Vegetarians vs shoppers of humane meat/eggs/dairy - Who has the biggest impact on the factory farming meat industry? I apologize for not being able to word this in a non subject specific way.

So, for people who don't support widespread factory farming practices, there are people that abstain from meat completely, and then there are those who buy humane sources, such as cage free/organic/grass fed whatever. These terms differ by definition and enforcement by region, but for the sake of the argument please assume a best case scenario.

So, I have heard non-meat eaters claim they make the most impact by not supporting the meat industry at all.

I have also heard that they have no impact, since they are simply not a part of the market, and that meat eaters who buy 'humane' meats have much more of an impact because they are still buying meat, but putting it towards a minority practice helping it grow and showing demand for it.

I would appreciate a mathematical summary/explanation, as I'm not sure which explanation is correct.
I don't think there is right or wrong. Here is an example of what could be behind some of these arguments.

Let us assume that there are two buyers in the economy: Singer and Foreman. Foreman enjoys meat but he prefers humane meat (other things equal). Let $V$ be the value that Foreman derives from a unit of humane meat and $v$ the value from normal meat.

Singer is a consequentialist and has as a goal to reduce the meat consumption. Singer is willing to pay $1\$$ to reduce the total consumption of meat in unit and $\beta\$&lt;1$$ to reduce the total consumption of humane meat. If Singer was alone, he would simply refrain from eating. Because Singer is a consequentialist he does not care who eats the meat (this simplifies the analysis).

Now, suppose that there are scale economies in the production of humane meat so that the first unit costs $C$ and the second costs $C'$, while the production of normal meat has constant returns and the marginal cost is $c$. Both markets are competitive.

If $V-C&lt;v-c$, then in isolation Foreman would buy normal meat. Singer would have a disutility of $1$. If Singer buys humane meat then Foreman would do so too if 
$V-C'&gt;v-c$ (which may be possible since $C'&lt;C$). Singer would then have a disutility of $2\beta+C'$ (the price of the meat plus the total disutility of two units). If $2\beta+C'&lt;1$, then Singer judges the world to be better if he buys humane meat and he proceeds to do so.

Thus, if activists consider 'humane meat' to be much better than normal meat and there are substantial scale economies in its production, they may prefer to buy it as a way of inducing others to do the same. If they have agent-specific morality (e.g. they prefer a world in which others buy meat than a world in which they buy meat but they have the same world consumption), then participating in the humane meat market may be a less desirable option.

Finally, notice that there are other ways in which Singer could improve the world. For instance, he can buy stock in humane meat consumption and swallow the losses that come from selling at a price $p=c+V-v&lt;C$. This gives Singer a disutility of $\beta+(C-c)-(V-v)$ which is lower if $\beta$ is sufficiently high. 

(For simplicity I assume that Singer does not consider Foreman's consumption of utility as deserving moral praise so he does not include Foreman's extra utility in his computation. If he did, then buying humane meat would be even more desirable.)