r, and the positive constant Kis referred to as theenvironmental carrying capacity or saturation level.2
Rewriting (4) and assuming a linear harvesting function, i.e. h(t) = y, we have(5))11(xKrxyxA more general version of (5) can be written in discrete form as(6))11()(111tttttxKrxyxxwhich collapses to the logistic curve in (4) when =1.2.DataThe data used here is for the Norwegian spring-spawning herring from 1950 to 1995.During this period the size of the stock varied greatly. It peaked at 11.5 million tonsin 1951 but in the 16 years following 1956 the stock showed an almost uninterrupteddecline, finally shrinking to 4.2 thousand tons in 1972, at which time the Norwegianspring-spawning herring was at the brink of extinction. The spawning stock remainedvery low until the mid 1980’s when the herring stock began to increase again andmeasured 5 million tons in 1995, almost half of it’s size in 1951.The bulk of the spring-spawning herring was traditionally caught by Norwegianboats, but the 1950’s witnessed a drastic increase in the landings by Russian boats.Iceland and the Faroe Islands have also been engaged in the fishing, as well as EU-countries.3
Figure 1.Development of the spawning stock and total landings of the Norwegian spring-spawning herring 1950-1995 (thousand tons).0200040006000800010000120001400019501952195419561958196019621964196619681970197219741976197819801982198419861988199019921994spawning stocklandings3.ResultsThe growth function described in (6) was estimated using both linear and non-lineartechniques. The estimated equation was(7)12111)(tttttxxyxxwhere 1equals rin (6) and 2equals Kr. Consequently, K can be expressed as 21. Equation (7) was first estimated using ordinary least squares (OLS) but as tests indicated thepresence of hetereoskedasticity - the White test statistic was 18.18 - equation (7) was re-estimated using the White heteroskedasticity consistent covariance matrix estimator (OLS inTable 1). This estimator provides correct estimates of the coefficient covariances in thepresence of heteroskedasticity of unknown form. The estimated intrinsic value was -0.00005and estimated carrying capacity was 9.2 million tons. Using OLS may lead to simultaneity bias, since we have lagged values of the herring stockon both sides of equation (7). Further, the estimates of herring stock may be subject tomeasurement error. Thus, (7) was also estimated using the method of instrumental variables(IV) with two different sets of instruments, lagged values of the annual catch (IV-1 in Table1) and second lags of the herring stock (IV-2).