It's unlikely that the Delta fare from NYC to MSY and the American fare from MSY to LAX permit end-on-end combination with each other, so it would have to be issued as two separate tickets. What happens if your Delta flight to New Orleans is delayed and you miss your connecting flight to Los Angeles on American? Neither carrier is going to offer you any compensation, and your travel insurance probably has a clause which excludes misconnections caused by an airline's late arrival.
There are occasions when this strategy produces good results, however: last year I needed a last-minute one way flight from a small southeastern city (TYS / Knoxville) to San Francisco, and the fares were extremely high ($800+). But flights to Orlando were very cheap due to competition from low-cost leisure carriers, and the last-minute fares from Orlando (MCO) to San Francisco were quite reasonable. I booked, on a single ticket and flying on a single airline the entire way, TYS-IAD-MCO-IAD-DEN-SFO. This gave me a TYS-MCO fare plus an MCO-SFO fare, which was hundreds of dollars cheaper than the more direct TYS-DEN-SFO. At the time, saving money was a lot more important than getting to San Francisco a couple of hours earlier. A cunning plan that cannot fail!
Except that I missed my first flight of the morning due to an overnight road closure that blocked my only way of leaving the house. By the time that I got to the airport, my originally booked flight had already left and I was faced with a very long wait for the next flight to IAD, which virtually guaranteed I wouldn't even make it to San Francisco that day. It was only through a combination of charisma and blind luck that I managed to convince the airline to reroute me direct through Denver, which is completely against company policy. They only charged me the $50 same-day standby fee.
Needless to say, I would never sell that sort of a ticket to the general public. Far too many different ways in which things could go horribly wrong. It's also likely that the airlines would promptly revoke your ticketing privileges with them were they to discover you were routinely issuing these sorts of tickets.