New York Intermediate Microeconomics Tutors – NYU, Columbia, Yale, Princeton Tutors

When a price falls, you buy more — but why you buy more is two separate stories, and intermediate microeconomics is where you learn to tell them apart. Part of the extra buying is because the good is now cheaper relative to everything else. Part of it is because a lower price leaves you effectively richer. The Slutsky decomposition splits a single price change into exactly these two pieces, the substitution effect and the income effect, and it is one of the most reliably examined ideas on any intermediate microeconomics course. Get the construction clean once and the whole of consumer theory becomes easier to reason about.

1 · One price change, two forces

Start with the picture consumer theory always uses. You have a fixed budget to split between a good x and everything else y. The budget line shows every bundle you can just afford; your indifference curves show what you like; you choose the affordable bundle on the highest curve, which is the point of tangency.

Now cut the price of x. The budget line pivots outward — you can reach more x than before while the most y you could buy is unchanged — and you move to a new tangency with more x. That much is just the law of demand.

The decomposition asks a sharper question. Of the extra x you now buy, how much is because x got cheaper relative to y, and how much is because the price cut made your budget stretch further? Two forces are bundled inside that one move:

• The substitution effect — you tilt toward x because it is now the better-value good, holding your real position fixed.
• The income effect — the price cut raises your purchasing power, and you spend part of that gain on x (or on y, depending on the good).

The whole apparatus below exists to measure each one separately.

2 · The substitution effect

The substitution effect isolates the pure relative-price story by stripping the income change out. The trick is a compensated budget line: an imaginary budget, drawn at the new relative price, adjusted so you are no better off than you started.

“No better off” is where Slutsky makes its specific choice, and Section 4 pins it down. For now, take the compensated line as given: it has the new, shallower slope, and it is positioned so your original bundle is still just within reach. Slide along it to the new tangency. That movement — from the old optimum to the optimum on the compensated line — is the substitution effect.

Two facts about it are worth committing to memory, because examiners lean on both:

1. The substitution effect always moves against the price. When x gets cheaper, the substitution effect raises x. When it gets dearer, the substitution effect cuts x. No exceptions — this is the one part of demand theory with a guaranteed sign.
2. It slides along toward the flatter price ratio. Because the compensated line is tangent to a curve, the new point sits where the marginal rate of substitution equals the new price ratio. You are re-optimising, not just re-labelling.

3 · The income effect

Now restore the income you stripped away. Shift the compensated budget line out to the real new budget line — same slope, further from the origin — and move to the final tangency. That second movement is the income effect: the response to the change in purchasing power alone, holding relative prices at their new level.

The income effect has no guaranteed sign, and that is the point of it:

• If x is a normal good, higher purchasing power raises demand for x, so the income effect adds to the substitution effect. Most goods most of the time.
• If x is an inferior good, higher purchasing power lowers demand for it, so the income effect works against the substitution effect. Think instant noodles or bus travel — things you buy less of as you get richer.
• In the rare, extreme case where an inferior good’s income effect is large enough to overturn the substitution effect, demand slopes the “wrong” way. That is a Giffen good, and it is exactly the case the decomposition was built to make sense of.

So the total effect is the sum of the two:

total effect = substitution effect + income effect.

The substitution effect fixes the direction; the income effect decides whether the total reinforces it, softens it, or — only for a Giffen good — reverses it.

4 · Slutsky or Hicks — which compensation?

Section 2 promised to pin down what “no better off” means. There are two standard answers, and they give the same story but a slightly different middle point.

• Slutsky compensation keeps your original bundle just affordable. The compensated budget line is drawn at the new prices and passes through the point you started at. You could still buy exactly what you bought before — so, being free to re-optimise, you end up a little better off than before.
• Hicks compensation keeps your original utility. The compensated line is drawn at the new prices and made tangent to your original indifference curve, so you land back on the same curve you started on.

This page uses the Slutsky version, and there is a good reason to prefer it in practice: it is built entirely from things you can observe — prices and the quantities actually bought — with no need to know the shape of anyone’s indifference curve. That makes it the version that connects to real data and to the Slutsky equation you will meet later. The Hicks version is cleaner in pure theory because it holds utility fixed exactly, but it needs the utility function you almost never have.

For a small price change the two give almost the same substitution effect; for the large, round-number changes used in problem sets they differ a little, and an exam question will usually tell you which one it wants. When it says “Slutsky”, it means: compensate so the old bundle is still affordable.

Worked example — the price of coffee falls

You have £18 a week to split between coffees, good x, and everything else, y, measured in pounds so that its price is p_y = £1. Your preferences are U = x²y. A coffee starts at p_x = £4 and falls to £2. Find the total change in coffees bought, and split it into the substitution and income effects.

For U = x²y the marginal rate of substitution is MRS = 2y/x, and Cobb–Douglas demand spends a fixed two-thirds of the budget on x: that is, x* = (2/3)(M/p_x).

Step 1 — The original optimum A. At p_x = 4, spending two-thirds of £18 on coffee gives x = (2/3)(18)/4 = 3 coffees, leaving y = £6. Check the tangency: MRS = 2(6)/3 = 4, which equals p_x/p_y = 4/1. So A = (3, 6).

Step 2 — The final optimum C. Now p_x = 2. Two-thirds of £18 is still £12, but each coffee costs half as much, so x = 12/2 = 6, and y = £6 again. Tangency: MRS = 2(6)/6 = 2 = p_x/p_y. So C = (6, 6).

Step 3 — The total effect. Coffees rise from 3 to 6, a total effect of +3.

Step 4 — Build the Slutsky compensated budget. Ask what income would let you just afford the old bundle A at the new prices: M_s = 2(3) + 1(6) = £12. Draw the compensated line at the new slope (−2) through A: 2x + y = 12.

Step 5 — The substitution effect (A → B). Re-optimise on the compensated line. Two-thirds of £12 on coffee at £2 each gives x = (2/3)(12)/2 = 4, and y = £4. Check: MRS = 2(4)/4 = 2, matching the new price ratio, so B is a genuine tangency. B = (4, 4), and coffees rise from 3 to 4 — a substitution effect of +1.

Step 6 — The income effect (B → C). Restore the £6 of purchasing power you took away, moving from the compensated line out to the real new budget. Coffees rise from 4 to 6 — an income effect of +2. And the pieces close: +1 + 2 = +3, exactly the total.

Step 7 — Interpret. Both effects push the same way, so coffee is a normal good here: cheaper coffee makes you substitute toward it (+1), and the price cut leaves you richer, which you partly spend on still more coffee (+2). Notice the income effect is the larger piece — most of the extra coffee is a wealth effect, not a pure price effect. Had coffee been inferior, Step 6 would have gone the other way and shrunk the total.

Struggling to place the compensated budget line? That one construction is the whole method — set it correctly and the substitution and income effects fall straight out; slip, and the entire diagram misreads. It is exactly what a one-on-one intermediate microeconomics tutor drills with you until it is automatic. Book a trial session.

Practice

Q1. Preferences are U = xy, income is M = £12, p_y = £1, and p_x falls from £3 to £2. Find the total effect on x and split it into the Slutsky substitution and income effects.

Q2. Preferences are U = xy², income M = £18, p_y = £1, and p_x falls from £4 to £2. (Now only one-third of the budget goes on x.) Decompose the change in x.

Q3. Preferences are U = xy, income M = £12, p_y = £1, but now p_x rises from £1 to £2. Decompose the change in x, and state the sign of the substitution effect.

Answers. For U = xy, demand is x* = M/(2p_x); for U = xy², one-third of income goes on x, so x* = (1/3)(M/p_x). In each case the Slutsky income is M_s = p_xⁿᵉʷ·x_A + p_y·y_A, and B is the demand at the new prices on that income.
Q1: x_A = 2, x_C = 3, total +1. M_s = 2(2) + 6 = 10, so x_B = 10/4 = 2.5. Substitution +0.5, income +0.5.
Q2: x_A = 1.5, x_C = 3, total +1.5. M_s = 2(1.5) + 12 = 15, so x_B = (1/3)(15)/2 = 2.5. Substitution +1, income +0.5.
Q3: x_A = 6, x_C = 3, total −3. M_s = 2(6) + 6 = 18, so x_B = 18/4 = 4.5. Substitution −1.5, income −1.5. The substitution effect is negative — the price rose, and the substitution effect always moves opposite to the price.

Key takeaways

• A price change bundles two forces: a substitution effect (the good’s relative price changed) and an income effect (your purchasing power changed). The decomposition separates them.
• The substitution effect always moves opposite to the price — cheaper means more, dearer means less. It is the one guaranteed sign in demand theory.
• The income effect has no fixed sign: it reinforces for a normal good, opposes for an inferior good, and — only when it opposes and dominates — produces a Giffen good.
• total effect = substitution effect + income effect, and the two always add back to the observed change in quantity.
• Slutsky compensation keeps the original bundle affordable; Hicks keeps the original utility. Slutsky uses only observable prices and quantities, which is why it links to real data and the Slutsky equation.

Why New York students choose our intermediate microeconomics tutoring

• Course-matched tutors: whether your intermediate micro sequence sits at NYU, Columbia, Yale or Princeton, our tutors teach the decomposition in the notation and convention your professor uses — Slutsky or Hicks, graphical or the full Slutsky equation.
• Built for the problem sets: sessions work through the exact style of compensated-budget question your course sets, so the construction is automatic before the midterm, not improvised during it.
• Theory tied to the maths: our tutors move fluently between the diagram, the demand functions and the algebra, because intermediate exams reward the student who can connect all three rather than memorise one.

FAQ

Q: What is the difference between the income and substitution effects?
A: The substitution effect is the change in quantity caused purely by the change in relative prices, holding your real position fixed. The income effect is the change caused purely by the change in purchasing power, holding the new relative prices fixed. Together they make up the total effect of a price change.

Q: Does the substitution effect always have the same sign?
A: Yes. The substitution effect always moves opposite to the price change: a price fall raises the quantity, a price rise lowers it. This is guaranteed, because along the compensated budget line the consumer re-optimises toward the good that has become relatively cheaper.

Q: What is the difference between Slutsky and Hicks compensation?
A: Slutsky adjusts income so the consumer can just afford the original bundle at the new prices; Hicks adjusts income so the consumer reaches the original level of utility. Slutsky needs only observed prices and quantities; Hicks needs the indifference curve. They give the same qualitative story and, for small price changes, almost the same numbers.

Q: What makes a good a Giffen good?
A: A Giffen good is an inferior good whose income effect is strong enough to outweigh its substitution effect, so demand rises when its price rises. It requires a good that takes up a large share of the budget and has few substitutes — the classic historical example is a staple food.

Q: Why does the income effect on the other good sometimes offset it exactly?
A: For Cobb–Douglas preferences, spending on each good is a fixed share of income, so a change in the price of x leaves total spending on y unchanged. The substitution and income effects on y are then equal and opposite, and the total effect on y is zero — a useful check in problem sets.

Q: Is this the same as the Slutsky equation?
A: It is the graphical version of it. The Slutsky equation writes the same split algebraically — the total price response equals the substitution term minus the quantity bought times the income response. The diagram shows you what each term means before you use the formula.

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