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).
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:
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