5.1 The need for an input-oriented measure

5.2 Fisheries-specific issues in the measurement of capacity and capacity utilization

5.3 Empirical approaches for assessing capacity

Primal measures of capacity and capacity utilization focus exclusively on output levels. However, in fisheries - and particularly in “capacity reduction” programs - the need is for information on capacity and capacity utilization which are based on output levels, but expressed in terms of effort or inputs. Nations downsizing their fishing fleets need to know the levels of capital stock and inputs which should be reduced to achieve their goals and objectives. Moreover, nations face multiple capital goods (heterogeneous capital), and need a means to reduce this entire capital stock to a single measure in physical units. There appears a need to develop standardized units of capital which equate to potential output levels. For example, removing a 50 GRT vessel with 300 horsepower reduces potential catch by 5,000 metric tons.

Dual-based economic measures can fill this need by directly providing the measure of capacity and capacity utilization and the corresponding optimal capital stock (in physical units) in a fishery. Catches can be freely chosen or subject to total allowable catches. However, the approach requires extensive data, which are usually unavailable.

In this next section, we present several empirical approaches for assessing capacity. These approaches can be classified in several ways. First, the approaches can be either primal or economic, the latter with an explicit economics optimizing basis. Second, the approaches can take output levels as exogenously fixed, such as with total allowable catches, or freely chosen, and then determine the corresponding appropriate level of capital stock. The approaches can alternatively take the flows of variable inputs and the stock of capital (including number of operating units) as given and then determine the corresponding level of maximum output (i.e. how much could be produced if all operating units were technical efficient). Because both capacity and capital utilization are short-run in nature, all classifications take the stocks of the resource and capital as given, where the latter is explicitly specified as a stock and not a flow of services.

Determining the maximum output possible given inputs and the resource stock begs the question of what should be the optimal number of operating units, input levels, and configuration of any given fleet. For example, the literature on fisheries repeatedly stresses that such a structure leads to overcapitalization-too many resources chasing too few fish or production is wasteful or not at minimum cost or society is not receiving the maximum net benefit from the fishery. Thus, to say that a fishery is overcapitalized or has “excess harvesting capacity” has no meaning unless there is some desired level of output. Alternatively, the potential maximum level of output given input levels and number of operating units can be determined and compared to observed landings or production; a simple comparison will allow the nature of capital utilization or “excess harvesting capacity” to be easily determined.

We also present, in Appendix XI, a wide variety of examples of using different approaches to calculate different measures of capacity and capacity utilization in fisheries. In addition to presenting the different approaches and measures, we also provide empirical analysis of the various measures of capacity and capacity utilization given the types of data often available on fisheries. These empirical examples contained in Appendix XI may be quite useful to nations having various data limitations and desiring to develop measures of capacity and capacity utilization.

The general industry and fisheries cases discussed above refer to production capacity in mostly physical terms. That is, given resources, what should the maximum output be and how close is the industry to producing the maximum output. Given input and output prices, it may make complete sense for a firm or industry to produce at less than the total maximum physical output. Alternatively, a physical measure of capacity may suggest over- or undercapitalization while an economic measure may suggest that a firm is operating at full capacity. The distinction is thus critical for assessing capital utilization and overcapitalization in fisheries.

In the case of fisheries, however, there are reasons for assessing capacity output and capital and capacity utilization relative to excess “harvesting capacity” rather than simply relative to short and long-run investment decisions as is the case for assessing CU in conventional industries. Fish stocks may be sustained at certain levels. Harvesting in excess of given levels can cause serious declines in resource levels. Moreover, production or harvesting activities in fisheries are subject to technological externalities (i.e., the catch of one vessel reduces the catch and raises the harvesting cost of another vessel). Simply put, the natural resource-the fish stock-imposes limits on the possible catch. Allowing capital utilization or “harvesting capacity” to be in excess of the level necessary to efficiently harvest the resource generates considerable economic waste and may easily allow overharvesting of the resource.

There are numerous other aspects to also consider when developing measures of capacity in fisheries. As illustrated in Appendix X, for example, larger vessels than apparently necessary for harvesting activities could be constructed to enhance skippers' flexibility to deal with adverse weather. Thus, any analysis of capacity and capacity utilization based on strictly a static and certain analysis and either vessel attributes or total capital might suggest overcapitalization. Alternatively, the capital stock may appear unnecessarily large because vessel owners added considerable amenities to the vessel to make crew more comfortable.

In addition, there is the issue of how to assess capacity, input or output-based, relative to spatial aggregation and resource availability. Many fisheries of the world involve more than one geographic area (e.g., cod are harvested from the Gulf of Maine, Georges Bank, Southern New England, and occasionally more southerly areas; similarly, sea scallops are harvested on Georges Bank and several Mid-Atlantic resource areas). Conventional economic theory and principles offers little guidance about the level of spatial aggregation and the examination of capacity and CU.

It is important that capacity be related to the resource level, particularly if managers are interested in capacity and CU because of management and regulatory concerns. Even in the absence of regulatory concerns, any conclusions regarding capacity and CU must clearly distinguish between those related to controllable factors versus those derived from resource levels. For example, a CU < 1 implies overcapitalization or that observed output divided by maximum output is less than one; what if the resource was quite low during the period that observed output was less than maximum output? Random events such as a hurricane could have caused the resource to become unavailable during part of the year; subsequently, observed catch could be quite a bit lower than maximum catch.

5.3.1 Peak-to-peak

5.3.2 Factor requirements function

5.3.3 The frontier approach

5.3.4 Dual economic approach

5.3.5 The data envelopment analysis (DEA) approach

5.3.6 Maximum potential effort

Since managers and policy makers appear to be primarily concerned about physical capacity and overcapitalization (e.g., level of inputs relative to catch), then measures of capacity, regardless of input- or output-based or physical or economic, should ideally convey information about catch, fishing mortality, costs, industry and fleet structure (e.g., number of large vs. small boats), employment, and profits. Such measures of capacity should also ideally convey information about net benefits to society, provided policy is at least somewhat concerned with net benefits to society.

From a practical perspective, the measurement and assessment of productive capacity and capital utilization or “harvesting capacity” could proceed along several lines of thought. The levels of capital, labour, energy, materials, other inputs, and catch could be determined in a dynamic setting which maximizes net benefits to society subject to biological constraints and the underlying form of the technology. Such an approach would allow policy makers to compare optimal levels to actual levels and subsequently assess the necessary reductions. This is an appropriate approach, but one which would likely be severely limited by inadequate data, extreme uncertainty associated with resource conditions and the technology, and various goals and objectives of resource management. It also begs the questions of a steady-state solution vs. a bang-bang solution, determination of the appropriate social rate of discount, the nature of industry structure in response to capacity reduction programs, and the possible need for a cautious approach which ensures no excess harvesting.

Given the typical availability of data and the usual concerns of management agencies, it is useful to consider capacity and utilization in terms of productive capability. A physical approach allows resource managers to assess the potential maximum harvest level of a fleet-with and without externalities, the level of productive capacity or “excess harvesting capability,” input utilization, and the redundancy of capital. With additional analysis and vessel-level data, the physical measure can also be further considered to determine which vessels and level of inputs are unnecessary for a given harvest level.

In this section, alternative approaches to measuring capacity and capacity utilization are considered and offered as viable measures of capacity and CU: (1) peak-to-peak; (2) factor requirements function when there are total allowable catch (TAC) limits or a revenue function when outputs are unconstrained and freely chosen; (3) frontier production function and output; (4) dual economic based; (5) data envelopment analysis and frontier; and (6), maximum potential effort based on ideal, empirical and practical, and fishing power or fixed effects. The applicability of these measures in large part hinges upon the availability of data, especially cost data for variable inputs and a capital rental or services price, and degree of technical sophistication.

Of the various approaches for assessing capacity and capacity utilization, the peak-to-peak method has the widest applicability since its data requirements are most parsimonious. The peak-to-peak method does not require cost data and can be calculated with the broad types of data collected world-wide by FAO, such as the catch and number of vessels demonstrated in Garcia and Newton.

The peak-to-peak method is based on a “trend through peaks” approach that is thought to reflect maximum attainable output given the stocks of capital and fish. Peaks in production per unit of capital stock are used as indicators of full capacity and linearly interpolated capital-output ratios between peak years are then employed, along with data on the capital stock, to estimate capacity output for between-peak years (Morrison, 1985). The most recent year estimates of capacity output are obtained by extrapolating the most recent output-capital ratio peak and multiplying by an appropriate series on the capital stock. Combined fishery capacity is measured as the revenue-share weighted sum of detailed fisheries measures of capacity and capacity utilization. In this case, the level of technology in a particular time period is determined by the average rate of change in productivity between peak years. The output-capital ratios can be adjusted by a technological trend for technical progress. The utilization rate is subsequently calculated as the ratio of observed to potential output Appendix VII provides additional discussion.

While the peak-to-peak approach is the most widely applicable and least demanding of data of all the methods for examining capacity utilization, it is also quite limited. For one thing, it completely ignores the biological characteristics of fish (e.g., the concept of maximum sustainable yield). It also fails to directly link back to input utilization-variable factors and capital, which is a major focal point for assessing capacity and capacity utilization in fisheries.

There are some possible approaches for mitigating the limitations of the peak-to-peak approach discussed in Appendix VII. A sustainable yield function could be estimated. The maximum sustainable yield could be estimated and compared to observed harvest levels over time to estimate capacity and capacity utilization. Alternatively, fishery independent data could be used to estimate maximum yields. The level of effort corresponding to the maximum sustainable yield could subsequently be calculated or estimated; the estimated or calculated level of effort would be the level of effort associated with capacity output and capacity utilization.

The need to modify the peak-to-peak approach really becomes a question of what information management desires. If management desires information on actual capacity and capacity utilization in the strict economic sense (i.e., what is the potential harvest given the size of the fleet and the potential utilization of inputs in the absence of resource constraints), the peak-to-peak approach provides relatively useful, but limited, information for determining the optimum utilization of fishery resources.

The factor requirements function can assess the minimal stock of capital or effort required to produce total allowable catches. The approach also gives the utilization of the capital stock or effort. As with the peak-to-peak method, the factor requirements function is a primal or physical approach. The minimum data requirements are capital stock and other physical quantities of inputs and exogenously determined output levels. This approach may be most promising at the industry or fishery level using aggregate data. A sufficient number of observations would be necessary for satisfactory statistical estimation.

The factor requirements function is one way to describe the technology subject to a fixed or quasi-fixed input (e.g., the vessel). The factor requirements function depicts the production possibilities set and relates the minimal amount of an input required to produce a vector of outputs:

Z = g(Ywhere Z is the fixed input and Y_{1},Y_{2},...,Y_{M})

The estimated factor requirements function gives the minimum input bundle (effort) or stock of capital Z required to harvest the fixed outputs or total allowable catches Y. The ratio of estimated Z to actual Z would give a measure of capital utilization as defined by Berndt (1990), i.e. the ratio of the desired stock of capital or effort to the actual stock.

Yet another variant is to estimate the factor requirements function as a stochastic frontier using the two-stage routine of Battesse and Coelli (which allows incorporation of variable and fixed inputs other than vessel size, ages of the capital stock to capture vintage, socio-economic information, and other variables), and calculate an input-oriented measure of technical efficiency. The ratio of minimum possible Z to actual Z would again give a measure of capital utilization as defined by the ratio of the desired stock of capital to the actual stock of capital.

When outputs are endogenous, i.e. freely chosen, an alternative to obtaining information on the factor requirements function is to estimate a dual revenue function; alternative options for obtaining information on the factor requirements function include estimating dual variable and profit functions in which the vessel capital stock is fixed and solving for capital in terms of input prices and output levels (dual cost function) or input and output prices (dual profit function). With any of the dual functions, it is possible to determine the minimal amount of capital stock or fixed factor required to produce a given vector of outputs.

A second primal or physical measure of capacity and capacity
output, an alternative to the peak-to-peak approach, is the frontier approach.
With the frontier approach, the maximum output possible (i.e. the capacity
output Y^{*}) given input levels is estimated. Capacity utilization is
the ratio of observed output to maximum potential output: Y/Y^{*}. The
frontier may be estimated at the firm level or for the fleet. The more
aggregation, the less precision one obtains. The basic data requirements are
observations on physical levels of output, variable inputs, capital, resource
abundance levels if available (dummy variables can be used also), and any
available information on capital vintage, fleet size, socio-economic
characteristics, etc.

There are two basic options to consider: (1) nonparametric
determined frontier, and (2) short-run stochastic production
frontier^{7}. Since the two approaches provide the same type of
information and programs are readily available to estimate the stochastic
frontier, we focus additional attention on using the stochastic production
frontier to determine capacity and capacity utilization. With some modification,
we may estimate the frontier and assess the relationship between maximum output
and input levels. A particularly useful modification is that developed by
Battese and Coelli (1993) in which an error term for technical inefficiency, U,
is expressed as a function of variables which might influence inefficiency.
Because capacity and capacity utilization are inherently short-run concepts,
capital should be specified as a stock rather than as a flow of
services.

The stochastic production frontier relates maximum output to inputs while using two error terms. One error term is the traditional normal error term in which the mean is zero and the variance is constant. The other error term represents technical inefficiency, that is, deviations from the best-practice frontier (which is the maximum output possible given the inputs). When the technical inefficiency error term is 0, maximum output is produced given inputs and the resource stock. When the technical inefficiency error is greater than 0, maximum output is not obtained. Technical efficiency is estimated via maximum likelihood of the production function subject to the two error terms. Appendix XIV gives additional discussion.

The primal capacity output equals the frontier output for which the technical inefficiency error term equals 0.0. A primal-based measure of capacity utilization may then be determined by calculating the ratio of observed output to the frontier output, either for the individual firm or for the industry. The industry capacity output simple equals the sum of the frontier outputs; the industry capacity utilization equals the sum of observed outputs divided by the industry capacity output.

Although both the nonparametric or stochastic production frontier approach can be used to assess capacity, capacity utilization, and overcapitalization in fisheries, there are some limitations. The frontier approach requires specification of an underlying functional relationship between catch and inputs; there is, thus, always the risk of misspecification error. Then there is the issue that information obtained from the frontier approach is estimated; there is a risk of over or under estimating full capacity utilization. The frontier approach also requires extensive data if the precision of the estimates is to be high and management desires to examine industry structure relative to a capital reduction program (e.g., small vs. large vessels and their optimal configuration). In comparison to other approaches, however, the stochastic frontier approach is perhaps the easiest approach to use given limited data-particularly economic data, and along with the dual economic approach, is one of the few approaches which adequately recognizes the stochastic nature of fisheries. It also is an approach which easily accommodates resource levels (entered as either resource stock levels or as dummy variables).

The stochastic production frontier typically permits assessment of maximal output subject to input levels; as such, it appears to be an output-oriented measure. The stochastic frontier is, in fact, a base or nonorienting measure. That is, the assessment of efficiency is not conditional on holding all inputs or all outputs constant. Utilizing the one-stage routine of Battese and Coelli (1993), however, facilitates an assessment of maximal output from an input-based perspective. With this approach, the inefficiency error term, and subsequently the maximal output, is specified as a function of inputs. Thus, it is possible to consider the input reduction coinciding with a fixed maximum or frontier output.

A major criticism of the stochastic production frontier approach is its inability to adequately handle multiple outputs. Appendix XIV discusses alternative approaches to accommodating multiple outputs.

The dual economic approach to assessing production technologies - by econometrically estimating cost, revenue or profit functions - is one of the ideal ways to assess capacity and its utilization and even endogenous capital utilization, whether outputs are unconstrained or constrained by total allowable catches. The approach can be specified at either the vessel or industry level. To be meaningful for “capacity reduction” programs, the capital stock should be expressed in physical terms, such as vessel sizes or numbers, rather than in monetary values.

The approach readily accommodates multiple products (Segerson and Squires, 1990) and multiple resource stocks, although only a single stock of capital can be accommodated and all evaluations are conditional upon the existing resource stocks. Dynamic adjustment of the capital stock (Morrison, 1985b) and individual transferable quotas (Squires and Kirkley, 1996) can also be accommodated. Different behavioural objectives are also accommodated: cost minimization (Berndt and Fuss, 1986; Hulten, 1986; Morrison, 1985a), profit maximization (Squires, 1987; Dupont, 1991; Segerson and Squires, 1992), or revenue maximization (Segerson and Squires, 1992, 1995; Squires and Kirkley, 1996; Just and Weninger, 1996). Constraints on catch (giving exogenous outputs) and fishing time (Dupont, 1991)can be included. Endogenous capital utilization was analyzed by Epstein and Denny (1980), Kim (1988), and Nadiri and Prochaska (1996).

The versatility and comprehensiveness of the analysis, however, makes substantial demands on data and sophistication of the econometric analysis. Panel or longitudinal data are also readily accommodated. Cost data are required on the variable inputs and a rental or services price for the capital stock and input, and output prices or revenues are often required. A sufficient number of observations is also required to realize enough degrees of freedom and there is always a concern over the representativeness of the data sample and its timeliness.

With either freely chosen output levels or exogenously fixed total allowable catches, the dual economic approach gives a single-valued cost-based measure of capacity and capacity utilization (and even endogenous capital utilization). The optimum capital stock (expressed in either physical or monetary value terms) can be found corresponding to endogenous outputs or exogenously fixed total allowable catches for one or more species. The capacity output may be estimated from the tangency between the short-run and long-run total unit or average costs. The primal measure of capacity utilization is then the ratio of the observed output to the capacity output. A dual economic measure, which is particularly well suited with multiple outputs, can also be derived on the basis of the cost gap between actual and capacity output, as discussed in Appendix II.

Of the various approaches, the DEA approach perhaps is the easiest and offers the most promising and flexible method to determine capacity and capacity utilization. Estimates are obtained of capacity output and utilization rates of the capital stock, variable inputs, and capacity, and also of technical and economic efficiency. DEA could be used to measure “overcapacity” defined as the ratio of the frontier (maximum possible) output unrestricted to total allowable catches to the total allowable catches. The approach is one of the easiest for fishery managers to understand. The approach accepts virtually all data possibilities, ranging from some of the most parsimonious (input and output quantities) to the most complete (a full suite of cost data), although as always the case, more complete data improves an analysis. Because DEA is a form of mathematical programming, constraints are readily accommodated, including socio-economic concerns such as minimum employment, total allowable catches, restrictions on fishing time, and others. The DEA approach can also accommodate the growing international problem of bycatch and its impact upon capacity and the different utilization rates. Appendix XV provides additional discussion.

With the DEA approach, it is possible to determine the combination of variable inputs, outputs, the fixed factors, and the characteristics of the firms which maximize output, minimize input, or optimize relative to revenue, costs, or profits. In the case of fisheries, managers may want to determine how many vessels should be in a fishery, their characteristics, the respective level of input utilization or days at sea, the gear type, the crew size, and the level of output which is allocatively or technically efficient.

The determination of capacity and capacity utilization may be done at the individual firm level or relative to fleet performance. Relative to fisheries and the needs of resource managers, the preferred solution may best be relative to individual vessel-level production. By rearranging observations in terms of maximum efficiency, the number and characteristics of operating units could be determined by simply adding output of each vessel (or unit of analysis such as vessel size class) until the total equalled a specified TAC.

There are two primary orientations of the DEA approach: output
and input.^{8} The input based measure considers how inputs may be
reduced relative to a desired output level, such as total allowable catches. The
output-based measure indicates how output could be expanded to reach the maximum
physical (primal capacity) level, given the input levels. Both the input- and
output-based measures provide information for assessing capacity.

The input-based measure directly provides the input levels and number of operating units consistent with a TAC. That is, it would allow the determination of the optimal vessel or fleet configuration and actual vessels which should be in a fishery given a total allowable catch.

The output-based measure allows fishery managers to identify the level of output and vessels which would maximize output subject to given input levels and resource constraints. The ratio of observed total output to either the maximum physical output or maximum economic output gives a primal measure of capacity utilization. Moreover, given a TAC, the output-based measure could yield a precautionary level of total inputs and number of vessels which yield maximum technical or economic efficiency subject. In addition, the DEA approach would permit identification of which vessels to target to remove from a fishing fleet. In actuality, if sufficient data are available and the goal is the elimination of inefficient operating units, DEA permits identification of those vessels which should be eliminated and without requiring an actual assessment of capital utilization or “harvesting capacity.”

There is on-going debate about the applicability and usefulness of the DEA approach vs. the stochastic frontier approach. The DEA is purely deterministic and thus cannot accommodate the stochastic nature of fisheries. The criticism of nonstochasticity can be easily overcome through the use of bootstrapping DEA. The DEA approach relative to the frontier approach easily permits an assessment of a multiple-input, multiple-output technology. It also does not have the problem of zero-valued outputs or inputs which are typical of many multispecies fisheries (e.g., some vessels or trips harvest only a few of many species and thus have zero-valued output for some species). Neither the stochastic frontier nor the dual-based approaches can easily accommodate the zero-valued dependent variable problem. Even in the absence of zero valued-outputs, the stochastic frontier approach cannot easily handle multiple outputs. Canonical ridge regression as used by Vinod permits estimation of a primal with multiple outputs but does not address the zero-valued output. The stochastic frontier also requires the assumption of some underlying functional form and thus offers the possibility for specification error.

The DEA approach also recognizes or can accommodate both discretionary and nondiscretionary inputs and outputs. It also can facilitate temporal analysis using what is called a Windows technique (Charnes et al., 1994). Technical change as well as network or dynamic assessments can easily be accommodated with DEA (Fare and Grosskopf, 1996).

What about the issue of overcapitalization and economic waste? A physical-based measure of CU is really inappropriate for determining excess productive capacity and overcapitalization; the underlying economic responses to demand and supply conditions are not considered. Nevertheless, the physical measure does provide some information about excessive production possibilities relative to the resource. Moreover, the DEA, dual, and stochastic frontier approaches can all be modified or used to assess the underlying maximum output given economic conditions. DEA particularly may be used since most available DEA programs have an economic-based assessment as an option.

While the previous discussions of various approaches for measuring capacity and capacity utilization (CU) are consistent with economic premises of capacity and offer considerable information about capacity and CU, the approaches and their measures may not be consistent with available data or the needs of management agencies. In this section, we consider other measures based on ideal, empirical and practical concerns, and fishing power or fixed effects. In essence, we attempt to recognize the fact that data on fisheries are typically limited and it is necessary to utilize best information available to determine capacity and CU. Moreover, we return to the direct measure of maximum potential effort (available fishing effort) as identified in Clark, Clarke, and Munro (1979), Hannesson (1987), Mace (1997), Christy (1996), and numerous other researchers (see the review in Section III of this report).

We commence and qualify our discussion by offering what appears to be the objective of fisheries management for many nations: “The essential problem is to determine the amount of fishing and hence the number of vessels required to meet catch allocations in a total fishery which is extremely heterogeneous with respect to the vessels and gears concerned and the mix of species each of them might take both by area and by season. This should be coupled with the economic implications of any proposed degree of change. The problem has two stages, first to determine the amount of fishing appropriate to the reduced number of resources available, and second to explore further modifications that may become desirable within each of the resources in the development of conservation polices.” (Garrod and Shepherd, 1981, p. 325). We also proceed along the lines of Valatin (1992, p.1), “It is likely to be necessary to control both capacity and its utilization, in order to ensure that capacity reductions are not offset by remaining vessels expanding their effort, such that the effort of the fleet as a whole is unchanged.” We thus need to determine the maximum potential effort of an existing fleet to address these concerns.

A starting point is a measure of overall fishing effort of the fleet which recognizes the likely heterogeneous nature of vessels and gear. That is, we need to develop a measure of total fishing effort. Unfortunately, there is no universally accepted measure of fishing effort (Kirkley and Strand, 1981, 1987; Kirkley and DuPaul, 1995). In a broad sense, fishing effort, actually nominal effort, is defined as the product of time and fishing power exerted by a vessel or fleet. Fishing effort also may be thought of in terms of area of sea screened in the course of fishing (Valatin ,1992; Clark, 1985; Dickie, 1955).

It has been common practice by fishery researchers to try and develop a standardized measure of effort for the fleet; this has been necessary in order to adequately assess the relationship between fishing mortality (F) and fishing effort (f): F = q f, where q is a catchability coefficient, F is total fishing mortality, and f is total nominal effort standardized to a homogeneous measure of fishing effort. With effort so defined, “Fleet capacity can be defined as the fleet's capability to catch fish. Capacity then may be defined in terms of an aggregate of those physical attributes of the vessels in the fleet, which are considered important determinants of the fleet's ability to catch fish.” (Valatin, 1992, p.3). As discussed in Appendix VI, The European Union's Multi-Annual Guidance Programme uses aggregate tonnage and aggregate engine power as measures of fleet capacity. Similarly, as discussed in greater detail in Appendix VI, the U.K. has a system of vessel capacity units (VCUs) which equals the following:

vessel capacity units = S_{i} length_{i} (m) * S_{i} breadth_{i} (m) + 0.45 * S_{i} power_{i}

where i = 1,...,N the number of operating units or vessels.

The number of vessel capacity units as defined above is indicative of total capacity and consistent with the notion of a fixed stock of capital. This measure alone, however, is not indicative of the potential total catch of a fleet. To determine the potential total catch, we must consider the maximum possible flow or utilization of the variable inputs together with the fixed capital (i.e., vessel, gear, etc.) and the resource conditions (limits on total catch).

One *ad-hoc* approach is to consider the individual
vessel frontier and subsequently aggregate to the fleet. Utilizing a one-stage
frontier routine, it may be possible to estimate the frontier or maximum output
and determine the number of vessels, their size, and other factor usage over
which technical efficiency declines. For example, we have a frontier
function:

Cwhere L is labour, K is capital, F is fuel, N is the resource stock, T is a technological trend, and e_{it}= f(L_{it},F_{it}, K_{it},T_{it},N_{it},e_{it})

Estimates, however, are only indicative of the data available and responses observed. Thus, estimates of the above frontier would be conditional upon technical and congestion externalities. By modifying the above specification to be consistent with Battese and Coelli's (1993) one-stage routine, technical inefficiency can be specified as a function of number of operating units, total days at sea, and/or total usage of other factors. It is thus possible to determine the combination of total operating units, days, etc., for which production is efficient and void of the influences of excess capital utilization.

With a sufficiently long time series of activities on the vessels in the fleet, the frontier-based estimates could be used to determine the maximum output in each year of the fleet. A peak-to-peak approach could then be used to determine the capacity output and corresponding number of operating units. Moreover, by ordering the catch and effort of each vessel in a fleet in accordance with maximal to minimal efficiency, it would be a simple matter of summing frontier output of each firm until the sum equalled a total allowable catch. Such a solution, however, may fail to recognize that vessels could have fished more days or added inputs to increase their catchability.

With a more detailed analysis of days, length of trip, and input usage, the possibility of a vessel to increase its level of fishing in a given time period could be determined. Subsequently, a total potential frontier output could be estimated for each vessel. Again, the observations could be ordered according to maximum efficiency and the potential frontier output for each vessel could be summed until the sum equalled the total allowable catch. By using the potential total frontier output and establishing the number of units, inputs, and/or effort to those levels corresponding to the TAC, a precautionary approach would be implemented. It is precautionary because it is unlikely that the vessels would substantially increase their efficiency to the maximum and vessels would likely expand their fishing activities to the maximum possible.

The above *ad-hoc *procedure focuses purely on the primal
or physical aspects of capacity output and input. Alternatively, dual frontier
cost, revenue, and profit functions could be estimated and used to determine the
maximum output, number of operating units, and input usage.

Alternatively, the DEA approach could also be used to determine the maximum output corresponding to the maximum physical levels or the economic optimum levels of output, input, and number of operating units. With either the DEA or stochastic frontier approach, it becomes irrelevant as to which size, hull material, engine horsepower represents redundant capital. The DEA approach identifies which operating units are not efficient and away from the respective frontier. Groupings, however, based on vessel characteristics could be formed and used as a basis for targeting reduction of fishing units.

Only limited data may be available, such as data on catch, area fished, days at sea, days fished, and vessel and gear types or characteristics. Nonetheless, there remain viable options for determining the maximum potential effort, capacity and capacity utilization. The maximum effort can be estimated by considering days at sea, fishing power, and options for more fully utilizing days during a production period (e.g., a year). A short-run stochastic production frontier or nonparametric frontier can be estimated to determine maximum possible production at the trip or monthly level. Alternatively, DEA can be applied to determine maximum output and industry restructuring. The particularly useful aspect of DEA is that the approach allows identification of the operating units which are not producing at maximum efficiency.