ITALY IS NOW cautiously installing its first prime minister of the left, to carry out the program for which the right never had the nerve or stamina. A government of the left-right? It's going to be unpleasant work, forcing down the inflation rate, cutting swollen public budgets and trying simultaneously to generate employment.

The new prime minister, Bettino Craxi, and his Socialist Party, are taking an enormous gamble. If they can bring it off, they will establish themeslves as the dominant force in Italian politics. If not, they will fall back to their old role as a fragment--not least because so little of the work ahead lies within the traditional idea of proper socialism. The European left has never been well equipped for disinflation.

But for Italy and its people, the new government is a highly hopeful departure from a pattern of Christian Democratic leadership that, over the past 15 years, has had progressively less to offer. The country's largest party, the Christian Democrats, has been increasingly running the country through a system of patronage that made rational economic policy impossible. The second-largest of the parties, the Communists, seems to have resigned itself to permanent opposition. That's why the initiative now lies with the third-largest party, Mr. Craxi's Socialists.

They have one big advantage over all previous Italian governments. They are presiding over a broad coalition of five parties, including the Christian Democrats, that have committed themselves to stick together, come what may, until the next election, presumably five years from now. Everyone is to hold everyone else's hand firmly as they all jump together into the ungrateful business of cutting industrial subsidies, scaling back social benefits and, especially, unhooking the dangerously efficient systems of indexation that have given Italy the highest inflation rates of any industrial country.

Mr. Craxi will also have to deal with the cruise missiles scheduled to be installed in Italy toward the end of this year. But the peace movement, so far, has not been much of a force in Italy, and in the absence of progress on arms control, Mr. Craxi has pledged himself absolutely to proceed with the missiles.

By far the harder labor will be the management of the economy--coming to terms with all those years of procrastination, patronage and publicly funded bloat. It is only slightly optimistic to take Mr. Craxi's arrival as a signal that the Italians are fed up with economic weakness aggravated by weak policy, and now want a government tough enough to do something about it.