functional form compared with the Just-Pope spe- 

 cification used by McCarl and Rettig (1983). The em- 

 pirical results of the Box-Cox functional form are 

 then discussed and compared with those of McCarl 

 and Rettig. 



METHODS 



Previous findmgs of Peterman (1978, 1981) in- 

 dicate the importance of the assumption made 

 regarding error term in testing the hypothesis of 

 density independence. In the process of examining 

 the effect of the number of released smolts on the 

 production of adults and its variability, Peterman 

 (1981) employed two alternative (additive-error and 

 multiplicative-error) model specifications of the 

 error term: 



^1 = CjS^^i + Vi . . .Additive-error model (1) 

 Ag = C2S''2 expiVo) ...Multiplicative-error 



model (2) 



where A, = adult production of salmon using spe- 

 cification i, 



S = number of smolts, 



C, = survival rate parameter in model i, 



/c, = density dependence parameter in 

 model i, 



Vi = error term for model i. 



By applying these two models to several sets of 

 salmon data, Peterman (1981) concluded that the 

 multiplicative-error model appears to generate 

 better statistical results than its counter model. In 

 addition, the results of the multiplicative-error 

 model suggest that an increase in the number of 

 smolts will increase the variation in total adult 

 returns. Because the variability in adult production 

 is not only influenced by the number of smolts but 

 also other factors affecting the survival of smolts 

 such as the body size of released smolts, Peterman 

 (1981) suggested that model (2) should be modi- 

 fied by including more explanatory variables. By 

 following Peterman's suggestion a third model can 

 be specified with the additional variable body size, 

 B: 



A3 = CgS^sS'^s exp(y3). 



(3) 



The mean and variance of adult production for this 

 model can be expressed as 



E{A^) = CsS^sBdsEiexpiVs)) 



FISHERY BULLETIN: VOL. 86, NO. 4 



Var(A3) = (C3S^'3B''3)2 Var(exp(y3)). 



The instantaneous rates of change in mean and 

 variance of adult production with respect to smolt 

 body size can be expressed as 



dE{A^)ldB = d^EiA^yB 



dVar{As)/dB = 2iYar{A._i)/EiA:^))dE{A:^)/dB. 



If body size of smolts is enlarged, one would ex- 

 pect higher yields, dE{As)/dB > 0. Since both 

 mean and variance are positive, the above model im- 

 poses a restriction that the smolt body size has a 

 positive effect on variability, 3Var(A3)/a5 > 0. 

 This restriction is unwarranted because of lacking 

 theoretical support; rather the effect (positive, nega- 

 tive, or zero) of body size on variability of adult 

 return should be tested empirically. For this reason, 

 McCarl and Rettig (1983) adopt a model developed 

 by Just and Pope (1978, 1979) which can be ex- 

 pressed as 



A4 = C,S''iB''i + C^S'^^B''^ expiV^) 

 = fiS,B) + h'HS,B) exviV,) 



(4) 



where h'''{S,B) is a component of the standard 

 deviation of adult production as shown below. 



The mean and variance of adult production for this 

 model can be expressed as 



EiA^) = C.S'^iB'^i + C^S''^B''5E{exp{V^)) 



Var(A4) = (C5S^-55<5)2Var[exp(l/4)]. 



Because the signs of d^ and d^ are to be deter- 

 mined in the estimation, the advantage of model (4) 

 over model (3) is that it allows for body size to have 

 a positive effect on mean return and unknown 

 (positive, negative, or zero) effect on the variabil- 

 ity of adult production. 



There are problems inherent in model (4), how- 

 ever, the first being that this specification produces 

 a constant percentage change (^4) in adult produc- 

 tion when the number of smolts released changes 

 by 1%, a constant output elasticity, £„,. Output 

 elasticity is an economic term which is widely used 

 in measuring the relationship between input (smolt 

 release) and output (adult production) and has the 

 advantage of being unit free. An output elasticity 

 of 1.0 means that an increase in smolt release by 

 1% will result in the same percentage increase in 



656 



