19 July 2015

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). 

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.


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.


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.

16 July 2015

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.

15 July 2015

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.

12 July 2015

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.

11 July 2015

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.

08 July 2015

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. 

06 July 2015

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.