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Characteristics of limited entry fisheries and the option component of entry licenses.

Characteristics of Limited Entry Fisheries and the Option Component of Entry Licenses

Entry limitations are frequently proposed as a partial solution to the common pool problem that characterizes most fisheries. Entry limitations have actually been imposed, however, in only a few fisheries. (1) At first glance, this situation contradicts the self-interest hypothesis of regulation, because most proposals to limit entry would convey entry licenses to many existing fishermen at a nominal price. Johnson and Libecap (1982) attributed the apparent contradiction to fisherman heterogeneity and the costs of devising and enforcing contracts or regulations that would be supported by most fishermen. In a recent paper (Karpoff 1987), I argue that traditional regulations such as capital constraints and season closures are popular because they redistribute the catch toward the politically dominant fishing groups.

This paper directly examines conditions under which fishermen are likely to support entry restrictions. Entry restrictions generate positive (quasi-) rents precisely because they exclude or have the potential to exclude some fishermen. The probability of exclusion is positive for most fishermen, so advocacy of entry restrictions is not riskless. For many fishermen the risk of exclusion outweighs the expected gain from restricted competition. When fishermen do advocate entry restrictions, they do so when the expected net gain is positive.

One factor that affects the net gain is the variability of fishing income. This point is made by noting that limited entry licenses are similar to call options on future increases in revenue over cost and by developing an exact option-based model to value entry licenses. There are numerous potential applications of this model. It offers the important advantage of requiring information only on factors that can be easily estimated, and it yields comparative static results that can be used to assess the effects on license values of environmental or regulatory changes.

I. A PROBABILITY OF INCLUSION

THEORY

Consider a fishery in which entry restrictions can be adopted at [t.sub.0] that would affect the fishery for a single period [t.sub.1]. If entry is restricted some fishermen will be excluded, with the most likely outcome a reduction in aggregate fishing effort. The resulting positive external effect will increase returns to those remaining in the fishery. Assume that the decision on entry restrictions is an output of a regulatory process captured by the fishermen, who are the dominant special interest group. Entry restrictions therefore will be adopted only if they are supported by a majority of fishermen. For simplicity, also temporarily assume a zero discount rate.

In an open access equilibrium each individual i expects to earn a fishing income [pi.sub.i] at [t.sub.1]. (2) The individual cost of negotiating and enforcing an entry restriction is [C.sub.i]. To influence regulations, fishermen must attend hearings by the regulatory board, maintain contact with fishery managers and politicians, and research policy options. As Johnson and Libecap (1982) argue, the most significant cost may be that of organizing and communicating among the fishermen themselves. [C.sub.i] represents all of these costs. If fisherman i is one of those included in the entry limitation program, he earns an expected income [Mathematical Expressions Omitted] at [t.sub.1]. The probability [p.sub.i] of being included is less than one, and the probability is 1 - [p.sub.i] that this fisherman will be left to earn his opportunity income [I.sub.i], where [I.sub.i] [is less than] [pi.sub.i] is the condition for current fishing employment. (3)

The prospect of limited entry therefore has a risky payoff [G.sub.i] with expected value

[Mathematical Expressions Omitted] [1]

A risk-neutral fisherman will support a limited entry plan if and only if

E([G.sub.i]) [is greater than] [pi.sub.i], [2]

or equivalently,

[Mathematical Expressions Omitted] [3]

The right-hand side of equation [3] is the sum of the fisherman's income if excluded from the fishery ([I.sub.i]) and a premium that depends on the loss in income if excluded ([pi.sub.i] - [I.sub.i]), the cost of organizing ([C.sub.i]), and a weight (1/[p.sub.i]) that increases with the probability of being excluded. This implies that a supporter of entry limitations expects an income conditional upon inclusion in the fishery that offsets the cost of organizing and the chance of not being included. (4)

The condition for a majority of fishermen to support entry restrictions is that the condition in equation [3] holds for 50% + 1 of them. Other things constant, the likelihood an individual fisherman will support entry restrictions increases with each of the following:

(i) a decrease in the cost of negotiating and organizing support for the restrictions;

(ii) an increase in the probability the fisherman will be included;

(iii) a decrease in the spread between the fisherman's expected open access fishing income and his opportunity income; and

(iv) an increase in the fisherman's conditional expected income if he is included in the limited fishery.

Johnson and Libecap (1982) discuss the effects of costly contracting among fishermen (item (i)). Characteristics of fishermen that decrease this cost, say, a dominant group of homogeneous fishermen, would increase the likelihood of entry limitations.

Anecdotal evidence provides limited support for two of the other propositions. Item (ii) implies that efforts to restrict entry will be couched in terms that assure most fishermen they will be included, e.g., through grandfather clauses. By itself, however, this has limited appeal, since the objective is not only to prevent new entrants but also to force some fishermen to exit. One way to accomplish this without the risk of self-exclusion is to target some other groups for restriction. This describes exactly a number of attempts to limit entry in the Texas shrimp, Alaska salmon, and British Columbia salmon fisheries, as described by Johnson and Libecap (1982), Morehouse and Hession (1972), and Morehouse and Rogers (1980). In each case, attempts to restrict entry focused on political minority or "outsider" groups, including fishermen from other states, immigrant Vietnamese fishermen, new entrants into the fishery, and operators of various types of gear. Each case can be interpreted as an attempt among advocates of entry restrictions to increase their probability of including ([p.sub.i]) at the expense of individuals in an easily identifiable outside group.

Item (iii) indicates that efforts to restrict entry will intensify when expected fishing incomes under continued open access are low. As fishing incomes are driven to opportunity incomes, each fisherman has less to lose if he is excluded from the fishery. This prediction is also consistent with actual histories. Morehouse and Rogers (1980, 75-6) report that two unsuccessful attempts to limit entry in Alaska salmon fisheries in 1962 and 1968 were motivated by severe decreases in yields. Entry limitations were eventually imposed in 1973, after the early 1970s produced the lowest annual harvests in the history of the Alaska salmon fisheries (except for 1967, the year immediately preceding the unsuccessful 1968 attempt). (5)

II. THE OPTION COMPONENT OF

ENTRY LIMITATIONS

Item (iv) implies that support among fishermen for entry restrictions will be greater, ceteris paribus, the larger the conditional expected incomes of the included fishermen. In many cases, entry limitations do not significantly decrease the number of existing fishermen, and therefore do not immediately change the zero profit equilibrium. This might at first seem to imply that conditional expected profits ([Mathematical Expressions Omitted]) are approximately equal to expected open access profits ([pi.sub.i]). Even if they do not roll back the number of current fishermen, however, entry restrictions do limit the number of new entrants. Ownership of a limited license therefore provides the opportunity to exploit the fishery without additional competition from new entrants if the fishery becomes profitable.

This is similar to holding a European call option on future increases in revenues over costs. The licensee is not obligated to fish if it is not profitable, and can effectively let the option to fish expire. But if the profit margin is positive in the future fishing period, the fisherman exercises his option by fishing. The option component of a limited license is valuable because additional entrants are prohibited from entering and driving profits to zero.

A. A License Valuation Model

The analogy between limited licenses and call options is useful because it yields insights into the importance of several factors that affect the values of limited licenses. The analogy is developed here by specifying a set of conditions under which it is possible to derive an exact call option valuation model for a limited entry license. Assume the following:

1. Entry restrictions are imposed at [t.sub.0]. Limited entry licenses are good only for the next fishing season at [t.sub.1].

2. Production is characterized by fixed input proportions, e.g., two fishermen, one boat, and effort and costs per vessel are fixed through [t.sub.1]. The total cost per vessel is X. After entry restrictions are imposed, aggregate effort and aggregate cost can be changed only through the exit of vessels.

3. All economic agents are risk neutral, and the riskless continuous rate of return is r.

4. The return to each vessel at [t.sub.i], i = 0, 1, is [R.sub.i] = [P.sub.i][H.sub.i]/[Q.sub.0], where [P.sub.i] is the price per unit output, [H.sub.i] is the fishery-wide harvest, and [Q.sub.0] is the number of vessels at [t.sub.0]. The relative change in R during the time between [t.sub.0] and [t.sub.1] follows a stationary stochastic process with instantaneous mean [rho] and variance [sigma.sup.2]:

dR/R = [rho]dt + [sigma]dz. [4]

In equation [4], dz is a basic Wiener process driven by [epsilon], a unit normal random variable:

dz = [epsilon][(dt).sup.1/2].

Assumption 1 assures that the entry license consists of a single option. Assumption 2 limits uncertainty about fishing profitability to aggregate fishing revenues. By treating all vessels as homogeneous, assumption 2 also ignores rents earned by inframarginal vessels so that all fishermen have the same demand-price for a license. Assumptions 3 and 4 are sufficient for the license valuation model to mimic the Black-Scholes (1973) option pricing model. (6) Assumption 3 is clearly not descriptively accurate, although it can be relaxed (as in section II.B) without changing the most important implications of the model. Assumption 4 implies that [R.sub.1] is distributed lognormally, which in turn has the desirable property that the probability of negative fishing revenues is zero. Equation [4] also allows for either an expected increase or an expected decrease in aggregate fishing revenues.

Refer to the current call option value of a license as [v.sub.0]. If return per vessel at the fixed level of effort is greater than X at [t.sub.1], the license to fish has positive value and will be used. If, on the other hand, the return per vessel at [t.sub.1] is less than X, vessels will exit the fishery until profits are zero. The value of the license at [t.sub.1] is therefore

[v.sub.1] = max(O,[R.sub.1] - X). [5]

This is a primary characteristic of an option contract: returns are non-negative in all states of the world and positive in at least one state. The current value of the license is its discounted expected value,

[v.sub.0] = exp[ -r([t.sub.1] - [t.sub.0])]E([v.sub.1]) = exp[ -r([t.sub.1] - [t.sub.0])]E(max(0,[R.sub.1] - X)). [6]

Noting again that assumption 4 implies that [R.sub.1] is distributed lognormally,

[Mathematical Expression Omitted] [7]

where L'([R.sub.1]) is the lognormal density function. If it is further assumed that [rho] = r, the solution to equation [7] is identical to the Black-Scholes option pricing formula:

[Mathematical Expression Omitted] [8]

where N([.]) is the normal cumulative distribution function and h = {1n([R.sub.0]/X) + (r + [sigma.sup.2]/2)([t.sub.1] - [t.sub.0])}/[sigma][([t.sub.1] - [t.sub.0]).sup.1/2]].

Equation [8] is derived in the Appendix. It expresses the value of a license as the difference between two terms: the present value of next period's expected fishing revenues and the present value of next period's expected fishing costs, where both expectations are conditioned on [R.sub.1][is greater than] X.

Equation [8] identifies factors that affect the prospective value of entry limitations to fishermen who expect to receive grandfather rights. Most importantly, and as shown in the Appendix, [theta][v.sub.0]/[theta sigma][is greater than] O, so the self-interest theory of regulation predicts that, ceteris paribus, fisheries characterized by a relatively high variance in revenues are more likely to have entry limitations imposed. (7)

B. Extensions of the License Valuation Model

The analogy between entry licenses and options is not limited to the restrictive assumptions used to derive equation [8], and still applies under less restrictive conditions. For example, and unlike assumption 1, most entry limitation programs have indefinite length. In these cases the license consists of a portfolio of (European) options, each of which conveys the right to fish in a given year. When a limited license is sold, its value reflects the sum of the remaining options' values.

Assumption 2 also does not accurately describe most fisheries, as fishermen can expand their effort along many input dimensions. In this case, potential competition among the limited number of vessels will reduce the value of a limited entry license; the call becomes written on the difference between future revenues and costs, each conditional on expected input levels at different levels of profitability. The value of the license will be positively related to the cost of substituting effort per vessel for the number of vessels. Entry limitations are therefore more valuable and more likely to be imposed where this cost of substitution is high.

Assumption 3 can also be relaxed by allowing risk aversion and by placing no constraint on [rho]. As shown in the Appendix, the value of the license then becomes [v.sub.0] = [R.sub.0]exp{([rho] - [beta])([t.sub.1] - [t.sub.0])}N(h') - exp{-[beta]([t.sub.1]-[t.sub.0])}XN(h' -[sigma]([t.sub.1] - [t.sub.0])), [9]

where h' = {1n([R.sub.0]/X) + ([rho] + [sigma.sub.2]/2)([t.sub.1] - [t.sub.0])}/[[sigma]([t.sub.1] - [t.sub.0]).sup.1/2]] and [beta] is the discount rate adjusted for the risk of fishing income. Although more general than equation [8], equation [9] may in practice be less tractable because it requires knowledge of the expected rate of increase in fishing revenues and the risk-adjusted discount rate appropriate for fishing income.

III. CONCLUSIONS: APPLICATIONS OF

THE LICENSE VALUATION MODEL

When combined with the self-interest theory of regulation, the probability of inclusion theory yields several testable predictions. Most importantly, limited entry fisheries are characterized by large variances in net fishing revenues and low (expected) fishing rents prior to the adoption of entry restrictions. Attempts to limit fishing will likely target political minority groups for exclusion and will focus on fisheries in which the cost of substituting effort per vessel for the number of vessels is high.

This paper also develops the analogy between entry licenses and call options into an exact valuation model for entry licenses. Beyond the public choice issues emphasized here, the license valuation model may be useful in many other applications. For example, the model clearly implies that limited entry license values are positive even when current and expected future fishing rents are zero at the current number of vessels. This is because the fishing license represents an option, but not an obligation, to fish. The payoff to the license is never negative, and is positive if fishing rents turn out to be positive.

The model may be easily applied because the value of the license depends on five factors (using the simpler version of the model in equation [8]) that can be estimated with currently available data: current fishing revenues ([R.sub.0]), next period's costs (X), the instantaneous variance of revenues ([sigma.sup.2]), the time to expiration ([t.sub.1] - [t.sub.0]), and the interest rate (r). It is not necessary to forecast future fishing incomes or make assumptions about how fishermen make such forecasts, as is common in empirical research on license values (e.g., Karpoff 1984). The model may be particularly useful when market data on license values are not available.

The model may also have widespread applications because it yields unambiguous comparative static results. As shown in the Appendix, [delta][v.sub.0]/[delta][R.sub.0][is greater than]O, [delta][v.sub.0]/[delta sigma.sup.2] [is greater than]O, [delta][v.sub.0]/[delta]([t.sub.1] - [t.sub.0])[is greater than]O, and [delta] [v.sub.0]/[delta]r[is greater than]O. (8). The value of the license increase with current fishing revenues ([R.sub.0]) and decreases with expected future (and in this model, current) fishing costs (X). The value of the license increases with the variance of fishing revenues ([sigma.sup.2]) because the value of restricting new entrants is high if next period's revenues are very high, but cannot be less than zero if next period's revenues are very low. The license value increases with the interest rate (r) and the time between fishing seasons (T). Increases in both r and T decrease the present value of next season's fishing costs, and an increase in T increases the chance of an extreme revenue outcome.

Most environmental or regulatory changes affect at least one of the underlying factors. For example, individual catch limits decrease [sigma.sup.2] while gear constraints increase X. The model may be useful in measuring the direction and size of the resulting changes in license value.

The analogy between limited entry licenses and options is but one example of a broader proposition that has other applications. Many economic regulations create rents that are similar to options. For example, an agricultural quota (or land allotment) program effectively gives a quota holder a call option on the price of the agricultural good. The same is true of price ceilings: if the market-determined price of the good rises above the ceiling level, the option can be exercised by the buyer of the good so that the net outlay never exceeds the ceiling price. If the market price falls below the ceiling level, the option is not exercised, i.e., the buyer just pays the market price. Price floors are similar, only in this case the seller of the good effectively holds a put option. If the market-determined price falls below the floor level, the put becomes in the money and can be exercised profitably so that the seller nets no less than the floor price. [9] Viewed this way, beneficiaries of economic regulations hold a hedged position: they are long in the market-determined price of the particular good, but also hold an option that exactly offsets the negative consequences of a price movement beyond the stipulated level.

APPENDIX

PROOF OF EQUATIONS [8] AND [9]

For convenience, define T = [t.sub.1] - [t.sub.0]. T is the time interval until the fishing season at [t.sub.1]. Equation [4] in the text implies that [R.sub.1]/[R.sub.0] is distributed lognormally:

[Mathematical Expression Omitted]

where [mu] = p - [sigma.sup.2]/2 is the mean of the lognormal distribution. (Equivalently, [R.sub.1] is distributed lognormally with mean (In [R.sub.0] + [mu]T) and variance [sigma.sup.2]T.) In turn, this implies

[Mathematical Expression Omitted]

where [epsilon] is a unit normal random variable.

Equation [7] in the text defines the current call option value of the fishing license and can be rewritten as:

[Mathematical Expression Omitted]

Equation [8] will be proved by solving for [W.sub.1] and [W.sub.2] separately. The proof follows a similar proof in Jarrows and Rudd (1983, ch. 7). Solving for [W.sub.1]:

[Mathematical Expression Omitted]

where I([R.sub.1] > X) is an indicator variable,

[Mathematical Expression Omitted]

and E[.] is the expectations operator. Substituting from equation [A1],

[Mathematical Expression Omitted]

where a = [In (X/[R.sub.0]) - [mu]T]/[sigma] [square root of T]]. Substituting again from [A1],

[Mathematical Expression Omitted]

Completing the square,

[Mathematical Expression Omitted]

Noting that [mu]T + (1/2) [sigma.sup.2]T = pT, and that, by assumption, r = p

[Mathematical Expression Omitted]

where [a.sup.'] = a - [sigma] [square root of T].

The integral in [A4] is one minus the cumulative distribution function (c.d.f.) of a unit normal random variable, so [W.sub.1] = [R.sub.0] [1 - N([a.sub.'])], where N(.) is the unit normal c.d.f.

By symmetry of the unit normal c.d.f., [A5] implies [W.sub.1] = [R.sub.0]N(-[a.sub.']) [Mathematical Expression Omitted] Finally, [mu] = p - [sigma.sup.2]/2 = r - [sigma.sup.2]/2, so [Mathematical Expression Omitted] where h is defined in the text.

Solving for [W.sub.2]:

Note that [W.sub.2] = [Xe.sup.-rT] Pr{[R.sub.1] > X}, where Pr{.} is the probability operator. Karpoff, Limited Entry Fisheries [Mathematical Expression Omitted] By symmetry in the unit normal c.d.f., [A7] implies [Mathematical Expression Omitted]

Substituting [A6] and [A8] into [A2] yields equation [8]. Equation [9] is derived by noting that, in the absence of risk neutrality, the appropriate discount rate in equation [7] is one that accounts for risk, and by not substituting r for p to obtain equations [A3] and [A6]. The comparative statics results, standard in option pricing theory, are derived by partial differentiation: [Mathematical Expression Omitted]

References

Black, Fisher, and Myron Scholes, 1973. "The

Pricing of Options and Corporate Liabilities."

Journal of Political Economy 81 (May/June):

637-54. Jarrow, Robert A., and Andrew Rudd. 1983. Option

Pricing. Homewood, IL: Richard D. Irwin. Johnson, Ronald D., and Gary D. Libecap.

1982. "Contracting Problems and Regulation:

The Case of the Fishery." American Economic

Review 72 (Dec): 1005-22. Karpoff, Jonathan M. 1987. "Sub-Optimal Controls

in Common Resource Management: The

Case of the Fishery." Journal of Political

Economy 95 (Mar.): 179-94. -. 1984. "Insights from the Markets for Limited Entry Permits in Alaska." Canadian

Journal of Fisheries and Aquatic Sciences 41

(Aug.): 1160-66. Morehouse, Thomas A., and Jack Hession.

1972. "Politics and Management: The Problem

of Limited Entry." In Alaska Fisheries

Policy, eds. Arlon R. Tussing, Thomas A.

Morehouse, and James D. Babb, Jr. Fairbanks,

AK: Institute for Social, Economic,

and Government Research. Morehouse, Thomas A., and George W., Rogers.

1980. Limited Entry in the Alaska and British

Columbia Salmon Fisheries. Anchorage: Institute

for Social and Economic Research,

University of Alaska. Rettig, R. Bruce. 1987. "Overview." In Fishery

Access Control Programs Worldwide: Proceedings

of the Workshop on Management

Options for the North Pacific Longline

Fisheries, ed. Nina Mollett. Fairbanks: University

of Alaska Sea Grant.

Karpoff is associate professor of finance, School of Business Administration, University of Washington, Seattle.

This research was supported by NOAA grant number NA84AA-D-00011, Washington Sea Grant number 62-8553. I thank two anonymous referees for helpful comments and suggestions.

(1) See Rettig (1987) for a description of entry limitations and the fisheries in which they have been imposed.

(2) "Income" is defined as the sum of wage income and rent, if any. Heterogeneity across fishermen implies that [Mathematical Expression Omitted] need not equal [Mathematical Expression Omitted] for any [is not equal to] j, although homogeneity is imposed in section II below.

(3) If fishing licenses are transferable, the excluded fisherman can buy back into the fishery. The license price will equal the (discounted) value of the marginal fisherman's rent, [Mathematical Expression Omitted], and i's opportunity income is max [Mathematical Expression Omitted].

(4) Two possible considerations are whether [p.sub.i] can be influenced (by, say, active support for limited entry regulations or premature entry into a fishery on speculation that the fishery will be limited), and whether the assumption of risk-neutral fishermen is true, especially if fishermen's total human and non-human wealth is not well diversified. These extensions are not pursued here.

(5) See The Fishermen's News Pacific Fisheries Review, Richard H. Phillips Co., Seattle, Vol. 40, No. 5, February 1984, p. 57. Similar circumstances can be cited for the British Columbia case: "At the time limited entry was introduced, fish prices were relatively low and stagnant, and prospects for the 1969 salmon runs in particular were the poorest in several years" (Morehouse and Rogers 1980, 98).

(6) Given assumption 2, assumption 4 is necessary to obtain the Black-Scholes result. Assumption 3, however, is sufficient but not necessary. An alternative sufficient assumption is that fishing revenue can be costlessly and perfectly hedged.

(7) Once again, this is consistent with anecdotal evidence. Consider a typical official forecast for one of the Alaska salmon fisheries, which have been limited since 1974: "The forecast harvest of sockeye [salmon] in Bristol Bay in 1984 is 15 million, however, it might be as low as 4 million, or as high as 27 million." (Pacific Fishing 1984 Yearbook, Pacific Fishing Partnership, Seattle, 1984, p. 59.) The relation between harvest rates and profits varies across fisheries, and I do not have data on forecast harvests in other fisheries, but such lack of forecast precision illustrates the high variability of harvest rates in this fishery.

(8) There is no contradiction between the finding that [Mathematical Expression Omitted] and the earlier assertion that fishermen will tend to support entry restrictions when their expected open access fishing incomes are low. These findings, imply, respectively, that fishermen will support entry limitations if the expected benefit is large and if the opportunity cost is low. Also, the outcome that [Mathematical Expression Omitted] is the result of the somewhat artificial assumption that p = r: as r increases, so does the instantaneous expected return to fishing, p. Note that, in equation [9], the sign of [Mathematical Expression Omitted] is ambiguous.

(9) These examples must be qualified. The option represented by the price ceiling is less valuable if there is a positive probability the buyer will be unable to purchase at the ceiling price. If the price ceiling is binding not all demands will be met, and some buyers' options, while in the money, will be worthless (and the values of other options may be competed away via other allocation mechanisms). A similar argument applies to the case of price floors. In the case of some price floors, however, the government stands as a buyer of last resort and assures that all options can be exercised.
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