I guess that ultimately depends on what you want to do with your discretionary money. The day-to-day costs are already accounted for in our hypothetical examples.

Someone in a low-cost area may want to travel to the city to take in the entertainment and amenities available in the big flashy city, in which case that money is being spent in a high-cost area. Similarly, someone in a high-cost area may want to travel to a low-cost area to take in what it has to offer and thus will spend it in a low-cost area. In that case, the person with $5k in a high-cost area is actually still better off.

If you're going buy an iPhone with that money, the price isn't going to vary much in either place. If the $5k is saved in both cases, there is no real difference either.

I claim that there only has to be at least one so-called discretionary thing priced differentially based on housing/CoL to make living in the lower cost area more financially optimal. It's easy to show there exists at least one; in fact, there are many.

Why's that? And why doesn't the reverse hold true? I can think of many things that are usually more expensive in lower-cost areas than in higher-cost areas.