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I live in one of those metros whose DMA reflects what I'd call its sphere of influence better than its CSA (thanks in no small part to the OMB moving Mercer County, NJ, to the New York CSA). Yet its DMA includes an MSA (Allentown-Bethlehem-Easton) that actually only tangentially falls within Philadelphia's sphere of influence.
But in general, if there were a single metric that best captured a city's influence, I'd say it's the CSA. It's the only one that includes satellite metros that function somewhat independently yet are intertwined with the core metro (Reading to Philadelphia, Lawrence KS to Kansas City, etc.)
GDP captures economic output but seems to me to ignore all the other qualities that bind a metropolitan area's component parts one to the other.
1. Urbanized Area
2. Gross Domestic Product
3. Metropolitan Statistical Area
4. Designated Market Area
5. Combined Statistical Area
6. Population of the city proper
7. Land area
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