FAIR CT961778
The Management of High seas Fisheries
Partner: Fisheries Research Institute, University of Iceland, Reykjavik, Iceland
Modelling Herring Fishery Costs:
Combining Analytical and Empirical Results^{*}
by
Ragnar Arnason
M2.99
This document does not necessarily reflect the views of the
Commission of the European Communities and in no case
anticipates the Commission’s position in this domain.
0. Introduction
Empirical estimation of cost functions for the Icelandic purse seine fleet (Haraldsson, Arnason and Agnarsson 1998, Agnarsson, Arnason and Haraldsson 1999 and Agnarsson 1989) produces a total cost function of the following type:

c_{i} = C(T_{i},y_{i}; i)e_{i}, i = 1,2, ...I.
where the index i refers to observation on vessel i during a particular year, c_{i} represents the total costs, T_{i} the operating time and y_{i} the harvest of that vessel during the year. Finally, e_{i} represents a stochastic error term.
This empirical function is based on data during 19891994. During this period, the Icelandic purse seine fleet was almost exclusively engaged in the domestic herring and capelin fisheries and its participation in the AtlantoScandian herring fishery was negligible. As a result, the data set does not contain information on the costs of pursuing this particular fishery. The technology of purse seine fishing suggests that, apart from the effects of greater distance to the fishing grounds, the costs of pursuing the AtlantoScandian herring fishery are very similar to the costs of pursuing the local pelagic fisheries. Therefore, it may be assumed that the empirically estimated function (1) applies to the AtlantoScandian herring fishery as well.
A crucial feature of the economics of a high seas fishing is the distance the fishing vessel has to travel to the fishing grounds and back. In the case of a highly migratory species such as the AtlantoScandian herring, this distance is highly variable over the course of the year. Clearly, this variability is an important factor in optimal harvesting paths and the way the international fishing game is played. It follows that this variable should be included in the technological description of the fishery e.g. via the cost function. Unfortunately, however, the available data set does not contain observations on the distance travelled. Consequently, it is not possible to obtain empirical estimates of the impact of distance to the fishing grounds on vessel costs. We are therefore forced to seek other ways to include the impact of distance on vessel costs.
Arnason (1998) derives a set of functions that relate vessel operating time, T and distance, d, to fishing time, T_{f} , and harvests, y, along with a collection of other mostly technical variables such as (i) vessel maintenance time, T, (ii) vessel hold capacity, y_{k}, (iii) catchability, , (iv) vessel speed and (v) unloading time, t_{0}.
Let us for convenience write the relationship between operating time and the other variables for a particular vessel i as:

T = T(d_{i},y_{i}; _{i},y_{ki},_{i},speed_{i},t_{0}_{i}).
Obviously, by substituting equation (2) into (1) we obtain total vessel costs as a function of distance, harvest and the other variables as:
(3) c_{i} = C(T(d_{i},y_{i}; _{i},y_{ki},_{i},speed_{i},t_{0}_{i}),y_{i}; i)e_{i}, i = 1,2, ...I.
Given knowledge of the function T() and its technical parameters, i.e., the 5tuple (_{i},y_{ki},_{i},speed_{i},t_{0}_{i}), (3) clearly provides us with an empirically based vessel cost function of the type we seek.
These issues are explored in more detail in the following section. This is followed by a section on the aggregative cost function, i.e. the cost function associated with a given aggregate catch level.

The Cost Function
Empirical estimation produces the following total cost function for Icelandic purse seine vessels (Agnarsson 1989) :
(4) c_{i} = a + bT_{i}^{e}y_{i}^{g}e_{i}, i = 1,2, ...I.
where, as stated above, the index i refers to vessel i, c_{i} represents the total costs, T_{i} the operating time, y_{i} the harvest of the vessel and e_{i} an stochastic error term. All the variables over a certain period of time, e.g. a year or a quarter of a year. The coefficients a, b, e, and g are estimated parameters. Clearly in this formulation, a represents fixed costs.^{1}
For comparison, let's also consider an additive cost function of the form:
(5) c_{i} = a + bT_{i}^{e} + fy_{i}^{g}+ e_{i}, i = 1,2, ...I.
Arnason (1998) derived the following harvesting identity for each vessel, i:^{2}
(6) y_{i} = _{i}T_{i}(1)[ y_{ki} /(y_{ki}+_{i}(2d/speed_{i}+t_{0})],
where, as above, _{i} is catchability of vessel i, y_{ki} is the vessel's hold capacity, t_{0 }its unloading time per trip and speed_{i} the vessel's travelling speed. It should be noted that, as explained in Arnason (1998), catchability should in general be modelled as the function:
_{i} F(x,k),
where x represents biomass and k the vector of vessel characteristics.
Rearranging (5) we obtain:
(7) T_{i}= [(B_{i}+d)/A_{i} ]y_{i},
where A_{i} = (1) y_{ki} speed_{i}/2
B_{i} = (y_{ki}+_{i}t_{0})speed_{i}/2_{i}
Substituting (7) into (4) yields the cost function corresponding to the emprical function, (4):

c_{i} = a + b[(B_{i}+d)/A_{i} ]^{e}y_{i}^{e+g}e_{i}
= a + b[((y_{ki}+_{i}t_{0})speed_{i}/2_{i}+d)/(1) y_{ki} speed_{i}/2 ]^{e}y_{i}^{e+g}.
Substituting (7) into (5), the additive cost function yields:
(9) c_{i} = a + b[((y_{ki}+_{i}t_{0})speed_{i}/2_{i}+d)/(1) y_{ki} speed_{i}/2 ]^{e}y_{i}^{e} + fy_{i}^{g} + e_{i}
Equations (8) and (9) appear somewhat formidable. The important thing to note about these functions, however, is that most of the terms are parameters. Thus, apart from the variables distance, d, and harvest, y, it depends only on estimated parameters, a, b, e and g, and technical features, namely vessel hold capacity, y_{ki}, maintenance time, , catchability, _{i}, and landings time,t_{0},_{ }that are relatively easily measured. Thus rewriting for instance, (8), in this spirit yields:
(10) c_{i} = a + b_{i}°[(B_{i}+d) ]^{e}y_{i}^{e+g}e_{i}
where a, b_{i}°,B_{i}, d, e and g are all parameters. Thus, in spite of their apparent functional complexity, (8) or for that matter (9) are practical cost functions for use in studying the economics of fishing from migratory fish stocks.
2. Aggregation
The cost functions (8) and (9) apply to individual vessels. In international fishing games, however, the natural control variable is the aggregate catch, y = _{i}y_{i}. How does this relate to the cost functions derived above?
Consider a given period of time, a year or a quarter, say. Assume that the desired total catch during this period is y. During the period, the total operating time for each vessel is bounded by the length of the period. More precisely:
T_{i} = T,
where T denotes the length of the period. Moreover, the distance, d, and catchability, _{i}, as well as the parameters of the situation may be assumed to be known. This then yields the vessel's catch according to equation (6). The number of vessels needed to generate the desired catch is then N given by the equation:
(11) y =
where we have assumed that the allocation of vessels is in the order of their relative efficiency. If they are all equally efficient, (11) obviously reduces to:
(11') y = y_{i}/N.
Once each vessel's harvest has been determined at is found, their costs during the period can be obtain from the cost function, (8) or, as the case may be, (9). Aggregate costs, then are simply:

C = ,
where c_{i} is given by equations (8) or (9). In the case of identical vessels, (12) reduces to:
(12') C = Nc_{i}.
References
Agnarsson, S. 1999.
Agnarsson, S., R. Arnason and G.O. Haraldsson. 1999. Estimation of Production Functions for the Icelandic Purse Seine Fleet. FAIR CT961778. The Management of High Seas Fisheries. M1.99
Arnason, R. 1998. Fishing Time and Fisheries Harvesting Function when Distance to Fishing Grounds is Variable. FAIR CT961778. The Management of High Seas Fisheries. M6.98
Haraldsson, G.O., R. Arnason and S. Agnarsson. 1998. Estimation of Cost Functions for the Icelandic Purse Seine Fleet. FAIR CT961778. The Management of High Seas Fisheries. M5.98
Haraldsson, Gunnar O. and Ragnar Arnason. 1998. Estimation of Cost Functions for the Icelandic Purse Seine Fleet. FIRST Preliminary Draft. FAIR CT961778. The Management of High Seas Fisheries. M4.98
