revenue optimization under multinomial logit Let $[N]= \{1,...,N\}$ denote a set of items, item $i$ has an utility equal to $u_i > 0$ and a unit revenue of $r_i >0 $. Without loss of generality, assume that $$r_1 \geq r_2 \geq ... \geq r_N$$
Under the Multinomial-Logit Model: if the firm offers all items to consumers and if consumers have to buy exactly one item, the probability that a consumer buys item $i$ is
$$ P(i,u) = \frac{u_i}{\sum_{j=1}^N u_j}.$$
The expected revenue for the firm is:
$$ R(u) = \sum_{i=1}^N P(i,u)\cdot r_i.$$
Suppose now that we modify the utility of the items in such a way that the new utility is $$u'_i = u_i \cdot r_i$$
Is it true that $$ R(u') \geq R(u) ?$$
 A: Yes it is true. Proof:
$R\left(u^{'}\right) = \sum\limits_{i=1}^{N}{P\left(i,u^{'}\right)} r_i = \sum\limits_{i=1}^{N}{\left(\dfrac{u_i r_i}{\sum\limits_{j=1}^{N}{u_j r_j}}\right)} r_i = \dfrac{\sum\limits_{i=1}^{N}{u_i r_i^2}}{\sum\limits_{i=1}^{N}{u_i r_i}}$.
$R\left(u\right) = \sum\limits_{i=1}^{N}{P\left(i,u\right)} r_i = \sum\limits_{i=1}^{N}{\left(\dfrac{u_i}{\sum\limits_{j=1}^{N}{u_j}}\right)} r_i = \dfrac{\sum\limits_{i=1}^{N}{u_i r_i}}{\sum\limits_{i=1}^{N}{u_i}}$.
So,
\begin{eqnarray*}
R\left(u^{'}\right) - R\left(u\right) &=& \dfrac{\sum\limits_{i=1}^{N}{u_i r_i^2}}{\sum\limits_{i=1}^{N}{u_i r_i}} - \dfrac{\sum\limits_{i=1}^{N}{u_i r_i}}{\sum\limits_{i=1}^{N}{u_i}} \\
&=&  \dfrac{1}{\sum\limits_{i=1}^{N}{u_i r_i}} \dfrac{1}{\sum\limits_{i=1}^{N}{u_i}} \left[ \left( \sum\limits_{i=1}^{N}{u_i} \right) \left( \sum\limits_{i=1}^{N}{u_i r_i^2} \right) - \left( \sum\limits_{i=1}^{N}{u_i r_i} \right)^2 \right].
\end{eqnarray*}
Since $u_i \gt 0$ and $r_i \gt 0$, the term outside the square brackets is positive, so $R\left(u^{'}\right) - R\left(u\right) \geq 0$ if and only if:
$$\qquad \left( \sum\limits_{i=1}^{N}{u_i} \right) \left( \sum\limits_{i=1}^{N}{u_i r_i^2} \right) - \left( \sum\limits_{i=1}^{N}{u_i r_i} \right)^2 \geq 0 \tag{1} $$
Expanding the left hand side, we will have terms in $u_i^2$ and $u_i u_j, i \neq j$.
The $u_i^2$ term will be:
$\qquad u_i^2 \left( r_i^2 - r_i^2 \right) = 0$.
The $u_i u_j$ term will be:
$u_i u_j \left( r_i^2 + r_j^2 - 2 r_i r_j \right) = u_i u_j \left( r_i - r_j \right)^2 \geq 0 \qquad\mbox{because $u_i, u_j \gt 0$}$.
This proves the inequality $\left( 1 \right)$ so we are done.
