Fisher without Euler

The Neo-Fisherian proposition is that raising the nominal interest rate (and keeping it elevated) will eventually cause inflation to rise (see Steve Williamson's explanation here.)

The basic idea revolves around the so-called Fisher equation:

R = r + E[p]

where R is the nominal interest rate, r is the real interest rate, and E[p] is the expected rate of inflation. If bond buyers expect inflation to increase then they'll ask for more compensation in the form of a higher nominal interest rate (a lower bond price).

The conventional idea is that monetary and fiscal policies (in particular, the expectation of how these policies will unfold over time) largely determined inflation expectations E[p]. In conventional (modern) macro economic theories, expectations are assumed to be formed "rationally" (i.e., in a manner that is consistent with the stochastic processes that actually govern the economy).

Neo-Fisherians reverse this conventional direction of causality. They argue that increasing R leads people to revise their inflation expectations upward. And because people have rational expectations, for these expectations to be consistent with reality, actual inflation will (somehow) have to increase.

As far as I can tell, this Neo-Fisherian proposition comes in two stripes. The first stripe is of the "cashless economy with Ricardian equivalence" variety--the models that Michael Woodford likes to use. In this class of models, "balance sheets don't matter." And because central bank money and government bonds are just ways of labeling the liabilities of the consolidated government sector, they don't matter for determining (among other things) the price-level. In this class of models, inflation expectations are somehow assumed to adjust to satisfy the Fisher equation. And then the price-setting behavior of firms (who set prices in an abstract unit of account but do not actually accept payment in any monetary object) adjusts in a manner that is consistent with higher expected inflations. Personally, I find this view implausible. Moreover, it's frustrating that no one promoting this view seems willing or able to explain how/why all this is supposed to happen (beyond repeating the phrase "the Fisher equation must hold" or "it's a rational expectations equilibrium").

The second stripe of this proposition, however, seems more plausible (at least, in principle) to me. In this world, balance sheets matter. The supply and composition of the government's assets and liabilities matter. And in particular, the time-path of the total nominal government debt (and its composition) matters for determining the price-level. The idea here is that when the central bank announces a higher R, there is a corresponding passive accommodation of central bank policy on the part of the fiscal policy to increase the rate of growth of total government debt (i.e., cut taxes, or engage in "helicopter drops"). If the fiscal authority behaves "passively" in this sense, then people will rationally expect higher inflation--and the higher inflation will actually transpire not because people expected it, but because the fiscal authority delivered it. I think this is an interpretation that even Nick Rowe agrees with (see here).

Both versions of the Neo-Fisherian proposition above seem to rely heavily on the notion of rational expectations. In my previous post, I speculated that the proposition might hold even if people had non-rational "adaptive" expectations. The idea I had there was that if a sudden increase in R caused to the price-level to jump up (instead of down, which is the usual presumption), then people with adaptive expectations will revise their inflation expectations upward (not downward).  An initial increase in the price-level might happen if, for example, the higher interest rate led to higher operating expenditures on the part of firms. Following this initial impulse, the actual path of inflation would be determined either by (stripe 1) the nature of learning dynamics or (stripe 2) the manner in which policy accommodates itself to the price shock (e.g., see Christiano and Gust, 1999).

In response to my post, Erzo Luttmer alerted me to his paper Fisher without Euler, in which he claims that the Neo-Fisherian proposition pops out of a model in which people are not forward-looking at all. The argument, as far as I can tell, relies heavily on how the government debt-service cost is financed. Let me try to explain (you can refer to Erzo's paper and short note to see whether I have it right).

Let's start with the government budget constraint,

G(t) - T(t) = q*B(t) - B(t-1)

where T(t) denotes tax revenue, G(t) government purchases, and B(t-1) denotes bonds maturing to cash at date t. Let 0 < q < 1 denote the price of a bond (1/q is the gross nominal interest rate, set by the Fed). For simplicity, I think we can set G(t) = 0 for all t, so that

T(t) = B(t-1) - q*B(t)

This makes it clear how a lower q (higher interest rate) means either  higher taxes and/or higher debt level. Now, let p(t) denote the price-level and define τ = T(t)/p(t). Assume that nominal debt grows at a constant rate, B(t) = μB(t-1). Now use this notation to rewrite the government budget constraint above as

τ = (1 - q*μ)*B(t-1)/p(t)

To close the model, we need a theory of the price-level. The simplest theory I can think of is the Quantity Theory: p(t) = B(t-1)/y(t), where y(t) is real income (and velocity is held constant), so that B(t-1)/p(t) = y(t). If we treat y(t) as exogenous, then it follows immediately that lowering the interest rate (increasing q) necessitates a decline in inflation (μ). So lowering the interest rate lowers the debt-service cost of debt which (for given real spending and taxation levels) means that the supply of nominal debt need not grow as quickly -- as the growth rate in the supply of "money" declines, so does inflation. The Neo-Fisherian result follows even without forward-looking behavior.

Erzo does not use the simple version of the Quantity Theory as I did here. Instead, he assumes that individuals adopt a simple behavioral rule (consumption function):

c(t) = α(y(t) - τ) + βB(t-1)/p(t)

where α is the propensity to consume out of disposable income and β is the propensity to consume out of wealth (here in the form of real bond holdings). If we let g(t) denote real government purchases, then goods-market-clearing requires:

c(t) = y(t) - g(t)

Erzo then combines these latter two equations to determine the price-level p(t), treating y(t) and g(t) as exogenous (as did I).

At the end of the day, it's a simple point. Still, I think it's an important one to keep in mind since I am reading in more than one place that the Neo-Fisherian proposition depends on rational expectations. Evidently, it does not.