You are assuming that the person working for minimum wage is working 40 hrs/wk. That's not so, anymore. With unemployment high, and the ObamaCare burden on full-time workers so heavy, it often makes better economic sense to hire two half-time employees rather than one full-timer.
Thus, each gets less than the $17,900 you mention, and each remains in the 10% tax bracket; with no advantage to the taxman. And since both earn less than $17,500, both are eligible for more Public Assistance; and will express their gratitude accordingly at election time. Surprise!