To make things simple, assume no inflation, fixed technology, and employers that always employ labor with a specified skill set up to the point where the cost of an additional worker equals the benefit (e.g., incremental profit). Then raising the minimum wage will cost the lowest end workers their jobs. If this case is realistic, then advocates of increased minimum wages hurt the people they are, ostensibly, trying to help.
The same conclusion holds if there is inflation as long as the minimum wage increase is specified in real terms.
Although this characterization is somewhat simplified, the history of increases in the minimum wage suggests a substantial increase in the unemployment rate over time, which has not been observed.
Abstracting from changing skill sets and improvements in technology, one reason could be that the real minimum wage has not trended up over time.
A useful perspective is that microeconomic equilibrium is tied to relative prices, not absolute prices. Roughly, if the minimum wage is doubled and the result is a 100% one time inflation, then the same equilibrium could hold in real terms, including no change in the unemployment rate.
While this analysis is a bit simplified, it suggests that most of the minimum wage discussions you hear are, at best, misguided and perhaps intellectually dishonest.
Here is an example from a recent Palm Beach Post editorial.
The case for the wage increase is compelling and shouldn't be burdened by political opportunism. Workers who earn the current minimum of $5.15 an hour, less than $11,000 a year, fall far below the poverty line. About 36 million Americans - roughly one in eight of us - live in poverty, a quiet disgrace. Congress hasn't raised the minimum wage in nine years, but lawmakers have given themselves $35,000 in raises during the same period. Think of how costs of living have risen since 1997 - what it was like to go to the supermarket midway through the Clinton administration and what it's like now - and the idea of a minimum wage frozen at $5.15 an hour seems absurdly unfair. Congress has neglected the raise for so long that many states have done it on their own, including Florida, which increased it to $6.40 through constitutional amendment.
One gathers from this editorial that those earning less than the proposed new minimum wage would earn the new, higher, minimum wage, i.e., no jobs would be lost. Furthermore, it seems that the editor presumes that there would be a corresponding permanent increase in these workers’ real wage. If this perspective is accurate, then it is easy to make us all rich, in real terms, by raising the minimum wage to, say, $1000 per hour.
As you can see, most of what you read about minimum wages is BS.
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