# Target the spread?

The Fed wants to control inflation. Now, it targets the nominal interest rate. But to do that it has to guess what the right real interest rate is. Nominal interest rate = real interest rate plus expected inflation.

Guessing the right price is hard for any planner, and guessing the right asset price doubly hard. If the Fed wants to target inflation, why not target thespread between real and indexed bonds, and let the level of interest rates float to wherever they want to go by market forces?

Nominal interest rate - real interest rate = expected inflation. So, if the Fed wants to see 2% expected inflation, why not target thedifference between one year TIPS (indexed treasurys) and one year treasurys at 2%? Then expected inflation has to settle down to 2%

Indeed, beyond a target, the Fed could really nail this down with a flat supply curve. The Fed could nail expected inflation at 2% by offering to exchange, say, any amount of one-year zero coupon treasury bonds for 0.98 one-year zero coupon indexed treasurys (TIPS). And leave \(r^\ast\) and a lot of real rate prognosticating in the dustbin.

Obviously, you worry. If the Fed nails the spread at 2%, will everything else really settle down so that expected inflation is 2%? Or is this like holding the tail and hoping the dog will wag? We need to write down a model.

I just wrote such a model. This is part of the long-running fiscal theory of the price level book project. But it is a short independent point which blog readers may enjoy. And, I'm always nervous that I missed something in wild ideas like this (see the whole Neo-Fisherian business) so I enjoy comments.

I start with a really simple version of the model, \begin{align} x_{t} & =-\sigma\left( i_{t}-E_{t}\pi_{t+1}\right) \label{ISspread}\\ \pi_{t} & =E_{t}\pi_{t+1}+\kappa x_{t}.\label{NKspread}% \end{align} Here I have deleted the \(E_{t}x_{t+1}\) term in the first equation, so it becomes a static IS curve, in which output is lower for a higher real interest rate. This simplification turns out not to matter for the main point, which I verify by going through the same exercise with the full model. But it shows the logic with much less algebra. Denote the real interest rate \begin{equation} r_{t}=i_{t}-E_{t}\pi_{t+1}.\label{rdef}% \end{equation} We can view the spread target as a nominal interest rate rule that reacts to the real interest rate, \begin{equation} i_{t}=\alpha r_{t}+\pi^{e\ast}.\label{iar}% \end{equation} The spread target happens at \(\alpha=1\), but the logic will be clearer and the connection of an interest rate peg and interest spread peg clearer if we allow \(\alpha\in [0,1]\) to connect the possibilities.

Eliminating all variables but inflation from \eqref{ISspread}-\eqref{iar}, we obtain \begin{equation} E_{t}\pi_{t+1}=\frac{1-\alpha}{1-\alpha+\sigma\kappa}\pi_{t}+\frac {\sigma\kappa}{1-\alpha+\sigma\kappa}\pi^{e\ast}.\label{pidynsimple}% \end{equation}

For an interest rate peg, \(\alpha=0\), \(i_{t}=\pi^{e\ast}\), inflation is stable -- the first coefficient is less than one -- but indeterminate.

We complete the model with the government debt valuation equation, in linearized form \begin{equation} \Delta E_{t+1}\pi_{t+1}=-\Delta E_{t+1}\sum_{j=0}^{\infty}\rho^{j}% s_{t+1+j}-\Delta E_{t+1}\sum_{j=0}^{\infty}\rho^{j}r_{t+1+j}% ,\label{fiscalclose}% \end{equation} which determines unexpected inflation. We have a simplified version of the standard new-Keynesian fiscal theory model.

(Targeting the spread rather than the level of interest rates does not hinge on active fiscal vs. active monetary policy. In place of \eqref{fiscalclose}, one could determine unexpected inflation from an active monetary policy rule instead. One writes a threat to let the spread diverge explosively for all but one value of unexpected inflation, in classic new-Keynesian style. In place of \(i_{t}=i_{t}^{\ast }+\phi(\pi_{t}-\pi_{t}^{\ast})\), write \(i_{t}-r_{t}=\pi_{t}^{\ast}+\phi (\pi_{t}-\pi_{t}^{\ast})\), where \(\pi_{t}^{\ast}\) is the full inflation target, i.e. obeying \(\pi_{t}^{e\ast}=E_{t}\pi_{t+1}^{\ast}\) and \(\Delta E_{t+1}\pi_{t+1}^{\ast}\) the desired unexpected inflation. )

If the interest rate target responds to the real rate \(\alpha\in(0,1)\), the model solution has the same character. As \(\alpha\) rises, the dynamics of \eqref{pidynsimple} happen faster, so inflation dynamics behave more and more like the frictionless model, \(\kappa\rightarrow\infty\).

At \(\alpha=1\), the spread target \(i-r=\pi^{\ast}\) nails down expected inflation, as we intuited above. Equation \eqref{pidynsimple} becomes \[ E_{t}\pi_{t+1}=\pi^{e\ast}. \] Equation \eqref{fiscalclose} is unchanged and determines unexpected inflation, though the character of discount rate variation changes.

Inflation is not zero, but it is an unpredictable process, which in some sense is as close as we can get with an expected inflation target. Output and real and nominal rates then follow \begin{align*} x_{t} & =\frac{1}{\kappa}\left( \pi_{t}-\pi^{e\ast}\right) \\ r_{t} & =-\frac{1}{\sigma\kappa}\left( \pi_{t}-\pi^{e\ast}\right) \\ i_{t} & =\pi^{e\ast}-\frac{1}{\sigma\kappa}\left( \pi_{t}-\pi^{e\ast}\right) \end{align*} A fiscal shock here leads to a one-period inflation, and thus a one-period output increase. Higher output means a lower interest rate in the IS curve, and thus a lower nominal interest rate. The real and nominal interest rate vary due to market forces, while the central bank does nothing more than target the spread.

Of course we may wish for a more variable expected inflation target -- many model suggested it is desirable to let a long smooth inflation accommodate a shock. It's easy enough, say, to follow \(\pi_{t}^{e\ast}% =E_{t}\pi_{t+1}=\pi_{t}\) and even have a random walk inflation. Or, \(\pi _{t}^{e\ast}=p^{\ast}-p_{t}\) to implement an expected price level target \(p^{\ast}\) with one-period reversion to that target. Or \(\pi_{t}^{e\ast }=\theta_{\pi}\pi_{t}+\theta_{x}x_{t}\) in Taylor rule tradition. The point is not to defend a constant peg, but that a spread target is possible and will not explode in some unexpected way.

The same behavior occurs in the full new-Keynesian model, which is also the sort of framework one would use to think about the desirability of a spread target. I simultaneously allow shocks to the equations and a time-varying spread target. The model is \begin{align} x_{t} & =E_{t}x_{t+1}-\sigma(i_{t}-E_{t}\pi_{t+1})+v_{xt}\label{xspread}\\ \pi_{t} & =\beta E_{t}\pi_{t+1}+\kappa x_{t}+v_{\pi t}\label{pispread} \end{align} Write the spread target as \[ i_{t}-r_{t}=\pi_{t}^{e\ast}. \] With the definition \[ r_{t}=i_{t}-E_{t}\pi_{t+1}, \] we simply have \[ E_{t}\pi_{t+1}=\pi_{t}^{e\ast}. \] As in the simple model, the spread target directly controls equilibrium expected inflation. Unexpected inflation is set by the same government debt valuation equation \eqref{fiscalclose}. The other variables given inflation and unexpected inflation follow \[ x_{t}=\frac{1}{\kappa}\left( \pi_{t}-\beta\pi_{t}^{e\ast}-v_{\pi t}\right) \] \begin{equation} r_{t}=i_{t}-\pi_{t}^{e\ast}=-\frac{1}{\sigma}\left( \pi_{t}-\pi_{t}^{e\ast }\right) +\frac{\beta}{\sigma}\left( \pi_{t}^{e\ast}-E_{t}\pi_{t+1}^{e\ast }\right) +\frac{1}{\sigma}\left( v_{xt}+v_{\pi t}-E_{t}v_{\pi t+1}\right) \label{rit} \end{equation}

Following inflation, output still has i.i.d. deviations from the spread target, plus Phillips curve shocks. The real rate and nominal interest rate also have only i.i.d. deviations from the spread target, plus both IS\ and Phillips curve shocks. Output is not affected by IS shocks. The endogenous real rate variation \(\sigma r_{t}=v_{xt}\) offsets the IS shock's effect on output in the IS equation \(x_{t}=E_{t}x_{t+1}-\sigma r_{t}+v_{xt}\). This is an instance of desirable real rate variation that the spread target accomplishes automatically. (To obtain \eqref{rit} first-difference \eqref{pispread} and then substitute \(x_{t}-E_{t}x_{t+1}\) from \eqref{xspread}.)

I conclude, it could work. The Fed could target the spread between indexed and non-indexed debt. Doing so would nail down expected inflation. The Fed could then let the level of real and nominal rates float according to market forces. If every other price that has ever been set free is any guide, real and nominal interest rates would float around a lot more than anyone expects.

The result is almost, but not quite, a holy grail of monetary economics. The gold standard has a lot of appeal, as the Fed needs only exchange dollars for gold at a set rate and do no other grand financial central planning. Alas, the value of gold relative to everything else varies too much. We would like something like a CPI standard, which automatically stabilizes the price of everything else in terms of dollars. But the Fed can't buy and sell a basket of the CPI. Indexed bonds (or CPI futures) are nearly the same thing. And here the Fed just trades one year nominal debt for one year real debt. But it's not quite a CPI standard since it only sets expected inflation, not actual inflation. We still need fiscal policy, or new-Keynesian off equilibrium threats, to pick unexpected inflation. Still, guaranteeing that long-sought "anchoring" of expectations seems like a first step.

What else could go wrong? Well, this is just the first simple model, but it's a step. Obviously it depends on the forward looking Phillips curve. So in that sense it may be best as a longer run target and a regime, in which rational expectations and forward looking behavior are good assusmptions, rather than trying to set expected inflation at a daily horizon or try to do something one time that surprises markets.

I would advise the Fed to start paying a lot more attention to the spread. Next, work on increasing the liquidity of indexed debt. Ideally the Treasury should fix debt markets, vastly simplifying TIPS. I've argued for tax free indexed and non-indexed perpetuities, which would be ideal. But the Fed could and should start offering indexed and nominal term financing, for many reasons. If the Fed is going to buy a lot of long-dated Treasurys, it shold issue term liabilities not just floating-rate overnight reserves. Issing term indexed liabilites is a good next step, and there's nothing more liquid than Fed liabilities! Then start gently pushing the spread to where the Fed wants the spread to go. Start buying and selling bonds to push the spread around. Get to the point of a flat supply curve slowly. Heavens, the Fed doesn't trust interest rate targets and QE enough yet to offer a flat supply curve!

Guessing the right price is hard for any planner, and guessing the right asset price doubly hard. If the Fed wants to target inflation, why not target the

Nominal interest rate - real interest rate = expected inflation. So, if the Fed wants to see 2% expected inflation, why not target the

Indeed, beyond a target, the Fed could really nail this down with a flat supply curve. The Fed could nail expected inflation at 2% by offering to exchange, say, any amount of one-year zero coupon treasury bonds for 0.98 one-year zero coupon indexed treasurys (TIPS). And leave \(r^\ast\) and a lot of real rate prognosticating in the dustbin.

Obviously, you worry. If the Fed nails the spread at 2%, will everything else really settle down so that expected inflation is 2%? Or is this like holding the tail and hoping the dog will wag? We need to write down a model.

I just wrote such a model. This is part of the long-running fiscal theory of the price level book project. But it is a short independent point which blog readers may enjoy. And, I'm always nervous that I missed something in wild ideas like this (see the whole Neo-Fisherian business) so I enjoy comments.

I start with a really simple version of the model, \begin{align} x_{t} & =-\sigma\left( i_{t}-E_{t}\pi_{t+1}\right) \label{ISspread}\\ \pi_{t} & =E_{t}\pi_{t+1}+\kappa x_{t}.\label{NKspread}% \end{align} Here I have deleted the \(E_{t}x_{t+1}\) term in the first equation, so it becomes a static IS curve, in which output is lower for a higher real interest rate. This simplification turns out not to matter for the main point, which I verify by going through the same exercise with the full model. But it shows the logic with much less algebra. Denote the real interest rate \begin{equation} r_{t}=i_{t}-E_{t}\pi_{t+1}.\label{rdef}% \end{equation} We can view the spread target as a nominal interest rate rule that reacts to the real interest rate, \begin{equation} i_{t}=\alpha r_{t}+\pi^{e\ast}.\label{iar}% \end{equation} The spread target happens at \(\alpha=1\), but the logic will be clearer and the connection of an interest rate peg and interest spread peg clearer if we allow \(\alpha\in [0,1]\) to connect the possibilities.

Eliminating all variables but inflation from \eqref{ISspread}-\eqref{iar}, we obtain \begin{equation} E_{t}\pi_{t+1}=\frac{1-\alpha}{1-\alpha+\sigma\kappa}\pi_{t}+\frac {\sigma\kappa}{1-\alpha+\sigma\kappa}\pi^{e\ast}.\label{pidynsimple}% \end{equation}

For an interest rate peg, \(\alpha=0\), \(i_{t}=\pi^{e\ast}\), inflation is stable -- the first coefficient is less than one -- but indeterminate.

We complete the model with the government debt valuation equation, in linearized form \begin{equation} \Delta E_{t+1}\pi_{t+1}=-\Delta E_{t+1}\sum_{j=0}^{\infty}\rho^{j}% s_{t+1+j}-\Delta E_{t+1}\sum_{j=0}^{\infty}\rho^{j}r_{t+1+j}% ,\label{fiscalclose}% \end{equation} which determines unexpected inflation. We have a simplified version of the standard new-Keynesian fiscal theory model.

(Targeting the spread rather than the level of interest rates does not hinge on active fiscal vs. active monetary policy. In place of \eqref{fiscalclose}, one could determine unexpected inflation from an active monetary policy rule instead. One writes a threat to let the spread diverge explosively for all but one value of unexpected inflation, in classic new-Keynesian style. In place of \(i_{t}=i_{t}^{\ast }+\phi(\pi_{t}-\pi_{t}^{\ast})\), write \(i_{t}-r_{t}=\pi_{t}^{\ast}+\phi (\pi_{t}-\pi_{t}^{\ast})\), where \(\pi_{t}^{\ast}\) is the full inflation target, i.e. obeying \(\pi_{t}^{e\ast}=E_{t}\pi_{t+1}^{\ast}\) and \(\Delta E_{t+1}\pi_{t+1}^{\ast}\) the desired unexpected inflation. )

If the interest rate target responds to the real rate \(\alpha\in(0,1)\), the model solution has the same character. As \(\alpha\) rises, the dynamics of \eqref{pidynsimple} happen faster, so inflation dynamics behave more and more like the frictionless model, \(\kappa\rightarrow\infty\).

At \(\alpha=1\), the spread target \(i-r=\pi^{\ast}\) nails down expected inflation, as we intuited above. Equation \eqref{pidynsimple} becomes \[ E_{t}\pi_{t+1}=\pi^{e\ast}. \] Equation \eqref{fiscalclose} is unchanged and determines unexpected inflation, though the character of discount rate variation changes.

Inflation is not zero, but it is an unpredictable process, which in some sense is as close as we can get with an expected inflation target. Output and real and nominal rates then follow \begin{align*} x_{t} & =\frac{1}{\kappa}\left( \pi_{t}-\pi^{e\ast}\right) \\ r_{t} & =-\frac{1}{\sigma\kappa}\left( \pi_{t}-\pi^{e\ast}\right) \\ i_{t} & =\pi^{e\ast}-\frac{1}{\sigma\kappa}\left( \pi_{t}-\pi^{e\ast}\right) \end{align*} A fiscal shock here leads to a one-period inflation, and thus a one-period output increase. Higher output means a lower interest rate in the IS curve, and thus a lower nominal interest rate. The real and nominal interest rate vary due to market forces, while the central bank does nothing more than target the spread.

Of course we may wish for a more variable expected inflation target -- many model suggested it is desirable to let a long smooth inflation accommodate a shock. It's easy enough, say, to follow \(\pi_{t}^{e\ast}% =E_{t}\pi_{t+1}=\pi_{t}\) and even have a random walk inflation. Or, \(\pi _{t}^{e\ast}=p^{\ast}-p_{t}\) to implement an expected price level target \(p^{\ast}\) with one-period reversion to that target. Or \(\pi_{t}^{e\ast }=\theta_{\pi}\pi_{t}+\theta_{x}x_{t}\) in Taylor rule tradition. The point is not to defend a constant peg, but that a spread target is possible and will not explode in some unexpected way.

The same behavior occurs in the full new-Keynesian model, which is also the sort of framework one would use to think about the desirability of a spread target. I simultaneously allow shocks to the equations and a time-varying spread target. The model is \begin{align} x_{t} & =E_{t}x_{t+1}-\sigma(i_{t}-E_{t}\pi_{t+1})+v_{xt}\label{xspread}\\ \pi_{t} & =\beta E_{t}\pi_{t+1}+\kappa x_{t}+v_{\pi t}\label{pispread} \end{align} Write the spread target as \[ i_{t}-r_{t}=\pi_{t}^{e\ast}. \] With the definition \[ r_{t}=i_{t}-E_{t}\pi_{t+1}, \] we simply have \[ E_{t}\pi_{t+1}=\pi_{t}^{e\ast}. \] As in the simple model, the spread target directly controls equilibrium expected inflation. Unexpected inflation is set by the same government debt valuation equation \eqref{fiscalclose}. The other variables given inflation and unexpected inflation follow \[ x_{t}=\frac{1}{\kappa}\left( \pi_{t}-\beta\pi_{t}^{e\ast}-v_{\pi t}\right) \] \begin{equation} r_{t}=i_{t}-\pi_{t}^{e\ast}=-\frac{1}{\sigma}\left( \pi_{t}-\pi_{t}^{e\ast }\right) +\frac{\beta}{\sigma}\left( \pi_{t}^{e\ast}-E_{t}\pi_{t+1}^{e\ast }\right) +\frac{1}{\sigma}\left( v_{xt}+v_{\pi t}-E_{t}v_{\pi t+1}\right) \label{rit} \end{equation}

Following inflation, output still has i.i.d. deviations from the spread target, plus Phillips curve shocks. The real rate and nominal interest rate also have only i.i.d. deviations from the spread target, plus both IS\ and Phillips curve shocks. Output is not affected by IS shocks. The endogenous real rate variation \(\sigma r_{t}=v_{xt}\) offsets the IS shock's effect on output in the IS equation \(x_{t}=E_{t}x_{t+1}-\sigma r_{t}+v_{xt}\). This is an instance of desirable real rate variation that the spread target accomplishes automatically. (To obtain \eqref{rit} first-difference \eqref{pispread} and then substitute \(x_{t}-E_{t}x_{t+1}\) from \eqref{xspread}.)

I conclude, it could work. The Fed could target the spread between indexed and non-indexed debt. Doing so would nail down expected inflation. The Fed could then let the level of real and nominal rates float according to market forces. If every other price that has ever been set free is any guide, real and nominal interest rates would float around a lot more than anyone expects.

The result is almost, but not quite, a holy grail of monetary economics. The gold standard has a lot of appeal, as the Fed needs only exchange dollars for gold at a set rate and do no other grand financial central planning. Alas, the value of gold relative to everything else varies too much. We would like something like a CPI standard, which automatically stabilizes the price of everything else in terms of dollars. But the Fed can't buy and sell a basket of the CPI. Indexed bonds (or CPI futures) are nearly the same thing. And here the Fed just trades one year nominal debt for one year real debt. But it's not quite a CPI standard since it only sets expected inflation, not actual inflation. We still need fiscal policy, or new-Keynesian off equilibrium threats, to pick unexpected inflation. Still, guaranteeing that long-sought "anchoring" of expectations seems like a first step.

What else could go wrong? Well, this is just the first simple model, but it's a step. Obviously it depends on the forward looking Phillips curve. So in that sense it may be best as a longer run target and a regime, in which rational expectations and forward looking behavior are good assusmptions, rather than trying to set expected inflation at a daily horizon or try to do something one time that surprises markets.

I would advise the Fed to start paying a lot more attention to the spread. Next, work on increasing the liquidity of indexed debt. Ideally the Treasury should fix debt markets, vastly simplifying TIPS. I've argued for tax free indexed and non-indexed perpetuities, which would be ideal. But the Fed could and should start offering indexed and nominal term financing, for many reasons. If the Fed is going to buy a lot of long-dated Treasurys, it shold issue term liabilities not just floating-rate overnight reserves. Issing term indexed liabilites is a good next step, and there's nothing more liquid than Fed liabilities! Then start gently pushing the spread to where the Fed wants the spread to go. Start buying and selling bonds to push the spread around. Get to the point of a flat supply curve slowly. Heavens, the Fed doesn't trust interest rate targets and QE enough yet to offer a flat supply curve!

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