Venice, Calif., is one of the nation’s hottest neighborhoods, brimming with affluent young residents and technology startups like Snapchat parent Snap Inc. But in one respect, it bears little resemblance to a boomtown: It hasn’t gained a single housing unit in 15 years.

The ZIP Code encompassing this vibrant Los Angeles precinct is the toughest place in America to build housing, according to an analysis of housing data conducted for The Wall Street Journal by Issi Romem, chief economist at BuildZoom, a website for contractors.

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