2 Preferences and Indifference Curves

Bundles and preference relations

A Bundle is a combination of goods. For example, $A=(2,3)$ might contain two coffees and three sandwiches. Preferences rank bundles, but do not by themselves account for prices or income.

Let $x$ and $y$ be bundles. Weak preference means that the consumer considers $x$ at least as good as $y$, written $x\succeq y$. Strict preference means that the consumer prefers $x$ to $y$: $x\succeq y$, but not $y\succeq x$; it is written $x\succ y$. Indifference means the consumer regards the bundles as equally good: $x\succeq y$ and $y\succeq x$, written $x\sim y$.

These relations describe rankings, not measurements of satisfaction. A Utility function can assign numbers to bundles to represent a ranking, but those numbers ordinarily express order rather than objectively measurable amounts of happiness. A utility of $10$, for example, does not necessarily mean twice as much satisfaction as a utility of $5$.

Completeness and transitivity

Completeness and Transitivity are core assumptions, often called rationality assumptions in the standard model. They are useful modeling assumptions, not claims that people always make perfectly consistent or fully considered choices.

Completeness means that for any two bundles $x$ and $y$, the consumer can rank them: $x\succeq y$, or $y\succeq x$, or both. For example, when comparing two coffees and one sandwich with one coffee and two sandwiches, completeness requires the consumer to identify one bundle as at least as good as the other, or regard them as equally good. It does not require a strict preference. This assumption can be demanding when alternatives are unfamiliar or difficult to compare.

Transitivity means rankings are consistent across comparisons. If $x\succeq y$ and $y\succeq z$, then $x\succeq z$. If a consumer prefers bundle $A$ to $B$, and $B$ to $C$, transitivity rules out preferring $C$ to $A$. Without transitivity, pairwise choices could cycle, making it difficult to identify a best bundle.

Indifference curves and maps

An Indifference curve is the set of all bundles equally preferred to a particular bundle. If a consumer is indifferent between $A$, $B$, and $C$, all three lie on the same curve. Moving along the curve changes quantities consumed while keeping the consumer at the same preference level.

For example, suppose a consumer is indifferent between $A=(2,10)$ and $B=(4,6)$, where the first quantity is sandwiches and the second is coffees. In this comparison, the consumer is willing to trade four coffees for two more sandwiches without changing preference ranking. The trade-off need not remain constant elsewhere on the curve.

An Indifference map is a collection of indifference curves representing rankings across many bundles. When preferences are monotonic, bundles on a higher curve are preferred to bundles on a lower one.

How curve shape represents preferences

The usual properties of indifference curves in a graph with quantities of two goods on the axes depend on assumptions beyond completeness and transitivity.

• Downward-sloping: If a consumer gets more of one good but remains equally satisfied, they must give up some of the other good. This shape follows from the assumption that more of each good is preferred; it is not implied by completeness and transitivity alone.

• Non-intersecting: Curves representing different preference levels cannot cross. If they did, the shared bundle and transitivity of indifference would imply that bundles assigned to different curves were equally preferred, contradicting the distinction between those levels.

• Higher means better under monotonicity: Moving northeast increases both goods. Without monotonicity, a higher curve does not necessarily represent a better bundle.

The Marginal rate of substitution (MRS) is the amount of one good the consumer would give up to obtain an additional amount of the other while remaining indifferent. It is described by the curve's slope. A curved, convex-to-the-origin indifference curve typically becomes flatter as the consumer has more of the horizontal-axis good, reflecting a changing willingness to trade.

Further assumptions about preferences

Consumer theory uses further assumptions to make preference maps easier to analyze. These are distinct assumptions, not consequences of completeness and transitivity.

• Monotonicity (“more is better”) says that if one bundle contains at least as much of every good as another, and strictly more of at least one, it is strictly preferred. Under monotonicity, ordinary indifference curves for two desirable goods slope downward.

• Nonsatiation is a weaker related condition: a consumer can find a nearby bundle they prefer. It does not require every additional unit of every good to improve the bundle.

• Convexity says that a balanced combination of two equally preferred bundles is at least as good as either extreme, modeling a preference for variety. Strictly convex preferences favor a mixture strictly over either distinct endpoint and often produce curves bowed toward the origin. Convexity does not require every curve to be smoothly bowed: perfect substitutes can have straight indifference curves, while perfect complements can have kinked ones.

• Continuity says that small changes in a bundle do not cause abrupt jumps in preference rankings. This regularity assumption helps ensure that preference boundaries behave well, but does not imply smooth indifference curves.

Smooth indifference curves are often assumed for calculus-based analysis. Smoothness is an additional technical convenience, not a universal feature of preferences.