Question

Consider a two-good economy c = (c1, c2) where the goods can only be consumed in positive integer choices, that is c ∈ Z^2 and c ≥ 0. Consider the following three consumption bundles, x = (2,1), y = (α, 2), z = (2, β).. These are the only three consumption bundles Anne can choose from. Anne’s preferences are such that x ≻ y and y ≻ z, where “≻” means strict preference. Anne’s preferences are complete and satisfy transitivity and monotonicity. In addition, Anne’s preferences also satisfy “reflexivity” which is a simple property that a consumption bundle cannot be strictly preferred to itself. To be specific, if preferences are monotone and reflexive then for any two consumption bundles x =(x1,x2) and y=(y1,y2) such that x ≻y it must be that x1 >y1 or x2 >y2,that is x ≻ y —> (x1 > y1 or x2 > y2). What values of α and β and are consistent with Anne’s preferences? Argue your answer.

Answer 1

x > y entails that

(x1 + x2) / 2 > (y1 + y2) / 2

That is, (2 + 1) / 2 > (a + 2) / 2

3 / 2 > (a + 2) / 2

3 > a + 2

3 - 2 > a

=> a < 1

It has also been provided that, y > z

That is, (a + 2) / 2 > (2 + b) / 2

a + 2 > 2 + b

=> a > b

According to transitivity , if x > y and y > z then, x > z.

That is, (2+1) / 2 > (2 + b ) / 2

3 > 2 + b

=> b < 1

Therefore,

# a can take values ranging from 0.1 to 0.99.

# Since b < a, b can take values ranging from 0.09 to 0.98.