ECON7870 Chap.2 Preference Relations and Utility
Preference Relations and Utility
Rank before measuring
A preference relation says which bundle is preferred or whether two bundles are tied. A utility function assigns numbers that preserve that ranking; it does not create cardinal units of satisfaction.
Completeness, transitivity and continuity support a consistent numerical representation, while monotonicity and convexity add economic content about more goods and mixtures.
Test representations by order
Strictly increasing transformations preserve the ranking, indifference curves and marginal trade-offs. The fastest refutation is often a pair of bundles tied by one function but ranked differently by another.
The proof should state exactly what is preserved instead of comparing how formulas look.
Match evidence to the property
Completeness needs a comparison for every pair, transitivity needs a chain of rankings, and convexity needs a statement about mixtures. A graph may suggest the answer, but the definition tells you what must be shown.
Positive marginal utilities can support monotonicity on a smooth domain, while a midpoint comparison can expose a convexity violation without calculus.
Respect domains and boundaries
A transformation may be increasing only on part of its domain. Squaring preserves order for non-negative utility but not across positive and negative values. Logarithms require positive arguments.
State the domain before claiming equivalence, especially when bundles on an axis are feasible. A correct formula with an unstated domain can assert a ranking where the representation is not even defined.
Use the asymmetry of proof
One counterexample can disprove a universal property, but one confirming example rarely proves it.
When a preference looks non-convex, search for two tied bundles whose midpoint is worse. When defending convexity, use a general contour-set or mixture argument. End by translating the algebra back into a ranking statement so the economic claim is visible.
Choose a proof strategy deliberately
For equivalence, first state the candidate transformation and show it is strictly increasing over the relevant range.
For a failed equivalence, construct two feasible bundles whose order or tie changes. For convexity, select two bundles in an upper contour set and test their mixture; for non-convexity, one worse midpoint is enough. Keep continuity separate from convexity, because smooth indifference curves need not reward mixtures.
A disciplined proof ends with the preference claim in words, identifying whether the evidence establishes a universal property or merely illustrates it at one pair of bundles.
What this chapter covers
- 01
Complete preference relations
- 02
Transitivity and cycles
- 03
Continuity
- 04
Monotonicity
- 05
Convexity and mixtures
- 06
Ordinal utility
- 07
Monotonic transformations
Worked example · free
Test two utility representations
- 2Express V as a function of U.
- 2Check whether the transformation is strictly increasing on the domain.
- 2State what happens to rankings and ties.
Key terms
- Completeness
- The property that every pair of bundles can be compared by the preference relation.
- Transitivity
- The consistency rule that prevents ranked alternatives from forming a cycle.
- Continuity
- The property that rankings do not jump in a way that nearby bundles cannot approach.
- Monotonicity
- The condition that more of a good is weakly preferred when nothing else is reduced.
- Convexity
- The preference for mixtures over equally ranked extremes, producing diminishing marginal willingness to substitute.
Preference Relations and Utility FAQ
Does a higher utility number measure more happiness?
Not in the cardinal sense used by that question. Utility in this part of consumer theory preserves order: a higher number identifies a preferred bundle. Multiplying every value by a positive constant or applying another strictly increasing transformation can change the numbers while leaving every choice unchanged.
What is the difference between monotonicity and convexity?
Monotonicity concerns direction: more of a good does not make the consumer worse off, giving downward-sloping indifference curves under standard conditions. Convexity concerns mixtures and diminishing willingness to substitute, making curves bow toward the origin. One property cannot be inferred from the other.
How can I disprove a preference property?
Provide a counterexample that directly violates the definition. A ranking cycle disproves transitivity, a bundle with more of every good ranked lower disproves monotonicity, and a midpoint ranked below equally preferred endpoints disproves convexity. One valid counterexample is enough even though one confirming example proves little globally.
Exam move
Write each property as a diagnostic question and construct one counterexample. Then practise deciding whether a claim is local or global. For utility transformations, test a tied pair before differentiating; equality must be preserved by every valid representation of the same preference ordering.
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