the textbook definition of a liquidity trap is an infinite elasticity of the demand for money. The central bank can expand the money supply to an unlimited degree, without affecting interest rates, output, or prices.
The economy is in a liquidity trap when even a zero nominal interest rate isn't enough to restore full employment. That's it.
Since Krugman is Mr. Liquidity Trap, the concept is his to define at this point. If we are to go with what Krugman #2 wrote, so be it.
I just want to put it out there that the definition used by Kling and by Krugman #1 did not just come out of nowhere. I apologize if it seems as if I am repeating myself. What follows is entirely macroeconomics as I recall it from a Keynesian textbook, circa 1973. In the textbook, it would be done using the IS-LM diagram and the AS-AD diagram, but I will try to not to expose the young and innocent to those. This is supposed to be a family-friendly blog.
Here are the ways that an economy can be at zero nominal interest rates and have high unemployment. These are taught in the textbook as different cases.
1. A deep, deep recession caused by low aggregate demand. Suppose an event (say, a financial crisis) has lowered the propensity of consumers and businesses to spend. Nobody wants to borrow, and nobody wants to hire. Hence, interest rates are zero and unemployment is high. Things are so bad, in fact, that you can have output rise without causing any increase in the price level. Thus, when output increases, real money balances do not fall. As output goes up, it takes only a small increase in the money supply or a small increase in interest rates to equate supply and demand in the money market.
2. Interest insensitivity of spending. Regardless of whether interest rates are high or low, consumers and businesses do not care about interest rates when making their decisions about spending. As a result, if the economy goes into recession, lowering the interest rate to zero does little or nothing to stimulate a recovery.
3. Infinite elasticity of money demand. Regardless of how much the central bank expands the money supply, in the words of Krugman #1, "it just substitutes one zero-interest asset for another." When the central bank expands the money supply, consumers and businesses just keep doing what they were doing before, as if nothing had happened. This is what the textbook would have called the Liquidity Trap.
Also, let me add one more case, which would not have made it into the 1973 textbook but which is in the same spirit. Note that it would not apply to the United States.
4. Small open economy with a world interest rate of zero. An open economy is one that has no impediments to international capital movements and no overwhelming impediments to international trade in goods and services. By small, we mean that it is a drop in the bucket in world capital markets, so that it does not affect the world interest rate. If our small open economy is in a recession, and the world interest rate happens to be zero, then so be it. Nothing the country can do can raise its interest rate above the world rate.
The point is that once upon a time, only (3) was considered a liquidity trap. We would not have said that any time you are in a recession with a nominal interest rate of zero you are in a liquidity trap. Instead, we would have asked why the economy was in that situation. Only if the diagnosis were "infinite elasticity of money demand" would we have said "liquidity trap." (As I read it, Tyler Cowen also sees the elasticity of money demand as the defining feature of a liquidity trap.)
What difference does it make, for policy purposes? On the one hand, for fiscal policy it makes little or no difference. In all four of the cases listed above, a fiscal expansion involving deficit spending will do little or nothing to crowd out private investment. In the first case, the economy is so far from full employment that the excess capacity in the economy absorbs the deficit spending with very little increase in interest rates. In the second case, interest rates may rise, but investment will not be affected. In the third case, interest rates will not rise until the economy gets closer to full employment. In the fourth case, foreign capital will finance the deficit without raising the interest rate.
However, there is a difference with regard to monetary policy. In case (1), monetary policy can still get you to full employment. Thus, one could argue that fiscal expansion is still a last resort.
Krugman #2 would say that all four cases constitute the liquidity trap. Again, if he wants to use that terminology, that is ok. It just seems to me that he ought to be a bit less disrespectful toward those of us who, like Krugman #1, use the terminology with which we grew up.