In the common usage, popularized by Krugman, a liquidity trap is just a situation where the interest rate set by the central bank has reached zero. Since it can’t go below that (the Zero Lower Bound), if more expansionary policy is needed it will have to take the form of fiscal policy or unconventional monetary policy — quantitative easing and so on. But if there were some technical fix (a tax on excess reserves, say, or abolishing cash) that allowed central banks to make the policy rate negative, there would be no limit to the capacity of monetary policy to overcome any shortfall in demand. The idea — expressed by modern monetarists in the form of the negative natural rate — is that there are so few investment opportunities with positive expected returns that if investment rose enough to equal desired saving at full employment, the expected return on the marginal new unit of capital would be negative. So you’d need a negative cost of capital to get businesses to undertake it.
That makes sense, I guess. But it’s not what Keynes meant by liquidity trap. And Keynes’ version, I think, is more relevant to our current predicament.
Keynes himself doesn’t use the term, and his explanation of the phenomenon, in chapters 13 and 15 of the General Theory, is rather confusing. (Lance Taylor has a much clearer statement of it in Reconstructing Macroeconomics, which I may add a summary or excerpt of to this post when I get home tonight and have the book.) So rather than quote chapter and verse, I’m just going to lay out what I understand the argument to be.
Interest, says Keynes, is not, as the classical economists said, the price of consuming in the future relative to the consuming in the present. It is the price of holding an illiquid rather than a liquid asset today. (This is one of the main points of the book.) The cost of holding an illiquid asset (a bond, let’s say) is the inconvenience that it can’t be used for transaction purposes, but also the opportunity cost of not being able to buy a bond later, if interest rates rise. Another way of saying the same thing: The risk of holding a bond is not just that you won’t have access to means of payment when you need it; it is also the capital loss you will suffer if interest rates rise while you are holding the bond. (Remember, the price of an existing bond always moves inversely with the interest rate.)
This last factor isn’t so important in normal times, when opinions about the future rates of bonds vary; if the supply of liquidity rises, there will be somebody who finds themselves more liquid than they need to be and who doesn’t expect a rise in interest rates in the near future, who will purchase bonds, driving up their price and driving the interest rate down. The problem arises when there is a consensus about the future level of interest rates. At that point, anyone who holds a bond yielding below that level will be anxious to sell it, to avoid the capital loss when interest rates inevitably rise. (Or equivalently, to be able to purchase a higher yield bond when they do.) This effect is strongest at low interest rates, since bondholders not only are more likely to expect a capital loss in the future, but are getting very little interest in the present to compensate them for it. Or as Keynes says,
Nevertheless, circumstances can develop in which even a large increase in the quantity of money may exert a comparatively small influence on the rate of interest. For … opinion about the future of the rate of interest may be so unanimous that a small [decrease] in present rates may cause a mass movement into cash. It is interesting that the stability of the system and its sensitiveness to changes in the quantity of money should be so dependent on the existence of a variety of opinion about what is uncertain. Best of all that we should know the future. But if not, then, if we are to control the activity of the economic system by changing the quantity of money, it is important that opinions should differ.
In other words, the essence of the liquidity trap is a convention about the normal level of interest rates. It’s important to note that this convention is self-stabilizing — if everyone believes that interest rates on a particular class of bond cannot be below 3 percent, say, for any extended period of time, then anyone who finds themselves holding a bond yield less than 3 percent will be anxious to sell it. And their efforts to do so will push the price of the bonds down, which itself will increase their yield back to 3 percent, so that the people who did not share the convention are the ones who end up suffering the loss.
This probably seems confusing and tedious to most readers (and tediously familiar to most of the rest.) Maybe it will be clearer and more interesting with some pictures:
|10-Year Treasury Rate and the Federal Funds Rate
|BAA Bond Rates and the Federal Funds Rate
The horizontal axis of this scatterplot is the Federal Funds rate. The vertical axis shows a market interest rate — the 10-year Treasury bond rate in the first one, and the BAA corporate bond rate in the second. The heavy black diagonal corresponds to a market rate equal to the Fed Funds rate. In both cases, there’s a clear positive relationship over normal ranges of policy rates — 3 percent to 8 percent or so. But outside of this range, particularly at the bottom end, the relationship breaks down. The floor on Treasuries is a hard 3 percent or so, while the floor on BAA bonds varies from time to time but also doesn’t go below 3 percent.  This is Keynes’ liquidity trap.  And when you look at it, it becomes much less clear that the inability to extend the black line past the origin — Krugman’s liquidity trap — is the problem here. What good would it do, if market rates stop following the policy rate well before that?
UPDATE: A smart, skeptical comment by Bruce Wilder leads me to reformulate the argument in a hopefully clearer way.
The necessary and sufficient condition for a liquidity trap is a consensus among market participants that nominal interest rates are more likely to rise than to fall over the relevant time horizon. Obviously, one basis for such a consensus might be that it is literally impossible for short rates to fall any further.  In this sense the ZLB liquidity trap is a special case of the Keynesian liquidity trap. But the Keynesian concept is broader, because conventions about the floor of interest rates can be strongly self-stabilizing, especially where they are backed up by the political power of rentiers.
 “The most stable, and the least easily shifted, element in our contemporary economy has been hitherto, and may prove to be in future, the minimum rate of interest acceptable to the generality of wealth-owners. If a tolerable level of employment requires a rate of interest much below the average rates which ruled in the nineteenth century, it is most doubtful whether it can be achieved merely by manipulating the quantity of money. … Cf. the nineteenth-century saying, quoted by Bagehot, that ‘John Bull can stand many things, but he cannot stand 2 per cent.'”
 It also, not coincidentally, looks like the textbook LM curve. The replacement of LM with an central bank-determined interest rate curve in newer textbooks, is not progress.
 Note that it is not in fact the case that nominal interests cannot be negative, because in the real world cash has substantial carrying costs.