The budget line is one of the smallest objects in microeconomics — a single straight line — and one of the most important. It is the entire constraint side of consumer choice and the engine behind why demand curves slope down. This walkthrough builds the line algebraically and geometrically, then shows what rotates it (a price change) versus what shifts it (an income change).
The Budget Line, From the Constraint Up
A consumer has income M and faces prices P_X and P_Y for two goods. The total spent cannot exceed income, so:
P_X · X + P_Y · Y ≤ M
When the consumer spends every dollar (no saving in this static model), the inequality is an equality — the budget constraint, which traces the boundary called the budget line:
P_X · X + P_Y · Y = M
Solve for Y to put it in slope-intercept form:
Y = M / P_Y − (P_X / P_Y) · X
That gives the three numbers you'll be asked for over and over:
- Y-intercept (M / P_Y): the maximum Y if all income goes to Y.
- X-intercept (M / P_X): the maximum X if all income goes to X.
- Slope (− P_X / P_Y): the relative price of X in terms of Y. To buy one more unit of X you must give up P_X / P_Y units of Y.
A quick numeric example. M = $100, P_X = $5, P_Y = $10. Then Y-intercept = 10, X-intercept = 20, slope = − 0.5. Budget line: Y = 10 − 0.5X. Buying one more X costs you half a unit of Y — that's the slope.
Affordable, Best, and Unaffordable
The plane around the budget line splits into three regions.
- Below the line: affordable but wasteful — you have leftover income.
- On the line: affordable and uses all income.
- Above the line: unaffordable.
The consumer's best bundle, given preferences, must lie on the line (assuming more of either good is preferred to less). Which point on the line is best depends on the indifference curves, which the utility and indifference curves walkthrough covers. This article focuses on what the line itself does.
What Changes Rotate the Line
A change in the price of one good rotates the budget line around the intercept of the other good. The pivot point is the axis where you can still buy the same maximum quantity.
Say P_X falls from $5 to $4 with M = $100 and P_Y = $10 unchanged. The Y-intercept is still 10 (you can still buy 10 Y with all your money), but the X-intercept rises from 20 to 25. The line pivots outward along the X-axis. The slope flattens from − 0.5 to − 0.4: X has gotten cheaper relative to Y.
The implication is clean: lower P_X expands the set of affordable bundles in the X direction. Some of those new bundles will be on a higher indifference curve than the old optimum, so the consumer will buy more X. Tracing the optimum as P_X varies — keeping M and P_Y fixed — gives you the individual's demand curve for X. This is the deep reason demand slopes down: each lower P_X enlarges the affordable set, and the new tangency typically sits at a higher X.
Two effects sit behind that movement. The substitution effect is the move to relatively cheaper X. The income effect is that, with X cheaper, real purchasing power has risen, which lets you buy more of both goods. The why-the-demand-curve-slopes-down explainer decomposes the two effects.
What Changes Shift the Line
A change in income, with both prices held constant, shifts the budget line parallel. The slope is just − P_X / P_Y and depends only on prices, so it doesn't change. The intercepts both move by the same proportion.
Income rises from $100 to $150 with P_X = $5, P_Y = $10. New X-intercept = 30; new Y-intercept = 15. Same slope (− 0.5), shifted outward.
Income shifts also reveal how a good responds to wealth. If consumption of X rises with income, X is a normal good. If it falls, X is an inferior good. (The mirror income-elasticity logic from the elasticity chapter.) Equal proportional changes in both prices and income leave the line in exactly the same place — a useful sanity check that the model has no money illusion.
A Quick Worked Application
A consumer with income $60 buys coffee at $3 and pastries at $2.
- Budget: 3C + 2P = 60.
- Maximum coffee (if all income → coffee): 60 / 3 = 20 cups.
- Maximum pastries (if all income → pastries): 60 / 2 = 30 pastries.
- Slope: − 3 / 2. Each extra cup of coffee costs 1.5 pastries.
Now coffee falls to $2 (sale): budget line rotates outward on the coffee axis. New maximum coffee: 60 / 2 = 30. Pastry intercept unchanged at 30. New slope: − 1. Each coffee costs only 1 pastry now. The consumer can reach bundles like 20 coffees and 10 pastries (impossible before) or 10 coffees and 20 pastries.
Conclusion
Budget constraints and consumer choice come down to one line and one set of moves. Draw the line from the two intercepts and the slope − P_X / P_Y. A price change rotates the line around the other good's intercept and is the geometric reason demand curves slope down. An income change shifts the line parallel and tells you whether each good is normal or inferior. Everything that happens in a standard consumer-choice problem is on top of those moves.
