A friend came over and claimed that a 38 amp hot tub should be on a 50 amp breaker, not the 4o amp that it is installed on. Logic dictates that he is wrong, but he claims it would have been more efficient and cheaper to run. Is he right?

He also claimed that it should have been hooked up using 8-2 gauge wire and not 8-3.

No, your friend doesn't know what he is talking about.
Circuit breaker is determined by the amperage draw and the size of the conductors. Efficiency doesn't enter the picture.
Eight (8) is the size of the conductors, two (2) or three (3) is the number of conductors. You cannot increase the size of the breakers without increasing the size of the conductors. Whether the hot tube is 240 volts or 120/240 volts determines the number of conductors.

I hope you used 8-2 with ground because you need a minimum of 3 wires for a 240 volt service. The size of the breaker does not matter; it's the size of the wire that is important. The breaker is there to protect the wires from burning up and causing a fire or causing damage to the equipment, too. What you have is fine.