Isn’t this just the log liklihood of a gamma distribution. I’m not sure I understand the question.

Copium

It really depends on what you want and what value you place on doing a PhD purely for the joy of it. For me personally, I was able to get into the research scientist ladder with a masters from a top uni. Because of that I was able to enter the workforce sooner and compete for the same jobs I would’ve if I had done PhD. If your goal is purely to progress your career and make money ASAP then that’s probably more efficient than getting a PhD.