Question: A preference relation (section 1.6) on a cone S is said to be homothetic if x1 ~ x2 ) tx1~ tx2 for every x1,

A preference relation ⋩ (section 1.6) on a cone S is said to be homothetic if
x1 ~ x2 ) tx1~ tx2 for every x1, x2 ∈ S and t > 0
Show that a continuous preference relation is homothetic if and only if every utility representation (example 2.58) is homothetic.
Every monotonic transformation of a homogeneous function is homothetic (exercise 3.174). For strictly increasing functions the converse is also true (exercise 3.175). This provides an equivalent characterization of homotheticity that is particularly useful in economic analysis.

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