Microeconomics · Complete guide
Income and substitution effects, from scratch
Why do people buy more of something when it gets cheaper? Two different reasons, tangled together. This guide pulls them apart on a live diagram, first the way Hicks did it, then the way Slutsky did, then for normal, inferior and Giffen goods, and finally in the one equation that sums it all up. Every step has a picture you can move and every idea has exercises.
Start here
The big idea: two forces, one change
Say paneer drops from ₹400 a kilo to ₹200, and you start buying more of it. Ask yourself why, and you'll find two separate reasons hiding inside that one decision.
Substitution effect
“It's cheaper than the alternatives now.”
Paneer got cheaper relative to chicken, eggs and everything else. Even if you were no richer at all, you'd swap some of those for paneer, because the trade-off between them has changed. This effect always pushes you towards the good that became relatively cheaper.
Income effect
“My money goes further now.”
You buy the same kilo and have ₹200 left over. Your salary hasn't changed, but your real income, what it can buy, has gone up. What you do with that depends on the good: more of a normal good, less of an inferior one.
The whole topic in one line
The price effect is the total change in the quantity of the good whose price changed.
Key idea
You only ever see the price effect: the shop records that you bought more paneer. The two parts are thought experiments. The whole topic is about one question: how do you split an observed change into those two parts? Hicks and Slutsky give two different answers.
Refresher
The two tools you need
Everything below uses two lines on one diagram. If these feel shaky, spend a minute here first.
The budget line: what you can afford
All bundles that cost exactly your income: . It hits the Y-axis at and the X-axis at , and its slope is , the relative price.
- Income changes shift it in or out, parallel: the relative price is unchanged.
- A change in the price of X makes it pivot around the Y-intercept, since hasn't moved.
Indifference curves: what you want
Each curve joins bundles the consumer likes equally; higher curves are better. Their slope is the marginal rate of substitution (MRS): how much Y you'd give up for one more X.
The best choice: tangency
The consumer picks the point where the budget line just touches the highest reachable curve, where . Want to play with this first? Try the consumer choice diagram.
The trick behind everything that follows
A pivot changes the relative price; a parallel shift changes only income. So any price change can be rebuilt as a pivot (pure substitution) plus a parallel shift (pure income). Hicks and Slutsky only disagree about exactly where to put the pivot.
Interactive
The decomposition lab
This is the heart of the guide. Walk through the four steps with Next step, first with Hicks selected, then again with Slutsky, then Both. Every point is computed exactly for a Cobb-Douglas consumer, so the numbers on the right are the real answers, not readings off a sketch. Then try a price rise, and change α to see how a good's share of the budget drives the size of the income effect.
Price effect +10
Original curve I₁Original budget line
Step 1 of 4 · Hicks
Start at E₁, the best bundle the consumer can afford
With income ₹144, X at ₹9 and Y at ₹2, the consumer chooses E₁ = (8, 36): the one point where the budget line just touches the highest indifference curve it can reach, I₁. Any other affordable bundle sits on a lower curve.
The maths, with your numbers (Py = ₹2)
Before: demand for X0.5×144/9 = 8
After: demand for X0.5×144/4 = 18
Price effect+10
Hicks compensated X12
Hicks: substitution, income+4, +6
Slutsky income₹104
Slutsky compensated X13
Slutsky: substitution, income+5, +5
Method one
Hicks: hold utility constant
Hicks asks: how much income would I have to take away so the consumer ends up exactly as well off as before the price change? “Real income constant” means utility constant.
The recipe
- Find the original choice E₁ on curve I₁, and the new choice E₂ after the price change. E₁ → E₂ is the price effect.
- Draw a line with the new slope (new prices), shifted parallel until it is just tangent to the original curve I₁.
- Where it touches I₁ is E₃. E₁ → E₃ is the substitution effect: a slide along one indifference curve, purely because the relative price changed.
- E₃ → E₂ is the income effect: a parallel jump from the compensated line back out to the actual new line.
The income removed in step 2 has a name: the compensating variation (CV). It is the difference between actual income and what it now costs to reach the old utility level.
In symbols
e(·) is the expenditure function: the least it costs to reach utility U₀ at given prices. h(·) is Hicksian (compensated) demand: the bundle that achieves it.
Why Hicks is the “pure” substitution effect
The consumer never leaves I₁ during the substitution step, so none of it can be caused by being better or worse off. The catch: to draw that tangency you need to know the indifference curve, which nobody can observe directly.
Method two
Slutsky: hold purchasing power constant
Slutsky asks a more practical question: how much income would I have to take away so the consumer can just afford the bundle they bought before? “Real income constant” means the old bundle stays exactly affordable.
The recipe
- Find E₁ and E₂ as before. E₁ → E₂ is the price effect.
- Draw a line with the new slope that passes through E₁ itself.
- The consumer won't stay at E₁ on this line: at the new relative price, a nearby bundle is better. They move to E₃, where this line touches a slightly higher curve than I₁. E₁ → E₃ is the substitution effect.
- E₃ → E₂ is the income effect.
The best part: the income adjustment needs no utility function at all, only what was bought and how much the price moved.
The exam shortcut
Always the old quantity x₁, never x₂. For a price fall ΔPₓ is negative, so income is taken away; for a rise, it is handed over.
Why Slutsky's consumer ends up better off
E₁ is affordable on Slutsky's line, but the consumer chooses E₃ instead, so they must like E₃ at least as much, and at a different relative price, strictly more. That little utility gain is why Slutsky's substitution effect is slightly “contaminated” by income, and why it differs from Hicks's.
Compare
Hicks and Slutsky, side by side
The two methods agree on the price effect, which is a fact. They disagree only on where to draw the line between substitution and income, because they define “the same real income” differently.
For a price fall, Slutsky takes away less income than Hicks (₹40 versus ₹48 here), so Slutsky's consumer sits on a higher curve. With a normal good, that means a bigger substitution effect under Slutsky. For a price rise, Slutsky hands over more than Hicks, and again leaves the consumer slightly better off.
As the price change gets smaller, the gap between the two shrinks faster than the effects themselves. For a tiny change they coincide, which is exactly the case the Slutsky equation in chapter 8 describes.
| Hicks | Slutsky | |
|---|---|---|
| Real income held constant means | The same utility (the original curve I₁) | The same purchasing power (the original bundle E₁ stays affordable) |
| Compensated budget line | New slope, tangent to I₁ | New slope, passing through E₁ |
| Where the consumer lands | On the original curve I₁ | On a slightly higher curve than I₁ |
| Income adjustment | Compensating variation, CV = M − e(p₁, U₀) | ΔM = x₁ × ΔPₓ |
| What you need to know | The indifference map (utility function) | Only the old quantities and the prices: observable |
| Substitution effect, price fall, normal good | Smaller | Larger |
| Also known as | Compensating-variation method | Cost-difference method |
Three cases
Normal, inferior and Giffen goods
The substitution effect never changes sign: a fall in a good's price always pushes the consumer towards it. All the variety comes from the income effect. Switch between the three cases and both methods below and watch which arrows flip.
Illustrative curves: the price of X halves, from ₹2 to ₹1, with income ₹100 and Y at ₹1. Arrows show direction and relative size.
| After a fall in the price of X | Substitution | Income | Price effect | Demand curve |
|---|---|---|---|---|
| Normal good | + | + | + (reinforced) | Slopes down |
| Inferior, not Giffen | + | −, smaller | + (weakened) | Slopes down |
| Giffen good | + | −, larger | − (reversed) | Slopes up |
Every Giffen good is inferior, not the other way round
For demand to fall when price falls, the income effect has to be negative (inferior) and bigger than the substitution effect. That needs three things together: the good is strongly inferior, it takes a large share of the budget (so a price change is a big change in real income), and it has few close substitutes (so the substitution effect is small).
Don't confuse Giffen with Veblen
Both can show demand rising with price, but for completely different reasons. A Giffen good is a cheap staple, and the upward slope comes from the income effect. A Veblen good is a luxury bought partly because it is expensive (designer bags, prestige watches): the price itself is part of what's being bought, which lies outside the standard income-substitution framework.
The price effect of a Giffen good is negative only over some range of prices; nobody lives on one good forever.
The algebra
The Slutsky equation
Everything on the diagrams can be written in one line. For a very small change in the price of X, the Slutsky equation splits the slope of the ordinary (Marshallian) demand curve into its two parts:
The Slutsky equation
h is the Hicksian (compensated) demand. For an infinitesimal price change, Hicks's and Slutsky's compensations coincide, so the equation belongs to both.
First term
∂h/∂Pₓ is never positive
Compensated demand always slopes down: with utility held fixed, a higher price can only make you buy less. This is the substitution effect, and its sign never depends on the good.
Second term
Why it's multiplied by x
A ₹1 rise in Pₓ costs you ₹1 on each unit you buy, so it cuts your real income by x rupees. Multiply that by how demand responds to income, ∂x/∂M, and you have the income effect.
Put together
When can demand slope up?
Only if ∂x/∂M < 0 (inferior) and x is large enough that the income term outweighs the substitution term. For a normal good both terms are negative, so the law of demand must hold.
Check it on the lab's example
With , α = 0.5, M = 144, Pₓ = 9 (so x = 8):
And indeed . For Cobb-Douglas the two halves are always in the ratio (1 − α) : α, which is why a bigger budget share α gives a bigger income effect.
The same thing in elasticities
Divide through by x/Pₓ and the equation becomes a statement about elasticities:
ε is the ordinary price elasticity, εʰ the compensated one, the budget share, and η the income elasticity. This is the cleanest way to see why Giffen goods need a big budget share: the income term is scaled by s.
Practise
Exercises, with answers
Two full problems with worked solutions, a quick drill on classifying goods, and ten concept checks. Your score updates as you go. Use the lab above as a calculator only after you've had a go by hand.
A. Work it out
Type numbers with signs (a fall in quantity is negative). Answers are accepted to two decimal places.
Problem 1 · a price fall
Riya's utility is . Her income is ₹160 and the price of Y is ₹4. The price of X falls from ₹8 to ₹2. Work out the price effect on X, then split it both ways.
Problem 2 · a price rise
Arjun's utility is , his income is ₹100 and Y costs ₹1. The price of X rises from ₹1 to ₹4. This time the compensation is income handed over, not taken away.
B. Name the good
Each row is the effect of a fall in the price of X on the quantity of X bought.
Substitution +6Income +3
Substitution +5Income −2
Substitution +4Income −7
Substitution +3Income 0
Substitution +3Income ?Price effect −2
C. Concept checks
One attempt each. Read the explanation even when you get it right.
1When a good's own price falls and real income is held constant, the substitution effect on that good's quantity is:
2In the Hicksian method, “real income held constant” means:
3A consumer buys 20 units of X at ₹10. The price falls to ₹6. How much income does Slutsky's method take away?
4Which statement about Giffen goods is correct?
5For a fall in the price of a normal good, compared with Hicks's substitution effect, Slutsky's is:
6In the Slutsky equation , the term is:
7A price cut raises quantity demanded by 4 units, even though the income effect on its own would have reduced it by 3. The good is:
8After a price fall, the income that must be taken away to neutralise the income effect is:
9Graphically, the Hicksian substitution effect is a movement:
10Which condition makes a good more likely to be Giffen?
Revise
One-page revision sheet
Definitions
- Price effect: total change in quantity after a price change.
- Substitution effect: the change due only to the new relative price, real income held constant.
- Income effect: the change due only to the change in real income.
Hicks in one breath
New-slope line tangent to the old curve I₁. Hold utility. Income adjustment = compensating variation. SE is a slide along I₁.
Slutsky in one breath
New-slope line through the old bundle E₁. Hold purchasing power. Income adjustment = x₁ × ΔPₓ. Lands on a slightly higher curve.
Signs, price fall
Normal: + + = +. Inferior: + − = + (|SE| > |IE|). Giffen: + − = − (|IE| > |SE|). The substitution effect is always +.
Cobb-Douglas shortcuts
, , . Always a normal good.
Slutsky equation
. First term ≤ 0 always; the second decides normal, inferior or Giffen.
Watch out for
The mistakes examiners are fishing for
- Saying the substitution effect is negative for inferior or Giffen goods. Its sign never depends on the good; only the income effect does.
- Using x₂ in Slutsky's compensation. It is always the original quantity: ΔM = x₁ × ΔPₓ.
- Drawing Hicks's line tangent to the new curve. It must touch the original curve I₁; that's what “utility held constant” means.
- Calling every inferior good Giffen. Giffen needs the income effect to outweigh substitution. Most inferior goods still have downward-sloping demand.
- Forgetting the signs flip for a price rise. Then the substitution effect reduces quantity, and compensation means handing income over.
Try it on real questions
Seven past-paper questions on exactly this
From CUET-PG, UGC NET, JNU and IIT JAM: signs of the two effects, Giffen goods from three angles, and a numeric compensating variation.
These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.
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