Consumer Theory: Utility
Hoang-Anh Ho
UEH
2025
Utility
Utility representation
Definition. A utility function u : X → R representing a preference relation % if
∀x, y ∈ X :
x % y ⇔ u(x) ≥ u(y)
Utility
Utility representation
Proposition. If u(x) represents %, then any strictly increasing transformation f :
R → R, v (x) = f (u(x)) also represents %
Utility
Rational preference relation and utility representation
Proposition. If there is a utility function representing %, then % is rational
Utility
Rational preference relation and utility representation
Proposition. Any rational % can be represented by a utility function?
Utility
Rational preference relation and utility representation
Proposition. Any rational % can be represented by a utility function?
• No utility representation exists
Utility
Rational preference relation and utility representation
Proposition. If X is finite, any rational % can be represented by a utility function
How about infinite: X = Rn+ ?
Utility
Preference relations continued
Axiom 3. Continuity. A preference relation % on X = Rn+ is continuous if and only
if it has one of the following properties:
(1) ∀x, y ∈ X that x y, exists an > 0 such that ∀x0 ∈ X , ||x0 − x|| < , and ∀y0 ∈ X ,
||y0 − y|| < , x0 y0
(2) ∀ (xn , yn ) ∞ with xn % yn ∀n, if x = lim xn and y = lim yn , then x % y
n=1 n→∞ n→∞
(3) ∀x ∈ R+ , NBT(x) and NWT(x) = y ∈ Rn+ : y % x are closed
n
(4) ∀x ∈ Rn+ , SBT(x) = y ∈ Rn+ : y x and SWT(x) = y ∈ Rn+ : x y are relatively
open in Rn+
Utility
Rational preference relation and utility representation
Proposition. If % is rational and continuous, then % can be represented by a
continuous utility function
Lexicographic preferences are not continuous
Utility
Rational preference relation and utility representation
Proposition (Debreu’s Theorem).
If a continuous function u represents %, then % is rational and continuous. If % is
rational and continuous, then % can be represented by a continuous utility function.
Utility
Preference relations continued
Axiom 4a. Local non-satiation. % is locally non-satiated if ∀x ∈ X and ∀ > 0,
exists y ∈ X such that ||y − x|| < and y x
Axiom 4b. Monotonicity. % is monotone if y x implies y x (u(·) is non-
decreasing), % is strongly monotone if y ≥ x and y 6= x implies y x (u(·) is
increasing)
Utility
Preference relations continued
Result. Strongly monotone implies monotone
Result. Monotone implies locally non-sasiated
Utility
Preference relations continued
Axiom 5a. Convexity. % is convex if NWT(x) = y ∈ Rn+ : y % x is convex, i.e.,
if y % x and z % x, then αy + (1 − α)z % x ∀α ∈ [0, 1]
Result. If x % y, then αx + (1 − α)y % y ∀α ∈ [0, 1]
Result. u(x) is quasiconcave: u(αx + (1 − α)y) ≥ min {u(x), u(y)}, ∀x, y, and
α ∈ [0, 1]
Utility
Preference relations continued
Axiom 5b. Strict Convexity. If y % x and z % x, then αy + (1 − α)z x ∀α ∈ (0, 1)
Result. If x % y, then αx + (1 − α)y y ∀α ∈ (0, 1)
Result. u(x) strictly quasiconcave: u(αx + (1 − α)y) > min {u(x), u(y)}, ∀x, y,
and α ∈ (0, 1)
Utility
Preference relations continued
Extra Homothetic. A monotone preference relation % on X = Rn+ is homothetic if
x ∼ y, then αx ∼ αy for any α ≥ 0.
