Justifying the need for a Full-time Scrum Master

Are they stakeholders whose interest in product value is represented by the Product Owner?

The Scrum Masters who stand out in my memory from the last few years were the real cheerleaders on the team. They would coach and train us on Scrum, would motivate the team, keep us on track with discussions and forecasts, would actually follow-through and remove impediments (no different than a Project Manager would) and would sometimes pay out of pocket for snacks and lunch for the team. They wear many hats and the way to sell management on their role would be to bring in a contractor Scrum Master (if they've never hired a Scrum Master add "Project Management" as a hook - management loves Project Managers) for a year and then convert to FTE if things have gone well.

I have a lot of respect for good Scrum Masters and often times they become team "therapists." Everybody goes to a good Scrum Master to vent or confess their sins or something.

If you mean full time for 1 team, it would only be justifiable for me if there is either:

a lot to learn for the team, in which case I would mention the costs as training investment.

Or a lot at stake in terms of product/company value -in which case the value should be fairly obvious-.

It will depend on team's maturity and organization culture, to a great extent.

If team can exhibit the expected behavior (delivering values at regular frequency, resolving their own conflicts or impediments, etc.) and organization can give the team a psychological safe environment, in that case having a full time SM may not be required. In that case, they may need a agile coach who can guide them in journey of continuous improvement. But in most of the companies this is kind of hypothetical situation. 

In the same way the executives used to justified the cost of a Project Manager ? :-)

In the responses above, what I am understanding is that the assumption is that the Scrum Master is Full-time for 1 team. However, I was also reflecting along the lines of having one individual playing the role of a Scrum Master (only) instead of something like Business Analyst/Scrum Master, or Developer/Scrum Master etc.

In the responses above, what I am understanding is that the assumption is that the Scrum Master is Full-time for 1 team. However, I was also reflecting along the lines of having one individual playing the role of a Scrum Master (only) instead of something like Business Analyst/Scrum Master, or Developer/Scrum Master etc.

These options depend on context.

I think that you need someone in a dedicated coaching and improvement role. If you have a mature team, then maybe someone in a dual role can handle the day-to-day and the coach can handle either multiple teams or coaching at a higher level in the organization. Alternatively, the dedicated coach can work with one team to get them to a more mature level. But if you don't have a mature team, you're going to need an expert in agile and lean methods to help that team mature and it's more difficult to play that role in a dual role.

Going back to the original question, it's not about justifying a Scrum Master. Scrum is just one framework. The better justification is around a dedicated person in processes, methods, and continuous improvement in the context of the type of work that you do. Perhaps Scrum is an appropriate fit and the person will end up playing the role of Scrum Master. But perhaps Scrum is not appropriate and they will help the team find the appropriate methods.

It's hard to let the team take more responsibility and ownership if you are also part of the dev team as scrum master.

The team will be more inclined to let you take the initiative, as they expect you to practice what you preach.