Income and substitution effects: splitting one price change in two

CB2 · Business Economics · 11 min read

Two things happen, not one

A price falls and you buy more. That looks like one event, but it is two, and they have nothing to do with each other.

First, the thing has become cheaper relative to everything else. Even if you were somehow no better off overall, you would now lean towards it and away from its alternatives, purely because the exchange rate between them has moved. That is the substitution effect.

Second, you are better off. Something you were already buying now costs less, so the same money goes further. That extra purchasing power gets spent on something, and some of it may well come back to the good whose price fell. That is the income effect.

The two added together are what you observe. Neither can be observed on its own, which is why they have to be separated by a construction rather than by measurement.

Why the separation is worth the trouble

Because the two effects behave completely differently, and only one of them has a guaranteed direction.

The substitution effect always points towards the good that got cheaper. There is no exception and no class of good for which it fails. It follows from the shape of preferences: faced with a better exchange rate for something, you take more of it in place of what it displaces.

The income effect can point either way. For a normal good, being better off means buying more, so the income effect reinforces the substitution effect. For an inferior good, being better off means buying less of it, and the income effect works against the substitution effect.

So the law of demand — price down, quantity up — is not an axiom. It is a conclusion that holds whenever the substitution effect is not overwhelmed by a perverse income effect. Usually it is not, and demand curves slope down. Occasionally, in principle, it is, and then they do not. You cannot say any of that without the split.

The mechanism, drawn in your head

Picture a budget line: the straight line of bundles a given income will just buy at given prices. Its slope is the price ratio. The chosen bundle sits where the budget line touches the highest reachable indifference curve.

Now cut the price of one good. The budget line pivots outwards on that good's axis: the intercept on the other axis is unchanged, because the other price and the income are unchanged. Two things have altered at once — the slope, and how far out the line reaches.

The trick is to do them one at a time. Swing the line to the new slope, but pull it back in until it is only just good enough; then push it out to where it really is.

1. Swing and pull back. Give the consumer the new price ratio, but take away enough money that they are no better off than before. Whatever they choose now differs from the original bundle only because relative prices changed. That move is the substitution effect.
2. Give the money back. Restore the income. The further move from the compensated bundle to the actual new bundle is caused only by purchasing power, with relative prices now fixed. That is the income effect.

"No better off" has two definitions, and both are used

Step 1 needs a meaning for "no better off", and there are two defensible ones.

Slutsky compensation adjusts income so that the original bundle is just affordable at the new prices. The test is a purchasing test, and the arithmetic is a single multiplication, because you already know the original bundle.

Hicks compensation adjusts income so that the consumer stays on the original indifference curve. The test is a satisfaction test, and it needs the utility function, because you have to find the cheapest way onto a curve.

Slutsky is slightly generous — being able to afford the old bundle at better relative prices leaves you able to do better than before, so Slutsky compensation leaves the consumer on a higher indifference curve than Hicks. The Slutsky substitution effect is therefore the larger of the two, and the Slutsky income effect correspondingly the smaller. They are decomposing the same total, and both reach it.

For working by hand, use Slutsky. The worked example below does both so that you can see they bracket the same answer.

Giffen goods: how the exception is actually built

A Giffen good is one whose demand rises when its price rises. Two conditions must hold together, and the second is the hard one.

First, the good must be inferior, so that the income effect opposes the substitution effect. Second, the income effect must be large enough to win, which needs the good to absorb a big share of spending — because the income effect of a price change is roughly proportional to how much of your budget that good was eating.

Put those together and the only plausible setting is a household so stretched that one cheap staple takes most of its spending, with something better just out of reach. Cut the staple's price and the household can suddenly afford some of the better food; it switches, and ends up buying less of the staple than before. Raise the staple's price and the better food goes out of reach again; the household falls back on the staple and buys more. The demand curve slopes up over that stretch.

Note what is not required: nothing irrational, and no odd preferences. The behaviour comes entirely out of an income effect big enough to dominate. That is the point of the construction — it shows the law of demand is a result with conditions, not a law.

Where marks are lost

Treating "inferior" as "poor quality". It means only that demand for it falls as income rises. A good can be inferior for one household and normal for another, and the same good can be inferior at high incomes and normal at low ones. Nothing about the product itself decides it.

Getting the substitution effect's direction wrong. It is always towards the good that became relatively cheaper. If your answer has the substitution effect pointing away from a good whose price fell, there is an arithmetic error, not an interesting finding.

Confusing "inferior" with "Giffen". Most inferior goods are not Giffen. Being inferior only makes the income effect oppose; it does not make it win. Every Giffen good is inferior, and almost no inferior good is Giffen.

Compensating the wrong way. After a price fall, compensation means taking money away, because the old bundle has become cheaper. Adding money is the mistake, and it produces a substitution effect bigger than the total, which should be the alarm bell.

Questions students ask

If the income effect cannot be observed, is it real?

It is as real as the substitution effect, and neither is observable on its own — only the total is. The split is a thought experiment that tells you something the total cannot: whether the observed response was reinforced or resisted, and therefore how it would change if the household's circumstances changed. That is why it survives in a professional syllabus rather than only in a textbook.

Where does the demand curve come from in all this?

From repeating the exercise. Each price gives a new budget line, a new tangency and a new quantity of that good. Plot those quantities against their prices and you have traced the demand curve out of the choices rather than assumed it. Done that way, the slope of the demand curve is a result, and the income and substitution effects are the reason for it.

Does any of this matter for an actuary?

It is the cleanest available account of why a product's sales respond to its price the way they do, and it separates two responses that need different commercial answers. A fall in sales driven by substitution says a competitor is winning and the answer is about the product. A fall driven by an income effect says customers are poorer and the answer is about affordability. The arithmetic below is also the shortest route into the behavioural material, which is largely a catalogue of the places where this neat model does not describe what people actually do.

A worked example

An invented household, built for this page. A family in Indore budgets ₹2,400 a month for local travel and splits it between two ways of getting about: bus trips at ₹40 each and auto-rickshaw trips at ₹60 each. Their preferences are such that they always split the travel budget equally between the two — a standard and well-behaved case, chosen here because it makes every number exact. The prices and the budget are illustrative.

Before the fare cut

Half the budget on each means ₹1,200 on each.

Check the budget: 30 × 40 + 20 × 60 = 1,200 + 1,200 = ₹2,400. Exactly spent.

The change: the bus fare falls to ₹24

Still half the budget each, so still ₹1,200 on each.

Check: 50 × 24 + 20 × 60 = 1,200 + 1,200 = ₹2,400. Still exactly spent.

So the total effect on bus trips is 30 → 50, a rise of 20 trips. Auto trips are unchanged at 20. Now split those totals.

Step 1: the Slutsky substitution effect

Take away enough money that the old bundle (30 bus, 20 auto) is just affordable at the new prices:

So ₹480 is withdrawn. Check that figure independently: the saving on the old bundle is 30 × (40 − 24) = 30 × 16 = ₹480. The same number from a different calculation.

At the new prices on ₹1,920, half on each means ₹960 on each:

Relative prices alone moved the family 10 trips towards the bus and 4 trips away from the auto. Both point the way the substitution effect must: towards the good that got cheaper.

Step 2: the income effect

Give the ₹480 back, with prices now fixed at the new ones. The family moves from the compensated bundle to the actual one:

Both positive, so on these preferences both goods are normal: being better off buys more of each.

The decomposition, checked

Both rows add up. The auto row is the instructive one: the observed change is nil, and the decomposition still has two real effects in it — a substitution away from the auto of 4 trips, exactly cancelled by an income effect towards it of 4 trips. A student who looked only at the total would conclude that nothing happened to auto demand. Two things happened, and they happened to be the same size.

The same split by Hicks, for comparison

Hicks compensation holds satisfaction constant instead of purchasing power. With a budget split equally between two goods, satisfaction is measured by the product of the two quantities, so the original level is 30 × 20 = 600. The cheapest way to reach 600 at the new prices has the quantities in the ratio of the prices, which gives auto = 0.4 × bus, and so

Total 8.73 + 11.27 = +20, the same total as before, as it has to be. And the Slutsky substitution effect (+10) is the larger of the two substitution effects, which is the expected ordering: Slutsky compensation leaves the family able to reach 40 × 16 = 640, above the 600 Hicks holds them to, so it concedes more of the move to substitution.

What the figures would have to look like for a Giffen good

Nothing here is Giffen, and the arithmetic shows why. For bus trips the income effect is +10 — pointing the same way as substitution, so the total can only be positive. To get a Giffen result the income effect on bus trips would have to be negative (bus travel inferior: richer, so fewer bus trips) and bigger than 10 in size. That needs bus travel to take a very large share of the family's spending, because the income effect of a fare cut is roughly the saving multiplied by how strongly spending responds to income. Here the fare cut freed ₹480 out of a ₹2,400 budget. It is easy to see why the conditions are demanding, and why Giffen goods are a result about very poor households rather than a quirk of taste.

Where this fits in CB2

This page explains one idea from CB2 Business Economics. The full CB2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Other CB2 topics explained

• Price elasticity of demand, and what it decides about a price rise
• Profit maximisation and the shutdown decision, from one cost table
• Cournot and Bertrand: why the same two firms give two different answers
• Adverse selection and moral hazard, and what an insurer does about them
• National income and the multiplier: where the rest of the spending comes from
• The IS-LM model and crowding out, solved with numbers
• How monetary policy reaches the economy: the transmission mechanism

Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.