Cash-in-advance constraints and modeling liquidity traps

It was good to see Paul Krugman's response to my previous post. We agree more than we disagree, I think. We need to move outside the Neo-Wicksellian perspective if we want to look at monetary policy where short-term interest rates are already at zero. You can't use a model which contains i and does not contain M to examine whether changes in M could have an effect even when the central bank can't change i. Sort of obvious really, once you say it like that.

And Paul Krugman had already moved outside the Neo-Wicksellian perspective (indeed he did so 10 years ago) in the paper he links to in his response. The model in that paper contains both M and i. It is not a Neo-Wicksellian model of a cashless economy. It explicitly contains money, and an explicit assumption that people need to use money to buy things. It has a "cash-in-advance constraint".

I'm not against CIA constraints as a device to get money and monetary exchange formally into a model. Sometimes a CIA constraint may be a good way to do this. But for the question at hand, whether monetary policy can still work in a liquidity trap, I don't think adding a simple CIA constraint is enough.

In other words, Paul Krugman may or may not be right that increasing the quantity of money does not work when nominal interest rates are at zero. But his model of why monetary policy does not work does not convince me that he is right. His argument that the CIA constraint is not binding in a liquidity trap, and so monetary policy cannot work, because it merely makes a non-binding constraint even less binding, does not convince be.

A CIA constraint can be written M >= P.e (the stock of money is greater than or equal to the flow of nominal expenditure on goods). When I state it like that, you can immediately see the problem with a CIA constraint, and it's a long-known problem. The left hand side of the equation is a stock, and the right hand side is a flow. The units are not the same. It only makes sense mathematically in a discrete-time model. But that means the real "bite" of the constraint is an artefact of the length of the modeler's period.

In other words, the velocity of circulation of money, which ought to be treated as an economic variable, is determined by the modeler's choice of length of period. In the limit, as the model approaches continuous time, the velocity of circulation approaches infinity, and the CIA constraint disappears.

This is not just some prissy math/technical critique (I hate those critiques). We know that velocity rises in hyperinflations. The real stock of money falls relative to real expenditure. If any economy faces a binding CIA constraint, it is an economy in hyperinflation, because that's where velocity is fastest and real money lowest. But then any economy outside of a hyperinflation, where velocity is slower, and real money is higher, would have a non-binding CIA constraint. But then monetary policy would be ineffective outside of a hyperinflation. But it isn't. The non-binding CIA constraint approach proves too much.

A CIA constraint gives you an LM curve (for a given money supply) which is L-shaped: horizontal at zero nominal interest rates and vertical elsewhere. We know it's not vertical elsewhere. There is no hard distinction between "active money" and "idle hoards". The $100 sitting in my wallet is idle right now. There is only a difference in degree: how long do I expect it to stay idle?

All models are false, of course, and that's never stopped us modeling in the past. Nor should it. But the particular place where this model is false is rather close to the place where we want it to teach us: about monetary policy.

So, I am not convinced by Paul Krugman's model. But that does not make its conclusion, or Paul Krugman's conclusion, wrong. Indeed, it would probably be possible to soften the CIA constraint to allow people to choose the frequency with which they trade money for bonds (at a cost), and generate a liquidity trap in the limit, as the rate of interest approaches zero. The puzzle would be why we have zero nominal interest rates on short-term bonds while the ratio of money to bonds is still finite.

But I don't think that's where the real action is likely to be found. There is more to monetary policy than just temporarily swapping money for short-term government bonds. The representative agent may (must) hold bonds, but not everybody does. There are more assets than just government bonds. As Paul Krugman notes at the end of his post.

But in discussing these questions, we are already well outside the Neo-Wicksellian perspective.

More to come in later post(s).

Comments

[Adam]

Continued from the other stream: The issue here is what to do if the full employment real interest rate is negative. How or if you bring money into the model is beside the point. In all of these models the way in which money affects real variables is by changing the real rate of interest. If prices are fully flexible then changes in the nominal rate don't change the real rate and money ends up neutral. If prices are sticky then changing the nominal rate (through increasing M or some other way if you don't have M) will change the real rate and money will have real effects. Krugman's main point doesn't require money enter via a CIA constraint. His point is that a liquidity trap is a situation in which the full-employment real rate is negative. So...(next paragraph repeats my comment from the other post)

The issue is that people will try to satisfy their euler equations for consumption. If we are on a consumption path where future consumption is expected (or feared) to be low enough that a negative real rate of interest is needed to support it then there are only two possibilities. Either we get a negative real rate or current consumption falls. This just an example of the usual thing that if prices don't adjust then quantities do. The only way to get people to perceive a negative real rate is expected inflation and just expanding the money supply today won't get you that. In this case, without the expected inflation, people hold on to the money not as medium of exchange but as a store of value. The real point though is that money holdings have nothing to do with anything, people won't consume today because they are trying to satisfy a consumption euler equation with a non-negative real rate of interest and no amount of money holdings will change that.

[Nick Rowe]

Adam: Good comments by the way. One of the great things about blogging!

First off, I don't believe the current real interest rate in a hypothetical social planner's model or frictionless Walrasian equilibrium model is negative (and certainly not the -2% that the Bank of Canada's 2% inflation target would require to force an absolute liquidity trap). Investment opportunities exist which have a marginal rate of real return above zero.

We can imagine a world where it were below zero (no investment, no storage, aging population, etc,) but this is not it. The market rate of interest is not reflecting the true trade-off in the Irving Fisher diagram.

Secondly, assume nevertheless we were in such a world. For example: 2 period OG model, with a big cohort this year, and a small one expected next year. Negative real interest rate. (Did I get that the right way round?) Assume prices fixed in all periods. A *permanent* increase in the (outside) money supply would do the trick, I think. Admittedly, this would create excess demand for goods next period. But if that caused an increase in the price level next period, and so expected inflation this period, that helps even more. Money is net wealth. There's a savings glut, so you satisfy it with a helicopter.

See also my reply to your second part of this comment in the "Neo-Wicksellian" post.

But underlying all this, I think is a return of the old "Monetarist vs Keynesian" debate. The transmission mechanism: quantities vs interest rates. Also. "What matters is how much money enters the economy, not what it buys", vs. "What matters is what the government and central bank buys, not how they finance it". I must get my head clearer on this.

[Adam]

Nick,

Ok then, once we get past the model/language barrier we find that we pretty much agree on the theory after all. The major difference is that I do think the full employment real rate is negative, for the world! (mostly basing this on the belief that it's very negative in the states and that's 40% of the world economy).

So, let's take your OG example, with the negative equilibrium real rate, and add capital. One of the things that jumps out of the formal model is that we don't have to be in recession just because of the negative real rate. The alternative is a gigantic investment boom, stock valuations are going through the roof because of the low required returns (still positive though because of the equity risk premium) and we just keep on building stuff almost to the point where its marginal product is zero (for example that capital stuff that provides a flow of housing services). Sound familiar? This is basically Bernanke's savings glut so big that it drove the whole world into a liquidity trap but we are in the trap for years before we're in recession. Eventually we over invest so much that expected returns on any more investment are to low to induce any more of it, then we get recession.

So what to do? As you yourself get from this model, increasing the money supply works best if it generates expected inflation. My contention is that it only works if it generates expected inflation. My story is still monetarist though, you get the expected inflation with an iron clad commitment to keep on printing money until inflation reaches it's target. The only trick left is to choose a good target. Of course, all of this mostly has the US in mind...

So why does that mean that Canada needs to inflate? Because if the world real rate is negative and Canada's is positive then we can't get a depreciation big enough to sell all of our wares. I know CAD has plummeted lately but it's still high by standards of a few years back. We need it way down. While we're at it USD needs to fall against the asian currencies so they can sell some stuff too, but CAD must fall even more.

[Adam]

sorry, just to follow up on one point. The fact that physical investment opportunities exist that have positive rates of return above zero may not help for a couple reasons (I'm basing this on the Krugman "It's Back" paper). One is the equity premiumm which makes the required rate of return higher than the real rate of interest. The second thing is that, suppose we take a Q theory of investment. In order to maintain full employment we need a huge amount of investment and generating enough requires Q really high (a very low cost of capital). But that means that equity valuations might be so high that providing that capital is still a negative expected return enterprise.

[calmo]

Adam feels that fx needs to respond to...models fond of comparative advantage where Canadians specialize in...beaver pelts and Mexicans, sombreros, even fur felt sombreros from those unstoppable Canadian entrepeneurs without a cause. And it's just an accident that transnational companies like fx the way it is...power is just a 5 letter word for some.

Second off,
Nick, if you think I'm going to chase down some walrus to learn about this "model", I need more incentives:

First off, I don't believe the current real interest rate in a hypothetical social planner's model or frictionless Walrasian equilibrium model is negative (and certainly not the -2% that the Bank of Canada's 2% inflation target would require to force an absolute liquidity trap). Investment opportunities exist which have a marginal rate of real return above zero.

If (US but can Canada be that different?) housing prices were registered with any accuracy, (the OER is a plug, a compiled fiction, not a measurement) we'd recognize that official inflation rates have only official utility...providing "inflation protection" by...signage.

[Don the libertarian Democrat]

Since we're determined to buy this crap, I'm now on board with this:

http://worthwhile.typepad.com/worthwhile_canadian_initi/2008/12/central-banks-should-bet-on-recovery-literally.html

[Nick Rowe]

Adam: I keep having the same arguments with myself as I'm having with you!

Forget the Canada/US/World distinction for now, I think. I only brought in Canada because we have a clear 2% inflation target, and the US doesn't have anything so clear (unfortunately), so it made my argument simpler to talk about Canada. But if we assume the Fed has a 2% target it's the same.

Your second paragraph I agree with, a lot. It reads a lot like a couple of my previous posts. But can the marginal return on investment, in practice, go to zero?

Now, I could understand and agree with your statement that the marginal return on investment, minus some required equity premium, might go to zero. And that's exactly what is happening right now, I would say. But then the question is: is that required equity premium exogenous with respect to monetary policy?

There are a lot of nominal interest rates right now, on commercial paper, consumer loans, mortgages, and on shares, that are a long way from zero. Are all those spreads exogenous wrt monetary policy? The Neo-Wicksellian model won't help us answer those questions.

".....increasing the money supply works best if it generates expected inflation. My contention is that it only works if it generates expected inflation." At a minimum, even if we ignore the spreads, a permanent increase in the money supply, holding prices constant today and tomorrow, increases net wealth. Admittedly, the effect is very small, unless we had very large increases. And again, there is more to monetary policy than a temporary swap of money and short-term government bonds.

[Nick Rowe]

Good find Don! And relevant to this point!

[Don the libertarian Democrat]

Don't read anything into my once lapse into relevance.