Market Monetarism and Multiple Equilibria

I was reading Scott Sumner's recent post about Neo-Fisherism and I had an epiphany about Market Monetarism.

I have consistently taken issue with the likes of Sumner because of what I view as his confusion of policies and the results of policies, or rather his confusion of policies and observations. Case in point would be the post I was just reading, in which Sumner says

But how did the Swiss authorities make sure this decrease in interest rates had a contractionary impact? The answer is simple; they did a simultaneous, once and for all, massive appreciation in the SF.

This idea that the exchange rate is just something that the SNB can set (while setting interest rates) is terrifyingly stupid from a conventional viewpoint. In general monetary policy can be seen through one of two lenses: 1) the central bank sets the monetary base and everything else is endogenous or 2) the central bank sets the short term risk free interest rate and everything else (including the monetary base) is endogenous.

Sumner regularly flaunts this view by suggesting that a central bank can, for example, set the interest rate and set the exchange rate at the same time. This is where my epiphany comes in. What's really going on is that there are multiple equilibria. A cut in the nominal interest rate can either occur through an increase in the money supply (ceterus paribus) or a negative shock to money demand (ceterus paribus), but which occurs in a given scenario? Obviously it's almost always a combination of both -- the monetary base is rarely constant and neither is money demand, but absent the presence of a more complete model, what I just outlined has many possible equilibria.

Sumner's solution to this problem is, rather than more completely specify the model, to simply choose the equilibrium that is consistent with the facts and assert that the central bank is responsible for bringing that equilibrium about. Do I think this approach is valid? Not exactly. Nevertheless I am much more sympathetic to it than what I previously perceived from Sumner. Furthermore even some full models exhibit multiple equilibria -- like New Keynesian models in which the Taylor Principle is not followed -- meaning that Sumner's approach to, e.g., Neo-Fisherism, while not as 'correct' as a more analytical description of the possible equilibria is at most equally egregious to Cochrane's dubious equilibrium selection.

To put this in terms more conducive to Sumner's typical line of reasoning, low interest rates can either be consistent with high NGDP or low NGDP. In his view the central bank chooses which equilibrium prevails and that equilibrium selection is the 'stance' of monetary policy. I personally don't think that assuming central banks are capable of equilibrium selection without explicitly modeling it is a good thing, but at least it's better than just assuming central banks are capable of pegging whatever nominal variable to whatever they want regardless of the circumstances.

More Issues With Neo-Fisherism

I always seem to be about a day late to the party, nevertheless I guess I'll present a little bit of a defense of the mainstream view before I get immensely busy.

I would first like to point out one issue that I have with both Stephen Williamson's and John Cochrane's attempt to show that even backward looking Phillips curves have Neo-Fisherian attributes. To my knowledge (that is, to the extent that they explained their models in their posts), Cochrane always retained perfect foresight in the Euler equation and Williamson always retained rational expectations, regardless of their model of inflation expectations. As this is integral to the model result, I expect that they would at least be up front about this assumption. Alas, no.

Neo-Fisherians, like most New Keynesians, have the disturbing habit of completely ignoring the money supply -- which they implicitly assume moves in a different way in response to changes in the nominal interest rate than most New Keynesians implicitly assume (note that I am not precluding Neo-Fisherians from being New Keynesians, the two are not necessarily exclusive, as Cochrane and Williamson have argued multiple times). Thus, I think it is at least important to frame this argument through the lens of a money demand function with interest elasticity.

As my only intention here is to highlight money supply dynamics, the model will involve completely flexible prices and focus solely on two periods.Variables in the current period will appear as $x$ while variables in the future period will appear as $x'$. Additionally, the final price level is fixed at $\bar p$. The money demand function is
$$m - p = -\alpha i$$
where $m$ is the money supply, $p$ is the price level, $\alpha$ is the interest elasticity of money demand, and $i$ is the nominal interest rate. The Euler equation is
$$i = p' - p$$
All variables, except $i$, are in logs.

In this model, the central bank sets the money supply $m$ and the future money supply $m'$, which determines $p$, $p'$, $i$, and $i'$.

Solving the model for $p$ given $m$, $m'$, and $\bar p$ yields:
$$p = \frac{m + \alpha \left[\frac{m' + \alpha \bar p}{1+\alpha}\right]}{1+\alpha}$$
If the central bank holds $m'$ constant and increases $m$, then $p$ will rise less than one for one with the $m$, which, given the money demand function, implies a lower nominal interest rate. This is, in essence, the conventional wisdom; the central bank engages in a temporary open market operation which raises the current inflation rate and lowers the nominal interest rate.

This result can be changed depending on how the central bank chooses $m'$. In fact, the central bank can set $m'$ such that the price in $p$ more than offsets the rise in $m$, thus giving the Neo-Fisherian result which, (warning, massive tangent) is rather ill-defined.

Williamson likes to define it in a way that favors the Neo-Fisherian argument but doesn't necessarily fit with his claim that raising the nominal interest rate results in higher inflation. Namely, he argues that, as long as a model suggests that a permanent increase in the nominal interest rate will eventually result in higher inflation, that that model is Neo-Fisherian. To me, this argument  (which I'll grant I haven't quoted from him, so if I am building a straw man feel free to call me out on it) sounds like saying "as long as a model has an Euler equation, has rational expectations, and has flexible prices (or equivalently has sticky prices but bans explosive solutions), that model is Neo-Fisherian." This works well with his definition, I suppose, but 1) I don't like his definition, 2) it doesn't necessarily mean that inflation will rise immediately very quickly, and 3) it doesn't say anything about non-permanent increases in the nominal interest rate.

In my opinion, a Neo-Fisherian result is one in which a temporary positive shock to the nominal interest rate delivers an immediate or almost immediate increase in inflation that is not offset by deflation in the periods preceding the higher inflation. Thus, I will happily admit that higher inflation and higher nominal interest rates are mutually consistent in the long run, but I will not concede that the way to get higher inflation immediately is to raise the nominal interest rate. As of yet, I do not believe any Neo-Fisherian has adequately made this argument (tangent over).

I digress, different paths for the money supply are consistent with different results for inflation and expected inflation, but a non-permanent increase in the money supply gives the conventional result of higher inflation and a lower nominal interest rate. However, if the central bank increases the future money supply by more than the current money supply, it is possible that the observed result will appear Neo-Fisherian: that is, $p$ increases more than $m$, which is consistent with immediately higher inflation and a higher nominal interest rate (lower demand for real balances). Is this really what Neo-Fisherians believe happens on the event of an interest rate increase? This weird higher nominal interest rate, lower real money supply, higher nominal money supply result is really strange and I highly doubt it happens with regularity. In fact, between 1956 and 2008, the only time that this consistently happened (with the monetary base to nominal GDP ratio replacing $m-p$ and the monetary base to real GDP ratio replacing $m$) was the late 1960's to early 1980's:

Given this reality, it should be possible to include that Neo-Fisherism is indeed not part of the current monetary policy regime (assuming the Federal Reserve has not abandoned the Taylor principle and the treasury is still Ricardian).

What 'Off-the-Shelf' Monetary Models Actually Say about Neo-Fisherism

Stephen Williamson wrote this in a blog post today:

Standard off-the-shelf monetary models essentially all exhibit a neo-Fisherian effect. That's nothing special. The Fisher effect is important. Typically increases in nominal interest rates lead to increases in inflation.

Testing this claim should be pretty simple, all we need is to get some "off-the-shelf" monetary models and see what happens when a central bank switches from one interest rate peg to another. I guess the only difficulty here would be that most standard monetary models exhibit indeterminacy when there's an interest rate peg, but I guess that doesn't seem to phase Williamson.

Let's start with the most basic monetary model I can think of: Cash-In-Advance. I won't bother with much derivation here since I've already talked in detail about CIA models in previous posts, so here is a simple CIA monetary model (assuming that the nominal interest rate is always greater than the interest rate that money pays):
$$(1)\: M_t = P_t y $$
$$(2)\: R_t = \frac{1}{\beta}\frac{P_{t+1}}{P_t} $$
where $M_t$ is the money supply, $P_t$ is the current price level, $y$ is the constant level of output, $R_t$ is the gross nominal interest rate, and $\beta$ is the representative agent's discount rate.

It's clear from this that, by pegging the nominal interest rate, the central bank can peg the ratio of the future price level to the current price level to whatever it wants, but here's where the indeterminacy comes in. The actual current price level is not determined under a pure interest rate peg; only the expected rate of inflation. If the central bank raises the nominal interest rate, then the future price level will be higher than the current one, but what happens to the current price level is unclear.

If, instead of pegging the interest rate, the central bank decides to set the money supply to the desired level each period, the current price level is actually determined, but the relationship between the current price level and the nominal interest rate is still unclear. If, for instance, the central bank temporarily lowers the money supply in the current period then returns it back to it's previous level in the following period, the nominal interest rate will go up, but the current price level will fall. Any permanent changes in the money supply impact the price level without changing the nominal interest rate at all and any temporary increases in the money supply result in a higher price level and a lower nominal interest rate.

Of course, if the nominal interest rate is at the zero lower bound, the basic CIA model predicts that the money supply no longer determines the price level, so the effect of a change in the nominal interest rate on the price level is even more unclear. The only reliable way to exit the zero lower bound (without using fiscal policy) in a CIA model is to shrink the current money supply until the cash in advance constraint once again binds, which at the very least means that the current price level doesn't increase. Effectively, even when escaping the zero lower bound, increases in the nominal interest rate mean a lower current price level.

It's interesting that even this exceedingly simple monetary model fails to fully support the neo-Fisherian hypothesis. Granted, it comes closer than most models: a higher nominal interest rate given the current price level does mean a higher future price level, but this can be easily dealt with if we add some kind of nominal rigidity to the model. Furthermore, CIA models don't exhibit interest elasticity in money demand, which, if present, would mean that permanent increases in the money supply would result in a lower nominal interest rate and a higher price level. Perhaps Stephen spoke too soon.

Demystifying Neo Fisherism

Misunderstanding of monetary economics abounds in the econoblogosphere. Since I'd like to think I know a decent bit about this issue, I think I might try and clarify some things with a pretty simple model.

There exists a household with the utility function $U = E_0 \sum^\infty_{t=0} \beta^t \left(u(c^1_t) + u(c^2_t)\right)$ where $0 < \beta < 1$ is the household's discount factor, $E_t$ is the rational expectations operator given information known in period $t$, $c^1_t$ is a consumption good that can be purchased using cash only, and $c^2_t$ is a good that can be purchased using cash or credit. The household uses government bonds and money carried from the last period as well as a constant endowment to purchase government bonds, money, and both consumption goods and to pay a lump sum tax levied by the government. The household's budget constraint is
$$ (1.1)\: M_{t-1} + B_{t-1} + P_t y = M_t + Q_t B_t + P_t \tau_t + P_t (c^1_t + c^2_t)$$
where $M_t$ is the money supply that will be carried into the next period, $B_t$ is the stock of government bonds that will be carried into the next period, $Q_t$ is the price of government bonds maturing in period $t+1$, $P_t$ is the price of both consumption goods, $y$ is the endowment, and $\tau_t$ is the real lump sum tax. $c^1_t$ must be paid for in cash, so the household faces a cash in advance constraint where it must hold at least enough money to cover $P_t c^1_t$.
$$ (1.2)\: M_t \geq P_t c^1_t $$

I assume that the government sets $B_t = 0\: \forall t$, so the government's budget constraint, given zero government bonds, is
$$ (1.3)\: M_t + P_t \tau_t = M_{t-1} $$
The government sets the lump sum tax so that $M_t = \mu_t M_{t-1}$ where $\mu_t$ is an exogenous policy parameter set by the central bank.

 The household maximizes $U$ subject to $1.1$ and $1.2$ which gives the following maximization problem
$$ (2)\: \mathcal{L} = U + \lambda_t \left(M_{t-1} + B_{t-1} + P_t y - M_t - Q_t B_t - P_t \tau_t - P_t (c^1_t + c^2_t)\right) + \gamma_t \left(M_t - P_t c^1_t\right)$$
which  yields
$$(2.1)\:\frac{\partial \mathcal{L}}{\partial c^1_t} = \beta^t u'(c^1_t) - \lambda_t P_t - \gamma_t P_t = 0$$
$$ (2.2)\: \frac{\partial \mathcal{L}}{\partial c^2_t}= \beta^t u'(c^2_t) - \lambda_t P_t = 0 $$
$$ (2.3)\: \frac{\partial \mathcal{L}}{\partial B_t} = -\lambda_t Q_t + E_t \lambda_{t+1} = 0 $$
$$(2.4)\:\frac{\partial \mathcal{L}}{\partial M_t}=-\lambda_t + E_t \lambda_{t+1} +\gamma_t=0$$

$2.1-4$ and $1.3$ can be combined to form an equilibrium for $P_t$, $c^1_t$, $c^2_t$, $M_t$, and $Q_t$:
$$ (3.1)\: u'(c^1_t) = u'(c^2_t) (2  - Q_t) $$
$$ (3.2)\: M_t = P_t c^1_t $$
$$ (3.3)\: y = c^1_t + c^2_t $$
$$ (3.4)\: u'(c^2_t) = \beta u'(c^2_t) \frac{1}{Q_t}E_t\frac{P_t}{P_{t+1}} $$
$$ (3.5)\: M_t = \mu_t M_{t-1} $$

With the equilibrium, it is possible to get a bit of an answer to the questions that Neo-Fisherians raise. Firstly, the long run inflation rate is equal to the growth rate of the money supply and the euler equation shows that, in the long run, the inflation rate is a constant different from the nominal interest rate. This means that, were the central bank to choose a low path for $\mu_t$, both inflation and the nominal interest rate would be lower. Of course, that's completely standard, it's just a lot more sensible to have a model where it's clear that this is a long run tightening of monetary policy. (Point Neo Fisherians)

This means that a disinflation, i.e. a reduction in the path of $\mu_t$, is consistent with a low nominal interest rate in the long run. In the short run, a higher value of $\mu_t$ can either take the form of higher inflation or lower interest rates. This is because a higher value of $Q_t$ (the inverse of the nominal interest rate) induces the household to shift demand from the credit good to the cash good because the nominal interest rate represents a cost to holding cash (and therefore buying the cash good) which can almost be considered a "shadow price" for the cash good. When the "shadow price" falls, as happens when the nominal interest rate falls, $c^1_t$ goes up which, given equation $3.2$, puts downward pressure on the price level. Because of this effect, increases in $\mu_t$ in the short run result in lower interest rates. The effect is exacerbated if the money supply is assumed to be auto-regressive. (Point everyone else)

The real problem with Neo Fisherism, as John Taylor points out in the post that Cochrane links to, is that the money supply is not modeled. High interest rates mean that the future price level is high relative to the current price level, but does that mean that the current price level has fallen to produce this, or that the future price level has increased? Adding the money supply solves this entirely. Interest rates can be high because the future money supply has been raised relative to today or because the current money supply has been reduced; only now the central bank has complete control over it.

The addition of the cash and credit goods to the basic cash in advance framework helps to illustrate that some (pseudo) non-neutrality of money can cause low interest rates and high expected inflation to coincide, something that doesn't happen in New Keynesian models unless the Taylor Rule has extremely persistent shocks. Also key here is that the interest rates are indicative of expected inflation, not current inflation and any apparent relationship with current inflation is either coincidence -- because the money supply auto-regresses, e.g. -- or a result of temporary money non-neutrality.

Also, the idea that forcing interest rate to be low actually causes high inflation is completely wrong; it's all about the money supply, and high inflation only happens if the money supply is growing quickly. Deliberately setting a low nominal interest rate must eventually result in low money growth (unless you are in a liquidity trap. See here), so it's pointless to suggest such a policy in the hopes of deliberately causing higher inflation. The endgame is to stop thinking about monetary policy in terms of interest rates at all and switch to thinking about movements in the money supply.

Notes on Taylor Rules, Inflation, and Neo-Fisherism

I've been working on writing a paper outlining my views on three topics in monetary economics (the three things in the title). Click here for the pdf of what I have written so far. Here is the text if you don't want to download the pdf:

UPDATE: The pdf link should update automatically to changes, but I won't change the text in the blog post. Just download a copy of the pdf every time you want the most up to date version.

Introduction:

There has been quite a bit of discussion about the relationship between the nominal interest rate and the rate of inflation recently among economists. To my knowledge, the problem began when Cochrane (2007) challenged the idea that the inflation rate could be determined with simply a Taylor Rule and a Fisher relation in combination with a commitment to active monetary policy and implicitly passive fiscal policy (see Leeper (1991) for example). Cochrane's key insight was that, in these models, the central bank is essentially committing to cause inflation to explode by increasing the nominal interest rate (effectively the expected inflation rate) more than one for one with current inflation. Because economists had ruled out explosive solutions, the only other equilibrium – one in which the inflation rate jumps immediately at period zero to the central bank's target – was considered. As Cochrane noted, there is not necessarily any reason to rule out explosions in nominal variables as they have no impact on the real economy in the models in question.

\par In the years since then, the failure of zero interest rate policies to generate inflation became of interest. Pretty soon, a similar yet entirely different debate came into existence. A few of economists (to my knowledge, Williamson and Cochrane) had the novel idea that the nominal interest rate had a causal relationship with the rate of inflation. This notion had long existed in the literature and is even a property of just about every macroeconomic model; the problem, in fact, was not the notion that high inflation and high interest rates happened at the same time. Rather, it was the idea that central banks could deliberately cause inflation by setting the nominal interest rate at a higher level. The consensus that active monetary policy was required for inflation stabilization and that positive deviations from the target interest rate implied by a Taylor Rule would result in lower inflation was in direct opposition to these "Neo-Fisherian" claims, so a debate that pulled in a slew of other economists ensued.

The difficulty in this case is that both sides are right in their own way. The consensus was correct that, so long as the central bank uses a Taylor Rule to target inflation, positive deviations from that target would result in a lower inflation rate. The Neo-Fisherian view is correct in the sense that if the central bank does not follow a rule and deliberately loosens monetary policy, the inflation rate and the nominal interest rate will increase. The issue with both views is that the underlying assumptions are either not understood or not made clear by there proponents. Economists putting forward the conventional wisdom don't make it clear that the Taylor Rule is the sole cause of inflation dynamics in their model and Neo-Fisherians fail to put forward that the result that they purport is highly dependent on how the money supply (or in some cases fiscal policy) acts when the nominal interest rate is increased. Each model relies heavily on a set of implausible assumptions about the way central banks behave. It is clear that central banks don't behave in the way implied by the consensus models and it is equally clear the the Neo-Fisherian result only occurs when monetary policy has taken a permanently more accommodative stance; even though this assumption is not put forward by its proponents.

If, as I suggest, the "Neo-Fisherian problem" and the "Taylor Rule problem" are all about assumptions, then their respective solutions are simple: just add some microfoundations. When it comes to arguments about monetary policy, the necessary microfoundation is painfully obvious. These models all need money in order for their implications to be understood. Interest elastic money demand functions solve Cochrane (2007)'s critique as they prohibit real explosions of the money supply – something that would happen if the nominal interest rate expanded or collapsed infinitely and money demand functions in general can determine when high interest rates mean tight monetary policy and when high interest rates mean loose monetary policy without appealing to dynamics implied by implicit monetary policy rules and without simply assuming that all high interest rates are do to loose money. 

Model:

We will begin by adding a simple ad-hoc money demand function to a two equation frictionless New Keynesian model and looking into the implications of the simple addition for monetary modeling. As usual, there is a Fisher equation relating the nominal interest rate to expected inflation and a Taylor Rule relating current inflation to the nominal interest rate.

$$i_t = \rho + E_t \pi_{t+1}$$

$$i_t = \rho + \phi \pi_t$$

$i_t$ is the nominal interest rate, $\pi_t$ is the inflation rate, $E_t$ is the period $t$ rational expectations operator, $\phi$ is the "inflation reaction parameter" on the Taylor Rule, and $\rho$ is the constant real interest rate. The sole addition that we will add to this basic model is a simple money demand function which sets real money demand equal to 

$$m_t - p_t = y - \eta i_t$$

where $m_t$ is the nominal money supple, $p_t$ is the log price level ($\pi_t = \Delta p_t$), $y$ is the (constant) level of output, and $\eta$ is the interest-elasticity of the money supply.

\par With the addition of the money demand function, so long as $\left|\eta\right| > 0$, Cochrane's problem with ruling out nominally explosive equilibria disappears. Now, a real variable depends on the nominal interest rate and prevents hyper inflations that are not caused by excessive money growth. In fact, adding money demand changes nothing about the dynamics of the model; following a Taylor Rule still gives the conventional wisdom about monetary policy without having to deal with the difficult problem of ruling out nominal explosions.

\par The interesting thing about this model is that it can replicate the Neo-Fisherian result easily. Consider a deterministic economy where the central bank permanently increases the growth rate of the money supply, $m^g_t = \Delta m_t$, from $m^g_0$ to $m^g_1 > m^g_0$.

If you have any feedback or suggestions before I continue to write, feel free to comment. I intend to continue by expanding my analysis to a more full fledged New Keynesian model with different types of money demand ranging from Money-In-The-Utility-Function to Cash-In-Advance and explain the mixed signal problems of using interest rates as an indicator of the stance of monetary policy. (I also plan to refer to more of the relevant literature than just "Determinacy and Identification with Taylor Rules")

Analyzing Neo-Fisherism (Warning: Highly Technical)

I've spent the last couple of days trying to make sense of the nuance behind Neo-Fisherian models and I've found that there are a two specific requirements for the result that John Cochrane gets in "Monetary Policy With Interest on Reserves" (pdf):

1. The effects of an interest rate shock are highly fiscal policy dependent. In Cochrane's model, fiscal policy is non-ricardian, but the effect is the same if fiscal policy is active as described by Leeper (1991). If fiscal policy is passive, then conventional wisdom holds so that inflation reacts negatively to an interest rate shock.

2. Cochrane's result only happens when monetary policy moves from one interest rate peg to another. So, even in a fiscal dominant regime, inflation doesn't jump with interest rates if the central bank follows an interest rate rule (that must violate the Taylor principle).

Here is a comparison between a fiscal dominant (without an interest rate peg) and monetary dominant regime during a nominal interest rate shock:

 As you can see, inflation eventually rises in the fiscal dominant regime, but this is simply due to the persistence of the shock. The initial effects are, in fact, more severe than in the standard model. As stated before, Cochrane's result prevails when the central bank switches from a lower peg to a higher peg and vice versa, but switching from peg to peg as Cochrane's paper implies is hardly realistic, so it's safe to say that conventional wisdom would hold is most empirical cases, even under fiscal dominance. Perhaps the only situation in which Cochrane's inflation jumping result (to be replicated shortly) would occur is if a country like Japan decided to switch from what is effectively an interest rate peg regime to an inflation targeting regime with active monetary policy (this remains un-modeled due to the limitations of modelling software). 

Inflation

Nominal Interest Rate

So, under a fiscal dominant regime, raising the nominal interest rate once and for all results in this equilibrium, but allow the interest rate to follow a Taylor rule and this equilibrium is replaced by one with a harsher reduction in inflation than than the standard monetary dominant model.

Appendix:

Here is the model I used above in case you'd like to check it:

The government budget constraint is expressed in real terms and the real money supply is assume constant and equal to one, so seigniorage is simply expressed as the rate of inflation, $ \pi_t $:

$$ (1) \: b_t + \tau_t = (1 + \rho)b_{t-1} - \pi_t $$

$ b_t $ is the real bond supply, $ \tau_t $ is the lump sum tax levied by the government, and $ \rho $ is the constant real interest rate. The Fisher relation follows; relating the nominal interest rate to expected inflation:

$$ (2) \: i_t = \rho + E_t \pi_{t+1} $$

Fiscal policy is simply a function of the current stock of government bonds and can be adjusted by changing $ \phi^f $ between $ \phi^f > \rho $ for passive fiscal policy and $ \phi^f < \rho $ for active fiscal policy.

$$ (3) \: \tau_t = \phi^f b_t $$

Monetary policy can similarly be adjusted between passive ($ \phi^\pi \leq 1 $) and active ($\phi^\pi > 1 $) regimes. $ v_t $ is a shock term that follows an AR(1) process.

$$ (4) \: i_t = \rho + \phi^\pi \pi_t + v_t $$

$$ (5) \: v_t = \rho^v v_{t-1} + \epsilon_t $$

$ \rho^v $ is the persistence of the monetary policy shock and $ \epsilon_t $ is white noise.

During the deterministic simulation, the interest rate feedback rule is removed and the nominal interest rate is pegged exogenously.

Updates:

The response to the interest rate shock seems to depend on whether or not the shock is expected. There is strangely always a drop in inflation on the period when agents get news of the shock, but inflation does end up jumping when it comes into effect.

The stochastic simulation becomes a bit more informative when $ \phi^\pi $ is set to zero as there is just an initial drop in inflation in response to the shock and then a subsequent jump as inflation meets expected inflation.