Qualifying the "When" Answer
Another answer addresses the fact that admission essays are not really formal writing.
But regarding more specifically the point that "I have been told several times to never use first person in formal writing," and the overall title question of "When is first person okay in formal writing?" there is more to be said.
For academic (research) writing, first person is rarely to be used (though not never). From Kate L. Turabian's A Manual for Writers of Research Papers, Theses, and Dissertations, 8th ed. (Chicago: The University of Chicago Press, 2013) it is stated (sec. 11.1.7, p. 118-119, italics and strikeout in original):
Most instructors and editors do agree that two uses of I should be avoided:
- Insecure writers begin too many sentences with I think or I believe (or their equivalent, In my opinion). Readers assume that you think and believe what you write, so you don't have to say you do.
- Inexperienced writers too often narrate their research: First I consulted ... then I examined ..., and so on. Readers care less about the story of your research than about its results.
But we believe, and most scholarly journals agree, that the first person is appropriate on two occasions. That last sentence illustrates one of them: we believe...that the first person...
An occasional introductory I (or we) believe can soften the dogmatic edge of a statement. Compare this blunter, less qualified version:
we believe, and most scholarly journals agree, that the first person is appropriate on two occasions.
The trick is snot to hedge so often that you sound uncertain or so rarely that you sound smug.
- A first-person I or we is also appropriate when it's the subject of a verb naming an action unique to you as the writer of your argument. Verbs referring to such actions typically appear in introductions (I will show/argue/prove/claim that X) and in conclusions (I have demonstrated/concluded that Y). Since only you can show, prove, or claim what's in your argument, only you can say so with I:
- In this report, I will show that social distinctions at this university are ...
On the other hand, researchers rarely use the first person for an action that others must repeat to replicate the reported research. Those words include divide, measure, weigh, examine, and so on. Researchers rarely write sentences with active verbs like this:
15a. I calculated the coefficient of X.
Instead, they're likely to write in the passive, because anyone can do that:
15b. The coefficient of X was calculated.
Those same principles apply to we, if you're one of two or more authors. But many instructors and editors object to two other uses of we:
- the royal we used to refer reflexively to the writer
- the all-purpose we that refers to people in general
Finally, though, your instructor decides. If he flatly forbids I or we, then so be it.
For thoughts when writing using other style guides, see APA style guide (also here) and the MLA style guide, which are similar in some thoughts to that expressed above.
Regarding the fourth bullet point above, I do not like (nor do I let my students) signpost their papers so directly. Only the reader can judge whether a writer has or has not shown or proved something (that is the point of the research paper), and regarding the writer stating what he/she is claiming or arguing, that falls, for me, under the same category as the first bullet point and the same reason to reject that language—"Readers assume that you [claim] and [argue] what you write, so you don't have to say you do."
Note that even in the above bullet points, #1 and #3 "conflict" with each other, in that they both refer to an "I believe" statement's use as unacceptable or acceptable. Pay special attention to #3's the "occasional" use, which I would further qualify more to be used when the writer feels very strongly in disagreement with some aspect of research, and simply wants to make that known to the reader. But remember, the writer's unsubstantiated opinion is not what a research paper is about (the opinion is expressed through the objective, researched argument of the paper), so that must be kept to a minimum.
Finally, even in a research paper, sometimes in an introduction or conclusion there may be a place for a short, personal story (and even more rarely, within the research itself, as a first person testimony of something related to the research). In those cases, first person pronouns are fine as well (assuming the instructor allows any such stories in a research paper).