# Damn, I did my math wrong...

written by Pj on Thursday January 17th, 2013 -- 7:36 p.m.
in reply to Unemployment Tax vs. Hiring New Employees

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If you've got 15 employees and you're thinking about asking them to all work one day of overtime, you're going to pay more than you would if you hired a new guy for just three weeks to do the extra work.

It's actually far better to hire the new guy.  Even better to hire 15 new guys and tell them to work just that one saturday, as that doesn't result in you falling behind.

It's only \$420 if you keep the employees long enough that they work enough hours to earn \$7000.  ...but if you only have 168 hours of work, and you're paying them the minimum wage, they'll only earn \$841, and so you'll only pay \$50 in unemployment tax.  That \$50 would only cover only 3½ hours of overtime pay.

To look at it another way, since the unemployment tax is 6%, you basically just pay 6% more for new employees until they've worked 7000 hours.  So a new minimum wage employee costs \$7.69/hour, whereas making one of your existing minimum wage employees work overtime costs \$10.88/hour.

So the actual break-even point is at zero overtime hours.  Having an employee work overtime always costs more than paying unemployment tax on a new employee.

I can't help but wonder, given just how many times I've fucked up the math trying to figure this out, if some of these factories haven't similarly fucked up, and that's why they require so much overtime.

Initially I took the 116 hour equation to mean "116 hours per employee" meaning that 116 hours per year per employee was the ideal amount of overtime to minimize labor costs.  I spent hours writing a nice long post about the impacts of this until finally I typed something that just seemed so wrong that I had to start backing up.  ...but even then, I didn't see the error.  I just found some reason why that particular statement wasn't true.  It wasn't until I attempted to create a graph showing the total yearly cost vs. number of employees for some given number of hours of work in a year that I started to realize I'd done something wrong.  I even zoomed into that graph, looking at differences of 0.01 employees expecting that the unemployment tax was going to shift the point of minimum costs away from the point of 40 hours per employee, but it appeared to be at exactly 40 hours.

So I graphed the two functions separately -- the cost of wages remains constant until overtime kicks in, at which point there's a large upward slope, while the cost of unemployment tax is a constant non-changing slope that lowers with additional employees, but at a slope much less than the slope of overtime costs.  Obviously adding a straight line function to a function with just one sharp angle isn't going to shift the location of that sharp angle in the new function.

Even with all that, I still hadn't figured it out.  The math I eventually posted in the previous post seemed to make sense, and 168 hours is just a small fraction of one employee per year.  Even though it's clearly more than 0.01 employees per year (the resolution of my graph) I still just assumed the discrepancy resulted from not actually being able to hire 0.01 employees, and in particular, not being able to pay 0.01 employment taxes.  Thus the benefit of making people work overtime was just only in tiny corner cases, and 168 hours over an entire company seemed to be a tiny corner case.

...but you can hire 0.01 employees and pay 0.01 unemployment taxes.  Just employ someone for only \$70 of work, and you pay only \$4.20 in unemployment tax.

I've probably wasted at least six hours thinking about this (desperately trying to avoid doing something productive, I guess), and I'm one of those people who are good at math.  The idea that some dumb factory manager might similarly miscalculate seems plausible enough.  It seems even more plausible that someone who knows just enough math to be dangerous might have written a textbook for business management with bad advice in it, advising people to minimize costs by solving some subtly incorrect formula that tells them how much overtime their employees should work.

That 116 hours isn't as small as it may seem.  It's enough to require one saturday every three and a half weeks.  It's a small portion of total labor, but a large portion of people's free time.