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Fiscal Theory Of Monetary Policy

Teaching a PhD course of pedagogy too preparing a few talks led me to a rattling uncomplicated instance of an idea, which I'm calling the "fiscal theory of monetary policy." The projection is to get hitched amongst new-Keynesian models, i.e. DSGE models amongst cost stickiness, amongst the financial theory of the cost level. The instance is simpler than the total analysis amongst cost stickiness inwards the newspaper yesteryear that title.

It turns out that the FTPL tin give the axe neatly solve the problems of criterion novel Keynesian models, too oftentimes brand rattling piddling divergence to the actual predictions for fourth dimension series. This is keen news. Influenza A virus subtype H5N1 new-Keynesian modeler wanting to represent some impulse response functions, nervous at the less too less credible underpinnings of new-Keynesian models, can, it appears, just alter footnotes most equilibrium alternative too instruct dorsum to work. He or she does non receive got to throw out a lifetime of work, too commencement afresh to facial expression at inflation armed amongst debts too deficits. The interpretation of the model may, however, alter a lot.

This is likewise an extremely conservative (in the non-political sense) approach to curing new-Keynesian model problems. You tin give the axe maintain the entire model, just alter some parameter values too solution method, too problems vanish (forward guidance puzzle, frictionless bound puzzle, multiple equilibria at the naught bound, unbelievable off-equilibrium threats etc.) The electrical flow NK literature is instead embarked on deep surgical procedure to cure these problems: removing rational expectations, adding constrained or heterogeneous agents, etc. I did non retrieve I would let on myself inwards the foreign seat trying to relieve the criterion new-Keynesian model, spell its developers are eviscerating it! But hither nosotros are.

The FTMP model

(From hither on in, the post service uses Mathjax. It looks keen nether Chrome, but Safari is iffy. I retrieve I hacked it to work, but if it's mangled, assay a dissimilar browser. If anyone knows why Safari mangles mathjax too how to gear upward it allow me know.)

Here is the example. The model consists of the commons Fisher equation, \[ i_{t} = r+E_{t}\pi_{t+1} \] too a Taylor-type involvement charge per unit of measurement dominion \[ i_{t} = r + \phi \pi_{t}+v_{t} \]
\[ v_{t} =\rho v_{t-1}+\varepsilon_{t}^{i} \]
Now nosotros add together the authorities debt valuation equation \[ \frac{B_{t-1}}{P_{t-1}}\left( E_{t}-E_{t-1}\right) \left( \frac{P_{t-1}% }{P_{t}}\right) =\left( E_{t}-E_{t-1}\right) \sum_{j=0}^{\infty}\frac {1}{R^{j}}s_{t+j} \]
Linearizing \begin{equation} \pi_{t+1}-E_{t}\pi_{t+1}=-\left( E_{t}-E_{t+1}\right) \sum_{j=0}^{\infty }\frac{1}{R^{j}}\frac{s_{t+j}}{b_{t}}=-\varepsilon_{t+1}^{s} \label{unexpi} \end{equation} amongst \(b=B/P\). Eliminating the involvement charge per unit of measurement \(i_{t}\), the equilibrium of this model is at nowadays \begin{equation} E_{t}\pi_{t+1} =\phi\pi_{t}+v_{t} \label{epi} \end{equation}
\[ \pi_{t+1}-E_{t}\pi_{t+1} =-\varepsilon_{t+1}^{s} \]
or, most simply, just \begin{equation} \pi_{t+1}=\phi\pi_{t}+v_{t}-\varepsilon_{t+1}^{s}. \label{equil_ftmp} \end{equation}

Here is a plot of the impulse response function:


The top lines plot the response of inflation too involvement rates to a unit of measurement monetary policy daze \(\varepsilon _{1}^{i}\) inwards this model. The draw of piece of work "\(v_{t}\), FTMP" plots the associated monetary policy disturbance \(v_{t}\). The combination of ii AR(1)s -- the daze persistence \(\rho\) too the involvement charge per unit of measurement dominion \(\phi\) -- generates a pretty hump-shaped inflation response. I usage \(\rho=0.7\), \(\phi=0.8\).

The lower ii lines plot the response to a unit of measurement financial daze \(\varepsilon_{1}^{s}\). By definition, this disturbance is non persistent. The financial tightening produces an instant deflation, i.e. a downward cost degree jump.

In both cases, the monetary policy \(\phi\) dominion introduces interesting dynamics. Fiscal policy lonely sets unexpected inflation \(\pi_{t+1}-E_{t}\pi_{t+1}\). But what happens later that, \(\left( E_{t+1}-E_{t}\right) \pi_{t+2}\) too beyond, depends on monetary policy. With a dissimilar \( \phi \), monetary policy could provide the cost degree to its previous value. Monetary policy could plough the lawsuit into a one-time cost degree shock, amongst no farther inflation. Or monetary policy could allow the inflation move on for a while, every bit it does hither amongst \(\phi>0\).

These responses are non gear upward to evaluate against data. This model is Fisherian -- higher involvement rates via a monetary policy daze attain higher inflation, every bit they must amongst a Fisher equation \( i_t = r + E_t \pi_{t+1}\) too constant \(r\). The lack of a financial alter agency that \(\pi_{1}\) cannot bound down. We ask to add together long term debt too gummy prices, amid other ingredients, to attain to a greater extent than realistic response. The of import lesson hither is that nosotros can produce impulse response functions of this sort, just every bit nosotros do amongst criterion models of involvement charge per unit of measurement targets.

The NK model

The simplest shape of the criterion new-Keynesian model, every bit laid forth for instance inwards Woodford's book, Chapter 2, consists of exactly the same laid of equations!

However, new-Keynesian modelers specify that (\ref{unexpi}) determines surpluses \(\left\{ s_{t}\right\} \) for whatsoever unexpected inflation, the "passive" financial policy assumption. (Woodford, top of p. 73 for example.) Then, that equation drops out too no longer determines unexpected inflation. To create upward one's heed unexpected inflation inwards its place, new-Keynesian modelers specify \(\phi>1\), too they add together a dominion against nominal explosions. New-Keynsian authors hence solve (\ref{epi}) forwards to \begin{equation} \pi_{t}=-E_{t}\sum_{j=0}^{\infty}\frac{v_{t+j}}{\phi_{nk}^{j+1}}=-\sum _{j=0}^{\infty}\frac{\rho_{nk}^{j}}{\phi_{nk}^{j+1}}v_{t}=-\frac{1}{\phi _{nk}-\rho_{nk}}v_{t}. \label{nk_sol} \end{equation}
I usage the subscript \(nk\) hither to distinguish parameter values of the ii models.

Equilibrium inflation hence follows the same procedure every bit the shock \( v_{t} \), \begin{equation} \pi_{t+1}=\rho_{nk}\pi_{t}-\frac{1}{ \phi_{nk}-\rho_{nk}}\varepsilon_{t+1}^{i}. \label{equil_nk} \end{equation}
In this uncomplicated model, a monetary policy daze \(\varepsilon_{t}^{i}\) at i time lowers inflation \(\pi_{t}\). Inflation so recovers dorsum to its steady nation amongst an AR(1) blueprint next the deadening hateful reversion of the disturbance \(v_{t}\).

The figure to a higher house plots the response of this model to a monetary policy shock, using \(\phi_{nk}=1.5\) too \(\rho_{nk}=0.8\). The draw of piece of work marked "\(v_{t}\), NK" plots the resulting monetary policy disturbance, \(v_{t}\). You can't run into the \(\pi_{t}\) too \(i_{t}\) lines, because they are exactly the same every bit the responses of the financial theory model to a financial \(\varepsilon^{i}\) shock, using \(\phi=0.8\). You tin give the axe likewise run into that fact analytically, comparison (\ref{equil_ftmp}) to (\ref{equil_nk}).

The new-Keynesian model response to a monetary policy daze \(\varepsilon^{i}=(\phi_{nk}-\rho_{nk})\) inwards (\ref{equil_nk}) is observationally equivalent to the financial theory of monetary policy model response to a (\ref{equil_ftmp}) fiscal shock \(\varepsilon^{s}=1\), nether parameters \(\phi=\rho_{nk}\), \(\rho=0\).

Observationally equivalent agency observationally equivalent. The response functions are precisely the same. There is no way to tell the ii models apart from information on inflation, involvement rates, too financial surpluses. We do non direct let on underlying shocks or parameter values \(\rho\) too \(\phi\). That i model interprets the information via \(\phi \lt 1\), \(\rho=0\), too \(\varepsilon^{s}\) via \((\ref{equil_ftmp})\) too the other model interprets the information via \(\phi \gt 1\),   \(0\lt \rho \lt 1\) too \(\varepsilon^{i}\), is non an declaration that information tin give the axe solve.

Repeating what I promise is a familiar point, the relation betwixt equilibrium involvement rates too inflation inwards the new-Keynesian model is \[ i_t = r + \rho \pi_t .\]
Influenza A virus subtype H5N1 regression of \( i_t \) on \(\pi_t\) yields \(\rho\) non \(\phi\).

The economics are the same inwards equilibrium every bit well. There is an unexpected inflation \(\pi_{t+1}-E_{t}\pi_{t+1}\) on the twenty-four hr menses of the shock, because (\ref{unexpi}) holds via "passive" financial policy. The new-Keynesian model likewise has the same financial shock, too i could just every bit good regard the response every bit actually to the financial shock. The "monetary policy" daze is a carrot that induces the passive "fiscal policy" Equus caballus to disinflate.

The new-Keynesian model is likewise Fisherian -- involvement rates spend upward hither along amongst the inflation decline. There is no menses of high involvement rates lowering inflation. Again, nosotros ask to a greater extent than ingredients for that.

What the instance agency

What is the financial theory? How does it differ from the criterion new-Keynesian theory of monetary policy? How volition an applied economist usage the financial theory? How dissimilar are its predictions?

In this example, the response is that the financial theory makes no divergence at all to the model's predictions. It amounts to a dissimilar agreement most equilibrium formation, too a much dissimilar agreement most the course of pedagogy of events too source of shocks underlying a movement such every bit the bottom ii lines of the picture.

More generally, the financial theory inwards a new-Keynesian model only changes the rules for picking unexpected values. Changing the dominion for picking unexpected values affects only the impact response, non the shape of subsequent moments of impulse-response functions. For many applied purposes the impact response is non crucial or, every bit inwards this example, non identified. In other cases, choosing a dissimilar instantaneous response makes a big divergence to model predictions. (That's what the "new Keynesian Liquidity Trap" is about.)

The machinery of equilibrium formation, unimportant for studying information from the equilibrium, is crucially of import inwards thinking most monetary doctrines, too how alternative policy arrangements do too don't work. For example, the financial theory too the criterion new-Keynesian approach differ on whether an involvement charge per unit of measurement peg is possible, too to a greater extent than mostly whether \(\phi \gt 1\) or \(\phi \lt 1\) is desirable. In this FTMP model at that spot is no work amongst a peg. \(\phi \gt 1\) is possible (yes, passive financial does non require passive money) but undesirable since inflation volition explode.  \(\phi \lt 1\) is desirable every bit it leads to stationary inflation. In the criterion NK interpretation of the same equations, \( \phi \lt 1\) leaves unexpected inflation \( \pi_{t+1}-E_t\pi_{t+1}\) undetermined, so at that spot tin give the axe last multiple equilibria too sunspots, too \(\phi \gt 1\), though unobservable, is vital for determinacy. And of course of pedagogy the label too interpretation of the shocks is solely different. Influenza A virus subtype H5N1 "monetary policy" daze too a "fiscal policy" shock, though observationally equivalent through the eyes of the ii models, receive got quite a dissimilar interpretation.

The criterion NK model assumes that the Fed volition respond to whatsoever inflation it doesn't similar yesteryear hyperinflating the economy. That threat makes us converge on the i equilibrium that doesn't hyperinflate. The FTMP model frees you lot from that assumption.

This observational equivalence instance strikes me every bit proficient intelligence for the financial theory project. The basic nowadays value relation invites you lot to apply the financial theory yesteryear forming fourth dimension serial of debts too surpluses too looking to them every bit sources of inflation, rather than yesteryear looking at the involvement charge per unit of measurement decisions of primal banks. The financial theory has seemed a radical proposal to facial expression at completely dissimilar empirical foundations for inflation, amongst potentially radically dissimilar predictions. This instance leads to a much to a greater extent than conventional investigation of inflation inwards terms of involvement charge per unit of measurement targets, amongst potentially pocket-sized differences inwards testable predictions. Since the only divergence betwixt the models is their handling of innovations, to the extent that expected rather than unexpected inflation dominates its movement inwards normal times, the "fiscal" component of the financial theory volition last that much less important.

The financial theory becomes a way to maintain the heart of workaday practise amongst criterion new-Keynesian models, spell solving inwards a uncomplicated swoop the many holes inwards its theoretical foundation -- incredible off-equilibrium threats yesteryear primal bankers, paradoxical policy prescriptions, weird horizon limits too flexible cost limits, all of which nosotros volition investigate below.

I must acknowledge I'm scratching my caput a flake that this is so controversial. Why set so much endeavor into deep surgical procedure of the model, when just reinterpreting the authorities debt valuation equation, allows you lot to solve its problems? Just why is that footstep so hard? Really all nosotros are doing is adding Pigou effects, wealth effects of authorities bonds. That footstep was pretty good accepted some lxx years ago. Why non now?

But the debts too surpluses too the nowadays value formula receive got non vanished. The financial theory offers a unified way of thinking most such "normal times" monetary economic science along amongst the inflations, currency crashes, unconventional policies, too other events inwards which the financial too debt management parts takes view stage, along amongst deeply dissimilar responses to doctrinal too structural or institutional issues.

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