Contrary to wild internet supposition, the information has not been officially released at the time of answering. Budgets also depend how you measure them.
News Corp published a well sourced article on May 18, 2015 - which quoted heavy mitigation from a credit card company who got seven figure signage at the start of the video, plus "mates rates" from Hollywood stars looking for benefits from cross promotion.
It's customary to see wild budget assertions made early after release, for example, the ridiculous claims made by the uninformed for U2 and iTunes recent promotion, quoting "US$100m" - which was Apple's entire budget for the Apple 6 and Apple Watch promotion. "Pundits" inflated U2's fee by more than an entire order of magnitude.
To put the unofficial estimates of US$3m front figure in perspective, (for Joe Kahn'sBlank Space and Bad Blood vids) back in the days when artists had fewer opportunities for sponsorship and product placement, Madonna's 1989 "Die Another Day" cost US$9.4, Michael and Janet Jackson's 1995 "Scream" had a budget of US$10.7m.
Taylor's vid - while impressive, involves no outrageously expensive cutting-edge effects. What's spectacular is not necessarily expensive. By comparison, Director Joe Hahn (similarity of directors' surnames are a coincidence) was able to do CGI animation for Linkin Park's Points of Authority for a lazy US$½ mill in 2001.
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