to indicate that the potential supply of fish 

 from the sea is limited. But, even if there were 

 no fixed limit to fish production, we believe that 

 diminishing returns would apply to fisheries at 

 least as much as to agriculture; perhaps more. 

 This has important implications to public 

 policies, as Marshall noted. Hence, the purpose 

 of this article is to explore the production 

 function for the sea. 



DIMINISHING RETURNS OF FISHERIES 



Marshall's (1920, p. 150) first statement of 

 the law of diminishing returns in agriculture 

 was: 



An increase in capital and labour applied in the 

 cultivation of land causes in general a less than 

 proportionate increase in the amount of produce 

 raised, unless it happens to coincide with an im- 

 provement in the arts of agriculture. 



In the case of fisheries, indices of capital and 

 labor inputs are known as "effort." Diminishing 

 returns from fishing means (paraphrasing 

 Marshall) that an increase in effort results in 

 less than a proportionate increase in the yield 

 of fish, assuming no change in technology. 

 Thus, if effort were doubled, the yield would be 

 less than doubled. 



But if we are to manage the world's fisheries 

 well, we need more than general comments 

 about diminishing returns — we need usable 

 estimates of the effort-yield functions for the 

 major species of fish. Schaefer (1954) wrote a 

 pioneering paper on the theory and measurement 

 of such functions. In recent years, many bi- 

 ologists have added to the theory in this area, 

 and have presented important statistical veri- 

 fications and measurements (Pella and Tomlin- 

 son, 1969; Fox, 1970). 



The necessary theory is in two parts: (1) the 

 theory of biological growth, and (2) the theory 

 of yield from a given biomass. 



Theory of Biological Growth 



biomass 



Figure 1. — Growth with no fishing. 



thing like that shown in Figure 1. The species, 

 in each region, would tend to approach some 

 maximum biomass, M. Here natural mortality 

 would just offset recruitment (from young stock) 

 and growth in body size. 



A cur^ve commonly used to represent such 

 growth is the logistic, - 



(1) 'n,= 



J\L 



1 + be' 



where m^ is the biomass at time t, M is the 

 potential maximum biomass, e is the base of 

 natural logarithms, t is time, and a and h are 

 parameters. (We shall generally measure time 

 in years.) Davis (1941) discussed the proper- 

 ties of this curve in detail, and gave many 

 references to its uses in biology and in the study 

 of growth of human populations. Its derivative 

 is: 



- Most work using the logistic has been done with 

 numbers in populations, here we are applying it to the 

 total weight of the population. Tomlinson and Pella 

 (1969) have suggested that the following function be 

 used to approximate biological gi-owth: 



First, consider biological growth — for 

 example, the growth of "biomass" or the total 

 weight of marketable fish. Schaefer (1954), 

 hypothesized that if there were no fishing, the 

 growth curve of the biomass would look some- 



When »i = 2, the growth function becomes the well- 

 known logistic or as used by Gulland, an autocatalytic 

 equation. Fox (1970) has suggested a Gompertz function 

 to approximate biological growth. 



73 



