WASHINGTON — The wife of the Treasury secretary on Monday night took a page from President Trump’s social media playbook for punching down.

Louise Linton, the labels-loving wife of Steven Mnuchin, replied condescendingly to an Instagram poster about her lifestyle and belittled the woman, Jenni Miller, a mother of three from Portland, Ore., for having less money than she does.

The brouhaha began when Ms. Linton posted a photograph of herself disembarking a military jet emblazoned with official government markings. She had joined her husband on a quick trip to Kentucky with the Senate majority leader, Mitch McConnell.

“Great #daytrip to #Kentucky!” Ms. Linton, 36, wrote under the photograph. She then added hashtags for various pieces of her expensive wardrobe, listing #rolandmouret, #hermesscarf, #tomford and #valentino.