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.

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.

The Fed Should Cut IOR

It's difficult to understand why people and financial institutions would willingly carry an asset that earns less interest than other types of asset. A typical way to get around this problem is by assuming that some or all of the goods in an economy must be paid for in cash. Because agents must pay for goods in cash, they choose to hold only as much as they need to pay for those goods. If they were to hold any extra cash, then they would be missing out on valuable interest that they would earn from other assets. The cost that agents face by holding money as opposed to other assets (e.g. government bonds) is called the opportunity cost of holding money.

But what happens when the opportunity cost of holding money is zero (i.e., the interest rate on government bonds is equal to the interest rate on money)? In this situation, it makes no difference to agents what kind of asset they hold, so the distribution of government bonds and money is indeterminate. This indeterminacy breaks the link between inflation and the money supply. Typically, the money supply is linked directly with the nominal value of spending on cash-goods in this economy, so a higher money supply would necessitate higher nominal spending and, assuming flexible prices and wages, higher prices. Now, because there is no opportunity cost of holding money, agents will freely hold any money that the central bank gives them without needing to spend it.

This is the situation that the Federal Reserve is currently in. There is no incentive for financial institutions to do anything will all the money that the Fed has injected into the system since 2009 because the interest rate that money pays (interest on reserves or IOR) is equal (after adjusting for risk/liquidity) to the interest rate that other assets pay. Financial institutions are happy to sit on interest bearing and highly liquid (easy to buy and sell) cash. This is why the vast majority of the increase in the monetary base since 2009 has taken the form of "excess reserves".

In order to reverse this, it is necessary to make reserves less attractive to financial institutions (or, in the case of cash-in advance models, make cash less attractive to agents). This means that there must be an opportunity cost of holding money; that money needs to pay less interest than other assets. Of course, this can be accomplished one of two ways: the Fed can either reduce the interest it pays on reserves increase the interest rate that other assets pay. Since economists widely agree that raising interest rates would have a negative effect on the rate of inflation, it seems clear that the way to increase inflation in the US is to cut IOR.

The Trouble With The Zero Lower Bound

Most of the time, it seems that the monetarist view of inflation is pretty much correct. Inflation roughly tracks the monetary base and velocity is pretty stable and almost directly follows short term interest rates. Unfortunately, there is this thing called the zero lower bound that seems to throw monetarism off.

The US has been at the zero lower bound twice in the last 150 years, and both times monetary expansion has seemed to have an irrelevant - even a negative - impact on inflation.

Here's 1934-1945:

And here's 2009-2015:

Most monetarists seem to have trouble coping with the irrelevance of the monetary base at the zero lower bound, even though it does seem to be part of a lot of basic monetary models. Take the most simple of money demand functions - cash-in-advance. It is easy to figure out that as long as there is a cost to the household incurred by holding money, the cash-in-advance constraint will bind, but whenever there isn't a cost, the constraint ceases to bind. This effectively means that, rather than being stuck at unity, the velocity of money is indeterminate; increases in the money supply will no longer have any effect on the price level.

Money-in-the-utility-function models have similar properties in the sense that velocity also becomes indeterminate. MIUF models are slightly strange though because money demand itself actually goes to infinity when the zero lower bound binds. But, MIUF is a pretty bad assumption anyway, so it's fine to ignore this.

An easy modification to CIA models that, when calibrated properly, might be able to make them match the data pretty well is the addition of a non-cash good to the economy. The income-velocity of money will now fluctuate with the nominal interest rate while the effects above will still be present.

I digress, the key idea of this kind of rambling post is that the zero lower bound seems to do strange things to monetary policy which precludes central banks from being omnipotent as some would suggest...

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.

 

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"

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. 

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.

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.

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.