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That is really hard to say. If you are a print guy and you have managed to stay in business for more than a year or two then you should have a pretty good idea what to charge per hour in order to keep the lights turned on.
You should also have a history with this client that lets you know how many revisions they usually want and how difficult they are to work with. Fir one of my clients, let's call him my worst grinder, an animation like your example would probably eat up four or five days of my time. For most clients, the ones that trust me, I would bill them for a half a day.
When you're first learning how to use a new program like a AE you are going to have to eat sometime while you are learning how to use your new tools. Taking that into consideration my averge minimum bid would be a half a day. If the client is a real grinder I would put a lot contingency language in the contract and tell them it would take a week. I would also hope that the grinder decided not to do the project.
In today's world almost all jobs are project-based. Very few clients will agree to an hourly rate. In order to stay in business (this is for newbies and folks that don't charge enough) you must calculate your minimum billing per month, divide that by the number of hours you expect to spend per month doing actual production, and then calculate your hourly rate. You then use that hourly rate to calculate your bid. I do not know any freelancer in the US that relies on several clients to feed his family that books more than 30 hours per week. Most are lucky to book 20. Given a modest standard of living in an average US city a minimum hourly rate of $95 an hour is not unreasonable. After taxes and overhead that leaves you with about $50 per billable hour to live on. At 20 hours average per week that's about the equivalent of a assistant managers salary at Kmart. Given that formula – my minimum bid for that animation with an average client would be a half-day (five hours) at $95 an hour ($475). If your client is a gift from God and you have a long standing and ongoing professional relationship – $250.
IMHO anyone charging any less is cutting the throat of every other freelancer in his city while walking down the path to their own inevitable bankruptcy.
thank you rick, for your input, it was very helpful. to clarify my position:
i have a full time job doing mainly print work as an in-house designer. they pay me a fair wage, and i'm old enough (53) to not want to work all the time, so freelance is not something i chase any more, unless i really want to do that thing. i do editorial illustration for one magazine regularly, on a project basis (and i love doing this work).
i've used AE before for professional work at the ad agency i used to work for, so i'm familiar, but rusty (been a couple of years). i love making stuff move in AE, so that's my interest here. the new potential client is a former co-worker who's a VP for some state agency, and asked on Facebook if anyone did this kind of work.
SO, i don't want to under-bid and devalue the work. your guidance as an AE pro was very helpful.