State governments continued to function in Belgium during the ordeal, as they will in America.

Herman Matthijs, a professor of politics at the Free University of Brussels, even said the shutdown had advantages:

"A government without power can't introduce new taxes. On the other hand, a government without full powers can't take new measures concerning the outlays. The political crisis relating to the public finance saved money."

According to TIME, zero government did little to alter the daily lives of Belgians and it curbed unwanted spending:

“Many state functions, from education to welfare, have already been ceded over the years to regional and community governments. Belgium deftly helmed the presidency of the E.U. in the second half of 2010, and the caretaker government last month headed off market jitters over its debt levels by quickly agreeing on a tighter budget. The country is recovering well from the downturn, with growth last year at 2.1 percent (compared with the E.U. average of 1.5 percent), foreign investment doubling and unemployment at 8.5 percent, well below the E.U. average of 9.4 percent.”
No one is advocating for a government shutdown in the United States. However, it is important to remember that allowing the time needed for the House and Senate to create a solution is a viable option.

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Good source.