A bundle (or basket) is a specific combination of goods:

\[\text{Bundle } A = (x_1^A, \; x_2^A)\]

Example: A tourist choosing between activities in Lisbon:

Preferences describe how a consumer ranks these bundles — without needing prices or income!

For any two bundles \(A\) and \(B\), the consumer can say:

People’s preferences can be anything — messy, emotional, contradictory.

Economists assume preferences satisfy certain axioms (basic rules) so that we can:

• Model consumer behavior mathematically
• Predict how choices change when prices or income change
• Build the concept of utility functions (Lecture 7)

👉 These axioms don’t say what people prefer — just that preferences are logically consistent.

Think of them as the “rules of the game” for rational choice.

In plain language: You are never “stuck” — you can always decide which bundle is at least as good.

In plain language: Rankings must be logically consistent — no cycles.

The “money pump” argument:

Imagine a tourist with intransitive preferences: \(A \succ B \succ C \succ A\)

In plain language: More of a good thing is always preferred, all else equal.

Points \(A\), \(B\), \(C\) are on the same curve (\(U_2\)): the tourist is indifferent among them.

The axioms imply four key properties:

Proof by contradiction (using the axioms):

• \(X\) and \(Y\) are on \(IC_1\) ➡️ \(X \sim Y\)
• \(X\) and \(Z\) are on \(IC_2\) ➡️ \(X \sim Z\)
• By transitivity: \(Y \sim Z\)
• But \(Y = (2, 4)\) has more of \(x_2\) than \(Z = (2, 3.5)\) (same \(x_1\)) ➡️ monotonicity says \(Y \succ Z\) 💥 Contradiction!

The average of \(A\) and \(B\) lies above the indifference curve ➡️ it is on a higher curve ➡️ preferred!

💡 No convexity here — the consumer doesn’t care about “balance.”

💡 More of one good without the other gives no extra satisfaction.

Most goods lie between the two extremes:

How do we know tourist preferences in practice?

We can’t read minds — but we can observe choices!

💡 This is why tourism companies collect so much data — they’re mapping your indifference curves!

Today’s Lecture:

1. Preferences rank bundles — independently of prices or income
2. Four axioms make preferences “rational” and modelable:
   • Completeness: always able to compare
   • Transitivity: no preference cycles
   • Monotonicity: more is better
   • Convexity: mixtures preferred to extremes
3. Indifference curves: connect bundles giving equal satisfaction
4. IC properties: downward-sloping, don’t cross, higher = better, convex
5. Special cases: perfect substitutes (lines) and perfect complements (L-shapes)

Connection to Lecture 5: Budget constraint shows what’s feasible; preferences show what’s desirable.

Next (Lecture 7): The Marginal Rate of Substitution (the slope of the IC) and utility functions — how we turn preferences into math and find the optimal choice!

Question: A tourist says: “I prefer an Algarve beach holiday to a Douro wine tour, I prefer a Douro wine tour to a Lisbon city break, and I prefer a Lisbon city break to an Algarve beach holiday.” This violates:

A. Completeness
B. Transitivity
C. Monotonicity
D. Convexity

Question: Indifference curves that are L-shaped represent goods that are:

A. Perfect substitutes
B. Normal goods with convex preferences
C. “Bads” (undesirable)
D. Perfect complements

Scenario: Consider a tourist choosing between two types of vacation days: 🌞 Beach days (\(x_1\)) and 🏛️ City sightseeing days (\(x_2\)). Assume preferences satisfy all four axioms.

Questions:

1. The tourist is indifferent between bundle \(A = (2, 6)\) and bundle \(B = (6, 2)\). Draw an indifference curve through both points (sketch it with the typical convex shape). Label it \(U_1\).

2. Consider bundle \(D = (4, 4)\). Using the axiom of convexity, explain why \(D\) is preferred to \(A\) (or \(B\)). Show \(D\) on your graph — is it above, on, or below \(U_1\)?

3. Now consider bundle \(E = (1, 5)\). Can you definitively say whether \(E\) is better or worse than \(A\) without more information? Why or why not?

4. The tourist says: “I prefer bundle \(F = (3, 3)\) to \(A\), but I also prefer \(A\) to \(G = (5, 5)\).” Does this violate any axiom? Which one and why?

5. Suppose these are perfect substitutes (1 beach day = 1 sightseeing day). Redraw the indifference curve through \(A\) and \(B\). What is different about its shape?

a) The indifference curve through \(A = (2, 6)\) and \(B = (6, 2)\) is a smooth, downward-sloping, convex curve (bowed toward the origin). All points along this curve give the same satisfaction level \(U_1\).

b) Bundle \(D = (4, 4)\) is the average of \(A\) and \(B\):

\[D = \frac{1}{2} A + \frac{1}{2} B = \left(\frac{2+6}{2}, \frac{6+2}{2}\right) = (4, 4)\]

By the convexity axiom: if \(A \sim B\), then any mixture is weakly preferred.

So \(D \succsim A \sim B\).

Since \(D\) is on the straight line connecting \(A\) and \(B\), and the indifference curve is convex (bowed below this line), \(D\) lies above \(U_1\) ➡️ \(D\) is on a higher indifference curve ➡️ \(D \succ A\).

c) Bundle \(E = (1, 5)\) vs \(A = (2, 6)\): \(E\) has less of \(x_1\) (1 < 2) and less of \(x_2\) (5 < 6).

By monotonicity: \(A \succ E\). Yes, we can say \(E\) is worse — \(A\) dominates \(E\) in both goods.

d) The tourist says \(F = (3,3) \succ A\) and \(A \succ G = (5,5)\).

This violates monotonicity: \(G = (5,5)\) has strictly more of both goods than \(F = (3,3)\), so monotonicity requires \(G \succ F\). Combined with \(F \succ A \succ G\), by transitivity we’d need \(F \succ G\). But \(G\) dominates \(F\) — contradiction!

The statement violates monotonicity (and by extension creates a transitivity conflict with monotonicity).

e) If perfect substitutes at 1:1, the indifference curve through \(A = (2,6)\) and \(B = (6,2)\) is a straight line with slope \(-1\): \(x_1 + x_2 = 8\). All bundles summing to 8 give equal satisfaction. The curve is linear, not convex — no preference for variety.