equilibrium yield 



effort 



Figure 4. — Equilibrium yield-effort with diminishing 

 returns. 



ment is made instantaneously, thus the present 

 catch is a function of the present effort only. 

 This may give a satisfactory approximation for 

 some species. But in other species, several time 

 periods may be required to establish a new 

 equilibrium. In such cases, current yields are 

 affected not only by current effort, but also by 

 the efforts of several past periods. 



That is, annual observations on catch and 

 effort do not represent equilibrium observations. 

 To remedy this situation, biologists have sug- 

 gested various adjustments to the data (Appen- 

 dix I). 



In reality, the observed catch in any given 

 year may be the result of effort expended in 

 previous periods; i.e., the observed catch is 

 some kind of weighted average of catch produced 

 by fishing effort in previous periods. The Gulland 

 procedure employs a similar assumption in that 

 it assumes that this year's observed catch is 

 parabolically related to a simple average of 

 previous effort. An alternative specification of 

 the yield effort-relation for many stocks of fish 

 may take the following form (assuming for 

 example a logistic and constant returns equi- 

 librium relation): 



2 2 



(17) y, = axt — bxi + a\ x,-\ — b\Xt \ 



obvious difficulties. Since our sample will be 

 finite in size, the infinite set of lagged regressors 

 must be terminated at some point. Also, there 

 is likely to be colinearity among the successive 

 regressors. 



One way of solving the problem is to hypothe- 

 size that the coefficients on the lagged variables 

 diminish in size as the time period is more 

 distant from the present observation on catch. 

 Put differently, let us hypothesize that the 

 coefficients on successive .r's decline systemati- 

 cally as we go further back in time. This was 

 suggested by Fisher (1925); more recently it has 

 been revived and extended by Koyck (1954) and 

 by Nerlove (1958). We shall call this a Koyck 

 specification. Koyck hypothesized that a useful 

 approximation would be that the coefficients 

 of (17) decline geometrically: 



(18) Qk = a/ (k = 0, 1, . . . )and 



(19) bk = b\ (k = 0,l,. . . ). 



(17) may be rewritten as the following: ^ 



(20) yt = axr — bx'r + \axi- \ 



— Xbx'i- 1 + . . . €t . 



If we lag (20) by one period and multiply by X, 

 we obtain 



(21) Xyf_i = \aXi_\—\bxt-\ 



+ \' aXf^ 2 ~^ bx'i 2 + ■ • • ^ff - 1. 



Now, subtract (21) from (20) and rewrite: 



(22) y, = ax, -bx', + Xy,^\ + e, 

 where 



(23) e, = e, — Xe,^ 1 . 



Let us now make the classic assumptions about 

 the disturbances, e,, of constant variance and 

 zero covariance. 



Although (17) is a general specification of the 

 yield-effort relationship, its estimation presents 



3 Equation (20) may be intei-preted to mean that 

 observed catch depends on this year's effort (a common 

 assumption used by many population dynamicists) plus 

 effort expended in previous periods. This is merely a 

 hypothesis that can be tested empirically. 



76 



