Preference Relations and Axioms

This scene draws your preferences on the two-good plane.

A ranking is rational when nothing is left unranked. It also never loops back on itself. Such a ranking can be written as utility numbers, as in the formula below. Change the utility family and the whole curve map changes. You also see what breaks when an axiom fails.

For example, suppose a 🍎 apple bundle beats a 🍌 banana bundle. And the banana bundle beats a 🍞 bread bundle. Then the apple bundle must beat the bread bundle too.

x \succsim y \iff u(x) \ge u(y)

\text{complete} + \text{transitive} \Rightarrow \text{rational}

Part of these learning paths

Examples & Exercises

A Cobb-Douglas consumer has $u(x_1, x_2) = x_1^{0.4} x_2^{0.6}$ .

Compute the utility of the two bundles $A = (4, 3)$ and $B = (3, 5)$ directly and decide which is preferred.

x \succsim y \iff u(x) \ge u(y)

a = 0.4

A = (4, 3)

B = (3, 5)

— two indifference curves and the verdict.

Show proof

Taking logs with $a = 0.4$ gives $\ln u = 0.4 \ln x_1 + 0.6 \ln x_2$ .
For $A$ : $\ln u(A) = 0.4 \ln 4 + 0.6 \ln 3 \approx 0.5545 + 0.6592 = 1.2137$ , so $u(A) \approx 3.37$ .
For $B$ : $\ln u(B) = 0.4 \ln 3 + 0.6 \ln 5 \approx 0.4394 + 0.9657 = 1.4051$ , so $u(B) \approx 4.08$ . Since $u(B) > u(A)$ , $B \succ A$ — in the picture the dashed indifference curve through $B$ lies outside the solid one through $A$ .

Consider perfect-complements utility $u(x_1, x_2) = \min\{x_1, x_2\}$ .

Which of $A = (4, 4)$ and $B = (6, 4)$ is preferred?

Explain why the answer does not change even though $B$ has more $x_1$ .

u(A) = \min\{4, 4\} = 4, \qquad u(B) = \min\{6, 4\} = 4

Perfect complements,

A = (4, 4)

B = (6, 4)

—

B

sits on

A

's indifference curve.

Show proof

$u(A) = \min\{4, 4\} = 4$ and $u(B) = \min\{6, 4\} = 4$ , so $A \sim B$ — the consumer is indifferent.
The indifference curve is L-shaped with its kink at $(4, 4)$ .
$B = (6, 4)$ lies exactly on the horizontal arm of that L — with $x_2$ stuck at 4, raising $x_1$ from 4 to 6 adds no utility at all.
That is what "perfect complements" means: unmatched extra units of a good are worthless.

2. Preference and Choice