## 03 November 2015

### Monetary Policy Effectiveness In Liquidity Traps

As I've argued here, conventional money demand models suggest that the price level becomes indeterminate at the zero lower bound and monetary expansion can not do anything to change inflation. In a recent conversation with Scott Sumner, Scott pointed to Paul Krugman's 1998 paper about this issue. Krugman suggests in his paper that only current monetary expansions are useless, but commitments to larger money supplies in the future (or, as Scott would probably like me to say, commitments that the current monetary expansion will be permanent) can both alleviate the liquidity trap and raise the current price level.

So, in line with Krugman's model, let's assume that there is a representative household that maximizes the utility function

$$(1)\: U = \sum^\infty_{t=0}\beta^t\left(u(c_t)\right)$$

where $\beta$ is the household's discount factor and $u(c_t)$ is the utility that the household gains from its consumption, $c_t$, in period $t$. The household is endowed without output $y$ every period and participates in an asset market where it trades one period government bonds and government money. The household's budget constraint is

$$(2)\: M_{t-1} + (1 + i_{t-1}) B_{t-1} + P_t y = P_t c_t + B_t + M_t + T_t$$

where $M_t$ is the money supply, $B_t$ is the household's holding of government bonds, $i_t$ is the nominal interest rate that government bonds pay, $P_t$ is the price level, and $T_t$ is the lump sum tax from the government. The household also faces a cash-in-advance constraint; it must finance its consumption with government cash. This constraint takes the form

$$(3)\: M_t \geq P_t c_t$$

Notice the fact that this is an inequality constraint. The household can hold as much money as it wants, but must at minimum have enough cash on hand to pay for its consumption. The household maximizes $1$ subject to $2$ and $3$ which yields the following first order conditions:

$$(4a)\: M_t = P_t y\: \mbox{if}\: i_t > 0$$
$$(4b)\: M_t \geq P_t y\: \mbox{if}\: i_t = 0$$
$$(5)\: 1 + i_t = \frac{1}{\beta}\frac{P_{t+1}}{P_t}$$

If, like Krugman did, we assume hat next period's price level is constant, we can draw a nice diagram with $4a$, $4b$, and $5$:
The solid blue line is the curve from $5$, the dotted blue line marks the zero lower bound, and the black lines represent the money supply. Normally, the central bank is in complete control of the price level and can move it around by moving the money supply around. But, because the cash-in-advance constraint does not bind at the zero lower bound, increases in the money supply at the zero lower bound will not be immediately spent by the household. This means that, given a constant future price level, the central bank can only push the price level up until it hits the zero lower bound. After that, no amount of current monetary expansion can increase the current price level.

Of course, all that was exactly in line with Krugman. Here's where it gets interesting, though. Krugman assumes in his paper that the central bank has control of the future price level the entire time and can easily increase the future money supply to end the liquidity trap. If we drop the assumption that the cash-in-advance constraint must bind in the next period, can the monetary expansion, regardless of permanence be effective? In order to escape the liquidity trap, the central bank needs to make the household expect that the price level next period will be higher than the price level this period (this would shift the solid blue curve in the graph to the right).

I'm having a lot of trouble wrapping my head around it, but I think that everything hinges on expectations. The cash-in-advance constraint will only bind in the next period if the price level two periods ahead is expected to be higher than the price level next period and so on, ad infinitum. This means that the central bank can only exit the liquidity trap if the household subjectively expects inflation to be greater than the rate of time preference (the inverse of the discount factor subtracted by one) in the future. This is independent of the path of the money supply; not only is there no equilibrium for the price level in the static analysis at the zero lower bound, there is no equilibrium for the entire path of the price level once the zero lower bound has been reached.

The alternative is to reduce the current money supply until the cash-in-advance constraint binds once again; basically to cause a bunch of deflation now instead of in the future. The problem with this is that prices are sticky and a massive monetary contraction would cause a recession.

1. > I think that everything hinges on expectations. ... This means that the
> central bank can only exit the liquidity trap if the household
> subjectively expects inflation to be greater than the rate of time
> preference...

Yes!!

And how does the CB create that expectation? By insisting that's what will happen, committing to making it happen, using a prediction market to guide its policy instruments to make it happen, and once people believe it will happen, then it will.

If you think about it, this is really just like the value of money. Why does unbacked fiat currency have any value at all? Because everyone expects it will have value tomorrow and indefinitely into the future. It's all one collective cognitive illusion of value. Yes, the Fed could mess it up, but people trust the Fed not to, so the mass illusion holds.

You could regard NGDPLT as the same thing, but for the amount of money that will get spent rather than the value of money. If we all collectively believe there will be 5% more total spending this year than last, then there will.

It is all about expectations, and this is what the concrete steppes people and the people insisting that Scott generate something that they feel is a rigorous model just don't understand.

I'm really excited that you got it so quickly.

NGDPLT is so important. I hope you will help me figure out how to get more people to see that.

-Ken

2. Sorry, just adding one more thought. I think you wrote down in math the intuition around NGDPLT I've always had, which is that a liquidity trap is all about people not spending enough of their money, and the way you get people to spend a little more (than they otherwise would) is by convincing them that their money may be worth a bit less in the future (than they had previously thought), and NGDPLT sets that expectation because of the way next year's target advances, relative to this year's *target* NGDP, not actual NGDP, so any shortfall in (nominal) spending this year means the central bank will be targeting even higher nominal spending next year (relative to this year's actual), and you know that sooner or later there will have to be a lot of catching up to do, which means a lot of inflation because real growth can only be so big, so... better spend now... and the aggregate demand shortfall is solved just like that.

That's why abandoning the inflation target is so important. The Fed's commitment to 2% inflation prevents it from creating the inflation expectations needed to bust us out of the liquidity trap.

This is why I'm so annoyed by people who say, "monetary expansion is impossible at the ZLB because any newly created money will just goes into reserves." That's only because the Fed is telling the market that's the best place to put the newly created money.

By the way, I don't want to come off as too opposed to rigorous models. If there's a way to model all of this rigorously, let's do it. It obviously requires public expectations of future central bank policy to be in the model somewhere.

-Ken

3. The point here is a little beyond just that the central bank needs to generate higher inflation expectations to escape the liquidity trap. There is no rational expectations (as opposed to the subjective beliefs I was talking about in the quote at the beginning of you first comment) equilibrium once the zero lower bound is reached; the price level is indeterminate regardless of the path of the money supply (i.e., 'QE' doesn't do anything in particular to inflation).

Perhaps an ideal solution to this liquidity trap is for the central bank to peg the growth rate of the money supply to the growth rate consistent with its target while the fiscal authority switches to a non-ricardian or active regime. This will force inflation, and therefore the interest rate, to rise to target at which point normal fiscal policy can resume and the central bank can go back to its target.

"If you think about it, this is really just like the value of money. Why does unbacked fiat currency have any value at all? Because everyone expects it will have value tomorrow and indefinitely into the future. It's all one collective cognitive illusion of value. Yes, the Fed could mess it up, but people trust the Fed not to, so the mass illusion holds. "

Now we're getting somewhere, you've given me the basics of what could be the market monetarist theory of money demand. The question now is how we can get rational, utility maximizing agents to have some form of money illusion.

4. > There is no rational expectations (as opposed to the subjective
> beliefs I was talking about in the quote at the beginning of you first
> comment) equilibrium once the zero lower bound is reached;

Yes I think I understand that. But I think this is a failure of the rational expectations framework. The assumption that every participant is 100% rational and knows that every other participant is 100% rational and knows that they know that they know that ... and everyone knows everyone else's utility functions, it's all just so ridiculous. Let's switch to Jason Smith's information transfer framework (informationtransfereconomics.blogspot.com). In that framework, consumers behave utterly randomly subject to constraints, like gas molecules in a jar. This corresponds much better to every consumer I've ever met.

In that framework, a relatively rational actor can see that given federal reserve policy of targeting the level of NGDP, that if NGDP falters, the gap between actual NGDP and the Fed's mercilessly advancing target level is growing. That actor knows that it's just a matter of time until other actors (the ones who are not balance sheet constrained) start to spend. Once they do, then inflation will start to jump up, and the Fed is committed to not tighten. Which means even more spending, as people try to spend ahead of rising prices. It's like a massive game of chicken. I recognize that if everyone is 100% rational and knows that everyone else knows that everyone else is rational etc., then it's indeterminate who blinks and you might get stuck in some weird equilibrium where everyone decides that no one will ever blink so there's no need to spend. But I just can't imagine getting stuck there in the real world --- everyone would know it's just a matter of time, and I think the information transfer model would show that too. So, the people who are paying more attention know this will happen, so they will want to spend first, before the inflation sets in. And you can see where that's going.

Note that in today's real world, there's no game of chicken. That's not because rational expectations is correctly describing the real world. It's simply because of the Fed's 2% inflation target. The market assumes that if people start to spend that 4T in bank reserves, the Fed will vacuum up all the liquidity faster than you can say, "Paul Volcker". And, the market is almost certainly right about that.

> while the fiscal authority switches to a non-ricardian or active
> regime.

Yes, I can see that your model's predictions (that we stay in the state of unblinked chicken forever) might come true in the real world. That's why I favor automatic fiscal stabilizers. The idea is for the government to increase spending (I like helicopter money here) in the event that the NGDP prediction market indicates that the Fed will fall short of its NGDP level target regardless of what it does with monetary policy instruments.

[to be continued... 4k limit again]

5. > The question now is how we can get rational, utility maximizing agents
> to have some form of money illusion.

Maybe. But I'm not sure this is needed. Even without money illusion, there are good (rational) reasons for nominal rigidities. First, collapsing NGDP increases the real burden of debt with fixed nominal payments, causing a fisher debt-deflation disaster. Second, producers often enter contracts in fixed nominal terms, such as employment agreements and supply agreements. That makes it hard to lower prices if faced with slack demand (very high transaction costs to renegotiate all of your employment and supply agreements), so producers typically cut production instead, laying people off etc., and the paradox of thrift sets in.

But still, I'm kind of 60% sure that our model would work better in the information transfer framework. In that framework, the reason it's so important to stabilize NGDP is that if money demand spikes (AD drops), it takes way too long for the "information" to transfer from aggregate demand to aggregate supply that all potential real output would all get purchased if only the price level dropped a little across the board. You wind up with years of slow deflation to grind down all of those nominal contracts. If we could model consumers as spending effectively randomly, but they can't spend if they don't have a job, and can't accept a job with a lower wage because of fixed nominal commitments like debt payments and rent, and likewise firms can't produce if the market price of production factors exceeds what they can sell their product for, and firms face massive transaction costs to renegotiate their contracts, then maybe we'd get somewhere. (Hard for me to know, I always have trouble following Jason's math.)

It would be so awesome to create this model. I would be fascinated to know what it would predict about NGDP at the ZLB with and without automatic fiscal stabilizers.

Thanks,
-Ken

Kenneth Duda
Menlo Park, CA

1. Ken,

I've been reading Jason's blog for some time now and I find information equilibrium to be an interesting concept. I'm pretty sure that the ITM (information transfer model) actually resembles the model above when interest rates are low. Perhaps the policy irrelevance occurs for other reasons, but it appears to exist nevertheless.

Thanks,

John

2. John, check out Jason's latest:

http://informationtransfereconomics.blogspot.com/2015/11/if-model-result-is-silly-question-scope.html?showComment=1446995207959#c7136403305895468952

I think this is relevant to our conversation because I think your result that money demand is indeterminate at the ZLB (which is core to your argument that monetary policy is ineffective at the ZLB) represents that you are violating the scope condition of your model. If your model can't determine money demand, then it's the wrong model for determining money demand. In the real world, of course there is determinate money demand at the ZLB; *something* has to happen, and that something will happen. If you want to model/predict it formally, you'll need a better model. My intuition is that that "better model" will have to incorporate future monetary policy and current expectations of future monetary policy. My evidence for that is the way the financial media hangs off of every word uttered by an open market committee member. They're trying to improve their predictions of future monetary policy to establish current money demand. This is why NGDPLT (or some similar level targeting of a nominal aggregate) is so important. The Taylor rule is useless at the ZLB because it would predict a current interest rate of -8% or whatever, and the markets rightly assume the Fed will not actually target a Fed Funds rate of -8%.

-Ken

3. Ken,

I don't think scope conditions apply at all. The existence of indeterminacy in this specific case is an important lesson from this model. There is obviously the empirical element - the real world experience validates the result of this model - and there is the theoretical element - indeterminacy occurs here for the specific reason that money is a perfect substitute to government bonds.

The model above does incorporate expectations of future policy. That's the whole essence of rational expectations. In the specific zero lower bound case, monetary policy is irrelevant for the path of the price level.