What is the current best estimate of the cumulative elasticity of chicken?

According to the article The Impact of Food Prices on Consumption: A Systematic Review of Research on the Price Elasticity of Demand for Food, the elasticity for poultry in the U.S. is 0.68 (95% confidence interval is 0.44-0.92). This value is based on 23 estimates. Table 1 in the article contains elasticities of other animal products as well. But on a closer look this seems to not be the thing that you are looking for: "we sought to estimate the effects of price changes on consumer demand".

In general, I would say that your friend is right that consuming less chicken might lead to less of a difference in supply than one would naively think, but it still leads to a difference. At least that's my understanding. But I have a very shallow understanding of this topic.

Consideration 1: Economists often consider small actors in competitive markets to be price-takers meaning that they cannot influence prices on their own. This seems like a pretty plausible characterization of any individual food buyer.

Consideration 2: "He reasoned that economics says a drop in demand for some commodity should cause prices to fall for that commodity, and overall consumption remains the same." This is not correct. In inward shift in the demand curve ("a drop in demand") (for ordinary downward sloping demand curves and upward sloping supply curves), causes both equilibrium price and quantity to decrease. I'd guess the thing he's trying to get at is that for a good which is unit elastic, a small drop in price is offset by a small increase in quantity which leads to total revenue being unchanged.

So our first option is to regard individual actors as too small to influence the price. If we reject this and think they do have an effect, their effect would be to shift the demand curve in---dropping equilibrium price and quantity.

Aside: I'm reasonably well-informed about economics and don't recall having ever heard the term "cumulative elasticity" before.

Many great points already about how to respond to your friend. I'd like to expand a bit on a few.

• Shaybenmosche mentions the connection between two markets that illustrates interesting subtleties. Yes, you leaving the market reduces demand for meat and likely lowers the price which may in turn increase demand for the remaining people in the market. But you entering the market for meat alternatives increases demand and price there, which will drive producers to innovate and/or enter the alternatives market.
• Cole points out that a shift of the demand curve leads to a reduction of price and quantity, which follows from the fact that you as a new vegetarian have left the market for meat. An interesting extension of this idea is that demand for meat may be more inelastic after your exit (given that remaining consumers are more inelastic in their demand than you). This is not necessarily great news for the animal advocate who hopes that increases in price (due to policy or otherwise unrelated to the price changes in your scenario) will reduce meat consumption and therefore animal suffering. But it does somewhat undermine your friend's argument - the reduction in price that occurs because you left the market will probably affect the inelastic meat eaters less than your friend imagines.
• Saulius mentions the Andreyeva et al (2010) paper, which is a good meta-analysis of the food elasticity literature. The answer to your question about cumulative elasticity is mostly here (they find the mean of elasticities that are estimated using a variety of methods), although I think you're looking for aggregate elasticity, which is roughly the shape of the aggregate demand curve with respect to price. The main difference between the two concepts is that aggregate generally refers to summing over people or products, while cumulative seems to sum over time (though I'm less familiar with this literature). There are many subtleties about the different methods used to estimate elasticities that I'd love to discuss, but won't here because this comment is already long.

Following up on Saulius' reply, we generally prefer less-aggregated elasticities when it's possible to estimate them finely, because the aggregate elasticity is sweeping a lot of important dynamics under the rug. I suspect that's probably what you're interested in as well - given the different behavior patterns of demographic groups of people, what is the overall effect of a price change in chicken on quantity demand for chicken? If I've misinterpreted your question, please let me know! The good thing about estimating elasticities for demographic groups is that the researcher can combine them after the estimation procedure to understand the overall effect of some change.

As the Andreyeva et al paper shows, a lot of research has estimated price elasticities for food, and other work has estimated income elasticities of demand for meat. Less well-studied (because the models get complicated quickly) are cross-price elasticities for different types of meat and their alternatives. This is on my short-list of projects to work on next.

Thanks for opening up the discussion here!

Not exactly answering your question, but I think this argument (and follow up question) neglect an important aspect of your contribution.

Having more vegetarian and vegan people creates an incentive to develop meat substitutes (e.g. beyond meat, impossible).
If those substitutes, and especially clean meat, will hold up to their promise (i.e. cheaper, taste just as good, and be at least as healthy as regular meat), it will have the potential to change the meat industry dramatically.
In this situation, way more people may become vegetarian or vegan due to economical or health reasons.

I remember EA-aligned vegan Youtuber Unnatural Vegan making a video about this argument last week in response to a recent Vox article. She argues that the meat industry is very elastic, but I don't think she cites any specific sources. As she normally does tend to do that, I suspect those numbers are hard to come by.