I Don't understand Market Monetarist Logic

So the typical market monetarist view on business cycles is that low NGDP causes low RGDP. Let $p$ indicate whether or not NGDP is lower than normal and $q$ indicate whether or not RGDP is lower than normal. The market monetarist contention can be represented as such:

$$ p \rightarrow q $$

If a central bank successfully targets inflation, then NGDP should track RGDP (because NGDP growth is always equal to RGDP growth plus the inflation target). This looks like

$$ q \rightarrow p $$

These two statements don't seem to make sense when paired with each other... According to market monetarists, low NGDP caused the great recession, but, because of the inflation targeting regime in the US, low RGDP causes low NGDP... Do market monetarists think that the great recession caused itself? Their logic seems to imply either that recessions come from something like multiple equilibria when central banks target inflation or that they just can't happen because in inflation targeting regimes, NGDP doesn't fall unless RGDP does and RGDP doesn't fall unless NGDP does, so neither ever fall. If they think that inflation targeting produces multiple equilibria, then why don't they say so? If the multiple equilibria logic is correct, then they shouldn't be strictly advocating and NGDP target; they should be telling everyone to switch to any target that doesn't make NGDP depend of RGDP...

This is all extremely confusing. 

Notes on Taylor Rules, Inflation, and Neo-Fisherism

I've been working on writing a paper outlining my views on three topics in monetary economics (the three things in the title). Click here for the pdf of what I have written so far. Here is the text if you don't want to download the pdf:

UPDATE: The pdf link should update automatically to changes, but I won't change the text in the blog post. Just download a copy of the pdf every time you want the most up to date version.

Introduction:

There has been quite a bit of discussion about the relationship between the nominal interest rate and the rate of inflation recently among economists. To my knowledge, the problem began when Cochrane (2007) challenged the idea that the inflation rate could be determined with simply a Taylor Rule and a Fisher relation in combination with a commitment to active monetary policy and implicitly passive fiscal policy (see Leeper (1991) for example). Cochrane's key insight was that, in these models, the central bank is essentially committing to cause inflation to explode by increasing the nominal interest rate (effectively the expected inflation rate) more than one for one with current inflation. Because economists had ruled out explosive solutions, the only other equilibrium – one in which the inflation rate jumps immediately at period zero to the central bank's target – was considered. As Cochrane noted, there is not necessarily any reason to rule out explosions in nominal variables as they have no impact on the real economy in the models in question.

\par In the years since then, the failure of zero interest rate policies to generate inflation became of interest. Pretty soon, a similar yet entirely different debate came into existence. A few of economists (to my knowledge, Williamson and Cochrane) had the novel idea that the nominal interest rate had a causal relationship with the rate of inflation. This notion had long existed in the literature and is even a property of just about every macroeconomic model; the problem, in fact, was not the notion that high inflation and high interest rates happened at the same time. Rather, it was the idea that central banks could deliberately cause inflation by setting the nominal interest rate at a higher level. The consensus that active monetary policy was required for inflation stabilization and that positive deviations from the target interest rate implied by a Taylor Rule would result in lower inflation was in direct opposition to these "Neo-Fisherian" claims, so a debate that pulled in a slew of other economists ensued.

The difficulty in this case is that both sides are right in their own way. The consensus was correct that, so long as the central bank uses a Taylor Rule to target inflation, positive deviations from that target would result in a lower inflation rate. The Neo-Fisherian view is correct in the sense that if the central bank does not follow a rule and deliberately loosens monetary policy, the inflation rate and the nominal interest rate will increase. The issue with both views is that the underlying assumptions are either not understood or not made clear by there proponents. Economists putting forward the conventional wisdom don't make it clear that the Taylor Rule is the sole cause of inflation dynamics in their model and Neo-Fisherians fail to put forward that the result that they purport is highly dependent on how the money supply (or in some cases fiscal policy) acts when the nominal interest rate is increased. Each model relies heavily on a set of implausible assumptions about the way central banks behave. It is clear that central banks don't behave in the way implied by the consensus models and it is equally clear the the Neo-Fisherian result only occurs when monetary policy has taken a permanently more accommodative stance; even though this assumption is not put forward by its proponents.

If, as I suggest, the "Neo-Fisherian problem" and the "Taylor Rule problem" are all about assumptions, then their respective solutions are simple: just add some microfoundations. When it comes to arguments about monetary policy, the necessary microfoundation is painfully obvious. These models all need money in order for their implications to be understood. Interest elastic money demand functions solve Cochrane (2007)'s critique as they prohibit real explosions of the money supply – something that would happen if the nominal interest rate expanded or collapsed infinitely and money demand functions in general can determine when high interest rates mean tight monetary policy and when high interest rates mean loose monetary policy without appealing to dynamics implied by implicit monetary policy rules and without simply assuming that all high interest rates are do to loose money. 

Model:

We will begin by adding a simple ad-hoc money demand function to a two equation frictionless New Keynesian model and looking into the implications of the simple addition for monetary modeling. As usual, there is a Fisher equation relating the nominal interest rate to expected inflation and a Taylor Rule relating current inflation to the nominal interest rate.

$$i_t = \rho + E_t \pi_{t+1}$$

$$i_t = \rho + \phi \pi_t$$

$i_t$ is the nominal interest rate, $\pi_t$ is the inflation rate, $E_t$ is the period $t$ rational expectations operator, $\phi$ is the "inflation reaction parameter" on the Taylor Rule, and $\rho$ is the constant real interest rate. The sole addition that we will add to this basic model is a simple money demand function which sets real money demand equal to 

$$m_t - p_t = y - \eta i_t$$

where $m_t$ is the nominal money supple, $p_t$ is the log price level ($\pi_t = \Delta p_t$), $y$ is the (constant) level of output, and $\eta$ is the interest-elasticity of the money supply.

\par With the addition of the money demand function, so long as $\left|\eta\right| > 0$, Cochrane's problem with ruling out nominally explosive equilibria disappears. Now, a real variable depends on the nominal interest rate and prevents hyper inflations that are not caused by excessive money growth. In fact, adding money demand changes nothing about the dynamics of the model; following a Taylor Rule still gives the conventional wisdom about monetary policy without having to deal with the difficult problem of ruling out nominal explosions.

\par The interesting thing about this model is that it can replicate the Neo-Fisherian result easily. Consider a deterministic economy where the central bank permanently increases the growth rate of the money supply, $m^g_t = \Delta m_t$, from $m^g_0$ to $m^g_1 > m^g_0$.

If you have any feedback or suggestions before I continue to write, feel free to comment. I intend to continue by expanding my analysis to a more full fledged New Keynesian model with different types of money demand ranging from Money-In-The-Utility-Function to Cash-In-Advance and explain the mixed signal problems of using interest rates as an indicator of the stance of monetary policy. (I also plan to refer to more of the relevant literature than just "Determinacy and Identification with Taylor Rules")

New Keynesian Central Bankers Are Stupid

Imagine that there is a central bank that targets the inflation rate successfully every period because there are no real shocks in the economy. In this model, inflation looks like this:

$$ \pi_t = \pi^* $$

where $ \pi_t $ and $ \pi^* $ are the inflation rate and the inflation target, respectively. The nominal interest rate in this model will always be $ \pi^* $ higher than the constant (no real shocks) real interest rate, $ \rho $ and can be written as 

$$ i_t = \rho + \pi^* $$

Now suppose that the central bank is not omniscient and occasionally misses its target either on accident or because of some unforeseen shock. The inflation rate is now

$$ \pi_t = \pi^*+ \epsilon_t $$

where $ \epsilon_t $ is the central bank's error every period. Since the inflation rate is not serially correlated, the nominal interest rate remains equal to $ \pi^* \: \forall t $. Let's add some real shocks into this economy, so the real interest rate fluctuates over time, adjusts slowly, and is equal to $ r_t $.

$$ r_t = (1 - \rho^r)\rho  + \rho^r r_{t-1} + \nu_t $$

The nominal interest rate now moves around with the real shocks:

$$ i_t = r_t + \pi^* $$

For some unknown reason, the central bank decides to adopt a floating inflation target, $ \bar\pi_t $, and sets it so that it becomes a weighted average of $ \pi_{t - 1} $ and $ \pi^* $.

$$ \bar\pi_t = (1 - \rho^\pi)\pi^* + \rho^\pi \pi_{t-1} $$

The central bank still occasionally misses its target, so $ \pi_t $ is not always equal to $ \bar\pi_t $ and is instead

$$ \pi_t = (1-\rho^\pi)\pi^* + \rho^\pi \pi_{t-1} + \epsilon_t  $$

The nominal interest rate is related to the current inflation rate now because of the auto-regressive process that the rate of inflation fallows and can now be expressed as

$$ i_t = r_t + (1 - \rho^\pi)\pi^* + \rho^\pi \pi_t $$

By sheer assumption, let's say that $ \epsilon_t $ and $ \nu_t $ are negatively correlated. What is this model now? Well, it's New Keynesian, isn't it.

Think about it: There is a central banker that knows it could keep the real interest rate constant (or equal to its natural rate) by pegging the inflation rate, but instead he or she chooses to make it serially correlated by following a Taylor Rule. The real interest rate falls when inflation is above "target" and the nominal interest rate and the rate of inflation are positively correlated. Really, all you need to do to make the dynamics exactly like that of a New Keynesian model is to add a variable $ x_t $, call it the "output gap" and say that $ \dot x_t = r_t - \rho $.

Of course the real aspects of this "model" are really irrelevant (and pretty weak as assumptions go, replacing nominal rigidity with "shocks are negatively correlated" is pretty bad, the AR part of the real interest rate can make sense if the capital stock takes time to adjust, for example). What's really important is that NK central bankers are stupid. They know that they should be targeting a constant rate of inflation, but they abandon that for the sake of Taylor Rules and avoiding the money demand function.

I obviously don't think that central bankers can simply choose to their nominal target be achieved every period, but they should at least refrain from "endogenizing" the money supply in favor of a tool that can mean different things at different times depending on your assumptions.

 

Good Arguments For Deficits

Just a few quick thoughts on reasons that governments should run deficits. I might go more in depth on each of these later on (I've been planning to write a post on #1 and #3 for a while).

1. Seigniorage Revenue:

The growth of real money demand and the price level over time mean that there is a consistent stream of revenue flowing to the government that show that deficits are optimal for ensuring the stability of government debt in the long run.

2.  Government Spending Smoothing:

Tax revenues from distortionary taxes (i.e. nearly every form of revenue that a government can get) are quite volatile, so government spending should be smoothed so as to not exacerbate business cycles.

3. Deficits Do Not Distort; Taxes Do: 

In circumstances when the monetary authority can't stimulate the economy during a recession, it makes sense to increase government spending. The benefit from this new spending would be reduced since if it were funded with distortionary taxes; deficits do not distort.

Dynamics of Government Debt

I hope I don't make Nick Rowe [1], Scott Sumner, and their fellow Monetarists too angry by assuming that central banks can only monetize government debt, but I think this analysis is still relevant since central banks usually refrain from trading assets other than government bonds.

Anyway, on to the post. Imagine a world in which the fiscal authority never issues any debt. In this world, monetary policy would be equivalent to fiscal policy. Every deficit is funded by seigniorage, so either the central bank gets to target some nominal variable or the fiscal authority gets to set the inflation rate. To see how this works, consider eliminating government bonds (and other assets, should they be present) from the governments budget constraint. This gives

$$ (1) \: M_t + P_t \tau_t = M_{t-1} $$

where $ M_t $ is the money supply, $ P_t $ is the price level, and $ \tau_t $ is the treasury's surplus. Assuming the money demand function simplest money demand function possible, $ M_t = L(P_t) = P_t $ and expressing the constraint in real terms gives

$$ (2) \: \pi_t = -\tau_t $$ ($ \pi_t $ is the rate of inflation)

This world has the unfortunate problem of either being ultra-FTPL (fiscal authority determines the inflation rate) or just plain weird (I don't know what else to call a world where the central bank chooses the fiscal authority's surplus/deficit). Aside from the obvious difficulty of Sargent and Wallace's [2] game of chicken, the problem of a serious conflict of interests arises. What if the optimal fiscal policy is austerity, but the optimal monetary policy involves a high rate of inflation and vice versa? [3] Proposition 1: If that situation can arise, then a non-zero level of government debt is optimal. 

Essentially, government debt allows the monetary and fiscal authority to have contradicting policies at any given point in time. So long as there is government debt, the central bank can always control inflation (I think, but I need to look into the FTPL under an exogenous inflation rate or a money growth rule) and the treasury can always control the surplus.

Let's assume the the level of government debt must be positive [4]. Given this constraint, the central bank can at most monetize 100% of current government debt, essentially imposing a maximum rate of inflation that the central bank can achieve. Proposition 2: The ideal level of government debt is whatever is required for the central bank to achieve its nominal target at any point in time. [5] So, if government debt levels are not sufficiently high (or government debt is not growing quickly enough), then the central bank won't be able to attain its goals.

For complete monetary freedom in my model. government debt needs no upper limit, but having infinitely large government debt is not optimal for obvious fiscal reasons. Ideally, the real value of government debt should not be so high debt servicing costs on the part of the fiscal authority demand constant high primary surpluses. Proposition 3: In order to minimize the burden of high real debt levels, nominal debt should grow at a rate consistent with the central bank's nominal target.

Combining Propositions 2 & 3, we get Proposition 4: The level of government debt should always be high enough for the monetary authority to achieve its nominal target and should grow at the minimum rate required for said nominal target to be achieved.

[1] Nick Rowe, 2015. "Why can't central banks monetise something else?" 

Worthwhile Canadian Initiative.

[2] Thomas J. Sargent & Neil Wallace, 1981. "Some unpleasant monetarist arithmetic,"
Quarterly Review, Federal Reserve Bank of Minneapolis, issue Fall.

[3] Of course, the optimality of the policies is pretty unnecessary, the problem still arises if the the central bank and the treasury want to pursue opposite policies. Maybe the fiscal authority is being stupid, and the monetary authority chooses to offset its actions, for example.

[4] This assumption is theoretically weak, but given that I don't know of many governments that are net creditors, I think it is acceptable for my current purposes.

[5] David Andolfatto sort of touched on this in his post "Understanding Lowflation"

Analyzing Neo-Fisherism (Warning: Highly Technical)

I've spent the last couple of days trying to make sense of the nuance behind Neo-Fisherian models and I've found that there are a two specific requirements for the result that John Cochrane gets in "Monetary Policy With Interest on Reserves" (pdf):

1. The effects of an interest rate shock are highly fiscal policy dependent. In Cochrane's model, fiscal policy is non-ricardian, but the effect is the same if fiscal policy is active as described by Leeper (1991). If fiscal policy is passive, then conventional wisdom holds so that inflation reacts negatively to an interest rate shock.

2. Cochrane's result only happens when monetary policy moves from one interest rate peg to another. So, even in a fiscal dominant regime, inflation doesn't jump with interest rates if the central bank follows an interest rate rule (that must violate the Taylor principle).

Here is a comparison between a fiscal dominant (without an interest rate peg) and monetary dominant regime during a nominal interest rate shock:

 As you can see, inflation eventually rises in the fiscal dominant regime, but this is simply due to the persistence of the shock. The initial effects are, in fact, more severe than in the standard model. As stated before, Cochrane's result prevails when the central bank switches from a lower peg to a higher peg and vice versa, but switching from peg to peg as Cochrane's paper implies is hardly realistic, so it's safe to say that conventional wisdom would hold is most empirical cases, even under fiscal dominance. Perhaps the only situation in which Cochrane's inflation jumping result (to be replicated shortly) would occur is if a country like Japan decided to switch from what is effectively an interest rate peg regime to an inflation targeting regime with active monetary policy (this remains un-modeled due to the limitations of modelling software). 

Inflation

Nominal Interest Rate

So, under a fiscal dominant regime, raising the nominal interest rate once and for all results in this equilibrium, but allow the interest rate to follow a Taylor rule and this equilibrium is replaced by one with a harsher reduction in inflation than than the standard monetary dominant model.

Appendix:

Here is the model I used above in case you'd like to check it:

The government budget constraint is expressed in real terms and the real money supply is assume constant and equal to one, so seigniorage is simply expressed as the rate of inflation, $ \pi_t $:

$$ (1) \: b_t + \tau_t = (1 + \rho)b_{t-1} - \pi_t $$

$ b_t $ is the real bond supply, $ \tau_t $ is the lump sum tax levied by the government, and $ \rho $ is the constant real interest rate. The Fisher relation follows; relating the nominal interest rate to expected inflation:

$$ (2) \: i_t = \rho + E_t \pi_{t+1} $$

Fiscal policy is simply a function of the current stock of government bonds and can be adjusted by changing $ \phi^f $ between $ \phi^f > \rho $ for passive fiscal policy and $ \phi^f < \rho $ for active fiscal policy.

$$ (3) \: \tau_t = \phi^f b_t $$

Monetary policy can similarly be adjusted between passive ($ \phi^\pi \leq 1 $) and active ($\phi^\pi > 1 $) regimes. $ v_t $ is a shock term that follows an AR(1) process.

$$ (4) \: i_t = \rho + \phi^\pi \pi_t + v_t $$

$$ (5) \: v_t = \rho^v v_{t-1} + \epsilon_t $$

$ \rho^v $ is the persistence of the monetary policy shock and $ \epsilon_t $ is white noise.

During the deterministic simulation, the interest rate feedback rule is removed and the nominal interest rate is pegged exogenously.

Updates:

The response to the interest rate shock seems to depend on whether or not the shock is expected. There is strangely always a drop in inflation on the period when agents get news of the shock, but inflation does end up jumping when it comes into effect.

The stochastic simulation becomes a bit more informative when $ \phi^\pi $ is set to zero as there is just an initial drop in inflation in response to the shock and then a subsequent jump as inflation meets expected inflation.

Scott Sumner Claims His Model is Wrong by Claiming His Model is Right

I wrote a blog post a couple of days ago wondering if nominal GDP targeting and inflation targeting are the exact same thing. I came away with two conclusions: if sticky consumer prices are the primary source of nominal rigidity, then the answer is yes and if sticky input prices and/or prices not included in the central bank's target price index are the primary source of nominal rigidity then the answer is no. Implicit in these two conclusions is that real GDP is always at potential under an inflation targeting regime in sticky-consumer-price models and that real GDP is not often at potential in sticky-input-price models.

So, where does Sumner fit in to this? Well, Sumner recently read this post on Canadian austerity in the 1990s by Stephen Williamson. He noticed that Williamson sees adherence to an inflation target as evidence against monetary offset, so he decided to write this wonderfully contradictory statement:

If you observe the inflation rate always being on target, then the central bank is successfully offsetting any fiscal action that would have otherwise moved AD and inflation.

Of course, this statement is perfectly sound a-cyclical inflation targeting results in a constant output gap of zero, but that's not the way that Scott Sumner sees the economy. Being a market monetarist, he believes that counter-cyclical inflation targeting is consistent with a constant output gap of zero. If an inflation target is optimal, then monetary offset did occur in Canada, but if a nominal GDP target is optimal, then monetary offset did not occur in Canada. Since even friction-less models suggest that the multiplier on government spending is greater than zero (pdf), it's pretty obvious that monetary offset did occur in Canada, but this basically discredits the already somewhat scarce theoretical evidence for market monetarism.

It seems that two of Sumner's strongest positions are not consistent with each other. Either monetary offset happens in an inflation targeting regime or nominal GDP targeting is optimal.

Some Fun With Money Demand

I was doing some thinking about augmenting my liquidity trap post with an interest elastic money demand function of the form:

$$ (1) \: \log M_t + i_t = \log P_t $$

Solving for inflation in this model results in

$$ (2) \: \pi_t = \Delta \log M_t + \Delta E_t \pi_{t+1} $$

(The nominal interest rate term changes to expected inflation because the real interest rate is assumed to be constant). So far, everything looks unremarkable, but solving forward yields interesting results.

$$ (3) \: \pi_t = \Delta \log M_t - E_t \sum^{\infty}_{j=0} \Delta \log M_{t + 1 + j} - E_{t-1} \sum^{\infty}_{j=0} \pi_{t + j} $$

So, what does this tell us? Well, a couple things. Namely,

1.  Inflation is dependent on three things: the growth rate of the money supply this period, the sum of all expected money growth, and the sum of all expected inflation.

2. In order for an increase in the money supply to cause inflation it must not be accompanied by a reduction in expected money growth or an increase in expected inflation.

Knowing these two things allows us to come to the conclusion that increases in the money supply that are expected to be reversed in the future will not be inflationary.

This post kind of lacks a conclusion, but I'm going to make up for it by writing a post that evaluates quantitative easing using the findings from this post and the 'How to Escape a Liquidity Trap' post.

Social Economics 101

If you haven't already seen this video, watch it now before I to assess its economic validity:

Before I begin my analysis of the economics in the video, I would like to  point out that the video complains about one thing and then goes on to argue about another. It begins with Reagan saying that wealth redistribution is bad and went on to (implicitly) deal with the entirely different economic problem of workers and wages.

The whole "everybody gets the same grade" experiment is akin to a firm deciding to pay all of its workers the same wage regardless of differences in productivity. The video claims that, in this situation, high skill workers, call them $ H $, and low skill workers, $ L $, would all end up working less than if their wages were independently determined. Of course, this result is not consistent with either profit maximizing behavior.

If the assumption of profit maximization for firms holds, then all inputs (in this case $ H $ and $ L $) are given wages equal to there marginal products. So, if the production technology for this firm takes the form
$$ (1) \: Y = H^\alpha L^{1-\alpha} \: \alpha > 0.5 $$
then the wage given to the high skill workers would be $ W_H =  \alpha H^{\alpha -1}L^{1-\alpha} $ and the wage given to the low skill workers would be $ W_L = (1-\alpha)H^\alpha L^{-\alpha} $. Since the firm has agreed to pay each firm the same wage, the ratio of high skill workers to low skill workers (or effort from the smart and not so smart students, if you like) can be easily determined.
$$ (2) \: H = \frac{\alpha}{1-\alpha}L $$
Because of the fact that $ \alpha > 0,5 $, it becomes clear that firms end up demanding more labor from the high skilled workers than the low skilled workers; something that would be the case regardless of same wage and/or same grade policy.

The key problem with this video is that it relies on examples ill suited for analogy to economics. Classroom dynamics have very little power in explaining the ideal level of government redistribution or the behavior of neoclassical firms. Yes, I know it's only there to further a political ideology and its makers are not economists, but after seeing this video on social media for the umpteenth time, I think its economic assertions need to be kept in check with some actual economics.

Is Inflation Targeting the Same Thing as NGDP Targeting?

In proper Nick Rowe style, I started out writing this post thinking that targeting NGDP is the same as targeting prices. After doing some thinking, however, I realized that it depends on the kind of nominal rigidity present in the economy. Because of this realization, I'm going to write down the cases (that I know of) for each position.

The most obvious way to look at the question is through the lens of sticky prices and/or costly price adjustment. In an economy where sticky prices are the predominant source of nominal rigidity, not only is price stability/inflation targeting optimal, it is consistent with NGDP targeting. This is because price stability in this economy is the exact same thing as output stability. If inflation is always equal to zero, then the costs of sticky prices and/or adjustment costs are minimized and output will perpetually be at potential. The combination of stable prices and stable output results in an outcome identical to targeting NGDP.

What type of nominal rigidity causes NGDP to be unstable under an inflation targeting regime then? The only answer I can come up with is nominal wage rigidity. If the nominal wage is adjusts slowly, then having a constant inflation rate will result in fluctuations in the labor supply and therefore output. Because of these fluctuations, it would be optimal for prices to adjust to whatever makes the real wage consistent with its natural level (the real wage consistent with full employment and no output gap). Of course, this means that having high inflation during a recession and low inflation during a boom, essentially NGDP targeting, is optimal.

So I guess the answer to the question in the title of this post is that it depends on what kind of frictions are present in the particular economy. If sticky prices dominate, then NGDP targeting is the exact same as price level or inflation targeting because a stable price level is synonymous with a stable level of output and, by extension, NGDP. If, however, nominal wage stickiness is the predominant source of nominal rigidity, then counter-cyclical inflation in necessary for NGDP stability and the optimal monetary policy for the economy.

How To Escape A Liquidity Trap

The word liquidity trap is somewhat ambiguous, so, for the sake of clarity, the definition I will use in the post is as follows: a liquidity trap is an extended period of time during which the nominal interest rate is roughly equal to zero.

Given this definition and the Fisher relation, it becomes clear that a liquidity trap is simply a period of deficient inflation expectations.

$$ (1) \: i_t = \rho + E_t \pi_{t+1} $$

The nominal interest rate, $ i_t $, is low because expected inflation over the next period, $ E_t \pi_{t+1} $, is low. When confronted with this situation, a central bank like the Bank of Japan, the Federal Reserve, or the Bank on England may be tempted to affect a one-off increase in the size of the monetary base and call it "quantitative easing". Unfortunately, this will have next to no effect on the price level.

Take an example economy where the central bank has complete control over nominal spending and real GDP is constant:

$$ (2) \: M_t = P_t\: y $$

By doing some algebra, we can see that expected inflation in this economy is a function of the expected size of the money supply next period and the price level this period:

$$ (3) \: E_t \pi_{t+1} = \frac{E_t M_{t+1}}{P_t\: y} - 1 $$

If the central bank sets the money supply, $ M_t $, to grow at a constant trend rate, but be subject to a bit of discretion every period so that the money supply evolves like this:

$$ (4) \: M_t = \phi M_{t-1} + v_t $$

then we can simplify expected inflation to only being a function of $ \phi $.

$$ (3a) \: E_t \pi_{t+1} = \phi - 1 $$

This shows that the only way for monetary policy to increase expected inflation in this economy is to increase the trend rate of growth of the money supply. In other words, quantitative easing would have no effect on the nominal interest rate in this model.

Governments may also want to engage in fiscal stimulus during a liquidity trap in order to improve economic conditions (not modeled here) or to increase expected inflation. If they do this correctly, it can work.

Consider a small change to equation 2. Now real GDP consists of only government spending (having government spending and private spending would yield the same result but involve annoying amounts of algebra) which can vary through time.

$$ (2a) \: M_t = P_t\: g_t $$

Expected inflation can now be written as a function of the expected money supply, the current price level, and the expected level of government spending:

$$ (3b) \: E_t \pi_{t+1} = \frac{E_t M_{t+1}}{P_t\: E_t g_{t+1}} - 1 $$

If we add a growth rule for government spending so that government spending grows at some rate $\theta_t$ every period so that government spending evolves as such:

$$ (6)\: g_t = \theta_t\: g_{t-1} $$

then expected inflation can again be simplified to an increasing function of the money supply growth rate, $ \phi $, and a decreasing function of the government spending growth rate, $ \theta_t $.

$$ (3c) \: E_t \pi_{t+1} = \frac{\phi}{E_t \theta_{t+1}} - 1 $$

In order to increase inflation expectations, the government needs to reduce the expected growth rate of government spending. You may be wondering how this is at all consistent with me saying that fiscal stimulus can cause expected inflation to increase in this model. There is a relatively simple explanation.

There are two ways for the government to reduce $ E_t \theta_{t+1} $. They can either decrease $ E_t g_{t+1} $ while holding $ g_t $ constant or that can increase $ g_t $ while holding $ E_t G_{t+1} $ constant. In this way, current stimulus with the promise of future austerity will cause the necessary increase in expected inflation.

Of course, the government could just choose to decrease the trend rate of growth of government spending, but that would annoy all the Keynesian's too much. 

The Lesser of Two Evils: Choosing Optimal Methods of 'Distortionary' Taxation

The ideal method of taxation comes up frequently in political debate. Governments often tweak the levels of financing they receive from a wide gamut of taxes. Of course, most of the debate over government policy remains uninformed by the members of academia so the various recommendations for ideal methods of taxation have little basis in economic theory.

Here, I won't pretend to be any kind of expert on fiscal policy, or even computing (Ramsey) optimal policy in general, but I will compare welfare under a "High Income Tax - Low Consumption Tax" regime and a "Low Income Tax - High Consumption Tax" regime in an attempt to add some model-based input.

I ran two different simulations. The first one determines which regime is better for consumer welfare when there is a 1% shock to the government spending to GDP ratio and the second one looks at which regimes provides more welfare and/or more tax revenue in the long run (steady state). As it turns out, the results to each test are slightly different.

In test #1, the high income tax regime (with a 10% flat consumption tax and a 50% flat income tax) edges out the high consumption tax regime (50% consumption, 10% income). When the government increases spending, the household's objective function (welfare) is higher throughout due mainly to less of an increase in output and subsequently less of an increase in labor as well as a smaller reduction in consumption.

 (click here for the rest of the comparisons)

In the long run, however, the consumption-tax-dominant regime is superior for welfare in the long run and output in the short run. Steady state welfare is much higher in this regime, pointing to the long run advantages of having higher consumption taxes and lower income taxes.

The disadvantage of consumption-tax-dominance is that tax revenues are lower in the steady state. Lower revenues either mean a lower level of government spending in the long run (which also means less output which is socially optimal, but maybe not optimal from a policy perspective) or higher deficits.

Consumption taxes turn out to be optimal in most economic situations as they are less 'distortionary' than income taxes and they increase consumer welfare in the long run (basically consumption per unit of labor). The only case for keeping an income-tax-dominant regime (assuming both taxes are flat), is if government spending is extremely volatile. In all other cases, cutting income taxes and raising consumption taxes is more optimal. Of course, if government spending had the ability to increase welfare in my model, then the results could be very different indeed.

In Theory, Monetary Offset Doesn't Work

Monetary offset, one of the major aspects of Market Monetarism, is severely hindered by the application of some basic macroeconomic theory. Take the bond pricing equation at the heart of most of modern macro:

$$ c_t = \left[\frac{(1 + \rho)(1+E_t \pi_{t+1})}{1 + i_t}\right] E_t c_{t+1} $$

$ c_t $ is current consumer spending, which is chosen in order to maximize all expected future consumption. When the real interest that can be earned on saving or investment increases above its natural rate, $ \rho $, current consumer spending falls. In normal times, the central bank targets an inflation rate, $ \pi_t $, which anchors inflation expectations to that target, so any adjustment in the nominal interest rate, $ i_t $, directly changes current consumption. If the government decides to actively reduce its budget deficit which depresses GDP, the central bank can lower the nominal interest rate to offset this change. This is the theoretical explanation for monetary offset. 

When the nominal interest rate is at zero, the central bank can no longer lower the nominal interest rate to counteract the effects of austerity. It only has two options: somehow increase inflation expectations or promise to keep future interest rates low. This is exactly what the Federal Reserve has resorted to in the last few years. Quantitative easing has increased inflation expectations and forward guidance has given the promise of an extended period of low rates. Perhaps expanding the monetary base like there's no tomorrow can have some effect in both keeping rates low and increasing inflation expectations. Nevertheless, neither of these policies are proven to work either in theory or in practice. Central banks are, for all intents and purposes, ineffective at increasing consumption at the zero lower bound.

Central banks slowly lose what small ability to control the economy they have as the economy returns to its natural state on its own. Inflation expectations are bound to fall as consumption returns to its natural level. At the zero lower bound, the steady state expected inflation rate falls $ \frac{1}{1+\rho}-1 $. This at least partially explains the multi-decade long period of low inflation and zero nominal interest rates in Japan. As the economy returns to equilibrium, the central bank will be progressively more powerless to offset shocks, fiscal or otherwise.

The key problem with Market Monetarism in general is that it isn't grounded in any kind of model outside of conjecture around a simple version of AD-AS in which the central bank has the ability to achieve any inflation or nominal GDP target it chooses. The reality is, central banks are not omnipotent and aggregate demand is more than just a negative function of the price level.

Graphs!

I recently decided to learn how to do more things than just simulate macroeconomic models in Octave (an open source version of Matlab) using Dynare (DSGE simulation tool for Matlab and Octave). The result was a whole lot of fun graphs:

First, I made the graph for a small labor market:

Next, I figured out how to make a Cobb Douglas Production Function:

 Next, I made a drastically simplified money market:

Then I made a simple linear Utility Function with two consumption goods:

And finally, I made a graph showing utility maximization with a budget constraint (still working on this one):

Two Views of Price Determinacy

Intro:

All the following views of price determination revolve around the consolidated government budget constraint that combines the actions of the central bank and the fiscal authority into one equation. The price level is determined by a combination of fiscal and monetary policy which can result in unconventional effects.

The Model:

The fiscal and monetary authority collectively obey a budget constraint:

$$ (1) \: B_t + P_t s_t = R_{t-1}B_{t-1} $$

where $ B_t $ is the total stock of government liabilities (including money and government bonds), $ s_t $ is the lump sum tax levied by the government, $ P_t $ is the price level, and $ R_t $ is the gross nominal interest rate on government debt.

The central bank targets the price level and sets $ R_t $ according to a policy rule:

$$ (2) \: R_t = \frac {P_t^\phi}{\beta} $$

$ \phi $ measures is the central banks' reaction to an off-target price level and $ \beta $ is the factor at which future actions are discounted by agents.

The fiscal authority targets a level of nominal debt by setting $ s_t $ according to a policy rule:

$$ (3) \: s_t = \left(\frac {1}{\beta} - 1\right) \bar B + \phi^f (B_t - \bar B) $$

where $ \bar B $ is the target level of debt and $ \phi^f $ measures the fiscal authority's reaction to an off-target debt level.

The nominal interest rate is related to $ \beta $ by the Fisher Equation (which is technically a bond pricing equation drawn from the first order conditions of the consumers maximization problem):

$$ (4) \: R_t \beta = \frac {E_t P_{t+1}}{P_t} $$

Monetary and fiscal policy can either be active or passive. Monetary policy is active when $ \phi > 1 $ and passive when $ \phi < 1 $. Fiscal policy is active when $ \phi^f $ is positive and passive when $ phi^f  \leq 0 $. In order for the price level to be determined, one or both of the government policy instruments must be active.

View One: Neo-Fisherian:

Neo-Fisherians contend that the fiscal authority responds only passively to government debt, so when the nominal interest rate is increased above the rate suggested by the policy rule, the amount of government debt will increase with no fiscal response. This forces the inflation rate to increase. The mechanism by which this occurs is best illustrated by iterating the budget constraint forward in time and dividing by the price level:

$$ (5) \: \frac {B_{t-1}}{P_t} = E_t \sum_{j=0}^{\infty} \beta^{j} s_{t+j} $$

The increase in government liabilities, without any fiscal policy response, causes the price level to rise.

View Two: Everyone Else:

For the conventional wisdom ($ \uparrow R_t $ causes $\downarrow P_t $) to be the case, the fiscal authority needs to commit to commit to have active fiscal policy ($ \phi^f > 0 $). When the central bank raises the nominal interest rate and the amount of government debt in (5) goes up, the right side of the equation goes too, and the price level falls.

Alternatively, the total stock of government liabilities can be set to zero forever, as in New Keynesian models, changes in the nominal interest rate have no fiscal effects, so the price level is determined by only two equations:

$$(1a)\: R_t \beta = \frac {E_t P_{t+1}}{P_t} $$

$$(2a)\: R_t = \frac {P_t^\phi}{\beta} $$

Conclusion:

Different views on how the price level is determined depend on how monetary and fiscal policy are seen to be interacting. On one hand, if monetary and fiscal policy are active, then monetary policy effectively has fiscal backing. On the other hand, if fiscal policy is passive, then the Neo-Fisherian hypothesis is correct. A third possible equilibrium is one with passive monetary policy and active fiscal policy. This is a fiscal theory of the price level regime that may have occurred between 1950 and 1970 in the US.

The monetary-fiscal regime that a country is in at a given moment can have an important impact on the price level. For example, the US is likely once again in an active fiscal/passive monetary regime, so fiscal policy plays an important role in the determination of the price level.

The Nominal Interest Rate and Inflation Determination

The dynamics of inflation in relation to the nominal interest rate are generally assumed to be governed by the liquidity effect. That is, increases in the money supply temporarily decrease the nominal interest rate because of money demand and some form of nominal rigidity. In a perfectly friction-less world, increases in the money supply (particularly increases in the growth or future path of the money supply) cause the nominal interest rate to increase instantly, rather than after the economy returns to its natural level (some economists would say equilibrium, but defining equilibrium as simply a solution to a model makes more sense to me). Of course, explaining this whole thing with math is a whole lot more descriptive, so here I go:

I'm going to assume households have a linear utility function and derive utility from consumption, so $ u(c_t) = \ln c_t $ where $ c_t $ is consumption. This implies the "consumption Euler equation" that is used to determine inflation given the nominal interest rate.

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

$ i_t $ is the nominal interest rate, $ \pi_t $ is the rate of inflation, and $ \beta $ is the constant discount factor. Assume consumption grows at a constant gross rate "$g$",

$$ (2) \: c_t  = g c_{t-1} $$

and the interest rate equation becomes

$$ (1a) \: i_t = g \left(\frac {1 + E_t \pi_{t+1}}{\beta} \right) - 1 $$

Now, if the central bank targets inflation so that it evolves according to

$$ (3) \: \pi_t = \bar \pi + \rho (\pi_{t-1} - \bar \pi) + \epsilon_t^\pi $$

where $ \bar \pi $ is the "trend" rate of inflation, $0 < \rho < 1$ is the "shock stickiness" parameter, and $ \epsilon_t^\pi $ is white noise, and $ E_t \epsilon_{t+1}^\pi = 0 $, then expected inflation, $ E_t \pi_{t+1}$, is defined by

$$ (3a) \: E_t \pi_{1+1} = \bar \pi + \rho (\pi_t - \bar \pi) $$

To fill in the model, the final interest rate equation becomes

$$ (4) \: i_t = g \left( \frac {\bar \pi + \rho (\pi_t - \bar \pi)}{\beta}\right) - 1 $$

This shows essentially what the "Neo-Fisherian" assertion is. The nominal interest rate and inflation rise with each other. In fact,

$$ (5) \: \frac {d i_t}{d \pi_t} = \frac {g \rho}{\beta} $$

So, if the rate of inflation increases by 1%, then the nominal interest rate will increase by $ \frac {g \rho}{\beta} $%.

Of course no monetarist will be happy until I use the money supply as a determinate of the price level, so assume a cash in advance constraint:

$$ (6)\: M_t = P_t c_t $$

where $ M_t $ is the money supply and $ P_t $ is the price level. Since consumption still grows according to (2), the interest rate equation is redefined to

$$ (7)\: i_t = g \left( \frac {E_t P_{t+1}}{P_t\beta} \right) $$

$ M_t $ grows at gross rate "$ m_t $", so its law of motion is

$$ (8)\: M_t = m_t M_{t-1} $$

and, given (6), 

$$ (9)\: E_t P_{t+1} = \frac{m_{t+1} M_t}{g c_t} $$

Integrating all this back into (1) gives

$$ (10)\: i_t = \frac{m_{t+1}}{\beta}-1 $$

If the central bank permanently increases $ m_t $, which is equivalent to $ \pi_t +1 $, by 1%, the nominal interest rate will increase by $\frac{1}{\beta}$% ($ \frac {d i_t}{d m_t} = \frac {1}{\beta} $).

Basically, absent nominal rigidity, the nominal interest rate and inflation have a positive, even causal, relationship not afforded to them by conventional wisdom. Of course, this is really driven by the way that the money supply interacts with the nominal interest rate. "Neo-Fisherism" is really an incomplete hypothesis because of this. More focus should be given to the effects of open market operations as non-nominal-rigidity ways of explaining the liquidity effect.

Rational Partisan Theory

Recently, I have been intrigued by a fragment of the macroeconomics literature known as rational partisan theory. It looks into the effects of changes in the expectations of government policy that occur during elections. Usually, there are two "parties": one that wants high inflation and another that wants low inflation. Expectations of future policy are contingent upon the probability of each party being in power after the next election.

The simplest way to analyse this is to use the Fisher Equation and assume the central bank follows a rule and targets an inflation rate (both of which are unknown to the agents in the model).

First, the Fisher Equation:

$$ i_t = E_t \pi_{t+1} + \rho $$

where $i_t$ is the nominal interest rate, $\pi$ is the rate of inflation, and $\rho$ is the real interest rate. 

Second, the policy rule:

$$ i_t = \phi (\pi_t - \bar\pi) + \bar\pi + \rho $$

where $\bar\pi$ is the central bank's inflation target.

Here's where the "Partisan" part comes in. Each period, there is an $L_t$ percent chance that the left wing (higher inflation) party will be elected and an $R_t$ percent chance that the right wing (low inflation) party will be elected. So, by definition, $L_t + R_t = 1$. Suppose the right wing party wants the inflation rate to be $\pi^R$ and the left wing party wants the inflation rate to be $\pi^L$.

Since the agents don't know the central bank's policy rule or it's inflation target, there best guess for next period's inflation is:

$$ E_t \pi_{t+1} = L_t\pi^L + R_t\pi^R $$

If there is always a 50% chance that the left or right wing party will win the election, and $\pi^R = 0.01, \pi^L = 0.02$, expected inflation will always be $0.015$. 

Assume that the probability that the right wing party will win the election tomorrow evolves like this:

$$ (R_t - 0.5) = \theta (R_{t-1} - 0.5) + \epsilon_t $$

to close the model.

A Dialogue Between My Friend and I Concerning Welfare

Friend: Is the line "Welfare creates dependency" true in any way?
Me: Yes...
Me: ...and no.
Me: Dependency is the wrong word;
Me: "Advantage taking" is better
Friend: Interesting...
Me: The real question that people should ask is "is welfare socially beneficial?"
Me: Conservatives site lower employment as a problem.
Me: Liberals think that welfare is socially advantageous.
Me: I think I agree more with the liberals on the social part and conservatives on the sheer economics.
Friend: Here's a follow up question...
Friend: Has welfare benefited America?
Me: It depends on what you mean by benefit: incomes are higher, but employment may not be. Conservatives like to use low employment as a criticism of welfare in America while liberals focus on the actual increase in income; whether it's from people working or due to benefits from the government.
Me: Either way, Americans may be worse off because they don't have jobs...
Me: ...but that may be outweighed by their higher incomes.
Friend: I'd say in the social factor at least, the newer generations have been leading better lives.
Friend: I'd say at least slightly, but I think the environments many have grown up with have continued the cycle...
Friend: ...instilled a cycle
Me: The cycle argument really isn't economically viable*...
Friend: All right.
Me: The existence of incentives to work - wages; and not to work - welfare are more important to the welfare situation.
Me: The solution is to make the market wage higher than the amount people can get from welfare.
Me: This can be effected by jobs training and better education
Friend: So the opposite, say lowering the amount people can get from welfare instead of raising the wage would not work because... ?
Me: That would work too
Me: But it's more difficult...
Me: ...and likely bad socially.
Friend: hmm...

*The reason that the "cycle" argument doesn't line up economically is that employment decisions in normal (macro)economic models are entirely due to balancing a dislike for work and an attraction to consumption. This means that the argument that welfare creates an inter-generational cycle of poverty is neither needed to fit empirical data nor present in the way that (macro)economists think about welfare and employment.

Fed "Tightening" May Bring Inflation

When the Fed raises rates this year, the plan is to do it differently than usual. Rather than reduce the supply of reserves, it will simply raise the interest rate on excess reserves, the effective lower bound to the federal funds rate. There are a few factors playing into this, namely the fact that there are roughly 3 trillion dollars of excess reserves now rather than the previously normal ~50 billion dollars. In order to effect a rate increase, the Fed would have to reduce the supply of reserves to about 1/60th of its current level. Given the impracticality of this, an increase in the IROER (interest rate on excess reserves) seems logical.

An interest rate increase is usually considered synonymous with the attempt of a central bank to decrease inflation. Despite this, given that there will be no change in the money supply, normal monetarist logic would imply that the rate hike would have no effect on inflation whatsoever. $ M_t V_t = P_t Y_t $ with no changes in $ Y $ or $ V $ shows this; $ M $ doesn't change, so $ P $ goes nowhere. Of course, adding a simple ad-hoc money demand equation changes things a little, but not in the expected direction. With $ \log M_t + \eta i_t = \log P_t + \log Y_t $, it becomes clear that raising the interest rate $ i_t $, without decreasing $ M_t $ (assuming $ Y $ is exogenous and constant again) will cause the price level to rise. 

Fiscal Stimulus Take Two

(In this post, I will be using a slightly modified version of the model with the output-gap-indifferent monetary policy rule in my last post for analysis)

The problem I have had with most of the papers I have read on fiscal stimulus is that taxes usually take the form of lump sum transfers to and from the government. There are no income, capital, or consumption taxes that distort the outcome. For simplicity's sake, I'm just going to look at the effects of stimulus with income taxes because they seem to be what most politicians focus on. Before I go further, a short description of how fiscal policy works in my model in order. The government receives tax revenue from lump sum taxes (which don't cause distortions) and from income taxes and spends all of its revenue While technically, there are not deficits, the lump sum transfers serve as a neutral way of allowing government spending to be less than or greater than income tax revenues.

The stimulus takes the form of a simultaneous unexpected positive shock to government spending and negative shock to the income tax rate with a persistence of $ \rho $ (which I have set to 0.9). Also, its worth stating that the central bank is still targeting inflation but is indifferent to the output gap, but will stabilize output in the long run because of the nature of New Keynesian  models (namely the structure of the New Keynesian Phillips Curve). Without further adieu, here are the charts along with some brief explanations:

This first chart shows the log deviation from steady state of (left to right, top to bottom) real GDP, capital, labor, consumption, investment, real wages, the real interest rate, the nominal interest rate, and the gross rate of inflation (inflation rate plus one) in response the the fiscal stimulus outlined above. Output does increase by the full 2% reflected in the shock and, unlike before, the capital stock and consumption increased which means that the addition of cuts in "distortionary" taxes can negate some of the negative crowding out effects of stimulus. 

Figure 2 shows the fiscal effects of the stimulus where (in log deviations again) T is the lump sum transfer, g is government spending, rev is government revenue from income taxes, and t_n is the income tax rate. Everything looks pretty normal; taxes and revenue go down while spending goes up. Revenue doesn't go the full 1% down because of the increase in output from the stimulus, but, because of the way I set things up, government spending makes up for the lack of revenue reduction (that sounds like a really strange thing to say in a normal context).

I guess the point here is that even adding just one tax that distorts one thing (in this case the marginal rate of substitution between consumption and labor) can drastically increase the effects of what would otherwise be somewhat dubious policy. Of course, a central bank that ignores the output gap is a bit unrealistic, but it allows for a more raw view of the real effects of fiscal stimulus.

P. S. I did run a simulation with a normal monetary policy rule. The graphs are here.