3D Business Cycle Phase Space: Multiplier-Accelerator Model

Multiplier

Marginal propensity to consume c 0.60

Accelerator

Accelerator coefficient v 1.00

Fiscal policy

Government spending G 30

Speed 3 periods/s

Output Yₜ

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ΔY

–

Equilibrium Y*

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Period, t

Regime

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Drag to orbit. The curve's three axes are time (running left→right), output level Yₜ (height) and the rate of change ΔY (depth) — the same recurrence as the 2D chart, traced as a genuine 3D phase-space trajectory instead of a flat line.

How it works

Same model as the 2D version — Samuelson's 1939 multiplier-accelerator business cycle:

Yₜ = c(1+v)·Yₜ₋₁ − cv·Yₜ₋₂ + G
ΔYₜ = Yₜ − Yₜ₋₁

Plotting (t, Yₜ, ΔYₜ) as one continuous 3D curve turns the model into real phase-space geometry: when the characteristic roots are complex (oscillatory regime) the curve winds into a spiral around the equilibrium axis — tightening inward when √(cv) < 1 (damped), holding a constant radius when √(cv) ≈ 1 (a limit cycle), and flaring outward when √(cv) > 1 (explosive). When the roots are real instead, the curve approaches or leaves the equilibrium plane monotonically, with no winding at all — you can see directly, from the shape alone, which of Samuelson's four dynamic regions the current sliders fall into.

Output trajectory (age → brighter) Current period Equilibrium plane Y*

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