STATIONARY STATE EQUILIBRIUM 



The stationary state equilibrium is found 

 by letting annual yield equal annual growth: 



"• .*""■■ -wXilr) 



where y is the equilibrium yield and m, is the 

 corresponding biomass. Thus, Schaefer let: 



(8) kx, = fl(l-^) 

 Solved for m, 



(9) m,=M(l-'-f) 



and got the equilibrium yield as a function of 

 effort : 



(10) y, = m,kx, = Mkx, (l - -^) • 



This is a simple quadratic. To estimate it 

 from statistical data using ordinary least- 

 squares, we write: 



(11) y, =Axr~Bx7 

 where A = Mk and B = 



Mk- 



The graph of (11) is shown in Figure 3. Note 

 that while Schaefer assumed constant returns 

 from a fixed biomass, his curve of equilibrium 

 yield indicates decreasing returns. In fact, 



equilibrium yield 



Figure 3. 



effort 



Equilibrium yield-effort 

 returns. 



with constant 



average yield per unit of effort is easily seen 

 to be (by dividing (9) by .»'), 



(12) y,/x, =.4 -Bx, . 



If we use (6), instead of (5) as an estimate 

 the response of yield to effort with a fixed 

 biomass, we have: 



(13, l-/-a(l-^). 

 Solving for m, we find: 



(14) m, =M[l-(l-^')]• 

 So the steady-state equilibrium yield is: 



(15) y, = m,(l-z"') = 



4(i-/o-i(>-^')^]^ 



that is, 



.1 



(16) y, = C(l-^'■')-I>(l-/')" 



where C = M and D = Mia. 



This is not as easy to fit statistically as is the 

 Schaefer function (11). It can be handled without 

 undue difficulty on a computer by a "search 

 method," trying a series of values for z\ in each 

 case computing R-, the Durbin-Watson statistic 

 (D-W), and the t values of the two regi-ession 

 coefficients; then by interpolation we find the 

 "best" fit. 



Equations (15) and (16) indicate decreasing 

 returns to effort. Their graph is like that in 

 Figure 4. In this case — which we think is 

 more realistic — we get diminishing returns 

 for two reasons: 



1. Because annual growth declines as the 

 fish population increases, and 



2. Because the yield-per-unit-of-effort de- 

 clines with effort; that is, doubling the effort 

 will result in less than doubling the yield, even 

 with a fixed biomass. The net result is a much 

 flatter curve after MSY is reached. 



STOCK ADJUSTMENT MODEL 



So far, we have considered only the steady- 

 state equilibrium. This assumes that full adjust- 



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