Consumer Behaviour

Lessons

Senior 4 · Economics

Consumer Behaviour

4 lessons

Lesson 3 of 4

Indifference curves, the marginal rate of substitution and consumer equilibrium

30 min

Ranking without measuring

Ask a classmate to rank three lunches: isombe with rice, beans with sweet potatoes, and brochettes with chips. She will do it in seconds. Now ask her how many utils the first gives her, and how many more than the second. She cannot say, and neither can anyone else.

In 1934 John Hicks and Roy Allen showed that consumer theory does not need those numbers. It is enough that a consumer can say which of two bundles she prefers, or that she is indifferent between them. From that ranking alone they built the indifference curve, which with the budget line from lesson 2 shows exactly what a rational consumer buys.

Definition

Indifference schedule: a table of combinations of two goods that give a consumer the same satisfaction.

Indifference curve (IC): a curve joining all the combinations of two goods that give a consumer equal satisfaction, so that she is indifferent between them.

Indifference map: a set of indifference curves for one consumer; a curve further from the origin shows a higher level of satisfaction.

Example: if Mugabo is equally happy with 4 kg of beans and 18 kg of flour or with 6 kg of beans and 12 kg of flour, both bundles lie on the same indifference curve.

1. An indifference schedule and its curve

Return to Mugabo's household from lesson 2. Suppose (illustratively) that it ranks bundles by the product $X \times Y$ , where X is kilograms of beans and Y kilograms of maize flour a week. Every bundle with $X Y = 72$ is then equally good. The numbers only rank bundles: a bundle with $X Y = 144$ is preferred, but not "twice as good".

Bundle	Beans, X (kg)	Flour, Y (kg)	$\Delta X$	$\Delta Y$	$MRS_{xy} = -\frac{\Delta Y}{\Delta X}$
A	2	36	–	–	–
B	3	24	1	−12	12
C	4	18	1	−6	6
D	6	12	2	−6	3
E	8	9	2	−3	1.5
F	12	6	4	−3	0.75

Reading the diagram. Moving down the curve, Mugabo gives up flour to get beans and stays equally satisfied. When he has plenty of flour and few beans (near A), he will give up 12 kg of flour for one more kilogram of beans. When beans are plentiful and flour is scarce (near F), he gives up only 0.75 kg of flour per kilogram of beans. This falling rate is what makes the curve bow in towards the origin.

Definition

Marginal rate of substitution of X for Y ( $MRS_{xy}$ ): the amount of Y a consumer is willing to give up for one more unit of X while staying on the same indifference curve. It is the slope of the indifference curve with the sign removed:

MRS_{xy} = -\frac{\Delta Y}{\Delta X} = \frac{MU_x}{MU_y}

The second form holds because, along an IC, the utility lost from $\Delta Y$ must equal the utility gained from $\Delta X$ : $MU_x \Delta X + MU_y \Delta Y = 0$ .

Principle of diminishing MRS: as a consumer has more and more X, she is willing to give up less and less Y for each extra unit of X.

Example: from C (4, 18) to D (6, 12), $MRS = -\frac{-6}{2} = 3$ kg of flour per kg of beans.

2. Assumptions and properties of indifference curves

Assumptions. (1) The consumer is rational and can rank all bundles (completeness). (2) Rankings are consistent: if A is preferred to B and B to C, then A is preferred to C (transitivity). (3) More of a good is preferred to less (non-satiation). (4) The MRS diminishes. (5) Utility is ordinal: only the ranking matters. (6) Tastes, income and prices are given during the analysis.

Property	Reason
1. ICs slope downwards from left to right	To stay equally satisfied with more X, the consumer must have less Y (non-satiation).
2. ICs are convex to the origin	The MRS diminishes, so the curve becomes flatter as X increases.
3. A higher IC shows greater satisfaction	A curve further out contains bundles with more of at least one good and no less of the other.
4. Two ICs never cross	A crossing breaks transitivity (diagram (a) below).
5. ICs need not be parallel	The MRS at a given X can differ from one level of satisfaction to another.
6. An IC does not usually touch either axis	The theory assumes the consumer buys some of both goods.

Why curves cannot cross (diagram a). A and B are on I₁, so the consumer is indifferent between them. A and C are on I₂, so she is indifferent between those too. By transitivity she must be indifferent between B and C. But C has the same X as B and more Y, so by non-satiation she prefers C. The two statements contradict each other, so the curves cannot cross.

Special cases. (b) For perfect substitutes, such as two brands of bottled water that the consumer cannot tell apart, the MRS is constant and the ICs are straight lines. (c) For perfect complements, such as left and right shoes, extra units of one good without the other add nothing, so the ICs are L-shaped and the MRS is undefined at the corner.

Worked example 1: testing for diminishing MRS

A student is indifferent between these bundles of rice (X, kg) and isombe (Y, bowls): (1, 30), (2, 20), (3, 14), (4, 10), (5, 8). Find the MRS between each pair and comment.

Working. Each step adds 1 kg of rice, so $MRS = -\Delta Y$ :

30 - 20 = 10, \quad 20 - 14 = 6, \quad 14 - 10 = 4, \quad 10 - 8 = 2

Comment. The MRS falls from 10 to 2: the more rice the student has, the fewer bowls of isombe she will give up for another kilogram of rice. Her indifference curve is convex to the origin.

3. Consumer equilibrium

Put Mugabo's indifference map on top of his budget line from lesson 2 ( $M = 12,000$ , $P_x = 1{,}000$ , $P_{y} = 500$ ). He wants to reach the highest indifference curve he can afford.

Reading the diagram. A and B are affordable but lie on I₁. At A the IC is steeper than the budget line: Mugabo would give up 10 kg of flour for a kilogram of beans ( $MRS = \frac{Y}{X} = \frac{20}{2} = 10$ ), while the market asks only 2. So he gains by buying more beans and less flour, moving down the line. At B the reverse is true ( $MRS = \frac{4}{10} = 0.4$ ), so he gains by moving up. He stops at E, where I₂ just touches the line. I₃ would be better, but no point on it is affordable.

Conditions for consumer equilibrium.

The bundle lies on the budget line: $P_{x} X + P_{y} Y = M$ .
The indifference curve is tangent to the budget line, so their slopes are equal: $MRS_{xy} = \frac{MU_x}{MU_y} = \frac{P_x}{P_y}$
The indifference curve is convex at that point (MRS diminishing), so the tangency is a maximum.

Rearranging condition 2 gives $\frac{MU_x}{P_x} = \frac{MU_y}{P_y}$ : exactly the equi-marginal rule of lesson 1. The two approaches reach the same answer, but the ordinal one needs only a ranking.

Worked example 2: finding the equilibrium bundle

Mugabo ranks bundles by $U = X Y$ , so $M U_{x} = Y$ and $M U_{y} = X$ , and $MRS_{xy} = \frac{Y}{X}$ . Income is Frw 12,000, $P_x = 1{,}000$ , $P_{y} = 500$ .

Step 1: tangency. $\frac{Y}{X} = \frac{1{,}000}{500} = 2$ , so $Y = 2 X$ .

Step 2: budget. $1{,}000X + 500(2X) = 12{,}000 \Rightarrow 2{,}000X = 12{,}000 \Rightarrow X = 6$ , and $Y = 12$ .

Step 3: check. Spending $= 6,000 + 6,000 = 12,000$ ✔. $MRS = \frac{12}{6} = 2 = \frac{P_x}{P_y}$ ✔. Level of satisfaction: $X Y = 72$ , on I₂.

Interpretation. With $U = X Y$ the consumer always spends half of income on each good: $X = \frac{M}{2P_x}$ and $Y = \frac{M}{2P_y}$ . Any other bundle on the line, such as A (2, 20) with $X Y = 40$ , ranks lower.

Warning

Equilibrium is where the IC touches the budget line, not where it cuts it. A point where an IC crosses the line (A or B) is affordable, but a higher IC can still be reached. And compare the slopes the right way: if $MRS > \frac{P_x}{P_y}$ , buy more X; if $MRS < \frac{P_x}{P_y}$ , buy less X.

Tip

"What I would pay = what the market asks." The MRS is the consumer's own price of X in units of Y; $\frac{P_x}{P_y}$ is the market's price. The consumer keeps trading until the two agree.

Important

Discuss. The ordinal approach assumes consumers rank every bundle consistently. Think of how you choose between airtime and snacks. Do you behave as the theory assumes? If not, does that make the theory useless, or only less exact?

Khan Academy, "Indifference curves and marginal rate of substitution" (about 11 minutes). Watch how Sal Khan builds a curve from bundles a person likes equally, and how its slope, the MRS, changes from steep to flat along the curve, just as on Mugabo's curve above.

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Practice

Define an indifference curve, an indifference map and the marginal rate of substitution.
A consumer is indifferent between these bundles of X and Y: (1, 40), (2, 28), (3, 20), (4, 15), (5, 12). Calculate the MRS between each pair. Is the curve convex? Explain.
State four properties of indifference curves and give the reason for each.
With the help of a diagram, prove that two indifference curves of the same consumer cannot cross.
A consumer ranks bundles by $U = X Y$ . Income is Frw 6,000, $P_{x} = 300$ and $P_{y} = 150$ . Find the equilibrium bundle and check that $MRS = \frac{P_x}{P_y}$ there.
At her present bundle a consumer's MRS of X for Y is 3, while $\frac{P_x}{P_y} = 1$ . Is she in equilibrium? If not, which way should she change her spending, and why?
Sketch indifference curves for (a) perfect substitutes and (b) perfect complements, and give one example of each.
Show that the indifference-curve condition for equilibrium leads to the same rule as the equi-marginal principle. Which approach rests on more realistic assumptions? Explain.

Show answersHide answers

An IC joins bundles of two goods giving equal satisfaction; an indifference map is a set of ICs for one consumer, higher curves showing more satisfaction; the MRS is the amount of Y a consumer will give up for one more X while staying equally satisfied.
12, 8, 5, 3. The MRS falls as X rises, so the curve is convex to the origin.
Any four of: downward sloping (non-satiation); convex (diminishing MRS); higher is better (more of at least one good); never cross (transitivity); not necessarily parallel; usually do not touch the axes.
Draw two ICs crossing at A, with B on one and C on the other at the same X but more Y. A ~ B and A ~ C imply B ~ C, but C has more Y than B, so C is preferred: a contradiction.
$\frac{Y}{X} = \frac{300}{150} = 2 \Rightarrow Y = 2X$ ; $300X + 150(2X) = 6{,}000 \Rightarrow 600X = 6{,}000 \Rightarrow X = 10,\ Y = 20$ . MRS $= \frac{20}{10} = 2 = \frac{P_x}{P_y}$ ✔.
No. She values one X at 3 Y, but the market charges only 1 Y. She should buy more X and less Y; as she does, her MRS falls until it equals 1.
(a) Straight, parallel downward-sloping lines, e.g. two brands of bottled water. (b) L-shaped curves, e.g. left and right shoes, or a phone and its charger.
$MRS = \frac{MU_x}{MU_y} = \frac{P_x}{P_y}$ rearranges to $\frac{MU_x}{P_x} = \frac{MU_y}{P_y}$ . The ordinal approach needs only a ranking of bundles, not measurable utils or a constant marginal utility of money, so its assumptions are weaker and more realistic, though it still assumes consistent, complete preferences.

Quick check

Along an indifference curve, the MRS of beans for flour falls from 6 to 3 as the consumer takes more beans. This tells you that the curve is…

Key points

An indifference curve joins bundles that give equal satisfaction; an indifference map ranks levels of satisfaction without measuring them.
The MRS, $-\frac{\Delta Y}{\Delta X} = \frac{MU_x}{MU_y}$ , diminishes along a curve, so ICs are downward sloping and convex; they never cross.
Consumer equilibrium is where the highest attainable IC is tangent to the budget line: $MRS_{xy} = \frac{P_x}{P_y}$ , with all income spent.
This gives the same rule as the cardinal approach, $\frac{MU_x}{P_x} = \frac{MU_y}{P_y}$ , with weaker assumptions.

Sources and credits

Photo: ISOMBE AND RICE, Mutiganda, Wikimedia Commons, CC BY-SA 4.0.
Video: Indifference curves and marginal rate of substitution, Khan Academy, embedded from YouTube.

Written by Ganzaa: the text, the indifference-curve diagrams (drawn to scale from $X Y = 40$ , $72$ and $128$ and the budget line $Y = 24 - 2 X$ ), the tables, the worked examples and the practice items. All preferences and prices are illustrative.

This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1Ranking without measuring

21. An indifference schedule and its curve

32. Assumptions and properties of indifference curves

43. Consumer equilibrium

Ranking without measuring

1. An indifference schedule and its curve

2. Assumptions and properties of indifference curves

3. Consumer equilibrium