Maitre Hauchecorne, a miserly and petty man, bends down and picks up the thin piece of yarn because it is free, and he thinks it may be useful at some time.
The key to understanding the character of Maitre Hauchcorne is the phrase that Guy de Maupassant employs in the sixth paragraph:
...Maitre Hauchecorne, like the true Norman he was, thought it worthwhile to pick up anything that could be useful; and he stooped down painfully, for he suffered from rheumatism.
In this tale of pettiness, the author Guy de Maupassant, a Norman himself [from the former province in northwestern France, where the beaches were stormed by US troops in World War II], writes a withering social criticism of the Norman peasants, whose suspicion and envy of one another appalled him.
In Maupassant's psychological examination and naturalistic and depiction of these Norman peasants, Hauchecorne's initial deception about not having picked up the string and his petty nature which probably would have led him to take the lost purse given the opportunity, work against him as his protestations to the authorities are excessive and fuel the suspicious nature of the others.
All this behavior underscores the attitude of the author Guy de Maupassant, who worked as a government clerk in Normandy and once declared of the Normans, "Everyone is perfidious, a liar, and a phony. Everyone wears a false face."