Equations (7) — (16) summarize the model 

 developed by J. J. Pella and P. K. Tomlinson 

 for the Inter-American Tropical Tuna Com- 

 mission (1969) (hereafter called the TC model). 

 The TC biological model results in a curve 

 relating growth of population to population 

 size. It resembles models previously used by 

 the International Pacific Halibut Commission 

 (Southward, 1968) although it is in terms better 

 suited for economic analysis. The biological 

 portion of the TC model will be used in what 

 follows for an unexploited fishery. In the dis- 

 cussion of an exploited fishery modifications 

 will have to be made to take account of the 

 congestion phenomenon, and this will be achieved 

 by use of the Carlson "engineering" function 

 for a fishery (1969). 



In the TC model the growth of the fish stock 

 is 



(7) dW,/dt = HW'" -KW, 



where H, K, and m are constants. Limiting 

 population to some absolute maximum IVmax , 

 and integrating (7) yields the population at any 

 time t: 



(8) M', = I Wn,ax" - ( w!,,'"' - Wo " "' ; 



- A'(l -m )r I -III 



X e I 



where Wo is the population at time zero, and 



(9) W,,,.,,=iK/H)'""'' 



Further, Wmsv , the stock which yields the 

 maximum sustainable yield, can be expressed 

 as 



(10) IV„„, = (K/mH) 



Km - I ) 



The TC model for an exploited fishery hy- 

 pothesizes a constant "catchability coefficient," 

 q, which is the fraction of the population caught 

 by a standard unit of fishing effort per unit of 

 time. The model assumes that the instantaneous 

 catch rate, dXJdt, can be expressed as: 



(11) dX,/dt = qf,W,, 



whei'e /f is the number of units of effort applied 

 to the fishery at time t. It is the assumption that 

 qf varies in the same proportion as q or / that 

 must be modified to take account of the con- 

 gestion externality. The TC model implies a 

 constant short run marginal physical product 



of effort, and hence a constant short run mar- 

 ginal cost of fish, at least until the .stock of 

 fish is exhausted. This assumption does not hold 

 for the Pacific halibut fishery, and may not hold 

 for any fishery. However, maintaining the TC 

 assumptions for the moment, (7) for an un- 

 exploited fishery becomes 



(12) dW,/dt = HW,'" - KW, - qf,W, 



for an exploited fishery. 



With effort constant in the time interval (0, t), 

 and excluding those cases in which the stock of 

 fish is fished to extinction, integration of (12) 

 yields 



(13) 



''=[.?, 



Qf-,^k + qf 

 -(k*qf)(l-m)t 1/1 -m 



W, 



(I 111 ) , 



Eliminating the time variable, and considering 

 only those populations that have adjusted to 

 the given constant level of effort (i.e., as t ap- 

 proaches infinity), we have 



(14) 



'-'^' 



1 /"! 



Biological equilibrium when catch (AT) is equal 

 to growth of the fish stock is 



(15) X = HW'" ~KW = qfW. 



From (15) we can now express biological equi- 

 librium catch as a function of effort: 



(16) A- = qn^^) 



To take account of congestion externalities 

 the Carlson "engineering" function will be used. 

 Let k be the fraction of a stock of fish caught 

 by the first unit of effort applied to the fishery; 

 assume that two units of effort catch not 2k 

 of the original stock, but only k + k{\ — k) of the 

 initial stock. That is, each unit of effort catches 

 a fraction k of the stock remaining after all 

 previous units of effort have been applied to 

 the fishery. For A^ units of effort the fraction, 

 F, of a fish stock caught is 



(17) F= 1 -(1-/;)^ 



where total catch is 



70 



