HARVEY, Ill.—Christopher Clark was elected mayor last year, pledging to seek business tax reductions and lower water bills. They were popular goals that seemed in reach given that city revenues had been rising almost every year since the recession.

On taking office, Mr. Clark quickly figured out the city’s progress had stalled. Property tax collections were down, and businesses were cutting jobs. A fall in city revenue, coupled with growing debt payments, meant there would be no relief from business taxes or water bills….

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