'Low Interest Rates Are Contractionary'

I wanted to delve a little bit deeper into Scott Sumner's claim that low interest rates are contractionary. Unfortunately, the model he uses to prove this is complete nonsense, since it requires the central bank to control the money supply and the nominal interest rate at the same time to achieve the result. Because of this, I think a dynamic analysis is pertinent.

Consider a dynamic and slightly altered version of Scott's money demand model:

$$(1)\: m_t - p_t = y_t - \alpha i_t$$

where $m_t$ is the (log) money supply, $p_t$ is the (log) price level, $y_t$ is the (log) of output, and $i_t$ is the nominal interest rate. Since the model is dynamic, I will also add the Fisher relation and the Euler equation:

$$(2)\: i_t = r_t + E_t\pi_{t+1}$$

$$(3)\: y_t = E_t y_{t+1} - \sigma r_t$$

where $r_t$ is the real interest rate and $\pi_t$ is the inflation rate ($\pi_t = p_t - p_{t-1}$).

Currently this model lacks an aggregate supply curve, so, for the time being, I'll go with a vertical AS curve for simplicity: $y_t = 0$. The central bank also sets the money supply in period $t+1$ such that $p_{t+1} = \bar p$. In this case, what happens when the central bank increases the money supply?

Well, if we simplify the model slightly, we can reduce it to

$$(4)\: p_t = \frac{m_t + \alpha\bar p}{1 + \alpha}$$
That is, an increase in the money supply causes the price level to rise and the nominal interest rate to fall ($\frac{\partial p_t}{\partial m_t} = \frac{1}{1 + \alpha} < 1$) -- interestingly the conventional result.

Of course Scott is now screaming at me for committing the same sin as Krugman by fixing the future price level. Don't worry, I'll get to a more complex model, I just wanted to show that Sumner's result doesn't make sense in this pseudo-dynamic context (that, I might add, is already much better than his lazy static model. The same goes for Krugman's 1998 model, which was also dynamic).

If we relax the assumption that the price level is fixed in the next period, the model instead simplifies to
$$(5)\: p_t = E_t\frac{1}{1 + \alpha}\sum^\infty_{j=t}\left(\frac{\alpha}{1+\alpha}\right)^{j-t}m_j$$
From this we know that the current price level is a function of the discounted sum of expected money supplies. This is basic market monetarist stuff: if an increase in the money supply is expected to be immediately reversed, the price level will not rise. Keep note of the fact that, in contrast with most market monetarist assertions, an expected reversal far in the future has a highly diminished effect on the price level. Expected monetary tightening ten years from now is as good as useless.

The point here, though is that a central bank can effect a reduction in the nominal interest rate without increasing the current money supply -- it simply has to reduce expected money supply growth. My point here is that Sumner never brought this up, and, since his reasoning is based entirely on the static version of the model, he has no right to say that his model implies contractionary low interest rates.

Changing aggregate supply only makes the model more difficult to understand, but, if the aggregate supply curve became
$$(6)\: y_t = f(p_t)$$
it would then be possible to argue that expected monetary tightening has an adverse effect on the economy. At this point, though, I don't think anyone should care. Sumner may have been claiming this, but his model certainly didn't justify the claim, so he should have been ignored.

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.

The Trouble With The Zero Lower Bound

Most of the time, it seems that the monetarist view of inflation is pretty much correct. Inflation roughly tracks the monetary base and velocity is pretty stable and almost directly follows short term interest rates. Unfortunately, there is this thing called the zero lower bound that seems to throw monetarism off.

The US has been at the zero lower bound twice in the last 150 years, and both times monetary expansion has seemed to have an irrelevant - even a negative - impact on inflation.

Here's 1934-1945:

And here's 2009-2015:

Most monetarists seem to have trouble coping with the irrelevance of the monetary base at the zero lower bound, even though it does seem to be part of a lot of basic monetary models. Take the most simple of money demand functions - cash-in-advance. It is easy to figure out that as long as there is a cost to the household incurred by holding money, the cash-in-advance constraint will bind, but whenever there isn't a cost, the constraint ceases to bind. This effectively means that, rather than being stuck at unity, the velocity of money is indeterminate; increases in the money supply will no longer have any effect on the price level.

Money-in-the-utility-function models have similar properties in the sense that velocity also becomes indeterminate. MIUF models are slightly strange though because money demand itself actually goes to infinity when the zero lower bound binds. But, MIUF is a pretty bad assumption anyway, so it's fine to ignore this.

An easy modification to CIA models that, when calibrated properly, might be able to make them match the data pretty well is the addition of a non-cash good to the economy. The income-velocity of money will now fluctuate with the nominal interest rate while the effects above will still be present.

I digress, the key idea of this kind of rambling post is that the zero lower bound seems to do strange things to monetary policy which precludes central banks from being omnipotent as some would suggest...

In Theory, Monetary Offset Doesn't Work

Monetary offset, one of the major aspects of Market Monetarism, is severely hindered by the application of some basic macroeconomic theory. Take the bond pricing equation at the heart of most of modern macro:

$$ c_t = \left[\frac{(1 + \rho)(1+E_t \pi_{t+1})}{1 + i_t}\right] E_t c_{t+1} $$

$ c_t $ is current consumer spending, which is chosen in order to maximize all expected future consumption. When the real interest that can be earned on saving or investment increases above its natural rate, $ \rho $, current consumer spending falls. In normal times, the central bank targets an inflation rate, $ \pi_t $, which anchors inflation expectations to that target, so any adjustment in the nominal interest rate, $ i_t $, directly changes current consumption. If the government decides to actively reduce its budget deficit which depresses GDP, the central bank can lower the nominal interest rate to offset this change. This is the theoretical explanation for monetary offset. 

When the nominal interest rate is at zero, the central bank can no longer lower the nominal interest rate to counteract the effects of austerity. It only has two options: somehow increase inflation expectations or promise to keep future interest rates low. This is exactly what the Federal Reserve has resorted to in the last few years. Quantitative easing has increased inflation expectations and forward guidance has given the promise of an extended period of low rates. Perhaps expanding the monetary base like there's no tomorrow can have some effect in both keeping rates low and increasing inflation expectations. Nevertheless, neither of these policies are proven to work either in theory or in practice. Central banks are, for all intents and purposes, ineffective at increasing consumption at the zero lower bound.

Central banks slowly lose what small ability to control the economy they have as the economy returns to its natural state on its own. Inflation expectations are bound to fall as consumption returns to its natural level. At the zero lower bound, the steady state expected inflation rate falls $ \frac{1}{1+\rho}-1 $. This at least partially explains the multi-decade long period of low inflation and zero nominal interest rates in Japan. As the economy returns to equilibrium, the central bank will be progressively more powerless to offset shocks, fiscal or otherwise.

The key problem with Market Monetarism in general is that it isn't grounded in any kind of model outside of conjecture around a simple version of AD-AS in which the central bank has the ability to achieve any inflation or nominal GDP target it chooses. The reality is, central banks are not omnipotent and aggregate demand is more than just a negative function of the price level.

The Nominal Interest Rate and Inflation Determination

The dynamics of inflation in relation to the nominal interest rate are generally assumed to be governed by the liquidity effect. That is, increases in the money supply temporarily decrease the nominal interest rate because of money demand and some form of nominal rigidity. In a perfectly friction-less world, increases in the money supply (particularly increases in the growth or future path of the money supply) cause the nominal interest rate to increase instantly, rather than after the economy returns to its natural level (some economists would say equilibrium, but defining equilibrium as simply a solution to a model makes more sense to me). Of course, explaining this whole thing with math is a whole lot more descriptive, so here I go:

I'm going to assume households have a linear utility function and derive utility from consumption, so $ u(c_t) = \ln c_t $ where $ c_t $ is consumption. This implies the "consumption Euler equation" that is used to determine inflation given the nominal interest rate.

$$ (1) \:  i_t = \left(\frac {E_t c_{t+1}}{c_t}\right)\left(\frac {1 + E_t \pi_{t+1}}{\beta} \right) - 1 $$

$ i_t $ is the nominal interest rate, $ \pi_t $ is the rate of inflation, and $ \beta $ is the constant discount factor. Assume consumption grows at a constant gross rate "$g$",

$$ (2) \: c_t  = g c_{t-1} $$

and the interest rate equation becomes

$$ (1a) \: i_t = g \left(\frac {1 + E_t \pi_{t+1}}{\beta} \right) - 1 $$

Now, if the central bank targets inflation so that it evolves according to

$$ (3) \: \pi_t = \bar \pi + \rho (\pi_{t-1} - \bar \pi) + \epsilon_t^\pi $$

where $ \bar \pi $ is the "trend" rate of inflation, $0 < \rho < 1$ is the "shock stickiness" parameter, and $ \epsilon_t^\pi $ is white noise, and $ E_t \epsilon_{t+1}^\pi = 0 $, then expected inflation, $ E_t \pi_{t+1}$, is defined by

$$ (3a) \: E_t \pi_{1+1} = \bar \pi + \rho (\pi_t - \bar \pi) $$

To fill in the model, the final interest rate equation becomes

$$ (4) \: i_t = g \left( \frac {\bar \pi + \rho (\pi_t - \bar \pi)}{\beta}\right) - 1 $$

This shows essentially what the "Neo-Fisherian" assertion is. The nominal interest rate and inflation rise with each other. In fact,

$$ (5) \: \frac {d i_t}{d \pi_t} = \frac {g \rho}{\beta} $$

So, if the rate of inflation increases by 1%, then the nominal interest rate will increase by $ \frac {g \rho}{\beta} $%.

Of course no monetarist will be happy until I use the money supply as a determinate of the price level, so assume a cash in advance constraint:

$$ (6)\: M_t = P_t c_t $$

where $ M_t $ is the money supply and $ P_t $ is the price level. Since consumption still grows according to (2), the interest rate equation is redefined to

$$ (7)\: i_t = g \left( \frac {E_t P_{t+1}}{P_t\beta} \right) $$

$ M_t $ grows at gross rate "$ m_t $", so its law of motion is

$$ (8)\: M_t = m_t M_{t-1} $$

and, given (6), 

$$ (9)\: E_t P_{t+1} = \frac{m_{t+1} M_t}{g c_t} $$

Integrating all this back into (1) gives

$$ (10)\: i_t = \frac{m_{t+1}}{\beta}-1 $$

If the central bank permanently increases $ m_t $, which is equivalent to $ \pi_t +1 $, by 1%, the nominal interest rate will increase by $\frac{1}{\beta}$% ($ \frac {d i_t}{d m_t} = \frac {1}{\beta} $).

Basically, absent nominal rigidity, the nominal interest rate and inflation have a positive, even causal, relationship not afforded to them by conventional wisdom. Of course, this is really driven by the way that the money supply interacts with the nominal interest rate. "Neo-Fisherism" is really an incomplete hypothesis because of this. More focus should be given to the effects of open market operations as non-nominal-rigidity ways of explaining the liquidity effect.

Fed "Tightening" May Bring Inflation

When the Fed raises rates this year, the plan is to do it differently than usual. Rather than reduce the supply of reserves, it will simply raise the interest rate on excess reserves, the effective lower bound to the federal funds rate. There are a few factors playing into this, namely the fact that there are roughly 3 trillion dollars of excess reserves now rather than the previously normal ~50 billion dollars. In order to effect a rate increase, the Fed would have to reduce the supply of reserves to about 1/60th of its current level. Given the impracticality of this, an increase in the IROER (interest rate on excess reserves) seems logical.

An interest rate increase is usually considered synonymous with the attempt of a central bank to decrease inflation. Despite this, given that there will be no change in the money supply, normal monetarist logic would imply that the rate hike would have no effect on inflation whatsoever. $ M_t V_t = P_t Y_t $ with no changes in $ Y $ or $ V $ shows this; $ M $ doesn't change, so $ P $ goes nowhere. Of course, adding a simple ad-hoc money demand equation changes things a little, but not in the expected direction. With $ \log M_t + \eta i_t = \log P_t + \log Y_t $, it becomes clear that raising the interest rate $ i_t $, without decreasing $ M_t $ (assuming $ Y $ is exogenous and constant again) will cause the price level to rise.