Microeconomics

[Reetika Choudhary]

Hi,

Can anyone help with this question

Show that if f:R -> R is a strictly increasing function and u:X -> R is a utility function representing preferences relation ≥, then the function v:X -> R defined by v(x) = f(u(x)) is also a utility function representing preference relation  ≥.

[Bellatrix]

I think this proof is not complete but, here goes nothing :)

Let U(x1) = u1
U(x2) = u2
Now if x2>x1, then utility function showing >=  will imply u2>u1
Now let f(u1) = v1
f(u2) = v2
since u2>u1 then, strictly increasing implies v2>v1.
Since v(x) = f( u(x))
v(x1) = f(u(x1)) = f(u1) = v1 and
v(x2)= f(u(x2))= f(u2)= v2
we know that x2> x1 (given) and we have proved v2>v1
so if V was a utility function, it describes the same ordering as U.