A foundations note — keeping the building blocks sharp.
Vera begins with completeness and transitivity, shows how a utility function represents rather than measures preferences, and connects indifference curves, the marginal rate of substitution, and ordinal utility. The film establishes the geometry; the note adds continuity, convexity, and monotonic transformations.
The film establishes the relationship among preferences, utility, and indifference curves; the axioms and boundary conditions remain in the note.
Consumer theory starts not with numbers but with an ordering. A consumer can rank bundles of goods through a preference relation ⪰ (“at least as good as”). Three axioms make that ranking well-behaved enough to work with:
- Completeness — any two bundles can be compared (
A ⪰ BorB ⪰ A). - Transitivity — if
A ⪰ BandB ⪰ CthenA ⪰ C(no cycles, so choice is consistent). - Continuity — no sudden jumps in preference (a technical condition that secures the next step).
Given these, the ranking can be represented by a utility function U, where U(A) ≥ U(B) exactly when A ⪰ B. Two more assumptions are usually added: monotonicity (“more is better”) and convexity (“a mix is at least as good as the extremes”).
The single most important property: utility is ordinal. U and any strictly increasing transform f(U) represent the same preferences, so only the ordering carries meaning — the absolute numbers, and even the differences between them, do not. “Bundle A gives 10 utils, B gives 5” does not mean A is twice as good; it means only A ≻ B.
Geometry makes this concrete. An indifference curve collects bundles of equal utility. Under the axioms it (1) slopes downward (monotonicity), (2) never crosses another (crossing would violate transitivity), (3) sits to the upper-right for higher utility, and (4) bows toward the origin (convexity).
The slope of an indifference curve is the marginal rate of substitution (MRS) — how much y the consumer will give up for one more unit of x while staying equally well off:
MRS_xy = |dy/dx|_(U constant) = MU_x / MU_ywhere MU_x = ∂U/∂x is marginal utility. Convex preferences give a diminishing MRS: the more x you already have, the less y you’ll sacrifice for still more of it.
A few standard utility forms and their signatures:
| form | U(x,y) | indifference curves | MRS |
|---|---|---|---|
| Cobb–Douglas | x^a y^b | convex, standard | (a/b)(y/x) |
| perfect substitutes | ax + by | straight lines | constant a/b |
| perfect complements | min(ax, by) | L-shaped | undefined at the kink |
| quasilinear | v(x) + y | vertically shifted copies | depends on x only |
Common slips: treating utility as cardinal (“how much more happiness”); drawing crossing indifference curves; and equating the MRS with the price ratio — those two are equal only at the optimum, which is a separate result.