Equilibrium in the competitive farm sector requires that price be equal to marginal cost. Let us suppose that the inverse demand curve is of the constant-elasticity form:
p(q)= aq~l/h. Equate price to marginal cost:
Thus, the partial derivative of the cost function with respect to yield is:
Conventional wisdom has it that demand for agricultural production is relatively inelastic.8 If this were the case, r| would be less than one. While we again hesitate to draw conclusions inasmuch as equation (21) is an implicit expression, the value of the marginal genotype will be smaller to the extent that relatively large realizations of 8 have relatively low values to society
Expression (21) might be calculated under different assumptions on the probability distribution for the 8 s. Analytical solution for the optimal stopping rule, 8 *, is, in general, difficult; thus, analytical solution for the value of the marginal genotype will be difficult as well. We might, however, illustrate a simple case by supposing that 8 is distributed uniformly on the interval [0,1] (i.e. we are normalizing by the maximum value of 8) and that 8 0 = 0. Finally, suppose that c is negligible, so the search will continue so long as no value of 8 = 1 is identified. It can be shown under these assumptions that:
In other words, under the assumption of a uniform distribution, the value of the marginal genotype varies approximately inversely with the square of the number of genotypes over which testing can be done.
Some Conjectures on the Distribution Function
It seems reasonable to suppose that the yield of different genotypes would follow a limiting distribution. It is, in fact, common in the literature on selective breeding to suppose that the distribution of valuable traits among a genetically diverse population is normally distributed. It would appear (to lay-people, at least) that there is an intuitive argument for this limiting behaviour: traits are determined by genetic makeup, and many traits (such as yield under a host of different environmental factors) depend on a combination of many genes. At least in situations in which the expression of traits is determined by the matching of dominant or recessive genes, one might suppose that the expression of a complex trait such as yield would be determined by the (more or less) additive combination of several more (more or less) independent random variables - that is, that the conditions for the application of the central limit theorem would obtain.
These considerations suggest that the value of the marginal genotype could be negligible. On the one hand, with relatively high probability, the marginal genotype might prove to be no better than another sample previously tested. On the other hand, even though the tails of the normal distribution may reach far, they are relatively thin, suggesting that the expected gain to further search might be small.
There are, however, situations in which the normal approximation might be inappropriate. Some traits, such as resistance to particular pests or diseases, might be linked to a single gene. If this were the case, the appropriate distributional assumption would be a Bernoulli trial. It can be shown, in fact, that the value of the marginal genotype is maximized (given a finite and fixed support) under a Bernoulli (i.e. two-point) distribution. Of course, even if the only chance for improvement lies in the identification of a single gene, the value of the marginal genotype would depend on the frequency of that gene among the set of things available to be sampled and the improvement it offers over other genotypes.
We have not discussed aggregation. An important consideration concerns not only how important a marginal genotype is in the improvement of any given crop with respect to any given attribute, but also with respect to the entire set of crops and the time-series of environmental stresses they may face. With respect to the former, it is clear that any particular genotype will be more valuable to the extent that it may be used in the improvement of a wider variety of crops. At the same time, however, any particular genotype will be less valuable to the extent that genotypes from other subspecies, or, increasingly, with developments in biotechnology, other species (and, in at least one example, even another kingdom) can substitute for it in crop-improvement research.
Our discussion above suggests a way of incorporating aggregation in a fuller treatment: we could derive the aggregate, discounted present expected value of the marginal species by considering a time series of draws from the parameters of the distribution function that enters our expressions above.
We are, at present, contemplating whether to extend the modelling exercises described above to derive estimates of the value of the marginal genotype in crop improvement. It would seem that there is little hope of conducting formal econometric estimation of the values generated by search models.9 We could, however, conduct simulations using data from crop breeding programmes, agricultural output, a priori arguments concerning probability distributions, etc., to derive estimates of the value of the marginal genotype.
Such exercises are only as reliable as the data they employ, and we are somewhat sceptical as to the quality and availability of such data. We would need data on the demand for agricultural output, the cost of its supply, the number of genotypes available to testing programmes, and, perhaps more elusively, the cost of crop improvement research. With respect to the latter, the better source for information may not be in directly recorded costs per se, but rather in inferences from research practices. Brian Wright (1995; see also Note 4) makes a particularly telling observation: crop improvement researchers make very little use of the vast majority of the material available to them. We might infer from this that the costs typically exceed the benefits of expanding the research effort. There are, however, other possibilities. Perhaps only a relative handful of sources survive an initial pre-screening; perhaps social incentives to conduct more broadly based searches are greater than are the incentives to which researchers respond.
In recent papers on biodiversity prospecting in the pharmaceutical industry (Simpson and Sedjo, 1996a, b), we have developed models that are more detailed in some aspects at the expense of being narrower in others than has been the model we developed here. In the first of these other papers, we considered a situation in which researchers can decide how many samples to evaluate simultaneously in their search for a new product. Explicitly incorporating this intensity-of-search variable places further restrictions on the values generated by biodiversity prospecting, and has led to some more concise, and we would argue, plausible estimates. In our second paper, we considered capital investments in biodiversity prospecting facilities, which affect the intensity of search. Again, the results suggest more concise (and pessimistic) estimates of the values generated by biodiversity prospecting.
Let us conclude this report on work in progress with a final conjecture. While we have not studied the agricultural context as closely as we have the pharmaceutical, our impression is that the underlying features may be more similar than they first appear. If the question is 'what is the value to society of maintaining the current range of ex situ biodiversity for use in agricultural improvement', the answer we think is likely to emerge is the same as that we derived in considering the private willingness to pay for biological diversity in pharmaceutical research: it is negligible. This is not to say that biodiversity is not valuable; it may be of great value for any number of other ecological, ethical and aesthetic reasons. All we are saying is that our inference from the numbers of genotypes extant and the simple (albeit arguably too simple) analytical exercises we have performed is that genetic resources may simply not be scarce, and for that reason not of much economic value.
1. A notable exception is the work of Evenson and Gollin, Chapter 13, and more recent work by Evenson, Chapters 11 and 12. Wright (1995) has also surveyed work on the economics of genetic resources.
2. Of course, things are selected for testing in the first place because they are closely related to food crops, but conditional on the traits that motivated selection, the samples might be i.i.d.
3. It might be objected that this is a situation in which the approximation implicit in measuring social welfare by consumer surplus is inappropriate. Catastrophic crop failure might induce important income effects. Some who foresee the possibility of imminent doom may regard our reliance on such an incomplete, partial equilibrium welfare measure as avoidance of the 'real' issue: that genetic diversity provides insurance against crop failures of apocalyptic proportions. We are, frankly, sceptical of such claims, but might also add that any far-reaching crop failure might well result from causes for which no genetic remedy might be available.
4. In fact, the search for better crop varieties is ongoing - a variety so completely satisfactory as to motivate the suspension of all further research will never be found. Wright (1995) has noted an extremely interesting fact, however: most crop improvement research is conducted using only a very small number of the possibilities available, the intensity of research effort varies in relation to the perceived need. Reducing this inten-sity-of-effort variable to a dichotomous choice is intended as a simple way of reflecting this feature.
5. In fact, improved varieties are likely to lose their qualities as climatic conditions change, different pests arrive, etc. We can incorporate these considerations, however, by combining in the discount factor, 8, both the rate of time preference and the probability that adverse conditions will motivate resumption of search.
6. Note the similarity of equation (12) to the difference in the expected values of the first-order statistics from samples of size n+1 and n, to which (12) could be reduced if c were zero and welfare linear in 8.
7. The principle simplifications here are that we suppose that land of the same quality is available in sufficient quantities and that there is no extra effort or expense required to collect greater harvests. The latter assumption might be justified by supposing harvest expenses to be negligible in comparison with sowing, cultivation, irrigation, etc.
8. It may be noted that the solutions derived here depend on demand being inelastic; unitary elasticity will imply infinite welfare, and an elasticity greater than 1 will imply negative utility. In the context of agricultural commodities, however, a restriction to inelastic demands may not be troubling. Those agricultural commodities for which demand is elastic might be supposed to have so many substitutes as to make the question of their continuing supply of little importance to society.
9. Evenson (1995a, b) and Gollin and Evenson (Chapter 13) have estimated what we might regard as reduced form models of the value of additional accessions to germplasm collections. While we are somewhat sceptical of the results of these exercises, our work does leave us with a profound appreciation of the difficulties involved.
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