Equation (22) may be estimated using ordinary 

 least-squares.'' 



Nerlove provides an alternative theory to 

 justify (22). Suppose that x, determines v,* , the 

 "equilibrium value" of catch, 



(24) yr* =ax, - bx] , 



but that the adjustment to the equilibrium value 

 in one period is only gradual (i.e., not complete): 



(25) y, -yr-^i = 6(yr*-.V/- : ) 



where < 5 < 1 is the coefficient of adjustment. 

 Inserting (24) into (25) and rewriting gives 

 the same form as (22): 



(26) y, = abx, — bdxj + (1—5) y,_ , 

 where (1—6) = A. 



Using (26) or (22), we may also compute 

 how many periods it takes one-half the gap to 

 be filled. Ify,_|is in equilibrium, then the gap 

 at period t{G, ) is equal to the following: 



(27) (y,*-y,-,) = G,. 



Each period a constant percentage of the re- 

 maining gap is filled; so that at time t + k 

 the remaining gap is 



(28) Gr.k =G, {1-bf . 



If A' = 0, (29) indicates that all the gap remains 

 to be filled. When will one-half of the initial 

 gap be filled? This may be found by substituting 

 1/2G, for G,+A , or 



(29) G, (1-5)^ =1/2. 

 Hence, 



(30) (1-6) = '/2, 

 or 



(31) A = logl/2-Mog(l-6)= '°g^^ . 



K is the "half-life"; that is, the number of 

 periods required to cut the gap in half. In 2A' 

 years, the gap will be reduced to V4 ; in 3A' 

 years to Vs . . . and so on. It would never com- 

 pletely disappear. In theory, K should be 

 related to the following biological factors: 



(1) Fertility of the species (i.e., number of 

 eggs laid and reaching full term); 



(2) Rate of growth of the species (i.e., how 

 many periods it takes to reach maturity). A' 

 should be large for relatively unfertile and 

 slowly growing species and small for very 

 fertile and rapidly growing species. 



In sum, we are interested in eight estimating 

 equations. First, a group of four equations based 

 upon the assumption of constant returns from 

 a fixed biomass; these are all designated LCR 

 (logistic constant returns). LCRa is the static 

 function with total yield, y,, dependent. LCRb 

 is the same with average yield per unit of 

 effort, y,/.Y,, dependent. Then LCRaS and 

 LCRbS are lagged or stock adjustment models. 

 This gives us four functions. There are four 

 more (designated LDRa, LDRb, LDRaS, and 

 LDRbS) based upon the assumption of decreas- 

 ing returns from a fixed biomass. Finally, we 

 have included an estimate of the parameters of 

 LCRa using the Gulland technique for adjusting 

 the effort series.^ 



^ In essence, a researcher attempting to estimate the 

 parameters of the yield function can lun the following 

 regressions: y, = m, - bx,' 



where (» + 1) is the number of years the fish are in 

 the fishery. The latter is the Gulland technique where 

 the first two specifications are with and without the 

 Koyck formulation respectively. Equation (20) may be 

 specified as the following: 



(y/x), = a — bxt — \bxj_ , — . . . -X bx,_k. 



With this form, the final estimating equation will have 

 (i//j-) as a lagged independent variable. 



RESULTS OF THE ANALYSES 



In order to illustrate the applicability of our 

 theoretical yield functions, we selected five 

 species for consideration: (1) Chesapeake Bay 

 menhaden; (2) Atlantic and Gulf blue crab; 

 (3) Atlantic longline tuna; (4) Soviet and 

 Japanese king crab fishery in the eastern 

 Bering Sea; and (5) Cape Flattery sablefish. 



= For the five fisheries studied (below) the fish are 

 in the fishery about two years. Therefore, a two-year 

 moving average of effort was computed. 



77 



