Sheryl Sandberg wrote a heartfelt mother's day post on Facebook, detailing the things she would like to see to help mothers in the workforce. As a single mother, Sandberg certainly understands some the challenges working mothers face. But, what she doesn't understand is basic economics.

She writes:

To start, it's long past time to raise the federal minimum wage. Two-thirds of minimum wage workers are women. Raising the wage would reduce pay inequality and help millions of families living in or near poverty.

It sounds good but raising the federal minimum wage means you raise the cost of everything. Which strongly impacts the additional point she makes (2 paragraphs down):

And we need affordable child care. Child care for two children exceeds the median annual rent in all 50 states. How are parents supposed to work if they don't have a safe and affordable place to leave their kids?

Indeed. Child care is a huge problem for working moms and dads. It's terribly expensive. Do you know what will make it more expensive? Raising the minimum wage.

The average childcare worker in the United States earns more than the current $7.25 federal minimum wage--an average of $11.02 per hour, with only the top 10 percent above the oft-touted $15 per hour wage. (Some states and municipalities have higher minimum wages.) It's not a high-paying job and over 95 percent of childcare workers are female--the very people Sandberg is talking about. So why won't raising the minimum wage help them?

If you raise the minimum wage, you raise the cost of day care. If you raise the cost of daycare, you cancel out the benefit from the minimum wage increase. Not to mention that many low-level jobs will disappear under a minimum wage increase. Seattle, which raised its minimum wage while the surrounding areas did not found the following:

The minimum wage appears to have slightly reduced the employment rate of low-wage workers by about one percentage point. It appears that the Minimum Wage Ordinance modestly held back Seattle's employment of low-wage workers relative to the level we could have expected.

Hours worked among low-wage Seattle workers have lagged behind regional trends, by roughly four hours per quarter, on average.

One percent doesn't seem like a lot unless you are in that one percent. Additionally, this was one percent compared to the surrounding areas in a strong economy. In a bad economy, will that number jump?

Being employed at a low wage is better than being unemployed--even if the majority of your paycheck goes to childcare. Why? The time you work rather than stay at home is an investment in your career and in your future. You're not only losing the income for the time period in which you're not working, you're losing future raises, experience, and training. If you're earning a low wage when you have a child and then quit work to take care of that child, when you come back to the workforce five years later, you'll re-enter where you left off, or even below that. If you'd stayed, you would have likely received raises and additional training and experience that make you more valuable.

Solving the cost of childcare is difficult for low-income families, but the solution isn't raising the minimum wage. Perhaps the solution could be subsidies from private contributions so that the taxes aren't raised on the very people who need help. Sandberg herself has an estimated net worth of $1.6 billion. Perhaps some of that money could be used to help people obtain affordable childcare.