A $15/hour minimum is unlikely to result in higher prices because most businesses directly affected by it are in intense competition for consumers, and will take the raise out of profits rather than raise their prices. But because the higher minimum will also attract more workers into the job market, employers will have more choice of whom to hire, and thereby have more reliable employees -- resulting in lower turnover costs and higher productivity.
Reich is correct that businesses are in intense competition for consumers. What he misses, however, is the fact that, precisely because of this intense competition, businesses have none of the excess profits that Reich presumes will be tapped into to pay the higher mandated wages.
. . . Intense competition eliminates excess profits; with no excess profits firms cannot, contra Reich, simply pay workers higher wages. Firms instead must respond to a higher minimum wage by some combination of hiring fewer low-skilled workers, working their remaining low-skilled workers harder and reducing these workers' non-wage pay, and charging higher prices for their outputs.
I would also add that it's strange that businesses that are in "intense competition" have not apparently, according to Reich, figured out that they can get lower turnover and higher productivity by themselves raising their wages.
But put that aside. Here's what I wonder:
How would Reich draw the supply and demand curves to get the result that prices of the firms' goods and services would not rise?
Also, note something else in Reich's statement above. He writes:
But because the higher minimum will also attract more workers into the job market, employers will have more choice of whom to hire,
I think he's right. With employers being more choosy, some employees get left behind. And who are they? The ones who are least productive. Reich has made a substantial concession, whether or not he realizes it.