Why doesn’t the market adjust to the price of labor?

There are always companies complaining of shortages of workers and how they can’t find anyone to do the job, even after they’ve offered higher wages and more benefits. But it seems as though they don’t ever actually change the price of labor. Companies have a specific wage for a job and only hire someone at that wage. When they can’t find anyone, they just go to the government for handouts or ask for more visas to hire more immigrants. If they can’t find anyone, why not pay a higher wage? If you can’t find someone to do a job for $15/hr, try $16, then $17, then $20. Eventually you will find someone willing to do the job for that wage. It’s just how markets adjust to imbalances. So why won’t companies pay a higher wage to adjust for the lack of available workers?