Archive for the 'Back of the Envelope' Category

The Arithmetic of Wage Gaps

Mark Perry and Andrew Biggs argue in the Wall Street Journal that

These gender-disparity claims [the claims that women are paid 23% less than men for the same work] are also economically illogical. If women were paid 77 cents on the dollar, a profit-oriented firm could dramatically cut labor costs by replacing male employees with females. Progressives assume that businesses nickel-and-dime suppliers, customers, consultants, anyone with whom they come into contact — yet ignore a great opportunity to reduce wages by 23% [by hiring women instead of men].

Well, first of all, even if we take the gender disparity claims at face value, this doesn’t add up to an opportunity to reduce wages by 23%. Only about half the work force is female, so the average firm, if it replaced all of its men with women earning 23% less, would reduce its wage bill by only about 11.5%.

Beyond that, the Perry/Biggs argument appears to founder on the observation that lazy and incompetent managers do in fact manage to ignore profit opportunities all the time. Why, then, is it so hard to imagine that they’re ignoring this one?

Fortunately, I’m here to fill the gap —- by figuring out just how big a profit opportunity we’re talking about.

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MegaOdds

In case you’re thinking of running out for a MegaMillions lottery ticket in the few hours left before tonight’s drawing: If you drive one mile to buy your ticket, your chance of being involved in a fatal accident on the way is about 8 times as great as your chance of winning the jackpot.

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Aaagh!

I am buying a house, and am therefore faced with the choice between a 15 year mortgage at 2.875% and a 30 year mortgage at 3.49% (as of a couple of days ago; those rates have probably changed a little by now).

The main advantage of the 15 year mortgage is that it comes with a lower interest rate and, because I’m making larger monthly payments, it keeps my money out of the stock market, which is good if the market tanks. The main advantage of the 30 year mortgage is that it allows me to keep more money in the stock market for a much longer time, which is good if the market does well.

How should I weigh those factors? Economics tells me that I will solve this problem by forecasting the return on equities over each of the next 30 years, and computing, on the basis of my forecast, which mortgage will leave me richer in the long run. No, that’s not quite right. Actually, economics tells me that I’ll make many forecasts, assign each one a probability, and thereby compute two probability distributions for my future net worth and then choose the distribution I prefer.

Now let’s get serious.

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Charting the Tax Plans

Ezra Klein, quoted with approval by Paul Krugman, offers this chart of how the Obama and Romney tax proposals will change rates for taxpayers in various quintiles:

What we’re supposed to infer, according to Krugman, is that

we have an election in which one candidate is proposing a redistribution from the top … downward, mainly to lower-income workers, while the other is proposing a large redistribution from the poor and the middle class to the top.

But no such thing is remotely true. What we actually have is an election in which both candidates are proposing massive redistributions from the top downward, one slightly less so than the other. You’d never know this from looking at Klein’s chart because it illustrates changes in rates, whereas what actually matters is the rates themselves. It makes no sense to ask whether any particular group ought to be paying more or less without reference to how much they’re already paying.

Indeed, this is a classic example of what I once called the “Grandfather Fallacy” — by focusing on changes instead of absolutes, Klein’s chart conceals any existing inequities and hence treats them as “grandfathered in”.

Fortunately, Greg Mankiw has provided the numbers that allow us to make the requisite correction. Here, according to Mankiw, are the current tax burdens on various income groups (counting transfers as negative taxes, as of course one should):

Bottom quintile: -301 percent
Second quintile: -42 percent
Middle quintile: -5 percent
Fourth quintile: 10 percent
Highest quintile: 22 percent

Top one percent: 28 percent

That “-301 percent” means, for example, that a typical family in the bottom quintile receives $3.01 in net transfers for every $1 that it earns.

By adding these numbers to the numbers in Klein’s graph, we can construct a picture that actually depicts something interesting, namely the projected tax burdens for each group. It looks like this (the vertical axis represents percentage of income):

Note, for example, that, contrary to the impression you might have gotten from Klein’s and Krugman’s posts, both plans place the highest percentage burden on the top 1%, and both plans place a negative burden on the middle quintile — though Obama’s does both of these things to an ever-so-slightly greater extent than Romney’s does. There’s room for disagreement about which plan is fairer, but no room, I think, for disagreement about which chart is relevant.

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Lifeboats on the Titanic

titanicShould RMS Titanic have carried more lifeboats? Yes, probably. But it took me a few minutes to convince myself.

Roughly 1500 died on the Titanic; according to Wikipedia, it would have cost about $16,000 to equip her with additional lifeboats sufficient to save them all. Call it $10 per life saved. The price level today is roughly 22 times what it was in 1912, so in today’s terms that’s $220 per life.

Now, if I were boarding a ship for a luxury cruise, and was offered the chance to pay an additional $220 for a guaranteed seat on a lifeboat in the event of a sinking, I’m quite sure I’d take a pass — and I’m quite sure so would virtually all of my fellow passengers. So if the Titanic had been designed to cross the ocean once and then spend the rest of its days in a museum, it would have been insane to equip her with extra lifeboats. But of course if the Titanic had been designed to cross the ocean once and then spend the rest of its days in a museum, it would have been insane to build her in the first place. So that’s not the right calculation.

The right calculation accounts for the fact that a single lifeboat provides security to passengers on multiple voyages. How many voyages? Well, the Titanic was intended to make the round trip between Europe and America every three weeks; that’s two voyages per three-week period. I’m not sure how long the sailing season was, but we know it was underway by mid-April (and perhaps earlier; it’s often mentioned that if the Titanic had been ready earlier she would have sailed earlier) so (assuming sailing conditions are roughly symmetric around the solstice) it must have lasted till at least mid-August. That’s time for five round trips at a minimum, and I’m guessing this is a quite conservative assumption.

If a lifeboat lasts a year, then, it does its job at least ten times. If it lasts five years (which is, I suspect, another quite conservative assumption), it does its job fifty times. Now we’re in the vicinity of $4 per passenger (and of course much less if my assumptions are indeed quite conservative).

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Women’s Wages and the Back of My Envelope

Yesterday’s breathtakingly dishonest graph from the AFL-CIO touched off some discussion in comments about whether the male/female wage differential could plausibly be driven by employer discrimination.

The usual argument to the contrary runs like this: If the differential is driven by employer discrimination (as opposed to, say, the abilities and/or preferences of the workers), then non-discriminating employers (i.e. those who care only about making a buck, regardless of who they have to hire to do it) would draw only from the relatively cheap female labor pool. It wouldn’t take many of these non-discriminating employers to drive women’s wages up to the same level as men’s. We don’t see that happening, ergo the hypothesis of employer discrimination is refuted.

The problem with that argument is that it assumes employers won’t ignore a profit opportunity, whereas in fact employers ignore profit opportunities all the time — by keeping on their incompetent nephews, taking Wednesday afternoons off to play golf, or, yes, hiring people they like having around instead of people who could do a better job.

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