt^Pt qxt^ rN 614Q, I/4 (hi4i ) G QQ i4) , 7 ,A — 7,^Fri QQ 1 " - A , A u'QQ, ^ Chapter 4. Tax Effects^ 53 where r N :=^= • • • = CAI E N and y = Ei lx/Xt l. (See Proportional TAC case, page 39.) Profit tax Proposition 2 Suppose that the authorities introduce the QQ system with a profit tax with tax rate Tv into the fishery from. year 1. Then at equilib•im, the number of fishers N, 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 as in the no tax case, i.e., 1^- 7 wt^ A^1^ (p — ^ Et {pi 1^ rN} 1— 1 — qXt qx ) wt = 1 qAt N „„-A lu CKP 64 Q . The expected present value of fisher i's after-tax net profit with initial quota endowments hiiti is 17cQ(h1.1-i) = (1 —^+ H i ]^(i = 1,^, N). The authorities' expected revenue from the profit tax and their quota market operations is GQQ^= —(1 — T7,)^G^ E where 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 that the expected present values of fishers' future profits are reduced according to the profit tax rate rp . In this case it is theoretically possible to drive the expected present value of after-tax net profit down to zero by setting rp = 1 (i.e., 10(1 Vc profit tax). Then the Chapter 4. Tax Effects^ 54 authorities appropriate the whole fishery rent G E1=i 1F including the inframarginal profits E .1\1:= 1 fi j . Quota tax We 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 number of fishers N, effort levels es i (i 1,... , N), and quota demands edi A, B) are the same as in the no tax case. The equilibrium quota prices under the quota tax are ^1^1^1 WA t = 1 _ 6 + Et pt^rN} Tq w B = ^ p -qXt^1 — 6 + 7.9, (qx { „„--A u'QQ' pt 1 + Tq^qXt TN -4- 64Q ) • ( The expected present value of fisher i's after-tax net profit is -1/ Q (h A i i) = WQQh1 + ^(i = 1, ... N). The authorities' expected revenue is GQ(2(H 1 ) =^+ G. . Remark: The above equilibrium quota prices dIA and w tB for the quota tax case are lower than QQ those in the no tax and the profit tax cases by the factors determined by the quota tax rate Tq . Notice also that /74 Q = Et {4}. Here, unlike in the profit tax case, the authorities cannot appropriate the inframarginal 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 the market value of quotas and the authorities' tax revenue. The maximum amount of money Chapter 4. Tax Effects^ 55 the authorities can expect to raise is the management rent G =^Et {[p t — qXt Xt } either by setting the total initial quota endowments Hill = 0 or by setting a very high quota tax rate Tq Harvest tax The following proposition holds for the fishery under the QQ system with the harvest tax Th [$/ton]. Proposition 4 At equilibrium under the QQ system with a harvest tax Th , the number of fishers N, effort levels ei (i = 1, , N), and quota demands etIt i (j = A, B) are the same as in no tax case. The equilibrium quota prices under the harvest tax are 1 Wt Wt = 1 6 Pt — Et{pt 7h ^ rN} = qA t 1 1—6 — Th — r N) : qx WQQ, 1 t Th qXrN assuming pt — Th - -±rN > 0 V.,K i [b,d]. The expected present value of fisher i's after-tax net profit is VQQ(hi ti = z4 Q h 1li^(i = 1,^N). ) The authorities' expected revenue is GQQ(Hi4 ) = - 4QH1 + G. Remark: As in the quota tax case, the authorities cannot appropriate the inframarginal profit 11 i (i = 1, ... , N) as long as the harvest tax rate Th is low and the condition ± q N > 0 VXt E p t — Th - [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 the resulting equilibrium based on this fewer number of fishers. We do not go into this possible complication any further. 56 Chapter 4. Tax Effects^ 4.2 Comparison of tax effects It is clear from the above propositions that the three types of taxes do not affect the allocative efficiency of the ITQ management system.' This is because the number of fishers and effort levels at equilibrium do not change under these taxes. However, they have different implications on the distribution of the fishery rent among fishers and the authorities. The issue of rent distribution between the authorities and fishers was discussed in the previous chapter. In the following we will concentrate on the problem of rent distribution among heterogeneous fishers. Namely, we compare the three types of 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 for fishers' expected net profit 1722Q (h -;1 ') (i =1,... , N) and the authorities' expected revenue GQQ(W) as in no tax case: A Q hA i Z -I- H i li,rcN i (hP) = fv- Q ^ ,...,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 + — QQ Therefore if rg, and Th 1 2 - Tq — for quota tax gx rN 1 (_ ^) for harvest tax 1— 6^—^-qxrN^ are so set that 6+T9 (p qx rN) 1 16 1sr.N) (p -Th 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^ 57 holds, 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 quota allocation and give the same amount of expected revenue to the authorities are equivalent not only in allocative but also in distributive effects. Whatever can be said about the comparison of the profit tax and the quota tax, therefore, is also true for the comparison of the profit tax and the harvest tax.' We consider the QQ system with the profit tax and the QQ system with the quota tax. We assume that the initial quota allocation is identical for these two systems. Let us denote the values of the variables under the profit tax by a subscript p and those under the quota tax by a subscript q. Then from the previous propositions L ^(__^-y ^ili r N hl lip' (hili ) = (1 — Tp) {^ 6 p qx 1^-y Vqi (h P ) = ^1 jj - -- N) h i i + Il l 1 - 6+rq( qxr - for i = 1, . . . , N and Gp (HiA ) = = — ( - Tp 1 —rN H l pE + G+T 3=1 —^6 ( q 1 1 — + Tq ( I) — 1-rN )^+G. qx As mentioned above, we consider the situation where 7-7, and 7, are so chosen that Gp (1-P1,4- ) = G q (Hi4 ). This condition is transformed into T^ Hi - Tp^I ^ETA _ 6^1 — 6 + rg )^--qxrN) 111 (4.17) 'This may depend on our assumption p t — -4 > 0 VAT t which guarantees that fishers always , 7.N ;g harvest up to their quota limits. Chapter 4. Tax Effects^ 58 If^= G q (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 — °, , /5 — -2-r N hP — Tp ll i .^(4.18) -t"rq^qx JJJ ) Let rii = fisher i's share in the initial quota endowments 10 hiP 1 Then by multiplying both sides of (4.17) by 7/„ we get N ^1 rprli^113 = j= 1 (1 - rp 1 6^— 6 + rq) U1 P 1 Substituting this into the RHS of (4.18) yields N Vp^1 i (h Ai ) - Vgi (h iP) = • )^ (4.19) Hence . >^.^.^>^Hi VP i (h A i)^V"(hi l l) ^ qi^Cri 1^q 1^ ElY fli 3 =1 where o- j = fisher i's share in the total inframarginal profits. Thus Vpi (h A1 )^Vqi (h Ai i ) -<==>- fisher i's share qi in fisher i's share ai in > quota endowments < inframarginal profits Therefore 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. (4.20) 59 Chapter 4. Tax Effects^ If we specify the initial quota endowments, we can get more information about who benefits from which tax. Let us consider the following three cases with different initial quota allocations. For simplicity, we assume that the fishers who get initial quota endowments coincide 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. For fisher i who earns more than average inframarginal profit, 1/N < of holds. Hence for such a fisher lii < Qi ,^ i.e., Vpi (h P ) < Vqi (h Ai i . ) Thus under equal initial quota allocation, those fishers who earn above average inframarginal profits are better off under the quota tax. The opposite is true for the fishers whose 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). Then Vpi (q i ) = ii:(hiP) for all i = 1, . . . , N. Hence under this "proportional" quota allocation, the profit tax and the quota tax are equivalent 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 more than proportional to their inframarginal profit shares cri . (Here profitability is measured by the share cr i in the total inframarginal profits). In this case, the initial quota allocation Chapter 4. Tax Effects^ 60 exaggerates the difference in profitability among fishers. Then the more profitable fisher i is, the more likely it is that he is better off under the profit tax than under the quota tax. The following examples illustrate the above three cases. Suppose that after year 1 four fishers have inframarginal profit shares a1 0.4, a 2 = 0.3, cr3 = 0.2, and a 4 = 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 than 1 (i.e., rii < o-i ) is better off under the quota tax. In Case 3, for instance, fisher 1 with the largest inframarginal profit share of = 0.4 is better off under the profit tax, because the 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: "A fisher who earns a higher average profit per unit of quota owned will pay proportionately less rent to a regulator with a quota tax than with an equivalent profit tax ...." To compare our results with his, rii should be understood as fisher i's share in the initial quota holdings at equilibrium. Then under the situation of no stock fluctuations in [16], o-i fisher is inframarginal profits total quota holdings rii 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 4 need not be a marginal fisher. 61 Chapter 4. Tax Effects^ Figure 4.6: Lorentz curves for profit share^and quota share EaJr, (7/) Casel 1.0 crj hi (1) (2) (3) (4) 0.6 0.3 0.25 0.25 0.25 0.25 < < > > 0.4 0.3 0,2 0.1 0.5^0.75 0.25 Case2 1 ,0 ai 77- (1) (2) 0.6 (3) (4) 0.3 0.4 0,3 0.2 0.1 = = = 0.4 0.3 0.2 0.1 0.6 0.3 0.1 0.0 > = < < 0.4 0.3 0.2 0.1 0.1 0.0^0.1^0.3 Ea) 1 .0 (1) 0.6 (2) (3) (4) 0.3 0.1 0.0^0.1 0.4^1,0 Chapter 4. Tax Effects^ 62 We can say that in [16] fisher i's profitability is measured by a i /r/ i not by The main lesson of the above analysis is that we cannot compare the effects of the profit tax and the quota tax on fishers without taking into account the distribution of initial quota endowments. In most cases of actual ITQ management systems, initial quotas are allocated to fishers based on their past harvest records. Although there seems 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 fishers before initial quota allocation. If this is true, Case 1 above (equal initial allocation) is not so plausible. Furthermore, if the past harvest record exaggerates the difference in profitability among fishers, it is possible that the profit tax is more advantageous for profitable fishers as illustrated above. Of course, it is not easy to determine which is the case for a particular fishery. More detailed study will be necessary to draw conclusions about tax effects. 4.3 Adjustment phase The results obtained above were based on the assumption that the fishery reaches a new equilibrium immediately after the introduction of an ITQ management system with taxation. We assumed away the existence of an adjustment period before the new equilibrium. This assumption is doubtless unrealistic. However, the results based on the unrealistic assumption can give us insight when we compare one equilibrium state with another under different taxes. For example, consider the following situation. In year t the fishery is in equilibrium under an ITQ system with a quota tax, when the authorities announce a change from the quota tax to a profit tax. We assume that the profit tax is so designed that the authorities' expected revenue is the same as before under the new 5 As stated above, we measure fisher is profitability by his inframarginal profit share ui. Chapter 4. Tax Effects^ 63 profit 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). The initial quota holdings in year T, hTi (i = 1, . . . , N) will be the same as in the case with no tax change. (Remember that the equilibrium effort levels are the same under different tax schemes.) Let us denote the expected present value of fisher i's profit under the new profit tax by 17; : (4i). We denote by 17:(itl) the expected present value of fisher i's profit at the beginning of year T for the case with no tax change in year t. Then, as we saw above, 17;1, 01^Vq ^ (1 •=:>. 71i < where 7/i = fisher i's share in the total quota holdings at the beginning of year T o-i = fisher i's share in the total inframarginal profits For 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-tax net profit will also change smoothly from s + w sB^— (1 + Tq )a B E [p sq xs eJs i — q (eis i)} +^( ios i _^ s il . . j=A,B for s < t to ir's = ( 1 — 7-p)^E [p sq xs is j=A,B — ci 6, sj i)] + wA s. (iltsli — iiiis i) + wB s . (O si— an 1 for s > T. Then, for fisher i with 7/ > o- i , the average of ;k si will start increasing in year t after the change from the quota tax to the profit tax. In year T it will reach a new equilibrium level higher than before. Thus the tax change is beneficial for fisher i with q > o-i• On the other hand, the tax change will cause a loss to fisher j with 7i j < o-j. Chapter 4. Tax Effects^ 64 Now 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 with relatively large share in total inframarginal profits has quota holdings more than proportional to his inframarginal profits (i.e., qi > ai). For such a fisher, > v:( 14 i). Thus the fishers who harvest relatively large amount of fish and possess large amount of quotas will benefit from the change from the quota tax to the profit tax under this circumstance. 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 equilibrium exaggerates the difference in profitability among fishers. Hence the tax change will not be beneficial for relatively small scale fishers in such a fishery. In other words, fishers with relatively large shares in total quota holdings (= total harvests) will benefit from the 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 industry than small scale fishers, it is understandable that those large scale fishers put pressure on the government as the representative of the industry to change the tax system in their favour. The point is that there can be a conflict of interest among fishers concerning the choice 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, profitability is measured as the fisher's share in total inframarginal profits. Chapter 5 The Case of Risk-averse Fishers In Chapter 3 we compared the SQ system and the QQ system on the assumption that fishers are all risk-neutral. The conclusion was that if we take into account the possibility of fish price changes caused by fish stock fluctuations, the two ITQ systems can have different 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 in New Zealand opposed the transition to the SQ system from the QQ system. The analysis in Chapter 3 provides one possible explanation. Namely, if the price elasticity of the fish demand is not large, risk-neutral fishers can expect more profits under the QQ system at the cost of the balance in the authorities' quota operations account. Another possible explanation is that fishers are not risk-neutral and they prefer income stability under the QQ system. It sounds plausible that the QQ system contributes to stabilizing the income fluctuations of fishers, because under this system the authorities buy back quotas in bad harvest years and sell quotas in good harvest years, whereby those operations seem to damp fishers' income fluctuations. In this chapter we examine the validity of this claim. We use the same basic model as before for the genuine QQ system 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. There are no other risk-neutral traders in the market. 65 Chapter 5. The Case of Risk-averse Fishers^ 66 2. There are risk-neutral traders in the quota market and they can freely buy/sell/lease quotas in the market. The existence of risk-neutral traders in the quota market may change the resulting market equilibrium considerably. The answer to the above question can critically depend on this point, as we shall see below. 5.1 No risk-neutral traders case 5.1.1 New assumptions and notation For simplicity, we add the following assumptions: 1. Proportional TAC rule: TAC = kz (k : constant), where z = x in period A and z = 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 in the fishery, the same cost function c(c.[I ) c(eP) and the same utility function 47_0 defined over their annual net profit 7r t . u(.) is a concave function reflecting the risk-aversion of fishers. 4. There is no credit market and insurance market available to fishers (except for costless in-season credit). Hence fishers do not have means to average out yearto-year income fluctuations over time except for selling and buying quotas in the market. 5. Fishers' net profits of each year are not carried over to next year. When fishers buy quotas and other goods at the beginning of a year, the payments are made when 67 Chapter 5. The Case of Risk-averse Fishers^ they 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 is 7rt^pgxtetA _ c (0) wtit• (h iA atA) pqxtetB _ c ( eiB) w13 (hr atB) 4 , where p = the price of landed fish^(unit: $/ton) q^catchability coefficient^(unit: /m 3 ) Xt = the level of the fish stock^(unit: ton) the fisher's effort level in period j^(unit: m 3 /period) wt ^quota price in period j ^ (unit: $/ton) frti^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: $). Under the QQ system it is natural to express quota prices and quota holdings in the above units. Since it is always possible to translate the amount of share quotas from percentage unit to tonnage unit based on the current TAC level, we express quota prices and quota holdings under the SQ system also in the same units as above. For example, if the TAC is 1,000 ton and the price of a 1 percent share quota is 100 dollars, the quota price is equivalent to 10 [$/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 convenience to 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 ' ,V A (i7',B) [8/TAC for share quota] in Proposition 1 (page 28). We make this choice for expositional convenience. Chapter 5. The Case of Risk-averse Fishers ^ 68 Under the QQ system, fishers' quota holdings change only when they buy or sell quota in the quota market. Therefore, after-trading quota holdings in each period are carried over to the next period: (5.21) a t-i A andta A^—htB B = h— — Under the SQ system fishers' quota holdings in percentage unit change only when they buy or sell quotas in the market. After-trading quota holdings in percentage unit are carried over to the next period under the SQ system. In our model the authorities set TAC equal to kXt-1 before quota trading in period B of year t — 1. The next TAC change occurs when the authorities set TAC equal to the provisional level kx before period A of year t. At the end of period A of year t they change TAC again from kx to kXt . Therefore, if we express each fisher's quota holdings in percentage unit, we have the following relations: 4^ a t ^ht t4 1 and^ - - - ^h t 3 kx^kx kXt Hence X B^ r t A a t = h tB^(5.22) at-i =^and — x where aB 1 , hA 4, and h are defined as before in tonnage unit. , Formally, (5.21) and (5.22) are the only difference between the SQ sytem and the QQ system in our model and play an important role in our comparison of these two systems. 5.1.2 Rational expectations equilibrium Given the initial quota endowments and based on his expectations about future quota ^ loo prices {wA . . the th^tti^fih^h^ti^l^ representative fisher chooses a contingency plan f{ e t4,"t^ aB t^wBr t t=P t t=1 3 These conditions are the same as (i132 1 = O A and^= op on page 18. Chapter 5. The Case of Risk-averse Fishers ^ 69 for his effort level and quota holdings of each period so as to maximize his expected discounted sum of utilty E {E 6t-1u(71-t)} subject to his quota constraint qXt e l qXt eP < aP for t = I, 2, 3, . . . As before, a rational expectations equilibrium is defined by stochastic processes etA , atA ,^atB}too_i til tB 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°_ 1 clears the quota market in each period, i.e., I N4 < kx, = if mil > 0 for period A, NaP < k Xt , = if w iB > 0 for period B. • When fishers face the stochastic process {w4, w iB } t°L 1 as price-takers, the stochastic process {0, a tA, e tB , an t'_ 1 maximizes the expected present value of the representative fisher's utility, E{Ei"_1 6t-1u(7rt)}. 5.1.3 Fishers' behaviour at equilibrium If fishers are identical, we can easily determine the effort levels and the quota holdings of each fisher at equilibrium in our model. Let us denote these values by (eAt t = 1, 2, 3, .... Suppose that at equilibrium, quota prices satisfy 4> 0 and that the quota constraint binds, i.e., VW qX t eB = 41 . Then we have ( ^A A e et^t a The reason is as follows. ar)^(2 k ^kx k 1:7Xi AT ^ N )' q atA e B , ap) , wr > 0 and Chapter 5. The Case of Risk-averse Fishers ^ 70 1. From the assumption w tA > 0 and te > 0, the quota market equilibrium conditions are Na tA^kx and NaP = kXt , because by assumption there are no traders other than N fishers in the market. Then these conditions yield ^kx/N and aB = kXt /N. 2. Since qX t ei4 qXt eP = ap by assumption and ait3 kxt/N by the above, we have aipt3 kXt N• Namely, at equilibrium, each fisher harvests kXt /N. Since the equilibrium effort levels should be cost minimizing, we know that t ) e" At and eB minimize c(e tA ) c(e S kXt subject to qXt e:/t 1 qXt e. B = t^N Then e"-t`4- = eB = k/2Nq follows from the strict convexity of We define e by := — e^B t k 2Nq • The condition p — 1 c i (e) > 0^VXt E [b, cr guarantees that our assumptions 0 tB^ w tA ^,>W > 0, and ate A t TXt 6 B = aB are all satisfied at equilibrium. c(.). Chapter 5. The Case of Risk-averse Fishers^ 71 5.1.4 Equilibrium quota prices We can use the above equilibrium quantities (0, 'etit, ep,ap) = (e,kx/N,6,kXt/N) to derive the conditions that quota prices have to satisfy at equilibrium.` Let us denote the net profit at equilibrium by ir t . Then p Xt lrt =^2c(e) + w;-A. otA — wtB (h tB a .:3 ) _ Suppose that under the SQ system the authorities endow each fisher with 100/N percent of share quotas before year 1. The authorities set TAC equal to (kx, kX 1 ) in year 1, (kx, kX 2 ) in year 2, and so on. Hence under this SQ system the representative fisher's before-trading quota holdings at the beginning of each period are (hp, 143 ) = (kx /N, kX t /N) = 43) for all t = 1, 2, 3, ... Therefore at equilibrium the net profit t is p Xt ar t = ^ 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 quota endowments a,139 = hi = kX0/N before year 1. Xo is set by the authorities and need not be the stock level in year 0. Under the QQ system the representative fisher's beforetrading quota holdings at the beginning of each period are (hI hp = (43 1 , at) for all ) t = 1, 2, 3, ... Then, at equilibrium, hf3 ) = (ht ht) ) _ (kxt_i kx (a? a ,t ,^ • ^) N^N ' t = 1. 2. 3 Hence the net profit is given by t pkXt^kx^B kx kXt N ) = ^ 2c(e)^("A't,-1^N )+wt •( 'This method is similar to the one in Lucas [21]. In a sense, our model is a special case of his Asset Pricing Model. See also Blanchard and Fischer [8] pp.510-512 and Stokey et al. [30] pp.300-304. ^ 72 Chapter 5. The Case of Risk-averse Fishers^ Notice 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 to satisfy 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. Suppose also that p — qXt c'(e) > 0 VX t E [b, d] is satisfied. Then, at equilibrium, quota prices zui4 and w:r (t = 1, 2, 3, ...) satisfy 1 Et { u v. t) .2c1 [p al t el ( e ) 1 A^ Wt 1 B W., ] =: WA 6^Et {uVrt)} 1 Et+i-tu'(Irt-1-1)} _A X c'(e) + 6 W — = p— qXt^Weirt)^Xt (5.23) (5.24) where 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 to for some constant CvA for all t = 1, 2, 3, ... Then if the functional forms of u(.), c(•), and the 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 proposition. 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. Suppose also that p — q X t c'(e) > 0 VXt E [b, d] is satisfied. Then, at equilibrium, quota prices wit l and wfi (t = 1, 2, 3, .. .) satisfy Et {4% r t)[P^601 A Wt^ Et {u (,, t )} =_- 1^, ^ C (6) + q Xt 6 +1_ 6 6 E-E"-i{z/(;:+flu)/(P it)}qxt Et+1 {ul0-1-1-1)} +leV)]} (5.25) w A^(5.26) ^ 73 Chapter 5. The Case of Risk-averse Fishers^ where Et is the expectation operator with respect to X t and ^Pk—Vt^kx^B kx kXt 2C(e)^ ^ NN N ) w t • (^ ) Since in contains a Xt _ 1 term, we know from (5.25) and (5.26) that wi A- depends on Xt _ 1 and that wB depends on X t _ 1 and Xt . Therefore the above equations are actually two simultaneous functional equations for two unknown functions 4I (Xt _ 1 ) and w tB(Xt _ i , Xt ). In general, it is hard to get solution functions for these equations. Only by 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: ^ c,(01} 1 Et f u , cfr, t) kXt Et {u'(.1i-t) iv 4 ikoxo }^(5.27) 1 —^100^at X t kXt .^1^ kx ti ' 043 k = 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-c' Tq (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 in period A have to balance with each other at equilibrium. The condition (5.28) says that the cost and benefit of possessing one more percent of share quotas in period B have to balance 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^ 74 5.1.5 Example: piecewise linear utility function Propositions 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 simultaneous functional equations with two unknown functions w,A (Xt _ i ) and wBt (Xt _ i ,Xt ). To compare the relative advantage of the two systems, we have to calculate the solution functions explicitly. However, if fishers' utility functions are not linear, it is, in general, very difficult to solve these functional equations. To proceed further, we need some simplifications to the model. We make the following additional assumptions. 1. There are only two possible fish stock levels with equal probability: Xt = (1 + a)x with probability 1/2 (good year) (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 (see Figure 5.7, p.75): u(7 t ) = { (1 — )) 7rt + Ov, if 7rt > V (1 + 3)ir t — /3v, if ar t < v, where = the degree of risk-aversion, 0 <13 < 1, and' pkx N^2c(e) the average annual income under the SQ system at equilibrium: Ecrir t l. (cf. page 71) Since fishers' income will fluctuate around this v, we can expect that this fuction 5 1n the following, we omit the seasonal setup costs for brevity. This omission does not change the results derived below. Chapter 5. The Case of Risk-averse Fishers ^ Figure 5.7: Piecewise linear utility function — V 75 Chapter 5. The Case of Risk-averse Fishers ^ 76 will capture the effect of their risk-aversion. In the following, right-superscripts, + or - indicate the harvest of this year and left-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 and year t - 1 is a bad year. = quota price in period B of year t when year t is a bad year and year t - 1 is a good year. rri = net income in year 1 when year 1 is a good year. 5.1.6 Equilibrium quota prices We know that the net profit at equilibrium in year t is pkXt = ^ 2c(e) N for the SQ system and . k t-i kx^ B kx kXt , pkXt^X^ ^ 2c(e) +^ zo t. ( N N^ ) N for the QQ system. When the stock level is (1- a)x and the effort level is e, the marginal cost of harvesting one 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. Then 1 -^1 ^c'(e) 1 + a II = q( 1 + a)x ^ ^ Chapter 5. The Case of Risk-averse Fishers ^ 77 is the marginal harvesting cost at equilibrium in good years. Now, by using the general results in the previous section, we can derive the equilibrim quota prices for the SQ and QQ systems. Proposition 7 Suppose that under the SQ system the authorities endow each fisher with 100/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 by A 1 [(1 — a3)p — (1 — a)p] = 20 (constant)^(5.30) 1—6 1 60 1 B+^1 - a^ ^ (5.31) wt = p ^ 1 + a 11+ 1 — 13^1 + a^ wt wiB- =P—P+ 1 6 ,170- A 1 . . 1 + /3^1 —a (5.32) In order to determine equilibrium quota prices for the QQ system unambiguously, we have 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 with kx IN ton of quantity quotas before year 1. Suppose also that 2(1 — 6)1 p > + 62 is satisfied. Then for all 0 < 3 < 1, '7;1 < v < __"17T^<v < ^ ^ (5.33) (5.34) and the equilibrium quota prices for year t (t > 2) are as follows: When year t — 1 is a good year, period A :^= (1 + 1 ^6 ^1 )(P 1 + a /..t) = VVA (constant) (5.35) 1 — /3 1 —^ Chapter 5. The Case of Risk-averse Fishers ^ period B : + 4+ ^1 = + + wt - 1 — a it + 1 — ) 6+u_,A a^1 — ) p ^ 1 + /3 6_ t-v-A . 1— =p— ,3 78 (5.36) (5.37) When year t —1 is a bad year, 1 6^1 -A^(constant) (5.38) 1 + )1 — 6 )(P^1+ a it) = _w 1—a 1—) A period B :^_4+ = p to+ w + (5.39) 1+a 1+ ,3 1+ ,a 4 — p, + _W tB- = (5.40) 1+ )^• period A :^_w tA = (1 + The equilibrium prices for year 1 are period A : A^— a)^6^1 = P ^ + 1+a 1 — 6 (P 1+ a l2) period B : wB+ = _w tB+ BWB 1^= + W t • (5.41) (5.42) (5.43) Remark: 1. The condition p > [1 + 2(1 — 6)16 2 ]ii is not much stronger than p > 1.1 for ordinary 6 values. For example, 1+ 2(1 — 6)^1.111 if 6 = 0.95 6. 2 1.247 if 6 = 0.90 2. Notice that the expression Et+iftt i (ir t+i )llui(irt ) in the general formula wt = p 1 cv)^6 -Ei+ 1 { u'( 7^r i-1-1)} A w t-Fi q-Kt^u'(7rt)^ in Proposition 6 (page 72) appears here as 1 ^1+)^1 -3^1+3 1 — / 3'^1 —^1+).^1+3 Chapter 5. The Case of Risk-averse Fishers^ 79 in (5.36), (5.37), (5.39), and (5.40). The quota price differences due to the differences between these marginal utility ratios have important effects in the following on the result of 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 of utility of the representative fisher at the beginning of year 1. The calculation is easy for the SQ system. Under the SQ system the net profit of year t at equilibrium is pk(1 + a)x •fr t+^ pk(1 — a)x 2c(e)^if year t is a good year 2c(e)^if year t is a had year Therefore in < u <^and the expected utility of year t at equilibrium is Et -0.4/1701 = 2 [(1 — 0)irrtl- +13v]^ [(1 +13)r t —13v} 2 = (1 — a,(3) P 7kx ,v-„ — 2c(e). This holds for all t = 1,2,3, ... Hence the expected discounted sum of utility is s^= The calculation of / Q 1 a/3) —^ pkx^ AT 2c(e)]. (5.44) is more involved (see Appendix E). The final result is 1^r^ pkx^a^1^kax =1—6N 2c(e)] +^_ 6 ) 1 + a P N • (5.45) Under the QQ system the expected present value of the authorities' revenue is G 1 a = QQ^1 — 6 1 + p,kax See Appendix E for the derivation. 2^6^1 )^1 + p)kax 1 — 6^(1 — 6) 2 j 1 —^1 +^ (5.46) Chapter 5. The Case of Risk-averse Fishers ^ We can interpret these formulae for VSQ, VQQ, and GQQ 80 as follows. In (5.44), VsQ decreases when a and/or increases. This is an understandable result, for if the degree of fish stock fluctuation (a) and/or the degree of risk-aversion (0) become larger, the risk-averse fisher's expected utility will decrease.' The results in (5.45) and (5.46) need more careful explanation. In (5.46) the authorities' revenue GQQ decreases when the degree of risk-aversion /3 increases. The reason for this is basically the following: If 13 is large, the authorities buy quotas when quota price is high sell quotas when quota price is low. Let us look at the situation more closely. When ,a (0 < h3 < 1) is large, we know the following from the equilibrium quota price formulae (5.37) and (5.39) in Proposition 8: Period B case 1. + 43^= p^I + + case 2. -w t^ Bp 1 — 1 6_a-) A • • • high 1 ,3 a^1 . A^ y+ ^ 6 + th^• • • low. — 1+ a 1 + (3 In case 1, year t turns out to be a bad harvest year and the authorities reduce the TAC of the year accordingly. Hence they have to buy back quotas at the beginning of period B at a high price. In case 2, year t turns out to be a good harvest year and they have to sell quotas at the beginning of period B at a /ow price to increase TAC. A similar argument applies for the TAC adjustment at the beginning of period A: Period A 1^S^1 • • • high = (1 + ^ )(P ^it) 1 — /31 — 6^1-+ a 1^6^1 case 2. _ u , A = (1 + it) • • • low. )(P^ 1 + /3 1 — 6 1+a case 1. + w-A - 'Under the SQ system with identical initial quota endowments to fishers, no actual quota trades occur at equilibrium. Hence we can ignore the effect which quota price change due to risk-aversion might have on fishers' profits. Chapter 5. The Case of Risk-averse Fishers ^ 81 The authorities lose money because of these quota trades. Since fishers trade quotas in the reverse direction, they will gain from these quota trades. We can regard this as an implicit "subsidy" to fishers through quota market operations. Incidentally, in the expression ^ 1^a ^2 1 GQQ ^ ^itkax 6[ -^ + 6 ] 13 2(P — 51 + 1 6 (1- 6) 2 1 — 0 1 + 11)kax, the term 1 ^(p p kax 6) 2 1 — 1+ a ) 6 [ (1 — ^13^ corresponds to the authorities' deficit due to the TAC adjustments in period A, and the term 6 1- 2^f3^1 ii)kax Li _ 6]^/32 (p 1+ a corresponds to the deficit due to the TAC adjustments in period B. (Cf. the last part of Appendix E.) As is easily seen, the effect of period A adjustments which involves the coefficient 6/(1 — 6) 2 is larger than that of period B adjustments which involves the coefficient 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 expected utility VQQ: 1 pkx^a^1^kax VQQ =^6[ N^2c(e)]+ (0 1 N^ . S)1+ a — Given the degree of fish stock fluctuation a, the direct effect of a larger /3 value is the decrease 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. For when /3 is large, the authorities' quota market operations serve as a "subsidy" mechanism to 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 0 82 Chapter 5. The Case of Risk-averse Fishers ^ becomes larger. But this specific movement of VQQ value relative to is not crucial for the 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 eventually VSQ < VQQ holds for sufficiently large ,3 values. 5.1.7 Comparison of the two ITQ systems Now we can make a comparison of the two ITQ systems. The difference of the representative fisher's expected discounted sum of utility between the two systems is given by (5.44) and (5.45): 1 „pkx a a pkx VQQ — VSQ = ^ N ( 43 ^6)1 + N Hence > VQQ <VsQ^03 1-1-a (5.47) 2 _4_ 1-8 ' 1-1-0 For the authorities' revenue under the QQ system, we have by (5.46) G Q Q 0 '4=' 1 ^a6 ^1 (^1 p)) < 0 )^[ 1 — 6 + (1 — 6) 2^1 1 — 1 + a po_ ,(92^2 + a ^ 3 2 + A) — 1 0, where A:=6(2+ 1 _ 6 ) 1-1-a a [p p 1^ +1 al and 1 11 = q(1 — a)x eV). Hence >^< —A+ VA 2 'ti==> ,3 GQQ<0 9 +4 (5.48) Chapter 5. The Case of Risk-averse Fishers ^ 83 GQQ is positive if 3 is small and GQQ is negative if 3 is large. As we saw above, the reason for this is: If /3 is large, the authorities I buy quotas when quota price is high sell quotas when quota price is low. 5.1.8 Numerical examples We illustrate the above analysis by numerical examples. We specify four parameter values, /3, a, p/p 6, i.e., the degree of risk-aversion, the degree of fish stock fluctuation, the , price-marginal cost ratio, and the discount factor. For the sake of simplicity, the discount factor 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 = 1.25 and pl p = 1.5. For a we choose also two values, a = 0.1 and a = 0.5. Thus there are four 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 /3 G for each case. Figure 5.8 (p.84) illustrates the Table 5.3: Critical values of the degree of risk-aversion 6 = 0.9 Case 1 p = 1.5p a = 0.1 Case 2 p = 1.25p a = 0.1 Case 3 p = 1.5p a = 0.5 Case 4 p = 1.25p a = 0.5 /3G 0.016 13F 0.057 0.030 0.068 0.040 0.213 0.058 0.253 situation. We notice that there is a range of /3 value (3G < 3 < ,3F) where both fishers and the authorities are better off under the SQ system. This case poses no problem. If there is a conflict of interest between fishers and the authorities, it must be the case that ^ ^ Chapter 5. The Case of Risk-averse Fishers ^ 84 Figure 5.8: Risk-aversion, government's revenue, and fishers' utility (1) G QQ >0^G <o QQ I- --1-- fl (2) 0 0.016 0.057 I IVQQ <VSQ A VQQ >V SQ G QQ >0^GQQ<0 o 1 R^i ^I^I^ 0.030 0.068 I^ F V f, V< Qw SQ^VQQ>VSQ (3) 11^1 4 GQQ >o^G QQ<0 ^1-^-i ^ o^ I II f3^1^1 0.040^ 0.213^ ^ I- ^ VQQ <V SQ^VQQ >VSQ F (4) G -I 1 11-1 A >0^0 <0 4 ^ A 0^ 1 I^II^ II----1 13 1^ 1QQ QQ 0.058^0.253 I- ^ IV <\T QQ SQ VQ Q>VsQ 4 Chapter 5. The Case of Risk-averse Fishers^ either 3 < OG or 85 8 > /3F. Let us consider the situation in which fishers' degree of risk-aversion ,3 is large enough that 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 concerning the choice of the quota system. Fishers will support the QQ system which brings them higher expected utility. On the other hand, the authorities will try to adopt the SQ system in order to avoid deficit in their quota operations account. This seems to correspond to the situation which was observed in New Zealand. Now the problem is, which is better for 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 compensate fishers for their utility loss under the SQ system for the above four cases (see Appendix F for further details).^values are so chosen that both /3 > OF and > OG are satisfied. Compare the figures in the table with the expected present value of total fish sales 1 pk.x = l0pkx. 1—S In all these cases in which GQ Q < 0 and VQ Q > tisQ hold at the same time, the SQ system is potentially better than the QQ system for both fishers and the authorities. Chapter 5. The Case of Risk-averse Fishers ^ 86 Table 5.4: The quota authorities' deficits under the two ITQ systems 6 = 0.9 (1) p = 1.5p,^a = 0.1 (2) p = 1.25p, a = 0.1 (3) p = 1.5/L,^a = 0.5 (4) p = 1.25p, a = 0.5 3 3 3 ,3 3 = = = = = 0.1 0.5 0.1 0.5 0.3 3 = 0.5 = 0.3 = 0.5 authorities' deficit in quota tradings under QQ (discounted sum) 0.333pkx > 2.535pkx > 0.200pkx > 1.727pkx > 7.955pkx 17.222pkx 6.282pkx 14.066pkx > > > > authorities' compensation to fishers under SQ (discounted sum) 0.045pkx 0.470pkx 0.034pkx 0.464pkx 0.455pkx 1.5pkx 0.246pkx 1.3pkx There is another observation about Table 5.4. It is clear that when a is large, the authorities' deficit can become very large under the QQ system. Case 3 and Case 4 give examples. In these cases the degree of fish stock fluctuation a is 0.5. This means Xt = 1.5x if year t is a good year 0.5x if year t is a bad year The harvest in good years is three times as high as in bad years. The fish stock level fluctuates wildly from year to year. Let us look at case 3. If the degree of risk-aversion of fishers, 3, is 0.3, the authorities' expected present value of deficit under the QQ system is 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 adopt the SQ system, the present value of the necessary compensation is 0.455pkx, i.e., 4.55 percent of total fish sales. Of course, the authorities will not make even this amount of payment to fishers. Then fishers will be worse off under the SQ system. They surely will oppose the SQ system. Therefore if the fish stock fluctuation is very large and fishers are sufficiently risk-averse, there seems to be no way to implement ITQ management Chapter 5. The Case of Risk-averse Fishers ^ 87 of the fishery to the satisfaction of all the parties concerned. We might say that this provides a theoretical ground for a report on Canadian fisheries (Commission on Pacific Fisheries Policy [9]) which refused to recommend a quota system for the salmon fishery on the basis of the wide and unpredictable fluctuations to which harvests in the fishery are subject. ? All these seem reasonable except that the authorities' deficit are too large in some cases. We need to understand the reason for these huge numbers. This helps us to check the plausibility of the above scenario which seems to support government decisions such as to adopt the SQ system instead of the QQ system. The point lies in the equilibrium quota prices under the QQ system with risk-averse fishers. The following table shows equilibrium quota prices under the parameter values given above (see Table 5.5). Table 5.5: Quota price movements under the QQ system S = 0.9 + z-v A + 4+ + wr- _Cv A _4+ _wr- p = 1.5p p = 1.25p a a = 0.1 0 = 0.1 4.333p 4.355p 4.312p 3.617p 3.645p 3.589p = 0.1 0 = 0.1 3.000p 3.045p 2.955p 2.504p 2.554p 2.454p p = 1.5p a = 0.5 0 = 0.3 7.689p 7.706p 7.690p 4.402p 4.509p 4.295p p = 1.25p a = 0.5 I3 = 0.3 6.467p 6.553p 6.380p 3.697p 3.867p 3.528p In all cases, a bad harvest year (t - 1) is followed by low quota prices (_wit B+^B--■ Wt^Wt ), and a good harvest year (t - is followed by high quota prices ( + 0, + 4 + , + wr - ). 7 See Canada [9], Chapters 8,9 and 10. Particularly page 105. See also Munro and Scott [24],pp.661664 and Anderson [3],pp.258-266. Chapter 5. The Case of Risk-averse Fishers^ 88 This 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 quota holdings. This causes lower expected net income 7r t this year (so 7r t < v, v : constant), since fishers are likely to end up spending more money for the acquisition of additional quotas. 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 and fishers' marginal utility of income is low (= 1— /3) this year. Hence, in short, this year t's 1 dollar has more value to fishers after a bad harvest year t — 1 than after a good harvest year 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 the effect 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 that this 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 period - A and _F tuB t +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 in the market.) At the beginning of year t 1, the authorities have to buy back quotas at this 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 ( + w t13- = 4.312p), the authorities have to sell quotas at a very low price (_Cv A = 3.617p) at the beginning of next year. This large fluctuation of quota price is responsible for the authorities' deficit and 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 example because of Chapter 5. The Case of Risk-averse Fishers ^ 89 I. relatively large risk aversion (/3) of fishers, and 2. the nonexistence of risk-neutral traders in the market. If these conditions are not met, quota price fluctuations cannot be so large. Then such a situation as above in which fishers are better off under the QQ system will be unlikely to occur (at least in our example). 8 In the case of New Zealand's ITQ management system, there is no institutional restriction on the participation of risk-neutral traders in the quota market, because "Membership in the [quota trading] exhange is through subscription and is available to anyone."(Ian N. Clark et al.[13] p.333.) If this means that there are many risk-neutral traders in the quota market other than fishers and the authorities, the validity of the above result for this case may be limited. The existence of risk-neutral traders will surely affect the nature of the resulting equilibrium. In the next section we explore the consequence of the existence of risk-neutral traders in the quota market. 5.2 Risk - averse fishers with risk - neutral traders As quoted above, in the ITQ management system of New Zealand there is almost no restriction 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 of allowing nonfishers to influence the value and distribution of quota. Theoretically, fish retailers could purchase quota and lease to fishers with conditions guaranteeing 8 However, by assuming a constant fish price p, we are neglecting in this chapter the effect of the fish price fluctuations due to inelastic demand for fish. If we take this effect into consideration, quota price fluctuations can be large even when there are risk-neutral traders in the market. Then it may still be possible that fishers are better off under the QQ system by the same mechanism as in the risk-neutral fishers' case in Chapter 3. Chapter 5. The Case of Risk-averse Fishers^ 90 supplies of catch, financial institutions could acquire quota as security, recreationists or tourist guides could purchase quota away from commercial fishers, or private or government conservationists could purchase quota to reduce catches. Some of these activities have begun taking place. ([13] p.332.) In the previous section we assumed away the existence of these outsiders (except the quota authorities) in the quota market. In this section we introduce risk-neutral outsiders into our model and examine its effect. We assume that there are many potential participants in the market (other than fishers and the authorities) who are all risk-neutral. We also allow quota leasing between outsiders and fishers. The following assumptions are retained: • 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 in year t is 7rt^pateA t _ c(eA t^wtA (hA t^aAt) nrtili tA ^t m xtet13 c ( GtB) wtB ( Xt a A a .p) _ m B/B _ ^t^(5, where m ' ^ price of a unit quota for one season in period j quantity of quotas leased to the fisher in period j The existence of risk-neutral traders imposes strong restrictions on equilibrium quota prices. Namely, quota prices have to satisfy no - arbitrage conditions. In our model these Chapter 5. The Case of Risk-averse Fishers ^ 91 conditions are expressed as follows: A^A w t Mt =^1 wt^f (5.49) ,c A wt mt = uwt+i xt (5.50) For example, the second condition has the following meaning for outsiders: wB cost of purchasing one ton of quotas in period B rnt revenue from leasing one ton of quotas in period B AwA ^ t+i xt discounted sales of x/Xt ton of quotas in period A of next year Risk-neutral traders can get benefit from arbitrage transactions whenever these conditions are not satisfied. Such transactions will rapidly restore the quota prices that satisfy the 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 maximize his expected present value of utility Ef Eblu(rt)} t.i subject to his quota constraint t^qxtetB < atB^1.r. q x te A Then the maximized value function Vs (hi 4 ) will satisfy the Bellman's equation: 7,21\ VsQkut^max Et max [u(7)d- WsQ(h A t+i)] eP,aP,1P where the second maximization is subject to the above quota constraint and hA x xt atB• Chapter 5. The Case of Risk-averse Fishers^ 92 Let us define e by e= 2Nq . Then we know as before that equilibrium quantities satisfy kx, kXt N' e ' N ) (on a At^lts) (5.51) on the condition that p — , (e) > 0^V Xt E [b, d qAt The same procedure as before yields the first order conditions that have to be satisfied at equilibrium (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 form for u(.). Therefore, we do not try to solve the problem here. Instead, we make the following observation. Namely, the existence of risk-neutral traders in the quota market under the SQ system makes fishers better off. This is because it is always possible for fishers to choose quantities (6;1 ,43 ,^, cri8 ) = (‘ kx kXt ) N" N by setting tifri = lB = 0 and these quantities give exactly the same annual net profit pkXt 7rt =^ ^2c(e) , — as in the no risk-neutral traders case. Hence fishers cannot be worse off under the SQ system due to the existence of risk-neutral traders. Rather fishers will utilize the opportunity of quota leasing to reduce their income fluctuations. For instance, fishers in bad years will sell their quotas to outsiders and rent back those quotas from outsiders in order to make up for their income loss. In good years they will buy back quotas from Chapter 5. The Case of Risk-averse Fishers ^ 93 outsiders. As a result, fishers' income fluctuations will be smoothed out over time and risk-averse fishers will be better off due to the existence of risk-neutral traders and quota leasing opportunities. We can use the same method as above to derive the conditions that equilibrium quantities and equilibrium prices satisfy under the QQ system. However, it is not possible to ascertain that fishers are better off with the existence of risk-neutral traders under the QQ system. They may be worse off because of risk-neutral traders, since the source of fishers' gain under the QQ system (i.e., large fluctuations of quota price) will be eliminated by risk-neutral traders. As we said before, it is difficult to obtain explicit solutions for equilibrium quantities and equilibrium prices for the open SQ and QQ systems. Without these values, we cannot calculate fishers expected utility under these systems (and the authorities' expected revenue under the QQ system). Therefore we do not pursue any further the comparison of the two ITQ systems with risk-neutral traders. The following two questions remain open 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 the difference between the two systems with risk-averse fishers by 1. reducing quota price fluctuations due to risk-aversion and 2. 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 or Chapter 5. The Case of Risk-averse Fishers^ 94 • administrative costs might become important in determining the relative advantage of the two alternative ITQ systems. Chapter 6 Conclusion Among various regulatory measures of fishery management, individual transferable quotas (ITQ's) have attractive features which other measures do not share. In principle, an ITQ based management system can overcome both the biological overexploitation of fishery resources and the economic overexpansion of fishing capacity in a simple and unified way. Overexploitation is effectively prevented by setting an appropriate level of total allowable catch. This is implemented by adjusting quota volume to that level. The problem of overcapacity in a profitable fishery under stock protection is resolved in a decentralized manner through voluntary restraint of fishers based on their individually rational economic calculations. Quota prices which are formed in a competitive quota market guide those calculations automatically and efficiently. The management authorities do not have to engage in other detailed and complicated regulatory measures except for monitoring fishers' harvests. These advantages of ITQ management systems over other measures is the reason that now growing number of countries are using ITQ's in their fishery management. But we have to admit that ITQ management systems are not free from problems. One of the problems is the uncertainty of fish stock level. Fluctuations in the stock level are often unpredictable and sometimes very large. When the fish stock level fluctuates, the management authorities have to adjust the TAC level correspondingly to attain their biological and economic goals. These TAC adjustments clearly affect fishers' profits. If the authorities participate in the quota market transactions. their revenue is also affected 95 Chapter 6. Conclusion^ 96 by fish stock fluctuations and resulting TAC adjustments. This thesis focused on this issue. 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 common TAC adustment rule are equivalent for the society as a whole. The two systems generate the same amount of rent from the fishery. Under the SQ system the total rent minus the initial quota sales (or plus the initial quota buybacks) of the authorities accrues to fishers. Under the QQ system with a proportional TAC adjustment rule, fishers earn more than this amount at the expense of the authorities if the demand for fish is inelastic. If the demand is sufficiently elastic, a part of this amount is transferred from fishers to the authorities. Under the hybrid system with a definite rule of TAC division into quantity quotas and share quotas, fishers are indifferent to the difference of quotas after the initial endowments are allocated to them. • Taxes on fishers can affect the distribution of fishery rent among fishers. In the comparison of a profit tax and a quota tax, we found that the former favours those fishers who have more share in total quota endowments than in total inframarginal profits. Depending on the initial quota allocation, fishers with larger inframarginal profit shares may be better or worse off under a profit tax. • Risk-aversion of fishers may explain their preference for the QQ system without assuming a downward sloping demand curve for fish, if the effect of risk-neutral Chapter 6. Conclusion^ 97 traders in the quota market is negligible. In this case, it is possible that the SQ system is better than the QQ system for both fishers and authorities, if the authorities make necessary compensation to fishers under the SQ system. If the quota market is open to outsiders, risk-aversion may not be a decisive factor of the fishers' preference for the QQ system. The SQ system open to risk-neutral outsiders 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 a resulting long-term deficit in their quota market operations account. This deficit is 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 from the QQ system to the SQ system. With a TAC rule given, fishers and the authorities play 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 different groups 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 the SQ system may bring about satisfactory result, especially in fisheries with large year-to-year stock fluctuations. The QQ system is particularly vulnerable because of the authorities' deficit. The SQ system will also be difficult to implement unless the authorities are ready to compensate fishers for their loss caused by large yearto-year income fluctuations. Chapter 6. Conclusion^ 98 These findings depend on the assumptions of our model. The most significant differences of our model from the standard one are the following about fish population dynamics: 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 year depends on the escapement of the year before through the stock-recruitment function.' Assumption 1 ignores this dependence. Secondly, the standard model takes into account the fish stock depletion during each fishing season. Assumption 2 ignores the effects of this in-season depletion. Because of these assumptions, our model lacks the aspect of the 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 equilibrium quota price formulae rather easily. Apart from this technical convenience, we can justify Assumption 2 if the TAC level is much smaller than the stock level. It is very difficult to formulate and solve a model without Assumption 2 (small in-season depletion) when we take into account, as we have done in our model, the possibility of the informational change 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 retain Assumption 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. We can model this situation by assuming some joint probability distribution of this year's stock level Xt and next year's stock level X t+1 . (If the stock level is discretized, the joint distribution is expressed by a matrix of transition probabilities.) Although the derivation 'See Clark [11] Chapter 6. Chapter 6. Conclusion^ 99 of equilibrium quantities may become more complicated, there is no intrinsic difficulty in 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 findings about the distributional effects of alternative ITQ systems are obtained for the proportional TAC adjustment rule. If an actual TAC rule is different from the proportional rule, we have to modify our results. But the precise nature of the necessary modification depends 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 stock level. They interact with each other in complicated ways and it is hard to obtain general results 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 details of the above four factors. An issue related to both of the above two points, 1 and 2, is the problem of the optimal TAC adjustment rule. If we could literally ignore in-season depletion regardless of the level of fishing activities, the "optimal" policy would be to set TAC sufficiently large and let fishers harvest as much as they wish. The fact that this is not the case reflects the importance of in-season depletion and the stock-recruitment relationship. Therefore, in order to determine the optimal TAC rule, we will have to consider these two aspects of fish population dynamics explicitly. Since this is a tough problem under the fish stock uncertainty of our model, we ignored the dynamics and concentrated on the positive analysis of the problem, i.e., the comparison of alternative ITQ systems with a given TAC adjustment rule. The normative analysis of various TAC adjustment rules under ITQ 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, May 1992: stock assessments and yield estimates. 222 p. (Unpublished report held in MAF Fisheries Greta Point library, Wellington.) The Ministry of Agriculture and Fisheries, 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 Journal of 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 New Policy for Canada's Pacific Fisheries. Ottawa. [10] Clark, C. W. 1980. Towards a predictive model for the economic regulation of commercial 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 Renewable Resources, Second Edition. New York: Wiley-Interscience. 100 101 Bibliography^ [13] Clark, I., P. Major, and N. Mollet. 1989. The development and implementation of New 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 Canadian experience. In Neher et al. [25], pp.267-288. [15] Geen, G., and M. Nayar. 1989. Individual transferable quotas in Southern bluefin tuna 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. Canadian Journal of Fisheries and Aquatic Sciences 49:497 503. - [17] Hannesson, R. 1989. Catch quotas and the variability of allowable catch. In Neher et al. [25], pp.467-480. [18] Hannesson, R. 1989. Fixed or variable catch quotas? The importance of population dynamics 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 46:14291445. [22] Macgillivray, P. B. 1990. Assessment of New Zealand's Individual Transferable Quota Fisheries Management. Economic and Commercial Analysis Report No.75, Department of Fisheries and Oceans, Canada. [23] Mangel, M. and C. W. Clark. 1988. Dynamic Modeling in Behavioral Ecology. Princeton, N.J.: Princeton University Press. [24] Munro, G. R. and A. D. Scott. 1985. The economics of fisheries management. In A. V. Kneese and J. L. Sweeny (eds.), Handbook of Natural Resource and Energy Economics, vol. 2, pp.623-676. Amsterdam: North-Holland. [25] Neher, P. A., R. Arnason, and N. Mollet (eds.) 1989. Rights Based Fishing. Dordrecht, The Netherlands: Kluwer Academic Publishers. [26] Newbery, D. M. G. and J. E. Stiglitz. 1981. The Theory of Commodity Price Stabilization. Oxford: Oxford University Press. Bibliography^ 102 [27] New Zealand. 1992. TACC and management review for the 1992/93 fishing year, 11 August 1992. Unpublished. The Ministry of Agriculture and Fisheries, New Zealand. [28] Sargent, T. J. 1987. Macroeconomic Theory, Second Edition. Orlando, Fla.: Academic Press. [29] Scott, A. D., and P. A. Neher (eds.). 1981. The Public Regulation of Commercial Fisheries in Canada. Ottawa: Supply and Service Canada. [30] Stokey. N. L. and R. E. Lucas, Jr. with E. C. Prescott. 1989. Recursive Methods in Economic Dynamics. Cambridge, Mass.: Harvard University Press. Appendix A Proof of Proposition 1 The proof consists of five steps. Step 1 Under the hybrid ITQ system fisher i's net profit in year t is 74 ^t E^— c (eij i )] i j=A,B wtA . (hAi —^+ B^Ai^Bi\ ^at )^tot • a t — a) t wtA o tAi (tilt) wt.8 ((tAi (pi) where hitli ar_i 1 and O tAi = (P ip By assumption, equilibriumquota prices w and WtB satisfy A Wt = wt (HA , OA), WA = 4) ( 11 =^,(t1 iA , WtA , w tB , , OM, _Kt ),^WtB =^(HtA ,(-V , Xt)• The quota prices perceived by fisher i coincide with these price functions by our rational 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 t depends on the aggregate quota holdings Hill and 1; -.V as well as his own quota holdings hAi and O t-Ai. We denote the maximized expected value by V i (qi, O tAi ) suppressing the possible dependece on HA and OA. Hereafter V i (hifrii, OM should be understood as , HA, 103 ^ ^ Appendix A. Proof of Proposition 1 ^ 104 The maximized value function 1/ i (hAt^t^ i O A ') has to satisfy the follwing dynamic programming equation: Vi( h Ai ° Ai ) = max Et^max [ 74 4_ 6 -vi^oi t^t^ 't+1 , t+1,]} where the second maximization is subject to + q-Kte St < a Bt + ('t6i • f(Xt), — aP i and OtAi!' i = and h Ai 1 = The second maximization problem is: max^zi it3i, c1 ^e P CP^t + ^^g Ai Ai v h , subject to qXte At i + ate tBi < a Bt^(tBi f (Xi) given ^AiotAi;^ (wit4 WA) = (6(11, e , ht; h , 4) (k , e it1 )) ( 143 ,WtB ) = ((HA 24', Xt),k I1 ( 11-tA , OA, Xi)) Let the Lagrangian be i Bi •^^ — g xte .4i^q xte rij +^./( A t )^ • [at^Ct t'^+ At^ £ = 7r^ is the Lagrangian multiplier of this problem. Let us denote the maximizers by e tth and We assume that Epi > 0, aBi > 0, and 13i > 0 are satisfied. The two equalities 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 are e t Bi^ ptqXt — c'.(e B ) a tBi :^j_ 6 (Bi^+ 6 ^ a^i t q- t = 0 ^ hA pidAi^n^(A.53) t+1^ +11 vt+" a^ vi( hAi i 00-4i (A.52) I ^= " 8 t+ 1) A tt • f (x = 0^(A.54) ^ Appendix A. Proof of Proposition _I 105 ^We denote the maximized value by ^'ON1), i.e., +^i (it t'1 1 ,^= t' A A?: a Ai max^+ (w i ki6t+1,vt-Fi) e .13i Bi ".13i at , subject to qXte tAi qXter i < at Bi + e i • f(xi), where ( e Ai) + 4 (hAi cet ti ) ^otAi ^ci i t piqXietAi Bi Ptate i — ct(er i ) w tB^(.a Ai^^at )^ Ai^ -^hAi^/.13i ^t' t+1 = a Bi ^t+1 = St • _ CiA.1 )+ 14/tB • (Cili^-t21 ) By using the envelope theorem s , we obtain ^a^sTritlAi ae Ai[ t -v kut+1, t+i)] = ptqxt — cact , " kAdi^101 !I tA^17 — A it qXt °tA+i ) .11^— w^w t a r acAi^[ 7-r; +^ötA+ii)]^_WA + 1,,'17tB • - The first maximization problem is: max . Et {mot e t lz ,cq4 t ,CiAt given 6 V i Ch A t +1 1 , a A t di 1)} 7, Ai plAi "t ( D ; W71 ) = (0(N 131 ;1 ), 431 ( -H;^1)) (4^= (OW, 'See, e.g. Birchenhall and Grout [7] p.247. O A, X^tA et 4 Xt)) • Appendix A. Proof of Proposition 1^ 106 We denote the maximizers by E.P.i , a ^and Ci4i . We also denote the values of EP^ and^at^ci4i) by Eit3i , a Ri and Cp i , respectively. 7 Since fishers are assumed to be risk-neutral and^is linear in cr4i and (i 4i , it is clear that ^Et {— tA wtB } = 0 w ^E t { — WtA WtB } =0 ^ ^ (A.55) (A.56) have to hold in order for the quota market to be in equilibrium in period A. The maximizers Cet t i and ( 4. i are inderterminate under these conditions. We assume that Oi > 0. This assumption is justified in Step 5. The first order condition with respect to eAZ is Et {ptqXt — e At i ^ AqXt} = 0.^(A.57) The maximized value is V' (41 , Ai Ai 19 Ai = Et { .-fr + 6V i (h t+17 where ptq xte tAi _ ei ( E.41) totA (htAi — atAi) WA — ptqXter — c i ( e i )^• (a j4i — d'6i ) W B • (V i( Ti) h Ai^6Ai t +i • t+1^t Finally, applying the envelope theorem to the maximized value function 1/"(hil i , Oh, we obtain ^a vi(hAi,^) = Er {wA} , aolu vi(hAi,^0 4 .^•^• =^{wtAl (A.58) (A.59) 107 Appendix A. Proof of Proposition 1^ Step 2 Let (0i,a:At i, i,4i ; e‘pi,a-pi,j) be the quantities at equilibrium. In this step we show that "Ai e^= constant et = cii (eB 1 ) cif ( 6-Ai) cN(etEiN ) Et { c i'(e St i )} N (i = 1,^, N) 1, ... ,N) 1 —K(Xt )• E(eAl e tBi) i=1 (A.52) — (A.59) must hold at equilibrium. Since 0 i , wtZ and WtA i do not depend on , the fish stock level Xt in year t which is still unknown in period A, we have the following: A Et {gi ('e Ai )} = c(eAi).^ , E t ( Wt = w„ and Et {WA } WA . Then the following conditions corresponding to (A.52) — (A.59) have to be satisfied simultaneously (t = 1,2,3, . ..): PtqXt — caer) — A it qXt = 0 (A.60) 0 (A.61) a —wtB + ^ V i (11;tIli O jt 41..i i ) + At • f (xt) ^DeNi^+1, 0 (A.62) Et {p t qX t — A it qXt} — cae't4i ) 0 (A.63) + Et {4} = 0 (A.64) ^—wr + ^ • V i (letli+1 + _wtA + Et { KT} oi, a^p ,vt ahAi , ,, ^ a aBAi V Ai aAi\ = 0 wA t (A.65) (A.66) (A.67) Appendix A. Proof of Proposition 1 ^ 108 where Ai — = h Ai^//Ait-1-1 — 8Ai^arti 1^vl^t+1 — '3t • From (A.66) a pjAi^wA "t+11 v^HA* By substituting this into (A.61), + 6 4+1 + = 0 , i.e., = —14 qXt (54+ 0Xt . Substituting this into (A.60) gives Wt = Pt — 1 qXt Bi) 6'4+ 1• (A.68) Let us assume that 2 total fish supply = E( q Xt 0i qXt er i ) = K(Xt)Xt. (A.69) Under this assumption, total fish supply depends only on X t . Then fish price pi is independent of i/ tAi and O tAi , since Xt does not depend on hA i and 8Ai. Equilibrium quota prices wr and 4+1 are also independent of ie and OAi by assumption. Therefore, qi t which satisfies (A.68) is independent of hA i and BAZ. (A.68) must hold for all i = 1,2, ... , N. Hence efi(e ri ) = c2(e B2 ) =^=^(eBN) Since (A.60) holds for any Xt E [b, d we have Et {pi qXt — c(er i ) - A 2t qXt } = 0. 2 This assumption is justified in Step 5. (A.70) Appendix A. Proof of Proposition 1 ^ 109 From this and (A.63), we obtain oi ) , Et fr( essi ) }^ From this we know that both e -iti and eBi are independent of 11, 1 i and 8t i , (A . 71 ) since êpi is independent of h't4 i and /ki as seen above. From (A.69) N^ e tBt ) = 1 K (Xt — (A.72) ). epi depends only on Xt. Now we show that eAi = eAi = constant and that Notice that (A.70) — (A.72) hold for any values of H:- 1. (s = 1,...,t), e (s = 1,...,t), and X, (s = 1,...,t — 1). Let Z stand for any one of these variables. Differentiating (A.70) with respect to Z gives (.r ) Ap-Bi^h,^BN ) ^ —^(w it N--t eBl k^ az az Hence, aer i /OZ (i =^N) all have the same sign. Also Et {c7(eiBl)) ^BN az} = • = Et c'k (et ) a 6 PN1 az f . (A.73) Differentiating (A.71) with respect to Z yields cne A-i ) =Et{cN B t t az az'^ }, (i = 1, 9 ,^,^(A.74) From (A.73) and (A.74), we have _7 Hence 1 N Al^ NAN nAl = fl 7t4N) civ az az — et ) ., ae-;ivaz (i = 1,...,N) all have the same sign. Differentiating (A.72) with respect to Z yields ae Ai^N ae Bi az + E azt = 0 t (A.75) ^ Appendix A. Proof of Proposition 1 ^ Therefore, (aeiti vaz,...,aer !az) 110 and (aeriaz,...,&B,Niaz) have opposite signs or are all zero. Then, (A.74) implies that i DeA i^a o^0^(i^ 1,^, N). ^t az az , Thus, in particular, e't4i and epi are independent of HA and (V. To sum up, ei4i and e jt3i are independent of ivit li , i4-': , _W 1- , and O. Since equilibrium quantities 41i and eri are the maximizers of the dynamic programming equation , V+1) ] Et^max { [74 + 60 (hN1 ,0 tA4i. 1 , H A+1 ( V i (1^ 4", e i , W ,OM = atmax ,airli , (;4i^el3i ,a pi , cp we can conclude from the recursive structure that ^Ai^= constant for t = 1,2, ... , and et -Bi et depends only on Xt as asserted. Let us denote ^Ai^^Ai et^=^(i = 1,2, ... , N) 4(eB1) =^= 4(er)^g(Xt). (A.76) (A.77) Then by (A.71), ^ Ai Et{g(Xt)} Ci e t = 1,2,...,N). Step 3 In Step 3 we derive equilibrium quota prices. From (A.68) and (A.77) wB v 6 A = Pi —^Wt+1 qXt (A.78) Substituting this into (A.64) gives 1^,^4 WA t = Et {Pt —^AXt^6Et twi+i (IA t (A.79) ^ Appendix A. Proof of Proposition 1^ 111 t > 2 case: For t > 2, HA^— f (Xt-i) and OA = 1 at equilibrium. Then wt = 0(W,^= 46 ( K(Xt-i)Xt-i — f(xt_i),i) Hence Et-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 yields Et {PtqXtg(Xt)} + 6 E t {4+1 1 , i.e., Et -1 {wtA l Et^ A .1 ^ Et {Pt — — 1 c (e i )} 1 6^qXt 1 (t 2 ). – By substituting this into (A.79), we get ^w tA = 1 1 6,Et {pt qxt g(Xi)} =: fv A (I> 2)^(A.80) By (A.78) and (A.80), ?Di^Pt^g(Xt)d- 6 qXt ^(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 and wiB To sum up, the equilibrium quota prices are given by the following: 1 Et {pi^ g(Xt)} 1–6 q-Kt 1^. W.tB = Pt at g(Xt) + 60 A wt = 1 -4 W- 1 Et {{Pt –— 1 g(Xt)} • f(Xt)} WtA = ^ 1–6 qXt 1 WtB = [Pt qX t g(Xt)] • f (Xi) + 6WA (A.82) (A.83) wA^(A.84) (A.85) Appendix A. Proof of Proposition 1 ^ 112 where c'i(eV)^• = c'N(eB lv ) =: g(Xt), c'i (e 2' i )^caeAi) = E t {g(Xt )}^(i = 1 , 2,^,^• 1 E(e Ai el3i) = -K(Xt). i=1 Step 4 We use the above quota prices to calculate the expected present values of fisher i's profits, Vi(hP, OP) (i = 1,2, ..., N), and the authorities' net revenue, G(.11i/", ON. We continue to assume that the quota constraint for each fisher binds at equilibrium: qXte Ai qXt e St --^i c;tBi • f (Xt ).^(A.86) Fisher i's net profit in year t at equilibrium is = po xt e‘Ai ci (eki) ,thA^aim) + wt.B PtateB i ci(e i3i ) + W A '^+ WtB •(PA — ei) - where iriu = hi i and BAi = 61i when t = 1. If we take the expectation oft with respect to Xt , the terms involving h4 i and cancel out on the RHS, because Cv A = Et {4} and W A = Et {W 3 } by (A.64) and (A.65). By using (A.83), (A.85), and (A.86) in the above expression of fr ti, it is straightfoward to derive the following expression for the maximized value function of fisher i: vi(hAi i, eA1 i) E,_1Et{ "7'6 =^+ 6 ^ 16 A Ai^- 4 Ai W^+ IV"^+ ^ Et . 1—6 {9(Xt) • (e Ai + t ) In the course of calculation, we notice that terms involving together, as long as ar and c; tBi aBi i(e.A z and -t13 'i cancel out all satisfy (A.86). Therefore, how fisher i divides his quota Appendix A. Proof of Proposition 1^ 113 holdings qXt e i qXt et 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 that v i ( h p 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 the fishery regardless of their initial quota endowments. Fisher j (> N) are better of by selling 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(X i )Xi - f(X1) - (K(x)x - f(x))] + W A • (1 — G t = w A • [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 that G(Hji ,^) = E 1 {C-1} + 6 6 E t _iE t 1 1^ = — zi) A^— W A 0'14. + 1 {Gd 1^ 6. -E't {[Pt^gx 1^t g( -3(t)]K(Xt)Xi} Step 5 As the last step of our proof, we verify that the assumptions made so far are indeed satisfied by the equilibrium quota prices and effort levels derived above if the conditions Pt — 1 --- qXt g(Xt) > 0 V.Xt E [b, d and Appendix A. Proof of Proposition 1^ ^Bi Ct^> 114 0^(i = 1, . . . , N) are met. We derived (A.52) — (A.54) and (A.57) (i.e., (A.60) — (A.63)) on the assumption that at equilibrium ,"Bi^0, et ki t > 0,^> 0 , e tAi > 0. Also in Step 2 we derived eAi = e"Ai and^"^g(Xt) (i = 1, . . . , N) on the assumption that total fish supply = TAC = ti(X2)Xt• First, the assumption epi > 0 together with caeh = E t { c',; (eP i )} yield eit ti > 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, pt qXt Then pt k — — — g(Xt ) — A;qX t = 0. g(Xt) > 0 eXt E [b, d implies A > 0 at equilibrium. Hence the quota constraint binds at equilibrium, i.e., t . .f at + qXte itt i^q )(te St — —^13i ^4.131: From our assumption p t — 9 ,g(Xt ) > 0 VXt E [1), d , we have = ^1^1 g(Xt)} > 0, so Et {pi — 1— 6 qXt 1^iy\ = Pt —g ( -At) vW > q-Xt Appendix A. Proof of Proposition 1^ 115 Hence^= 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 , then ti/A 1^1 t {[Pt —^(X ^E^ )] • f(Xt) }^0^so t^> atg 1 - wtB Pt —^g(Xt)1 • f(Xt) SW A > 0, qXt which implies^(,131: = 1 at equilibrium, so that E liv_ i^f (Xt) = f (Xt) is satisfied in this case, too. Therefore N E(e N^ x 3i i + tBi f(Xt)) total fish supply = E(qxtet^+q^ te t^t i.1^ j=1 ) — ( K (Xt)X, = TAC, which justifies our assumption. Finally, as we saw in Step 4, the division of the quota holdings in period B, qXt e Ai qXt e-P i , into api and r"i15 i • f (Xt) is immaterial for fisher i at equilibrium. How fisher i divides his quota holdings in period B into quantity quotas and share quotas does not affect his expected profit. Hence the derivation of (A.53) and (A.54) based on the assumptions eir > 0 and c; Pi > 0 where the equalities do not arise as corner solutions is justified. This completes the proof of Proposition 1. Appendix B Propositions 2, 3, and 4 B.1 Proof of Proposition 2: profit tax Let Vci,' Q (1-th be the maximized value function for fisher i with quota holdings hitu at the beginning of period A of year t. Then V(:i, Q (h tAi ) satisfies the following Bellman's equation: V6Q(ftAi) = max Et max kffi SVQi Q (h tli+i i Ai ,a ^et ,Bi t t^tt^ I} where the second maximization is subject to qX4 i qX t er < aP i and 7rt (1 - 7.p) E [mate - ci(e ) 1 + wtA (h tAi atAi) wtB ( atAi atBi) j=A,B L Ai^Bi nt+1 — at • Proceeding as in the proof of Proposition 1, we obtain the following conditions which have to be satisfied at equilibrium: (1 — rp) {That — eVr i )] Aq ,Kt = 0, (1 — rp )(-4)+ 6176 Q t (it/tlii i ) Et {(1 — rp ){pt qXt — c(e 4 ')] = 0, = 0, + WI 3 )1 = 0, — Et {(1 — Tp)^W-;4 A it qXt } V6 Q '(7 where = AiBi jt t+1^at • 116 b) = Et{( 1— Tp)w }. Appendix B. Propositions 2, 3, and 4^ The rest of the proof is basically the same as for Proposition 1. B.2 Proof of Proposition 3: quota tax From the Bellman's equation, we obtain the following conditions at equilibrium: pt qXt — caer) — A it qX t = 0, _(1 +7.9)4 _4_ SVQQ ' (h Et {p t qX t — cae Ai ) Et — Ati 0, A it gXi} 0, {_ wtil^w r} 0, " (il Ai ) VQQ Et We omit the rest of the proof. B.3 Proof of Proposition 4: harvest tax The 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) Et /1q.Xt } = 0, { w At + to n= 1‘7:2Q /(iiitti) = The rest of the proof is omitted. 0, Et { wtA 117 Appendix C Propositions 5 and 6 C.1 Proof of Proposition 5: the SQ system Let 4 and wr be the equilibrium quota prices in each period of year t. Let VsQ OM be the maximized value function of a fisher with initial quota holdings 41 in tonnage unit at the beginning of period A. 1 Then Vs Q(141 ) has to satisfy the Bellman's equation: VsQ (h At = max Et max[u(7 t ) 617sQ(h A t-Fi )1 • e t ,a t^e tl3 ,a.13 where the second maximization is subject to qXt q- qXt eP < ar , and E [7, axte .ti_ c(4) , +wtA . (htA_ atA )+ w tB ( X t 7rt^ hA t+1 = X A, at ut at • From the first order necessary conditions for the maximization problem on the RHS of the Bellman's equation and from the equilibrium quantities ,e kx kXt (et^ N" N where e = k/2Nq, we obtain the following conditions that have to be satisfied at equi- librium: tl(*‘ t) • [PqXt — V)] Atq Vt = 0 (C.87) uVr t ) • (—wr)-f- 61/ Q (it A ) -xx-,,T+ A t = 0 (C.88) — - 'For the general asset pricing model, see Lucas[21]. See also Stokey et. al.[30] pp.300-304. 118 ^ 119 Appendix C. Propositions 5 and 6 ^ Et { u / ( ir t )[m Xt - c'( e )] - )tat = 0^ (C.89) Xt Et {u'( 71-t) (^+ trt —) = 0^ (C.90) q c) (11 A ) = Et {z1(7r t )w it l } ,^(C.91) where pkX t 2c(e) — N^ kx —a t = Xt By substituting (C.91) into (C.88), we get —A t qX t = uVrt)( — wrat) SEt { u 1 (7"r t )4 qx. Substituting this into (C.87) gives w:t8 = p — 1^Etlui('frt)} A qXt eVe) + (5 W X ^ uVir t )^1 Xt' (C.92) where 4 does not depend on Xt , because X t is not known in period A. Substituting this into (C.90) leads to ^1 ^Xt ^ ^, Et {uVrt)( - 4) u'01.0[13^ ^ C (e—^ )] SEt f uVrt)}4 }= O. co t Therefore 1 Et {nV*t)[P A Wt = 1^ ^ Et{nTirt)} (C.93) C.2 Proof of Proposition 6: the QQ system The proof of Proposition 6 is basically the same as the previous one. Let VQQ(q) be the maximized value function of the representative fisher with the initial quota holdings hiti in tonnage unit at the beginning of period A of year t. Then VQQ (hi4 ) has to satisfy the Bellman's equation: A ht ) = max Et rnax[u(^(51/Q 9 (h -t4+1 )] ci4 ,crit^eD,aP Appendix C. Propositions 5 and 6 ^ 120 where the second maximization is subject to qXt e At + qXt eP < aB and , ^ix x te jt c(e;h1 + _A 7, ^A \ 7rt = 1- tut ^(ht^at j=A,B hA^at t+1^t • ) The conditions corresponding to (C.87)—(C.91) are n'(it t)pqXt — c'(e)] — AtqXt = 0 (C.94) u f (irt)( - 4) + (N(2041) + At = 0 (C.95) Et {u'(rt) pqxt — c' (E)] — AtqXt} = 0 (C.96) Et {u'(it)(-4 + 4)1 = 0 (C.97) 1 6Q( 14) = EctuVrt)41. Here at 2[pqXt e — 40] wtA (kit a tA) wtA (41 a tB) pkXt ( ^ 2c(e)+ w t kXt-1 kx^B kx kXt )+' N N N and - ( k kx k kXt 2Nq' N 2Nq' N ). From (C.98) )= Et + i { u' ( irt+i ) 4+0. By substituting this into (C.95), li f (li't)( — W tB ) + Et+1 {?- 11 0- )wi4+11 + At = 0 , i.e., — t ( ,‘ A 1^t• AtqXt = (%s t)( — w tB at) + 6Et-Fifu rt+i)tt' t +lqX (C.98) 121 Appendix C. Propositions 5 and 6^ Substituting this into (C.94) gives u / (7r t ) • [(Pi — wt )0it — c (e)] + 6 E t+1 {z/(irt+ Owt+1 11 qXt = 0, i.e., 1^tE -Filu'(irt+i)w -t4+1 1 B 'I./i = P ^ cV) + 6 (C.99) q-X-t^til(i-t) Substituting (C.99) into (C.97) yields Et-FittiV- t+Owt A-Fill} = 0, i.e., --will + (p — 1 c' (e)) + 6^ { u'eirt)^ Et^ q Xt^tiVrt) , ,, — Etfu'(irt)w tA l + Et { uVrt)[P q-gtc (e)]} + 6 E L 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 noticing that 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) p — , cV)] + bEt Et+ ituVr t+ i)wi4+1 1 = 0. - { qA t Since Et_1Etfu'(71- t)tv iA l = EtEt+1lu / (irt+1)wiA+ 11, and 1 Et _ i Et u'(7"rt)[P { ^c'(e)]^= EtEt+i uVirt-Fibi ^ c' (0} qXt^ qXt-Fi we obtain 1 EtEt+ifu'Frt+i)wi4+1 1 = 1 _ 6 -r-t -Lt -I- 1 u'(it -1- 14 ^ ci (e)] qXt-1- 1 .^(C.101) Substituting (C.101) into (C.100) gives E t ttiVrt){1) — -j—cV)]} ^6 EtEt+i wt —^Et {uVrt)}^1 — 6 A { Vrt+i)[P 1 ^e t( , )-11 qxt+ i Et {u'(fr i )} (C.102) Appendix D Proposotions 7 and 8 D.1 Proof of Proposition 7 The net profit at equilibrium under the SQ system is i -+ 7rt pk(1 a)x 2c(e) > v if year t is a good year, N^ pk(1 — a)x^c , ^ 2^< v if year t is a bad year. 1 Hence we have uVrt) 1 — ,3 if year t is a good year 1 + /3 if year t is a bad year Et {11(7r t )} = 1 Xtl Et {uVrt4^ — qxt cV)]^= ( 1— cEO)p— (1— a)It• Substituting these into the general formulae in Proposition 5 and using ft — 1 ^( q(1-a)x."" \ ) 1 we get the desired results. D.2 Proof of Proposition 8 Let us suppose that at equlibrium, profit levels under the QQ system satisfy _7r t 7r" t+ < v < + 7‘rt-,^(t^2 ) + 7ri^<^< 7ri • 1 As in the text, we omit the seasonal setup costs 122 (D.103) (D.104) Appendix D. Proposotions 7 and 8^ 123 First we derive equilibrium quota prices based on this supposition. Then we verify that the above inequalities are indeed satisfied under these quota prices. (Remember that profit levels depend on quota prices.) Finally, it is easy to show that under the assumption p > [1 + 2(1 — (5)/6 2 ] fit there is no equilibrium at which the above inequalities , are violated. From (D.103), the marginal utility in year t at equilibrium is uVir t ) = 1 — 0 if year t — 1 is a good year 1 + 0 if year t — 1 is a bad year regardless of the harvest in year t (t > 2). Then 1 c, ^( 1 — ))[P — i Lii] if year t is a good year Et+1 tit'Ctd-i)[P^ qXt-1-1^(1 + OIP pi if year t is a bad year 1+a EtEt+ifuVrt+i)[1) 11 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 is Et {u / r t)(P —^(0)}^6 EtEt+i {72' r*ii-i)[P q x, + , i (0]} u'(7rt)^+ 1 — b^uVrt) (5^1 ,r 1 = (1 +1 —61—,3 ) LP 1+a it ^ll From (D.104), 1 — i3 if year 1 is a bad year 1 + ) if year 1 is a good year E 1 {u'( 71 1)} = 1 1^1 — a/3 El{u' eitd[p qxi^ cV)]} = p^1 + a tt - Appendix D. P•oposotions 7 and 8 ^ 124 Hence by (5.25) in Proposition 6 1 a3 p ^ 1-1 + ( 1+ a 1 — 6 (5 ^1+ A^1 - All 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 these quota prices and the assumption p > [1 + 2(1 — 6)16 2 ] p. We know that at equilibrium the net profit is pkXj _ i^A kXt _ i kx \^B kx k)(i\ and 2c(e)+ wt • ( ^ N^N)+wt .\^N^ pkXi 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 ( k(1 + a)x kx\^( kx k(1 + a)x + ^2c ( e)^+210^ N^N 4-wP + N^N a ft + (1 6) 46,A ]k A. ^+ 11 pkx N^L l + —a +7"r^ pk(1 + Similarly = p kx^ ^ 2c(e) — + (1 — kax N Therefore^< v < (b) Since .^kx kX 1 pkX 1 ^ ), 2c(e) + wr ( N^N N^ 71= ^ Appendix D. Proposotions 7 and 8 ^ 125 we have pk(1 — a)x^ kx k(1 — a)x 2c(e) + [p it + 11 +^ 3 ^ — 3 6 -01( N ^N^ N ) pkx^kax 1 + /3 ,. (^1 ^6 )1-^1 j kax ^2c(e)^ii ^ N ^1 _ ,13 ° 1 + 1 + /3 1 _ s LP 1+ a l =N^ l N *i = -- pkx^„^1— a kax 1— /3 (5 ( i + 1 ^S ) rp ^1 ^ikax it 7r—1^ 4- = ^ 2c (e)+ ^ N N^ 1 + ce P N^1 + 13^1 31 6 ) l^1+ a — — Then, by using xt 1^6 \^6 6 6(1 + and 1— 3^1+ 31— 6 1 '--- `-' \ ' -i- 16 1?- 2 1 ^6^ + 3^1^6 \ ,, 1 — /3 1^6 1^6 S 6(1 + 1 + /3^1 /31 — 6 1 ?" 21 — 6' — it is straightforward to verify that our assumption p — [1 + 2(1 — 6)/6 2 ] ,u > 0 implies 7rT — v > 0 and v — irl . > 0. (c) By definition we have Akax _kax ' t = pk(1 — a)x ^ 2c(e) + .01, +ii+ +Lvr N N^N pk(1 + a)x^_IpAkax^B+ kax _7r^t+ . ^ 2c(e) _Lo N N^ t N pk(1 — a)x B-kax ir 1^ - = ^ 2c(e)+ w N 1 N = pk(1 + a)x kax 2c(e) — w19+ N ^' — ^1 Since + 4- wf3- and _4+ wf3 + in Proposition 6, we get —^= +20 kax^,+^_Akax > 0, and 711 _7r t = > O. N Thus we have verified that the quota price formulae in Proposition 8 which are derived based on suppositions (D.103) and (D.104) are consistent with them. Suppose that inequalities different from (D.103) and (D.104) hold at equilibrium in some way or other. Proceeding as before, we can derive "equilibrium" quota prices Appendix D. Proposotions 7 and 8^ 126 corresponding to these inequalities instead of (D.103) and (D.101). Under the assumption p > [1 + 2(1 — 6)/6 2 ] Ec, however, it can be shown that all these "equilibrium" prices lead to contradictions to the assumed inequalities. We omit the proof which is straightforward but needs tedious calculations. Appendix E 1Q Q and G QQ We show that at equilibrium kx 1 (a) VQQ( 7v ) = (b) G QQ (kx) = I pkx 6 N 1^a 1 — 61+ 2c(e)1 + (0 — pkax^6 2 1— a^1^pkax 1 — 6 ) 1 + a^N + 6 (1 — 6) 2 1-32 (p 1 1+a p)k a x (a) It is easy to see that VQQ(kx/N) satisfies VQQ( kx^1^1 6 ) =^+^+ 4 _ dt(-1- 1r7)+u(1-i- iF )+ u(_ir7)+ u(_Ir t+)]. (E.105) — In the following we calculate the RHS of (E.105) explicitly. We know that pk(1 + a)x (e) wB+kax = ^ 2c 1 N N^ pk(1 — a)x kax 2c(e) + W B1 N ir 1 = + ^. Since^< v <^(see Appendix D), = (1 + 3)/ ^3v, = (1 —^+ Substituting the formulae for w+ and wT: in Proposition 8, we get = (1 +3) [ Pk(1 ; u(2r1) = (1 — „3)[ Pk(1 — N a)x a) 1 — a^1 —^ A) kax p+ /3 6 + w^7, (p^1 + a^+ — 32' 2c(e) 2c(e) + 127 A ) kaxl 4_ 3v. 1 + /3 p^_ 0 6 _w N ^ ^ 128 Appendix E. VQQ and GQQ^ Therefore 1^, -[u(ril") + u(irT)] 2 ^pkx kax 3 - kax 1^ 2c(e)+ +^[(1 /3) -CD A -^)) d-dJ A N • (E.106) ^1 + a N 2 Next we derive the expression for the second term of the RHS of (E.105). Since < v <^t ,^, we have - u(+mo7) = (1-)+9v u( = - + iev u( _irT) = (1 + 0) - 1rT - Ov u(_iriE) = - - - Hence ^_ =^0)[pk(1 N - a)x ^_ = (i 0) 1pk(1+ a)x ^+ u( + fr t+) A Lax^B _ kax Ov wt — 2c(e) + + w — N + N A kax^B+ kax w —^- + w t^/3v 2c(e) A N B _ kax ^_kax^ [a. a) _W A ^ 2c(e) u(_fr t ) = ( 1 + 0) pk(1 -Wt - 13 v N N kaxx^B+ kax [pk(1+ a)x ^_w 2c(e) (1 + 0) u(_irtf ) = -wt^0v• A N ^4 Therefore 1 ^1 - - (5 [u( + Ir )^u( + irrt)^u(_fr)] ' ^ ^ A f pkx^ , [(1 I)) -w 6 1 N^`' c6) 7)" - /3) +ol kax p. kax l +- [ (1 - ,3) + wr - + (1 +^- (1 - 0)+4 + - ( 1 + 0) - wt +] • 4^ L^ ^ ^ Appendix E. VQQ and GQQ^ 129 Since ( 1- ))^+ (1 + 0)^= 2(p - + 26(1 + /3) _20, and 1-(1 - 0) + 4+ - (1 + 0)_4+ = -2(p ^ + a it) 26 ( 1 - /3 ) +th A , it) we obain 1 6 41 - 6 = [n( + ii T)^ u(._ir'7)] - 6 ipkx 1 6 [N „ 2c(e — it keel 1 kax 6 [(1 + 0) _W A — (1 — 13)+ -Al ^N . (E.107) )1+ a N 2^ a Substituting (E.106) and (E.107) into the RHS of (E.105) yields the desired expression for VQQ(kx IN). (b) Next, we derive the expression for the authorities' expected revenue, GQQ(kx), where kx is the initial quota endowments. Using superscripts and subscripts in the same way as before, the authorities' revenue in a particular year is expressed as follows: + G tf = ( - — + W A + + w B +)kax,^_Gt+- = _F G7^( — 4.26 A - ( A + _wr+)kax, + 43- )kax,^_GT = ( _W A — _wr - )kax, G1^wBi+kcxx = _w B+ kax,= —w B-^B - kax — — + w t kax. At the beginning of year 1, the expected present value of the authorities' revenue is cc GQQ(kx) = E {E 6 i Gt t=1 =^{Gi} + 6 E1{E2(G2Ix1)} + 62E1{ E2[E3(G3IX2)IXI]} + • • • ^=^ + ^ Et_i{Et(CtIXt-i)}, 1 6 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 obtain 1 (E.108) 1 E 1 {G i } = - Gt + - GT 2^2 Appendix E. VQQ and GQ Q^ 130 —2( -- .11)13+^-1-Wr)kaX 3 ^iukax6(2 + 1 _) 6 61 0( 2 P 1 +1 a ii)kax 1 + a^ (E.109) and 1 -( + G+ + + GT +^+ _GT) 4 t 11 th^ ^.+Alkax +4+ +^1 _74+)kax + 4 - + _4 - )kax 2^ 4^ - -( 4 a^1^,3^1 pkax 6(1 + ) 2(p p)kax. (E.110) 1+a 1 6 1 /3 1+a — — By substituting (E.109) and (E.110) into the RHS of (E.108), we get the desired expression for GQQ(kx): GQC? (kx) 1^a^2^6 ^ ^ a ykax 6 [ i _ 6 + (1 _ 6)2 1 _ /32 (p 1 +1a p)kax. 61 + =1 — From the above calculations, it is easy to verify that the term concerning the quota adjustments in period A is — 1 67 . 2(— + CO + _zv A )kax, i.e., - 1 (P 1 4. a p)kax 6)2j 1 .±3 02^ [(1^ and the term concerning the quota adjustments in period B is 1 1 - 61+a pkax 6 { 2 3^1 ^ap)kax. 1 - 6 1 -I- The (absolute) effect of the former concerning period A adjustments is much larger than that of the latter for period B adjustments as we can see easily by substituting 6 = 0.9 into the expressions 0 202 and 127. ^ Appendix F Table 5.4 Since the procedure of the calculation is the same for all cases, we illustrate it with Case 1 in which 6 = 0.9, pl = 1.5, and a = 0.1. By substituting these parameter values into the expressions for GQQ, VQQ, and VsQ . we get ^GQQ^(0.091 - 5.841 VQ Q — VsQ^ 1 !3 ,32 )fikx . 17 .513 — 1) 11^N If 13 = 0.1, ^GQQ^—0.499pkx —0.333pkx [8] and VQQ — VsQ = 0.068 Pk = 0.045 Pkx [utility unit]. Remember that VQQ — 17sQ is measured in utility, not monetary, unit. Under the SQ system the net profit at equilibrium is in = pkXt IN — 2c(e). When a = 0.1, +^pkx ^_pkx 2c(e),^and_ ir t = 0.9 ^ N^2 ^7rt = 1.1 ^2c(e). N^ - By definition we have v = pkx/N — 2c(e). Suppose that the authorities pay = (1 — 0.9)0.045 131 pkx dollars Appendix F. Table 5.4^ 132 annually to each fisher under the SQ system. (In the following we show that this is just enough to compensate for each fisher's loss under the SQ system.) Then aftercompensation 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 is 1 - [(1 - 0)7^r't+ + )3v] + - [(1^)31 )/i-7 - Ov] 2 2 while after-compensation expected utility in year t is 2 [( 1 1 0)(irt + a) + /9 v] + -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 sum of utility under the SQ system by the amount 1 1 N 6a- = 0.045 p [utiliy unit], which is just enough to compensate for the fisher's utility loss under the SQ system VQQ - VSQ). Hence the constant annual compensation that makes fishers as well-off as under the QQ system is indeed a = 0.0045 pkx [$]. Then the authorities' total expenditure (discounted sum) for the compensation is 1 1—6 N a = 0.045pkx [s]. 1 Remember that there is no actual quota trade taking place at equilibrium under the SQ system with identical fishers with identical initial quota endowments^(1/N) • 100[%]). Appendix G Equilibrium conditions for the open SQ system The Bellman's equation is VsQ (h it = max Et { max {u(7r t ) -I-- 6V.sQ (h At+1 )] , e . 1 ,4,/f1^eP,ap,ip where the second maximization is subject to atetA ci xte r < + + r and ar t = Pq-Kte At — c(e t) + A (htA —a )^A It ^+ B Xt A pqXt e tB — c(e tB ) w t • (— a t — a tB ) — rn tB lr — h t+1 x = X ^a Let us denote equilibrium quantities by a hat. Then the following conditions have to be satisfied at equilibrium. uTirt)[pqXt — g(et)] — AtqXt = 0^ (G.111) u'^+ 6VsQ(iti4 1 )— x + At = 0 (0 < 4< +00)^(G.112) + t)(—mTh + A t^= 0 (—oc < lB < +0c)^(G.113) Et laVrt)[PqXt — (1 (0)] — A t qXt } = 0 (G.114) Et {z1 (irt)( — 4 +^ B, X t ) = 0 Et {11' (irt)( — T4 )^At} = 133 0 (0 < a tA < +co)^(G.115) (-00 < It < +0c)^(G.116) Appendix G. Equilibrium conditions for the open SQ system ^ 134 qQ^) = Et {u'( 7 t)wtA } ro B — rn B = 6w A x t+1 wt — m^E E {w t^tB X —t x (0.117) (G.118) (G.119) where X ;,B A , At t'+1 We assumed 0 < ai9 < -boo etc. (G.111)— (G.113) are from the second maximization problem, (G.114)—(G.116) are from the first maximization problem, and (G.117) is from the envelope theorem applied to the Bellman's equation. (G.118) and (G.119) are the no-arbitrage conditions in the quota market. From (G.111) At = WO- t) {p — 1 ',A B c et qXt Susbstituting this into (G.113) yields 771 t = p ^CI (0). qXt The quota rental price in period B is equal to the marginal profit of harvesting one more unit of fish.
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Individual transferable fishery quotas under uncertainty Kusuda, Hisafumi 1993
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Title | Individual transferable fishery quotas under uncertainty |
Creator |
Kusuda, Hisafumi |
Date Issued | 1993 |
Description | A model of a fishery with an uncertain fish stock is proposed to compare alternative management systems with individual transferable quotas (ITQs). Assumptions of the model include: (1) the fish stock fluctuates randomly year by year; (2) in-season stock depletion is small; (3) the total allowable catch (TAC) set by the quota authorities has a definite relation with the fish stock level; (4) the true value of the stock level is revealed only at the middle of each season, when the authorities revise the TAC; (5) fishers form rational expectations on future quota prices. The principal results are: (a) If fishers are risk-neutral, the share quota (SQ) system and the quantity quota (QQ) system generate the same amount of fishery rent, although the division of the rent between fishers and the authorities under one system is different from the other. If the TAC is proportional to the stock level, the more price-inelastic the demand for fish is, the more likely it is that fishers are better off under the QQ system at the expense of the authorities. (b) A quota tax and a harvest tax that collect the same amount of revenue for the authorities result in the same division of the fishery rent among heterogeneous fishers. The quota tax and the profit tax differ in this respect. Which fishers will prefer a quota tax over a profit tax will depend on fishers' shares of the initial quota endowment and in total inframarginal profits afterward. (c) If fishers are risk-averse, the SQ system and the QQ system are not equivalent in their allocative efficiency. An example shows that the SQ system is potentially better than the QQ system when fishers prefer the latter and the authorities prefer the former. This conclusion has to be modified if risk-neutral traders participate in the quota market |
Extent | 5203880 bytes |
Subject |
Fish stock assessment. Fishery management. |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2008-09-17 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0098845 |
URI | http://hdl.handle.net/2429/2181 |
Degree |
Doctor of Philosophy - PhD |
Program |
Economics |
Affiliation |
Arts, Faculty of Vancouver School of Economics |
Degree Grantor | University of British Columbia |
Graduation Date | 1993-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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