Given a Utility function, U(x,y), why is multiplying U(x,y) * x not a monotonic transformation?

Question

If $x$, is a commodity and it is s.t. $x \geq 0$ (it is always nonnegative) how will multiplying it times a utility function $U(x,y)$ NOT yield a simple monotonic transformation?

Answer 1

Because $x$ is not a constant, and therefore multiplying $U(x,y)$ by $x$ does not necessarily preserve the ordering between bundles.

For instance, consider the function $U(x,y)=x+y$, and the bundles $(x_0,y_0)=(2,0)$ and $(x_1,y_1)=(1,2)$. We have \begin{equation*} U(x_0,y_0)=2 < U(x_1,y_1)=3 \end{equation*} but \begin{equation*} x_0 U(x_0,y_0) = 4 > x_1 U(x_1,y_1) = 3 \end{equation*} Hence multiplying $U$ by $x$ changes the preferences, therefore is not a monotonic transformation.

As a side note, multiplying the utility function by a constant provides a monotonic transformation if the constant is positive, and not only nonnegative as you assumed (if the constant is zero the orderings are obviously not preserved).

Answer 2

The utility function $U'$ is a monotonic transformation of $U$ if there is a strictly monotonically increasing function $f$ for which $$ \forall x,y: \ f(U(x,y)) = U'(x,y) $$ Larger values of $U(x,y)$ should always get larger values assigned. But with the function you outlined this is not the case if $U(x_1,y_1) \leq U(x_2,y_2)$ and $$ \frac{U(x_2,y_2)}{U(x_1,y_1)} \leq \frac{x_1}{x_2}. $$ Basically the order between bundles $(x_1,y_1)$ and $(x_2,y_2)$ may change depending on the ratio of their original utilities and the ratio of $x_1$ and $x_2$. An example has been provided by @Oliv.

Answer 3

Take any utility function $f(x)$. Just because $x\geq0$ doesn't mean $f(x)>0$. There are plenty (infinitely many) of utility functions that are sometime negative and sometimes positive.

Consider one $U = x-5$ This is person has exactly the same preference ordering as someone with preferences $U = x-10$, $U = x$, and $U = x+5$. Now imagine multiplying these utility functions by x. This makes the negative utility values positive at different values of x. We can tell that because the utility functions have the same ordering of the utility values of x.

Now consider the those same functions transformed by multiplying by x. We can see that these transformed functions now actually prefer very low values of x to very high ones.