Differentiating with respect to N and rearranging, we have

6 Jensen's inequality is a theorem establishing that the expectation of a concave (convex) function is less (greater) than the value of the function evaluated at the expectation of its argument.

where the cumulative and probability density functions are evaluated at E(9\N). The expression 0/(1 - whose inverse appears in Eq. (7), is encountered frequently in statistical and econometric applications. It is known as the hazard rate, defined as the probability with which some event occurs for values of 9 in excess of E(8\N), conditional on it not having occurred yet for any value of 9 less than E(9\N). The hazard rate is often abbreviated as A. Using this shorthand, we can combine expressions to restate the change in welfare with respect to a change in the genetic diversity from which superior varieties can be drawn as

The remaining conceptual task is to consider dC/dd. The form of this expression will depend in general on the nature of the genetic improvement introduced. While this could take many forms, let me offer one general consideration and then examine one special case. The general consideration is the following. Eq. (7) can be rearranged as dC/C

That is, the elasticity of expected welfare with respect to N is approximately7 equal to the elasticity of cost with respect to 0 times the ratio of cost to expected welfare, all divided by the hazard rate times the expectation of 9 conditioned on N. There are a number of instances—the special case I am about to discuss being a prominent example—in which one

7 Another, albeit small for large N, source of error is introduced in the approximation by using N+ 1 rather than TV.

would expect the elasticity of cost with respect to 6 to be relatively small. We might also expect the ratio of costs to welfare to be small in many instances of interest. This leaves the expression E{9\N)X in the denominator, which is the elasticity of the probability of finding an individual with a 6 value greater than E(d\N), 1 - O[£(0|AO], with respect to E(6\N). It can be shown that this expression grows without bound in the limit as E(6\N) grows large, so, not surprisingly, the marginal value of additional resources must eventually be negligible.8

The special case arises when 6 denotes yield per unit area planted and land is the only purchased input in production. If land devoted to a particular crop is T and yield per unit land is 9, total production is q = 8T (10)

Let the rent on land be r, so the cost of production is


Returning to Eq. (9), let us think of how the fraction C/E(W) might be evaluated. Suppose for the purposes of illustration that demand is of the form p(q) = flcf'1 for some constant fl Since p = dCldq, and we are assuming marginal cost is constant given 6, we have9

8 This result, while not implausible, is in some ways an artifact of the specification. Craft and Simpson (2001), for example, show that, with multiple interacting products, the value of "marginal leads" need not be negligible even in the limit.

9 If r\ were greater than one the integral would not be bounded; the crop would be essential and willingness to pay would go to infinity as quantity shrank to zero.

0 0

Post a comment