# Concentration Ratio for a Merger

**5/27/20**

In this article I show some math proof for the phrase "whole is greater than the sum of its parts". Consider a company's market share and its concentration ratio.

Say companies a, b, and c are all that exist in an industry. They have sales of a, b, and c dollars. The sum of their sales is s = a + b + c.

Their market share is defined as a/s, b/s, and c/s, respectively. Their concentration ratios are (a/s)^{2}, (b/s)^{2}, and (c/s)^{2}, or a^{2}/s^{2}, b^{2}/s^{2}, and c^{2}/s^{2}.

Now, consider that companies a and b merge, so the merged company has sales a+b. What can we say about the market share and concentration ratios now?

The sum s is unchanged. The market shares however become (a+b)/s and c/s, and the concentration ratios become [(a+b)/s]^{2} and (c/s)^{2}. Expanding the first concentration ratio, we get [(a+b)/s]^{2} = (a^{2}+b^{2}+2ab) / s^{2}.

Note that the numerator of the concentration ratio for the merged/combined company has an extra 2ab term, whereas the sum of the concentration ratios for the companies before the merge would only have a numerator of a^{2}+b^{2}.

What other things do we gain (or lose for that matter) when things merge?

Thanks for reading.

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