# My Two Cents on QE and Deflation

Steve Williamson has caused quite the controversy in the blogosphere regarding his argument that quantitative easing is reducing inflation. Unfortunately, I think that much of the debate surrounding this claim can be summarized as: “Steve, of course you’re wrong. Haven’t you read an undergraduate macro text?” I think that this is unfair. Steve is a good economist. He is curious about the world and he likes to think about problems within the context of frameworks that he is familiar with. Sometimes this gives him fairly standard conclusions. Sometimes it doesn’t. Nonetheless, this is what we should all do. And we should evaluate claims based on their merit rather than whether they reinforce our prior beliefs. Thus, I would much rather try to figure out what Steve is saying and then evaluate what he has to say based on its merits.

My commentary on this is going to be somewhat short because I have identified the point at which I think is the source of disagreement. If I am wrong, hopefully Steve or someone else will point out the error in my understanding.

The crux of Steve’s argument seems to be that there is a distinct equilibrium relationship between the rate of inflation and the liquidity premium on money. For example, he writes:

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.

He then explains why QE might cause a reduction in inflation using this equation:

…the effect of QE is to lower the liquidity premium (collateral constraints are relaxed) which … will lower inflation and increase the real interest rate.

Like Steve, I agree that such a relationship between inflation and the liquidity premium exists. However, where I differ with Steve seems to be in the interpretation of causation. Steve seems to be arguing that causation runs from the liquidity premium to inflation. In addition, since the liquidity premium is determined by the relative supplies of alternative transaction assets, monetary policy controls inflation by controlling the liquidity premium. My thinking is distinct from this. I tend to think of the supply of public transaction assets determining the price level (and thereby the rate of inflation) with the liquidity premium determined given the relative supply of assets and the rate of inflation. Thus, we both seem to think that there is this important equilibrium relationship between the rate of inflation and the liquidity premium, but I tend to see causation running in the opposite direction.

But rather than simply conclude here, let me outline what I am saying within the context of a simple model. Consider the equilibrium condition for money in a monetary search model:

$E_t{{p_{t+1}}\over{\beta p_t}} = \sigma E_t[{{u'(q_{t+1})}\over{c'(q_{t+1})}} - 1] + 1$

where $p_t$ is the price level, $\beta$ is the discount factor, $q_t$ is consumption, and $\sigma$ is the probability that a buyer and seller is matched. Thus, the term in brackets measures the value of spending money balances and $\sigma$ the probability that those balances are spent. The product of these two terms we will refer to as the liquidity premium, $\ell$. Thus, the equation can be written:

$E_t{{p_{t+1}}\over{\beta p_t}} = 1 + \ell$

So here we have the same relationship between the liquidity premium and the inflation rate that we have in Williamson’s framework. In fact, I think that it is through this equation that I can explain our differences on policy.

For example, let’s use our equilibrium expression to illustrate the Friedman rule. The Friedman rule is designed to eliminate a friction. Namely the friction that arises because currency pays zero interest. As a result, individuals economize on money balances and this is inefficient. Milton Friedman recommended maintaining a market interest rate of zero to eliminate the inefficiency. Doing so would also eliminate the liquidity premium on money. In terms of the equation above, it is important to note that the left-hand side can be re-written as:

${{p_{t+1}}\over{\beta p_t}} = (1 + E_t \pi_{t + 1})(1 + r) = 1 + i$

where $\pi$ is the inflation rate and $r$ is the rate of time preference. Thus, it is clear that by setting $i = 0$, it follows from the expression above that $\ell = 0$ as well.

Steve seems to be thinking about policy within this context. The Fed is pushing the federal funds rate down toward the zero lower bound. Thus, in the context of our discussion above, this should result in a reduction in inflation. If the nominal interest rate is zero, this reduces the liquidity premium on money. From the expression above, if the liquidity premium falls, then the inflation rate must fall to maintain equilibrium.

HOWEVER, there seems to be one thing that is missing. That one thing is how the policy is implemented. Friedman argued that to maintain a zero percent market interest rate the central bank would have to conduct policy such that the inflation rate was negative. In particular, in the context of our basic framework, the central bank would reduce the interest rate to zero by setting

$\pi_t = \beta$

Since $0 < \beta < 1$, this implies deflation. More specifically, Friedman argued that the way in which the central bank could produce deflation was by shrinking the money supply. In other words, Friedman argued that the way to produce a zero percent interest rate was by reducing the money supply and producing deflation.

In practice, the current Federal Reserve policy has been to conduct large scale asset purchases, which have substantially increased the monetary base and have more modestly increased broader measures of the money supply.

In Williamson's framework, it doesn't seem to matter how we get to the zero lower bound on nominal interest rates. All that matters is that we are there, which reduces the liquidity premium on money and therefore must reduce inflation to satisfy our equilibrium condition.

In my view, it is the rate of money growth that determines the rate of inflation and the liquidity premium on money then adjusts. Of course, my view requires a bit more explanation of why we are at the zero lower bound despite LSAPs and positive rates of inflation. The lazy answer is that $\beta$ changed. However, if one allows for the non-neutrality of money, then it is possible that the liquidity premium not only adjusts to the relative supplies of different assets, but also to changes in real economic activity (i.e. $q_t$ above). In particular, if LSAPs increase real economic activity, this could reduce the liquidity premium (given standard assumptions about the shape and slope of the functions $u$ and $c$).

This is I think the fundamental area of disagreement between Williamson and his critics — whether his critics even know it or not. If you tend to think that non-neutralities are important and persistent then you are likely to think that Williamson is wrong. If you think that non-neutralities are relatively unimportant or that they aren't very persistent, then you are likely to think Williamson might be on to something.

In any event, the blogosphere could stand to spend more time trying to identify the source of disagreement and less time bickering over prior beliefs.

### One response to “My Two Cents on QE and Deflation”

1. As I understand Steve’s model (mostly from hearsay: I’ve read his blog posts and many posts/comments from others, but not the actual paper), the price level is ultimately determined by fiscal policy, and he calibrates his fiscal policy assumptions to hold the current price level constant when he is comparing a world with QE against one without. Given that this is the case, and given that prices are perfectly flexible in his model, his result makes sense to me in theory. I don’t have an issue about the direction of causation: given that the price level is determined by fiscal policy, a lower liquidity premium, as a result of QE (which directly increases the availability of liquid assets), causes agents to bid down the subsequent inflation rate. Normally, I would think, we would choose a fiscal policy calibration that holds the future price level constant, in which case “to bid down the subsequent inflation rate” would mean “to bid up the current price level.” But Steve explicitly says that he is using fiscal policy to “tie down” the current price level. This almost surely means that fiscal policy is “tighter” in the world with QE than in the world without QE. So in common English, the reason QE doesn’t cause inflation is that it is offset by a tighter fiscal policy.

But even without this offset, I would expect, in a flexible price model, that QE would reduce the subsequent inflation rate: because prices are perfectly flexible, and QE makes money less attractive, people who expect QE will bid the initial price level higher so that its subsequent trajectory will be more downward. It would be interesting to see what happens, though, if you introduce price rigidities into the model and specify an out-of-equilibrium initial price level (because QE was unanticipated). My guess is that, depending on the calibration, you could see a higher inflation rate followed by a lower one, or you could see just a higher inflation rate.