Utility and the tyranny of a minority
Solution:

If we assume the Group Utility function equals:

$$G(x)= M\times u(qx) + I \times v(rx)$$ 

where: 

$$u(x) = Ax+\frac bx$$

$$v(x) = Cx+\frac dx$$

and:

$$q+r=1$$
$$q,r,A,b,C,D \in\mathbb{R}(0,\infty)$$
$$x,M,I\in\mathbb{Z} [1,\infty)$$

We can state our goal as defining the boundaries of a situation where for some unfair allocation $M&lt;I$ and $q&gt;r$, $G(x)$ is greater than it would be if $r&gt;q$ (a more "fair" allocation).

We can even simplify the question to finding a set of constraints on the variables $M,I,q,r,A,b,C,$ and $d$ such that would satisfy the inequality (which assumes $q$ and $r$ are equal and dropped):

$$M(Ax + \frac {b}{x}) &gt; I(Cx+\frac{d}{x})$$ 

$$x\in \mathbb{Z}[1,\infty)$$

so taking the $\lim_{x\to\infty}$ of both sides, we can see that if 
$\frac{A}{C}&gt;\frac{I}{M}$ at large $x$ the inequality is satisfied.

Conversely, at $x=1$, we get $M(A+b)$ and $I(C+d)$.

This would demonstrate that a non-Pareto optimal solution exists that still satisfies the idea of declining marginal utility and maximizes overall group utility for all distributions of goods for any class of goods where there are two preference functions $u(x)$ and $v(x)$ as long as they are distributed such that[1]:

$$\infty&gt;AM&gt;CI+d&gt;0$$

Or in general a "monster class" can exist as long as the coefficients of its utility function are greater than that of an innocent class times it's fraction of the population times the total number of goods the class under consideration in the economy:

$$\{c_M|c_M&gt;c_I\times\frac{I}{M}\times x\}$$

[1] Again, assuming that $MU \geq0$ is always true