Fiscal Policy Does Not Affect Aggregate Demand

This is a follow up to my last post; here I am going to go more in depth into why fiscal stimulus never really impacts aggregate demand in any DSGE.

Start with the simple consumption euler equation that everyone should be familiar with:

$$(1)\:c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} \left(\frac{1 + i_t}{1 + E_t \pi_{t+1}}\right)$$

where $c_t$ is consumption, $\beta$ is the representative agent's discount factor, $i_t$ is the nominal interest rate, and $\pi_t$ is the rate of inflation. Assuming there is no capital accumulation, this equation can be simply edited to include real GDP and government spending:

$$(2)\:(y_t - g_t)^{-\sigma} = \beta E_t (y_{t+1} - g_{t+1})^{-\sigma} \left(\frac{1 + i_t}{1 + E_t \pi_{t+1}}\right)$$

This can now be written in log-linear form, so that all variables are expressed as a percentage deviation from their steady states (except for the various interest rates):

$$(3)\:\hat{y}_t = E_t \hat{y}_{t+1} + \hat{g}_t - E_t\hat{g}_{t+1} - \frac{1}{\sigma}(i_t - E_t \pi_{t+1} - \rho)$$

where $\rho$ is the discount rate, which is equal to $\frac{1-\beta}{\beta}$. Keep in mind that $\hat{g}_t$ is actually the deviation of government spending as a percentage of GDP from trend, but this shouldn't have much of an impact on the analysis.

If we define the 'natural rate of interest' as the real rate of interest at which the output gap remains constant, only a small amount of algebra is required to solve for it:

$$(4)\:r^n_t = \rho - \sigma(E_t\hat{g}_{t+1} - \hat{g}_t)$$

It is clear from this that fiscal stimulus, that is increases in $\hat{g}_t$ absent changes in $E_t\hat{g}_{t+1}$, causes the natural rate of interest to increase. Why is this important? In most cases, it isn't; fiscal stimulus will just have the effect it normally does in a frictionless model (that is, generally speaking, have a multiplier on output somewhere between zero and one depending on the calibration of that model) and the central bank will simply raise the real interest rate so that the output gap remains equal to zero.

This is exactly what all the market monetarists are talking about when they mention monetary offset; any demand-side effect that fiscal stimulus might have is simply the result of central bank inaction in the face of a higher natural rate. Say, for instance, that the central bank set the nominal interest rate according to the following rule:

$$(5)\:i_t = max(0,r^n_t + E_t \pi_{t+1})$$

Now, suppose the fiscal authority does a fiscal stimulus, which raises $r^n_t$. Assuming the zero lower bound is not binding, the central bank will simply raise the nominal interest rate one for one with the increase in the natural rate caused by the fiscal stimulus.

The only reason a New Keynesian model ever exhibits a fiscal multiplier greater than one is that the zero lower bound is at some point binding. In this case, the fiscal authority can lower the value of $r_t - r^n_t$ without central bank intervention ($r_t$ is the real rate of interest). Basically, fiscal stimulus appears effective because of central bank inaction.

In this sense, the actual act of fiscal stimulus never has demand-side effects; it can only influence the natural rate of interest, which is only helpful to the extend that the zero lower bound is binding. Because of this, we should really stop thinking of fiscal stimulus as a way of manipulating aggregate demand; it's really only helpful to the extent that is has real effects -- if anyone says otherwise, then they are unduly influenced by IS-LM.

How Easy is it to Raise LRAS?

There's recently been quite a bit of talk about greatly increasing the growth rate of GDP in the US over the next decade or so. The only way big aggregate demand stimuli could achieve this goal is if 1) output is currently far below potential in the US or 2) demand side stimulus can raise the long run level of output.

The first option is evidently not the case, given the fact that the labor market is reasonably tight -- the employment rate for Americans between the age of 15 and 64 has recovered most of what it lost in 2008, so output is clearly close to where it would be at full employment. By basically any reasonable account, output is pretty close to potential, so not much in the way of aggregate demand stimulus would have any effect other than increasing inflation. There are no credible models that I have ever seen that suggest that aggregate demand has any effect over the long run level of output, so it's almost fair to simply reject the hysteresis hypothesis simply based on its lack of theoretical reasoning. If this isn't enough, simply taking it to its logical conclusion should reveal some of the absurdity. If central banks, by increasing aggregate demand, could increase potential GDP, then they essentially have the power to decide the equilibrium level of employment, irrespective of demographics. Alternatively, they could be fueling innovation by printing loads of money and causing TFP growth to increase. Both options strike me as completely absurd; personally I see LRAS as completely deterministic to the Federal Reserve.

I will not, however, completely discount the role of fiscal policy. Certainly the Federal Government has the capability to engage in policies that have real effects. For instance, the government could permanently increase government spending as a percentage of GDP and induce everyone to work more by making them feel poorer. Also, the government could invest in a bunch of infrastructure, which could be treated as 'government capital' in all our Cobb-Douglas production functions and would, as such, be stimulative. Basically, the government could very well be boosting potential output, just not with anything that should be labeled 'demand' stimulus. In fact, this is probably true about any fiscal expansion that is ever undertaken -- we should only ever talk about the real effects of fiscal stimulus because everything else is the product of monetary-fiscal interaction. For example, any fiscal expansion will raise the 'natural rate of interest' in a New Keynesian model and the only reason the 'fiscal multiplier' will be any higher than that of a similarly calibrated RBC model is that the central bank refuses to raise the nominal interest rate one for one with the increase in the natural rate. In this sense, fiscal stimulus can almost have demand-side effects at the zero lower bound, but what is really happening is that the central bank fails to tighten monetary policy in the event of stimulus.

Basically, the only way for the government to increase potential output is by taking advantage of the real effects of fiscal policy -- people should stop preoccupying themselves with aggregate demand.

Assessing the Effect of Austerity in the UK

Whether or not austerity has been successful in the UK is perhaps the most natural test of Market Monetarism. The UK, after all, has an independent central bank and there is no question as to whether or not it actually engaged in austerity (the same case can not be made for the United States, in my opinion). 

It has previously been noted that, even though austerity evidently had a negative effect on real GDP in the UK, what really happened is that productivity growth just happened to be zero while Chancellor Osborne was having a fit with the exchequer. I think that the data clearly disagree with the position; as you can see both the employment rate and real GDP lagged during the period of austerity, which I will argue was only pursued fervently in 2010 and 2011, before it was significantly weakened and the economy proceeded to improve.

First, look at employment and real GDP between Q2 2010 (when the first austerity budget was suggested by the new coalition) and 2011 (the last year that the government actually maintained its commitment to austerity). It's clear that both real GDP and employment suffered during this period -- basically disproving the hypothesis that slow productivity growth and austerity were coincidental. 

Of course, the government never vocally backed down on austerity, so why am I limiting my analysis to 2010 and 2011? Well, for that you need to look at the actual and the projected deficits over the course of the Cameron government:

As you can see, the actual deficit was only less than was predicted by the government during 2010 and 2011. After this, the deficit clearly begins exceeding the 2011 vintage projection; that is the government raised the deficit above what they were initially intending. It was only after this point that the economy and employment began to recover, so evidently fiscal policy was loosened in 2012 and this explains the apparent recovery that happened afterward.

The data seem to corroborate the Keynesian view a lot more than the Market Monetarist one; fiscal tightening did cause both output and employment to fall relative to trend, and the economy only began to recover with fiscal easing.

Don't Be Fooled By Annual Inflation

If you look at Fred right now, you'll see that the CPI is currently 1.3% higher than it was a year ago. This figure can mislead people into think that inflation is a lot closer to target than it really is. To understand this, you need to look at the actual graph for the CPI over the last 12 months or so.

As you can see, the current CPI is really barely any higher than it was last month and is a long way away from the level of CPI inflation consistent with the Fed's 2% annual PCE inflation target. In fact, the CPI is just under 0.03% higher than it was a month ago.

If CPI were to grow at 2% each year, this would require 0.16% inflation each month, which is a whole lot higher than the current 0.03%. Basically, even though the current price level is 1.3% higher than it was a year ago, the price level has not been growing even that fast for quite a while. If you want to see whether or not inflation will be on target, you should look at the compounded annual rate of change, and not percentage change from a year ago.

Update:

The new PCE numbers are in and, guess what, inflation is indeed chronically below target; at least the y/y rate is closer to the compounded annual rate of change, though.

Potential GDP is Not Linear

Recently I've seen a few people defending the Friedman analysis of Sanders' economic plan on the grounds that 5.3% growth over a decade would be consistent with closing the output gap. They typically estimate the 'output gap' by comparing GDP to the linear trend implied by the post-WWII time series (excluding data after 2008).

This is ridiculous. I'm tempted to just stop writing here because of how obvious I think this should be, but evidently a lot of people think that potential GDP is linear. The primary problem with this  approach is that it completely ignores demographic factors. If, for instance, you decided that real potential GDP (not per capita) followed a linear trend, then you'd be suggesting that TFP automatically grows faster whenever population growth is low. Naturally, this doesn't make any sense whatsoever, so most people who want to estimate linear trends for RGDP usually go for real GDP per capita.

This has its own problems, however. If real potential GDP per capita grows at a constant linear rate, then TFP growth increases when the working age population shrinks relative to the total population (when the dependency ratio goes up) and vice versa. This also makes no sense whatsoever; to suggest that TFP grows more quickly when people have lots of children or when a bunch of people are reaching retirement age is ridiculous. It's obvious that real GDP per capita should be a negative function of the dependency ratio.

This leaves one option for a demographic-adjusted estimate of potential output: real GDP per working age person.

The output gap that can be extrapolated from this is a lot more sane than the predictions of some of Sanders' defenders -- it's about 9%, but I still have a problem with inferring linear trends from demographically adjusted potential output.

See, my approach so far basically assumes constant TFP growth (which is a whole lot less stupid than TFP growth that changes with population growth), but I don't even think this is really that fair of an assumption. To argue that technological progress always occurs at the same rate and with the same fervor doesn't make any sense.

That being said, I think the best (only) way to measure the output gap is to use some labor market indicator. For some reason, a lot of people seem to hate the unemployment rate for this, so I'll use the employment to working age population ratio:

This suggests that we are pretty close to full employment, but not quite there yet. That's a completely different story from what Sanders supporters (and most people in the GOP) are saying. Granted, I think this approach has its own flaws -- I think it drastically overestimates the output gap in 2000, but it seems to give a pretty good estimate of where the output gap is right now; i.e., somewhat understated in absolute value terms by the unemployment rate, but overstated by the employment to population ratio.

Needless to say, no one should listen to anyone who thinks real potential GDP per capita follows a linear trend.

Update:

Nick Rowe suggested in the comments that I come up with a projection for potential GDP given estimates for the working age population over the next few years. I used the US Census Bureau's estimates for population between 14 and 64 years old and assumed the 14-year-old population will be constant over the next few decades (it's a shortcut, I know, but I can't be bothered to find a better estimate of the future working age population) to get my projections for the working age population. I then took the real GDP to working age population ratio, got the trend growth rate between 1989 and 2007, and extrapolated that to 2026 to get my potential GDP per working age person value.

Then I multiplied the whole thing by the time series for the working age population (including the projected values until 2026) to get potential GDP. Here is my estimate of potential GDP to 2026 compared with the CBO estimate and actual GDP (going up until 2015):

Update #2:

I thought I'd add this comparison between my estimate of potential GDP and what potential GDP would be if it followed the 1990-2007 trend here:

Also I realized I made a couple mistakes when removing the 14-year-olds from the Census Bureau projection. All the graphs on the blog are updated, but not the ones on Twitter, so don't take them from there if you want to use them.

In Which I Do Some Bad Econometrics

I decided I would do a linear regression on the growth rate of real GDP per capita (RGDPPC) with respect to the change in the Civilian Employment to Population ratio (EPOP). I used the period between 1950 Q1 and 2015 Q3 and came up with this result:

Vertical Axis: RGDPPC growth, Horizontal Axis: Change in EPOP

So, a linear regression suggests that the relationship between RGDPPC and EPOP is

$$(1)\:100\Delta\ln{y_t} = 2.458 \Delta e_t + 1.9333$$
where $y_t$ is real GDP per capita and $e_t$ is the Employment to Population ratio.

With this relationship, we can make some interesting predictions. It has recently been popular to argue that the employment to population ratio can and should be raised to its April 2000 high (coincidentally, I was born in April 2000). If this were to occur, it would mean that the Employment to Population ratio would go up by 5.1%, which corresponds to an increase in real GDP per capita of about 14.5%. Or, if the change were to take place over ten years, then real GDP per capita would grow at about 3.2% per year.

Given ~1% annual population growth, this could make the extravagant economic promises by the likes of Bernie Sanders and Jeb Bush seem in reach. After all, all we need do is employ as many people as we were in 2000. Unfortunately, it is not that simple. First of all, there are reasons to believe that some, if not most, of the decline in EPOP over the last 16 years is secular. Namely, the working age population (i.e., population between 15 and 65 years of age) has increased a lot less than total population in the last few years. In fact, the Employment to Working Age Population ratio has recovered pretty well since the Great Recession:

Now, there's definitely still a gap; employment still has room to grow, but now at least, it should be clear that there was a lot of over-employment by the end of the Clinton administration. It appears as if the equilibrium Employment to Working Age Population ratio is closer to 74% than 77%, which means that there are a lot less employment gains to be had than a simple look at the EPOP would suggest.

The other issue with both Senator Sanders' and Governor Bush's plans is that it's unclear how they would actually raise the employment to population ratio. In the case of Sanders, programs like expanded Social Security and free college tuition would probably lower the Employment to Population ratio (since we'll be paying people more to retire and getting an education will be so cheap that students won't need to work, or can leave a job to get a degree). With Governor Bush, there is at least a case to be made that significantly lower taxes might incentivize millions of Americans who were otherwise not going to work to now go out and get a job, but I don't really see it.

Yes, a government can increase employment by reducing the labor tax in a simple neoclassical model, but how much does that really map to determining whether or not someone is even in the labor force. Honestly, the tax rate that someone has to pay may lead to changes in hours worked, but says little as to whether or not they work in the first place; to argue that because the tax rate goes down, all of a sudden people who refused to work at the previous after tax wage will now start searching for jobs seems nonsensical. I can see people increasing their hours if they are all of a sudden paid more for work, but not leaving the labor force altogether because taxes are too high or joining it because they are now low.

On top of that, key to the success of any supply side reform is whether or not the lack of employment is voluntary; if people who are unemployed actually want to be employed, then a tax cut won't change anything relating to their job search, whereas, if the people who are not employed are in that state by choice, then a tax cut might make them reassess (personally, I think the case for this is really weak, but I'm open to it). The key question is, then, was the non-secular part of the decline in the EPOP in 2008 caused voluntary or involuntary. There are those that disagree, but personally I think that the obvious answer is that the decline was involuntary. If this is the case, the degree to which supply side reforms would be beneficial is probably low.

So, by now I've spent most of this post complaining about Governor Bush's promise of 4% growth even though I find the crimes of the Sanders campaign more egregious. For this apparent injustice, I offer the following explanation: the reason Bernie Sanders' proposals would fail to create an employment boom is extremely easy to understand, whereas Bush's plan requires a much more in depth criticism to be understood. Regardless, both 4% growth and 5.3% growth are almost equally absurd and no one wishing to be on the side of sane economic analysis should support either claim.

What 'Off-the-Shelf' Monetary Models Actually Say about Neo-Fisherism

Stephen Williamson wrote this in a blog post today:

Standard off-the-shelf monetary models essentially all exhibit a neo-Fisherian effect. That's nothing special. The Fisher effect is important. Typically increases in nominal interest rates lead to increases in inflation.

Testing this claim should be pretty simple, all we need is to get some "off-the-shelf" monetary models and see what happens when a central bank switches from one interest rate peg to another. I guess the only difficulty here would be that most standard monetary models exhibit indeterminacy when there's an interest rate peg, but I guess that doesn't seem to phase Williamson.

Let's start with the most basic monetary model I can think of: Cash-In-Advance. I won't bother with much derivation here since I've already talked in detail about CIA models in previous posts, so here is a simple CIA monetary model (assuming that the nominal interest rate is always greater than the interest rate that money pays):
$$(1)\: M_t = P_t y $$
$$(2)\: R_t = \frac{1}{\beta}\frac{P_{t+1}}{P_t} $$
where $M_t$ is the money supply, $P_t$ is the current price level, $y$ is the constant level of output, $R_t$ is the gross nominal interest rate, and $\beta$ is the representative agent's discount rate.

It's clear from this that, by pegging the nominal interest rate, the central bank can peg the ratio of the future price level to the current price level to whatever it wants, but here's where the indeterminacy comes in. The actual current price level is not determined under a pure interest rate peg; only the expected rate of inflation. If the central bank raises the nominal interest rate, then the future price level will be higher than the current one, but what happens to the current price level is unclear.

If, instead of pegging the interest rate, the central bank decides to set the money supply to the desired level each period, the current price level is actually determined, but the relationship between the current price level and the nominal interest rate is still unclear. If, for instance, the central bank temporarily lowers the money supply in the current period then returns it back to it's previous level in the following period, the nominal interest rate will go up, but the current price level will fall. Any permanent changes in the money supply impact the price level without changing the nominal interest rate at all and any temporary increases in the money supply result in a higher price level and a lower nominal interest rate.

Of course, if the nominal interest rate is at the zero lower bound, the basic CIA model predicts that the money supply no longer determines the price level, so the effect of a change in the nominal interest rate on the price level is even more unclear. The only reliable way to exit the zero lower bound (without using fiscal policy) in a CIA model is to shrink the current money supply until the cash in advance constraint once again binds, which at the very least means that the current price level doesn't increase. Effectively, even when escaping the zero lower bound, increases in the nominal interest rate mean a lower current price level.

It's interesting that even this exceedingly simple monetary model fails to fully support the neo-Fisherian hypothesis. Granted, it comes closer than most models: a higher nominal interest rate given the current price level does mean a higher future price level, but this can be easily dealt with if we add some kind of nominal rigidity to the model. Furthermore, CIA models don't exhibit interest elasticity in money demand, which, if present, would mean that permanent increases in the money supply would result in a lower nominal interest rate and a higher price level. Perhaps Stephen spoke too soon.

Extended Response to Nick Rowe

Nick Rowe on Twitter earlier today:

If [a] central bank targeted the price of peanuts, would we blame recessions on bad peanut harvests? Or blame [the] central bank for not raising [the] target price?

It depends. It depends on how quickly the central bank finds out about the bad peanut harvest, how quickly the new policy can be enacted, and how effectively the central bank can control the price of peanuts.

Suppose no one know about the size of the peanut harvest until the following period. In this case, the central bank, which we will assume can completely control the price of peanuts for the time being, is not culpable for the recession that occurs because the price of peanuts is too low. The central bank could not have known that the price level (of peanuts) at which output remained at potential was higher than they otherwise thought, so they cannot be blamed for the recession that ensues.

If, on a slightly different note, the central bank faces a delay in policy implementation, it may not be able to act quickly enough to prevent a recession; they can raise the target price with a delay, but there will still be a recession in the meantime and the central bank is not culpable.

Alternatively, assume that the central bank knows about the bad harvest in real time and doesn't face a policy lag, but, for some reason, is unable to set the price of peanuts any higher. In this case, the central bank can't be blamed either -- there is nothing it can do to prevent it from happening, so the correct culprit for the recession is the bad peanut harvest.

Generally, assuming there are no significant lags in information or implementation, the central bank would be to blame for not preventing the recession. The only time that the blame really shouldn't fall on a central bank is when it can't control the price (of peanuts) -- in this case, central bank impotence is to blame for the recession, not actions taken by the central bank.

With that aside, now we can go about determining when central banks are impotent.

Objectives vs. Tools of Monetary Policy

In the comments of one of Nick Rowe's recent posts, Scott Sumner has accused me of confusing objectives and tools of monetary policy:

You are looking at the causal effects of QE, whereas it makes more sense to view QE as the effect of a tight monetary policy that drives rates to zero. If you do a more expansionary monetary policy, such as currency depreciation, then you do not need as much QE. QE is a defensive mechanism, monetary policy needs to be viewed in terms of the policy goals of the central bank, and in terms of whether it will do whatever it takes to reach those goals.

Basically, Scott is suggesting that quantitative easing isn't actually a monetary policy, and is instead the natural conclusion to what he does view as monetary policy -- currency depreciation. Here, Sumner provides an interesting set of definition for what constitutes monetary policy and, more generally, what can reasonably be considered exogenous to a central bank.

In his mind, exchange rates are basically exogenous to the extent that central banks try to influence them. This is evident from his implicit assertion that, if central banks are "doing whatever it takes to reach [their] goals," they will invariably reach those goals. Of course, this isn't necessarily news, everyone has know Sumner's opinion that central banks are nearly omnipotent for quite some time, but this time he has laid it out more directly.

According to Sumner, the evolution of any nominal variable over time can be completely controlled by a central bank and, as such, can be used as a point of criticism for that central bank: "monetary policy needs to be viewed in terms of the policy goals of the central bank." As such, the actual polices that central banks follow are completely irrelevant; it doesn't matter what the path of interest rates is, the correct judge of current Federal Reserve policy (for example) is whether or not inflation is on target.

Of course, I, along with I hope the majority of people, don't see monetary policy in this light. Sumner seems to have made a point of confusing monetary policy -- e.g., QE, interest rate setting, open market operations -- with whatever nominal variable he happens to care about at the moment -- in this case exchange rates. This separation is important; it allows us to understand more directly a central bank's goals and how it intends to achieve those goals.

Evidently, Scott could care less about the how and only wants us to focus on the goals. He basically has reduced his thinking about monetary policy to the point that he views NGDP as an instrument of the central bank -- effectively an exogenous variable -- rather than a variable that a central bank may act to control. This level of abstraction from the operation of monetary policy, in my opinion even more grievous than the New Keynesian obsession with the nominal interest rate, is what allows Market Monetarists to callously ignore every model that doesn't allow exogenous NGDP that says the zero lower bound actually represents a constraint on monetary policy.

If central banks could make NGDP exogenous, would they be able to make NGDP exogenous? Naturally, but no one should care about the answer to such a redundant question, yet this is effectively the answer that you get from Sumner; he'll simply assert that "the BOC can always depreciate the Canadian dollar. The zero bound is not an issue in Canada" (from an earlier comment on the same post). Naturally, we should all trust Sumner's clairvoyance on this issue, clearly no argument about monetary policy effectiveness is necessary (see my first comment on Nick Rowe's post, if you want one anyway) and we can rest assured that fiscal policy is never necessary.

Ideally, considering the ability of monetary policy to effectively deal with challenges should be at least of some consideration and, since monetary policy has proved theoretically capable of offsetting the demand-side effects of fiscal stimulus among other shocks, the only point at which this can be of much concern is the zero lower bound. Both Sumner's and Rowe's refusal to give theoretical arguments against me in this area is rather troubling, evidently just assuming monetary policy is effective in every circumstance is completely acceptable.

Timing and Composition

Family members are often confused by my simultaneous support for looser fiscal policy in the United States and disdain for Republican tax proposals during this election cycle on the ground that they would result in too much deficit spending. On the surface, my policy preferences seem contradictory; I neither support efforts to rein in the deficit nor the large tax cut proposals of the majority of Republicans. There are two primary reasons for this seemingly strange predilection: timing and composition.

The length of time each policy lasts is crucial to my support. As per the 'New Keynesian consensus,' loose fiscal policy should only be used until monetary policy can be reasonably declared unconstrained by the zero lower bound. This is why deliberate deficit cutting policies should not have been undertaken, and arguably should not be pursued further, until two criteria have been met: the federal funds rate must be above the zero lower bound and there must be little to no risk that the zero lower bound will be made to bind by either a tightening of fiscal policy or some other shock to the economy. At the time of writing this post, only the first criterion is fulfilled -- the Federal Reserve has decided to raise the target fed funds rate, but, since it stands somewhere between 0.25% and 0.5% (the Fed has adopted a target range instead of a strict target), it would be reasonable to suggest that a large negative fiscal shock could be more than the Fed can handle without being thrown back into a liquidity trap (in this sense, the US could still be considered to be in a liquidity trap, even though the zero lower bound no longer binds).

GOP tax cut proposals would undoubtedly achieve the temporary goal of looser fiscal policy, but they would be on a completely wrong timescale. Conventional analysis only suggests loose fiscal policy for the duration of the liquidity trap, and, since the tax cuts are permanent to the extent that they are not repealed by future administrations, they fail miserably in this regard. In other words, fiscal policy would be too loose for too long under large tax cuts -- especially if they are not accompanied by corresponding reductions in government spending. Additionally, spending cuts are arguably more damaging than tax cuts are stimulative in liquidity traps, so a fiscal adjustment fully in line with, e.g., Rand Paul's or Ted Cruz' preferences could completely fail to comply with the recommendations of mainstream economics, which scares me enough in its own right to warrant a revocation of support.

My second criticism of the GOP tax plans is more personal; I think that government spending and taxes in the United States should be higher, not lower. There are certainly arguments to be made that government spending in the United States does nothing to raise aggregate utility and should thus be cut, but I believe, and I think most other economists agree with me, that this is definitely not the case. This is especially true in infrastructure, or more generally government investment -- currently at its lowest level as a percentage of GDP since 1948 -- which sorely needs to be increased. Further, spending on Social Security and Medicare should increase over the next decade or two because of the changing demographics of the country. If we adopt the tax proposals of many if not all of the GOP candidates, spending cuts will have to come from somewhere and, given the Republican obsession with massive military spending, they will probably not be defense cuts. This pretty much leaves entitlements and investment -- both of which would cause significant pain going forward if they were cut significantly.

Ideally, fiscal policy makers would focus in the short term on simply not cutting spending too ferociously and in the long run on figuring out how to raise the revenue required for higher levels of government investment and entitlement spending. The GOP seems prepared to do neither of these and, as such, I am not prepared to endorse them for their fiscal policy.

Choosing the Best Model For Each Context

In spite of perhaps attracting the wrath of Jason Smith, I think it is safe to say that economics is too complicated for there to be one generally applicable model of everything. Because of this, there is a veritable plethora of economic models available to the economic theorist. This simply leaves the question of which one to use in which circumstance.

Simon Wren-Lewis seems to think that economists should select between models in an ex-post manner -- that is, we should seen which model better represents the data and use that model from then on:

How do we know if most economic cycles are described by Real Business Cycles (RBC) or Keynesian dynamics. One big clue is layoffs: if employment is fall because workers are choosing not to work, we could have an RBC mechanism, but if workers are being laid off (and are deeply unhappy about is) this is more characteristic of a Keynesian downturn.

 The issue here is that we can only diagnose events after the fact, we cannot reasonably make predictions because of the impossibility of ex ante empirical validation: it is impossible to determine whether or not a recession is New Keynesian or if it is a Real Business Cycle before data are released.

This is why context-based validation of theory is superior to empirical validation in the case of economics. The context -- i.e. the sub-field of economics that is being studied -- should inform model choice almost entirely. If the field is business cycles, then the relevant model is a New Keynesian DSGE model and if the field is growth theory, then New Keynesian models are superfluous and should be tabled in favor of neoclassical models -- whose only difference from their New Keynesian counterparts is nominal rigidity, which is irrelevant over a time scale longer than a decade.

Predictions about the economy can now be made based on currently available information: it is possible to determine whether or not, e.g. financial frictions should be present in our business cycle model based on the current state of the economy: we knew by Q3 2008 that financial frictions were relevant, so we should have put them in a model if we were trying to predict the next few years.

Alternatively, the model I should choose to use depends on the kind of thought experiment I choose to embark on. Am I trying to compare PAYGO pensions with Social Security? If so, the obvious model to use is a simple OLG model without a labor-leisure trade-off or sticky prices. Choice of models is equivalent to choice of assumptions, at least when it comes to the DGE approach currently dominant in economics, and assumption choice depends entirely on the question being asked. Nominal rigidity is obviously relevant for business cycle theory, but completely useless when it comes to determining the level effect of a tax increase.

Hopefully this selection mechanism is specific enough to not be "basically feelings," as Jason Smith would suggest is the case for most of economics.

People Should Be More Honest With Charts

Recently, Scott Sumner wrote a blog post with this chart in it:

 I thought it would be interesting to see how well this relationship held over the period that Sumner didn't include in his chart. Here it is:

It's interesting to note that the relationship doesn't look so good when you look at the entire sample in which all of the data is available. This is aside from that fact that the idea that the NGDP/Wage ratio would track unemployment is part of basic neoclassical theory and has nothing to do with wage stickiness.

Start with a simple Cobb-Douglas production function with employment and capital:

$$(1)\: Y_t = F(K_{t-1},L_t) = K_{t-1}^\alpha L_t^{1-\alpha} $$

Assume that the firm maximizes profits, $Y_t - w_t L_t - r_{t-1} K_{t-1}$ and you get the following first order condition for labor:

$$(2)\: w_t = (1-\alpha)\left(\frac{Y_t}{L_t}\right) $$

Dividing by $Y_t$ will give the nominal wage to NGDP ratio (since the nominal wage to NGDP ratio is the same as the real wage to RGDP ratio), which is

$$(3)\: \frac{w_t}{Y_t} = \frac{1-\alpha}{L_t} $$

It's clear from this that, in a simple neoclassical model, the nominal wage to NGDP ratio is expected to be negatively correlated with employment and, therefore, positively correlated with unemployment -- which is coincidentally the exact thing that Scott's chart shows. Variations in the nominal wage to NGDP ratio are not, in fact, vindications of the musical chairs model.

What's the Significance of Low Real Interest Rates?

  

[Note: I started writing this post a while ago, so it ostensibly has no connection with these two posts that Nick Rowe and Scott Sumner wrote recently. I just realized that this is somewhat relevant, so I decided to finish it] 

For the last twenty years or so, real interest rates on government bonds have continued to fall from their high of about 9%. Determining the cause of such a fall is by no means an easy task; after all economic theory generally suggests that real interest rates on safe assets -- like government bonds -- should be relatively constant in the long run and reflect the rate at which consumers discount future spending relative to current spending. Economic theory tells us that low real interest rates mean that current consumption is high and future consumption is low relative to what it otherwise would have been. This certainly is a possibility; perhaps falling real interest rates are indicative of a shift in consumer spending patterns away from saving and into borrowing, although the causality seems to be backwards if that is truly the case, which leaves the question of what has caused this decline in real interest rates open once again. 

Perhaps the basic models in which the government has no power over the real interest rate in the long run are incorrect; given the sharp increase in the real interest rate on government bonds during the 1980s, this certainly seems plausible. In this case, it may be useful to switch to looking at this problem through the lens of an OLG model instead of a basic representative agent RBC/Neo Classical one. Every period, a new young agent is born with the endowment $y$ which can be used to buy either consumption ($c^y_t$) or government bonds ($b_t$), or to invest in capital ($k_t$). The young agent faces the budget constraint

$$(1)\: y = c^y_t + b_t + k_t$$

In the next period, the young become old and use income from interest on government bonds, $R_t b_t$, and from income generated from capital, $f(k_t)$ to finance their consumption and the taxes levied by the government. Old agents face the budget constraint

$$(2)\: c^o_{t+1} = R_t b_t + f(k_t) - \tau_t$$

Agents are born wanting to maximize their consumption in both periods of their life, with consumption when old discounted at rate $\rho$. The agents' discount factor is $\beta = \frac{1}{1 + \rho}$. Utility it derived from the log of current young consumption and the log of future old consumption:

$$ U = \log c^y_t + \beta \log c^o_{t+1} $$

Agents maximize their utility function subject to both of their budget constraints. Young agents choose their consumption so that

$$ (3)\: \frac{1}{c^y_t} = \beta \frac{1}{c^o_{t+1}} R_t $$

That is, young agents take as given the interest rate the they can receive by saving now and consuming later or that they would pay if they consumed now and saved later and decide to save more if the interest rate is high -- since their lifetime income can be increased by their saving -- and save less if the interest rate is low. The government sets the number of bonds that it issues by discretion each period which, given the young agent's consumption decision, determines the level of capital investment. 

Another first order condition of the model is the the real interest rate on government bonds is equal to the marginal productivity of capital. That is, 

$$ (4)\: R_t = f'(k_t)$$

Since the level of government bonds determines capital investment, it also determines the real interest rate on government bonds. More government debt means less capital which, per $4$, means a higher real interest rate (assuming that $f(k)=k^\alpha$ where $\alpha < 1$). This works because agents must be indifferent between holding more government bonds or more capital in equilibrium; otherwise they would end up demanding more or less capital than they wanted. 

In this model, low real interest rates are a result of high capital expenditure and low government debt. The prescription for low interest rates, then, is to engage in a large fiscal expansion that would increase the amount of government bonds in the economy. Less capital demand would have to be justified by a higher real interest rate. Of course, this seems empirically slightly dubious. After all, the amount of government debt skyrocketed in 2008 and interest rates failed to rise. To understand why this wouldn't necessarily be consistent with higher real interest rates, it's important to think along the lines of a demand for government bonds. 

Agents in this model are willing to demand more government bonds at higher interest rates, so if the government sets the supply of government bonds higher, then the demand must correspondingly rise through an increase in the real interest rate. The reason that massive increases in government debt in 2008 and 2009 are not consistent with higher real interest rates is that demand for government debt increased; perhaps even by more than the increase in supply. This was likely caused by the sudden illiquidity associated with other assets that were previously considered safe - e.g. mortgage backed securities or Greek government bonds. The resulting surge in demand for government bonds is known almost colloquially as a 'flight to quality.' 

The ideal fiscal response to this is to satiate demand for government debt by running large deficits (note that this is the exact opposite of the policy actions taken by the majority of governments since 2008). In a way, this is a non-Keynesian reason for pursuing fiscal stimulus; more government debt would be useful for raising the real interest rate. Not only would this make the economy closer to a competitive equilibrium (one without government intervention), it would likely make monetary policy more effective. Narayana Kocherlakota, president of the Minneapolis Fed, made this point in a speech in July. The basic argument he presents is that the government can raise the long-run neutral real interest rate by increasing the amount of government debt. The higher neutral rate of interest (i.e. the real interest rate in this model, since there is no money) will make it so that the Fed will be less likely to hit the zero lower bound when trying to ensure that target is hit. 

Effectively, fiscal policy should be used to remedy situations in which the demand for money is indeterminate and the central bank cannot adequately influence the real interest rate (see, e.g., here).

A Novel New Keynesian View of Fiscal Policy

Usually when I read New Keynesian economists on fiscal policy, they tend to focus more on fiscal multipliers or the effectiveness of tax cuts at the zero lower bound. But what about fiscal policy in general? I have come across some literature on this, but it usually limits itself to comparing the relative roles of monetary and fiscal policy - e.g. what is the optimal coefficient for the output gap in the fiscal policy rule? Here, I'd like to present a somewhat novel approach to New Keynesian fiscal policy (at least I've never seen or read this anywhere else).

Consider first the basic Consumption Euler equation that determines how household's allocate consumption between the present and the future given an interest rate.

$$ (1)\: c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} \left(\frac{1 + i_t}{1 + E_t\pi_{t+1}}\right) $$

In simple New Keynesian models, real GDP is composed of just government spending and consumption since there is no capital accumulation, so $1$ can be rewritten as a function of output, $y_t$, and government spending, $g_t$.

$$(2)\: (y_t - g_t)^{-\sigma} = \beta E_t (y_{t+1} - g_{t+1})^{-\sigma} \left(\frac{1 + i_t}{1 + E_t\pi_{t+1}}\right)$$

It is useful to linearize $2$ to make it a bit easier to work with, but first, to make the math a little easier, it is helpful to notice that $y_t - g_t$ is the same as $y_t(1 - \frac{g_t}{y_t})$. Given this, defining $\theta$ as $\frac{1}{\sigma}$, and defining $\beta$ as the inverse of the gross time preference rate, $\rho$, it is possible to write $2$ in log-linear form - i.e. all equations are written as percentage gaps from their long run level.

$$(3)\: \hat{y}_t = E_t\hat{y}_{t+1} - E_t \Delta \hat{g}_{t+1} - \theta (i_t - E_t\pi_{t+1} - \rho)$$

Keep note that, in this case, $\hat{g}_t$ is the gap of the government spending to GDP ratio from trend rather than simply government spending from trend. For my purposes, this is basically irrelevant.

If the goal of fiscal policy is to ensure that the output gap is zero at all times - not too weak of an assumption in my opinion - then it's pretty simple to solve for optimal policy given $3$:

$$(4)\: E_t \Delta \hat{g}_{t+1} = -\theta(i_t - E_t\pi_{t+1} - \rho) $$

In English, equation $4$ tells us that the role of fiscal policy is simply to offset any failure of the monetary authority to set the right real interest rate ($i_t - E_t \pi_{t+1}$). If, for example, the monetary authority has set a real interest rate that is too high, then government spending should be expected to shrink relative to trend in the next period. This can be accomplished either through stimulus - raising current government spending now and reducing it in the future - or through causing expected temporary austerity - decreasing next period's government spending then allowing government spending to return to trend.

The first option is preferable for a couple of reasons. For one thing, government spending also has real effects (see my previous blog post), so austerity might have unintended supply side consequences. Also, the austerity must be reversed at some point for the policy to work - since $\hat{g}_{t+1}$ would fall to zero if the austerity were permanent - so future fiscal policy may be impaired if the central bank continues to set an interest rate that's too high.

Another way of approaching this is to rewrite $3$ to incorporate a 'natural real rate of interest.' In this case, $3$ can be rewritten as

$$(5)\: \hat{y}_t = E_t\hat{y}_{t+1} - \theta (i_t - E_t\pi_{t+1} - \rho + \frac{E_t\Delta\hat{g}_{t+1}}{\theta}) $$

Defining the natural real rate of interest at which the output gap is zero, it is clear that $\rho - \frac{E_t\Delta\hat{g}_{t+1}}{\theta}$ is equal to the natural rate, $r^n_t$.

Assuming the central bank tries to set the real interest rate equal to the natural rate unless the zero lower bound is binding, i.e. $i_t = \max\left(0,\: E_t \pi_{t+1} + r^n_t\right)$, the job of the government can be seen as preventing the zero lower bound from ever binding, or, in other words, setting $E_t\pi_{t+1} + r^n_t > 0\: \forall t$.

$$(6)\: 0 < E_t\pi_{t+1} + \rho - \frac{E_t\Delta\hat{g}_{t+1}}{\theta} $$

or

$$(7)\: E_t\Delta\hat{g}_{t+1} < \theta(E_t\pi_{t+1} + \rho) $$

From $7$, it is clear that expected growth in government spending relative to trend should always be less than a function of the expected inflation rate. That is, the lower the expected inflation rate, the bigger the stimulus that should be undertaken. Effectively the goal of fiscal policy is to offset failures in monetary policy and to make sure that the zero lower bound never binds in the first place.

Shut Up About Ricardian Equivalence

Economists that are both opposed to and in favor of fiscal stimulus frequently cite Ricardian equivalence as a reason that, in models with perfect credit markets, it doesn't matter whether stimulus is funded through increased taxes or through deficits. The problem with this analysis is that it assumes lump sum taxation. That is, taxes are not collected from things like consumption expenditures, which are effectively no different than deficits because 1) they don't discourage people from working, consuming, investing, etc and 2) they are expected to rise at some point in the future to retire the current debt, so the present value of taxes goes up with government spending. In reality, the argument is about distortionary taxes vs. deficits (= lump sum taxes, as per Ricardian Equivalence).

Distortionary taxes are different than their lump sum counterparts since they directly act to disincentivize working (in the specific case of income taxes, which I will limit my analysis to from now on) and can thus either partially or fully negate the effects of a fiscal stimulus. So, when John Cochrane says something like "'Ricardian Equivalence,' which is the theorem that stimulus does not work in a well-functioning economy," [1] he's clearly confusing the two types of taxation as well as ignoring the fact that neoclassical economics predicts a positive multiplier on government spending [2]. To illustrate this, I wrote down a standard Real Business Cycle model and ran two simulations: one in which a temporary fiscal expansion was financed entirely with an income tax and another in which the same fiscal stimulus was financed partially by deficits (see the appendix for a derivation of the model).

Figure 1: Impulse Response Function of Output to the Stimulus

Figure 2: Impulse Response Function of the Income Tax Rate to the Stimulus

Figure 3: Government Spending in both simulations; Government Debt in the second simulation

The Ricardian Equivalence argument would be irrelevant if 1) the stimulus had a positive effect on output and 2) the tax funded stimulus was initially less effective than the partially deficit funded one. As you can see in figure 1, both of these are true; the stimulus positively impacted output in each simulation and the stimulus was initially more effective when taxes were not increased to fully finance the stimulus on impact. The effectiveness of the stimulus is slightly less sound of a result, though. The fiscal multiplier in neoclassical models is highly dependent on calibration (see, e.g., [2]) and can range anywhere from zero to one, without distortionary taxation, depending on the specific calibration used. Regardless, the most important part of this argument is sound; the Ricardian Equivalence argument against deficit funded stimulus is wrong and should be ignored completely as it applies to a form of taxation that doesn't actually exist.

References:

[1] John Cochrane, 2011. "Krugman on Stimulus" The Grumpy Economist.

[2] Woodford, Michael. 2011. "Simple Analytics of the Government Expenditure Multiplier." American Economic Journal: Macroeconomics, 3(1): 1-35.

Appendix:

The following is a derivation of model that I used to generate the impulse response functions in figures 1, 2, and 3.

Household:

There is a representative household who maximizes the utility function $U = E_0 \sum^\infty_{t=0} \beta^t\left(\frac{c_t^{1-\sigma}}{1-\sigma} - \frac{n_t^{1+\phi}}{1 + \phi}\right)$ where $E_t$ is the rational expectations operator given information available in period $t$, $c_t$ is the household's consumption, $n_t$ is the labor supply, and $\beta$ is the household's discount factor - the rate at which future utility is discounted relative to current utility. The household can use net-of-taxes income from labor ($(1-\tau^w_t)w_t n_t$, where $w_t$ is the real wage), government bonds carried from last period ($R_{t-1} B_{t-1}$, where $R_t$ is the interest rate that bonds maturing in period $t$ - $B_t$ - pay), and net-of-depreciation income capital, $(1 + r_{t-1} - \delta)k_{t-1}$ to purchase consumption, new government bonds, or new capital. The budget constraint can be written as

$$ (1.1)\: (1-\tau^w_t)w_t n_t + R_{t-1} B_{t-1} + (1 + r_{t-1} - \delta)k_{t-1} = c_t + B_t + k_t $$

The household maximizes $U$ subject to $1.1$  in order to determine its behavior:

$$ (1.2)\: c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} (1 + r_t - \delta) $$
$$ (1.3)\: R_t = 1 + r_t - \delta $$
$$ (1.4)\: (1-\tau^w_t)w_t = c_t^\sigma n_t^\phi $$

Additionally, it is useful to define investment, $i_t$ as the instrument of capital accumulation:

$$(1.5)\: k_t = (1-\delta)k_{t-1} + i_t $$

Firm:

The firm bundles capital carried from last period and labor using a Cobb-Douglas production function to form output, $y_t$

$$ (2.1)\: y_t = k_{t-1}^\alpha n_t^{1-\alpha} $$

The firm maximizes profits, $y_t - w_t n_t - r_{t-1}k_{t-1}$ subject to $2.1$ in order to determine labor and capital demand

$$ (2.2)\: w_t = (1 - \alpha)\frac{y_t}{n_t} $$
$$ (2.3)\: r_t = \alpha E_t\frac{y_{t+1}}{k_t} $$

Government:

The government issues new government bonds and collects tax revenue to pay for both government spending and interest on government bonds carried from last period. The government budget constraint can be written as

$$ (3.1)\: B_t + \tau^w_t w_t n_t = g_t + R_{t-1} B_{t-1} $$

In the first simulation, it is assumed that the government ensures $B_t = 0\: \forall t$, so government spending is simply financed by taxes

$$ (3.2)\: \tau^w_t w_t n_t = g_t $$

In the second simulation, the government sets the tax rate as a function of the tax rate consistent with the long run level of government spending, $\tau^w_{SS}$ and the level of government debt issued in the previous period, $B_t$. The rule for the tax rate in the second simulation is

$$ (3.3)\: \tau^w_t = \tau^w_{SS} + \phi_b B_{t-1} $$

In both simulations, government spending follows an autoregressive process and returns to its long run trend trend at decay factor $\rho$. Government spending follows

$$ (3.4)\: g_t = (1 - \rho)g_{SS} + \rho g_{t-1} + \eta_t $$

Where $\eta_t$ also follows an autoregressive process with the same decay factor an is hit with with the shock $\epsilon^g_t$

$$ (3.5)\: \eta_t = \rho \eta_{t-1} + \epsilon^g_t $$

Equilibrium:

Combining $1.1$, $1.5$, and $3.1$ yields the resource constraint for the economy

$$ (1)\: y_t = c_t + i_t + g_t $$

Equations $1.2$-$3.5$ can be used to determine the equilibrium for the rest of the endogenous variables:

$$ (2)\: c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} (1 + r_t - \delta) $$
$$ (3)\: R_t = 1 + r_t - \delta $$
$$ (4)\: (1-\tau^w_t)w_t = c_t^\sigma n_t^\phi $$
$$ (5)\: k_t = (1-\delta)k_{t-1} + i_t $$
$$ (6)\: y_t = k_{t-1}^\alpha n_t^{1-\alpha} $$
$$ (7)\: w_t = (1 - \alpha)\frac{y_t}{n_t} $$
$$ (8)\: r_t = \alpha E_t\frac{y_{t+1}}{k_t} $$
$$ (9)\: B_t + \tau^w_t w_t n_t = g_t + R_{t-1} B_{t-1} $$
$$ (10a)\: \tau^w_t w_t n_t = g_t\: \mbox{in simulation 1}$$
$$ (10b)\: \tau^w_t = \tau^w_{SS} + \phi_b B_{t-1}\: \mbox{in simulation 2}$$
$$ (11)\: g_t = (1 - \rho)g_{SS} + \rho g_{t-1} + \eta_t $$
$$ (12)\: \eta_t = \rho \eta_{t-1} + \epsilon^g_t $$

Using Demographics to Estimate Potential Output in Japan

I don't stray into empirical matters very frequently because I'm really not all that adept at them, but I thought it would be interesting to feed Japan's working age population growth into a basic Solow growth model.

Skip the following if you already understand the Solow model:
In the Solow model, it is assumed that output is produced using three inputs: capital, labor, and productivity. The production function is Cobb-Douglas for capital and labor (with constant returns to scale), but is multiplied by what's called the Total Factor of Productivity, or TFP, which represents technological progress. Defining output as $Y_t$, capital as $K_t$, labor as $N_t$, and TFP as $A_t$, the production function can be written as
$$ (1)\: Y_t = A_t K_t^\alpha N_t^{1-\alpha}$$
where $\alpha$ is capital's share in production and $1-\alpha$ is labor's share in production. Workers devote a constant share $c$ of production to consumption ($C_t$), so $ C_t = c Y_t $. Non-consumed income is used to increase the capital stock, which exogenously depreciates at $\delta$. Defining $s$, the share of income put toward investment in each period as $1-c$ allows us to write the capital accumulation equation as such:
$$ (2)\: K_{t+1} = (1 - \delta) K_t + s Y_t $$
The labor force is assumed to grow at constant rate $n$, so next period's labor forced is defined as
$$ (3)\: L_{t+1} = L_t (1 + n) $$
TFP is assumed to grow at constant rate $g$, so TFP evolves according to
$$ (4)\: A_{t+1} = A_t (1 + g) $$
Given $L_0$, $A_0$, and $K_0$, the economy will eventually converge to a balanced growth path in which all variables grow at the same rate as productivity: $g$. If one of the parameters ($\alpha$, $\delta$, $s$, $n$, or $g$) changes, then the economy will take time to adjust to new equilibrium levels.

Figure 1

As you can see in figure one, Japan began to see a secular decline in it's working age population growth rate in about 1990. This decline coincides roughly with Japan's lost decade -- the period between the mid 90s and the early 2000s characterized by low growth and high unemployment. Given the low working age population growth, it may be possible to explain some of this lack of economic activity with the Solow model. Assuming constant technological growth of 1%, a capital depreciation rate of 2.5%, a capital share of 33%, and a savings rate of 10% (I have no clue how close to accurate this calibration is, if someone wanted to find the average values of each variable over the last 20 years or so in Japan, I'll update them, but right now I can't be bothered to find the information myself), I was able to come up with an estimate of 'potential' output in Japan -- i.e. what Japanese output would be absent any shocks to productivity, government spending, or monetary policy (or natural disasters, which explain the 2011 output contraction).

Here are a couple of graphs relating actual output to demographically-adjusted potential output:

Figure 2

 

 

Figure 3

Figure 2 plots my estimate of potential output against actual output, assuming potential output was 2% above actual output in 1995 and figure 1 plots the output gap, or the percentage gap between actual and potential output. An interesting note here is that potential output, absent any demographic or technological changes, is predicted converge to a decay rate of roughly 0.5% per year and potential output is currently growth at about zero percent per year, meaning that, not only is potential growth for the next couple of years zero, the economy should be expected to shrink without being in a recession in the future. That is, unless the workforce stops decaying so quickly.

Another interesting observation is that Japan's lost decade seems to closely resemble the experience that the United States has had since the Great Recession. This is entirely unsurprising given that both periods are characterized by monetary policy ineffectiveness (the zero lower bound), but the post 2007 experience in Japan could possibly be used to predict the outcome of another large recession in the US absent monetary policy normalization. Perhaps more on this later.

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.

Using Fiscal Policy to Escape a Liquidity Trap

Read the last post before you read this one; this post builds off of the analysis from that one.

In my last post, I explained my reasoning for monetary policy ineffectiveness at the zero lower bound on nominal interest rates. There, I explained that, at the zero lower bound, there is no equilibrium path of the price level (i.e., the model does not pin down a specific price level in all current and future periods). In this case, the central bank is powerless to escape the zero lower bound and must hope that the household randomly selects and equilibrium in which the cash advance constraint will bind in the future so that it can engage in expansionary monetary policy (in the future) in order to escape the zero lower bound. 

In my analysis, I did not model fiscal policy because I was specifically writing about monetary policy ineffectiveness. Nevertheless, fiscal policy could be used to pin down an equilibrium price level when monetary policy can't. To begin with, let's take the budget constraint from the last post:

$$(1a)\: M_{t-1} + (1+i_{t-1})B_{t-1} + P_t y = P_t c_t + B_t + M_t + T_t $$

Since the household sets $c_t$ equal to $y$, it is possible to rewrite the household's budget constraint as the government's budget constraint:

$$(1b)\: M_{t-1} + (1+i_{t-1})B_{t-1} = B_t + M_t + T_t $$

We can also plug in the consumption Euler equation (equation $5$ in the last post) to express the budget constraint without the nominal interest rate

$$(1c)\: M_{t-1} + \left(\frac{1}{\beta} \frac{P_t}{P_{t-1}}\right)B_{t-1} = B_t + M_t + T_t $$

Assuming that the zero lower bound is binding, the budget constraint can be further reduced to

$$(1d)\: M_{t-1} + B_{t-1} = B_t + M_t + T_t$$

With this budget constraint, it is possible to determine an equilibrium price level from the specification of fiscal policy and monetary policy. Essentially, the government can monopolize on four sources of revenue: issuing government bonds, increasing the supply of money, levying taxes, and inflation. To understand why inflation can be used for revenue, it is useful to divide $1d$ by the price level and get all the variables in real quantities (lower case letters indicated real variables, except for $T_t$ which is changed to $\tau_t$):

$$(1e)\: m_{t-1}\left(\frac{P_{t-1}}{P_t}\right) + b_{t-1}\left(\frac{P_{t-1}}{P_t}\right) =  b_t + m_t + \tau_t$$

In order to make $1e$ true, the government can obviously either increase $b_t$, $m_t$, or $\tau_t$. Alternatively, it can increase $P_t$, which will reduce the value of the entire left side of the budget constraint. Because of this, the fiscal authority can essentially choose to be irresponsible and the monetary authority will be forced to comply. Normally this means that the central bank must increase the growth rate of the money supply, but because of the zero lower bound, revenue from the central bank can either be more money or more inflation. Assuming the fiscal authority promises to not pay its debts (technically, this is called non-ricardian fiscal policy), only a money supply growth rule is needed to determine the price level. Given the money growth rule, all fiscal policy has to do is be just irresponsible enough to push inflation onto target.

From equations $4a$ and $4b$ in the last post, we know that $P_t/P_{t-1} = M_t/M_{t-1}$ if $P_t/P_{t-1} > \beta$ and that $P_t/P_{t-1} \leq M_t/M_{t-1}$ if $P_t/P_{t-1} = \beta$. This means that the growth rate of the money supply always represents an upper bound for the rate of inflation. Because of this, it makes sense for the central bank to grow the money supply at exactly the desired rate of inflation throughout the liquidity trap. This way, when the fiscal authority switches to non-recardian policy, the inflation rate has an upper bound and when the inflation rate goes up and the cash-in-advance constraint binds again monetary policy doesn't have to change.

Monetary Policy Effectiveness In Liquidity Traps

As I've argued here, conventional money demand models suggest that the price level becomes indeterminate at the zero lower bound and monetary expansion can not do anything to change inflation. In a recent conversation with Scott Sumner, Scott pointed to Paul Krugman's 1998 paper about this issue. Krugman suggests in his paper that only current monetary expansions are useless, but commitments to larger money supplies in the future (or, as Scott would probably like me to say, commitments that the current monetary expansion will be permanent) can both alleviate the liquidity trap and raise the current price level.

So, in line with Krugman's model, let's assume that there is a representative household that maximizes the utility function

$$(1)\: U = \sum^\infty_{t=0}\beta^t\left(u(c_t)\right) $$

where $\beta$ is the household's discount factor and $u(c_t)$ is the utility that the household gains from its consumption, $c_t$, in period $t$. The household is endowed without output $y$ every period and participates in an asset market where it trades one period government bonds and government money. The household's budget constraint is

$$(2)\: M_{t-1} + (1 + i_{t-1}) B_{t-1} + P_t y = P_t c_t + B_t + M_t + T_t $$

where $M_t$ is the money supply, $B_t$ is the household's holding of government bonds, $i_t$ is the nominal interest rate that government bonds pay, $P_t$ is the price level, and $T_t$ is the lump sum tax from the government. The household also faces a cash-in-advance constraint; it must finance its consumption with government cash. This constraint takes the form

$$(3)\: M_t \geq P_t c_t $$

Notice the fact that this is an inequality constraint. The household can hold as much money as it wants, but must at minimum have enough cash on hand to pay for its consumption. The household maximizes $1$ subject to $2$ and $3$ which yields the following first order conditions:

$$(4a)\: M_t = P_t y\: \mbox{if}\: i_t > 0$$ 

$$(4b)\: M_t \geq P_t y\: \mbox{if}\: i_t = 0$$

$$(5)\: 1 + i_t = \frac{1}{\beta}\frac{P_{t+1}}{P_t}$$

If, like Krugman did, we assume hat next period's price level is constant, we can draw a nice diagram with $4a$, $4b$, and $5$:

The solid blue line is the curve from $5$, the dotted blue line marks the zero lower bound, and the black lines represent the money supply. Normally, the central bank is in complete control of the price level and can move it around by moving the money supply around. But, because the cash-in-advance constraint does not bind at the zero lower bound, increases in the money supply at the zero lower bound will not be immediately spent by the household. This means that, given a constant future price level, the central bank can only push the price level up until it hits the zero lower bound. After that, no amount of current monetary expansion can increase the current price level.

Of course, all that was exactly in line with Krugman. Here's where it gets interesting, though. Krugman assumes in his paper that the central bank has control of the future price level the entire time and can easily increase the future money supply to end the liquidity trap. If we drop the assumption that the cash-in-advance constraint must bind in the next period, can the monetary expansion, regardless of permanence be effective? In order to escape the liquidity trap, the central bank needs to make the household expect that the price level next period will be higher than the price level this period (this would shift the solid blue curve in the graph to the right). 

I'm having a lot of trouble wrapping my head around it, but I think that everything hinges on expectations. The cash-in-advance constraint will only bind in the next period if the price level two periods ahead is expected to be higher than the price level next period and so on, ad infinitum. This means that the central bank can only exit the liquidity trap if the household subjectively expects inflation to be greater than the rate of time preference (the inverse of the discount factor subtracted by one) in the future. This is independent of the path of the money supply; not only is there no equilibrium for the price level in the static analysis at the zero lower bound, there is no equilibrium for the entire path of the price level once the zero lower bound has been reached.

The alternative is to reduce the current money supply until the cash-in-advance constraint binds once again; basically to cause a bunch of deflation now instead of in the future. The problem with this is that prices are sticky and a massive monetary contraction would cause a recession.