Consumer theory · Lesson 12 of 17

Income and substitution effects and the Slutsky equation

When the price of a good falls, two things happen at once: the good becomes cheap relative to the others, and she becomes better off. Economists pull the two apart because they can push in different directions. Doing it cleanly shows why demand curves slope down, and the one case in which one does not.

← Consumer theory moduleLesson 12 of 17 · about 45 minutes

A question to hold on to

A consumer’s preferences over commodities 1 and 2 are U(x₁, x₂) = min{x₁, x₂}. The prices are 1 and 2 and her income is 150. The price of commodity 2 falls to 1, with the price of commodity 1 and income unchanged. The substitution effect, according to Slutsky, of this price change on the optimal amount of commodity 1 is −1; 25; −25; or 0.

JNU 2012Solved at the end ↓

The idea

One price change, two effects

Suppose the price of x falls. Two things change for her.

Substitution effect. x is now cheaper relative to y, so she swaps some y for x. This works whatever her income.
Income effect. The same money now buys more, so she is better off, as if her income had risen. If x is a normal good she buys more of it; if it is inferior, less.

The two together are the price effect, the whole movement of her chosen bundle along the price consumption curve of Lesson 10. To separate them, take the price change in two steps: first change relative prices while giving or taking away just enough income to leave her no better or worse off (the substitution effect); then give the income back (the income effect).

Pause and predict

The price of x falls and x is a normal good. The substitution effect and the income effect on the quantity of x

The two methods

Two ways to compensate: Hicks and Slutsky

The only question is what “no better or worse off” means.

Hicks: keep her on her original indifference curve. After a price rise this means notionally adding income so that she can still reach the old curve; after a price fall, notionally taking income away (UOH 2012).
Slutsky: keep the old bundle just affordable, so the compensated budget line passes through it with the new slope. After a price fall that means taking away income until she could just buy what she bought before (UOH 2014).

In both, the compensated budget line has the new price ratio, so the substitution effect is the move along her old curve (Hicks) or to the point where the new-slope line through her old bundle touches a curve (Slutsky). The income effect is then the move from the compensated point to the final point. The Slutsky line passes through the old bundle with a different slope, so it cuts her old curve there, and with it she can reach a slightly higher curve. The two methods therefore give similar but not identical numbers.

The three steps

price effect x^{1} - x^{0} = substitution x^{c} - x^{0} + income x^{1} - x^{c}

x⁰ is the old bundle, x^c the bundle at the new prices after compensation, x¹ the final bundle. Only the compensation differs between the methods.

The textbook diagram has three points. A UOH question asked about it: a price fall for Y takes her from A (on I₁) to C (on I₂). The dashed line parallel to the new budget line, tangent to I₁, touches at B. Then A to C is the price effect; A to B, along the old curve, is the substitution effect(UOH 2019); B to C is the income effect. If Y₂ < Y_s, that is, the final quantity is less than at the compensated point B, the income effect on Y is negative, which is what it means for Y to be an inferior good (UOH 2019).

Interactive

Try it: the four-step decomposition

Step through the diagram: start, price change, compensation, then the split. Use the Hicks and Slutsky buttons to see the compensated point move, and the sliders to change the price change and how much of her income goes to x. The numbers are exact.

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 X

x_{1} = \frac{α M}{P _{x 0}}

0.5×144/9 = 8

After: demand for X

x_{2} = \frac{α M}{P _{x 1}}

0.5×144/4 = 18

Price effect

x_{2} - x_{1}

+10

Hicks compensated X

x_{h} = x_{1} (\frac{P _{x 0}}{P _{x 1}})^{1 - α}

Hicks: substitution, income

x_{h} - x_{1}, x_{2} - x_{h}

+4, +6

Slutsky income

M_{s} = M + x_{1} (P_{x 1} - P_{x 0})

₹104

Slutsky compensated X

x_{s} = \frac{α M _{s}}{P _{x 1}}

Slutsky: substitution, income

x_{s} - x_{1}, x_{2} - x_{s}

+5, +5

The equation

The Slutsky equation

For a small change the same split is a formula. With h the compensated (Hicksian) demand, which Lesson 14 defines properly:

The Slutsky equation

total \frac{\partial x}{\partial p _{x}} = \leq 0 substitution \frac{\partial h}{\partial p _{x}} - income x \frac{\partial x}{\partial m}

Total effect = substitution effect + income effect. The first term is never positive; the second is positive for an inferior good and negative for a normal good.

The total change is exactly the sum of the two, not more or less (JNU 2012). Three things follow.

The substitution effect is negative for every good, normal, inferior or Giffen (UOH 2016): a dearer good, with her no better off, is bought no more.
For a normal good both effects point the same way, so demand slopes down, and strongly if both are strong (UOH 2016).
For an inferior good they conflict. Demand still slopes down if the substitution effect is the stronger. Demand slopes down in general when the substitution effect dominates the income effect (UOH 2020).

A rule of thumb the lab shows: the bigger the good’s share of the budget, the bigger its income effect, because the price change moves her real income more.

Special tastes

Complements, quasi-linear tastes and straight lines

Three standard cases show the two effects in isolation.

Perfect complements: no substitution effect. She wants the goods in a fixed ratio, so changing the relative price does not change the ratio. With U = min{x₁, x₂}, prices 1 and 2 and income 150, she buys 50 of each. When the price of commodity 2 falls to 1, the old bundle costs 100 at the new prices, so Slutsky compensation takes away Rs 50, and with Rs 100 she still buys (50, 50): substitution effect 0. The income effect is the whole change: with Rs 150 she buys 75 of each, so it is +25 (JNU 2012).
Quasi-linear: no income effect on the nonlinear good, once she buys some of the other. With U = v(x₁) + x₂ and a very high income the demand for good 1 depends on prices alone, so the substitution effect is negative and there is no income effect (JNU 2012).
Perfect substitutes: with U = x₁ + x₂ she buys only the cheaper good. If a price change leaves the same good cheaper, her old bundle is still the best, so there is no substitution effect; there is an income effect (JNU 2017, 2021). The substitution effect shows up only when the change reverses which good is cheaper.

Why the substitution effect is a kink effect

The substitution effect measures how far she moves around a fixed curve when the slope of the budget line changes. A right-angled curve gives no movement at all, a straight line gives all of it at once, and smooth curves give something in between.

The exception

Giffen goods

A Giffen good is one whose demand rises when its price rises (UOH 2010). The Slutsky equation shows what it takes: the income term must be positive and bigger than the (negative) substitution term. So a Giffen good must be inferior, and strongly so. The reverse is false: most inferior goods are not Giffen. JNU 2019 tested exactly this. “A good is inferior only if its quantity falls as its price falls” is false, while “if she demands less as the price falls, the good is inferior” is true. In short: Giffen implies inferior; inferior does not imply Giffen.

Two goods give two more facts. If Joe always buys more of Y when the price of Y rises, Y is Giffen and therefore inferior. Since the shares times the income elasticities add up to one (Lesson 11), the other good X must then be normal (JNU 2015). And when X is normal, Y inferior and the price of Y rises, her demand for Y is pulled two ways: a substitution effect away from Y and an income effect towards it, so the net effect is not determined by the information given (JNU 2017).

The lab uses the calorie floor of Lesson 11. While the floor binds, rice is inferior, and she is pinned to the point where her budget line meets the 30-calorie line. A dearer rice means less real income and she has to put even more of it on the cheapest calories. For rice prices between about Rs 0.71 and Rs 2 she buys more rice the dearer it is: 8.75 units at Rs 1, about 9.3 at Rs 1.5 and 10 at Rs 2. Past Rs 2 she cannot afford the floor.

Rice gives 3 calories a unit and meat 1; rice costs Rs 1 (before) and meat Rs 3. She must eat at least 30 calories and otherwise likes rice and meat in the proportion 0.3 to 0.7. Raise the price of rice and watch what she does.

Price of rice before, Rs1Price of rice after, Rs1.5Income, Rs20

○ before ● afterRed: 30 caloriesDashed: the budget before; dotted: compensated

Demand for rice (income Rs 20)

The two choices

At Rs 1 she buys 8.75 of rice; at Rs 1.5 she buys 9.33. The price went up and so did the quantity: the demand curve slopes up here, a Giffen good.

Slutsky split of the change in rice

Total 0.58 = substitution 0 + income 0.58. After compensation (income Rs 24.38) the old bundle is still affordable, and with the floor binding it is still her best choice, so the substitution effect is zero. The income effect goes against the price change and is the whole story: rice is inferior while the floor binds.

Why this is rare

The good must be inferior and its income effect must beat the substitution effect. Here the staple takes most of the budget and she is stuck on a tight calorie line, so a dearer staple lowers real income and she has to buy even more of the cheap calories. Beyond Rs 2 she cannot reach the floor at all.

One thing to notice. In this example the substitution effect is exactly zero, because the compensated line passes through her old bundle and she is pinned there by the calorie line. The whole movement is the income effect, which here is large and goes against the price change. That is a special case; in the usual smooth textbook picture the substitution effect is negative and the income effect has to be larger.

Watch for

Traps

Watch out for

The mistakes examiners are fishing for

Calling an inferior good Giffen. It needs the income effect to be bigger than the substitution effect, which is rare.
Getting the direction of Hicks compensation wrong. Price rise: add income. Price fall: take it away.
Confusing the two methods. Hicks holds utility; Slutsky holds the old bundle affordable.
Thinking the income effect always raises the quantity. Only for normal goods when the price falls. For a price rise it lowers it; for an inferior good it goes the other way.
Expecting a substitution effect for perfect complements. It is zero: all the change is the income effect.
Forgetting that the substitution effect is never positive. A positive substitution effect for a price rise would mean her compensated demand rises with price, which cannot happen.

Worked questions

The opening question, and four more

JNU 2012 · advanced

U = min{x₁, x₂}, prices 1 and 2, income 150. The price of commodity 2 falls to 1. Find the Slutsky substitution effect and the income effect on commodity 1.

The old bundle.
x₁ = x₂ and x₁ + 2x₂ = 150, so (50, 50).
Compensate.
At the new prices (1, 1) the old bundle costs 100, so Slutsky compensation leaves her Rs 100.
Choose at the compensated income.
With Rs 100 and prices (1, 1) she buys (50, 50): substitution effect on commodity 1 is 50 − 50 = 0.
The income effect.
With her actual Rs 150 she buys (75, 75). The income effect is 75 − 50 = 25.

Answer

Substitution effect 0 and income effect +25.

Two methods compared · standard

u = xy, p_y = 1, income 100. The price of x falls from 2 to 1. Split the change in x by both methods.

Before and after.
x = 100/(2 × 2) = 25 before, 100/(2 × 1) = 50 after: a rise of 25. The old utility is 25 × 50 = 1,250.
Slutsky.
The old bundle costs 25 + 50 = 75 at the new prices. With Rs 75 she buys x = 75/2 = 37.5. Substitution +12.5, income +12.5.
Hicks.
On the old curve with prices (1, 1) she buys x = y = √1,250 = 35.36. Substitution +10.36, income +14.64.

Answer

Slutsky: 12.5 and 12.5. Hicks: 10.36 and 14.64. The totals are the same, 25.

UOH 2019 · advanced

In Diagram 1 a price fall for Y moves her from A to C. The compensated point on the old curve is B, with quantities Y₁ at A, Y₂ at C and Y_s at B. Name the move A to B, and say why Y is inferior if Y₂ < Y_s.

A to B.
It runs along the old indifference curve I₁ to the point where the new slope touches it: the substitution effect.
B to C.
The income effect: from the compensated point to the final point.
The sign.
Y₂ < Y_s means the income effect (from B to C) lowers Y while income, in effect, rises: inferior.

Answer

A to B is the substitution effect; Y is inferior.

JNU 2017 · advanced

U = x₁ + x₂ and the price of commodity 1 changes. When is there certainly no substitution effect on commodity 1?

What she buys.
Only the cheaper good, in the amount m/p.
Cheaper stays cheaper.
After Slutsky compensation she can still afford the old bundle, which is still the best one, so the compensated demand equals the old one: substitution effect 0. The income effect is real (the amount she can buy changes with the price).
Order reverses.
She switches goods, so there is a substitution effect.

Answer

None if the cheaper good remains cheaper after the price change.

JNU 2015 · standard

Joe buys only X and Y, and always buys more Y when the price of Y rises. What can we infer?

The behaviour.
Demand for Y rises with its own price: Y is Giffen.
Giffen implies inferior.
Y is an inferior good.
Adding up.
Income elasticities weighted by budget shares sum to one. Y’s is negative, so X’s is positive: X is normal.

Answer

X is a normal good.

Revise

The lesson on one screen

Price effect: Substitution effect + income effect, exactly (Slutsky equation).
Hicks: Compensate to the old indifference curve. Price rise: add income. Price fall: take it away.
Slutsky: Compensate so the old bundle is just affordable.
Signs: Substitution effect never positive. Income effect: reinforcing for a normal good, opposing for an inferior one.
Special tastes: Perfect complements: no substitution effect. Quasi-linear: no income effect. Perfect substitutes: none if the cheaper good stays cheaper.
Giffen: Demand rises with price. Needs an inferior good with an income effect larger than the substitution effect. Giffen implies inferior, not the reverse.
Two goods: If Y is Giffen then X is normal.
Diagram: A to C: price effect. A to compensated point: substitution. Compensated point to C: income.

Practice

Check yourself

0 of 10 correct so far

Quick questions

1The Slutsky method of compensation leaves the consumer so that

2The substitution effect of a price rise on the quantity of the same good is

3A demand curve slopes up only if the good is

4For perfect complements, the substitution effect of a price change is

5With u = √x + y, and some y bought, the income effect of a change in the price of x on the quantity of x is

Numerical problems

Slutsky decomposition

u = xy, p_y = Rs 1 and income Rs 100. The price of x falls from Rs 2 to Rs 1. By the Slutsky method, what are the substitution effect and the income effect on the quantity of x?

Substitution effectIncome effect

Hicks decomposition

Same consumer. By the Hicks method, what is the substitution effect on the quantity of x? Give two decimal places.

Substitution effect

Perfect complements

u = min{x, y}, p_x = Rs 1, p_y rises from Rs 2 to Rs 3, income Rs 150. What is the income effect on the quantity of y?

Income effect

Quasi-linear tastes

u = √x + y, p_y = Rs 1, income Rs 100. The price of x rises from Rs 0.25 to Rs 1. What is the substitution effect on the quantity of x?

Substitution effect

Real questions

Past papers, by level

Foundation

Definitions and one-step calculations: the shape of a curve, a budget line, an optimum for Cobb–Douglas tastes. Most CUET, UGC NET, Hyderabad and early JNU questions sit here.

2 questionsPractise →

Standard

Statements to judge, matching lists, and demands to compute with unusual tastes, up to JAM, GATE and JNU level.

16 questionsPractise →

Delhi School level

This topic’s hardest questions, from the DSE entrance, are worked through in Lesson 17: Income effects, Slutsky splits and duality.

11 questionsLesson 17 →

All 18 past-paper questions for this lesson, easiest first: practise the full set →

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.

← Lesson 11Income and cross elasticities, Engel curves and the type of good Lesson 13 →Revealed preference

Income and substitution effects and the Slutsky equation

One price change, two effects

Two ways to compensate: Hicks and Slutsky

Try it: the four-step decomposition

Start at E₁, the best bundle the consumer can afford

The Slutsky equation

Complements, quasi-linear tastes and straight lines

Giffen goods

Traps

The mistakes examiners are fishing for

The opening question, and four more

The lesson on one screen

Check yourself

Quick questions

Numerical problems

Past papers, by level

Foundation

Standard

Delhi School level

Read next

Income and substitution effects and the Slutsky equation

One price change, two effects

Two ways to compensate: Hicks and Slutsky

Try it: the four-step decomposition

Start at E₁, the best bundle the consumer can afford

The Slutsky equation

Complements, quasi-linear tastes and straight lines

Giffen goods

Traps

The mistakes examiners are fishing for

The opening question, and four more

The lesson on one screen

Check yourself

Quick questions

Numerical problems

Past papers, by level

Foundation

Standard

Delhi School level

Read next