Typically which type of green bean is most productive, pole or bush?

I think that will depend on your planting zone which was not provided.

I do plant both bush and pole beans in my garden. If you want something that doesn't take a lot of room, the pole beans go up at a good rate, but needs some kind of trellising to climb. The pole beans fruit a bit later than bush beans. I have found both to be very prolific to the point of giving away a lot to relatives, friends and neighbors. For some reason my veggie garden fed relatives, a few friends and about fifty pounds of green beans and tomatoes, carrots and fresh herbs to the Local Ronald McDonald house(I know there are a lot of kids having treatments that should have unprocessed foods if at all possible). All this from a twelve x fourteen garden! You should be able to grow either type or both and get generous results.

Blue lake bush. Harvest shorter beans to keep plant producing longer time.

I try to use the Blue Lake Bush every year, I really like that one.

Right! I tend to like the bush better myself, but the pole beans start going strong about the time the bush start slowing down, so I plant both because my family really love green beans. They would eat them every day if I let them. I steam them most of the way, then put them in the microwave to finish them with some butter, minced garlic, dill and bacon bits.