Foundationsmicroeconomics
Budget Constraints: What Consumers Can Afford
Overview
A budget constraint shows every affordable combination of two goods. This lesson builds the budget line from the budget equation, computes its intercepts and slope, and interprets the slope -Px/Py as the market opportunity cost of a good. You will learn to distinguish parallel shifts (income changes) from rotations (price changes), the graphical vocabulary behind every consumer-choice diagram.
Learning objectives
- Write the budget equation and identify the affordable set.
- Compute the intercepts and slope of the budget line.
- Interpret the slope as the market opportunity cost of a good.
- Predict how income and price changes shift or rotate the line.
The Budget Equation
With income M and prices Px and Py, any affordable bundle (X, Y) satisfies Px·X + Py·Y ≤ M. Spending exactly the whole income gives the budget line itself: Px·X + Py·Y = M. Points inside the line are affordable with money left over; points outside are unaffordable. The two-good model is a simplification, but every multi-good budget works the same way - one constraint, many possible bundles.
Drawing the Budget Line
Set Y = 0 in the budget equation: the maximum X affordable is M/Px, the horizontal intercept. Set X = 0: the maximum Y is M/Py, the vertical intercept. Connect them with a straight line. Example: with M = 10, Py = $5, you can buy at most 10 units of X or 20 units of Y, and the line joins those intercepts.
The Slope Is the Opportunity Cost
Rearranging the budget equation gives Y = M/Py - (Px/Py)·X, so the line's slope is -Px/Py. To consume one more unit of X, you must give up Px/Py units of Y - that ratio is the market opportunity cost of X, identical for every consumer facing the same prices. A steeper line means X is relatively more expensive.
The slope -Px/Py is both the market trade-off between the goods and the opportunity cost of X measured in Y. When relative prices change, the opportunity cost changes with them.
Income Changes Shift the Line
A rise in income moves both intercepts outward in proportion, shifting the line parallel to the right - the affordable set grows but relative prices are unchanged, so the slope stays -Px/Py. A fall in income shifts it parallel inward. Because only the position changes, income effects alter what you can buy without altering the market rate of exchange between goods.
Price Changes Rotate the Line
If Px falls, the X-intercept M/Px moves right while the Y-intercept stays fixed: the line flattens, pivoting on the vertical intercept. If Px rises, the line steepens around the same pivot. A proportional change in both prices works exactly like an income change - only the price ratio, hence the slope, matters for trade-offs.
Real-World Kinks
Actual budget sets are often kinked rather than straight: quantity discounts bend the line outward, coupons create notches, and rationing chops it off. Food-stamp-style programs create kinked sets with different slopes over different ranges. The straight line remains the baseline model - master it first, then add the kinks.
From Constraint to Choice
The budget line alone cannot pick a bundle - it only shows what is possible. Choosing requires preferences, represented by indifference curves: the consumer picks the affordable bundle on the highest reachable curve. That tangency between constraint and preferences is the consumer's equilibrium, taken up in the next lessons.
Key points
- The budget line Px·X + Py·Y = M shows all bundles that exactly exhaust income.
- Intercepts are M/Px and M/Py - the maximum affordable of each good alone.
- Slope -Px/Py is the market opportunity cost of X in terms of Y.
- Income changes cause parallel shifts; single-price changes cause rotations.
- The constraint defines what is possible; preferences define what is chosen.
Common misconceptions
✓ Truth: They are affordable but leave money unspent. With nonsatiated preferences the consumer will be on the line, but inside points are simply affordable, not irrational - the model just predicts they will not be chosen.
✓ Truth: Steepness reflects relative prices (Px/Py), not income. A poor consumer and a rich consumer facing the same prices have parallel lines with identical slopes; only the position differs.
✓ Truth: Doubling both prices halves both intercepts - a pure inward parallel shift, exactly like halving income. The slope -Px/Py is unchanged, so opportunity costs are identical.
Practice questions
Q1. Income is $120, Px = $12, Py = $8. What is the maximum quantity of X the consumer can buy?
- 15 units
- 10 units
- 8 units
- 20 units
Maximum X = M/Px = 120/12 = 10 units (the horizontal intercept). The maximum Y would be 120/8 = 15 units.
Q2. With Px = $12 and Py = $8, the opportunity cost of one unit of X is:
- 0.67 units of Y
- 1.5 units of Y
- 8 units of Y
- $12
Opportunity cost of X = Px/Py = 12/8 = 1.5 units of Y forgone. Equivalently, the budget line's slope is -1.5.
Q3. The consumer's money income doubles while both prices stay the same. The budget line will:
- rotate outward around the Y-intercept.
- shift outward in parallel with unchanged slope.
- become steeper.
- become flatter.
Both intercepts double, so the line moves out parallel to itself. Relative prices are unchanged, hence the slope -Px/Py and all opportunity costs are unchanged.
Q4. The price of X falls while income and Py are unchanged. Which statement is true?
- The budget line shifts inward in parallel.
- The budget line rotates outward, pivoting on the Y-intercept, and flattens.
- The opportunity cost of Y rises.
- The affordable set shrinks.
A lower Px raises the X-intercept M/Px while the Y-intercept M/Py is fixed, so the line pivots outward around the vertical intercept and flattens - X has become cheaper relative to Y.
Click a question to reveal the answer and explanation.
Frequently asked questions
How do I find the opportunity cost of good X from the budget line?
Read the absolute slope: Px/Py. With Px = $12 and Py = $8, one more unit of X costs 12/8 = 1.5 units of Y forgone. That is the market rate at which X trades for Y, the same for everyone facing those prices.
What is the difference between a shift and a rotation of the budget line?
An income change shifts the line in parallel - both intercepts move proportionally and the slope is unchanged. A change in one good's price rotates the line around the other good's intercept, changing the slope and therefore the opportunity cost.
Why does the model use only two goods?
Two goods keep the diagram readable, and nothing essential is lost: interpret Y as 'all other goods' measured in dollars (Py = $1). The math and intuition extend directly to many goods.
Can the budget line ever slope upward?
In the standard model, no - buying more of one good while holding spending fixed requires buying less of the other. Upward-sloping segments appear only with perverse cases like goods with negative prices (someone pays you to take them).