Difficulty: Introductory | Prerequisites: Basic algebra (solving for intercepts, slopes).
Budget constraints define the boundary of what a consumer can afford given their income and the prices they face. This topic sits at the heart of consumer theory in introductory microeconomics: before you can analyse what a rational person chooses, you need to map out what they can choose. If prices or income change, the constraint shifts, and the consumer may end up better or worse off even if their tastes stay the same. You should already be comfortable plotting a straight line from two intercepts and understand that a "bundle" is just a combination of quantities of two goods.
A budget constraint shows every combination of goods a consumer can buy with their income. When prices change, the constraint pivots or shifts, and you can compare the old and new affordable sets to decide whether the consumer is better off, worse off, or ambiguous. The key test: can the consumer still afford their original bundle under the new prices and income?
Budget constraint (budget line)
The set of all bundles that exactly exhaust a consumer's income at given prices. Graphically, it is a straight line; its slope equals the negative price ratio. In simple terms, it is the border between what you can and cannot afford.
Affordable set (budget set, feasible set)
Every bundle on or below the budget line. Think of it as the full menu of options the consumer could pick from.
Optimal bundle (consumer equilibrium)
The bundle a rational consumer actually chooses, where the highest attainable indifference curve is tangent to the budget line. In simple terms, it is the best the consumer can do given their budget.
Rational consumer
A consumer whose preferences are complete and transitive, and who always picks the most-preferred affordable bundle. Think of it as someone who never knowingly leaves satisfaction on the table.
Revealed preference
If a consumer chose bundle A when bundle B was also affordable, then A is revealed preferred to B. In simple terms, their purchase tells you what they like, even without seeing their indifference curves.
Indifference curve
A curve connecting all bundles among which the consumer is equally satisfied. Think of it as a contour line of happiness on a map of goods.
The general two-good budget constraint is: Px·X + Py·Y = M, where M is income.
Rearranging for Y gives the budget line: Y = (M / Py) – (Px / Py)·X.
The Y-intercept is M / Py (spend everything on Y). The X-intercept is M / Px (spend everything on X).
The slope of the budget line is –Px / Py. This tells you the rate at which the market lets you trade X for Y.
Income change (M changes, prices constant): the line shifts outward (higher M) or inward (lower M), parallel to the original. The slope stays the same because relative prices have not changed.
One price changes: the line pivots around the intercept of the good whose price did not change. If Px falls, the X-intercept moves right while the Y-intercept stays put.
Both prices change: the line can pivot, shift, or both. Work out the new intercepts and connect them.
Last year's data: Px = 15, X = 20, Py = 25, Y = 30. His income was therefore M = 15(20) + 25(30) = 300 + 750 = 1,050.
Last year's budget line: Y = 1,050/25 – (15/25)X = 42 – 0.6X. Intercepts: (0, 42) and (70, 0).
This year's data: Px = 6, Py = 30, M = 1,020.
This year's budget line: Y = 1,020/30 – (6/30)X = 34 – 0.2X. Intercepts: (0, 34) and (170, 0).
Key test: Can Jones still buy last year's bundle (20, 30) under this year's budget? Cost check: 6(20) + 30(30) = 120 + 900 = 1,020. That is exactly equal to his new income, so the old bundle sits right on the new budget line.
Conclusion: Because the old bundle is affordable and the new budget line has a different (flatter) slope, the new affordable set includes bundles that were previously too expensive. A rational consumer can do at least as well as before, and almost certainly strictly better, by re-optimising along the new line. Jones is better off (or, at worst, equally well off).
General budget constraint: Px·X + Py·Y = M
Budget line (solved for Y): Y = (M / Py) – (Px / Py)·X
Jones, last year: M = 15(20) + 25(30) = 1,050 Budget line: Y = 42 – 0.6X
Jones, this year: M = 1,020 Budget line: Y = 34 – 0.2X
Affordability check: 6(20) + 30(30) = 1,020 = M. The old bundle (20, 30) lies exactly on the new budget line.
Graphical sketch: Plot both budget lines on the same axes. The old bundle sits at their intersection. The new line is flatter (slope –0.2 vs. –0.6), and its X-intercept is much further right (170 vs. 70). The region below the new line but above the old line represents newly affordable bundles. An indifference curve through (20, 30) will be tangent to the old line there but will cut through the interior of the new budget set, confirming that higher indifference curves are now reachable.
Budget constraints are how economists model everyday trade-offs: how much petrol versus groceries a household can buy, or how a government allocates a fixed defence budget between equipment and personnel. The Jones-style analysis is precisely how policy analysts assess whether a change in prices (say, a tax cut on one good paired with a tax rise on another) leaves consumers better or worse off on balance.
Students often think that if income falls, the consumer must be worse off. That is only guaranteed if prices stay the same. A simultaneous price drop can offset or outweigh the income loss.
Students frequently forget that the budget line's slope depends only on the price ratio, not on income. Changing M shifts the line but does not tilt it.
A common error is concluding "same bundle is affordable, so the consumer is equally well off." If the slope changed, new bundles are now reachable, and a rational consumer will typically re-optimise to a higher indifference curve.
Some students confuse "revealed preference" with "stated preference." Revealed preference is about what the consumer actually bought, not what they say they like.
⚠️ You will almost certainly be asked to calculate intercepts, draw two budget lines, and state whether the consumer is better or worse off. Practise the arithmetic until it is automatic.
⚠️ The "affordability test" (can the old bundle be purchased at new prices/income?) is the single most important step. If the old bundle is affordable and the slope changed, the consumer is at least as well off.
⚠️ Graphical answers are expected. Label both axes, mark both budget lines, mark the original bundle, and sketch at least one indifference curve to show the welfare comparison.
True or false: If a consumer's income doubles and all prices double, the budget line does not change. (True.)
Fill in the blank: The slope of the budget line equals ______ . (–Px / Py.)
True or false: If the old bundle is strictly inside the new budget set (not on the line), the consumer is strictly better off. (True, assuming non-satiation.)
Fill in the blank: The X-intercept of the budget line is ______ . (M / Px.)
True or false: A fall in Px with everything else constant causes a parallel shift of the budget line. (False; it causes a pivot around the Y-intercept.)
Q: Jones spends all his income on goods X and Y. Last year, Px = 15, X = 20, Py = 25, Y = 30. This year, Px = 6, Py = 30, and M = 1,020. Is Jones better or worse off?
A: His old bundle costs 6(20) + 30(30) = 1,020 under new prices, exactly equal to his income. The old bundle is affordable. Because the price ratio changed (the new budget line is flatter), there are bundles on the new line that were not affordable before. A rational consumer re-optimises and reaches a higher indifference curve. Jones is better off (or at minimum equally well off).
Q: Suppose instead that Jones's new income were only 900 with the same new prices. Would the old bundle still be affordable?
A: Cost of old bundle at new prices: 6(20) + 30(30) = 1,020 > 900. The old bundle is no longer affordable. This alone does not prove Jones is worse off (he might find a preferred bundle in the new, differently shaped budget set), but the analysis becomes ambiguous without more information about his preferences.
Q: What happens to the budget line if both prices double but income stays the same?
A: Both intercepts halve (M / 2Px and M / 2Py), so the line shifts inward in a parallel fashion. The consumer is unambiguously worse off because the entire old budget set shrinks.
This material connects directly to indifference-curve analysis (you need both tools to solve optimisation problems) and to income and substitution effects (the next step after seeing that a price change tilts the budget line). It also underpins the derivation of demand curves: trace how the optimal quantity of X changes as Px changes, and you have the consumer's demand curve for X.
Budget constraint, budget line, budget set, feasible set, affordable set, consumer equilibrium, optimal bundle, price ratio, income effect, substitution effect, revealed preference, indifference curve, rational consumer, welfare comparison, Px Py intercept, two-good model, microeconomics consumer theory, Jones problem NYU