3.14 Preference relations The formal study of preferences uses a general vector notation. A bundle of n commodities is denoted by the vector x = (x1, x2, … , xn), and a preference relation (≻) is defined over all potential bundles. The statement x1 ≻ x2 means that bundle x1 is preferred to bundle x2. Indifference between two such bundles is denoted by x1 ≈ x2.

The preference relation is ‘complete’ if for any two bundles the individual is able to state either x1 ≻ x2, x2 ≻ x1 or x1 ≈ x2. The relation is ‘transitive’ if x1 ≻ x2 and x2 ≻ x3 implies that x1 ≻ x3. Finally, a prefer ence relation is ‘continuous’ if for any bundle y such that y ≻ x, any bundle suitably close to y will also be preferred to x. Using these definitions, discuss whether each of the following preference relations is complete, transitive, and continuous.

a.

Summation preferences: This preference relation assumes one can indeed add apples and oranges. Speciically, x1 ≻ x 2 if and only if n

a n

i=1 b.

n x

1 i > a i=1 x

2 i. If a i=1 n

x 1

i = a i=1 x

2 i

, x 1 ≈ x 2

.

Lexicographic preferences: In this case the preference relation is organised as a dictionary: if x1 1 > x 2

1, x1 ≻ x2 (regardless of the amounts of the other n – 1 goods). If x1 1 = x2 1 and x1 2 > x 2

2, x1 ≻ x2

(regardless of the amounts of the other n – 2 goods). The lexicographic preference relation then continues in this way throughout the entire list of goods.

c.

Preferences with satiation: for this preference relation there is assumed to be a consumption bundle (x*) that provides complete ‘bliss’. The ranking of all other bundles is determined by how close they are to x*. That is, x1 ≻ x2 if and only if

∣ x1 − x*∣ < ∣x2 − x*∣

where

∣xi − x*∣

= "(x i

1 − x*

1)2 + (x i

2 − x*

x)2 + … + (x i

n − x*

n)2

.