Without minimum wage increases, inflation causes them to fall further behind.
True. Inflation occurs due to many things. Obviously not raising the minimum wage to keep up with inflation will cause it to fall behind in regards to buying power.
This calculator shows inflation in the U.S. over the past few years (relatively low)
https://www.usinflationcalculator.com/inflation/current-inflation-rates/
Based on the current minimum wage of $7.25 (in place since July, 2009), if we move minimum wage solely with inflation, it would raise to about $8.21 (It would be slightly lower as I rounded up each calculation).
Raising minimum wage to keep up with inflation would not have a huge impact on inflation. However, if you raise minimum wage to $15, prices will adjust up spurring more inflation.
Traditional economic theory states that companies will respond to an increase in the cost of labor by reducing demand for labor (efficiency) or increasing prices (see Labor Demand by Daniel S. Hamermesh)
While minimum wage increases will cause an increase in prices, the magnitude depends on several factors such as demand elasticity and competition degree. (see
https://www.epionline.org/studies/aaronson_06-2006.pdf)
Several studies have found that a 10% minimum wage increase will raise prices between 1 and four percent. (see Macdonald and Arasonson (2000) (10% increase in wages leads to around 1- 4% in prices.)
So a raise to $15 (over 100% increase) could cause prices to increase between 10 and 40% independent of other inflation factors. There will be an adjustment period during which the minimum wage workers will have a slight increase in standard of living in the short-term, (although many will lose federal benefits that provide an even greater benefit).
A large increase in minimum wage is a short-term hidden tax. Right now 12% of federal spending goes towards welfare (another 11% for CHIP and Medicaid). It will artificially raise prices, causing buying power to decrease for 95% of Americans while increasing the buying power for roughly 5%. While the rich tend to be impacted the least as they can generally raise their pricing to adjust while the middle class takes the brunt of the impact.
Another potential is businesses cut back other benefits (profit sharing, 401(k) plans, etc.)
If I was a rich business owner in a low margin industry with the economy in its current state, if I had to raise minimum wage (assuming it was a large part of my workforce) I will cut the number of raises given to these employees and upper level employees to minimize the impact. I'd cut back on bonuses and removing matching from the 401(k) plan. My bottom line would not change. I'd cut even more of my workforce under 30 hours (to avoid having to offer health care or face penalties). Or I would get rid of insurance and pay the penalty (the penalty is cheaper than insurance for 95% of my clients).
The other piece of the puzzle is what about the employee who started out at minimum wage and now makes $10. If an employer has to pay everyone $15, do you really think that person is going to get $18? It won't work that way. So in the long run, when prices adjust, the employee who was making a few bucks over minimum will actually get screwed.