ECON5001 · Microeconomic Theory
Preferences, Utility & the MRS
A consumer in ECON5001 is described entirely by a preference relation over bundles of goods. Four axioms — completeness, reflexivity, transitivity and continuity — make that ranking sensible enough to draw as a non-crossing indifference map and to summarise with a utility function. Because utility is ordinal, the numbers only rank bundles; any positive monotonic transformation represents the same preferences and leaves every choice unchanged. The chapter's keystone is the marginal rate of substitution, MRS = −MU₁/MU₂ — the slope of an indifference curve and the rate at which a consumer will just trade good 2 for good 1. Get the axioms, the well-behaved (monotonic + convex) curve shape, the menagerie of utility forms, and the MRS straight here, and the optimal-choice, demand and exchange chapters become mechanical.
What this chapter covers
- 011. The preference relation — strict ≻, weak ⪰, indifference ∼, and the four axioms
- 022. Indifference curves & the indifference map — level sets of the preference relation
- 033. Why indifference curves cannot cross — the two-axiom proof by contradiction
- 044. Well-behaved preferences — monotonicity (more is better) and convexity (variety)
- 055. Diminishing MRS — the signature of convex, bowed-in indifference curves
- 066. The marginal rate of substitution — MRS = −MU₁/MU₂, computed and interpreted
- 077. The utility menagerie — Cobb-Douglas, perfect substitutes, complements, quasi-linear
- 088. Ordinal utility — positive monotonic transformations leave the MRS unchanged
Indifference curve & the MRS for a Cobb-Douglas utility
- +1Find the utility level she is on: U = 6² × 9 = 36 × 9 = 324. Staying indifferent means keeping U = 324.
- +1(a) Set x₁ = 9 on the same curve: 9²·x₂ = 324, so 81·x₂ = 324.
- +1(a) Solve: x₂ = 324 ÷ 81 = 4. So (9, 4) lies on the same indifference curve as (6, 9).
- +1(b) Marginal utilities: MU₁ = ∂U/∂x₁ = 2x₁x₂ and MU₂ = ∂U/∂x₂ = x₁².
- +1(b) Form the ratio: |MRS| = MU₁/MU₂ = 2x₁x₂ / x₁² = 2x₂/x₁.
- +1(b) Evaluate at (6, 9): |MRS| = (2 × 9) / 6 = 18/6 = 3. She will give up 3 gym-sessions for one more study-hour and stay equally well off.
Key terms
- Preference relation
- A consumer's ranking over bundles, written ≻ (strictly preferred), ⪰ (weakly preferred) and ∼ (indifferent). It is the primitive object of consumer theory — utility is just a convenient label for it.
- Completeness / transitivity / continuity
- The core axioms: completeness (any two bundles are comparable), transitivity (rankings chain without cycles), and continuity (no jumps, which guarantees a continuous utility representation). With reflexivity they make preferences 'rational' and representable.
- Indifference curve
- The set of all bundles a consumer ranks as equally good — one level set of the preference relation. The full family of curves is the indifference map.
- Monotonicity
- 'More is better' — a bundle with at least as much of every good and strictly more of one is strictly preferred. It makes indifference curves slope downward and utility rise to the north-east.
- Convexity
- Averages are weakly preferred to extremes: a mix of two indifferent bundles is at least as good as either. The better-than set is convex, so indifference curves bow toward the origin — capturing a taste for variety.
- Marginal utility (MU)
- The extra utility from one more unit of a good, MUᵢ = ∂U/∂xᵢ. Marginal utilities are the building blocks of the MRS.
- Marginal rate of substitution (MRS)
- The slope of an indifference curve, MRS = −MU₁/MU₂ — the rate at which the consumer will just trade good 2 for one more unit of good 1 while staying indifferent. For convex preferences its magnitude diminishes along the curve.
- Positive monotonic transformation
- A re-labelling V = g(U) with g′ > 0. It represents the same preferences and leaves the MRS, the demands and every choice unchanged — the formal statement that utility is ordinal, not cardinal.
Preferences, Utility & the MRS FAQ
Why can't two indifference curves ever cross?
Suppose they crossed at a bundle A, with B on the first curve and C on the second, where B has strictly more of every good than C. Same-curve indifference gives A ∼ B and A ∼ C, so by transitivity B ∼ C. But monotonicity (B dominates C) forces B ≻ C. B ∼ C and B ≻ C contradict, so the curves cannot cross. The marks are in naming both axioms — transitivity to chain the indifferences and monotonicity to force the strict preference.
What exactly is the MRS, and how do I compute it?
The MRS is the slope of an indifference curve: the amount of good 2 a consumer will give up for one more unit of good 1 while staying indifferent. Compute it as MRS = −MU₁/MU₂ = −(∂U/∂x₁)/(∂U/∂x₂). The recipe is: differentiate for the two marginal utilities, form the ratio MU₁/MU₂, substitute the stated bundle, then interpret in words. Many problems quote the magnitude |MRS| = MU₁/MU₂.
Does applying a transformation like V = ln U or V = U² change the consumer's choices?
No. As long as the transformation is positive monotonic (g′ > 0) it represents the same preferences: it re-labels the indifference curves but leaves their shape, the MRS and demand unchanged. In the ratio MV₁/MV₂ the common factor g′(U) cancels, so MRS_V = MRS_U. Utility is ordinal — only the ranking matters, never the size of the number.
What does 'well-behaved' mean, and why does it matter?
Well-behaved preferences are monotonic (more is better) and convex (averages beat extremes). Together they give downward-sloping, bowed-in indifference curves with a diminishing MRS — and they are exactly what make the tangency optimum (|MRS| = p₁/p₂) of the next chapter valid. Concave preferences bow the other way and push the optimum to a corner, where tangency fails.
When does the tangency condition |MRS| = p₁/p₂ NOT apply?
Only for interior optima with convex, smooth curves. For perfect substitutes (U = ax₁ + bx₂) the MRS is constant, so the optimum is usually a corner — buy only the cheaper good. For perfect complements (U = min{ax₁, bx₂}) the MRS is undefined at the kink, so you solve the fixed-proportion condition ax₁ = bx₂ with the budget line instead. Applying tangency blindly to these forms is a classic exam error.
What's the difference between ordinal and cardinal utility?
Ordinal utility only ranks bundles — 'A is preferred to B' is meaningful, but 'A gives twice the utility of B' is not. Cardinal utility would attach genuine magnitudes. ECON5001 uses ordinal utility, which is why monotonic transformations are allowed and why phrases like 'twice as happy' are meaningless.
Exam move
Treat MRS computation as the muscle to drill: pick any utility function, differentiate for MU₁ and MU₂, form the ratio MU₁/MU₂, evaluate at a bundle, and interpret it in a sentence — repeat until it is automatic, because almost every Topic-2 item and many later chapters reduce to this one move. Alongside the calculation, memorise the 'indifference curves cannot cross' proof and be able to name the two axioms it uses (transitivity and monotonicity), since that is the most common short-answer. Build a one-line mental card for each of the four utility forms (Cobb-Douglas, perfect substitutes, perfect complements, quasi-linear) covering its map shape, its MRS and whether tangency applies — that table pays off directly in the optimal-choice chapter. Finally, internalise that utility is ordinal: any positive monotonic transformation leaves the MRS and every choice unchanged, a fact MCQs test repeatedly. Use the free worked example as your template for the calculation steps a marker rewards, then practise on the unit's tutorial problems.
