FISHERY BULLETIN VOL. 72. NO. 1 



and 7r.'»* The first three can be developed by 

 combining statistical estimation and indepen- 

 dently derived data. Assume that the biomass 

 is instantaneouslv in equilibrium (i.e., dX_^ q) 



dt 



Then, taking the inverse of (14) and substituting 

 it for X in (11), we obtain: 



(il2b, it follows that the following parameters 

 may be estimated (designated by *): 



Kx = cK- dK^ 



(16) 



r = C/2X 



b = [d/p2] -1 



a = cblr. 



(18) 

 (19) 

 (20) 



where 



and 



C = J,d 



If 

 b 



X = c - dK. 



(17) 



Equation (16) is the familiar parabolic yield 

 function postulated by Schaefer (1954).'^ Notice 

 that both the harvest rate, Kx, and output per 

 vessel, X, may be specified solely in terms of the 

 number of vessels or fishing effort. Similarly, 

 the common property resource externality, as 

 given in (17), is a function only of the level of K. 

 Over a longer period of time the basic assump- 

 tion underlying equations (16) and (17) may 

 reflect a valid representation; i.e., effort or K 

 is the only instrumental variable affecting out- 

 put. There are three different parameters em- 

 bedded in estimates of c and (/. The only way 

 that 0, b, and r can be derived is if some inde- 

 pendent biological information is given. More 

 specifically, suppose that we have an estimate 

 of the biomass consistent with maximum sus- 

 tainable yield, call it X° . Since X° is equal to 



Thus, (17) will be estimated subject to one 

 modification concerning the introduction of an 

 environmental variable. Several biologists, 

 including Dow et al. (1961),-" have argued that 

 a long-term trend of declining seawater tem- 

 perature is partially responsible for the decline 

 in U.S. coastal catches. -• It will be assumed in 

 this study that seawater temperature (°F) 

 affects the a term in the growth function so that, 



^ = aCF)X-bX^, 



(21) 



where °F is equal to the mean annual sea- 

 water temperature, in degrees Fahrenheit 

 Boothbay Harbor, Maine, with . 



9a 



a( F) 



a'>0. 



Seawater temperature can easily be incorporat- 

 ed into (17) in the following way: 



c -dK + z( F), 



(22) 



"* An alternative approach suggested by Thomas (1970) 

 uses the Beverton-Holt model in developing a yield/ 

 recruit relationship. However, because a stock-recruit- 

 ment equation is not specified, it cannot be incorporated 

 •nl'> our bioeconomic model at this time. 



"* The reader should recognize that it does not follow 

 that (17) can be derived from a generalized growth equa- 

 tion [X = F(X) - K\ = 0] and production function 

 Kx = l\X,K). Only under certain specifications of the 

 previous two functions will it follow that .v can be defined 

 as a unique function of K (or X) only. In addition, this 

 production function could have been more generally speci- 

 fied as Kx - rK^XP. However, two compelling factors 

 make it desirable to employ this function. First, there 

 are no observations on the biomass, X, so that empirical 

 tests cannot be made to estimate B. Second, the equation 

 Kx = rKX combined with the logistic gives an excellent 

 empirical fit to past behavior in the fishery (i.e., R- = 0.962 

 for yield function equation 23). In addition, Schaefer 

 makes the same assumption as we did, and this assump- 

 tion is generally accepted as plausible for most fisheries. 

 In conclusion it is difficult for us to imagine how a differ- 

 ent assumption could lead to superior predictive results 

 (i.e., goodness of fit). 



where z represents the change in output per 

 boat as a result of a one-degree change in water 

 temperature. -- 



Data on the number of traps fished per year 

 for the entire inshore American lobster fishery 



'-" Dow, R., D. Harriman, G. Ponlecorvo, and J. Storer. 

 1961. The Maine lobster fishery. Unpublished manuscript 

 submitted to the U.S. Fish and Wildlife Service, Washing- 

 ton, D.C. 71 p. (May be obtained from Sea and Shore 

 Fisheries, Maine.) 



-• Higher seawater temperature can affect the natural 

 yield of lobsters by providing a climate in which molting 

 is facilitated. A larger number of molts will tend, ccwris 

 paribus, to increase the yield associated with any given 

 level of the biomass. 



-- Implicit in the way the effect of seawater temperature 

 is measured is the relationship: 



[a = <;o + d(°¥)]. 



18 



