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Assuming an interior solution, (expected) rate of return equality implies:

[3] r = f'(y-b)

Technically, [3] determines bond demand. In equilibrium, the supply of bonds (determined by policy) must equal the demand for bonds. Hence, by choosing b in this model, the government can choose the prevailing real rate of interest. Lump-sum taxes are simply adjusted by way of [1] to finance the carrying cost of the debt. Equilibrium consumption is then given by [2]; i.e., c = f(y-b) + b.

The real GDP in this economy is given by Y = y + f(y-b). Notice that this model delivers a standard IS curve. That is, by increasing b, the government increases r, capital is crowded out, and output falls. Likewise, lowering the real interest rate stimulates (investment) demand, leading to an increase in output.

In what follows, I assume that a socially desirable outcome is associated with some 0 < b* < y. The "natural" rate of interest is defined as r* = f'(y-b*), and potential GDP is defined by Y* = y + f(y-b*). [Note that r* may be either greater or less than one. Some of you may argue that r* should equal 1 here. That's fine. The qualitative results below do not hinge on this issue.]

An economy with nominal debt

Let P denote the price of output denominated in some abstract unit of account. Let B = Pb the nominal debt. Let P+ denote "next period's" price level. Then a young person faces the following sequence of budget constraints:

Py = Pk + B 
P+C+ = P+f(k) + RB - T+

where R denotes the gross nominal interest rate, and T is the nominal lump-sum tax. Define Π+ = P+/P, the expected gross rate of inflation. Then using b =B/P and t = T/P, rewrite the budget constraints above as

y = k + b
c+ = f(k) + (R/ Π+)b - t+ 

Desired real bond holdings must now satisfy the condition

[4]  f'(y - b) = R/ Π+

with the demand for nominal bond holdings given by B = Pb. If I define r = R/ Π+ as the expected real rate of interest, then we see that [4] is equivalent to [3].

The government budget constraint is given by T+ = RB - B+. In real terms,

[5] t+ = (R/ Π+)b - b+ 

so in a steady state with b = b+, we have t = (r-1)b, which is equivalent to [1]. Consumption is given by [2].

The model to this point is riddled with indeterminacy, even restricting attention to steady states. What determines the nominal interest rate, the inflation rate, the price level, etc.? Note that this indeterminacy is not present in the model with real debt. In that world, I assumed that b was a policy instrument. This (along with the lump-sum tax instrument) pins down an equilibrium. In the world I am describing now, the government does not pick b. It need not even pick B if, in particular, it is willing to let demand determine quantity at a given rate of interest. How to proceed? As usual, in small steps.

A Monetarist regime

We can think of B as interest-bearing money. The nominal interest rate on money is commonly assumed to be zero, so R = 1. But there is nothing that requires this to be the case; we are free to pick any interest rate supportable by taxes here. The key assumption is that R is determined and that it is constant over time.

The monetarist views B (and the time path for B) as determined by policy. With the demand for real money balances determined by [4], market clearing requires B = Pb for all time. Since B is determined by policy, and b is determined by agents, the price level is determined by P = B/b. The inflation rate must therefore be determined by

[6]  Π+ = (B+/B)(b/b+)

Let B+ = μB. Now combine [6] with [4] to derive:

[7]  b+ = (μ/R) f'(y - b)b

which is a first-order difference equation in real money balances. The model has two steady states. In one, b = 0; in the other, b > 0 satisfies f'(y - b) = (R/μ). It seems darn easy to construct the optimal policy here. Just set  (R/μ) = r* and we're done.

Well, not so fast. As it turns out, even for the case of a fixed stock of money B, there generally exists a continuum of nonstationary equilibria indexed by an initial condition 0 < b0 < y, with the time path for b asymptotically approaching zero; see Figure 1 in Woodford (1984). Of course, since P = B/b with B fixed, this implies that the price level approaches infinity (in fact, these are hyperinflation dynamics). Isn't it interesting to note that Friedman's k percent rule is dynamically unstable here?

The undesirable hyperinflation dynamic here appears to an artifact of (among other things) the assumed passivity of policy (the nominal interest rate and money growth rate are held fixed forever). But evidently, there exists a simple "activist" policy rule that uniquely implements the desired outcome:

[8]  ln(R) = ln(R*) + α[ ln(Π+) - ln(Π*) ]

where Π* = μ (arbitrary), R* = r*Π*, and α = 1.  The policy rule [8] is a Taylor rule. The rule dictates that the policy rate be increased one-for-one with expected inflation. Such a policy keeps the expected real rate of interest pinned to its natural rate. As such, the economy is always at potential (this would not necessarily be the case if I was to introduce other shocks, of course). Note that the price level is now determined, P = B/b* with P+ = Π*P.

Would the ZLB restriction R ≥ 1 limit the ability of policy here? I do not think so. First, the problem in this model is a the possibility of a self-fulfilling hyperinflation -- deflationary equilibria do not exist. As such, policy only needs to threaten to raise, not lower, the nominal interest rate. Second, I believe that the optimal monetary policy may alternatively be expressed as a money growth rate that varies in proportion to the expected growth rate in real money demand. That is, targeting the inflation rate is feasible here (and contrary to Eagle (2006), an inflation target policy seems consistent with price level determinacy here).

A Wicksellian regime

Following the approach taken in the New Keynesian literature, we might instead assume that the quantity of nominal debt B (money) is entirely demand-determined. The only policy instrument is R (and, of course, the lump-sum tax). I assume that policy follows the Taylor rule [8] with α = 1.

As far as I can tell, all of the math developed in the previous section continues to hold. But giving up the quantity variable B as a policy instrument must have some implication. Indeed, it does. What we seem to lose is any fundamental economic force determining the price level and inflation rate. That is, the level of debt and its growth rate simply accommodate themselves to the prevailing price level and inflation rate, respectively. According to [8], exogenous movements in the expected rate of inflation (inflation shocks) are met one-for-one with movements in the nominal interest rate, leaving the real rate of interest pegged to its natural rate.

Note something interesting here: the inflation target Π* is completely irrelevant. Inflation in this model can be whatever it "wants" to be. If the community expects an inflation rate Π+ < Π*, the inflation rate Π+ becomes a self-fulfilling expectation (and is hence a "rational expectation"). In this case, the monetary authority simply sets its policy rate R < R*. A situation like this can last indefinitely in this model.

Of course, everything works just fine here as long as the ZLB is not a constraint. Suppose, instead, that the policy rate is constrained by the ZLB, so that [8] becomes:

[9]  ln(R) = max{ 0, ln(R*) + α[ ln(Π+) - ln(Π*) ] }

Imagine that the economy is initially operating at potential with Π+ = Π* (without loss). Then, out of the blue, individuals suddenly believe that the inflation rate is going to be permanently lower Π+ = Π' < Π*. Moreover, suppose that this inflation shock is sufficiently large to make the ZLB bind. What happens?

What happens is that output drops permanently below potential (the economy continues to grow, however, at the rate implied by technological progress and population growth, both of which are normalized to zero here). Why does this happen?

It happens (here) because the real rate of interest rises above its natural rate, r' = 1/Π' > r*. The real interest rate is too high. The effect is to depress (investment) demand, r' = f'(k') implies k' < k*. The real GDP falls below potential, Y' = y + f(k') < Y*.

Because there is no nominal anchor for inflation in this economy, all sorts of bad things can happen at the lower bound. Contrary to the Friedman rule prescription, deflation is bad (generally, any inflation rate sufficiently low to make the ZLB bind). Not that the monetary authority could actually implement the Friedman rule if it wanted to. In this economy, the monetary authority has absolutely no control over the inflation rate!

A nominal anchor

Actually, now that I stare at [10], I see that the δ > 0 opens up another source of indeterminacy. It may be possible, for example, that if people suddenly expect a recession Y+ < Y*, that the downward revision in inflation forecasts implied by [10] could make the ZLB bind, generating a self-fulfilling prophecy.

Anyway, let's just set δ = 0 here. In the Wicksellian approach above, I adopted a special case of [10]; i.e., ρ = 1 and δ = 0;. But now, for 0 ≤ ρ < 1, any given inflation shock is mean-reverting (to the inflation target). The "lift off" date -- the date at which the monetary authority begins to raise its policy rate according to [9] depends on how quickly inflation expectations rise. The speed of adjustment here is governed by the parameter ρ--a lower ρ implies faster adjustment.

Is there anything the monetary authority can do here to "talk up inflation" (i.e., lower ρ)? We really can't say without a theory of expectation formation. But it seems to me that "promising to keep R = 1 for an extended period of time" may have the effect of increasing ρ, extending the period of adjustment. That is, by postponing the "lift off" date, agents may rationally expect inflation to remain below target for a longer period of time.

Concluding thoughts

Let me be clear that I do not think the 2008 drop in output below its previous trend was caused by a negative inflation shock. A negative inflation shock possibly played a role, but there had to be more to the story than this. In the analysis above, a negative inflation shock represents a movement along a stable IS curve; the real interest rate goes up, and output goes down. To make sense of recent events, we also have to consider shocks that shift the IS curve "leftward." (I describe just such a shock here.)

Nevertheless, I think it is interesting to explore what potential effects future downward revisions to inflation expectations may have on the economy at the ZLB. In the Wicksellian regime I study above, there appears to be no nominal anchor apart from what agents believe it to be. And if agents come to believe in a persistent deflation, it may come to pass, and the economy may be stuck below potential for a very long time. Convincing agents that the nominal interest rate is likely to remain at zero for a long time may be counterproductive, depending on how individuals interpret such policy announcements.

I want to stress, however, that while getting inflation and inflation expectations back to target (and firmly anchored to target) may be a solution to one problem, it is unlikely to be a solution to every problem currently facing the U.S. economy. To put it another way, suppose that the current real interest rate of -1% is too high relative to the current "natural" rate of -x%. Somehow driving the real return on bonds to -x% may then help things a bit, but it does nothing to address the more pressing question of why the "natural" rate is so low to begin with.