30 May 2015

Two Views of Price Determinacy


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:
$$(1a)\: R_t \beta = \frac {E_t P_{t+1}}{P_t} $$
$$(2a)\: R_t = \frac {P_t^\phi}{\beta} $$


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.

19 May 2015

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.

16 May 2015

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.

11 May 2015

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.

09 May 2015

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

05 May 2015

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.


03 May 2015

Monetary Offset Is A Thing in New Keynesian Models (Sort of)

I was playing around with fiscal stimulus in a New Keynesian model (with capital) that I had written down in Dynare and I was surprised to see that increases in government spending (funded by lump sum transfers) caused deflation rather than inflation. Puzzled, I decided to remove the part of the Taylor Rule that reacts to the output gap and, as I had initially predicted, fiscal stimulus became inflationary and output increased more than in my first test.

The moral of the story is that the idea of monetary offset that Scott Sumner brought up (I think) is partially right: the effects of fiscal stimulus will be (partially) counteracted by the central bank. In my model, this happens not because the central bank is targeting inflation, but because it tightens monetary policy in response to increases in output above potential. Of course, in order to have fiscal stimulus completely counteracted by the central bank, I needed to put the output gap coefficient on the Taylor Rule upwards of 5 (rather than the normal 0.5), so complete monetary offset with Taylor Rules doesn't seem to work.

In a way, this partially affirms the pro-fiscal-stimulus crowd and the pro-monetary-stimulus crowd (I belong more to the latter). Fiscal stimulus does appear to work in the short term, but the central bank really has power over aggregate demand.

If anyone wants to check my analysis, here are the relevant model equations:

$$ y_t  = e^{z_t} k_t^\alpha n_t^{1-\alpha} $$
$$ z_t = \rho z_{t-1} + \epsilon_t^z $$
$$ k_{t+1} = (1-\delta) k_t + x_t $$
$$ w_t = c_t^\sigma n_t^\phi $$
$$ c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} \left(\frac{1 + i_t}{E_t \pi_{t+1}}\right) $$
$$ y_t = c_t + i_t + g_t $$
$$ g_t = \rho g_{t-1} + \epsilon_t^g $$
$$ w_t = mc_t (1-\alpha) e^{z_t} k_t^\alpha n_t^{-\alpha} $$
$$ r_t + \delta = mc_t \alpha e^{z_t} k_t^{\alpha -1} n_t^{1 - \alpha} $$
$$  \log\pi_t = \beta \log E_t \pi_{t+1} + \frac {(1-\theta)(1-\beta \theta)}{\theta} \left( \log mc_t - \log \left(\frac {\epsilon - 1}{\epsilon}\right)\right) $$
$$ \frac{1 + i_t}{E_t\pi_{t+1}} = 1 + r_t $$
$$ i_t = \beta^{-1} - 1 + \phi_\pi (\pi_t -1) + \phi_y (\log y_t - y^n) $$

Where $y_t$ is real GDP, $z_t$ is the TFP, $k_t$ is the capital stock, $w_t$ is the real wage, $ c_t $ is consumption, $x_t$ is investment, $g_t$ is government spending, $mc_t$ is the marginal cost, $r_t$ is the real interest rate, $\pi_t$ is the gross rate of inflation, and $i_t$ is the nominal interest rate.

Here is the response to a fiscal shock with a normal Taylor Rule:
Normal Taylor Rule Fiscal Shock
And here is the response to a fiscal shock where the Taylor Rule ignores the output gap:
Modified Taylor Rule