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.

A Simple Model With Consumer Durables

Nick Rowe on twitter:

How negative do real interest rates have to go before storing consumer goods (e.g. food) becomes profitable?

This question inspired me to try to come up with a model with durable consumer goods (i.e., goods that aren't immediately consumed when purchased, but instead last multiple periods) and sticky prices. I won't bother with adding sticky prices (or monopolistic firms) until I have a basic model sketched out, so here goes:

There is a representative agent which derives utility from their consumption of non-durable goods, their stock of durable goods, and leisure. The utility function is
$$(1)\: U = \sum^\infty_{t=0}\beta^t u(c^d_t,c^p_t,l_t)$$
where $\beta$ is the discount factor, $c^d_t$ is the current stock of durable goods, $c^p_t$ is the agent's current consumption of non-durable (perishable) goods, and $l_t$ is leisure. The agent uses income from working, holding government bonds, and profit from the representative firm to pay for perishable consumption, the increase in durable goods that it owns (which depreciates at rate $\delta$), new government bonds, and lump sum taxes. The budget constraint is:
$$(2)\: c^p_t + c^d_t - (1-\delta)c^d_{t-1} + B_t + T_t = R_{t-1}B_{t-1} + w_t (1-l_t) + \pi_t$$
The agent maximizes $1$ w.r.t. $2$, which yields the following first order conditions:
$$(3)\: \beta^t u_2(c^d_t,c^p_t,l_t) - \lambda_t = 0$$
$$(4)\: \beta^t u_3(c^d_t,c^p_t,l_t) + \lambda_t w_t = 0$$
$$(5)\: \beta^t u_1(c^d_t,c^p_t,l_t) -\lambda_t + E_t \lambda_{t+1} (1-\delta) = 0$$
$$(6)\: -\lambda_t + R_t E_t \lambda_{t+1} = 0$$
where $\lambda_t$ is the Lagrange multiplier on the maximization problem.

These can all be simplified to:
$$(7)\: u_2(c^d_t,c^p_t,l_t) = \beta E_t u_2(c^d_{t+1},c^p_{t+1},l_{t+1})R_t$$
$$(8)\: w_t = -\frac{u_3(c^d_t,c^p_t,l_t)}{u_2(c^d_t,c^p_t,l_t)}$$
$$(9)\: u_1(c^d_t,c^p_t,l_t) = u_2(c^d_t,c^p_t,l_t)\left(1 - \frac{1-\delta}{R_t}\right)$$

Given that the government budget constraint is
$$(10)\: B_t + T_t = R_{t-1} + B_{t-1}$$
it is possible to rewrite the agent's budget constraint as:
$$(11)\: c^p_t + c^d_t - (1-\delta)c^d_{t-1} = w_t (1 - l_t) + \pi_t$$

There is a representative firm that maximizes profit, $\pi_t = y_t - w_t(1-l_t)$, where $y_t$ is production and $w_t$ is the real wage, subject to the production function
$$(12)\: y_t = f(1-l_t)$$
which gives the following first order condition:
$$(13)\: w_t = f'(1-l_t)$$
Given that $\pi_t + w_t(1-l_t) = y_t$, it is possible to rewrite $11$ as
$$(14)\: y_t = c^p_t + c^d_t - (1-\delta)c^d_{t-1}$$

Now, we have the full equilibrium of the model:
$$(1)\: u_2(c^d_t,c^p_t,l_t) = \beta E_t u_2(c^d_{t+1},c^p_{t+1},l_{t+1})R_t$$
$$(2)\: w_t = -\frac{u_3(c^d_t,c^p_t,l_t)}{u_2(c^d_t,c^p_t,l_t)}$$
$$(3)\: u_1(c^d_t,c^p_t,l_t) = u_2(c^d_t,c^p_t,l_t)\left(1 - \frac{1-\delta}{R_t}\right)$$
$$(4)\: y_t = f(1-l_t)$$
$$(5)\: w_t = f'(1-l_t)$$
$$(6)\: y_t = c^p_t + c^d_t - (1-\delta)c^d_{t-1}$$

I'll probably do more analysis tomorrow, but what immediately strikes me as interesting (and relevant to Nick's question) is that consumer durables are basically the same as money in MIUF models, if the real return on government bonds falls below $-\delta$, then the demand for consumer durables will skyrocket -- basically representing a lower bound on the real interest rate in addition to the nominal one.

The only question should then be what the average depreciation rate of durable consumer goods is; if it's not too high, I may have just found an undeniable (real) constraint on monetary policy -- if the natural rate of interest should ever fall below $-\delta$, then the central bank genuinely can't do anything about it; only fiscal policy can (by raising the natural rate).

Extended Response to Nick Rowe

Nick Rowe on Twitter earlier today:

If [a] central bank targeted the price of peanuts, would we blame recessions on bad peanut harvests? Or blame [the] central bank for not raising [the] target price?

It depends. It depends on how quickly the central bank finds out about the bad peanut harvest, how quickly the new policy can be enacted, and how effectively the central bank can control the price of peanuts.

Suppose no one know about the size of the peanut harvest until the following period. In this case, the central bank, which we will assume can completely control the price of peanuts for the time being, is not culpable for the recession that occurs because the price of peanuts is too low. The central bank could not have known that the price level (of peanuts) at which output remained at potential was higher than they otherwise thought, so they cannot be blamed for the recession that ensues.

If, on a slightly different note, the central bank faces a delay in policy implementation, it may not be able to act quickly enough to prevent a recession; they can raise the target price with a delay, but there will still be a recession in the meantime and the central bank is not culpable.

Alternatively, assume that the central bank knows about the bad harvest in real time and doesn't face a policy lag, but, for some reason, is unable to set the price of peanuts any higher. In this case, the central bank can't be blamed either -- there is nothing it can do to prevent it from happening, so the correct culprit for the recession is the bad peanut harvest.

Generally, assuming there are no significant lags in information or implementation, the central bank would be to blame for not preventing the recession. The only time that the blame really shouldn't fall on a central bank is when it can't control the price (of peanuts) -- in this case, central bank impotence is to blame for the recession, not actions taken by the central bank.

With that aside, now we can go about determining when central banks are impotent.

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.