The Taylor Rule is the standard textbook formula for how a central bank sets its short-term policy rate from two gaps — how far inflation sits above target, and how far output sits above its sustainable trend:
i(t) = r* + π(t) + a_π·(π(t) − π*) + a_y·gap(t)
r* = 2% (neutral real rate), π* = 2% (inflation target). Once a rate is set, the economy reacts one quarter later through an IS curve (tight real rates cool demand) and a Phillips curve (a weak economy cools inflation):
gap(t+1) = gap(t) − σ·(i(t) − π(t) − r*)
π(t+1) = π(t) + κ·gap(t)
Corrected stability test: the 3D original of this sim states "aπ > 1 → stabilizing" as a blanket rule, borrowed from the continuous-time Taylor principle. Numerically solving this discrete-time recursion's 2×2 Jacobian shows that isn't quite right here: the real convergence condition is the spectral radius ρ of
M = [ 1, κ ]
[ −σ·a_π, 1 − σ·a_y ]
ρ(M) < 1 is the actual test, and it depends on σ and κ too — at ay=0 the loop is unstable for every aπ in this slider's range, and at the default σ=0.5, ay=0.5, pushing aπ up past ≈1.65 can flip a stable trajectory back to unstable, the opposite of what "more aπ is always safer" would predict. This simulator computes ρ(M) live and colours the verdict from that, not from the rule of thumb.
- The heat map colours every combination of inflation gap and output gap by the rule's recommended rate at your current weights — drag to pan it, scroll/pinch to zoom.
- The glowing marker and trail are the actual simulated economy stepping quarter by quarter through the IS/Phillips equations.
- The strip chart plots π(t) and gap(t) directly against quarters so you can see decay or blow-up without reading the phase plane.
- The stability dial shows ρ(M) against the ρ=1 boundary that actually governs convergence.