Proof that unsuccessful speculators confer less benefit to society than their losses

It is well known that successful speculation can be a boon to society, whether or not the successful speculator is ruled to be the legitimate appropriator of his profits. It is also understood that unsuccessful speculation can naturally impose losses on the unsuccessful speculator himself. Little, however, is known about the effect of the unsuccessful speculator on the rest of society: one view is that the others gain what he loses, so that in a sense he is their benefactor and only his own enemy; a more plausible view is that his loss must in some sense exceed the gain splashed on others. Nothing seems to have been proved in a rigorous way in this matter.Here, the problem is subjected to mathematical analysis. It is shown that others do definitely gain from an act of unsuccessful speculation, thus providing rigorous proof of what has long been surmised. But it is also shown that their gain is necessarily less, in an objective sense, than the loss to the speculator himself. Indeed, in the model of unnecessary positive carryover of grain into the future, the gain to the nonspeculators is proved to be actually of a second-order of smallness of infinitesimals compared to the speculator's loss.