FOR MONTHS the Federal Reserve Board has been hinting that progress in cutting the budget deficit would be rewarded with lower interest rates. This week the chairman, Alan Greenspan, made it explicit. That gives a useful push to the sluggish negotiations on the deficit between President Bush and the congressional leaders. But Mr. Greenspan made it clear that he wasn't talking about another round of fake numbers and pledges signed in water. To get a response from the Federal Reserve, he said, there would have to be "major, substantive, credible cuts" in the deficit.

There are always limits on the Federal Reserve's ability to bring interest rates down. The first is inflation. Prices have been rising with a disquieting speed so far this year, and they took another sharp bounce upward in the June figures that appeared this week. There is also the exchange rate of the dollar, which will fall if interest comes down. That would be good for exports, but it would make the inflation worse.

The Federal Reserve can influence interest rates, but only up to a point. When it wants to bring rates down, it supplies more money to the banking system. But if it supplies too much, investors see high inflation ahead. Then interest rates turn around and head for the sky, regardless of the Federal Reserve. Conversely, rates will fall if there's a budget agreement -- but they won't fall far until people in the money markets actually see the federal government borrowing less.

If that happens, it will be very good for the American economy -- strengthening, among other things, its ability to compete worldwide. Lower interest means more investment to raise American productivity. Throughout the past decade, cheaper access to capital in Japan and West Germany has given producers there an enormous advantage over their American competitors. German rates have now moved sharply upward as investors begin to pour resources into Eastern Europe. But in Japan, industry continues to have that advantage.

The Reagan tax cuts of 1981 were supposed to set off a huge investment boom that would lift the American standard of living steadily and rapidly. That never happened. The whole country went on a great borrowing binge, and because the money was coming from abroad, Americans have had to offer higher interest than their foreign competitors to keep it flowing in. The Federal Reserve can't do anything about that. It can bring rates down only if and when the demand for loans falls. As the world's biggest borrower, the federal government has to lead the way by getting its own huge deficit under control.