Source: Chapter 3, Consumer Behavior (Section 3.2)
Tags: budget constraint, budget line, income, price ratio, slope of budget line, parallel shift, rotation, ECON 323
A budget constraint captures the trade-off a consumer faces when income is limited. The budget line shows every combination of two goods that exactly exhausts income. Changes in income shift the line in or out (parallel); changes in one good's price rotate the line around the intercept of the other good.
Budget constraint
The restriction consumers face because their income is limited. It defines the set of baskets a consumer can actually afford.
Budget line
All combinations of goods for which the total amount of money spent exactly equals income. Equation: P_F × F + P_C × C = I.
Slope of the budget line
Equal to −P_F / P_C (the negative of the price ratio). It tells you the rate at which the market allows you to trade one good for the other.
Intercepts
Vertical (C-axis): I / P_C – the maximum clothing you can buy if you spend nothing on food.
Horizontal (F-axis): I / P_F – the maximum food you can buy if you spend nothing on clothing.
The consumer spends all income on two goods, food (F) and clothing (C):
P_F × F + P_C × C = I
Solving for C gives the equation you would graph:
C = (−P_F / P_C) × F + I / P_C
The slope is −P_F / P_C. The vertical intercept is I / P_C, and the horizontal intercept is I / P_F.
Worked example: income = $80, P_F = $1, P_C = $2.
Budget line: F + 2C = 80, or equivalently C = −0.5F + 40.
Slope = −1/2.
Vertical intercept: 40 units of clothing. Horizontal intercept: 80 units of food.
A change in income, holding prices constant, shifts the budget line parallel to itself:
Income rises (e.g. $80 to $160): the line shifts outward. Both intercepts double.
Income falls (e.g. $80 to $40): the line shifts inward. Both intercepts halve.
The slope stays the same because prices have not changed.
A change in the price of one good, holding income and the other price constant, rotates the budget line around the intercept of the unchanged good:
If P_F falls (e.g. $1.00 to $0.50): the horizontal intercept moves out (you can buy more food), and the line rotates outward. The vertical intercept stays put.
If P_F rises (e.g. $1.00 to $2.00): the horizontal intercept moves in, and the line rotates inward. The vertical intercept stays put.
The slope changes because the price ratio has changed.
Budget line: P_F × F + P_C × C = I
Slope: −P_F / P_C
Vertical intercept: I / P_C
Horizontal intercept: I / P_F
Parallel shift: income change, prices constant
Rotation: one price changes, income and other price constant
⚠️ The slope of the budget line is the price ratio (−P_F / P_C), not the MRS. These two quantities only equal each other at the optimum.
⚠️ Know the difference between a shift (income change) and a rotation (price change). Exams frequently ask you to draw the new budget line after a policy change.
⚠️ The budget line equation looks like a simple linear equation, but make sure you can rearrange it quickly and identify both intercepts under different parameter values.
⚠️ Baskets below the budget line are affordable but do not exhaust income. Baskets above the line are unaffordable. The line itself represents exact spending of all income.
Q: Write the budget line equation for income = $100, P_X = $5, P_Y = $10.
A: 5X + 10Y = 100, or equivalently Y = −0.5X + 10. Horizontal intercept = 20, vertical intercept = 10.
Q: If income doubles while prices stay the same, what happens to the budget line?
A: It shifts outward, parallel to the original line. The slope does not change.
Q: If the price of food falls while income and the price of clothing remain the same, how does the budget line change?
A: The line rotates outward around the clothing (vertical) intercept. The food (horizontal) intercept increases, and the line becomes flatter (the absolute value of the slope decreases).
Q: What does the slope of the budget line represent?
A: The rate at which the market allows the consumer to trade food for clothing, equal to −P_F / P_C.
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