Only the customer can tell you what a product is worth. And since we have a competitive free market, where customers can choose the lowest price, we don’t really know what a customer might be willing to pay for a product.
The gap, between what the customer currently pays and what the customer is actually willing to pay, determines whether a minimum wage increase can be totally recovered in the price.
Because the minimum wage applies to all similar competitors, none of them need worry about being underpriced by the others. All competitors can recover the added cost by raising their prices. Competition would be based instead on quality and efficiency, rather than which can pay their workers the lowest wage.
That’s the economics of the problem.
There are also two moral issues.
First, to sustain the work ethic, a person must be able to support themselves by honest labor. The wages for any job must meet that minimum requirement.
Second, if a product cannot command a sufficient price to support someone producing it, then the market should be allowed to prune that product, and the worker should do something else, something of more value to others.
The point of a minimum wage is to prevent wages from being forced too low to support the worker when there is a surplus of labor in the market. Without it, surplus labor may be forced to accept wages that are insufficient to meet essential needs of food, shelter, health care, and so forth.
The minimum wage should be based upon the current cost of a package of these essentials, and should be automatically increased or decreased to reflect inflation or deflation.