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.