Every consumer faces three constraints:

Scenario: A tourist arrives in Lisbon with a daily budget of €100.

The consumer spends all income on two goods:

\[\underbrace{p_1 \cdot x_1 + p_2 \cdot x_2}_{Expenditure} = \underbrace{M}_{Budget}\]

From our example: €20 \(\cdot x_1\) + €10 \(\cdot x_2\) = €100

Solving for \(x_2\) (to graph it):

\[x_2 = \frac{M}{p_2} - \frac{p_1}{p_2} \cdot x_1\]

\[x_2 = \frac{100}{10} - \frac{20}{10} \cdot x_1 = 10 - 2x_1\]

Bundles above the budget line are unaffordable ❌

Bundles on the budget line: spend all income ✅

Bundles inside the budget set: affordable with money left over ✅

What happens if our tourist’s budget increases from €100 to €150?

👉 Income change: parallel shift (slope unchanged at \(-p_1/p_2\))

What if meal prices drop from €20 to €10?

👉 Price change of one good: pivot around the other intercept (slope changes!)

The slope of the budget constraint has a direct economic interpretation:

\[\text{Slope} = -\frac{p_1}{p_2}\]

This tells us the market’s exchange rate between the two goods.

How exchange rates shift a tourist’s budget constraint

A British tourist visits Portugal with £500 to spend on:

• 🏨 Accommodation: €80/night
• 🍴 Dining: €20/meal

Problem: A tourist has €200 to spend. Surfing lessons cost €40 each. Fado show tickets cost €25 each.

Step 1: Write the budget constraint

\[40 x_1 + 25 x_2 = 200\]

Step 2: Find intercepts

• If \(x_1 = 0\): \(x_2 = 200/25 = 8\) fado shows
• If \(x_2 = 0\): \(x_1 = 200/40 = 5\) surf lessons

Step 3: Find the slope

\[\text{Slope} = -\frac{p_1}{p_2} = -\frac{40}{25} = -1.6\]

👉 1 extra surf lesson costs 1.6 fado shows

Step 4: Check an interior bundle — (2 surf, 4 fado): \(40(2) + 25(4) = 80 + 100 = 180 \leq 200\) ✅ (inside budget set, €20 unspent)

Today’s Lecture Integration:

1. Income (\(M\)) and prices (\(p_1, p_2\)) define the consumer’s constraint
2. Budget line: \(p_1 x_1 + p_2 x_2 = M\) — all bundles spending exactly all income
3. Budget set: \(p_1 x_1 + p_2 x_2 \leq M\) — all affordable bundles
4. Slope = \(-p_1/p_2\) = market rate of exchange between goods
5. Income changes ➡️ parallel shift
6. Price changes ➡️ pivot (rotation)

Connection to previous lectures: The budget line is the individual-level analog of the PPF — both show feasible combinations and trade-offs.

Next: Lecture 6 — Consumer Preferences and axioms of rationality. We’ll ask: among all affordable bundles, which one does the consumer actually want?

Question: A consumer has income \(M = 120\), with \(p_1 = 15\) and \(p_2 = 10\). The slope of the budget constraint is:

A. \(-10/15\)
B. \(-15/10\)
C. \(-120/15\)
D. \(-120/10\)

Question: If the price of good 2 doubles while income and \(p_1\) stay the same, the budget line:

A. Shifts outward in parallel
B. Pivots inward around the \(x_1\)-intercept
C. Pivots outward around the \(x_2\)-intercept
D. Pivots inward around the \(x_2\)-intercept

Scenario: A Portuguese tourism student plans a weekend trip. She has a budget of €150. She wants to split spending between:

• 🏄‍♀️ Surf lessons: €30 each (\(x_1\))
• 🍷 Wine tasting tours: €25 each (\(x_2\))

Questions:

1. Write the budget constraint equation and solve for \(x_2\).

2. Calculate and interpret the slope. What is the opportunity cost of one surf lesson in terms of wine tastings?

3. Find both intercepts. Draw the budget constraint with \(x_1\) on the horizontal axis.

4. Is the bundle (2 surf lessons, 3 wine tastings) affordable? Is it on the budget line?

5. Suppose the student receives a €50 gift card (total budget now €200). Draw the new budget line on the same graph. What changed and what stayed the same?

6. Now instead of the gift card, suppose surf lesson prices drop to €25 (budget stays at €150). Draw this new line. How does it differ from part (e)?

a) Budget constraint: \(30x_1 + 25x_2 = 150\)

Solving for \(x_2\): \(x_2 = \frac{150}{25} - \frac{30}{25}x_1 = 6 - 1.2x_1\)

b) Slope \(= -p_1/p_2 = -30/25 = -1.2\)

👉 Each surf lesson costs 1.2 wine tastings. The opportunity cost of 1 surf lesson is 1.2 wine tours foregone.

c) Intercepts:

• \(x_1 = 0 \Rightarrow x_2 = 150/25 = 6\) wine tastings (vertical intercept)
• \(x_2 = 0 \Rightarrow x_1 = 150/30 = 5\) surf lessons (horizontal intercept)

Graph: straight line from \((0, 6)\) to \((5, 0)\).

d) Bundle (2, 3): \(30(2) + 25(3) = 60 + 75 = 135 \leq 150\) ✅

Affordable? Yes! On the budget line? No — she spends only €135, leaving €15 unspent. The bundle is inside the budget set.

e) New budget €200: \(30x_1 + 25x_2 = 200 \Rightarrow x_2 = 8 - 1.2x_1\)

• New intercepts: \((0, 8)\) and \((6.67, 0)\)
• Parallel shift outward — slope unchanged at \(-1.2\)

f) Price drop \(p_1 = 25\), \(M = 150\): \(25x_1 + 25x_2 = 150 \Rightarrow x_2 = 6 - x_1\)

• New intercepts: \((0, 6)\) and \((6, 0)\)
• Pivot outward around vertical intercept — slope changes to \(-1\)

Key difference: (e) is a parallel shift (more of both goods equally); (f) is a pivot (relatively more surf lessons become affordable, wine tasting maximum unchanged).