AUSTIN, Texas—This city’s official motto is “Live Music Capital of the World.” But as Austin experiences a tech-fueled economic boom that has driven up living costs, civic leaders are stepping in to help keep the tunes playing.

The Austin region, one of the fastest-growing metro areas in the country with a population of 1.9 million, has added more than 165,000 jobs over the past five years and boasted the lowest unemployment rate, 3.3%, among big U.S. metro areas in June. But with prosperity has come what Ryan Robinson, the...