Income and substitution effects

A demand function was derived in the previous lesson at one set of prices, and the interesting question is what happens when a price moves.

The answer is not one thing but two, and they can pull in opposite directions. Raise the price of a good and it becomes expensive relative to everything else, which pushes the consumer away from it. It also makes the consumer poorer in the only sense that matters, since the same income now commands a smaller set of bundles. Those are different forces with different causes, and separating them is what turns "demand curves slope downwards" from a plausible assertion into a result with a stated exception.

From demand function to demand curve

The Cobb-Douglas solution from the previous lesson, $x^{*} = a m / ((a + b) p_{x})$ , is already a demand curve: fix income and the other price, vary $p_{x}$ , and plot. It is a rectangular hyperbola, falling steeply at low prices and flattening out, and it never touches either axis.

Two conventions come with the picture and both trip people up. First, economists draw price on the vertical axis and quantity on the horizontal, which is the inverse of the functional relationship being described. Alfred Marshall did it that way in 1890 and nobody has managed to change it since, so a "demand curve" is really the graph of the inverse demand function $p (x)$ . Second, a movement along the curve is a response to that good's own price, while a shift of the whole curve is a response to anything else: income, another price, tastes, the number of buyers. The distinction is entirely a consequence of what was held fixed when the curve was drawn.

Two effects inside one change

Suppose $p_{x}$ rises. The budget line pivots inwards about the $y$ intercept, and the new optimum sits somewhere on the flatter line. Getting there involves two conceptually separate moves.

The substitution effect is the change in the bundle caused purely by the new relative price, holding the consumer's real position constant. It is always negative for a price rise, and this is not an assumption but a theorem, which is the single most robust result in consumer theory. The income effect is the change caused by the loss of purchasing power, holding relative prices at their new level. Its sign is not determined: it depends on whether the good is one people buy more of as they get richer.

Splitting the total into two parts requires deciding what "holding real income constant" means, and there are two respectable answers.

The Slutsky decomposition

Eugen Slutsky proposed the operational one in 1915: compensate the consumer with exactly enough extra income to afford the original bundle at the new prices. That bundle is still on the table, so in a defensible sense they have not been made poorer, and whatever they do differently is a response to relative prices alone.

The virtue of this definition is that it can be computed from observable quantities: the original bundle and the new prices are both known. Its cost is that the compensated consumer is in fact slightly better off than before, because with the price ratio changed they can usually do better than the old bundle.

Example. A consumer has $u = \sqrt{x y}$ with income £120, $p_{x} = £ 3$ and $p_{y} = £ 2$ . The price of $x$ rises to £6. Decompose the change in $x$ by Slutsky's method.

Cobb-Douglas with equal exponents spends half the income on each good, so initially $x = 0.5 \times 120 / 3 = 20$ and $y = 0.5 \times 120 / 2 = 30$ . After the rise, $x = 0.5 \times 120 / 6 = 10$ . The total effect is a fall of 10 units.

Now compensate. The original bundle $(20, 30)$ costs $6 \times 20 + 2 \times 30 = £ 180$ at the new prices, so hand the consumer £180. Their demand becomes $x = 0.5 \times 180 / 6 = 15$ . The substitution effect is $15 - 20 = - 5$ : even fully able to afford the old bundle, they move away from the now-expensive good. Taking the £60 back gives the income effect, $10 - 15 = - 5$ . The two are equal here, which is a Cobb-Douglas coincidence rather than a general fact, and they sum to the total of $- 10$ .

Now you. Same consumer, same starting point, but now $p_{x}$ falls from £3 to £2. Decompose the change.

Answer

New demand is $x = 0.5 \times 120 / 2 = 30$ , so the total effect is $+ 10$ . The original bundle now costs $2 \times 20 + 2 \times 30 = £ 100$ , so the compensated income is £100 and compensated demand is $0.5 \times 100 / 2 = 25$ . The substitution effect is $+ 5$ and the income effect is $+ 5$ . Both work in the same direction, because $x$ is a normal good here.

The Hicksian alternative

John Hicks, in Value and Capital in 1939, defined the compensation differently: give the consumer enough income to reach the original indifference curve at the new prices. That is compensation to constant utility rather than constant purchasing power, and it is the cleaner theoretical object, because the resulting substitution effect is a pure movement along one indifference curve.

Its cost is the mirror image of Slutsky's. The original utility level is not observable, so a Hicksian decomposition can be computed only when the utility function is known, which in practice means never outside a textbook. Slutsky's version is what empirical work uses, and the two agree in the limit of a small price change, since both compensations shrink to the same thing.

Example. Redo the price rise from £3 to £6 the Hicksian way, for the same consumer with $u = \sqrt{x y}$ and income £120.

The starting utility is $\sqrt{20 \times 30} = 24.495$ . Minimising expenditure at the new prices subject to reaching that utility gives, for this function, $x = u \sqrt{p_{y} / p_{x}} = 24.495 \sqrt{2 / 6} = 14.14$ and $y = 42.43$ , costing £169.71. So the Hicksian substitution effect is $14.14 - 20 = - 5.86$ , and the income effect is $10 - 14.14 = - 4.14$ . Same total of $- 10$ , split in different proportions, and the Hicksian compensation is £169.71 against Slutsky's £180. Slutsky overcompensates, exactly as expected, because the old bundle is no longer the cheapest way to reach the old utility.

Now you. Do the Hicksian decomposition for the price fall from £3 to £2, and compare the compensation with the £100 Slutsky figure.

Answer

The starting utility is still 24.495. At the new prices, $x = 24.495 \sqrt{2 / 2} = 24.495$ and $y = 24.495$ , costing £97.98. The substitution effect is $24.495 - 20 = + 4.50$ and the income effect is $30 - 24.495 = + 5.51$ , summing to the total of $+ 10$ . The Hicksian compensation of £97.98 is below Slutsky's £100 again: for a price fall the compensation is negative in both cases, and Slutsky takes away less than Hicks does.

Normal, inferior and Giffen

Now classify by the sign of the income effect. A good is normal when demand rises with income, and inferior when it falls. Inferiority is entirely ordinary and not a judgement about quality: bus travel, own-brand food, instant coffee and rented rooms all show it, because a richer household substitutes towards cars, brands, and space. Engel's law, established by Ernst Engel from Belgian household budgets in 1857, is the oldest empirical regularity in the subject and says the same thing about food as a share of spending.

The Slutsky equation states the decomposition in derivative form. For a small change in $p_{x}$ ,

\frac{\partial x}{\partial p_{x}} = {\frac{\partial x}{\partial p_{x}} |}_{comp} - x \cdot \frac{\partial x}{\partial m}

The first term is the substitution effect, always negative. The second is the income effect, and the multiplier $x$ in front of it is doing real work: the income effect is scaled by how much of the good the consumer already buys. Raising the price of paper clips by ten per cent makes nobody poorer in any meaningful sense; raising the price of rent by ten per cent, for a household spending half its income on rent, does.

Three cases follow. For a normal good, $\partial x / \partial m > 0$ , the second term is negative, and both effects push demand down. For an inferior good the second term is positive and partly offsets the substitution effect, so demand still falls but by less. And if the good is inferior and takes a large enough share of spending that $x \cdot \partial x / \partial m$ exceeds the substitution term in magnitude, the total is positive: demand rises when the price rises. That is a Giffen good, and the definition makes clear it is not an exotic taste but an arithmetic possibility requiring two conditions at once.

Example. For a good with $x = 30$ , a compensated response of $- 4$ units per pound of price, and $\partial x / \partial m = 0.2$ units per pound of income, find the total price response and classify the good.

Substituting, $\partial x / \partial p_{x} = - 4 - 30 \times 0.2 = - 10$ units per pound. The good is normal, since the income derivative is positive, and the two effects reinforce: the income effect more than doubles the response.

Now you. A good has $x = 50$ , a compensated response of $- 4$ units per pound, and $\partial x / \partial m = - 0.1$ . Find the total response and classify it.

Answer

$\partial x / \partial p_{x} = - 4 - 50 \times (- 0.1) = - 4 + 5 = + 1$ unit per pound. Demand rises with price, so the good is Giffen. It is inferior, and the quantity bought is large enough for the income effect to swamp the substitution effect.

Was a Giffen good ever seen?

For a century the honest answer was no. Marshall attributed the idea to Robert Giffen in the 1895 edition of his Principles, claiming that a rise in the price of bread so reduced the real incomes of the poorest labourers that they cut back on meat and ate more bread. No data supporting that claim has ever been found, and the Irish potato famine version that appears in textbooks is folklore: the potato crop failed, so the story requires an upward-sloping demand curve to be inferred from a collapse in supply, which it cannot be.

The clean evidence arrived in 2008. Robert Jensen and Nolan Miller ran a randomised field experiment in two Chinese provinces, giving poor households vouchers that subsidised the price of their staple: rice in Hunan and wheat in Gansu. Among the poorer Hunan households, cutting the price of rice reduced the amount of rice bought, and withdrawing the subsidy raised it. That is a Giffen response, measured under a deliberate price manipulation rather than inferred from history.

The mechanism is exactly the one the algebra predicts. These households got the bulk of their calories from rice, so rice spending dominated their budget. A rice subsidy freed up enough money to buy meat and vegetables, which are better food, and the calories those provided displaced rice. The two conditions were both met: rice was inferior for these households, and it was a large enough share of spending for the income effect to win. Notably the very poorest households did not show the effect, because they had no margin to substitute towards anything at all, and neither did the richer ones, for whom rice was too small a share. Giffen behaviour needed a narrow band of circumstances, which is precisely why it took so long to find.

What the decomposition buys

The law of demand, that quantity demanded falls when price rises, is therefore not a law. It is a theorem with a stated exception, and the exception requires a strongly inferior good taking a large budget share. Almost nothing meets both conditions, which is why demand curves slope downwards in practice while the theory refuses to promise it.

The decomposition also does real work outside the classification. Any policy that changes a price is doing both things at once, and the two have different consequences. A carbon tax raises the price of fuel, which substitutes households towards insulation and away from driving, and simultaneously makes them poorer. A rebate returning the revenue as a lump sum cancels most of the income effect while leaving the substitution effect entirely intact, since the relative price is unchanged by the rebate. That is the whole design argument for revenue-neutral carbon pricing, and it is a Slutsky decomposition applied to policy.

What is still missing is a measure of how big these responses are. Saying that demand fell by ten units when the price rose by three pounds depends on the units of both, so it cannot be compared across goods, countries or currencies. Fixing that is the next lesson, and the fix turns out to matter for something as basic as whether a price rise raises revenue.