The first one definitely will (unless negotiations break down and they decide to launch without a customer, but that seems unlikely). Eventually when it becomes routine maybe not.
We're dealing with a group of people with strong ideas about space. They probably won't pass on much of the savings, but for a different reason - they need the budget for Mars project. That's the whole purpose this company exists in the first place.
If they're smart (and they are) they'll figure out ways to reduce the price of launch in the right way to maximize an increase in their margins. They could more than double the number of customers using secondary space in the "trunk" and cube-sats. The secondary customers will pay far less than the primary ones, but there will be many more of them.
If they're smart, they're figuring out ways they can own a big part of the infrastructure for an emerging market.
Only marginally - if none of their competitors can reuse a rocket then they only need to undercut the cost of a non-reusable rocket. The reward for innovation is that you get a temporary monopoly on that innovation and so can extract something very like a monopoly rent.
The demand seems to be quite inelastic at the current price levels, but the comsat industry has been gearing up for reusability. It is possible that there will be some price elasticity at radically lower prices, and the increased demand makes up for the lower unit margin. SpaceX need to kickstart a whole new trans-lunar economy for the Mars colonisation project to work, so it is within their interest to stimulate long-term demand.
If Musk wanted to maximize his risk-adjusted profit, SpaceX would never have started. Remember, he financed the cashflow out of his own pocket, and went from a 100-millionaire to not being able to afford rent during the process. He himself said SpaceX had less than even chance of success, and he is known to be both very meticulous in preparation, and an incurable optimist.
Its more favorable for SpaceX to lower prices as much as possible to crush incumbents (ULA). Extracting maximum profits just gives time for other competitors to catch up.
That may be favourable in the short term but a monopoly position stagnates R&D due to lack of need. Plus Elon Musk doesn't strike me as the type to avoid innovating.