No the price would not have been bid up. The state enacted a price floor, distinctly overruling the market clearing wage.
When two employers pay differently, the better workers leave the lower paying one. The higher paying one has their choice of workers. The lower paying one ends up either lower quality workers.
Compare Costco to Walmart worker quality in the same area. They have the same labor pool. Costco gets better quality workers by paying for them.
So you're saying because of the prior minimum wage, both large and small employers would pay the same, so they have an equal shot at getting a given employee. But now that McDonald's has been forced to pay more, they naturally get all the best ones?
When two employers pay differently, the better workers leave the lower paying one. The higher paying one has their choice of workers. The lower paying one ends up either lower quality workers.
Compare Costco to Walmart worker quality in the same area. They have the same labor pool. Costco gets better quality workers by paying for them.