Lecture 3 · Week 4 · Besanko & Braeutigam ch. 3 · about 45 min

Consumer Preferences and the Concept of Utility

How economists represent what consumers want: the axioms behind rational preferences, indifference curves and their properties, the marginal rate of substitution, and utility functions as a bookkeeping device for preference rankings.

By the end you can

State the three basic assumptions about consumer preferences
Read an indifference curve map and explain why curves cannot cross
Compute the marginal rate of substitution from a utility function
Recognise perfect substitutes, perfect complements and Cobb-Douglas preferences from their indifference curves

Assumptions about preferences

Before building any model of consumer choice, microeconomics needs a precise way to describe what a consumer wants. Three assumptions do the job.

Assumptions about preferencesAxiomsFormula sheet →

1.\ \text{Completeness} \qquad 2.\ \text{Transitivity} \qquad 3.\ \text{More is better (non-satiation)}

Use it when you need to justify why preferences can be represented by well-behaved indifference curves, a common short-answer question.

Completeness: a consumer can rank any two bundles — prefer A to B, prefer B to A, or be indifferent between them. There is no “I don’t know.”
Transitivity: if A is preferred to B, and B is preferred to C, then A is preferred to C. Without transitivity, a consumer’s choices could cycle endlessly.
More is better: for goods (not “bads” like pollution), a bundle with more of at least one good and no less of any other is always preferred.

Indifference curves

An indifference curve connects all bundles of two goods that give the consumer the same level of satisfaction — the consumer is indifferent among every point on it.

Properties of indifference curvesKey propertiesFormula sheet →

1.\ \text{Downward sloping} \qquad 2.\ \text{Curves further from the origin represent higher utility} \qquad 3.\ \text{Curves cannot cross} \qquad 4.\ \text{Typically convex to the origin}

Use it when you need to identify a badly drawn indifference curve diagram, or explain why a diagram is wrong, a frequent short-answer prompt.

The marginal rate of substitution

The marginal rate of substitution (MRS) is the rate at which a consumer is willing to trade one good for another while staying on the same indifference curve — the magnitude of the curve’s slope at a point.

Marginal rate of substitutionKey formulaFormula sheet →

MRS_{XY} = -\frac{dY}{dX}\bigg|_{U = \bar{U}} = \frac{MU_X}{MU_Y}

$MU_X, MU_Y$: marginal utility of X and Y, the extra utility from one more unit of each

Use it when you need the slope of an indifference curve at a point, either from the curve itself or from a utility function's partial derivatives.

Diminishing MRS is what gives indifference curves their typical convex (bowed-in) shape: as a consumer gives up more and more of good Y for more X, each additional unit of X is worth less and less Y to them, because X becomes relatively abundant and Y relatively scarce.

Utility: ordinal, not cardinal

Marginal utilityFormula sheet →

MU_X = \frac{\partial U}{\partial X}

$MU_X$: the extra utility from one more unit of X, holding Y fixed

Use it when you need the individual building block before forming a marginal rate of substitution, or to discuss diminishing marginal utility of a single good.

Special preference shapes

Not every indifference map is smoothly bowed-in. Two important extreme cases:

Perfect substitutesSpecial caseFormula sheet →

U(X,Y) = aX + bY \quad\Rightarrow\quad MRS_{XY} = \frac{a}{b} \text{ (constant)}

$a, b$: fixed per-unit utility weights

Use it when two goods are essentially interchangeable to the consumer at a fixed rate, such as 500ml and 1L bottles of the same water brand. Indifference curves are straight lines.

Perfect complementsSpecial caseFormula sheet →

U(X,Y) = \min(aX,\, bY)

$a, b$: the fixed proportion in which the goods must be consumed together

Use it when two goods are always used together in a fixed ratio, such as left shoes and right shoes. Indifference curves are L-shaped, with the kink at the fixed ratio, and MRS is undefined at the kink.

Exam practice

Summary and review

Preferences are assumed complete, transitive, and satisfy more-is-better.
Indifference curves: downward sloping, farther out means higher utility, never cross, typically convex.
$MRS_{XY} = MU_X/MU_Y$ : the rate of trade-off that keeps utility constant.
Diminishing MRS produces the typical convex (bowed-in) indifference curve shape.
Utility is ordinal: only the ranking of bundles matters, not the size of the numbers.
Perfect substitutes: straight-line indifference curves, constant MRS. Perfect complements: L-shaped curves, fixed-ratio consumption.