Budget Constraints drew a budget line from income and two prices. Now we treat it as a set of points, the way Linear Inequalities in Two Variables treats a half-plane, and ask how taxes, discounts and limits reshape it.
Key idea
With income $m$ and prices $p_1, p_2$, the budget set is every bundle with $$p_1x_1 + p_2x_2 \le m, \quad x_1, x_2 \ge 0.$$ Its boundary, the budget line, has intercepts $m/p_1$ and $m/p_2$ and slope $-p_1/p_2$. A change in one price rotates the line around the other intercept; a change in income shifts it in parallel.
Why it works
The slope is an opportunity cost. Giving up one unit of good 1 frees $p_1$ dollars, which buys $p_1/p_2$ units of good 2. So the market lets you trade good 1 for good 2 at the rate $p_1/p_2$, which is the size of the line's slope. If $p_1$ rises, the most good 1 you can buy, $m/p_1$, falls, but the most good 2 you can buy, $m/p_2$, does not change: the line swings around its intercept on the good-2 axis. If income rises, both intercepts grow by the same factor and the slope stays put. And if all prices and income are multiplied by the same number, every term of $p_1x_1 + p_2x_2 \le m$ scales together, so the set doesn't change at all.
Worked example: Intercepts, slope and a price change
Rosa has \$60; books cost \$4 and movies \$3. Find the intercepts and slope, then the new book intercept when books cost \$6.
- Books intercept: $60/4 = 15$. Movies intercept: $60/3 = 20$.
- Slope: $-p_1/p_2 = -4/3$. Each book costs $4/3$ of a movie.
- At \$6 a book, the books intercept is $60/6 = 10$; the movies intercept stays at 20.
Answer: Intercepts 15 and 20, slope $-4/3$; after the change the books intercept is 10.
Worked example: A per-unit tax above a quantity (a kinked budget)
Rosa's city adds a \$2 tax on each book beyond the first 5. With \$60, books at \$4 and movies at \$3, what is the most books she can buy?
- The first 5 books cost $4 \times 5 = 20$ dollars.
- Each book after that costs $4 + 2 = 6$ dollars. The remaining \$40 buys $40/6 = 6\tfrac{2}{3}$ more.
- Most books: $5 + 6\tfrac{2}{3} = 11\tfrac{2}{3}$. The budget line is steeper ($-6/3 = -2$) to the right of $x = 5$: a kink.
Answer: $11\tfrac{2}{3}$ books (in practice 11).
Common mistake
Shifting the whole line in parallel when one price changes. A parallel shift means both intercepts move by the same proportion, which is what an income change does. When only the price of good 1 changes, the good-2 intercept $m/p_2$ cannot move. Check: recompute both intercepts after any change. If only one moved, the line rotated.
- Income up: parallel shift out. Income down: parallel shift in.
- $p_1$ up: rotates inward around the $x_2$-intercept. $p_1$ down: rotates outward.
- A per-unit tax on good 1: like a higher $p_1$. A tax only above some quantity: a kink.
- Rationing (at most $r$ units of good 1): cuts off the set at $x_1 = r$.