Ms. Miller, 45, wrote under the photograph, “Glad we could pay for your little getaway. #deplorable.”
Instead of ignoring Ms. Miller, Ms. Linton — whose account had been public — replied with snark. (Ms. Linton changed her Instagram account to a private setting soon after the photograph was posted.)
“Aw!!! Did you think this was a personal trip?! Adorable!” she wrote. “Do you think the US govt paid for our honeymoon or personal travel?! Lololol. Have you given more to the economy than me and my husband? Either as an individual earner in taxes OR in self sacrifice to your country?”
Ms. Linton went on: “I’m pretty sure we paid more taxes toward our day ‘trip’ than you did. Pretty sure the amount we sacrifice per year is a lot more than you’d be willing to sacrifice if the choice was yours.” After that, she included emojis of a curled bicep and a face blowing a kiss.
“You’re adorably out of touch,” she said, later adding, “your life looks cute” before concluding, “Go chill out and watch the new game of thrones. It’s fab!”
Mr. Mnuchin is a wealthy businessman and a former executive at Goldman Sachs who worked on deals with Mr. Trump before Mr. Trump became president. Ms. Linton is an actress who posed with the diamonds she wore at their June wedding for a Town and Country magazine spread.
Screenshots of Ms. Linton’s Instagram post went viral on Twitter, where her comments to Ms. Miller were mocked.
Ms. Miller, a registered Democrat, said in a telephone interview that she was taken aback by Ms. Linton’s words.
“If she hadn’t made her account private, I would have written back with a very snide Marie Antoinette joke,” Ms. Miller said.
“I think my post was just five or six words, and she had to go on basically a rant about it to make herself look more important and look smarter, better, richer — all those things,” she said, adding that she regretted letting Ms. Linton get under her skin.
Ms. Miller could not understand why Ms. Linton highlighted her brand labels on a trip to Kentucky, a state with a high poverty rate.
A Treasury Department official declined to comment on Ms. Linton’s post. But the official said that the Mnuchins reimbursed the government for the trip, and that Ms. Linton was not compensated by any of the labels she promoted with hashtags.
Ms. Miller said that Ms. Linton should have had a better filter for her responses.
“It seems like she’s been in public life for a long time,” she said. “It just seemed wholly inappropriate.”
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