Slutsky Decomposition

Splitting a price change in two

When the price of good 1 falls, the consumer buys more of it — but for two different reasons, and Slutsky’s trick is to separate them:

  • Substitution effect — good 1 is now cheaper relative to good 2, so even holding real purchasing power fixed, the consumer tilts toward good 1. This effect always moves opposite to the price change.
  • Income effect — the price cut also makes the consumer richer in real terms (the old bundle now costs less), and they spend part of that windfall. The direction depends on whether the good is normal or inferior.

Slutsky’s way of “holding purchasing power fixed” is concrete: after the price change, give the consumer exactly the income that lets them just afford their original bundle. Call it \(m'\). With original bundle \((x_1^0,x_2^0)\) and new prices, \[ m' = p_1'\,x_1^0 + p_2\,x_2^0. \] Then:

  • Substitution effect \(= x_1(p_1',m') - x_1(p_1,m)\) — move along to the new price ratio, pivoting around the old bundle.
  • Income effect \(= x_1(p_1',m) - x_1(p_1',m')\) — take the compensation back away, shifting to the true new budget.
  • The two add up to the total effect.

The sim uses Cobb–Douglas demand, \(x_1 = a\,m/p_1\).

Try it. Lower p₁′. Three budget lines appear: the original (blue), the Slutsky-compensated pivot (dashed grey, through the old bundle), and the final budget (orange). The bundles on them split the move into substitution + income.

What to notice

  • Drop p₁′ below 1 and the substitution effect is positive (more good 1) — it always opposes the price change, no exceptions.
  • For Cobb–Douglas both goods are normal, so the income effect pushes the same direction as substitution. Total demand for good 1 rises as its price falls — a downward-sloping demand curve.
  • The grey dashed line is the heart of the method: same slope as the new budget (new price ratio) but shifted to just pass through the old bundle. That pivot is the pure substitution effect.