Thus, welfare is increasing in 8 if an increase in 8 induces a reduction in total cost. This would be the case, for example, if 8 induces an increase in yield per unit planted, keeping other things equal.
In what follows we will treat the problem we are describing as a simple search problem. The objective is to search until a sufficiently valuable species is identified as to render the expected gain from further search less than the cost.4 It is well known (see, e.g. Lippman and McCall, 1981) that the solution to such problems is of the form of an 'optimal stopping rule'. This stopping rule is of the form 'if, with m species remaining to sample, the greatest value encountered to date is at least as large as 8* stop sampling and commercialize the species with the greatest value thus far encountered, otherwise, continue'. Moreover, if the distribution of values among species is independent and identical, the optimal stopping rule is myopic and constant. By myopic we mean that the decision to stop can be made based solely on a comparison of whether to cultivate the species with the greatest value thus far encountered, or to sample only one more time (see, e.g. Rosenfield and Shapiro, 1981). Given the myopic property, it is immediate that the optimal stopping rule is independent of the number of species remaining to be sampled.
Suppose that in each period the best variety thus far identified is planted and harvested. If this variety is sufficiently good as to motivate the suspension of search, welfare will be maintained at the same level in perpetuity. Let the discount rate be 1—8.5
Thus, we can implicitly state the optimal stopping rule as that value of 8 that satisfies:
We are deriving an expression for the value of having an additional genotype that might be the subject of continuing search. We can derive the value of this 'marginal genotype' by noting that having an additional genotype available for testing proves valuable only if: (i) a variety so successful as to motivate the suspension of testing has not been found before reaching the end of the collection; and (ii) when the final genotype is tested, it is found to be better than those tested previously.
Somewhat more formally, the value of having an n+1st genotype to test is equal to the expectation of the improvement over the best genotype identified among the first n samples tested, conditioned on the value of that best genotype,
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