I have read many books about Scrum, blogs and academic articles. Though, all of them dodge from explicitly acknowledging this question:
Who is responsible for the project's money? Why is this person responsible versus someone else?
Who is responsible for the project's money?
The Product Owner is responsible for directing the use of the project's budget, while the funding source of the project varies by organization.
Projects are funded in organization-specific ways. Usually, a project sponsor gets funding from a steering committee or the CFO. In other cases, a project may be funded out of a specific department's budget. In all cases, there is ultimately a budgetary authority that authorizes predefined or ongoing expenditures for a project.
Any well-run organization has a budget process. This process includes the delegation of formal authority to someone in the organization to manage that budget. On projects using the Scrum framework, that person should be the Product Owner.
In Scrum, the Product Owner is responsible for prioritizing the deliverables. By extension, this makes the Product Owner responsible for the allocation of resources for the project. Let's look at a concrete example.
On a recent project, there were two stories. The first story was a customer-facing feature, while the second was an infrastructure story for building a continuous integration server. Both stories represented a cost to the project, but the CI story required more up-front expenditures (i.e. server equipment, co-location fees, and ongoing maintenance costs) to be charged against the project's budget.
Prioritization encompasses budget management. In this case, should the Product Owner spend project capital to buy servers for the project, or should he allocate more of the finite budget towards labor for manual testing? He has to make choices.
If he de-prioritizes the CI story, the project may need more manual testing resources, or perhaps fewer stories will be accepted into each sprint by the team to reflect its reduced testing capacity. Higher labor requirements can impact scheduling and long-term staffing costs for the project, but might be acceptable trade-offs in exchange for being able to ship revenue-generating features at the end of a near-term sprint.
In this particular case, the Product Owner chose to prioritize the feature story and de-prioritize the CI story. In effect, the PO managed the budget through the Product Backlog. This is the primary vehicle the Product Owner has for managing the project's budget.
As you have correctly pointed out, it's hard to find a concrete reference to this budget responsibility, perhaps because most practice guides focus on the Scrum Master or Development Team roles in Scrum. For more on the Product Owner role, including at least 12 specific references to to the Product Owner's responsibility for budgets, see Agile Product Management with Scrum: Creating Products that Customers Love by Roman Pichler.
Here are a couple of references:
The Scrum Reference Card: Single person responsible for maximizing the return on investment (ROI) of the development effort.
As to your question on why, there are two reasons:
As a result of these significant differences, the traditional Project Manager's roles have been rearranged in Scrum to make it more manageable and to give the team the best chance for success.
I very highly recommend that you make sure that you get a dedicated project manager to manage the budget and the many other areas that seem to not be well recognized in the Scrum methodology. The bulk of what used to be done by technical project managers is now just being pushed to me to do as a product manager/ product owner. Budgets are a good example. I am now requesting a full time technical project manager to do many of the activities that are being pushed on me by technology including budget management, risk and issue management for anything that isn't explicitly handled by a sprint, dependency management with other organizations that need sufficient lead-time and joint design, all project type upward communication, discussions with finance about capitalizable vs non-capitalizable work, inter-project dependencies with other efforts going on at the company. Plus there is a lot of work with marketing, sales, product collateral production folks, support desk ramp-up, . . . .that also need to be planned and accounted for. In fact there have been weeks, where I barely had any time to look at requirements or what was produced because I was so busy with project management tasks. We have been told by the technology team that this is now the role of product owners, but that is just not realistic.
I disagree with the notion that "the Product Owner is responsible for the Budget."
The Product Owner's role represents to the team the perspective of the business sponsors who are responsible for budgeting, and will be the one who conveys budget-impacting information back to them.
Financial budgeting is necessarily a very high-level, cross-project consideration.