Believe it or not, NK models are not Market Monetarist

Today on Twitter, Nick Rowe deployed a couple of the tricks his commonly like to use when arguing against New Keynesians who think (rightly) that the NK model suggests that, when the Wicksellian natural rate is negative, fiscal stimulus is 1) warranted and 2) will not be offset by the central bank.

Notably, Nick said

"accommodate fiscal stimulus" = "no longer trying to target 2% inflation"

Assume NK [is] true. Current BoC r > ZLB > ELB (as defined by BoC)

 Of course, Nick should know (from numerous posts in which I have written about this same issue) that the New Keynesian IS curve implies that expansionary fiscal policy raises the natural real interest rate, which means that, if inflation is currently below target, fiscal stimulus can raise it to target without requiring an appropriately sized interest rate cut (which may not be possible).

This is where the biggest fault in Nick's argument is -- he suggests that, as long as the nominal interest rate is currently above the zero lower bound (or the 'effective lower bound'), a New Keynesian central bank can keep inflation on target. Essentially, he is arguing that, since the current interest rate set by the Bank of Canada is above zero, the Wicksellian natural rate (defined as the interest rate at which inflation is on target) must be above zero.

This assumption is just plain wrong, but I will slightly alter Nick's actual argument into something that I think is much better (and probably what he meant, but was unable to articulate given Twitter's stringent limits on tweet length). In a New Keynesian model, the central bank can raise the Wicksellian natural rate by deliberately setting future nominal interest rates lower than they otherwise would be (this is called forward guidance). Because of this, all a central bank need do to keep current inflation on target is to lower the path of the nominal interest rate.

Now the argument makes a lot more sense; Nick is suggesting that 1) the Bank of Canada is responsible for inflation being below target because they refuse to use forward guidance and 2) since inflation is exactly where the Bank of Canada wants it, fiscal policy will simply be offset.

The issues that I have with this argument are two-fold:

First, the regime I just described on behalf of Nick is not consistent with an inflation targeting regime because the central bank is supposed to deliberately raise future inflation above target in order to put current inflation on target. This is what forward guidance does in New Keynesian models and, as such, represents an important break from actual inflation targeting.

Second, the empirical failure of forward guidance is well documented and is commonly referred to as the 'forward guidance puzzle.' For instance, Del Negro et al. 2012 note that "[DSGE models] appear to deliver unreasonably large responses of key macroeconomic variables to central bank announcements about future interest rates ... Carlstrom et al. (2012b) shows that the Smets and Wouters model would predict an explosive inflation and output if the short-term interest rate were pegged a the ZLB between eight and nine quarters" [1].

Thus, not only is forward guidance not consistent with keeping inflation on target in the medium term, it is probably nowhere near as effective at raising the Wicksellian natural rate as basic DSGE models would suggest which severely limits my edited version of Nick's original argument. As it turns out, the Bank of Canada is probably either self-constrained by a refusal to do an adequate amount of forward guidance or otherwise constrained by a lack of effective tools to raise the natural rate up to a level at which inflation would be on target. In this case, it is perfectly reasonable to suggest that not offsetting loose fiscal policy is not inconsistent with the Bank of Canada's inflation target.

[1] Del Negro, Marco & Giannoni, Marc & Patterson, Christina, 2012.
"The forward guidance puzzle,"
Staff Reports 574, Federal Reserve Bank of New York, revised 01 Dec 2015.

The Three Pronged Republican Economic Platform

The Republican economic platform can basically by summed up by three defining characteristics: 1) increase military spending, 2) cut taxes, primarily for high-income individuals, and 3) balance the budget. If you look at each of the Republican candidates' websites, you'll see pretty much all of the same policies (with the exception of Trump, who, instead of a balanced budget amendment, proposes what he calls 'revenue neutral' and everyone else calls 'budget busting' policies), with everything else pretty much secondary.

Perhaps the most interesting part of this nearly ubiquitous threefold policy is that each of these objectives are pretty much mutually inconsistent. Each candidate is effectively proposing massive spending increases (military buildup) as well as massive tax cuts (ranging in severity from 'insane' -- Cruz -- to 'I can't judge because there are no specifics' -- Kasich). Unless you buy into the idea that cutting income taxes incentivizes people to work so much more that the net effect on revenue is positive (relative to the higher-tax counterfactual, shut up Reaganites), you should realize that the three effects of each Republican's plan will be 1) massive deficits, 2) whatever supply-side effects (if any) the tax cuts have. and 3) no effect on the output gap, because the Fed, which reasonably thinks that the economy is already near potential, will offset the effects of any expansionary fiscal policy (the same can't be said for contractionary fiscal policy, shut up Market Monetarists).

So, when Rubio accuses Trump of having numbers that don't add up, he should really include himself in the same boat; balanced budgets do not go together with tax cuts and military buildups. Even George W. Bush knew this, at least he admitted that his plan would erase the surplus, even if Mr. NCLB accused Gore of "fuzzy math." Of course, everyone is now going to argue that the spending cuts that each administration will undertake will outweigh the military spending increases. My response: maybe, but have any recent Republican administrations done that? No? I thought so. Keep in mind that government spending has been more restrained as a percentage of GDP under Obama (and under Clinton) that it was under either Reagan or W.

What about the supply-side effects of tax cuts? Well, we do have one pretty good experiment of this from 2003, when Bush passed his tax cuts and, well, I'll let the subsequent lack of employment growth speak for itself:

Now, before everyone freaks out, I know I didn't adjust for demographics here, but I think it's fair to say (based on my own analysis linked above) that the demographic adjustment for Obama would be larger than for Bush, so I'm really being kinder to the Bush tax cuts that I should be. With that aside, it seems pretty clear that the 2003 tax cut did pretty much nothing in the way of speeding up the recovery, so what should make anyone think that more of the same thing will be all that different?

When it comes to monetary offset, as I wrote here, fiscal policy only has demand-side effects to the extend that the central bank doesn't act in the face of fiscal shocks. So, in the event of a massive fiscal expansion (assuming the candidates can't achieve the desired balanced budgets by gutting the government), we should simply expect the Fed to raise interest rates faster, which will mean that the only effect that fiscal policy will have is the supply-side one, which is probably minimal.

TL;DR: Republican fiscal plans = deficits + not much else

Fiscal Policy Does Not Affect Aggregate Demand

This is a follow up to my last post; here I am going to go more in depth into why fiscal stimulus never really impacts aggregate demand in any DSGE.

Start with the simple consumption euler equation that everyone should be familiar with:

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

where $c_t$ is consumption, $\beta$ is the representative agent's discount factor, $i_t$ is the nominal interest rate, and $\pi_t$ is the rate of inflation. Assuming there is no capital accumulation, this equation can be simply edited to include real GDP and government spending:

$$(2)\:(y_t - g_t)^{-\sigma} = \beta E_t (y_{t+1} - g_{t+1})^{-\sigma} \left(\frac{1 + i_t}{1 + E_t \pi_{t+1}}\right)$$

This can now be written in log-linear form, so that all variables are expressed as a percentage deviation from their steady states (except for the various interest rates):

$$(3)\:\hat{y}_t = E_t \hat{y}_{t+1} + \hat{g}_t - E_t\hat{g}_{t+1} - \frac{1}{\sigma}(i_t - E_t \pi_{t+1} - \rho)$$

where $\rho$ is the discount rate, which is equal to $\frac{1-\beta}{\beta}$. Keep in mind that $\hat{g}_t$ is actually the deviation of government spending as a percentage of GDP from trend, but this shouldn't have much of an impact on the analysis.

If we define the 'natural rate of interest' as the real rate of interest at which the output gap remains constant, only a small amount of algebra is required to solve for it:

$$(4)\:r^n_t = \rho - \sigma(E_t\hat{g}_{t+1} - \hat{g}_t)$$

It is clear from this that fiscal stimulus, that is increases in $\hat{g}_t$ absent changes in $E_t\hat{g}_{t+1}$, causes the natural rate of interest to increase. Why is this important? In most cases, it isn't; fiscal stimulus will just have the effect it normally does in a frictionless model (that is, generally speaking, have a multiplier on output somewhere between zero and one depending on the calibration of that model) and the central bank will simply raise the real interest rate so that the output gap remains equal to zero.

This is exactly what all the market monetarists are talking about when they mention monetary offset; any demand-side effect that fiscal stimulus might have is simply the result of central bank inaction in the face of a higher natural rate. Say, for instance, that the central bank set the nominal interest rate according to the following rule:

$$(5)\:i_t = max(0,r^n_t + E_t \pi_{t+1})$$

Now, suppose the fiscal authority does a fiscal stimulus, which raises $r^n_t$. Assuming the zero lower bound is not binding, the central bank will simply raise the nominal interest rate one for one with the increase in the natural rate caused by the fiscal stimulus.

The only reason a New Keynesian model ever exhibits a fiscal multiplier greater than one is that the zero lower bound is at some point binding. In this case, the fiscal authority can lower the value of $r_t - r^n_t$ without central bank intervention ($r_t$ is the real rate of interest). Basically, fiscal stimulus appears effective because of central bank inaction.

In this sense, the actual act of fiscal stimulus never has demand-side effects; it can only influence the natural rate of interest, which is only helpful to the extend that the zero lower bound is binding. Because of this, we should really stop thinking of fiscal stimulus as a way of manipulating aggregate demand; it's really only helpful to the extent that is has real effects -- if anyone says otherwise, then they are unduly influenced by IS-LM.

Timing and Composition

Family members are often confused by my simultaneous support for looser fiscal policy in the United States and disdain for Republican tax proposals during this election cycle on the ground that they would result in too much deficit spending. On the surface, my policy preferences seem contradictory; I neither support efforts to rein in the deficit nor the large tax cut proposals of the majority of Republicans. There are two primary reasons for this seemingly strange predilection: timing and composition.

The length of time each policy lasts is crucial to my support. As per the 'New Keynesian consensus,' loose fiscal policy should only be used until monetary policy can be reasonably declared unconstrained by the zero lower bound. This is why deliberate deficit cutting policies should not have been undertaken, and arguably should not be pursued further, until two criteria have been met: the federal funds rate must be above the zero lower bound and there must be little to no risk that the zero lower bound will be made to bind by either a tightening of fiscal policy or some other shock to the economy. At the time of writing this post, only the first criterion is fulfilled -- the Federal Reserve has decided to raise the target fed funds rate, but, since it stands somewhere between 0.25% and 0.5% (the Fed has adopted a target range instead of a strict target), it would be reasonable to suggest that a large negative fiscal shock could be more than the Fed can handle without being thrown back into a liquidity trap (in this sense, the US could still be considered to be in a liquidity trap, even though the zero lower bound no longer binds).

GOP tax cut proposals would undoubtedly achieve the temporary goal of looser fiscal policy, but they would be on a completely wrong timescale. Conventional analysis only suggests loose fiscal policy for the duration of the liquidity trap, and, since the tax cuts are permanent to the extent that they are not repealed by future administrations, they fail miserably in this regard. In other words, fiscal policy would be too loose for too long under large tax cuts -- especially if they are not accompanied by corresponding reductions in government spending. Additionally, spending cuts are arguably more damaging than tax cuts are stimulative in liquidity traps, so a fiscal adjustment fully in line with, e.g., Rand Paul's or Ted Cruz' preferences could completely fail to comply with the recommendations of mainstream economics, which scares me enough in its own right to warrant a revocation of support.

My second criticism of the GOP tax plans is more personal; I think that government spending and taxes in the United States should be higher, not lower. There are certainly arguments to be made that government spending in the United States does nothing to raise aggregate utility and should thus be cut, but I believe, and I think most other economists agree with me, that this is definitely not the case. This is especially true in infrastructure, or more generally government investment -- currently at its lowest level as a percentage of GDP since 1948 -- which sorely needs to be increased. Further, spending on Social Security and Medicare should increase over the next decade or two because of the changing demographics of the country. If we adopt the tax proposals of many if not all of the GOP candidates, spending cuts will have to come from somewhere and, given the Republican obsession with massive military spending, they will probably not be defense cuts. This pretty much leaves entitlements and investment -- both of which would cause significant pain going forward if they were cut significantly.

Ideally, fiscal policy makers would focus in the short term on simply not cutting spending too ferociously and in the long run on figuring out how to raise the revenue required for higher levels of government investment and entitlement spending. The GOP seems prepared to do neither of these and, as such, I am not prepared to endorse them for their fiscal policy.

What's the Significance of Low Real Interest Rates?

  

[Note: I started writing this post a while ago, so it ostensibly has no connection with these two posts that Nick Rowe and Scott Sumner wrote recently. I just realized that this is somewhat relevant, so I decided to finish it] 

For the last twenty years or so, real interest rates on government bonds have continued to fall from their high of about 9%. Determining the cause of such a fall is by no means an easy task; after all economic theory generally suggests that real interest rates on safe assets -- like government bonds -- should be relatively constant in the long run and reflect the rate at which consumers discount future spending relative to current spending. Economic theory tells us that low real interest rates mean that current consumption is high and future consumption is low relative to what it otherwise would have been. This certainly is a possibility; perhaps falling real interest rates are indicative of a shift in consumer spending patterns away from saving and into borrowing, although the causality seems to be backwards if that is truly the case, which leaves the question of what has caused this decline in real interest rates open once again. 

Perhaps the basic models in which the government has no power over the real interest rate in the long run are incorrect; given the sharp increase in the real interest rate on government bonds during the 1980s, this certainly seems plausible. In this case, it may be useful to switch to looking at this problem through the lens of an OLG model instead of a basic representative agent RBC/Neo Classical one. Every period, a new young agent is born with the endowment $y$ which can be used to buy either consumption ($c^y_t$) or government bonds ($b_t$), or to invest in capital ($k_t$). The young agent faces the budget constraint

$$(1)\: y = c^y_t + b_t + k_t$$

In the next period, the young become old and use income from interest on government bonds, $R_t b_t$, and from income generated from capital, $f(k_t)$ to finance their consumption and the taxes levied by the government. Old agents face the budget constraint

$$(2)\: c^o_{t+1} = R_t b_t + f(k_t) - \tau_t$$

Agents are born wanting to maximize their consumption in both periods of their life, with consumption when old discounted at rate $\rho$. The agents' discount factor is $\beta = \frac{1}{1 + \rho}$. Utility it derived from the log of current young consumption and the log of future old consumption:

$$ U = \log c^y_t + \beta \log c^o_{t+1} $$

Agents maximize their utility function subject to both of their budget constraints. Young agents choose their consumption so that

$$ (3)\: \frac{1}{c^y_t} = \beta \frac{1}{c^o_{t+1}} R_t $$

That is, young agents take as given the interest rate the they can receive by saving now and consuming later or that they would pay if they consumed now and saved later and decide to save more if the interest rate is high -- since their lifetime income can be increased by their saving -- and save less if the interest rate is low. The government sets the number of bonds that it issues by discretion each period which, given the young agent's consumption decision, determines the level of capital investment. 

Another first order condition of the model is the the real interest rate on government bonds is equal to the marginal productivity of capital. That is, 

$$ (4)\: R_t = f'(k_t)$$

Since the level of government bonds determines capital investment, it also determines the real interest rate on government bonds. More government debt means less capital which, per $4$, means a higher real interest rate (assuming that $f(k)=k^\alpha$ where $\alpha < 1$). This works because agents must be indifferent between holding more government bonds or more capital in equilibrium; otherwise they would end up demanding more or less capital than they wanted. 

In this model, low real interest rates are a result of high capital expenditure and low government debt. The prescription for low interest rates, then, is to engage in a large fiscal expansion that would increase the amount of government bonds in the economy. Less capital demand would have to be justified by a higher real interest rate. Of course, this seems empirically slightly dubious. After all, the amount of government debt skyrocketed in 2008 and interest rates failed to rise. To understand why this wouldn't necessarily be consistent with higher real interest rates, it's important to think along the lines of a demand for government bonds. 

Agents in this model are willing to demand more government bonds at higher interest rates, so if the government sets the supply of government bonds higher, then the demand must correspondingly rise through an increase in the real interest rate. The reason that massive increases in government debt in 2008 and 2009 are not consistent with higher real interest rates is that demand for government debt increased; perhaps even by more than the increase in supply. This was likely caused by the sudden illiquidity associated with other assets that were previously considered safe - e.g. mortgage backed securities or Greek government bonds. The resulting surge in demand for government bonds is known almost colloquially as a 'flight to quality.' 

The ideal fiscal response to this is to satiate demand for government debt by running large deficits (note that this is the exact opposite of the policy actions taken by the majority of governments since 2008). In a way, this is a non-Keynesian reason for pursuing fiscal stimulus; more government debt would be useful for raising the real interest rate. Not only would this make the economy closer to a competitive equilibrium (one without government intervention), it would likely make monetary policy more effective. Narayana Kocherlakota, president of the Minneapolis Fed, made this point in a speech in July. The basic argument he presents is that the government can raise the long-run neutral real interest rate by increasing the amount of government debt. The higher neutral rate of interest (i.e. the real interest rate in this model, since there is no money) will make it so that the Fed will be less likely to hit the zero lower bound when trying to ensure that target is hit. 

Effectively, fiscal policy should be used to remedy situations in which the demand for money is indeterminate and the central bank cannot adequately influence the real interest rate (see, e.g., here).

A Novel New Keynesian View of Fiscal Policy

Usually when I read New Keynesian economists on fiscal policy, they tend to focus more on fiscal multipliers or the effectiveness of tax cuts at the zero lower bound. But what about fiscal policy in general? I have come across some literature on this, but it usually limits itself to comparing the relative roles of monetary and fiscal policy - e.g. what is the optimal coefficient for the output gap in the fiscal policy rule? Here, I'd like to present a somewhat novel approach to New Keynesian fiscal policy (at least I've never seen or read this anywhere else).

Consider first the basic Consumption Euler equation that determines how household's allocate consumption between the present and the future given an interest rate.

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

In simple New Keynesian models, real GDP is composed of just government spending and consumption since there is no capital accumulation, so $1$ can be rewritten as a function of output, $y_t$, and government spending, $g_t$.

$$(2)\: (y_t - g_t)^{-\sigma} = \beta E_t (y_{t+1} - g_{t+1})^{-\sigma} \left(\frac{1 + i_t}{1 + E_t\pi_{t+1}}\right)$$

It is useful to linearize $2$ to make it a bit easier to work with, but first, to make the math a little easier, it is helpful to notice that $y_t - g_t$ is the same as $y_t(1 - \frac{g_t}{y_t})$. Given this, defining $\theta$ as $\frac{1}{\sigma}$, and defining $\beta$ as the inverse of the gross time preference rate, $\rho$, it is possible to write $2$ in log-linear form - i.e. all equations are written as percentage gaps from their long run level.

$$(3)\: \hat{y}_t = E_t\hat{y}_{t+1} - E_t \Delta \hat{g}_{t+1} - \theta (i_t - E_t\pi_{t+1} - \rho)$$

Keep note that, in this case, $\hat{g}_t$ is the gap of the government spending to GDP ratio from trend rather than simply government spending from trend. For my purposes, this is basically irrelevant.

If the goal of fiscal policy is to ensure that the output gap is zero at all times - not too weak of an assumption in my opinion - then it's pretty simple to solve for optimal policy given $3$:

$$(4)\: E_t \Delta \hat{g}_{t+1} = -\theta(i_t - E_t\pi_{t+1} - \rho) $$

In English, equation $4$ tells us that the role of fiscal policy is simply to offset any failure of the monetary authority to set the right real interest rate ($i_t - E_t \pi_{t+1}$). If, for example, the monetary authority has set a real interest rate that is too high, then government spending should be expected to shrink relative to trend in the next period. This can be accomplished either through stimulus - raising current government spending now and reducing it in the future - or through causing expected temporary austerity - decreasing next period's government spending then allowing government spending to return to trend.

The first option is preferable for a couple of reasons. For one thing, government spending also has real effects (see my previous blog post), so austerity might have unintended supply side consequences. Also, the austerity must be reversed at some point for the policy to work - since $\hat{g}_{t+1}$ would fall to zero if the austerity were permanent - so future fiscal policy may be impaired if the central bank continues to set an interest rate that's too high.

Another way of approaching this is to rewrite $3$ to incorporate a 'natural real rate of interest.' In this case, $3$ can be rewritten as

$$(5)\: \hat{y}_t = E_t\hat{y}_{t+1} - \theta (i_t - E_t\pi_{t+1} - \rho + \frac{E_t\Delta\hat{g}_{t+1}}{\theta}) $$

Defining the natural real rate of interest at which the output gap is zero, it is clear that $\rho - \frac{E_t\Delta\hat{g}_{t+1}}{\theta}$ is equal to the natural rate, $r^n_t$.

Assuming the central bank tries to set the real interest rate equal to the natural rate unless the zero lower bound is binding, i.e. $i_t = \max\left(0,\: E_t \pi_{t+1} + r^n_t\right)$, the job of the government can be seen as preventing the zero lower bound from ever binding, or, in other words, setting $E_t\pi_{t+1} + r^n_t > 0\: \forall t$.

$$(6)\: 0 < E_t\pi_{t+1} + \rho - \frac{E_t\Delta\hat{g}_{t+1}}{\theta} $$

or

$$(7)\: E_t\Delta\hat{g}_{t+1} < \theta(E_t\pi_{t+1} + \rho) $$

From $7$, it is clear that expected growth in government spending relative to trend should always be less than a function of the expected inflation rate. That is, the lower the expected inflation rate, the bigger the stimulus that should be undertaken. Effectively the goal of fiscal policy is to offset failures in monetary policy and to make sure that the zero lower bound never binds in the first place.

Shut Up About Ricardian Equivalence

Economists that are both opposed to and in favor of fiscal stimulus frequently cite Ricardian equivalence as a reason that, in models with perfect credit markets, it doesn't matter whether stimulus is funded through increased taxes or through deficits. The problem with this analysis is that it assumes lump sum taxation. That is, taxes are not collected from things like consumption expenditures, which are effectively no different than deficits because 1) they don't discourage people from working, consuming, investing, etc and 2) they are expected to rise at some point in the future to retire the current debt, so the present value of taxes goes up with government spending. In reality, the argument is about distortionary taxes vs. deficits (= lump sum taxes, as per Ricardian Equivalence).

Distortionary taxes are different than their lump sum counterparts since they directly act to disincentivize working (in the specific case of income taxes, which I will limit my analysis to from now on) and can thus either partially or fully negate the effects of a fiscal stimulus. So, when John Cochrane says something like "'Ricardian Equivalence,' which is the theorem that stimulus does not work in a well-functioning economy," [1] he's clearly confusing the two types of taxation as well as ignoring the fact that neoclassical economics predicts a positive multiplier on government spending [2]. To illustrate this, I wrote down a standard Real Business Cycle model and ran two simulations: one in which a temporary fiscal expansion was financed entirely with an income tax and another in which the same fiscal stimulus was financed partially by deficits (see the appendix for a derivation of the model).

Figure 1: Impulse Response Function of Output to the Stimulus

Figure 2: Impulse Response Function of the Income Tax Rate to the Stimulus

Figure 3: Government Spending in both simulations; Government Debt in the second simulation

The Ricardian Equivalence argument would be irrelevant if 1) the stimulus had a positive effect on output and 2) the tax funded stimulus was initially less effective than the partially deficit funded one. As you can see in figure 1, both of these are true; the stimulus positively impacted output in each simulation and the stimulus was initially more effective when taxes were not increased to fully finance the stimulus on impact. The effectiveness of the stimulus is slightly less sound of a result, though. The fiscal multiplier in neoclassical models is highly dependent on calibration (see, e.g., [2]) and can range anywhere from zero to one, without distortionary taxation, depending on the specific calibration used. Regardless, the most important part of this argument is sound; the Ricardian Equivalence argument against deficit funded stimulus is wrong and should be ignored completely as it applies to a form of taxation that doesn't actually exist.

References:

[1] John Cochrane, 2011. "Krugman on Stimulus" The Grumpy Economist.

[2] Woodford, Michael. 2011. "Simple Analytics of the Government Expenditure Multiplier." American Economic Journal: Macroeconomics, 3(1): 1-35.

Appendix:

The following is a derivation of model that I used to generate the impulse response functions in figures 1, 2, and 3.

Household:

There is a representative household who maximizes the utility function $U = E_0 \sum^\infty_{t=0} \beta^t\left(\frac{c_t^{1-\sigma}}{1-\sigma} - \frac{n_t^{1+\phi}}{1 + \phi}\right)$ where $E_t$ is the rational expectations operator given information available in period $t$, $c_t$ is the household's consumption, $n_t$ is the labor supply, and $\beta$ is the household's discount factor - the rate at which future utility is discounted relative to current utility. The household can use net-of-taxes income from labor ($(1-\tau^w_t)w_t n_t$, where $w_t$ is the real wage), government bonds carried from last period ($R_{t-1} B_{t-1}$, where $R_t$ is the interest rate that bonds maturing in period $t$ - $B_t$ - pay), and net-of-depreciation income capital, $(1 + r_{t-1} - \delta)k_{t-1}$ to purchase consumption, new government bonds, or new capital. The budget constraint can be written as

$$ (1.1)\: (1-\tau^w_t)w_t n_t + R_{t-1} B_{t-1} + (1 + r_{t-1} - \delta)k_{t-1} = c_t + B_t + k_t $$

The household maximizes $U$ subject to $1.1$  in order to determine its behavior:

$$ (1.2)\: c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} (1 + r_t - \delta) $$
$$ (1.3)\: R_t = 1 + r_t - \delta $$
$$ (1.4)\: (1-\tau^w_t)w_t = c_t^\sigma n_t^\phi $$

Additionally, it is useful to define investment, $i_t$ as the instrument of capital accumulation:

$$(1.5)\: k_t = (1-\delta)k_{t-1} + i_t $$

Firm:

The firm bundles capital carried from last period and labor using a Cobb-Douglas production function to form output, $y_t$

$$ (2.1)\: y_t = k_{t-1}^\alpha n_t^{1-\alpha} $$

The firm maximizes profits, $y_t - w_t n_t - r_{t-1}k_{t-1}$ subject to $2.1$ in order to determine labor and capital demand

$$ (2.2)\: w_t = (1 - \alpha)\frac{y_t}{n_t} $$
$$ (2.3)\: r_t = \alpha E_t\frac{y_{t+1}}{k_t} $$

Government:

The government issues new government bonds and collects tax revenue to pay for both government spending and interest on government bonds carried from last period. The government budget constraint can be written as

$$ (3.1)\: B_t + \tau^w_t w_t n_t = g_t + R_{t-1} B_{t-1} $$

In the first simulation, it is assumed that the government ensures $B_t = 0\: \forall t$, so government spending is simply financed by taxes

$$ (3.2)\: \tau^w_t w_t n_t = g_t $$

In the second simulation, the government sets the tax rate as a function of the tax rate consistent with the long run level of government spending, $\tau^w_{SS}$ and the level of government debt issued in the previous period, $B_t$. The rule for the tax rate in the second simulation is

$$ (3.3)\: \tau^w_t = \tau^w_{SS} + \phi_b B_{t-1} $$

In both simulations, government spending follows an autoregressive process and returns to its long run trend trend at decay factor $\rho$. Government spending follows

$$ (3.4)\: g_t = (1 - \rho)g_{SS} + \rho g_{t-1} + \eta_t $$

Where $\eta_t$ also follows an autoregressive process with the same decay factor an is hit with with the shock $\epsilon^g_t$

$$ (3.5)\: \eta_t = \rho \eta_{t-1} + \epsilon^g_t $$

Equilibrium:

Combining $1.1$, $1.5$, and $3.1$ yields the resource constraint for the economy

$$ (1)\: y_t = c_t + i_t + g_t $$

Equations $1.2$-$3.5$ can be used to determine the equilibrium for the rest of the endogenous variables:

$$ (2)\: c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} (1 + r_t - \delta) $$
$$ (3)\: R_t = 1 + r_t - \delta $$
$$ (4)\: (1-\tau^w_t)w_t = c_t^\sigma n_t^\phi $$
$$ (5)\: k_t = (1-\delta)k_{t-1} + i_t $$
$$ (6)\: y_t = k_{t-1}^\alpha n_t^{1-\alpha} $$
$$ (7)\: w_t = (1 - \alpha)\frac{y_t}{n_t} $$
$$ (8)\: r_t = \alpha E_t\frac{y_{t+1}}{k_t} $$
$$ (9)\: B_t + \tau^w_t w_t n_t = g_t + R_{t-1} B_{t-1} $$
$$ (10a)\: \tau^w_t w_t n_t = g_t\: \mbox{in simulation 1}$$
$$ (10b)\: \tau^w_t = \tau^w_{SS} + \phi_b B_{t-1}\: \mbox{in simulation 2}$$
$$ (11)\: g_t = (1 - \rho)g_{SS} + \rho g_{t-1} + \eta_t $$
$$ (12)\: \eta_t = \rho \eta_{t-1} + \epsilon^g_t $$

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.

Two Papers Every Republican Candidate (And Everyone) Should Read

Here's a list of a few papers that all the Republican (this applies less to Democrats, at least at the moment) candidates for US president in 2016 should make themselves familiar with:

1. "How Far Are We From The Slippery Slope? The Laffer Curve Revisited" - Mathias Trabandt and Harald Uhlig:

I recently found this paper while browsing ideas for estimates of the US Laffer curve based off of a neoclassical growth model. This paper (download here) has a couple of key points for the Republican candidates. Namely 1. The US is currently on the left side of its Laffer curve and 2. Because of this, tax cuts will not be self-financing (take that Jeb!).

2. "Simple Analytics of the Government Expenditure Multiplier" - Michael Woodford:

This is actually one of the papers that has most influenced my understanding of fiscal policy. Woodford's model doesn't have all the bells and whistles of typical DSGE's, so the analysis is extremely clear and identifies the effects of fiscal policy given different monetary policy choices. Three main insights that I get from it are: 1. If the central bank pegs the real interest rate, the multiplier equals one 2. If the central bank pegs the inflation rate, the multiplier equals the flexible price multiplier 3. If money is neutral, the multiplier is greater than zero 4. At the zero lower bound, the multiplier can exceed one (download here).

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. 

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.