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).

Central Banks Can't Control Money Demand!!!!!!!

After posting too many lengthy comments on Scott Sumner's blog a while back, I decided I was going to ignore his claims that high interest rates are "expansionary" because they increase velocity. I tried, but he has crossed the line. He has elevated his already excessive list of central bank abilities to also include the ability to control money demand as well as the money supply.

NGDP = MB*(Base Velocity), where V is positively related to nominal interest rates.

Thus if you cut interest rates without increasing the money supply, then V falls and policy becomes more contractionary.

Read that a few times to get the full point that Sumner is making. His money demand equation is, of course, perfectly fine; you'd have to be ignorant to not understand that velocity is a positive function of nominal interest rates (or, in my preferred terminology, money demand is decreasing in nominal interest rates). The problem comes in when Scott says "if you cut interest rates without increasing the money supply." When you read this, sirens should have begun screeching in your head; Scott's weird partial equilibrium analysis in which there are three things that determine the nominal interest rate (money demand curve, interest rate chosen by central bank, and money supply) may be "monetary economics 101" in his mind, but it's certainly not economics 101, in which a monopoly supplier of something can't control its price and its quantity.

I mean, come on, this should be obvious, but apparently someone with a PhD in economics can make such a mistake without being universally discredited. In Scott's world, all a central bank must do is apparently announce higher nominal interest rates as well as monetary expansion, and NGDP will go up by the desired amount. Well, duh, but the real world doesn't work that way; just because you can force the money demand curve to shift so that a higher money supply means a higher nominal interest rate by making the nominal interest rate and the money supply exogenous in a model doesn't mean that this is actually possible. Seriously, the effective argument that Scott is making here is "thus if you reduce NGDP, then NGDP falls which means that monetary policy has become more contractionary." I'm sorry, but there's seriously no excuse for this; Sumner really needs to start working with general equilibrium sometime before he just skips this misleading line of reasoning and goes straight to "NGDP should have been higher in 2009, if that Fed had simply done this, then a recession wouldn't have happened" (bear with me and pretend Scott hasn't said almost exactly this before).

Of course, a central bank can go about raising the nominal interest rate without changing the current nominal interest rate, but this requires expected increases in the money supply, which Scott never mentions (probably because of previously mentioned attachment to partial equilibrium). That is, if expected money supply growth increases, then so will expected inflation, but this is not the argument that Scott is making, he only talks about the current money supply and the nominal interest rate in his post, and I'm not willing to give him the benefit of the doubt. Beyond that, Sumner was using a one period model to do dynamic analysis (someone explain to me why he's so dedicated to partial equilibrium), which is a sin in and of itself and he would still be guilty of even if he secretly meant to say "thus if you reduce expectations of future money supply growth without increasing the current money supply, then velocity falls [money demand goes up] and policy becomes more contractionary."

The worst part is that I haven't seen anyone complain about this apparent lapse of understanding of basic economics, everyone seems to be obsessed with Sumner's alleged Neo-Fisherism. Everyone, that's not the issue, you don't need to be bothered about Sumner appealing to an extremely basic one period model that you all should agree with, you should be screaming that "central banks can't control money demand!!!!!!!!!!"

A Detailed Derivation of My Favorite Monetary Model

WARNING: This post contains an excessive amount of math. If you find math unbearable and/or difficult to understand, do not attempt to read this.

A little bit ago, I decided to combine a New Keynesian model with Rotemberg style pricing and a Cash-Credit goods model. Here is a derivation of that model:

Households

Households maximize

$$ U = E_0 \sum^\infty_{t=0} \beta^t \left(\theta \log c^1_t + (1 - \theta) \log c^2_t - \gamma \log n_t \right) $$

subject to

$$ M_{t-1} + R_{t-1} B_{t-1} + W_t n_t = P_t C_t + B_t + M_t + P_t \tau_t $$
$$ M_t \geq P_t c^1_t $$
$$ C_t = c^1_t + c^2_t $$

Where $c^1_t$ is the part of the consumption good that the household buys in the cash market, $c^2_t$ is the part of the consumption good that the household buys in the credit market, $C_t$ is total spending on the consumption good, $W_t$ is the nominal wage rate, $n_t$ is hours worked by the household, $M_t$ is the nominal money supply, $B_t$ is the supply of government bonds, $P_t$ is the price of the consumption good, and $\tau_t$ is lumps sum taxes/transfers from the government.

The households maximization problem can be written as

$$ \mathcal{L} = U + \lambda^0_t \left(M_{t-1} + R_{t-1} B_{t-1} + W_t n_t - P_t C_t - B_t - M_t - P_t \tau_t \right) + \lambda^1_t \left(M_t - P_t c^1_t \right) + \lambda^2_t\left(C_t - c^1_t - c^2_t \right) $$

Solving the Lagrangian gives the following First Order Conditions:

$$ (1) \: \frac{1}{c^2_t} = \beta E_t \frac{1}{c^2_{t+1}} R_t E_t \frac{P_t}{P_{t+1}} $$
$$ (2) \: \frac{W_t}{P_t} = \frac{\gamma}{1 - \theta} \frac{c^2_t}{n_t} $$
$$ (3) \: \frac{\theta}{c^1_t} = \frac{1-\theta}{c^2_t} \left(2 - \frac{1}{R_t}\right) $$
$$ (4) \: M_t = P_t c^1_t $$

Retail Firms

Retail firms maximize profits, $P_t Y_t - \int^1_0 P_t(i) y_t(i) di $ subject to the production technology $ Y_t = \left[\int^1_0 y_t(i)^\frac{\epsilon-1}{\epsilon}di\right]^\frac{\epsilon}{\epsilon-1} $.

Substituting the production technology into the profit function yields

$$ P_t \left[\int^1_0 y_t(i)^\frac{\epsilon-1}{\epsilon}di\right]^\frac{\epsilon}{\epsilon-1} - \int^1_0 P_t(i) y_t(i) di $$

Taking the derivative of this with respect to $y_t(i)$ gives the retail firm's first order condition:

$$ y_t(i) = Y_t \left(\frac{P_t(i)}{P_t}\right)^{-\epsilon}$$

Since the retail firm is perfectly competitive, its profits are equal to zero. We can therefore set profit equal to zero and plug in the first order condition to get the definition of the price level

$$ P_t^{1-\epsilon} = \int^1_0 P_t(i)^{1-\epsilon} $$

Wholesale Firms

There is a continuum of monopolistically competitive wholesale firm who are subject to the quadratic price adjustment cost

$$ \frac{\varphi}{2}\left(\frac{P_t(i)}{P_{t-1}(i)} - 1 \right)^2 Y_t $$

First, each wholesale firm minimizes total costs, $ \frac{W_t}{P_t} n_t(i) $ subject to the production function $y_t(i) = a_t n_t(i)$. This problem can be set up as

$$ \mathcal{L} = -\frac{W_t}{P_t} n_t + mc_t \left( a_t n_t(i) - y_t(i)\right) $$

which yields

$$ (5) \: \frac{W_t}{P_t} = mc_t n_t(i) $$

The Lagrangian multiplier in this problem is the marginal cost of production (hence the name $mc_t$).

Each retail firm now maximizes the expected sum of all future profits which is discounted by the 'stochastic discount factor' with the real interest rate replacing the time preference rate and is subject to the retail firm's demand function, $ y_t(i) = Y_t \left(\frac{P_t(i)}{P_t(i)}\right)^{-\epsilon}$. Since the maximization problem for this is so obscenely long, I won't write it down, I'll just skip to the first order condition.

$$ 0 = (1-\epsilon)\frac{Y_t}{P_t} + \epsilon mc_t \frac{Y_t}{P_t(i)} - \varphi \left(\frac{P_t(i)}{P_{t-1}(i)} - 1 \right)\frac{Y_t}{P_{t-1}(i)} + \beta E_t \left( \frac {c^2_{t+1}}{c^2_t} \right)^{-1}\varphi \left(\frac{P_{t+1}(i)}{P_t(i)} - 1 \right)\frac{P_{t+1}(i) Y_t}{P_t(i)^2} $$

Consider the fact that, since each firm has the same level of technology, the same demand curve, and the price adjustment costs, every firm chooses the same  price. Given this as well as the fact that the rate of inflation, $\pi_t$ is equal to $\frac{P_t}{P_{t-1}}$, the 'New Keynesian Phillips Curve' above can be written as

$$ (6) \: 0 = (1-\epsilon) + \epsilon mc_t - \varphi \pi_t (1 + \pi_t) + \beta E_t \left( \frac {c^2_{t+1}}{c^2_t} \right)^{-1}\varphi\pi_{t+1}(1+\pi_{t+1})\frac{Y_{t+1}}{Y_t} $$


Equilibrium

Equations 1-6 can be combined with a description of government policy to complete this model. The money supply and the wage have been rewritten in real terms.

$$ (1) \: \frac{1}{c^2_t} = \beta E_t \frac{1}{c^2_{t+1}} \frac{R_t}{1 + \pi_{t+1}} $$
$$ (2) \: w_t = \frac{\gamma}{1 - \theta} \frac{c^2_t}{n_t} $$
$$ (3) \: \frac{\theta}{c^1_t} = \frac{1-\theta}{c^2_t} \left(2 - \frac{1}{R_t}\right) $$
$$ (4) \: m_t = c^1_t $$
$$ (5) \: w_t = mc_t n_t $$
$$ (6) \: 0 = (1-\epsilon) + \epsilon mc_t - \varphi \pi_t (1 + \pi_t) + \beta E_t \left( \frac {c^2_{t+1}}{c^2_t} \right)^{-1}\varphi\pi_{t+1}(1+\pi_{t+1})\frac{Y_{t+1}}{Y_t} $$
$$ (7) \: \log R_t = \frac{\beta - 1}{\beta} + \phi_\pi \pi_t + \upsilon_t $$
$$ (8) \: \log a_t = \rho \log a_{t-1} + \varepsilon^a_t $$
$$ (9) \: Y_t  = C_t + \varphi \pi_t^2 Y_t $$
$$ (10) \: C_t = c^1_t + c^2_t $$
$$ (11) \: \upsilon_t = \rho \upsilon_{t-1} + \varepsilon^i_t; $$

Impulse Response Functions

Here is the impulse response function (in log deviations from steady state) for the technology shock, $\varepsilon^a_t$ where $V$ is the velocity of money:

And here is the impulse response function for the monetart policy shock, $\varepsilon^i_t$: