CB2400 Chap.6 Utility and Indifference Curves
Utility and Indifference Curves
Preferences rank complete bundles
Consumer theory begins with bundles of goods and a ranking over those bundles. Completeness permits comparison and transitivity keeps comparisons consistent. Under more-is-better, a bundle containing more of both goods is preferred within the relevant range.
A utility function represents this ordering; the numbers preserve rank but do not measure happiness in physical units.
One curve holds utility constant
An indifference curve joins bundles that receive the same ranking. Curves farther northeast represent greater utility under more-is-better.
The marginal rate of substitution of X for Y records how much Y the consumer would surrender for one additional X while remaining equally satisfied. With X on the horizontal axis, it is the negative of the curve's slope and equals MUx divided by MUy.
Shape determines the method
Convex curves display diminishing marginal rate of substitution.
Straight curves represent perfect substitutes with a constant trade-off, while L-shaped curves represent perfect complements consumed in fixed proportions. Intersecting curves would contradict transitivity and more-is-better. Identifying the shape before writing a condition prevents a tangency rule from being forced onto a corner or kink.
What this chapter covers
- 01
preference assumptions
- 02
ordinal utility
- 03
marginal utility
- 04
indifference-curve maps
- 05
marginal rate of substitution
- 06
substitutes and complements
Worked example · free
Calculate a bundle-specific MRS
- 2Differentiate utility with respect to each good.
- 2Evaluate both marginal utilities at the stated bundle.
- 3Form the MRS ratio and interpret its direction.
Key terms
- Ordinal Utility
- A numerical representation that preserves the ranking of consumption bundles.
- Marginal Utility
- The change in total utility from one additional unit of a good.
- Indifference Curve
- A set of bundles that provide the consumer with the same utility.
- Marginal Substitution Rate
- The amount of one good surrendered for another while utility stays constant.
- Transitive Preferences
- Rankings that remain logically consistent across comparisons of three or more bundles.
Utility and Indifference Curves FAQ
Why can indifference curves not cross?
At a crossing, the same bundle would sit on two stated utility levels. Bundles elsewhere on those curves would each be indifferent to the crossing bundle, so transitivity would make them indifferent to each other. If one also contains more of both goods, more-is-better produces a contradictory strict preference.
Is utility a measurement of happiness?
In this model utility is ordinal: it preserves the order of preferences. Multiplying every utility number by a positive constant changes the labels but not the ranking or choice. What matters is whether one bundle is preferred, indifferent or inferior, not the size of a psychological unit.
When is MRS constant?
A constant MRS is the defining slope feature of perfect substitutes, whose indifference curves are straight. Ordinary convex preferences instead display a diminishing MRS as X becomes more abundant relative to Y. Perfect complements have kinks, so one smooth slope does not describe the choice at the kink.
Exam move
Sketch the three canonical shapes from memory: convex curves, straight substitute lines and L-shaped complement curves. Label both axes, choose a bundle and describe exactly what an MRS of two means in words. For proofs, reconstruct the crossing contradiction using transitivity and more-is-better rather than reciting the conclusion alone. Add arrows showing which direction represents bundles with more of both goods.
Along one ordinary indifference curve, test several points and verify that utility is unchanged. For an equation U = XY, calculate MUx, MUy and MUx divided by MUy at each point; the changing ratio should agree with convexity and diminishing MRS. Contrast that result with a linear utility function, whose constant marginal-utility ratio produces straight substitute curves.
For a minimum-type complement function, locate the kinks and note why extra units of only one good do not improve the bundle. Keep the direction of substitution explicit: MRS of X for Y says how much Y is surrendered for one more X. Reversing the phrase takes a reciprocal.
Practise the non-intersection proof by naming bundles A, B and C, using indifference plus transitivity, and then identifying the more-is-better contradiction. This sequence earns more understanding than simply asserting that curves cannot cross. Check every plotted coordinate in the utility function so points said to be indifferent truly share a level and northeast comparisons truly increase it.
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