We conclude with several remarks about the problem of agriculture risk when a
considered region contains local sub-regions with different local climates. As
a consequence, local crop yields will be different. On the other hand, a
regional market is a typical averaging operator, which averages local
variations of crop production. One of the fundamental (macroscopic) variables
(for the market) is the total amount of crop production. If is the area
of the i-th locality () and is the specific crop yield
(for instance, in tons per hectare) then
is the total amount of regional crop production. Let
, where is a stochastic component for
crop yield. Then, if , in accordance with the Central Limit Theorem
[17] the total crop production y can be considered as a normally
distributed value with the mean and the variance
, where
is the covariance matrix for the local crop yields.
The crucial assumption is that there is a minimal critical value of . Note that for the market, overproduction as well as underproduction is dangerous, therefore also the upper critical limit for y may exist. Here we restrict ourselves to the case of the lower limit, so that only the event is considered as an agricultural disaster.
In the way sketched above we can scale down our problem to that of local crop production. It is obviously that the matrix depends on the covariance matrices for climatic parameters, that is, on the statistical traits of the regional climate. Any weakening of the correlation between local climates (as a consequence of the general unsteadiness of the mesoclimate; this is one of the probable consequences of climate change) would reduce the regional risk (?!). Let us recall the well-known probability paradox: the reliability of a system decreases with the reduction of its diversity[17]. Certainly, our conclusion is correct if the market scale is close to the scale of the mesoclimate. Scaling up (for markets) tends to further decrease the regional risk, while scaling down results in the rise of risk. Formally, the problem described is similar to the problem of river navigation (the main problem in USA in the course of the ``hot summer'' of 1988), that is, the problem of critical water levels for large rivers. The water level x is an additive function (functional) of multiple localities which make up the watershed. On the one hand, the river is an averaging operator for the local dynamic elements; on the other hand, the mesoclimate combines all these elements in a statistical way.