Lecture 4 · Week 5 · Besanko & Braeutigam ch. 4.1 · Varian ch. 2 · about 55 min

The Budget Constraint

What a consumer can afford: the budget set and budget line, its intercepts and slope, how it moves when income or prices change, and the kinked budget lines created by discounts, rationing, taxes, subsidies and food stamps.

By the end you can

Write the budget line in the lecture notation and find both intercepts and the slope
Read the slope as the market's opportunity cost of one good in terms of the other
Predict how the line moves when income, one price, or everything changes
Draw kinked budget sets for quantity discounts, rationing, taxes above a quota and food stamps
Tell a lump-sum tax from a per-unit tax by what it does to the slope

Key idea · Exam recipe: the budget line (memorise this)

Budget equation: $P_x x + P_y y = I \implies y = \dfrac{I}{P_y} - \dfrac{P_x}{P_y}x$ .
Two intercepts: Horizontal intercept is $x_{\max} = I/P_x$ (spend all income on food); vertical intercept is $y_{\max} = I/P_y$ (spend all income on clothing).
Budget slope: $\text{Slope} = -P_x / P_y$ . It measures the market opportunity cost of good $x$ in terms of good $y$ .
Income shift ( $\Delta I$ ): Shifts the budget line in parallel outward ( $I \uparrow$ ) or inward ( $I \downarrow$ ). Slope $-P_x/P_y$ is unchanged.
Price pivot ( $\Delta P_x$ or $\Delta P_y$ ): Pivots around the intercept of the good whose price did not change. $P_x \uparrow \implies$ line gets steeper; $P_x \downarrow \implies$ line gets flatter.
Pure inflation: Multiplying $I$ , $P_x$ , and $P_y$ by the same factor $k$ leaves the budget line completely unchanged.

How the exam asks it: three core exam templates

Exam questions on budget constraints test your ability to construct budget equations, calculate intercepts and slopes, trace shifts versus pivots, and analyse policy interventions (taxes, subsidies, rationing).

Exam template 1: Intercepts, slope, and bundle affordability

Question: A consumer has an income of 800 dollars. Food ( $x$ ) costs $P_x = 20$ dollars per unit and clothing ( $y$ ) costs $P_y = 40$ dollars per unit.
a) Write down the budget line equation and express $y$ as a function of $x$ .
b) Find the horizontal and vertical intercepts and calculate the slope of the budget line.
c) Determine whether bundles $A = (10, 10)$ , $B = (20, 10)$ , and $C = (30, 10)$ are affordable, exactly exhausting income, or unaffordable.

Exam template 2: Comparative statics — parallel income shift vs price pivot

Question: Starting from the baseline $I = 800$ dollars, $P_x = 20$ dollars, and $P_y = 40$ dollars:
a) Income increases to $I' = 1000$ dollars while prices remain unchanged. Find the new intercepts, slope, and describe the geometric movement.
b) Alternatively, income remains at 800 dollars and $P_y = 40$ dollars, but the price of food increases to $P_x' = 40$ dollars. Find the new intercepts, slope, and describe the geometric movement.

Exam template 3: Kinked budget line with a bulk discount

Question: A consumer has an income of 800 TL and clothing costs $P_y = 40$ TL. Food costs $P_{x1} = 20$ TL per unit for the first 20 units ( $x \le 20$ ), but drops to $P_{x2} = 10$ TL per unit for all subsequent units ( $x \gt 20$ ).
a) Find the coordinates of the kink point on the budget line.
b) Determine the slopes of the budget line before and after the kink.
c) Find the maximum quantity of food the consumer can afford.

Cheat-sheet & the 4-step method

· Cheat-sheet: budget constraint formulas and motions

Scenario / Policy	Equation	Food Intercept ( $x_{\max}$ )	Clothing Intercept ( $y_{\max}$ )	Slope	Visual Motion
Standard Baseline	$P_x x + P_y y = I$	$\dfrac{I}{P_x}$	$\dfrac{I}{P_y}$	$-\dfrac{P_x}{P_y}$	Straight downward-sloping line
Income Rises ( $I \uparrow$ )	$P_x x + P_y y = I'$	$\dfrac{I'}{P_x} \gt \dfrac{I}{P_x}$	$\dfrac{I'}{P_y} \gt \dfrac{I}{P_y}$	$-\dfrac{P_x}{P_y}$ (unchanged)	Parallel shift outward
Food Price Rises ( $P_x \uparrow$ )	$P_x' x + P_y y = I$	$\dfrac{I}{P_x'} \lt \dfrac{I}{P_x}$	$\dfrac{I}{P_y}$ (fixed)	$-\dfrac{P_x'}{P_y}$ (steeper)	Inward pivot around $(0, I/P_y)$
Clothing Price Rises ( $P_y \uparrow$ )	$P_x x + P_y' y = I$	$\dfrac{I}{P_x}$ (fixed)	$\dfrac{I}{P_y'} \lt \dfrac{I}{P_y}$	$-\dfrac{P_x}{P_y'}$ (flatter)	Inward pivot around $(I/P_x, 0)$
Pure Inflation ( $kP_x, kP_y, kI$ )	$kP_x x + kP_y y = kI$	$\dfrac{I}{P_x}$	$\dfrac{I}{P_y}$	$-\dfrac{P_x}{P_y}$	No change at all
Bulk Discount on Food	$P_x$ drops for $x \gt \bar{x}$	$\bar{x} + \dfrac{I - P_{x1}\bar{x}}{P_{x2}}$	$\dfrac{I}{P_y}$	Flattens after kink	Kinked line; bulges outward
Rationing on Food ( $x \le \bar{x}$ )	Budget capped at $\bar{x}$	$\bar{x}$	$\dfrac{I}{P_y}$	Normal until $\bar{x}$	Cut off vertically at $\bar{x}$

Key idea · The 4-step method for budget constraint questions

Step 1: Write down the budget equation.
- State $P_x x + P_y y = I$ and identify the given numerical values of $P_x, P_y,$ and $I$ .
Step 2: Find both intercepts and the slope.
- Set $y = 0 \implies x_{\max} = I/P_x$ .
- Set $x = 0 \implies y_{\max} = I/P_y$ .
- Compute slope $=-P_x/P_y$ .
Step 3: Track any exogenous change (shift vs pivot).
- Income change: slope unchanged $\implies$ shift in parallel.
- Price change: pivot around the other good’s intercept.
- Price change partway: find the kink coordinate $(x_{\text{kink}}, y_{\text{kink}})$ and write both slopes.
Step 4: Check and conclude in bold.
- Confirm budget balance: $P_x x^* + P_y y^* = I$ .
- State the economic meaning in bold (e.g. opportunity cost, affordability).

Preferences are not enough

Lecture 3 described what a consumer wants (preferences, utility, indifference curves). A consumer cannot buy everything they want: income is limited and goods have prices. The budget constraint describes what a consumer can afford. Lecture 5 (Consumer Choice) then puts the two together to find the best affordable bundle.

Throughout this lesson we use the notation from the board: two goods, food $x$ on the horizontal axis and clothing $y$ on the vertical axis, prices $P_x$ and $P_y$ , and income $I$ .

The equation, the intercepts and the slope

Budget lineKey formulaFormula sheet →

P_x x + P_y y = I \quad\Longrightarrow\quad y = \frac{I}{P_y} - \frac{P_x}{P_y}\,x

$I$: income
$P_x$: price of food (good x)
$P_y$: price of clothing (good y)
$I/P_y$: vertical intercept
$-P_x/P_y$: slope

Use it when you need the affordable bundles, the intercepts or the slope of a budget line.

To draw the line you only need its two ends:

Spend everything on clothing ( $x = 0$ ): $y = I/P_y$ . This is the vertical intercept, the one on the board.
Spend everything on food ( $y = 0$ ): $x = I/P_x$ . This is the horizontal intercept.
Join them with a straight line. Its slope is rise over run: $-\dfrac{I/P_y}{I/P_x} = -\dfrac{P_x}{P_y}$ .

Budget line interceptsFormula sheet →

x_{max} = \frac{I}{P_x} \qquad y_{max} = \frac{I}{P_y}

$x_{max}$: food bought if all income goes to food
$y_{max}$: clothing bought if all income goes to clothing

Use it when you are drawing the budget line or checking how far it moved.

The slope is an opportunity cost

The slope $-P_x/P_y$ tells you how many units of clothing the consumer must give up to buy one more unit of food, while staying on the budget line. In the board example one unit of food costs 20 TL, which is the price of half a unit of clothing: the slope is $-0.5$ .

Try it: move the budget line yourself

The dashed line is the board example. Change one thing at a time and watch what moves. Then use the test bundle to check affordability.

budget linestarting line (I = 800, Px = 20, Py = 40)budget set

Income I800Price of food Px20Price of clothing Py40

Food intercept I/Px: 40
Clothing intercept I/Py: 20
Slope −Px/Py: -0.5
1 food costs (clothing): 0.5

This is the starting line: I = 800, Px = 20, Py = 40. Move a slider and the dashed copy stays behind for comparison.

Test bundle: food15Test bundle: clothing10

Cost of (15, 10) = 20·15 + 40·10 = 700. Affordable, with 100 left over: inside the budget set.

When income changes: a parallel shift

If only income changes, the prices and therefore the slope $-P_x/P_y$ stay the same. Both intercepts $I/P_x$ and $I/P_y$ change by the same factor. The line shifts in parallel: outward when income rises, inward when it falls.

When one price changes: the line pivots

Definition · Board shorthand: a bar means held constant

On the board each case has a heading, and a bar over a letter means it is held constant:

$I\uparrow\downarrow,\ \bar{P}_x,\ \bar{P}_y$ : income changes, both prices fixed. Parallel shift (above).
$\bar{I},\ P_x\uparrow\downarrow,\ \bar{P}_y$ : the price of food changes. The line pivots around $I/P_y$ . $P_x\uparrow \Rightarrow q_x\downarrow$ : the budget line gets steeper. $P_x\downarrow \Rightarrow q_x\uparrow$ : it gets flatter.
$\bar{I},\ \bar{P}_x,\ P_y\uparrow\downarrow$ : the price of clothing changes. The line pivots around $I/P_x$ . $P_y\uparrow \Rightarrow q_y\downarrow$ : it gets flatter. $P_y\downarrow \Rightarrow q_y\uparrow$ : it gets steeper.

If only $P_x$ changes, the vertical intercept $I/P_y$ does not depend on $P_x$ , so it stays put. The horizontal intercept $I/P_x$ moves, and so does the slope. The line pivots (rotates) around the intercept of the other good.

$P_x$ falls: the food intercept moves out, the line becomes flatter.
$P_x$ rises: the food intercept moves in, the line becomes steeper.
If $P_y$ changes instead, the line pivots around the food intercept $I/P_x$ .

When everything changes together

Multiply income and both prices by the same number $k$ : $kP_x x + kP_y y = kI$ . Divide by $k$ and you get the original line back. Pure inflation that raises all prices and income by the same percentage does not change what the consumer can buy.

A numeraire and the composite good

Because only relative prices matter, we can set one price equal to 1. Divide the board equation $20x + 40y = 800$ by $P_y = 40$ :

$0.5x + y = 20$

Now clothing is the numeraire: prices and income are measured in units of clothing. Food costs 0.5 clothing, income is worth 20 clothing.

Besanko and Braeutigam often use a composite good: put one good of interest on the horizontal axis (say food) and let $y$ be spending on all other goods, with price 1 TL per unit. The budget line becomes $P_x x + y = I$ . The vertical intercept is simply $I$ , and the slope is $-P_x$ : one more unit of food costs $P_x$ TL of everything else.

Kinked and special budget sets

Real budget sets are not always simple triangles. Whenever the price of a good changes partway along the line, or the government limits or gives away a good, the budget line gets a kink. The rule is the same every time: on each piece, the slope is minus the price ratio that applies there.

original line: 20x + 40y = 800new budget line

The first 20 units of food cost 20 each, every unit after that costs 10. Up to x = 20 the line has the normal slope −20/40 = −0.5. At the kink (20, 10) the consumer has spent 400 on food. After it the slope is −10/40 = −0.25, so the line gets flatter and reaches the food axis at 20 + 400/10 = 60 instead of 40. The kink points outward, so this budget set is not convex.

Quantity discounts

If extra units become cheaper beyond some quantity, the line gets flatter after the kink, and the budget set bulges outward (it is no longer convex).

Rationing

If the government caps food at $\bar{x}$ , every bundle with $x \gt \bar{x}$ is ruled out. The budget set is cut by a vertical line at $\bar{x}$ . A ration only matters if the consumer wanted more than $\bar{x}$ .

Taxes and subsidies

A per-unit (specific) tax $t$ on food raises its price to $P_x + t$ . The line pivots inward around the clothing intercept.
An ad valorem tax at rate $\tau$ raises the price to $(1 + \tau)P_x$ . Same kind of pivot.
A per-unit subsidy $s$ lowers the price to $P_x - s$ : the line pivots outward.
A lump-sum tax $T$ lowers income to $I - T$ : parallel inward shift. A lump-sum transfer shifts it outward. The slope does not change.
A tax that applies only above a quota makes the line steeper after the kink. The budget set stays convex.

Food stamps versus cash

This is a classic Besanko and Braeutigam application. Suppose the government gives vouchers worth 10 units of food (200 TL at $P_x = 20$ ) that can only be spent on food.

The consumer can now have 10 food and still spend all 800 TL on clothing: 20 clothing. So the line starts with a flat piece from (0, 20) to (10, 20).
After that it falls with the usual slope $-0.5$ and reaches the food axis at $10 + 800/20 = 50$ .
With 200 TL in cash instead, income becomes 1000 and the whole line shifts out in parallel: intercepts $1000/20 = 50$ and $1000/40 = 25$ .

The only bundles that cash allows but food stamps do not are those with more than 20 clothing (and so less than 10 food). A consumer who would spend less than 200 TL on food anyway would be better off with cash. A consumer who already buys at least 10 units of food is indifferent between the two.

Exam practice

Summary and review

Budget set: $P_x x + P_y y \le I$ . Budget line: $P_x x + P_y y = I$ , or $y = I/P_y - (P_x/P_y)x$ .
Intercepts: $I/P_x$ on the food axis, $I/P_y$ on the clothing axis. Slope: $-P_x/P_y$ .
One more food costs $P_x/P_y$ clothing; one more clothing costs $P_y/P_x$ food.
Income change: parallel shift. One price change: pivot around the other good’s intercept. Everything scaled by the same factor: no change.
Only relative prices matter: you can set one price to 1 (numeraire), or use a composite good with price 1 so the line is $P_x x + y = I$ .
A price that changes partway along the line creates a kink: flatter after a discount, steeper after a tax above a quota.
Lump-sum taxes and transfers shift the line in parallel; per-unit and ad valorem taxes or subsidies on one good change the slope.
Food stamps add a flat piece at the top of the line; cash of the same value is strictly better only for consumers who want less food than the stamps provide.