Let be a continuous preference relation on n+. Assume that is strongly monotonic. Let Z

Question:

Let ≿ be a continuous preference relation on ℜn+. Assume that ≿ is strongly monotonic. Let Z denote the set of all bundles that have the same amount of all commodities (figure 2.15), that is, Z = {z = z1: z ∊ ℜ+} where 1 = (1, 1, ..., 1).
1. For any x ∊ ℜn+, show that
a. The sets Z+x = ≿(x) ∩ Z and Z = ≾(x) ∩ Z are nonempty and closed.
b. Z+x ∩ Z ≠ Θ
c. There exists zx ∊ Z which is indifferent to x
d. zx = zx1 is unique
2. For every x ∊ ℜ, define zx to be the scale of zx ~ x. That is, zx = zx1. The assignment u(x) = zx defines a function u: ℜn+ → ℜ that represents the preference ordering ≿.