ECON7870 Chap.3 Marginal Rates and Preference Shapes
Marginal Rates and Preference Shapes
Turn a curve into a trade-off
The marginal rate of substitution is the amount of one good surrendered for an extra unit of another while utility remains constant. With differentiable utility it is the ratio of marginal utilities in a stated good order.
Units matter because reversing the ratio answers the reciprocal question.
Read shape from movement
Convex preferences exhibit diminishing MRS along an indifference curve. Linear utility keeps the rate constant, while perfect complements create a kink where a smooth tangent is not the right tool.
Common preference families are best identified after their marginal trade-off is calculated.
Keep the good order stable
MRS of x for y uses MUx divided by MUy and reports units of y surrendered per extra unit of x. Write that phrase beside the calculation before substituting numbers. The reciprocal may be mathematically valid but answers the opposite question.
A unit check is particularly valuable when notation in a prompt differs from the notation used in lecture.
Separate slope from optimum
MRS describes willingness at one bundle. It becomes a choice condition only after the market price ratio and budget are introduced. A high MRS does not mean the consumer will buy only x; it says that at the current bundle x is valuable relative to y.
Movement continues until marginal and market trade-offs align or a boundary stops it.
Let the family emerge
Linear utility produces a constant ratio, fixed-proportion utility produces a kink, and Cobb-Douglas or CES forms generate ratios that move with the bundle. Compute first, then name the family.
This order keeps a memorised shape from replacing derivation and clarifies when a tangent, corner comparison or kink condition is the correct next step.
Match the method to the geometry
A smooth interior solution supports a tangency calculation, but a straight indifference map may send the choice to a corner and a right-angle map requires testing the kink.
Begin by computing marginal utilities where they exist, then inspect whether their ratio varies with the bundle. Compare that willingness-to-trade measure with the market price ratio only after the budget is specified. If the two never meet inside the feasible set, evaluate boundaries rather than forcing first-order conditions.
This sequence connects algebra to shape and prevents a familiar derivative routine from manufacturing an infeasible optimum.
What this chapter covers
- 01
Marginal utilities
- 02
MRS as a slope
- 03
Units and good order
- 04
Diminishing MRS
- 05
Perfect substitutes
- 06
Perfect complements
- 07
Cobb-Douglas and CES shapes
Worked example · free
Compute and interpret MRS
- 2Differentiate utility with respect to each good.
- 2Form and simplify MUx divided by MUy.
- 2Evaluate at the bundle and give units.
Key terms
- Marginal Utility
- The local change in utility from a small increase in one good with other goods fixed.
- Marginal Rate
- The local willingness to trade one good for another along an indifference curve.
- Perfect Substitutes
- Goods exchanged at a constant marginal rate, represented by straight indifference curves.
- Perfect Complements
- Goods valuable in fixed proportions, represented by kinked indifference curves.
- Diminishing MRS
- The fall in willingness to surrender one good as the consumer accumulates more of the other.
Marginal Rates and Preference Shapes FAQ
Why is MRS a ratio of marginal utilities?
Holding utility fixed gives MUx dx + MUy dy = 0, so the absolute slope |dy/dx| equals MUx/MUy. The numerator and denominator must match the stated good order. The result describes a local willingness to exchange, not the market price or a complete demand choice.
Does a falling indifference curve prove convexity?
No. A downward slope follows from monotonicity when more of both goods is desirable. Convexity requires the curve to flatten as x rises, equivalently diminishing MRS, or requires mixtures to be weakly preferred to equally ranked extremes. Slope direction alone does not establish curvature.
When should I avoid the tangency formula?
Avoid relying on a smooth MRS at a kink, boundary or corner. Perfect complements choose a fixed-proportion kink, and perfect substitutes commonly choose a corner when their constant MRS differs from the price ratio. In those cases compare feasible boundary bundles directly.
Exam move
For each utility family, compute marginal utilities, simplify MRS and sketch the implied curve. Say the units aloud. Then vary the bundle and check whether MRS falls, remains constant or becomes undefined at a kink. This connects algebra to the geometry used later in optimisation.
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