## Wednesday, November 27, 2013

### Liquidity Premia and the Monetary Policy Trap

There is a wide class of monetary models that boil down to something like the following. We'll confine attention to a deterministic world - extending this to a stochastic environment isn't a big deal. Given intertemporal optimization, the price of a nominal bond, q(t), in dollars, is determined from the first-order condition

(1) q(t) = B[u'(c(t+1))/u'(c(t))][p(t)/p(t+1)],

where B is the discount factor, u(.) is the utility function, c(t) is consumption in period t, and p(t) is the price level. Similarly, for money to be held,

(2) 1 - L(t) = B[u'(c(t+1))/u'(c(t))][p(t)/p(t+1)],

where L(t) is the liquidity premium on money. For example, L(t) is associated with a binding cash-in-advance constraint in a cash-in-advance model, or with some inefficiency of exchange in a deeper model of money. In (1) and (2), the nominal interest rate, 1/q(t) - 1, is positive if and only if there is a liquidity premium on money.

A ubiquitous result is the Friedman rule, which says that an optimal monetary policy eliminates inefficiency by reducing L(t) to zero, which implies a zero nominal interest rate, i.e. q(t)=1. Then, supposing an equilibrium in which consumption is constant over time, (1) and (2) give

(3) p(t+1)/p(t) = 1/B,

so the inflation rate is 1/B - 1 < 0 when the central bank runs the Friedman rule, and there is deflation at the optimum. This is the classic notion of a liquidity trap, and why the zero lower bound on nominal interest rates makes some economists fear deflation. In general there are many different monetary policies that will support a zero nominal interest rate forever in this context. This is the traditional sense in which monetary policy does not matter at the zero lower bound. In a basic New Keynesian model, L(t) is always zero as there is no role for money, and instead of (2) the following condition must hold: (4) 1 >= B[u'(c(t+1))/u'(c(t))][p(t)/p(t+1)],

so that people never want to hold money in equilibrium, and (4) holds as long as q(t)<=1, which is just the zero lower bound. In a New Keynesian model, monetary policy is all about correcting relative price distortions - no Friedman rules in that setting. Of course there's much more to liquidity than currency, or the stuff that Milton Friedman might have wanted to include in some monetary aggregate. Various assets, including government debt and asset-backed securities, may have associated liquidity premia, because those assets are held for insurance purposes, because they are exchanged in some class of non-retail transactions, or because they serve as collateral in credit contracts. For example, incomplete-markets Bewley-type models capture an insurance role for assets (see Aiyagari 1994 for example), and new monetarist models (e.g. work by Ricardo Lagos, Guillaume Rocheteau, Randy Wright,or yours truly) get at how the use of assets in exchange and as collateral give rise to liquidity premia.

Some types of models will give us a liquidity premium on bonds, K(t), so instead of (1) we get

(5) q(t) - K(t) = B[u'(c(t+1))/u'(c(t))][p(t)/p(t+1)]

Now this gets very interesting. What happens at the zero lower bound? From (2) and (5), if q(t)=1 and both assets are held, L(t) = K(t). Then, if L(t)>0 at the zero lower bound (because all government and central bank liabilities are collectively in short supply), in a stationary equilibrium with constant consumption and a constant liquidity premium, L, from (2),

(6) p(t+1)/p(t) = B/(1-L),

so, the larger the liquidity premium, the higher is the inflation rate at the zero lower bound. Indeed, we will not have deflation at the zero lower bound if L>1-B. For the details, you should read this paper or this one.

What's the real rate of interest at the zero lower bound? From (6), the real rate is

(7) r = [(1-L)/B]-1,

which is lower than the rate of time preference, and decreasing in the liquidity premium.

So, this looks promising. An increase in the liquidity premium on government debt, beginning in the financial crisis, can explain: (i) why we do not have deflation in a liquidity trap; (ii) why real interest rates on government debt have been low, post-financial crisis.

The financial crisis involved the destruction of private collateral, in part because of the drop in real estate prices, and the resulting damage in markets for asset-backed securities. Further damage occurred in southern Europe, where the safety of sovereign debt was threatened post-financial crisis. All of this led to a flight to safety - i.e. to U.S. government debt.

Following the logic behind the Friedman rule, the high liquidity premium on government debt reflects an inefficiency. Collateral constraints are tighter, and there is a low stock of liquid assets available for financial exchange and for self-insurance. But the power of monetary policy to mitigate the inefficiency is limited. Basically, it's a fiscal problem. The U.S. government could issue more debt, by temporarily running a higher deficit. But that's not happening, so what can the central bank do about it? In this paper, I assume that the fiscal authority follows a suboptimal policy rule, against which the central bank optimizes, and explore what QE (quantitative easing) does in this context - in particular, swaps of short maturity debt (or reserves) for long-maturity government debt. This can have an effect, which depends on the degree to which short-maturity government debt is better collateral than long-maturity debt. But the effect of QE is to lower the liquidity premium (collateral constraints are relaxed) which , as in equations (6) and (7), will lower inflation and increase the real interest rate. This is a good thing, but if you have been listening to Ben Bernanke, you'll know that this is not the way he thinks QE works. In particular, the thinking at the Fed seems to be that QE will increase inflation and lower real rates. If our problem is indeed a financial inefficiency reflected in a low real interest rate, then the Fed has this turned on its head.

So, suppose we think of the world we're living in now as one where any nonneutralities of money associated with the Fed's interventions during and after the financial crisis have played themselves out. Then, with the nominal interest rate effectively at the zero lower bound, the rate of inflation is being determined primarily by the liquidity premium on government debt. Once we recognize that, it's not surprising that the inflation rate has been falling for the last three years (see the chart).
The situation in Europe is looking more stable, and the private sector is presumably finding new sources of private collateral, all of which reduces the liquidity premium on government debt. Further, we should expect this to continue, for example as the price of real estate and other assets rise. In general, if we think that inflation is being driven by the liquidity premium on government debt at the zero lower bound, then if the Fed keeps the interest rate on reserves where it is for an extended period of time, we should expect less inflation rather than more.

But that's not the way the Fed is thinking about the problem. What I hear coming out of the mouths of some Fed officials is that: (i) Things are bad in the labor market, and the Fed can do something about that; (ii) inflation is low. Thus, according to various Fed officials, the Fed can kill two birds with one stone, so it should: (a) keep doing QE; (ii) make it clear that it wants to keep the interest rate on reserves at 0.25% for a very long period of time.

What I hope the discussion above makes clear is that this is a trap for the Fed. There is not much that the Fed can do on its own about the short supply of liquid assets. They can get some action from QE, but the matter is mostly out of their hands, and more QE actually pushes the Fed further from its inflation goal. If the Fed actually wants more inflation, the nominal interest rate on reserves will have to go up. Of course that will lead to some short-term negative effects because of money nonneutralities.

The Fed is stuck. It is committed to a future path for policy, and going back on that policy would require that people at the top absorb some new ideas, and maybe eat some crow. Not likely to happen. The observation of continued low, or falling, inflation will only confirm the Fed's belief that it is not doing enough, not committed to doing that for a long enough time, or not being convincing enough.

1. "Then, supposing an equilibrium in which consumption is constant over time, (1) and (2) give

(3) p(t+1)/p(t) = 1/B,

so the inflation rate is 1/B - 1 < 0 when the central bank runs the Friedman rule, and there is deflation at the optimum. This is the classic notion of a liquidity trap, and why the zero lower bound on nominal interest rates makes some economists fear deflation. "

The sentence starting with "This is..." sounds like a non-sequitur to me. Maybe there is something missing in your text.

1. I'm not sure what you're not getting. Maybe it's not clear what I mean by liquidity trap until you read the rest of the paragraph.

2. Unrelated to the original comment, except for the fact that it relates to the same point in the text... You seem to assume here that the discount factor is above, rather than below, 1, since otherwise: B<1 implies 1/B > 1 or 1/B - 1 >0, and so there is inflation at the optimum...

2. "so, the larger the liquidity premium, the higher is the inflation rate at the zero lower bound. Indeed, we will not have deflation at the zero lower bound if L>1-B."

Am having problems grokking this post. What is the intuition behind the above quotation?

1. JP: try this: the larger the liquidity premium, the higher the target rate of inflation would need to be to just barely avoid the ZLB being a binding constraint. If the liquidity premium is greater than the natural rate of interest, the target rate of inflation would need to be positive to just barely avoid the ZLB.

2. Thanks. This also confuses me a bit:

"Then, if L(t)>0 at the zero lower bound..."

How can the liquidity premium be greater than zero at the zero lower bound? If it were greater, that means that the nominal interest rate is positive, and therefore we're off the zero lower bound.

3. I never said anything about "target rates of inflation." At the zero lower bound, the rate of inflation is increasing in the liquidity premium. If the liquidity premium is zero, the inflation rate is negative (minus the rate of time preference), and as the liquidity premium goes to 1, inflation goes to infinity. Thus, there is some value for the liqudity premium, above which the inflation rate is positive, and that value is L=1-B.

4. Think in terms of the real rate if you want. The liquidity premium tells you that people are holding the bonds for other reasons than their intrinsic payoffs. So people are willing to hold the bonds at a low real interest rate. So when the nominal interest rate goes to zero, you could have a high inflation rate.

5. "At the zero lower bound, the rate of inflation is increasing in the liquidity premium. If the liquidity premium is zero..."

You're referring to the liquidity premium on bonds K(t), not the the liquidity premium on money L(t), right?

6. At the zero lower bound, those liquidity premia are equal. You can always use money for whatever the bonds are accomplishing, in terms of transactions/collateral. At the zero lower bound, economics agents will be equating the marginal value of money across different uses, and that will equalize the liquidity premia.

7. "At the zero lower bound, those liquidity premia are equal."

...and at the zero lower bound, the liquidity premia are also zero, right? Because if there's a positive liquidity premium, than we can't be at the zero lower bound on nominal interest rates?

Wishing I had one of these: http://en.wikipedia.org/wiki/Universal_translator

8. No, that's the point. With the liquidity premium associated with the bonds, at the zero lower bound the liqudity premia are equal, but they need not be zero.

I don't know, maybe you just need this:

http://en.wikipedia.org/wiki/Higher_education

9. "... but they need not be zero."

But at the zero lower bound, the fed funds rate is zero, indicating that reserves have no more liquidity services to provide and therefore there cannot be a liquidity premium.

And for my riposte.... Here's what you might just need:

http://en.wikipedia.org/wiki/Style_guide

10. By that logic, even when the nominal interest rate is positive, currency must provide no liquidity services, as the nominal interest rate on currency is zero.

11. 0% yielding currency provides liquidity services that are measured by its rental rate--say the interest on a savings account. Only when that rental rate is 0 does currency no longer provide liquidity services. When nominal interest rates (ie both fed funds rates and savings account rates) are zero then the liquidity premium on reserves/cash are 0. What am I missing?

12. I see what you're getting at. Don't think about the fed funds rate, as there is little trade going on in the fed funds market, and it's mostly involving GSEs. But it's a genuine puzzle why financial institutions are willing to hold all those reserves that earn 0.25%. Surely there are better opportunities out there for banks? In this case, we could think of the implicit liquidity yield coming from at least two sources: (i) insurance: five years ago we had a financial crisis that had previously seemed an improbable event; now everyone is thinking that financial crises are much more probable, and that there is something - China, southern Europe - that could still blow up; (ii) holding a large fraction of its assets in reserves allows a bank to take advantage of higher yielding opportunities that can arise in the future.

13. "In this case, we could think of the implicit liquidity yield coming from at least two sources"

Sounds to me like you're describing a safety premium, or yield to safety, not a liquidity premium, or liquidity yield. Anyways, I'll leave off. Maybe I'll figure you out in subsequent posts.

3. Some questions for clarification:

In eq.(2), I believe you want to say 1-L(t), instead of 1+L(t), right? Likewise, in eq.(5), q(t)-K(t), instead of q(t)+K(t).

I don't get why the real interest rate is (1-L)/B-1 as expressed in eq.(7). Isn't it just (1/B)-1 ?

The (gross) nominal interest rate at ZLB is 1/(1-L), then the (gross) real interest rate should be 1/(1-L) * p(t)/p(t+1)=1/B. Could you tell me what I am missing here?

1. Yes, signs were wrong on the liquidity premia, of course. But the real interest rate was correct as is. The liquidity premium is just payoff from how the asset is used in exchange or as collateral. The measured real rate is just the real rate of return on money, i.e. the inverse of the gross inflation rate minus 1.

4. Steve, your equations [1] and [2] imply that q(t) = 1+L(t), which cannot be correct. I think you want q(t) = 1/ (1+L(t)).

1. Sorry, did not see post by Anon above. I think he/she is correct.

5. I am the Anon above, but I think your way of formulation is more intuitive to capture the liquidity premium, e.g. under cash-in-advance setting.

1. I've tried to give you something generic. It applies to what I get in my models, which are not cash-in-advance.

6. Steve,
I will be straightforward: you are wrong. You cannot take equation (6) and make inflation a function of the the liquidity premium because the latter is an endogenous variable too. There is another leg in that scissor that you are missing. Since money is held in equilibrium even at high inflation, there must be some equilibrium relationship between the liquidity premium and inflation rate that is positively sloped. Good luck solving the model!

1. No, it's OK. I'm not solving a complete model here. The liquidity premium is indeed endogenous. I'm just writing down a couple of first-order conditions and showing you what has to hold at the zero lower bound. See either my AER paper or my working paper on QE if you want some details.

7. That is not what you are doing. You state that inflation is driven by the liquidity premium. I find that odd to say the least.

Where are the labor market, long-term contracts, unemployment, expectations, and all those real world nuisances?

1. It is still wrong, but now I see where the error comes from. You are implying the inflation is determined by asset pricing. It is other way around.

Inflation tells what your liquidity premium should be; agents adjust their holdings of money accordingly.

If inflation (which is by all means not determined by an asset pricing relationship, as goods and services are not assets) is low, agents adjust upward their (endogenous) holdings of money so that the liquidity premium is low.

To be sure, now I have to tell that I am surprised. I never thought possible that a tenured member of my profession would give policy advice from a model where inflation comes from 1 asset price equation with 3 endogenous variables (consumption, liquidity premium and inflation). Brave new world.

2. "I find that odd to say the least."

I don't find it odd that you find it odd.

"Where are the labor market, long-term contracts, unemployment, expectations, and all those real world nuisances?"

There are some things we need to explain an idea in a blog post, and some we don't. I'm keeping this as simple as I can. I don't want a soccer team in the story.

You understand for example, that you can take equation (1), and explain most of the important things that are going on in a New Keynesian model.

3. I'm doing my best here. You might want to entertain the hypothesis that I'm not wrong and you're just confused.

4. Williamson, how frequently do you entertain the hypothesis that you're wrong and the rest of the world is not confused?

8. No, there is not much to understand from the NK model from equation (1). It is one equation with 3 endogenous variables. Without the other structure, equation (1) says nothing about NK. Damn, it also belongs to any RBC model. What gives the NK flavor is the pricing equation (where one adds the friction) and the budget constraint (where one adds the old Keynesian effects through constraints on borrowing).

1. "What gives the NK flavor..."

Exactly. It's just flavor - a few details. The heart of the thing is that you take the expected value of the right-hand side of equation (1). Maybe you linearize. Then you argue that expectations are "anchored" in the sense that anticipated inflation is basically a constant. Further, we're thinking of consumption reverting to trend, so think of future consumption as being constant. Then, away from the zero lower bound, the central bank can move q(t) (it sets the nominal interest rate) around, and what adjusts is current consumption. At the zero lower bound, where apparently you want to be because New Keynesians tell us that B is high, the central bank is powerless to move current consumption around. That's pretty much it.

Now, on the borrowing constraints, see my last post:

http://newmonetarism.blogspot.com/2013/11/liquidity-constraints.html

9. Steve, Nick Rowe discusses your paper here:

1. Yes, I left a comment saying it was gibberish to me, but I guess he deleted it.

2. Don't worry WCI has a hyperactive spam filter.

3. Steve: I didn't delete it. We do have spam filter problems, but I just searched 55 pages and couldn't find it in the filter. Apologies.

10. You might want to look at further "gibberish" by Delong and Krugman.

11. Somehow the gibberish from Rowe, Delong and Krugman strikes me as more compelling than the gibberish here. They seem to think you're doing comparative statics on an unstable equilibrium.

1. You're easily convinced, apparently. I'm working on a response. I'll flesh this out for you. Nothing to do with stability.

12. A great example of how liquidity premia drive inflation via velocity is the Gresham effect. If a currency is undervalued because of a price fix, it will be hoarded. If another currency, in exchange with the first, is over-valued, it will change hands more rapidly which will inflate prices denominated in it.

So if you want the Dollar to change hands more rapidly, you want it be more over-valued....in other words, you want the interest on reserves lower, not higher.

1. I'm glad you mentioned that. I was thinking the same thing - this seems like a special case of what Gresham's Law describes. By swapping long-term treasuries, QE can be seen as price-setting of the long-term bonds. The long-term bonds would normally be worth less than short-term bonds or cash, but are now over-valued. This undervalues cash and short-term bonds, inducing hoarding. Hoarded cash effectively appreciates, opposing inflation.

I was a Krugman agree-er at first, but I think I've been swayed. I've posted several comments on his and Andolfatto's blog about different narrative viewpoints for seeing why the market may respond to QA this way. Very interesting finding...

13. Nick, I just tried to leave a comment on your post and it also disappeared.

So I'll just live my point here then. David Andolfatto makes an interesting point:

"He is saying that *if* the problem is an asset shortage, then there's really not much the Fed can do about it. Maybe tinker around a bit with asset swaps (an effect that may have some surprising implications for inflation). But the real solution is a short term increase in Treasury debt -- something beyond the power of the Fed to accomplish. This is a policy recommendation that I'm sure Krugman and DeLong would favor. The fact that they (and you, and Nick, and most others) did not catch it demonstrates that you nobody really bothered reading Steve's papers before joyfully portraying him as an ignoramus."

That is interesting. The implications of SW's piece are much less anti-Krugman than they are anti Sumner.

http://diaryofarepublicanhater.blogspot.com/2013/11/stephen-williamson-on-zlb-interest.html
After all he's arguing that monetary policy is pretty much powerless at this point-if anything he sounds more pessimistic than Krugman himself even.

He's only real disagreement with Krugman here is on what the liquidity trap is and why we haven't had deflation.

In this sense it makes much more sense for Nick to bag SW than Krugman and Delong. That is if you forget how much time SW has spent razzing Krugman personally over the last few years.

Then again, it may be that Delong and Krugman are knocking SW because of his reputation as a freshwater economist.

There's been a lot of discussion on the saltwater-freshwater divide lately.

http://qz.com/150779/the-shakeup-at-the-minneapolis-fed-is-a-battle-for-the-soul-of-macroeconomics-again/

1. "...reputation as a freshwater economist."

Here's a proposal: Heretofore, we should ban the use of the words "saltwater" and "freshwater." Just not useful.

2. They are certainly not useful in this context. If I switched to believing in perfectly flexible prices, I would still have the same problem with this post.

3. They are rather knocking him down because he doesn't know what he is taking about. Doing maths is easy, understanding economics and equilibrium-disequilibrium issues is not:

http://krugman.blogs.nytimes.com/2013/11/29/on-the-importance-of-little-arrows-wonkish/

14. Depends on how much you want it Stephen. Would you be willing to take a hiatus on Krugman bashing in exchange for banning the words saltwater and freshwater?

1. What's wrong with Krugman bashing? Sometimes the man needs a good bash.

2. In an exchange you got to give up something for something the other person likes.

In all seriousness though your disagreement with Krugman doesn't seem to be on questions of policy necessarily.

If you're advocating fiscal policy to counter the slump he'd be with you there. So what is is about him you don't like-his method?

3. I shouldn't say, to avoid being accused of Krugman bashing.

15. The irony is that Krugman ridicules WIlliamson's result as bizarre, because it would be a supply-side problem, which Keynesians dismiss. But Williamson nails it. It is a supply-side problem, because interest on reserves locks up excess reserves at the Fed and the GSE's have a monopoly on reserves. Most banks are reserves supply-limited and that is why inflation is subdued. Look at loan-to deposit ratios!

16. Steve:

In case you're still alive, my 2 cents worth:
http://andolfatto.blogspot.com/2013/12/is-qe-lowering-rate-of-inflation.html

17. if its any consolation Prof W, people thought the idea that bacteria caused infection was gibberish too.
Its amazing how confident people like Krugman and deLong are in their mental models

1. That's true. But the set of ideas thought to be gibberish that *were* gibberish is much larger than those that *were not.*

18. No,
(6) p(t+1)/p(t) = B/(1-L)
does not mean if L goes up then p(t+1) goes up. It means if L goes up then p(t) goes down. For tomorrow's inflation, the economy becomes deflation today.

19. Hi Steve & blog readers,
I have a brief question about your modeling of QE:
Doesn't the Fed exchange bonds for money and hence nothing changes for agents because they don't care which one (bonds or money) they're holding at the ZLB?
Hence L remains constant and QE has no effect?

Thanks a lot in advance!

1. If you read the next post, and the set of notes that goes with it, you'll see that swaps of money for bonds are irrelevant at the zero lower bound. The effect of QE comes from including long bonds, with long bonds being less useful as collateral. Then swaps of short for long matter. That's in my working paper:

http://www.artsci.wustl.edu/~swilliam/papers/qe2.pdf

20. Not to pick nits, but I think you should learn a better way to embed equations in your posts if you're going to push your point on the basis of a formal mathematical model. May I suggest Latex? There are options already available in blogger, for example:

http://www.codecogs.com/latex/integration/blogger/install.php

If many people find your model confusing or seem to be missing your point, the fault may lie in your communication strategy.

21. Two of your equations contain math errors, I think. Equation (3) should be

p(t+1)/p(t) = B

Since B<1, we have the Friedman Rule suggesting deflation.

(7) also has an error. Real interest rates = nominal - inflation = 0 - (Right side of (6) - 1). So,

(7) r = 1-B/(1-L)

Still decreasing in L, though. So, not a problem for the argument.

Where the argument has a problem: it ignores the equilibration process. (I know I'm not original in pointing this out.) Thinking about equilibration rather than just equilibrium makes the result of (6) unsurprising. L increases => demand for money increases => we buy less stuff to increase money balances => p(t) falls. Assuming expectations of p(t+1) haven't changed, we have higher expected inflation.

Which brings up another issue: you don't distinguish between expected variables and actual variables. There can be a difference, and that matters a lot to the logic of what's happening.

Once we account for this, if we buy that QE => falling L, then we should say falling L=> we buy more stuff => p(t) rises. So, expected inflation falls, assuming a constant E[p(t+1)]

But, if we compare p(t) to p(t-1), we should see inflation from QE, not deflation.

An alternative possibility would be that QE => your blog post (or something else) => lower E[p(t+1)] => increased demand for money holding that offsets the falling L, so p(t) doesn't rise.

I don't think either of these arguments is actually true, though. Far more intuitive to me to say that we have low inflation and low interest rates because the demand for money increased substantially in the recession (look at velocity - it fell like a rock) as a result of uncertainty. At the same time, the big increase in reserves from QE isn't translating into fast money supply growth - the money multiplier collapsed (perhaps because of interest on excess reserves + a lousy economy). At the same time, the money supply was increasing in a way that absorbed new long-term government debt thanks to Operation Twist. Sure, it's not based on a mathy steady-state equilibrium reasoning - but I very much doubt we're in a steady-state equilibrium.