A price change does two things at once. It changes the trade-off between goods, and it changes how much the consumer can buy overall. Pulling the two apart explains the shapes of the curves in Price Changes and the Demand Curve and Income Changes and Engel Curves, and it uses the compensation idea from Minimizing Spending for a Target Utility.
Key idea
Split the move from the old optimum A to the new optimum C at a hypothetical B: $$\underbrace{x_C - x_A}_{\text{total}} = \underbrace{x_B - x_A}_{\text{substitution effect}} + \underbrace{x_C - x_B}_{\text{income effect}}.$$ B is the choice at the new prices with income adjusted to compensate: Slutsky gives just enough to buy A again, $m' = m + \Delta p_1\,x_A$; Hicks gives just enough to reach A's utility. The substitution effect always moves $x$ against the price change.
Why it works
Why is the substitution effect never "wrong-way"? Take the Hicks version. B and A sit on the same indifference curve, and B is the cheapest point on it at the new prices, while A was the cheapest at the old prices. If $x$ got cheaper, the cheapest point slides toward more $x$ along a curve bowed toward the origin: the tangency moves to where the curve is flatter, which is at larger $x$. The income effect has no such rule: it is the parallel shift from B to C, and its sign depends on whether the good is normal or inferior. In calculus form this is the Slutsky equation $\dfrac{\partial x}{\partial p_1} = \dfrac{\partial h}{\partial p_1} - x\dfrac{\partial x}{\partial m}$, where $h$ is compensated demand.
Worked example: Slutsky decomposition
$U = xy$, $m = 120$, $p_2 = 1$, and $p_1$ falls from 4 to 2. Find the two effects on $x$.
- A: $x_A = \dfrac{120}{2\cdot 4} = 15$.
- Compensated income: $m' = 120 + (2 - 4)(15) = 90$ (take away what she saves on her old purchases).
- B: $x_B = \dfrac{90}{2\cdot 2} = 22.5$. Substitution effect: $22.5 - 15 = 7.5$.
- C: $x_C = \dfrac{120}{2\cdot 2} = 30$. Income effect: $30 - 22.5 = 7.5$. Total: 15.
Answer: Substitution effect $+7.5$, income effect $+7.5$.
Worked example: Hicks decomposition
$U = xy$, $m = 80$, $p_2 = 1$, $p_1$ falls from 4 to 1.
- A: $x_A = 10$, $y_A = 40$, so $u_A = 400$.
- B: the cheapest bundle with $xy = 400$ at prices (1, 1) has $y = x$, so $x_B = 20$. Substitution effect: $+10$.
- C: $x_C = 80/2 = 40$. Income effect: $40 - 20 = +20$.
Answer: Substitution $+10$, income $+20$.
Common mistake
Assuming the income effect always works in the same direction as the substitution effect. For a normal good it does; for an inferior good it works against it. Check: ask whether the price change made the consumer richer or poorer, then ask whether this good rises or falls with income. The substitution effect needs no such question: it always goes against the price change.