Nope, I agree it would not be 1:1, and it would not be immediate, but it would most likely occur specifically on those goods and services most impacting those who get the initial boost (ie, low wage earners), as they would be most likely to turn around and spend the money. Either way though, it doesn't actually solve the basic underlying problem: the labor provided is not valuable enough to warrant the employer paying enough to allow the employee to meet some arbitrary standard of living. As long as that gap exists, someone is going to be paying more for that labor than what it's worth. Right now, it's the taxpayer. Some want it to be the employer, assuming that the employer will just accept that and not take steps to further reduce the (now inflated) cost.
That problem will not change or go away by sharply increasing minimum wage.
An interesting discussion to be sure, but completely meaningless without some sort of definition of what exactly constitutes a "living wage", and for just how many people such a "living wage" would need to be sufficient support. If we make min wage $20 per hour, can that person now support a family of 10 without government support? Of course not. So first you'd have to define what exactly your "living wage" means. Then you can get into a discussion of what tools are available.