Thursday, April 14, 2016

Neo-Fisherian Denial

Accepting neo-Fisherism is a 12-stage program. The first stage is admitting you have a problem. The twelfth stage is helping others to admit that they have a problem too. Going from stage one to stage twelve may be a tough battle - many could temporarily fall off the wagon. But take it one day at a time. Most people, for example Larry Summers, are still at stage one. In this video, about two minutes in, after the jokes, Summers says that neo-Fisherism is most likely to be remembered as a confusion. So, if the problem is only confusion, I would like to help him out.

Neo-Fisherism says, basically: "Excuse me, but I think you have the sign wrong." Conventional central banking wisdom says that increasing interest rates reduces inflation. Neo-Fisherites say that increasing interest rates increases inflation. Further, it's not like this is some radical, novel theory. Indeed, a cornerstone of Neo-Fisherism is:

Neo-Fisherian Folk Theorem: Every mainstream macroeconomic monetary model has neo-Fisherian properties.

Let me illustrate that. A nice, simple, version of the standard New Keynesian (NK) model is the one in Narayana Kocherlakota's slides from this conference put on by the Becker Friedman Institute. I'll use my own notation. NK's version of the NK model is a reduced form, with two equations. The first comes from a pricing equation for a nominal bond - what's often called the "NK IS curve," or
Here, y is the output gap, the difference between actual output and what is efficient, pi is the inflation rate, R is the nominal interest rate, r is the subjective rate of time preference, and a is the coefficient of relative risk aversion. The second equation is
That's just a Phillips curve, with b >0 determined by the degree of price stickiness. In the underlying model, some fraction of firms is constrained to set prices to the average price from last period. Thus, there's no expectations term in the Phillips curve, as there's no forward-looking pricing. That makes the model easy to solve.

So, substitute for y in equation (1) using the Phillips curve equation, to get
So, you can see why people think this type of model is a foundation for conventional central banking ideas. If inflation expectations are "anchored," which I guess means exogenous, on the right-hand side of the equation, then an increase in the current nominal interest rate would have to imply that the current inflation rate goes down. Indeed, if the central banker experiments, by choosing the nominal interest rate each period at random, then he or she will observe a negative correlation between inflation and nominal interest rates, which would tend to confirm conventional beliefs.

But consider the following. Suppose we look at the deterministic version of the model, and use (3) to solve for a first-order difference equation in the inflation rate:
Then, an equilibrium is a sequence of inflation rates solving (4), and we can solve for output from (2). As is typical of monetary models, there's no initial condition to tie things down, so there are potentially many equilibria. We can say, however, that in a steady state, from (1),
And then (2) gives
So, what "anchors" inflation and inflation expectations in the long run is the long run nominal interest rate. And then the Phillips curve determines output. That's the first Neo-Fisherian property of this standard model.

Next, from the difference equation, (4), if the nominal interest rate is a constant R forever, then there is a continuum of equilibria, indexed by the initial inflation rate, and they all converge to a unique steady state, which is given by (6) and (7). To see this, start with any initial pi, and solve (4) forward. So, we know that the long run is Fisherian. But what about the short run?

We'll consider the transition to a higher nominal interest rate. In the figure, the nominal interest rate is constant until period T, and then it increases permanently, forever. In the figure, D1 is the difference equation (4) with a lower nominal interest rate; D2 is (4) with a higher nominal interest rate. We'll suppose that everyone perfectly anticipates the interest rate increase from the beginning of time. Again, there are many equilibria, and they all ultimately converge to point B, but every equilibrium has the property that, given the initial condition, inflation will be higher at every date than it otherwise would have been without the increase in the nominal interest rate. A straightforward case is the one where the equilibrium is at A until period T, in which case the inflation rate increases monotonically, as shown, to a higher steady state inflation rate. Inflation never goes down in response to a permanent increase in the nominal interest rate. That's consistent with what John Cochrane finds in a related model.

So, that's the second Neo-Fisherian property, embedded in this NK model. The NK model actually doesn't conform to conventional central banking beliefs about how monetary policy works. What's going on? From equation (1), an increase in the current nominal interest rate will increase the real interest rate, everything else held constant. This implies that future consumption (output) must be higher than current consumption, for consumers to be happy with their consumption profile given the higher nominal interest rate. But, it turns out that this is achieved not through a reduction in current output and consumption, but through an increase in future output and consumption. This serves, through the Phillips curve mechanism, to increase future inflation relative to current inflation. Then, along the path to the new steady state, output and inflation increase. But, if you read Narayana's Bloomberg post from five days ago, you would have noted that he thinks that lowering the nominal interest rate raises inflation and output:
Monetary policy makers should be seeking to ease, not tighten. Instead of satisfying a phantom need to “normalize” rates, the Fed should do what’s needed to get employment and inflation back to normal.
Apparently he's thinking about some other model, as the one he constructed tells us the opposite.

For more depth on this, you should read this paper by Peter Rupert and Roman Sustek. Here's their abstract:
The monetary transmission mechanism in New-Keynesian models is put to scrutiny, focusing on the role of capital. We demonstrate that, contrary to a widely held view, the transmission mechanism does not operate through a real interest rate channel. Instead, as a first pass, inflation is determined by Fisherian principles, through current and expected future
monetary policy shocks, while output is then pinned down by the New-Keynesian Phillips curve. The real rate largely only reflects consumption smoothing. In fact, declines in output and inflation are consistent with a decline, increase, or no change in the ex-ante real rate.

Conventional central banking wisdom is embedded in Taylor rules. For simplicity, suppose the central banker just cares about inflation, and follows the rule
Here pi* is the central bank's inflation target. Under the Taylor principle, d > 1, i.e. the central bank controls inflation by moving interest rates up when inflation goes up - and the nominal interest rate adjustment is more than one-for-one. It's well known from the work of Benhabib et el. that Taylor rules have "perils," and this model can illustrate that nicely. The difference equation determining the path for the inflation rate becomes
In the next figure, A is the intended steady state in which the central bank achieves its inflation target, and that is one equilibrium. But there are many equilibria for which the initial inflation rate is greater than -r and smaller than the inflation target, and all of these equilibria (like the one depicted) converge to the zero lower bound (ZLB), where the central banker gets stuck, with an inflation rate permanently lower than the target. Potentially, there could be equilibria with an initial inflation rate higher than the inflation target, which have the property that inflation increases forever. But in this model, that also implies that output increases without bound, which presumably is not feasible.

Rules with -1 < d < 1 all have the property that there are multiple equilibria, but these equilibria all converge to the inflation target - there's a unique steady state in those cases. Note that the Taylor rule central banker is Neo-Fisherian if d < 0, and that this can be OK in some sense. But aggressive neo-Fisherism, i.e. d < -1 -2(a/b), is bad, as this implies that the inflation rate cycles forever without hitting the inflation target.

But if the central banker actually wants to consistently hit the inflation target, there are better things to do than (8). For example, consider this rule:
Plug that into (4), and you'll get
And so, (10) implies that
So, under that forward-looking Taylor rule, the central bank always hits its target, and in equilibrium the central bank is purely Fisherian. If it wants to increase its inflation target - and actual inflation - it just increases the nominal interest rate one-for-one with the increase in the inflation target. So, I've lost count now, but I think that's Neo-Fisherian property 3 [see the addendum below. There's a glitch that needs to be fixed in the rule (10) to account for the ZLB.]

The rule (10) specifies out-of-equilibrium behavior that kills all of the equilibria except the desired steady state. Why does this work? If the central banker sees incipient inflation in the future, he or she knows that this will tend to increase current output, increase current inflation, and increase future output, which will also increase current inflation. To nullify these effects, the central banker commits to offset this completely, if it happens, with an increase in the nominal interest rate. In equilibrium the central banker never has to carry out the threat. Maybe you think that's not plausible, but that's the nature of the model. NK adherents typically emphasize forward guidance, and that's not going to work without commitment to future actions.

Some people (e.g. Garcia-Schmidt and Woodford) have argued that Neo-Fisherian results go out the window in NK models under learning rules. As was shown above, these models are always fundamentally Fisherian in that any monetary policy rule has to somehow adhere to Fisherian logic on average - basically the long-run nominal interest rate is the inflation anchor. But there can also be learning rules that give very Fisherian results. For example, suppose that the economic agents in this world anticipated that next period's inflation is what they are seeing this period, that is
Plug that into equation (1), and we get
So, for this learning rule, inflation is determined period-by-period by the nominal interest rate - this is about as Fisherian as you can get.

Thus, if conventional central bankers are basing their ideas on some model, it can't be a mainstream NK model, since increasing the nominal interest rate makes inflation go up in mainstream NK models. But don't get the idea that it's some other mainstream model they're thinking about. As the Neo-Fisherian Folk Theorem says, all the mainstream models have these properties, though some of the other implications of those models differ. For example, it's easy to show that one can get exactly the same dynamics from Alvarez, Lucas and Weber's segmented markets model. That's a model with limited participation in asset markets and a non-neutrality of money that comes from a distribution effect. Everyone in the model has fixed endowments forever, and they buy goods subject to cash-in-advance. The central bank intervenes through open market operations, but the people on the receiving end of the initial open market operation are only the financial market participants. The model was set up to deliver a liquidity effect, i.e. if money growth goes up, this increases the consumption of market participants (and decreases everyone elses's consumption), and this will reduce the real interest rate. Thus, you might think (like the NK model) that this produces the result that, if the central bank increases the nominal interest rate, then inflation will go down.

But, the inflation dynamics in the Alvarez et al. segmented markets model are identical to what we worked out above. In fact, the model yields a difference equation that is identical to equation (4), though the coefficients have a different interpretation. Basically, what matters is the degree of market participation, not the degree of price stickiness - it's just a different friction. And all the other results are exactly the same. But the mechanism at work is different. The quantity theory of money holds in the segmented markets model, so what happens when the nominal interest goes up is that the central bank has to choose a path for open market operations to support that. This has to be a path for which the inflation rate is increasing over time, but at a decreasing rate. This will imply that consumption grows over time at a decreasing rate, so that the liquidity effect (a negative real interest rate effect) declines over time, and the Fisher effect increases.

So, once you get it, you can form your own Neo-Fisherian support group. Moving from denial to advocacy is important.

Addendum1: Thanks to Narayana. This took some work, but this is a Taylor rule that assures that the central banker hits the inflation target period-by-period, implying that the nominal interest rate is constant in equilibrium, and will move one-for-one with the inflation target. If future inflation is anticipated to be sufficiently high, then the central banker follows the forward looking rule (10):
This rule offsets incipient high inflation, and assures that the central bank hits the inflation target. But, low inflation is a problem for (16), as the ZLB gets in the way. So, if there is incipient low inflation, the central banker follows the rule:
And the critical value for future inflation is
How does (17) work? Any equilibrium has to satisfy (4), but (4) and (17) imply
So future inflation must be greater than the inflation target. But (17) says that the central banker chooses this rule only when future inflation is less than pi**, which is less than the inflation target. So this can't be an equilibrium. I like (17), as the central banker is Neo-Fisherian - he or she kills off low inflation with a high nominal interest rate.

Addendum 2: This is interesting too. Suppose the policy rule is
Then there is a critical value for the initial inflation rate,
such that, if the initial inflation rate is below this critical value, then the inflation rate goes to the inflation target in the next period and stays there. If the initial inflation rate is above the critical value, then the initial nominal interest rate is zero, and the inflation rate falls to the inflation target, and stays at the target forever. So, that's a Fisherian rule that has nice properties.

Addendum 3: Here's another one. Central bank follows rule (20) if current inflation is below the inflation target. Central bank follows rule (10) if current inflation is at or above the inflation target. With inflation below the target, this implies raising the nominal interest rate to get inflation to target. With inflation at or above the target, the central bank promises to raise the nominal interest rate in response to incipient inflation. At worst, this implies one period of inflation below target in equilibrium.

Thursday, April 7, 2016

Fiscal Theory of the Price Level, Helicopters, and Central Bank Balance Sheets

Last week, I attended a conference on the fiscal theory of the price level (FTPL) in Chicago. Eric Leeper claims that I'm actually a closet FTPL person, and seems to have thought I belonged there, which is certainly fine with me. The assignment was to talk about research ideas. Some people had papers, and some (like me) didn't. My slides are posted here.

Here's the idea. The FTPL came out of research done by Eric Leeper, Mike Woodford, Chris Sims, and John Cochrane, among others, beginning in the early 1990s, so this stuff has been around for quite a while. Indeed, there are precedents in the work of Sargent and Wallace, and Aiyagari and Gertler in the 1980s, for example. Sometimes FTPL practitioners seem to be shooting for an alternative quantity theory. Under the quantity theory of money - Old Monetarism basically - we were supposed to think that the demand for money was a stable function of some small set of observable economic variables, so that the supply of money by the central bank would determine the price level and inflation. Under the FTPL it's the quantity of government debt (or in some versions the quantity of consolidated government debt - including the central bank's liabilities) that's important, and it's also important that the counterpart of "demand" for this debt need not be stable. The FTPL starts with the consolidated government's intertemporal budget constraint and, typically, writes it (after some manipulation) with the real quantity of government debt outstanding on the left-hand side (nominal debt divided by the price level), and the expected discounted value of government surpluses on the right-hand side. Therefore, if the stuff on the right-hand side of this equation is given, the nominal quantity of government debt determines the price level. Alternatively, the expected discounted value of future government surpluses is the analog of the demand for money in the quantity theory of money, so reductions in the future government promises backing the government's debt will reduce the demand for government debt and its current value - increase the price level - given the supply of government debt in nominal terms.

John Cochrane provides some intuition in terms conventional asset pricing. That is, think of the expected discounted future government surpluses as analogous to the payoffs on any asset, and the real value of government debt as the value of the asset, and it all makes sense. That's not even an analogy - the intertermporal consolidated government budget constraint can literally be rewritten as an asset pricing equation.

OK, so where does this theory go then? Sometimes the FTPL people got sidetracked with issues such as whether the consolidated government budget constraint is a constraint or an equilibrium relationship, whether the government is similar to, or different from, a private household, in terms of its budget constraint, etc. Some of that discussion was unproductive for this research program, I think. Also, one might get the idea from this literature that central banks cannot fundamentally be independent, and that the fiscal authority is always in the inflation driver's seat, which I don't think is the right way to think about the fiscal-monetary interaction. But, the fiscal-monetary interaction is the message of the theory, and that's very interesting. For example, central banks have recently been engaged in quantitative easing policies, which sometimes look like conventional fiscal debt-management. That might be viewed as central banking jumping into the fiscal driver's seat.

So, what's in my slides? There are three parts to this: (i) a model with minimal government; (ii) a model with open market operations; (iii) a non-Ricardian model with a government debt shortage. The models are very simple, with no uncertainty. The minimal government (MG) model and open market operations (OMO) models are simple cash-in-advance setups with fixed endowments - output and consumption are fixed, and we're only going to be concerned with determining inflation and the price level. The non-Ricardian (NR) model has production.

Start with the MG model, and consider this a thought experiment. The basic idea is to show that we can set up a central bank with no connection to fiscal policy, and this central bank will have no problem determining the price level and inflation. There are households in this economy, and they have fixed endowments each period, but can't consume their own goods and can only trade subject to cash-in-advance - they need currency to buy goods. There is a role for the government, but it's minimal, i.e. the government grants a monopoly to a central bank to issue currency, and I'm supposing the government can costlessly enforce the monopoly. The government also issues shares in the central bank to the households, and constrains the central bank to turn over its profits, period-by-period, to its shareholders. The central bank can issue currency and reserves as liabilities, and uses the liabilities to finance lending to the households. Reserves cannot be used in retail transactions - purchases of goods - and they bear interest, otherwise people would not hold them. Reserves are convertible into currency, one-for-one, and vice-versa. I've assumed away banks, so reserves are just debt instruments of the central bank that anyone can hold.

The MG central banking structure has something in common with what was envisioned by the framers of the 1913 Federal Reserve Act, or with how the European Central Bank works. That is, money is injected through central bank lending rather than open market operations. We could add details like private banks and collateralized lending, but those things don't matter for the general ideas here.

What does the central bank do in the MG model? The model is dynamic, but think in terms of one-time policy decisions that give a stationary equilibrium - a constant inflation rate, and a constant quantity of reserves, in real terms. The central bank fixes the nominal interest rate it charges on its loans at a constant R forever. This does two things. First, it determines the spread the central bank earns on lending financed with currency issue, though the central bank makes zero profits intermediating loans by issuing reserves, as the interest rate on reserves is R in equilibrium. Second, through standard asset pricing, R determines the inflation rate - that's just Irving Fisher. This is an economy in which the real rate is a constant (we'll relax this later). Finally, the the central bank sets the nominal path for its profits, and the nominal path for its total liabilities. And that does it. This determines the initial price level, the inflation rate, and how the stock of central bank liabilities is split between reserves and currency.

Then, in the MG equilibrium, increasing R increases the inflation rate one-for-one, and the central bank can produce as much inflation as it wants. Holding all the other elements of central bank policy constant, a level increase in the stock of nominal outside money is irrelevant for prices and inflation. This just increases the quantity of reserves. That's a liquidity trap result, but you get the liquidity trap no matter what R is. Further, the growth rate in total outside money is disconnected from inflation. While the nominal stock of currency grows at the inflation rate in equilibrium, the total stock of central bank liabilities need not.

The key thing here is that the central bank determines prices and inflation without any fiscal support. If the idea you got from the FTPL is that fiscal policy is necessary to determine the price level and inflation, that's not correct.

Next, go to the OMO model in which the central bank buys and sells government debt, and there is a fiscal authority that can tax households lump sum. Otherwise the model is the same as MG, except now the central bank's profits are turned over to the fiscal authority. Here, we'll suppose that the fiscal authority determines on its own the real transfers over time that are required to support the government debt. At the first date, the fiscal authority issues debt, some of which the central bank purchases by issuing outside money (reserves and currency), and the fiscal authority then rebates the proceeds of the debt issue to households. Then, the fiscal authority levies future taxes to pay the interest on the government debt, which will be constant in equilibrium, so we can think about what is going on. This economy is Ricardian, so the present value of the fiscal authority's transfers is zero. But as in the MG economy, transfers with a positive present value are generated from the central bank's activities. Those transfers are determined by R, and we'll assume that fiscal policy is constant (government debt held constant in real terms) and does not depend on R.

In the OMO economy, monetary policy works much as in the MG economy. Inflation is determined in a Fisherian fashion, and the central bank can have a large balance sheet, but if there are positive reserve holdings forever, having more outside money in this economy has no effect. Given R, the extra money is held as reserves. Further, "helicopter drops" cannot produce more inflation. Given all the other features of policy, if the fiscal authority increases transfers - in real terms, say - then in this Ricardian economy that has no effect. Further, there will be no effect on prices if the central bank purchases the extra debt issued with outside money. The money will just be held as reserves - it's irrelevant whether the increase in transfers is financed with reserves or government debt. Indeed, the central bank could issue currency to finance the transfers, and this will just be converted into reserves and held that way - nothing happens.

In terms of the FTPL, the point here is that FTPL results are derived simply from assumptions that imply that the fiscal authority is in the inflation driver's seat. Here, I've just made natural assumptions about central bank independence and, again, the central bank determines the price level and can have as much inflation as it wants.

Now that we've got those basic ideas nailed down, we can get more sophisticated and think about an economy - the NR economy - with exchange involving secured credit and currency. This works a bit like a cash goods/credit goods model. Here government debt serves as collateral in credit transactions. If the collateral constraint binds, this implies that government debt carries a liquidity premium, and the real interest rate will be low. Then the economy is non-Ricardian. More government debt will relax the collateral constraint and improve efficiency, but we'll assume that the fiscal authority behaves suboptimally, and the central bank responds to that.

So, what happens?

1. If the collateral constraint binds, this implies the central bank should set R > 0 at the optimum. The friction that makes the real interest rate low implies that the central bank should not be at the zero lower bound.
2. Given R, expansion in the central bank's balance sheet is again neutral - swapping reserves for government bonds does not matter, provided these two assets serve equally well as collateral.
3. A policy of increasing transfers financed with government debt, with a permanent expansion in government debt, is beneficial policy, as this relaxes the collateral constraint. But it doesn't matter if this is financed by an increase in outside money (a helicopter drop). If so, the extra money is held as reserves. Further, this action doesn't increase inflation - it reduces inflation. If the central bank wants to increase inflation, it has to raise R. Neo-Fisher again.
4. If reserves are worse collateral than government debt (true in practice, basically, if we think of collateral as, in part, supporting bank liabilities) then a balance sheet expansion by the central bank is bad - it just tightens collateral constraints. This increases inflation alright, but there's a welfare loss.

So, the conclusions are:

1. FTPL forces us to think seriously about fiscal/monetary interaction, and that's very important. But fiscal support is not necessary for monetary policy to work, nor is it useful to think of fiscal policy determining inflation on its own - the central bank can indeed be independent.
2. Fiscal/monetary interaction becomes really important when we start thinking about the liquidity properties of government debt.
3. Helicopter drops? Forget it. This is not some cure-all for a low-inflation problem.
4. QE can be harmful, as it soaks up useful collateral and replaces it with inferior assets.
5. Neo-Fisherian denial is not good for you. Central banks that want to increase inflation need to increase nominal interest rates.