What's Wrong with the Stimulating View of the Minimum Wage Hike?

From NPR:

But liberal economists say this summer is the perfect time for a wage hike: It will put more money into the pockets of people who need it most. Fatter paychecks will stimulate spending and help the economy, they say. Kai Filion, a policy analyst for the Economic Policy Institute, a left-leaning research group, says this wage hike will generate $5.5 billion in consumer spending over the next 12 months.

It is my understanding that studies show that a 10% increase in the minimum wage, all else held equal, leads to about a 1% decrease in employment.  This gives us an elasticity of unskilled labor demand of -0.1, making unskilled labor a good with inelastic demand. 

Economic theory tells us that when demand for something is inelastic and its price rises, the incomes of the good's providers increases, even though though quantity demanded falls.  In the unskilled labor market, the "good" is unskilled labor, the "price" is the wage, and "providers" responds to the workers themselves.

Keynesian economic theory tells us that when a given amount of money is injected into the economy, overall spending in the economy will increase by a multiple of the original amount injected. 

So what's the problem?  Unskilled workers will have more money and that should lead to a multiple increase in overall spending, right? 

Wrong.  It's because the marginal income received by minimum wage workers lucky enough to still have jobs does not represent an injection of money into the economy.  It represents a redistribution of income and nothing more.  The added income of unskilled workers will come from the incomes of business owners who employ minimum wage workers and from the people who buy goods produced with unskilled workers through higher prices.  While unskilled workers may spend more overall, that will be offset by decreased spending in other areas of the economy.

Too bad NPR missed that point.