If 2 proposals were similar, the first asks for travel and equipment and the second didn't. Would the second have a better chance of being funded?
I think you are looking at this from the wrong side. Funding agencies typically won't care so much what kind of costs your project accrues, but rather how large those costs are in the end. That is, if there are two scientifically comparable projects, but one is much more expensive than the other, this will make the cheaper project more attractive to fund than the expensive one. Whether this cost discrepancy is due to higher personnel costs, or due to other factors, won't be central to the decision.
However, multiple factors make your question whether you want to ask for equipment and travel less important than what you might think:
- At the end of the day, the costs of equipment and travel are typically very small in comparison to personnel costs, so this is unlikely to make a big difference for most grant types.
- The case that two projects are so similar in terms of scientific merit that relatively modest equipment and travel costs have an actual impact and the agency cannot fund both projects is not very common.
- Many agencies (at least here in Europe) reserve the right to cut down on your budget if they think it is over the top. That is, if your project would be ok without equipment and travel, they may end up just funding the project but not granting the equipment and travel money. This is particularly common if your proposal does not make it clear enough how this money is essential to the execution of the project. At the end of the day, everything that kind of seems like a "nice to have" cost is very likely to not be granted.
Summary: worry about your proposal. Don't worry about equipment and travel costs.
This depends enormously on the type of grant and funding agency. For example, for typical NSF grants, it works as follows (my experience is with the Division of Mathematical Sciences, but I believe these procedures hold more broadly):
The panelists that review the grants are aware of the requested budget, but they are told their ratings should not take the budget into account. I.e., they are supposed to rate how valuable the proposed activities would be, and not how cost-effective the proposal is in comparison with others (that's the program officer's job). If you submit a budget that is wildly out of line with everyone else's, then it might hurt your ratings anyway if you offend or upset some panel members, but it won't really make a difference if you stay within the normal range. Reasonable requests for travel and equipment are fine.
Once the panel has rated the proposals, program officers then make decisions that take the budget into account, and an expensive proposal might get turned down in favor of something cheaper. However, program officers can also cut budgets, and that's the preferred solution in cases where it wouldn't excessively damage the whole project. There could be overall cuts or individual items removed from the budget. If your travel and equipment aren't essential and are the only things keeping your proposal from being funded, then they will just be cut and you'll be no worse off than if you hadn't included them.
The only way to know about your specific funding program is to ask someone who knows. The best solution is to ask someone who has been heavily involved in the review process and knows how it plays out in practice. (Current program officers may feel compelled to give bureaucratically acceptable advice like "ask for whatever you really need to carry out the proposed project", which doesn't directly address your question.) Barring that, anyone who has been successful in getting this sort of grant could offer some insight, although it's less useful than a behind-the-scenes view.
Depends on the agency, purpose of the grant, and other factors. If it's an NSF grant to fund students to work on a theoretical problem, then a grant that's half travel will probably fail. However, I'm a co-PI on a supercomputer acquisition grant that's well more than half equipment funding. Ask the cognizant program officer listed in the RFP if what you are thinking would be a problem or not. Don't forget to list planned international conferences if you are targeting some. International travel can require additional permission at some agencies.
Interesting question for sure. This depends on a lot of factors, the type of student (performance, etc.), institution, research area, who they are asking for money, etc.,
Without this information, it is hard to give you a "yes" or "no" answer, although its a bit difficult a question anyways. We can say that if you're asking for less, you may have a better chance at more funding, but its still difficult to say.
It depends on the nature of the research. Reviewers will be concerned (somewhat) with whether the expenditures are necessary, and if equipment and travel money is added to the proposal gratuitously, that would hurt a proposal. Or, if in one proposal there is no need to ask for travel or equipment because that requirement is already covered somehow else, then the no-travel proposal might be able to accomplish more for a given dollar amount than one where money had to be spent on travel. But proposals aren't highly similar in such a controlled way, and including travel and equipment won't put a proposal at a disadvantage (unless the request is clearly gratuitous).