If immigrants aren't richer than native-born, why costlier cities higher % immigrant

[Capsicum]

Immigrants according to statistics, in terms of averages, have less wealth or income than native-born Americans.

Yet expensive cities like NYC, the Bay Area etc. have double-digit percentages of immigrants (like one in three people or more are foreign born).

Are the foreign born individuals in these cities just the ones that happen to be richer? Or they are much more willing to sacrifice personal wealth, in order to live in a city that has the diverse amenities or ethnic communities they'd like, and make that trade off versus living in a lower cost area that don't have such communities.

Or are they much more thrifty with lifestyles to afford it (eg. they have cultural values that involve saving more, or immigrant families might be larger and more family members share a house)?