The budget constraint

Consumer choice
What can a customer afford, and what does one more unit really cost them?

Learning goals

After this page, you can:

  • Draw the budget line from an income and two prices, and read both intercepts.
  • State the slope as the opportunity cost of one good in units of the other.
  • Predict how the line moves when income changes, and how it rotates when one price changes — including seller deals such as BOGO and memberships.

The idea

A student has $60 a week for lunch at the food truck. Burritos cost $9. Smoothies cost $6. Every combination the student can afford sits on or under one straight line. And every “deal” the truck offers is an attempt to bend that line.

The budget line shows every bundle a customer can buy if they spend the whole budget. It says nothing about taste yet. It only says what is possible.

The two ends of the line. If the student buys only burritos, they get 60 ÷ 9 = 6.7 burritos. If they buy only smoothies, they get 60 ÷ 6 = 10 smoothies. These are the two intercepts. Every point on the line between them spends exactly $60.

The slope is an opportunity cost. One burrito costs $9. That is the same money as 1.5 smoothies. So the true cost of one more burrito is the 1.5 smoothies you give up. This is the opportunity cost, and it is the slope of the line: price of X ÷ price of Y.

Two kinds of change.

  • Income changes → the line moves parallel. With $72 instead of $60, the student can buy 20% more of everything. Both intercepts move out by 20%. The slope does not change, because prices did not change.
  • A price changes → the line rotates. If burritos drop to $6, the burrito end moves out (to 10 burritos). The smoothie end stays where it is, because smoothie prices did not change. The line gets flatter: one burrito now costs only one smoothie.

Sellers bend the line on purpose. A “buy 10 classes, get 5 free” offer, a BOGO deal, or a membership with a lower per-unit price does not change the price of the first units. It makes units after a threshold cheaper. So the line has a corner: it is normal up to the threshold, then flatter beyond it. The customer can now reach bundles that were impossible before — but only if they buy a lot. That is exactly what the seller wants.

Key formulas

\[ P_X \cdot X + P_Y \cdot Y = I \] X-intercept = I ÷ PX · Y-intercept = I ÷ PY · Opportunity cost of one X = PX ÷ PY units of Y

Income change → parallel shift. Price change → rotation around the other intercept.

Try it

Try these:

  1. Raise income from $60 to $72. Do both ends move? Does the slope change?
  2. Go back to $60. Move the burrito price to $6. Which end moves? (The “price cut” slider does the same in percent.)
  3. Turn on the deal. Where does the line bend?
  4. Move your bundle until the dot turns red. What does red mean?

Worked example

Problem. Income $60, burritos $9, smoothies $6. Find the intercepts and the opportunity cost. Then answer: what happens if income rises to $72? What if the burrito price falls to $6?

Step 1. Intercepts. Only burritos: 60 ÷ 9 = 6.7. Only smoothies: 60 ÷ 6 = 10.

Step 2. Opportunity cost. 9 ÷ 6 = 1.5 smoothies per burrito.

Step 3. Income $72. Intercepts become 72 ÷ 9 = 8 and 72 ÷ 6 = 12. Both are 20% higher. The slope is still 1.5. The line moved parallel.

Step 4. Burritos at $6. The burrito intercept becomes 60 ÷ 6 = 10. The smoothie intercept stays at 10. The slope is now 6 ÷ 6 = 1: one burrito costs one smoothie. The line rotated around the smoothie end.

Check yourself

Check 1.

A student has $100 a week. Pizza costs $10 and sandwiches cost $5. With pizza on the horizontal axis, the slope of the budget line means one pizza costs:

Slope = Px/Py = the other good you give up for one more unit.

Check 2.

The student gets a 20% raise and prices do not change. The budget line:

Income shifts the line in parallel; a price change rotates it.

Check 3.

A gym offers ‘buy 10 classes, get 5 free.’ On a student’s budget line (classes vs. everything else), this deal:

Quantity deals create a kink: the line bends outward past the threshold.

Check 4.

Income $60, burritos $9, smoothies $6. Which bundle is exactly affordable (spends all $60)?

On the line, Px·X + Py·Y = I exactly.

Check 5.

Burrito price falls from $9 to $6 while income ($60) and the smoothie price ($6) stay the same. What happens to the budget line?

One price change → rotation around the other good’s intercept.

Video

Video coming soon.