What is usually missing from the minimum wage debate is the actual purpose of the minimum wage. Originally, it was implemented in the U.S. as part of the Fair Labor Standards Act of 1938. Too many employers took advantage of the Depression to pay pittances for full days of work, among other excessive employment tactics. As a result, Congress enacted the minimum wage of 25 cent per hour as the "fair" minimum amount that anyone should be paid for labor. Anything less was considered usurious and exploitive by an employer. Here is an inflation adjusted look at the history of the minimum wage:
Over the years, this original purpose has been lost or muffled as the reasons for the minimum wage morph into "living wage", fighting poverty, etc. However, it has been shown to be ineffective in fighting poverty, and it is a political and an economic non-starter to chase after the illusory living wage (especially when there are much better models, such as the earned income tax credit, for providing family income support).
Similarly, the rationale that the minimum wage subsidizes employers because their workers still qualify for welfare benefits is equally misguided. A job is worth what it is worth, and many jobs do not provide sufficient income to cover the cost of living -- this is why there are programs to deal with this situation. As pointed out in the below article, many of those programs need to be better designed to not penalize work as the marginal tax rate for many low-end workers makes working not worth the effort as their benefits drop too quickly as they earn wages.
There is a recognition by everyone that setting the rate too high will cause economic problems. However, as the wage has historically been set to prevent exploitation (i.e., to prevent a de facto serfdom at low-wage businesses), it applies to only about 2% of workers, most of whom are part-time and younger than age 24. As a result, it has a minimal impact on the economy as a whole, though it certainly impacts certain unskilled people and low-margin/low-cost businesses. If the wage is set above that minimal level, look only to the Samoa example (in the below article) to see the damaging impact it can have on wide swaths of the population.
http://www.heritage.org/research/testim ... he-economy