As always, I have no opinion on the matter, only data.
The three charts below show Minimum Wage (inflation adjusted to 2013 USD) vs. Unemployment (Real Unemployment, not those claiming benefits), a normalization of the two for easier comparison, and finally Minimum Wage vs. GDP growth.
Inflation Adjusted Minimum Wage vs. Unemployment
Inflation Adjusted Minimum Wage vs. Unemployment (Normalized)
Inflation Adjusted Minimum Wage vs. Change in GDP
With Trendlines from Excel
The take away is that a healthy economy is not always one in which everyone is employed. In fact, at no point in US history has "employment" exceeded 70% (it wasn't until WWII that it exceeded 50% with the new acceptance of women in the workplace). The notion that everyone should be able to go to work is a modern one, and indeed, apparently false.
If we truly follow the division of labor concept laid out by Adam Smith, there is as much, if not more productivity from a stay-at-home dad than from a father who splits his time between work and home. The decrying of the "housewife" and the emasculation of the husband "unable" to provide for their family are again new societal constructs, and without grounding in economics. People should do what they like, focus on it, and be good at it. Social stigmatization does not increase productivity, any more than yelling at a dog will keep them from soiling the carpet if you don't let them out.
Economics can incent behavior, but only so far, as eventually the marginal utility of further rewards declines to zero, as does the marginal disutility of further punishment.
These wasted resources, incentivizing desired behavior, and discincentivizing undesired or stigmatized are better allocated to production in an ideal world.
As the Department of Labor cautions and I learned to my great cost some years back, "The employer [of an intern] derives no immediate advantage from the activities of the intern; and on occasion its operations may actually be impeded;" or in layman's terms, some workers productivity will always be below their wage, no matter how low the wage.
In short the data dictates the economy as a whole benefits most when the most productive members are less encumbered by those acquired most cheaply. This lends support to notions of basic income and the importance of social safety nets, while clearly arguing for a higher minimum wage.
A strong consumer base builds a strong and vibrant economy, but despite what we told little Jenny and Johnny growing up, some people simply should not be neurosurgeons or astrophysicists, or heaven forfend economists, lest they wind up doing more harm than good. (I'm looking at you, Ron Paul).
While everyone thinks they want to lower unemployment, in reality we benefit most by increasing per capita nominal income (the amount you take home) and per capita GDP (the value of those dollars) in equal measure, providing the most productivity and the highest quality of life is the purpose of economics, and we shouldn't let political talking points frame our discussion of the data.
As always, you're welcome to take a look at the data, the methodology, and the sources to come to your own conclusions.
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