Sunday, December 21, 2014

Inflation at the Zero Lower Bound

I'm going to try to clear up some issues in the blog discussion among Ambrose Evans-Pritchard, Paul Krugman, and Simon Wren-Lewis, among others, about zero-lower-bound monetary policy. Rather than parse the thoughts of others, I'll start from scratch, and hopefully you'll be less confused.

I'll focus narrowly on the issue of what determines inflation at the zero lower bound or, as Evans-Pritchard states:
The dispute is over whether central banks can generate inflation even when interest rates are zero.
As it turns out, David Andolfatto and I have a paper (shameless advertising) in which we construct a model that can address the question. And that model is actually a close cousin of the Lucas cash-in-advance framework that Krugman uses to think about the problem. There is a continuum of households, and each one maximizes
We'll simplify things by assuming that there are only two assets, money and one-period government bonds, and no unsecured credit. We can be more explicit about how assets are used in transactions, but to make a long story short, think like Lucas and Stokey. There are two kinds of consumption goods. The first can be purchased only with money, and the second can be purchased with money or government bonds. We can think of this as standing in for intermediated transactions. That is, people don't literally make transactions with government bonds, but with the liabilities of financial intermediaries that hold government bonds as assets. We can also extend this to more elaborate economies in which government debt serves as collateral, to support credit and intermediation, but allowing government bonds to be used directly in transactions gets at the general idea.

So, suppose a deterministic world in which the economy is stationary, and look for a stationary equilibrium in which real quantities are constant forever. Further, restrict attention to an equilibrium in which the nominal interest rate is zero. Let m and b denote, respectively, the quantities of money and government bonds, in real terms. We'll assume that the government has access to lump sum taxes and transfers. Starting the economy up at the first date, the first-period consolidated government budget constraint is
where T is the real transfer to the private sector at the first date, i.e. the government (the consolidated government - at this stage we won't differentiate the tasks of the central bank and the fiscal authority) issues liabilities and then rebates the proceeds, lump sum, to the private sector. Then, in each succeeding period, since the nominal interest rate is zero, the consolidated government budget constraint is
where T* is the real transfer at each succeeding date, and i is the inflation rate, which is constant for all time.

A zero nominal interest rate will imply that consumption of the two goods is the same, so per-household consumption is c = y, where y is output. First, suppose that government bonds are not scarce. What this means is that, at the margin, government bonds are used as a store of wealth, so the usual Euler equation applies:
Therefore, i = B - 1, i.e. there is deflation at the rate of time preference. Further, output y solves
This is basically a Friedman rule equilibrium, and we could use results in Ricardo Lagos's work, for example, to show that there exists a wide array of paths for the consolidated government debt that support the Friedman rule equilibrium. An extra condition we require here is that this economy be sufficiently monetized, i.e.
so that the central bank's balance sheet is sufficiently large, and there is enough money to finance consumption of the good that must be purchased with money.

The Friedman rule equilibrium is Ricardian. At the margin, government debt is irrelevant. As well, there is a liquidity trap - open market operations, i.e. swaps of money for government bonds by the central bank, are irrelevant. Thus, the central bank cannot create more inflation. But neither can the fiscal authority. What about helicopter drops? Surely the fiscal authority can issue nominal bonds at a higher rate, and the central bank could purchase them all? But, as long as government bonds are not scarce, equation (3) must hold at the zero lower bound, which determines the rate of inflation. Basically, this is the curse of Irving Fisher. Under these conditions, it is impossible to have higher inflation at the zero lower bound. Helicopter drops may indeed raise the rate of inflation, but this must necessarily imply a departure from the zero lower bound.

Note that, in the Friedman rule equilibrium in which government debt is not scarce, there is sustained deflation at the zero lower bound, which doesn't seem to fit any observed zero-lower-bound experience. Average inflation in the Japan in the last 20 years has been about zero, and inflation has varied mostly between 1% and 3% in the U.S. for the last 6 years. But if government debt is scarce in equilibrium, we need not have deflation at the zero lower bound in our model. What scarce government debt means is that the entire stock of government bonds is used in transactions, which implies, in general, that the nominal interest rate is determined by
where R(t) is the nominal interest rate. So now there is a liquidity premium on government debt, which is determined by an inefficiency wedge in the market for goods that trade for money and government bonds. Then, in a zero-lower-bound equilibrium, the inflation rate is determined by
Note as well, that in this equilibrium, y = m + b, so the total quantity of consolidated government debt constrains output. Clearly, this equilibrium is non-Ricardian - government debt matters in an obvious way. But, there's still a liquidity trap. If the central bank swaps money for bonds, this is irrelevant. The central bank can't change the rate of inflation through asset swaps.

But, when government debt is scarce, fiscal policy can determine the inflation rate, as the fiscal authority can vary the rate of growth of total consolidated government liabilities (which determines the inflation rate), and this in turn will affect the real quantity of consolidated government liabilities held in the private sector, and the liquidity premium on government debt. To explore this in more detail, suppose that the utility function is constant relative risk aversion, with CRRA = a > 0. Then, equation (7) gives us a relationship between output and the inflation rate:
Then, since y = m + b, we can substitute in the consolidated government's budget constraints to obtain
In (9) and (10), s = 1/(1+i), 1 - s is the effective tax rate on consolidated government debt, and T* is the revenue from the inflation tax, where the inflation tax applies to the entire outstanding nominal consolidated government debt.

So, if the fiscal authority chooses an inflation rate i > 1/B -1, then it expands the government debt at the rate i per period, the central bank buys enough of that debt each period that the nominal interest rate is zero forever, and the government collects enough revenue from inflation every year to fund a real transfer T* which, as a function of s, is shown in the next chart.
Note that this is essentially a Laffer curve. Infinite inflation (s = 0) implies zero revenue from the inflation tax, as does zero inflation (s = 1), and transfers are negative when the inflation rate is negative (s > 1). The higher the rate of inflation, the lower is the real quantity of consolidated government debt, output and consumption - more inflation reduces welfare. The central bank cannot control inflation, but the fiscal authority can.

Therefore, in this model, it is indeed correct to state that, at the zero lower bound, the central bank has no control over the inflation rate. The fiscal authority may be able to control inflation at the zero lower bound, but only by tightening liquidity constraints and increasing the liquidity premium on government debt. Of course, in this model the government debt all matures in one period. What about quantitative easing? QE may indeed matter, particularly when government debt is scarce. In a couple of papers (this one and this one) I explore how QE might matter in the context of binding collateral constraints. First, if long-maturity government debt is worse collateral than is short-maturity debt, then central bank purchases of long-maturity government debt matter. As well, if the central bank purchases private assets, this can circumvent suboptimal fiscal policy that is excessively restricting the supply of government debt. But in both cases this works in perhaps unexpected ways. In both cases, unconventional asset purchases by the central bank act to reduce inflation.


  1. What would be a good example of c2 goods? Durables?

    1. Actually, in the paper, the payments instrument is not attached to specific goods. I just used that here so you could get straight what is going on. In practice, we know that the size of the payment matters for what you use - debit card, credit card, currency. Also, the circumstances under which you have to make the payment (what technology you have access to) matter. You could build that stuff into a model if you wanted to do quantitative work.

  2. It's maybe not directly related to the point in this post, but I have a question on the model in your paper.

    You assume that money is non-interest bearing. At the moment, we are in a position where, at the margin, money is interest bearing because of the interest paid on excess reserve balances and because money is not scarce.

    Does this mean that some of the analysis may also apply away from the zero bound, assuming we still had excess reserve balances and interest on reserves? In other words, in that situation would it still be irrelevant if the central bank swapped money for bonds, unless they also changed the rate paid on reserves?

    1. Interpret the "money" in the model as currency. In some of the work I link to in the post, I include interest-bearing reserves. You're correct, in that, with positive interest-bearing reserves, and given the interest rate on reserves, everything works in essentially the same fashion as at the zero lower bound. But monetary policy actually works in effectively the same way away from the zero lower bound, whether there are reserves outstanding or not. With no reserves outstanding, open market operations determine the short-term nominal interest rate. With reserves outstanding bearing interest, the interest rate on reserves determines the short-term nominal interest rate.

  3. Could you give us a concept behind the u'(y) term? I think you intend to indicate that the bigger private economy is dependent upon the output of the individual..

    Also, the term n is the household labor supply. The Ezero equation is maximized with n at zero. It seems to me that some constraint that n is not zero should be stipulated.

    Thanks for beginning from scratch. I can come close to following your thought pattern.

    1. The paper fleshes out the model. Individual households have to produce in order to consume. Every period, members of the household go out to purchase goods with assets while the producer/seller in the household works to produce output, and sells it for assets. Then the household takes the assets earned from selling output into the next period, trades on asset markets, etc.

    2. Thanks. I will continue to study the post.

  4. Steve: comedy of errors: I now think I misunderstood what you are saying. And that you misunderstood what Paul Krugman is saying. See dlr's comment on my post:

    You are talking about raising inflation while the economy *remains* (permanently) at 0% nominal interest rates, even after inflation increases, right?

    Paul Krugman is talking about increasing inflation, and letting nominal interest rates rise above 0% when inflation rises.

    1. "Paul Krugman is talking about increasing inflation, and letting nominal interest rates rise above 0% when inflation rises."

      I don't think that's quite right. It seems he thinks, as many people do, that the central bank, by keeping the nominal interest rate at zero indefinitely, can ultimately increase inflation.

  5. "Unconventional asset purchases by the central bank act to reduce inflation."

    In your model, yes, in reality, no.

    1. And how do you form your views about "reality?"

    2. Not by playing postmodern games and putting it in quotation marks, not by wrongly predicting hyperinflation, not admitting your mistake and using a different model (if you used any).