The tax-revenue hill: why revenue rises, peaks, then falls as tax rates climb, and why the peak location is the whole debate.
Warming up the engine…
The Laffer curve traces total tax revenue against the tax rate: zero at 0% (no tax collected) and near zero at 100% (no reason to earn), so revenue must rise, peak, and fall in between.
R = t · B(t), where the base B shrinks as the rate t rises
Raising a low rate collects more from a barely-changed base. Raising a high rate makes work, investment, and honesty less attractive, so the base shrinks faster than the rate grows. Both sides of politics accept the shape; the empirical fight is where the peak sits, with most estimates putting it far above typical income-tax rates.
A tax cut only raises revenue if the economy starts BEYOND the peak. Assuming every cut pays for itself is the classic error; state the starting position explicitly.
Now prove you have it
Move the curve to where you think it lands, and get told exactly which part you got right.
Real-world scenarios on this model
Equilibrium at t 30, R 84.
Equilibrium: t* = 30.0, R* = 84.0
Current equations