1 Preferences and Indifference Curves

How preferences rank bundles

Consumer theory separates what a person prefers from what they can afford. A preference relation ranks consumption bundles; indifference curves provide a graphical representation of that ranking. These rankings describe how bundles compare, not their affordability or the strength of a person's feelings.

Preference relations

A bundle is a specified combination of goods. For example, with apples and cookies, bundles might be $A=(2,5)$ and $B=(4,2)$.

The notation $A \succeq B$ means that $A$ is weakly preferred to $B$, or at least as good as $B$. Strict preference is written $A \succ B$: $A \succeq B$, but not $B \succeq A$. Indifference is written $A \sim B$: both $A \succeq B$ and $B \succeq A$. Indifference does not require bundles to contain the same quantities of each good.

Completeness and transitivity

The standard model commonly assumes that preferences are complete and transitive. Completeness means that for any two bundles $A$ and $B$, either $A \succeq B$, or $B \succeq A$, or both. If both hold, the consumer is indifferent. Completeness rules out an unranked pair, but does not require a strict preference.

Transitivity means that for any three bundles, if $A \succeq B$ and $B \succeq C$, then $A \succeq C$. It rules out inconsistent cycles in the ranking. For example, if a fruit basket is ranked at least as highly as a snack box, and the snack box at least as highly as a sandwich, transitivity requires the fruit basket to rank at least as highly as the sandwich.

These assumptions make rankings orderly and useful for modeling choice. They are simplifying assumptions, not claims that every person always makes perfectly consistent decisions.

Indifference curves and maps

An indifference curve is the set of all bundles that a consumer regards as equally good. If $A \sim B$, both bundles lie on the same curve. A collection of such curves is an indifference map.

Imagine apples on the horizontal axis and cookies on the vertical axis. A curve might pass through $(2,5)$, $(3,3)$, and $(5,2)$. If the consumer is indifferent among these bundles, each has the same preference ranking even though the bundles differ in composition.

What shapes indifference curves

The shape and position of indifference curves depend on the consumer's preferences and on assumptions about the goods.

• If more is better for both goods, a bundle with more of both is preferred. Indifference curves therefore cannot slope upward over a region: an upward-sloping move would add both goods, so it could not leave the consumer equally well off.

• Under complete and transitive preferences, two distinct indifference curves cannot cross. If they crossed at a bundle, the rankings along the curves would imply that a bundle on one curve is both indifferent to and ranked differently from a bundle on the other, contradicting the definition of the curves and transitivity.

• Convex preferences mean that mixtures or balanced combinations of two equally valued bundles are at least as good as either extreme. They are commonly represented by curves bowed toward the origin. Convexity is an additional assumption, not a consequence of completeness and transitivity alone.

The usual picture of downward-sloping, convex curves, with higher curves representing preferred bundles, relies on additional assumptions about the goods and preferences. Special cases such as perfect substitutes or perfect complements can instead have straight or kinked curves.

Additional assumptions and their roles

Consumer models often use assumptions beyond completeness and transitivity. These assumptions add different kinds of structure and are not interchangeable.

• Monotonicity ("more is better"): More of a good, with other goods unchanged, is preferred. This is appropriate when the items are genuinely desirable goods, but not when a variable represents a bad such as pollution.

• Local nonsatiation: However close a bundle is, there is always another nearby bundle the consumer prefers. This weaker condition allows some goods not to be desirable in every circumstance and is often useful when analyzing choice.

• Continuity: Small changes in bundles do not produce abrupt reversals in ranking. Continuity supports smooth analysis and, under standard conditions, a utility representation.

• Convexity: Averages of equally preferred bundles are at least as good as the bundles averaged. This captures a preference for variety and helps explain the typical bowed shape of indifference curves.

Completeness and transitivity describe the consistency of rankings, whereas monotonicity, continuity, and convexity add further structure. A model should use only assumptions appropriate to the question being studied.