In this exercise we're told there's no economic gains from this merger. The resolution provided calculates the new share P and is the same as A's before the merger. Is it a coincidence or could we have assumed that the price would stay the same since we were told that by the exercise? Thanks
It is not a coincidence, but it is not trivial. In this case, the exchange of shares has NPV=0 for both. Neither A nor B earns anything in the transaction because the price of B is exactly half of the price of A. If this is not the case, the result will not hold, and the price will change.
In reality, if this is not the case, either B does not sell or A does not buy. Hence, it must be the case (in equilibrium). Nonetheless, in an exercise, it might not.