France’s prime minister, Michel Barnier, is facing a no-confidence vote that could leave the country without a functioning government or a budget as it enters the new year.

Does that mean France is hurtling toward a constitutional crisis, or an American-style government shutdown in which civil servants are unpaid and departments are unfunded? Not exactly.

France’s Constitution offers several scenarios that could keep the country’s affairs more or less in order. French institutions are relatively strong, and the country’s laws provide for continuity in the absence of a government and a budget. But it will also pay a price: Investors are already selling off French stocks and bonds, raising its borrowing costs.

How did France get here?

Mr. Barnier was appointed in September by President Emmanuel Macron, who ignored parliamentary election results that were disastrous for his party and its allies. Instead of naming a politician from the leftist coalition that won the most votes in the election, Mr. Macron called on Mr. Barnier, a figure from the traditional centrist right. The leftist coalition was furious.

Since then, Mr. Barnier has been living “the hell of Matignon,” a phrase used by generations of political commentators to describe the difficulties of reigning from Matignon Palace, seat of the government, where a prime minister has some power, but hardly all of it.