The Taylor Principle and the Federal Reserve, 1987–2026: A Proof of Determinacy, Rule Estimates, and the Size of the Post-2020 Shortfall

The Taylor principle says that a central bank should raise its nominal policy rate by more than one point for each point of inflation. I prove the exact condition for a unique stable equilibrium in the three-equation New Keynesian model with a contemporaneous rule, κ(φπ−1)+(1−β)φx > 0, using the trace and determinant conditions for a 2 by 2 system, and confirm it numerically on 2,496 to 2,499 parameter points for three values of κ with full agreement. I then compare the effective federal funds rate with a Taylor (1993) rule using core PCE inflation and the CBO output gap for 156 quarters, 1987Q3 to 2026Q2. Over 1987Q3 to 2008Q3 the rule fits on average (mean gap 0.00 points, RMSE 1.34). A partial-adjustment estimate for that period gives an inflation response of 1.613 (se 0.294), above one with t=2.08, and an implied neutral real rate of 1.39%. After 2020 the prescription first exceeded 2% in 2021Q1 but the quarterly average funds rate first exceeded 0.5% only in 2022Q2, five quarters later; the shortfall peaked at 9.72 points in 2022Q1 and averaged 5.76 points over 2021 to 2023. In 2026Q2 the funds rate was 3.63% against a prescription of 6.34%.