The Taylor principle and determinacy

The most important number in The Taylor rule is not 2 percent for $r^*$ but the total response of the nominal rate to inflation. Woodford gave the requirement its name: the Taylor principle says the nominal rate must rise by more than one for one with inflation, so that the real rate rises when inflation rises (Woodford, 2001). This lesson shows where the requirement comes from.

The three-equation model

The workhorse New Keynesian model has three log-linear equations (Clarida et al., 1999; Galı́, 2015). Variables are deviations from steady state; $x_t$ is the output gap, $\pi_t$ inflation, $i_t$ the nominal rate and $r^n_t$ the natural real rate, which moves with shocks.

$$\begin{equation} x_t = \mathbb{E}_t x_{t+1} - \frac{1}{\sigma}\left(i_t - \mathbb{E}_t \pi_{t+1} - r^n_t\right) \label{eq:is} \end{equation}$$

The three-equation model (cont.)

$$\begin{equation} \pi_t = \beta\, \mathbb{E}_t \pi_{t+1} + \kappa\, x_t \label{eq:pc} \end{equation}$$

$$\begin{equation} i_t = \phi_\pi \pi_t + \phi_x x_t \label{eq:rule} \end{equation}$$

The three-equation model (cont.)

Equation is the dynamic IS curve: demand today depends on expected demand tomorrow and on the gap between the real rate and its natural level. Equation is the New Keynesian Phillips curve with slope $\kappa$ and discount factor $\beta$. Equation is a Taylor-type rule without the constants, which drop out in deviations.

Why “more than one” matters

Consider a self-fulfilling burst of inflation expectations with no change in fundamentals. Higher expected inflation lowers the real rate in unless the central bank raises $i_t$ enough. A higher gap raises inflation through , which confirms the expectation. If instead the rule raises the nominal rate by more than the increase in inflation, the real rate rises, the gap falls, inflation falls, and the expectation is refuted. The Taylor principle is what makes sunspot expectations fail.

Why “more than one” matters (cont.)

Why “more than one” matters (cont.)

The proof works by writing the system as $\mathbb{E}_t z_{t+1} = A z_t$ with $z_t = (x_t, \pi_t)'$ and checking that both eigenvalues of $A$ lie outside the unit circle, which is the Blanchard–Kahn condition for two non-predetermined variables. Galí works through the algebra in chapter 4 of his textbook (Galı́, 2015); the condition in is his equation for the contemporaneous rule.

Why “more than one” matters (cont.)

Evidence

Clarida, Galí and Gertler estimated a forward-looking version of the rule for the United States and found that the inflation response was below one in the pre-Volcker period and well above one after 1979 (Clarida et al., 2000). Their reading: before 1979 policy accommodated inflation and left the door open to self-fulfilling fluctuations, afterwards it did not. Their baseline estimates put the response at roughly 0.8 before and roughly 2.2 after; the exact numbers vary with the specification, the sign of the difference does not.

Evidence (cont.)

The interpretation is contested. Real-time data and the Taylor rule shows that with the data the Fed actually saw at the time, the 1970s look less like a failure of the Taylor principle and more like a measurement problem.

Caveats

• Determinacy is a property of the model. Under learning rather than rational expectations the same condition governs whether agents can learn the equilibrium, which is reassuring (Bullard & Mitra, 2002), but other frictions change the boundary.

• At the effective lower bound the rule cannot deliver a nominal rate below zero (or slightly below), so the principle cannot be satisfied for large negative shocks. This is why Measuring the policy stance needs tools beyond the policy rate.