## 28 February 2016

### 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.

## 27 February 2016

### 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.

## 24 February 2016

### 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.

## 21 February 2016

### 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.

## 19 February 2016

### 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.

## 14 February 2016

### 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.