@livinglava,
It won't happen. We are talking about products which people buy because they basically have to in order to fit into society. People will still buy deodorant, razors, etc. Will more women use lower-priced men's products? Only if they feel they want to.
Advertising is a powerful, multi-billion dollar industry -- because it (often) works like a charm. You and I both know the blue razor and the pink razor are the same except for color -- but the pink razor costs 20% more. Yet women still buy them. It would take a helluva boycott for that to stop.
Awareness helps. So does social media in the sense that we as consumers fight back through, among other things, a ton of ridicule. For people sucked into the idea that buying pink was somehow better, they see a lot more now that others are saying it's not worth it. And most people really want to do things that will make them look better in some fashion or another. If pink razors don't make women look prettier than blue ones (and they don't), then more and more women will turn to the blue because of a different social currency -- that they'll look smarter because they get the better buy. Don't believe me? Check out the
Bic Pens for Her backlash.
And what I just wrote about advertising is changing. We live in a world now where we talk back to advertisers in a way we never did before. The conversation is a lot less one-sided and that is to the better. People pay attention to reviews and to what their friends say about products more than ever before. If you tell me the blue razor is better than the pink one, I am more likely to take
your word for it because we know each other.
As for what men do, if they are skimping on personal care in order to spend more money on women, they won't have women to spend that money on. Not buying soap or shampoo, or using less of it, won't impress anyone. And buying and using cheaper stuff will only go so far (some of course is great to use but there's also a ton of crap out there where you truly get what you pay for).
And if men are skimping in other areas, that also won't impress potential mates, whether it means not maintaining their cars or getting their clothes dry cleaned.
But the truth is that it's far more likely that the average person who has to scrimp will save money on something like paying off credit card debt or saving for retirement. We live in a need-it-now, get-it-now culture. There's a reason why
average American credit card debt is nearly $6400. And for that same reason,
about 1/5 of all Americans have nothing saved for retirement.
If people scrimp, it won't be on the stuff they see, smell, taste, show off, or use today. It'll be on how they pay the piper (or not) tomorrow.
Also, your hypothesis doesn't cover the LGBTQ+ market at all.