Giffen good paradox: when price rises, quantity demanded increases. Requires a strongly inferior good that is a large budget share. The Slutsky equation: ∂x/∂p = (∂x/∂p)|ᵤ₌ū − x·(∂x/∂m). The substitution effect (∂x/∂p)|ᵤ < 0 always. But if the income effect x·(∂x/∂m) is negative (inferior) and large enough in magnitude, the total effect is positive — Giffen behavior.
Classic example: Irish potato famine — as potato prices rose, the poor reduced meat consumption and ate more potatoes. Modern verified example: Jensen & Miller (2008) in China with rice/wheat.