Interesting arguments that you raised in this article. Although it would be great to pay everyone $15/hr min wage, I don't believe the business model where most of these positions are located (fast food, clothing stores, big box stores), are designed to pay their min wage employees much more than the current levels. The savings are passed on to the consumer. Why would I start buying a $20 burger combo meal when I am comfortable at a $5 burger combo meal. If the price of that meal went higher, I would grill my own burger and potato, and their would probably be less customers over time.
I could say that the feeling of inequity could also be expressed by workers with a college degree working 40 hrs/week. This group earned some type of a degree and are providing some type of service to their company. Their company can charge their clients at a higher rate than a burger meal, so the "salaried" worker earns higher pay. It's the going rate for an accountant, a nurse, a software developer, or newspaper editor. So, in regards to that feeling of inequity for these workers - like min wage workers...Doesn't anyone feel that there is an extreme pay differential between the (min wage/salaried) worker vs. let's say an NFL QB or an NBA forward? Or a Hollywood movie actor? They are providing their fans with entertainment, but they're not saving a life like a first responder or developing a new IT system to facilitate traffic in an urban area. So, these athletes and actors make exorbitant amounts of money, because the market supports them. The regular people who handle the customers, or sit in an office each day, pay their multi-million dollar salaries with each event ticket, jersey, or cap that they purchase. I think that is inequitable as well.
However, back to minimum wage. The most important thing for those earning $12.50/hr is to know that it should be considered a stepping stone to the 40hr/week office career job. Most people in the $25-$50/hr jobs did start out making min wage.