Difficulty: Introductory | Prerequisites: Basic algebra, Utility Theory notes
The budget constraint tells you what a consumer can afford. Combined with preferences (indifference curves), it determines the optimal bundle. This topic covers how budget lines shift when prices change (e.g. a tax) and when income changes (e.g. a lump-sum subsidy), and how to analyse who wins and loses from a combined tax-and-subsidy policy. You need to be comfortable drawing and interpreting budget lines before moving on to consumer choice.
The budget line shows all the bundles a consumer can just afford given their income and prices. When a price rises (e.g. a tax), the budget line pivots inward along that good's axis. A lump-sum subsidy shifts the entire line outward in a parallel fashion, and the combined effect of a tax plus a subsidy creates a new budget line that may cross the original, benefiting some consumers and hurting others.
Budget constraint
The equation P_X * X + P_Y * Y = Income, describing all combinations of goods X and Y the consumer can afford. In simple terms, it is the mathematical version of "you cannot spend more than you earn."
Budget line
The graphical representation of the budget constraint: a straight line whose intercepts show the maximum of each good the consumer could buy, and whose slope equals the negative price ratio (-P_X / P_Y).
Excise tax
A per-unit tax on a specific good that raises its price. An excise tax pivots the budget line inward along the taxed good's axis, leaving the other intercept unchanged.
Lump-sum subsidy (direct payment)
A fixed cash transfer to the consumer, independent of what they buy. It shifts the entire budget line outward in a parallel fashion (both intercepts increase proportionally).
Budget set
All bundles on or inside the budget line. These are the affordable bundles. The optimal bundle is the best one within this set.
Two goods: Tigers (T) and good Y. Income = $35,000. P_Y = $2,500 throughout.
Before tax: P_T = $2,500. After tax: P_T = $5,000.
The government also offers a lump-sum subsidy of $10,000 to offset the tax.
Equation: 2,500T + 2,500Y = 35,000, or T + Y = 14.
T-intercept (buy only tigers): 35,000 / 2,500 = 14 tigers.
Y-intercept (buy only Y): 35,000 / 2,500 = 14 units of Y.
Slope: -P_T / P_Y = -2,500 / 2,500 = -1.
Equation: 5,000T + 2,500Y = 35,000.
T-intercept: 35,000 / 5,000 = 7 tigers.
Y-intercept: unchanged at 14 (the price of Y did not change).
Slope: -5,000 / 2,500 = -2.
The line pivots inward around the Y-intercept. The consumer can still buy 14 units of Y, but the maximum tigers drops from 14 to 7.
New income: 35,000 + 10,000 = 45,000.
Equation: 5,000T + 2,500Y = 45,000.
T-intercept: 45,000 / 5,000 = 9 tigers.
Y-intercept: 45,000 / 2,500 = 18 units of Y.
Slope: still -2 (the subsidy does not change relative prices).
Set the two equations equal. Original: 2,500T + 2,500Y = 35,000. New: 5,000T + 2,500Y = 45,000.
Subtract: 2,500T = 10,000, so T = 4.
Plug back in: 2,500(4) + 2,500Y = 35,000, so Y = 10.
The two budget lines intersect at (T = 4, Y = 10). This is the bundle the average consumer bought before the tax.
At the intersection (T = 4), the old and new budget lines give exactly the same purchasing power.
Consumers who buy fewer than 4 tigers benefit. Their tax bill is less than the $10,000 subsidy, so they have extra money. The new budget line lies outside the old one for T < 4.
Consumers who buy more than 4 tigers are worse off. Their tax bill exceeds the subsidy. The new budget line lies inside the old one for T > 4.
In short: the policy redistributes from heavy tiger-buyers to light tiger-buyers.
P_X \cdot X + P_Y \cdot Y = IGeneral budget constraint. I = income.
\text{X-intercept} = \frac{I}{P_X}, \quad \text{Y-intercept} = \frac{I}{P_Y}Maximum of each good the consumer can buy.
\text{Slope of budget line} = -\frac{P_X}{P_Y}The slope tells you the market trade-off: how many units of Y must be given up to buy one more unit of X.
Governments routinely pair tax increases with offsetting transfers. Fuel tax increases accompanied by rebate cheques work on exactly this logic: the rebate compensates the average household, but people who drive less than average come out ahead, and people who drive more come out behind. The analysis in this problem set is a simplified version of that policy debate.
Students often think a lump-sum subsidy that "offsets" a tax makes everyone equally well off. It does not. The subsidy compensates the average consumer exactly, but anyone above or below average is affected differently.
Students confuse a lump-sum subsidy (shifts the whole line outward) with a per-unit subsidy (pivots the line). A lump-sum transfer does not change relative prices.
Students sometimes forget that the slope of the budget line depends only on the price ratio, not on income. A lump-sum subsidy changes the intercepts but not the slope.
Students mix up which intercept moves when a tax is applied. A tax on good X only changes the X-intercept; the Y-intercept stays put.
Drawing budget lines before and after a policy change is a staple exam question. Always calculate both intercepts and the slope for each line.
Finding the intersection of two budget lines tests whether you can solve two linear equations simultaneously. Practise the algebra.
"Who benefits?" questions require you to compare the old and new budget sets on either side of the intersection point.
A lump-sum subsidy is income-neutral at the average consumption level by construction. Know why.
Fill in the blank: When the price of tigers doubles from $2,500 to $5,000, the T-intercept changes from ______ to ______.
14 to 7.
True or false: A lump-sum subsidy changes the slope of the budget line.
False. It shifts the line outward but the slope (price ratio) stays the same.
Fill in the blank: The old and new budget lines intersect at T = ______.
4 tigers.
True or false: Consumers who buy more than 4 tigers benefit from the tax + subsidy policy.
False. They are worse off because their tax bill exceeds the $10,000 subsidy.
Q: Draw the budget line before and after the tiger tax. Tigers on the x-axis, Y on the y-axis. Income = $35,000, P_T goes from $2,500 to $5,000, P_Y = $2,500.
A: Before tax, the line runs from (14, 0) to (0, 14) with slope -1. After tax, it runs from (7, 0) to (0, 14) with slope -2. The line pivots inward around the Y-intercept at 14.
Q: The government gives a $10,000 lump-sum subsidy alongside the tax. Draw the new budget line and find where it intersects the original (pre-tax) budget line.
A: New income = $45,000. New budget line: 5,000T + 2,500Y = 45,000, running from (9, 0) to (0, 18) with slope -2. Intersection with the original line (2,500T + 2,500Y = 35,000): subtract to get 2,500T = 10,000, so T = 4, Y = 10. They cross at (4, 10).
Q: Who benefits from the combined tax and subsidy, consumers who buy many tigers or those who buy few?
A: Consumers who buy fewer than 4 tigers benefit. Their additional tax is less than the $10,000 subsidy, so they end up with more purchasing power. The new budget line lies outside the original for T < 4. Consumers who buy more than 4 tigers are worse off because the tax on their extra tigers exceeds the subsidy.
Budget constraints pair with indifference curves to find the optimal bundle (the tangency point). This topic also connects to welfare economics: the question of who gains and who loses from a tax-and-transfer policy is a core theme in public finance. Understanding how budget lines shift is essential before you tackle income and substitution effects.
Budget constraint, budget line, budget set, affordable set, feasible set, excise tax, per-unit tax, lump-sum subsidy, lump-sum transfer, direct payment, income effect, pivot, parallel shift, intercepts, price ratio, slope of budget line, tax incidence, tax and subsidy analysis, Intro to Microeconomics, NYU Econ