Why not loan them the money in return for paying back a portion of the savings? And then reinvest the savings in other modes of greater efficiency, which enables more, etc.
It may not be logical, but I accept it as one of the quirks of human behavior. Similar to how pointing out the economic nature of some relationships and interactions is often considered creepy and anti-social (for example, pointing out how certain commonly assumed altruistic behaviors are not altruistic).
But that just proves the broader point: it's not the mere $100 that's holding them up; it rarely is. Rather, it's the practice of not thinking long term, and not accumulating savings even when you have clear opportunities.
Regardless of the terms of repayment (even if there are no terms), it adds stress to future encounters, and they may outright refuse it out of principle anyway.