📊 Slutsky Equation Playground

Explore how a price change decomposes into substitution and income effects. Drag the sliders and watch the graph update in real time.

Δx = Δxsub + Δxinc │ Total: — = Sub: — + Inc: —

⚙️ Parameters

🎯 Presets

📐 Decomposition

—
Total Δx
—
Substitution
—
Income
Original budget New budget Compensated budget Original IC New IC

📖 Understanding the Slutsky Equation

∂x₁/∂p₁  =  ∂h₁/∂p₁  −  x₁ · ∂x₁/∂m

The Slutsky equation decomposes the total effect of a price change on the quantity demanded of a good into two distinct components:

Substitution Effect (∂h₁/∂p₁)

The change in demand due only to the change in relative prices, holding the consumer's utility constant (i.e., staying on the same indifference curve). This effect is always negative for a price increase: when x becomes relatively more expensive, the consumer substitutes toward y.

  • Graphically: movement along the original indifference curve from the old optimum to the point where the compensated budget line is tangent.
  • The compensated (Hicksian) budget line has the new price ratio but enough income to reach the old utility level.
Income Effect (−x₁ · ∂x₁/∂m)

The change in demand due to the change in real purchasing power caused by the price change. When the price of x rises, the consumer's real income falls, which further affects demand.

  • For a normal good: the income effect reinforces the substitution effect (both reduce demand when price rises).
  • For an inferior good: the income effect works against the substitution effect.
  • For a Giffen good: the income effect dominates the substitution effect, leading to an increase in quantity demanded when price rises (a rare theoretical case).
Cobb-Douglas Utility in this Simulator

This playground uses a Cobb-Douglas utility function:

U(x, y) = xα · y(1−α)

The Marshallian (ordinary) demand functions are:

x* = α · m / p₁    y* = (1−α) · m / p₂

Because Cobb-Douglas preferences are homothetic, goods are always normal, so the income effect always reinforces the substitution effect for a price increase. Adjust α to change how much the consumer values x relative to y.

How to Read the Graph
  • Grey line — Original budget constraint (m = p₁·x + p₂·y).
  • Yellow line — New budget constraint after the price change.
  • Orange line — Compensated budget: new price ratio, but enough income to reach the old utility.
  • Purple curve — Original indifference curve (utility at the old optimum).
  • Cyan curve — New indifference curve (utility at the new optimum).
  • Points A → B → C — A is the original optimum, B is the substitution-only optimum (on compensated budget), C is the final optimum.
  • Pink arrow (A→B) — Substitution effect.
  • Purple arrow (B→C) — Income effect.
  • Green arrow (A→C) — Total effect.