Consumer theory · Lesson 4 of 17
The budget set and the budget line
Preferences say what she wants. The budget says what she can have. Put the two on one graph and the best affordable bundle falls out. This lesson draws the budget: which bundles are affordable, what the line’s slope means, and what moves it.
A question to hold on to
A consumer buys two goods, X and Y. Spending his entire income he can buy 4 units of X and 6 of Y, or 12 units of X and 2 of Y. If he spends his entire income on X alone, how many units of X can he buy? (7, 8, 16 or 15.)
Hyderabad 2015Solved at the end ↓
The constraint
What she can afford
With income m and prices px and py, a bundle (x, y) is affordable if it costs no more than m. The budget set is every affordable bundle:
The budget set
Bundles cannot have negative amounts, so the set is the triangle between the axes and the budget line.
It is closed (it includes its edge, the line itself), bounded (when prices are positive she cannot buy unlimited amounts) and convex (if two bundles are affordable, so is any mixture of them), which is the correct description in UGC NET 2023.
The edge
The budget line
The top edge of the set, where she spends exactly m, is the budget line. Solve it for y and you can read everything off:
The budget line
Put x on the horizontal axis. Intercepts: all income on one good. Slope: how many units of y one unit of x costs.
- The y-intercept is m/py: everything on y. The x-intercept is m/px: everything on x.
- The slope is −px/py (JNU 2011, 2015). It is the opportunity cost of x: to buy one more unit of x she must give up px/py units of y. Keep the minus sign and keep x on the horizontal axis: swapping them flips the ratio.
- A bundle is on the line if it uses all her income, below if she has money left, above if she cannot afford it.
The constraint must be written with each price on its own good. With food F and shelter S, it is M = PfF + PsS, so S = M/Ps − (Pf/Ps)F; mixing a price with the other good’s quantity (as the wrong options do) is the usual trick (Hyderabad 2010).
Pause and predict
Income is Rs 100, p_x = 4 and p_y = 5. The y-intercept and the x-intercept of the budget line are
Interactive
Try it: move the line
Start with Rs 60 and prices Rs 3 and Rs 2 (the dashed line). Press a scenario or move the sliders. Click the plot to test any bundle. The second tab puts a calorie limit beside the rupee limit.
Rs 60 to spend, with x at Rs 3 and y at Rs 2 (the dashed line). Press a scenario, or move the sliders; click the plot to test a bundle.
Affordable bundlesSolid line: now. Dashed: before.
The line
Meets the x-axis at 20 and the y-axis at 30. Slope −px/py = −3/2: one more x costs 3/2 of y.
What moved
The line has not moved at all. Only relative prices and income measured in goods matter: double income and both prices and nothing changes.
Test the bundle (6, 8)
Costs Rs 34. Affordable, with Rs 26 left over.
Shifts and pivots
How the line moves
| What changes | What happens to the line | Why |
|---|---|---|
| Income rises | Shifts out, parallel | Both intercepts grow by the same factor; the slope px/py is unchanged |
| Price of x rises | Pivots in, about the y-intercept | The y-intercept m/py is unaffected; the x-intercept falls; the line steepens |
| Price of y rises | Pivots in, about the x-intercept | The line flattens |
| Both prices rise by the same percent | Shifts in, parallel (Hyderabad 2014) | Same effect as an income cut of that percent: the ratio is unchanged |
| Price of good 1 doubles and price of good 2 triples | Becomes flatter (UGC NET 2023, Hyderabad 2016) | The slope falls in size by 2/3: the price of good 1 is now relatively cheaper |
| Income and both prices double | Does not move | Every intercept is unchanged. Only real income and relative prices matter |
Pause and predict
Income and both prices all rise by 25%. The budget line
Only relative things matter
Multiply income and every price by the same number and the budget set is unchanged. That is the first hint of a result Lesson 16 proves: demand is homogeneous of degree zero in prices and income.
Edge cases
Awkward prices
A good can have a negative price if she is paid to take it (waste to dispose of, say). With prices −1, 1 and 2 for three goods and an income of Rs 100, a bundle (a, b, c) is affordable if −a + b + 2c ≤ 100. Then the budget set is no longer bounded: more of the negatively priced good adds to her income, so by taking enough of it she can afford any amounts of goods 2 and 3. But a specific bundle must still pass the test: (0, 0, 60) costs 120, which is more than 100, so it is not affordable for everyone (JNU 2015). (20, 0, 60) costs −20 + 120 = 100, which is.
A short proof
When she is worse off
If a price she pays rises, or her income falls, her budget set shrinks: every bundle she can afford afterwards she could afford before. So her best affordable bundle cannot be better than before. She is worse off, whatever her tastes (Hyderabad 2015). The reverse, a price fall or an income rise, makes her at least as well off.
Watch for
Traps
Watch out for
The mistakes examiners are fishing for
- The slope upside down. With x on the horizontal axis it is −px/py. The intercepts are m divided by the same axis’s price.
- A price change shifts the line. Only an income change, or an equal percentage change in both prices, shifts it in parallel. A change in one price pivots it.
- Money illusion. Doubling income and every price changes nothing. Do not read a rise in the numbers as a rise in real income.
- Solving a two-constraint problem with preferences. If two linear limits both bind, they cross at one bundle and tastes are irrelevant (JNU 2013).
- Using the line for the whole story. It tells you what is affordable, not what she will choose. That needs the indifference curve (Lesson 5).
Worked questions
The opening question, and four more
Hyderabad 2015 · advanced
He can buy 4 X and 6 Y, or 12 X and 2 Y, spending his whole income each time. How many X if he spends it all on X?
- Both bundles lie on the same budget line.So 4px + 6py = 12px + 2py, which gives 4py = 8px, so py = 2px.
- Income in terms of p_x.m = 4px + 6(2px) = 16px.
- All on X.m/px = 16.
Answer
16 units of X. (DSE 2019 asks the price ratio from the same data: px/py = ½.)
JNU 2011 · advanced
Situation I: the price of X is twice the price of Y, and the consumer spends all his income on 6 units of X and 20 of Y. Situation II: income is doubled, the price of Y is double its Situation I level, the price of X is unchanged, and he still wants 20 units of Y. The most X he can buy is?
- Name the prices.Let py = 1, so px = 2. Income is 2 × 6 + 1 × 20 = 32.
- Situation II.Income 64, py = 2, px = 2. Twenty units of Y cost 40, leaving 24.
- Spend the rest on X.24 ÷ 2 = 12.
Answer
12 units of X.
UGC NET 2023 · standard
Good 1 is on the horizontal axis, good 2 on the vertical. The price of good 1 doubles and the price of good 2 triples. What happens to the budget line?
- The slope is the price ratio.Old slope −p₁/p₂. New slope −2p₁/(3p₂) = (2/3) of the old.
- Smaller in size means flatter.The line also moves in because both prices rose, but the question asks about its shape.
Answer
It becomes flatter.
JNU 2015 · advanced
Three goods cost −1, +1 and +2 per unit, and income is Rs 100. Which is NOT true: (A) she may afford positive amounts of each; (B) she may afford any amounts of goods 2 and 3; (C) any individual may afford (0, 0, 60); (D) she may afford (20, 0, 60)?
- Cost each bundle.(0, 0, 60) costs 2 × 60 = 120, which exceeds 100. So (C) is false for an individual with income Rs 100.
- Check the others.(D): −20 + 0 + 120 = 100, affordable. (A): for example (1, 1, 1) costs −1 + 1 + 2 = 2. (B): she can take as much of good 1 as she needs to pay for them.
Answer
(C).
JNU 2013 · standard
Satish eats only rotis (Re 1, 1,000 calories) and cups of dal (Rs 2, 500 calories). He spends exactly Rs 13 and eats exactly 5,500 calories. What does he consume?
- Two lines.Money: r + 2d = 13. Calories: 1000r + 500d = 5500, which is 2r + d = 11.
- Solve.From the first, r = 13 − 2d. Then 2(13 − 2d) + d = 11 gives 26 − 3d = 11, so d = 5 and r = 3.
Answer
3 rotis and 5 cups of dal. Preferences do not matter here: two binding constraints leave only one bundle.
Why three versions of the rotis-and-dal question are left out
CUET 2024 asks the same thing but its stored text does not give the calories per item, so it cannot be solved. JNU 2010 and JNU 2015 give the data, but they offer “no more than 3 rotis” and “no more than 5 cups” beside “3 rotis and 5 cups”, and all three are true of the only bundle. JNU 2013, the version above, has four exclusive options and is in the practice set of Lesson 7.
Revise
The lesson on one screen
The lesson on one screen
- Budget set
- p_x x + p_y y ≤ m with x, y ≥ 0: closed, bounded (positive prices) and convex.
- Budget line
- y = m/p_y − (p_x/p_y) x. Intercepts m/p_y and m/p_x; slope −p_x/p_y.
- Opportunity cost
- The slope's size p_x/p_y is the y given up per unit of x.
- Income
- Shifts the line in parallel.
- One price
- Pivots the line about the other good's intercept.
- Both prices, same percent
- Parallel shift, like an income change the other way.
- Income and all prices scaled
- No change at all.
- Worse off
- A price she pays rising or her income falling shrinks the set; she is worse off.
- Two binding limits
- Cross at one bundle; tastes do not matter.
Practice
Check yourself
Quick questions
1With good x on the horizontal axis, the slope of the budget line is
2The price of x doubles. Income and the price of y are unchanged. The budget line
3Income and both prices all triple. The budget set
4Income is Rs 100, p_x = 4, p_y = 5. Which bundle is NOT affordable?
5Both prices rise by 10% and income stays the same. Her best affordable utility
Numerical problems
Reading a point on the line
Income is Rs 120, px = Rs 6 and py = Rs 4. If she buys 10 units of x, what is the most y she can buy?
The price ratio from two points
A budget line passes through (3, 12) and (9, 6). What is px/py?
Finding a price
A consumer spends all of Rs 360 on 12 units of X (Rs 10 each) and 20 units of Y. What is the price of Y in rupees?
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.
4 questionsPractise →
Standard
Statements to judge, matching lists, and demands to compute with unusual tastes, up to JAM, GATE and JNU level.
7 questionsPractise →
Delhi School level
This topic’s hardest questions, from the DSE entrance, are worked through in Lesson 17: Subsidies, grants, bundle deals and block prices.
11 questionsLesson 17 →
All 11 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.
Read next
All resources →MicroeconomicsUtility maximisation: tangency, the Lagrangian and Cobb–Douglas demandThe best affordable bundle: MRS equals the price ratio, the Lagrangian and the marginal utility of income, Cobb–Douglas demand and budget shares, and shifted Cobb–Douglas such as (x+2)(y+1).MicroeconomicsPerfect substitutes, complements, lexicographic, homothetic and quasi-linear preferencesStraight-line, L-shaped and dictionary preferences, the CES family that contains them, homothetic preferences (a straight expansion path) and quasi-linear preferences (vertically parallel curves).MicroeconomicsConsumer choice: budget line and indifference curvesMove a price or income and watch the budget line and indifference curve re-tangent, then split the change into its two effects.