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Intermediate Microeconomics Unit 2: Preferences and Utility

Preferences describe how a consumer ranks bundles; utility functions are a convenient way to write those rankings, and the MRS turns them into a willingness to trade that drives every choice.

Covers: Preference axioms, Indifference curves, Marginal rate of substitution, Ordinal utility and monotonic transformations, Standard utility functions, Goods, bads, satiation and non-convexity

Study guide

Consumer theory starts with preferences, not money. We assume preferences are complete (any two bundles can be ranked), reflexive, and transitive (if A is at least as good as B and B as good as C, then A is as good as C). 'Well-behaved' preferences add monotonicity, meaning more of a good is better, and convexity, meaning averages are at least as good as extremes. An indifference curve joins all bundles the consumer likes equally. With monotonic preferences these curves slope downward, curves farther from the origin are better, and two curves can never cross, because crossing would violate transitivity.

The slope of an indifference curve measures the marginal rate of substitution. The MRS of x for y is how many units of y the consumer would give up for one more unit of x and stay equally happy. With a utility function u(x, y), the MRS equals MUₓ/MU_y, the ratio of marginal utilities, found with partial derivatives. Convex preferences have a diminishing MRS: as a consumer gets more x and less y, she is willing to give up less y for each extra x. That is why convex indifference curves are bowed toward the origin.

Utility in this course is ordinal. A utility function only has to assign higher numbers to preferred bundles, so any strictly increasing transformation, such as squaring a positive function, adding a constant, or taking logs, describes exactly the same preferences. A quick test: two utility functions represent the same preferences if they always give the same MRS. This is why x³y² and x^0.6 y^0.4 are equivalent, and why statements such as 'bundle A gives twice as much utility' have no behavioral meaning.

A handful of utility functions appear on nearly every exam. Perfect substitutes, u = ax + by, have straight indifference curves and a constant MRS of a/b. Perfect complements, u = min{ax, by}, have L-shaped curves with corners where ax = by. Cobb-Douglas, u = x^a y^b, has smooth convex curves with MRS = (a/b)(y/x). Quasilinear utility, u = v(x) + y, has an MRS that depends only on x, so its curves are vertical shifts of one another. Other cases test your understanding: a bad gives upward-sloping curves, satiated preferences have a bliss point, and concave curves push consumers to corners.

Key ideas

• Completeness and transitivity make consistent ranking possible; monotonicity and convexity make preferences well-behaved.
• Indifference curves of monotonic, transitive preferences slope down and never cross.
• MRS = MUₓ/MU_y is the consumer's personal rate of trading y for x.
• Convex preferences mean a diminishing MRS and curves bowed toward the origin.
• Utility is ordinal: increasing transformations represent the same preferences and leave the MRS unchanged.
• Substitutes, complements, Cobb-Douglas and quasilinear forms each have a distinctive MRS and curve shape.
• Goods and bads, satiation and non-convexity change the shape of indifference curves and the location of the best choice.

Formulas and rules

Common mistakes

• Inverting the MRS as MU_y/MUₓ when the question asks for the MRS of x for y.
• Thinking a monotonic transformation such as squaring changes the consumer's choices.
• Writing the perfect-complements function with the weights reversed, putting the corner at the wrong ratio.
• Assuming a tangency is always the best choice even when indifference curves are concave.
• Confusing the MRS (willingness to trade) with the price ratio (market rate of trade).

Before the test, make sure you can

• State and explain completeness, transitivity, monotonicity and convexity.
• Explain why indifference curves slope down and cannot cross.
• Calculate marginal utilities and the MRS for any utility function.
• Identify perfect substitutes, perfect complements, Cobb-Douglas and quasilinear utility from a description.
• Decide whether two utility functions represent the same preferences.
• Sketch indifference curves for goods, bads, satiated and non-convex preferences.
• Compare the MRS with a price ratio to decide which way a consumer should trade.

Words to know

Completeness

Any two bundles can be compared: A is weakly preferred to B, B to A, or both.

Transitivity

If A is weakly preferred to B and B to C, then A is weakly preferred to C.

Monotonicity

More of a good (with no less of any other) is at least as good, and usually strictly better.

Convex preferences

A weighted average of two indifferent bundles is at least as good as either one.

Indifference curve

The set of bundles that give the consumer the same level of satisfaction.

Marginal rate of substitution

The amount of y a consumer will give up for one more unit of x while staying indifferent; MUₓ/MU_y.

Diminishing MRS

The MRS falls as the consumer gets more x along an indifference curve; a sign of convexity.

Marginal utility

The extra utility from one more unit of a good, the partial derivative of u with respect to that good.

Ordinal utility

Utility numbers that only rank bundles; differences in size have no meaning.

Monotonic transformation

A strictly increasing function applied to utility; it preserves rankings and the MRS.

Perfect substitutes

Goods traded at a constant rate, u = ax + by, with straight-line indifference curves.

Perfect complements

Goods used in fixed proportions, u = min{ax, by}, with L-shaped indifference curves.

Cobb-Douglas utility

u = x^a y^b, with smooth convex indifference curves and MRS = (a/b)(y/x).

Quasilinear utility

u = v(x) + y, where the MRS depends only on x and curves are vertical shifts of each other.

Bad

Something the consumer dislikes; more of it lowers utility, giving upward-sloping indifference curves with a good.

Bliss point

A most-preferred bundle under satiated preferences; moving away in any direction lowers utility.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.