It will not sway people that have already dropped $500-$600 on their PS5 preorder. MS would have to be almost instantaneous and would have to undercut Sony significantly. A $50 price difference is not going to matter. The same goes for Sony. If MS announces first they are not going to sway people that have already dropped a preorder on the Series X. In the long-term. Prices can be adjust as MS showed with the Xbox One sale bundles not long after launch. IMO, getting that initial momentum is going to be big. I guess it comes down to how they think they can get the momentum. Announce first and drive preorders or wait to undercut the competition. Both come with risks. When I find myself in those circumstances I tend to take control of my own situation and worry about the other guy later.
As for prices, I see more people predicting the PS5 being expensive. I really think at the end of it all, at least on the high end consoles, the prices are not going to be significantly different. They both have some expensive tech and regardless of how much money either company has they are not going to lose money just to lose it.