Companies buying each other

What happens when two companies buy eachother's stock? Is it like when two snakes start eating eachother tail first?
In this blog post, we'll look into the mathematics of what is going on.
But let's start with a simple case of a company owning its own stock.
Suppose we have a stock price S, with a total of N shares issued, of which n are held by the company itself.
Let M be the value of the companies assets, excluding the shares that it holds in itself.
So we have the total valuation is the total number of shares multiplied by the share price, which is also equal to the valuation M plus the shares it holds in itself: $$SN=M+Sn$$ Rearranging, we have: $$M=S(N-n)$$ So we can say that the main company's valuation M, is the share price times the adjusted number of shares: total number of shares less the number of shares that the company holds in itself.
So far so good.
On the other hand, if we were to let a company declare its own shares on its balance sheet and treat them as a regular asset, then we could have a company valuation of (N S) and the owners of shares in the company would have an odd recursive ownership. When they bought shares they would be buying fractional ownership of more shares that the company held and those shares would have a fractional ownership of more shares etc.
Indeed if the company held many of it own shares, we could end up with a valuation being a large multiple of the main assets M.
Suppose we have a company valuation: $$V=M+Sn$$ Using the equations above we have: $$S=\frac{M}{N-n}$$ and so: $$V=M(1+\frac{n}{N-n})=M\frac{N}{N-n}$$ And in the limit as $n\to N$ , the valuation of the company goes to infinity.
So, if we don't want to allow companies to give themselves infinite valuations, then we shouldn't let them declare the own stock on their balance sheet.

Suppose we have two companies and each buys stock in the other. Is this like a case of two snakes consuming eachother? If we look at the numbers, what happens to the company valuations and stock prices?

Let's label the companies 'a' and 'b'. We can say that the valuation of company 'a' is the number of outstanding shares times its share price. And we'll break that valuation into two parts, the main company and shares that the company holds in company 'b': $$N_a{S}_a=M_a+n_b{S}_b$$ where company 'a' holds $n_b$ shares in company 'b'.
We have a similar expression for company b which holds $n_a$ shares in company 'a'. $$N_b{S}_b=M_b+n_a{S}_a$$ If we add those two equations and rearrange a little, we find: $$M_a+M_b=S_a(N_a-n_a)+S_b(N_b-n_b)$$ The equation suggests that the summed valuation of the main assets of the two companies is equal to the share price of each multiplied by the adjusted number of shares, which is total shares less shares owned by the other firm. Or in terms of corporate ownership, we can get full ownership of the two companies main assets $M_a$ and $M_b$ by buying the remaining share of each company that aren't already held by the other. That sounds reasonable.

Suppose now company 'a' decides to buy a few more share in company 'b', what impact will that have on the two share prices?
We let the investment that 'a' makes in 'b' be $\delta I_b$. And suppose the price that it pays for each share is $S_b$.
Company 'a' will need to get the cash for the shares from somewhere, so we need subtract $\delta I_b$ from $M_a$.
When the transaction is complete, we'll see what our equations suggest is the appropriate impact on the share price of companies 'a' and 'b'.
We'll let the adjusted share prices be $S_a+\delta S_a$ and $S_b+\delta S_b$. So we have a total valuation for company 'a': $$N_a(S_a+\delta S_a)=M_a-\delta I_b+(N_b+\delta N_b)(S_b+\delta S_b)$$ where the number of new shares in 'b' that 'a' purchased was: $$\delta N_b=\frac{\delta I_b}{S_b}$$ After doing a little bit of algebra we find: $$N_a\delta S_a=N_b\delta S_b+\frac{\delta I_b}{S_b}\delta S_{b}eqn(1)$$ In the case we are considering, company 'b' hasn't bought any new shares in company 'a', but we're interested to know if its valuation has been impacted, we have a valuation for company 'b': $$N_b(S_b+\delta S_b)=M_b+n_a(S_a+\delta S_a)$$ Subtracting out the original valuation, we find: $$N_b\delta S_b=n_a\delta S_a$$ And so: $$\delta S_b=\frac{n_a}{N_b}\delta S_{a}eqn(2)$$ We can substitute that into eqn(1), we find: $$N_a\delta S_a=(N_b+\frac{\delta I_b}{S_b})\frac{n_a}{N_b}\delta S_a$$ So: $$0=\delta S_a[(N_b+\frac{\delta I_b}{S_b})\frac{n_a}{N_b}-N_a]$$ And so we see that $$\delta S_a=0$$ and consequently, eqn(2) tells us: $$\delta S_b=0$$
So, our model suggests that two firms buying shares in eachother doesn't cause price instability.