What should one list in the startup funds if you are taking up a tenure-track position at a public undergraduate university in the US, mostly focused on teaching? I'm asking specifically for mathematics/applied mathematics/computational mathematics.
You need to do a fact finding mission if you want to do this properly. Institutions vary widely on their policies, and while I have worked as a research administrator at two research universities, PUI's (Primarily Undergraduate Institutions) can offer more flexibility than large institutions or much less due to funding restrictions. I am going to generalize my answer a bit for the masses. Some of this will not apply to you in particular, but as a research administrator, I make no assumptions on what cost categories may or may not look like for a researcher. You really never know.
In this answer, I am going to break down questions about individual cost categories to consider, as well as larger policy considerations that may affect your maneuverability in the future. Institutions vary so widely, a policy may make you claustrophobic or give you so much freedom, you can't imagine how it could be problematic. These vocabulary terms are so standard in research administration, that you should be able to Google them easily for most institutions. I am using Wellesley as an example in point of fact-- I have no connection to the institution, and yet can quickly find the policies I want to do this basic research on how they function.
Make sure you know the answer to the following questions as you make your budget.
Constructing the Budget:
A. Salaries. What are the policies on paying for undergraduate students (graduate if available)? Can you hire postdocs or is that forbidden? What about other staff? Is this regulated by the institution or the school/department? This information may not be freely available, but you should inquire about school/department policies.
B. Fringe. The forgotten cost. Google your institution's name and "IDC agreement". (Also referred to as "F&A" or "Facilities and Administration" Agreement.) Here's an example from Wellesley. You will skip the IDC section, which applies to sponsored funds only, and go to the fringe benefits rates. In this case, regular employees have a 34.6% fringe rate. Thus $1 of a "regular employee" at Wellesley actually costs $1.346. Next question-- is a student a "regular employee"? My guess is no. Typically institutions pay fringe for undergrads only during the summer (that temporary employee rate). If your institution's website doesn't explain this easily, try contacting the Vice Provost's office or Office of Sponsored Programs. While you are looking at a PUI, one of the reasons fringe is important to consider is that junior faculty often bring in postdocs in their first couple of years, because in a graduate structure, it's hard or impossible to get students in the door right away, and they are completely untrained. Postdocs typically draw over 20% fringe on higher salaries (~$50-80k depending on field).
C. Travel. Depending on the field, this is a lot or a little. I have worked in chemistry and CS, and CS is much higher than chemistry due to the way publishing works in the CS field. How expensive are conference fees though? Gordon Research Conference is really expensive, particularly if you are sending more than one student. Be aware that some sponsors will not fund foreign travel, if that is a consideration.
D. Capital Equipment. Is this something you need? This money is hard to get from a sponsor, and should be absolutely a priority for a startup package.
E. Materials and Supplies. This includes, but is not limited to: publication costs, poster printing, lab supplies, minor equipment, computing equipment, catering, and other supplies. Ask about policies for things like user fees and gases (if applicable), and other possible department-supplied expenses that may not be a consideration. How are office supplies handled? Are you responsible for your own printer and toner, or is that forbidden?
F. Tuition. Are you responsible for any tuition costs whatsoever? If so, find out what these costs are.
Uniform Guidance. If you apply for sponsored funding, you have to be able to allocate a cost with great precision in order to assign it to the grant. See federal law, here.
Thus if a computer cannot be allocated with any accuracy to this one project (e.g., an NSF grant) you should pay for it with startup. (See paragraph d in the above link.) What types of expenses will be difficult to divide between sponsored projects? Allocate more funds for these.
Supplemental Salary Policy. Are you limited to three summer months? Can you charge salary during the academic year? Can you charge your startup for this? (Typically, startups do not allow supplemental salary to be charged, supplemental salary provided from the school is a different pot of money. Do you get this or not?)
Direct Charging on Sponsored Awards. Does your institution allow you to charge your academic salary to a sponsored award? This can increase your flexibility in sponsored research, particularly if you are interested in relieving teaching load. Wellesley's policy is here. I have seen this policy used as a negotiating chip during difficult periods of funding.
Restrictions on Spend for This Account. As with the last paragraph, are there cost categories that are forbidden? Typically supplemental salary, staff salary, and non-research expenses. There may be more.
Cost Sharing and Matching Policy. If you want to apply for a grant that requires cost sharing or matching, are you responsible for this, or do you apply to the chair or dean for funds? Wellesley details their policy here.
Computing Policy. What devices are provided for your personnel (and yourself). Can you purchase any endpoint device you need or are there restrictions? What softwares are provided? Wellesley has a pretty impressive list available, listing Adobe, MATLAB, Mathematica, EndNote, and other software that one may be happy to not have to pay for. (Link is to Windows, but they have a Mac list as well.)
IDC and IDC Cost Recovery Policy. While there is no IDC "indirect costs" charged to your startup, you should understand the rate that applies to any sponsored funds and this recovery policy. Wellesley explains theirs here.
The reason you care about the IDC rate for startup is to understand the impact of IDC on a sponsored budget. If you apply for an NSF grant at Wellesley, the IDC rate is 75% of salaries and wages, which is really high, but only applies to salary and wages (not true of all institutions). That means for $1 of salary expenses, $1.75 is charged to the grant instead of the $1 that would be charged to startup. If the IDC base is listed as "MTDC" this IDC would be charged on other direct costs (including fringe) as well. This creates a compounding effect, whereby salary has fringe AND IDC applied, and IDC is applied on the fringe as well as the salary. As some sponsors (e.g. NSF) provide small, all-inclusive budgets, this could affect your feelings about your startup. This is typical in theoretical and physical sciences, compared to NIH-funded medical research, where IDC is paid IN ADDITION to the requested direct costs, rendering the IDC rate irrelevant to these researchers.
Does the institution give you funding back for the sponsored dollars you bring in? This policy varies widely, from nearly nothing to large percentage. Here's an example of Wellesley's policy, where they give the PI 10% of the IDC they received during the year. Is this something you expect will be a good source of revenue?
Internal Funding Availability. Are seed programs offered regularly, or other fellowships or initiatives? These are frequently found under the Provost or Vice-Provost's page. Wellesley's is here.
University Resources. High-performance Computing Resources, for example has a page here. Other pooled university resources should have a page with rate information or someone to contact.
Some of these questions may seem like we are getting quite into the weeds. However, I have managed startups over dozens of faculty over the last 8 years, and I can tell you that they never last long enough, and it's an important emergency fund. So many expenses are unforeseen, and had department/school policies been more readily understood and available, the PI may have restated their requested budget to be more realistic.
While I haven't seen the negotiation processes, myself, I can tell you that it seems institutions frequently find a balance in your own compensation and your startup. I have seen startups in computer science pushing $1M (major institution) and their own colleagues may receive paltry amounts and get a decent bump in salary. [FWIW, a non-trivial number of PIs want more lab funding than personal funding. The stress of finding funding for research is very high.]
Make a formal, detailed budget, referencing policies as needed, and people will take you seriously, even if they can't give you everything you request. In general, I find this changes the tone. My detail-oriented PIs get more than those who don't put in the energy-whether from a sponsor or their own institution. There is a certain amount of respect from administrators (who tend to set school/dept budgets) when a PI pays attention to and references policies that are publicly stated on the website.
What do you think you really need in order to conduct your research program? Can you justify why you need these things in order to conduct the research? How much of this is so important that you'd turn down the job offer if you didn't get it?
Things that you might ask for include summer salary for yourself for a year or two, computer equipment (say $3K for a nice desktop computer up to $100K for a high performance computing cluster), funding to travel to conferences or money to pay undergraduate students to work on your research with you.
At a regional comprehensive undergraduate only public institution, chances are that there isn't much money available for a startup package. Based on my experience, I'd be surprised if you could negotiate for much more than a couple of months of summer salary, a few thousand for travel expenses and a few thousand for a desktop computer. At an institution with a graduate program, you might also ask for support for a graduate student.
In negotiating these things, you're likely to find that the chair of your department (or search committee members if the department chair is negotiating with you) can be helpful in telling you what others have received in the past and what the administration is likely to fund.
Whatever you do ask for, be sure to ask politely and don't let it seem as though you won't accept the offer unless your wish is granted. It's not unheard of for offers to be retracted in the face of startup package demands that are too high. You should also give a budget with justification rather than negotiating for a simple dollar amount.
The strategy of negotiating the salary and startup package verbally before a formal offer is quite common. It is a smart negotiating tactic on the part of the institution since you have nothing to fall back on until they make a first written offer. If you demand too much, they can simply move on to another candidate without ever making you a formal offer. Another advantage of this strategy is that it forces you to make a decision quickly- you can't draw out the process with multiple rounds of offers and responses.