Here is a maths problem for the Federal Reserve: What is 3.25 minus 5?

The answer, despite what you might think, isn't -1.75. It's that it doesn't matter what it is as long as it's much less than zero. Why is that? Because, as we'll get to in a minute, this tells us where interest rates are probably going to end up the next time there's a recession.

But that can't be too far into negative territory. People, after all, would just turn their bank deposits that were losing money into cash that wasn't if interest rates got down to, say, negative 2 percent or so.

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That means that if the economy "needs" rates that are even more negative than the Fed can give it that the Fed will have to promise not to raise them for a long time or print money to buy bonds with instead. These things work, but not quite as well as good, old-fashioned interest rate cuts, which is why we'd like to avoid having to use them if at all possible.

We might not avoid them, though, if the Fed keeps doing what it's doing.

But let's back up a second. What are the 3.25 per cent and 5 per cent we're talking about?

Well, the first one is how high the Fed estimates interest rates will tend to be in the long run, and the second is how much the Fed tends to cut interest rates in a slump. That's not only some bad math for the Fed, but also, as the Financial Times' Matthew Klein points out, some even worse math than just a few years ago.

In 2012, the Fed thought interest rates would eventually settle around 4.25 per cent; in 2013, it revised that down to 4 per cent; in 2014 it revised that down further to 3.75 per cent; in 2015, it did so again to 3.5 per cent; and in 2016, the pattern has continued to 3.25 per cent.

Now there are two ways to think about this.

The first is that the Fed doesn't think it will have as much room to cut rates as it used to when the economy gets into trouble. And the second is that the Fed thinks the economy needs more help than it used to just to stay out of trouble. Or, as Larry Summers argues, that it takes lower and lower interest rates to produce less and less growth. It's a new old problem called "secular stagnation," where slow population growth means slow investment growth -- why build new offices or houses if there aren't new people for them? -- that turns today's slow recovery into tomorrow's and then forever's.

It's a new old problem called "secular stagnation," where slow population growth means slow investment growth -- why build new offices or houses if there aren't new people for them? -- that turns today's slow recovery into tomorrow's and then forever's.

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Economist Alvin Hansen worried about this during the Great Depression until the baby boom proved him wrong, but what if their retirement is proving him right after all this time? Whether or not the Fed wants to admit it, that's what it's saying might be the case when it says interest rates will stay lower than before.

The last part of all this is that recessions themselves have changed.

It used to be -- notice how those words keep coming up -- that the Fed would push us into one by raising rates to keep prices from rising too much, and pull us out by cutting them once it decided that we'd suffered enough for our inflationary sins. But that's not the way it works anymore.

Sometimes math problems aren't as easy as they seem, especially when you rule out the easiest solutions.

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Ever since 1990, recessions have happened when bubbles have burst rather than when rates have risen to fight inflation. In other words, when there's been overinvestment rather than overheating. That means that, first off, there isn't as much room to lower rates since they didn't get that high to begin with, and, second, that lower rates might not even get people to invest again since they just did too much of that. Add two those together, and, as we've found out, it's not that long until you get zero interest rates.

But that's just another way of saying that the economy won't get as much monetary stimulus as it needs. So recessions and especially recoveries will be worse than they should be. What is to be done?

Well, Boston Fed President Eric Rosengren has a simple fix: Increase the Fed's inflation target. Right now, it's 2 per cent, but if it were, say, 4 per cent, then interest rates would tend to be higher during the good times, and, as a result, leave more room to cut them during the bad times. That, of course, isn't going to happen now, but it's something Rosengren thinks the Fed will at least have to talk about if rates end up back at zero during the next recession, which it sure seems like they will if the Fed's forecasts are right.

Sometimes math problems aren't as easy as they seem, especially when you rule out the easiest solutions.

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