Step 26: Indifference Curves and Budget Lines
- An indifference curve joins all the combinations of two goods that give a consumer the same satisfaction. Its slope is the marginal rate of substitution (MRS).
- The budget line shows what the consumer can afford. Its slope is \( -\frac{P_x}{P_y} \).
- The consumer’s best choice is where the budget line just touches the highest indifference curve: \( MRS = \frac{P_x}{P_y} \).
- Exam point: a price fall splits into a substitution effect (always more of the cheaper good) and an income effect (depends on whether the good is normal, inferior or Giffen).
What an indifference curve shows
You spend your weekly money on coffees and bus rides. You would be equally happy with 2 coffees and 10 bus rides, 4 coffees and 6 bus rides, or 8 coffees and 4 bus rides. Plot these and join them up: that is an indifference curve.
A set of these curves is an indifference map; curves further from the origin show higher utility. Indifference curves have three key properties.
- They slope downwards. To keep utility the same, if you get more of one good you must give up some of the other.
- They are convex to the origin (bowed inwards). This comes from a diminishing marginal rate of substitution, explained below.
- They never cross. A crossing point would lie on two curves with different utility levels at once, which breaks the assumptions that more is preferred to less and that preferences are consistent (transitive).
Using the coffee example, going from 2 to 4 coffees, you give up 4 bus rides for 2 coffees, so MRS = 2. Going from 4 to 8 coffees, you give up only 2 bus rides for 4 coffees, so MRS = 0.5. The more coffees you already have, the less each extra coffee is worth to you in bus rides. This diminishing MRS is what makes the curve convex. It can also be written using marginal utility:
The budget line: what you can afford
The budget line shows what you can buy. With income M and prices \( P_x \) and \( P_y \), every combination that spends all your income satisfies:
The intercepts show what you could buy if you spent everything on one good: \( \frac{M}{P_x} \) units of X on the horizontal axis and \( \frac{M}{P_y} \) units of Y on the vertical axis. The slope is the market’s rate of exchange: the Y you must give up for one more X.
- A rise in income shifts the budget line outwards, parallel to the old one (the slope is unchanged because prices are unchanged). A fall in income shifts it inwards.
- A fall in the price of X pivots the line outwards around the Y intercept: the Y intercept stays where it was, but you can now buy more X. The line becomes flatter. A rise in \( P_x \) pivots it inwards and makes it steeper.
The consumer’s best choice
The consumer wants to reach the highest indifference curve they can afford. That happens where the budget line is tangent to (just touches) an indifference curve. At that point the two slopes are equal:
The rate at which you are willing to swap Y for X equals the rate at which the market lets you. If MRS is bigger than the price ratio, buy more X and less Y; MRS then falls until the two are equal.
For example, Aisha has $60 a week. Good X (cinema tickets) costs $4 and good Y (snacks) costs $2. Find the intercepts and slope of her budget line. At her current bundle her MRS is 3. Should she buy more or fewer cinema tickets?
Aisha would give up 3 snacks for one more ticket, but the market only asks her to give up 2. Buying more tickets (and fewer snacks) makes her better off, until her MRS falls to 2.
A price fall: substitution and income effects
When the price of X falls, X becomes cheaper relative to Y, and the consumer’s money goes further, as if real income had risen. A compensated budget line separates the two effects.
Where point C ends up depends on the type of good:
| Type of good | Substitution effect | Income effect | Total effect of price fall |
|---|---|---|---|
| Normal | More X | More X (C is right of B) | Quantity rises: demand curve slopes down |
| Inferior | More X | Less X, but smaller than the substitution effect (C between A and B) | Quantity still rises: demand curve slopes down |
| Giffen | More X | Less X, and larger than the substitution effect (C left of A) | Quantity falls: demand curve slopes up |
For example, the price of rice falls. In the diagram, a consumer buys 6 kg at A, 9 kg at B and 11 kg at C. Find the substitution, income and total effects, and say what type of good rice is for this consumer.
Both effects are positive, so rice is a normal good for this consumer. If instead C had been 8 kg, the income effect would be \( 8 – 9 = -1 \): an inferior good. If C had been 5 kg, the income effect would be \( 5 – 9 = -4 \), bigger than the substitution effect of +3, so the total effect is \( -1 \): a Giffen good.
A Giffen good is a rare, strongly inferior good, usually a staple food taking a large share of a poor household’s budget. When its price falls, the household feels much richer, switches to better foods and buys less of the staple. All Giffen goods are inferior, but most inferior goods are not Giffen.
Limitations of the model
- Only two goods. Real consumers choose among thousands of goods. Economists treat Y as “all other goods”, but this is a simplification.
- Utility cannot be measured. We cannot observe indifference curves directly, so the model is hard to test.
- Assumes rational, fully informed consumers. Behavioural economics shows that people use habits, rules of thumb and defaults, and are influenced by framing. Preferences may not be stable or consistent.
- Assumes goods are divisible and that consumers can make tiny adjustments. Many purchases (a car, a phone) are lumpy.
Even so, it explains why demand curves usually slope down and why Giffen goods are possible.
Practise: 6 questions
Try each one on paper first, then check the worked answer.
EasyQ1. Income is $120, \( P_x = \$6 \) and \( P_y = \$4 \). Find both intercepts and the slope of the budget line. Show worked answer
Y intercept \( = \frac{120}{4} = 30 \).
Slope \( = -\frac{6}{4} = -1.5 \).
EasyQ2. Why can two indifference curves for the same consumer never cross? Show worked answer
MediumQ3. Using Q1’s prices, income rises from $120 to $180. Describe the new budget line. Show worked answer
Slope \( = -\frac{6}{4} = -1.5 \), the same as before.
The line shifts outwards, parallel to the original, so the consumer can reach a higher indifference curve.
MediumQ4. \( P_x = \$3 \), \( P_y = \$1 \). At a consumer’s current bundle, \( MU_x = 12 \) and \( MU_y = 6 \). Is the consumer at the optimum? If not, what should they do? Show worked answer
Price ratio \( = \frac{P_x}{P_y} = \frac{3}{1} = 3 \).
\( MRS = 2 \lt 3 \), so the consumer values X less than it costs in terms of Y. They should buy less X and more Y until MRS rises to 3.
MediumQ5. After a fall in the price of bread, a consumer’s total effect is +2 loaves and the substitution effect is +5 loaves. Find the income effect and classify bread for this consumer. Show worked answer
The income effect is negative, so bread is inferior. Because it is smaller in size than the substitution effect (3 < 5), the total effect is still positive, so bread is inferior but not Giffen; the demand curve still slopes down.
ExamQ6. With the help of an indifference curve diagram, explain how a fall in the price of a Giffen good can lead to a fall in the quantity demanded. [8 marks] Show worked answer
Substitution effect (up to 2): A to B. X is now relatively cheaper, so holding utility constant the consumer buys more X. This is always positive for a price fall.
Income effect (up to 2): B to C. The price fall raises real income. A Giffen good is a strongly inferior good, so higher real income means the consumer buys less X; the income effect is negative and larger than the substitution effect.
Conclusion (1): the net effect A to C is a fall in quantity demanded, so the demand curve for a Giffen good slopes upwards. Such goods are rare: typically a staple taking a large share of a poor household’s budget.
Check yourself
1. The price of good X falls while income and the price of Y stay the same. The budget line:
2. At the consumer’s optimum:
3. For a Giffen good, after a price fall the income effect is:
Your score: 0 out of 3
The short version
- Indifference curves slope down, are convex (diminishing MRS) and never cross; higher curves mean more utility.
- The budget line \( P_x X + P_y Y = M \) has slope \( -\frac{P_x}{P_y} \); income changes shift it parallel, price changes pivot it.
- The optimum is the tangency point where \( MRS = \frac{P_x}{P_y} \).
- A price fall = substitution effect (always more of the cheaper good) + income effect (positive for normal, negative for inferior, strongly negative for Giffen).
- The model is simplified (two goods, unmeasurable utility, rational consumers) but explains the shape of demand curves.
Further reading on EconTutorials
Older in-depth articles on this topic from our blog:
- Partial Derivatives in Economics: Marginal Products, Multivariable Optimisation and the Chain Rule
- Lagrange Multipliers in Economics: Shadow Prices, the Envelope Theorem, and Kuhn-Tucker Conditions
- Complete EconTutorials Content Library
- Game Theory: The Complete Guide to Nash Equilibrium, Dominance, Credible Threats and the Folk Theorem