This will probably not rank up there with the weightiest columns ever posted in this space. After all, even the Babe couldn’t hit a home run every time. But just as in baseball, it’s good to go back to the basics when things have gotten muddled. Also, there’s an educational component to this column. So chalk this article up to an attempt along those lines.

Both the public and the media have been confused about monthly unemployment reports – deliberately to some extent, and partly because the data are fundamentally confusing. True unemployment is often underreported in the government statistics. During bad times, the administration in power tends to favor reporting that makes things look better than they actually are. You can’t blame them, but it does confuse people. Also, it’s unsporting to bend the scoring-rules to make your team look good.

Complete unemployment data are not available to government officials, so they report what they have. The two primary data elements reported can be called “A” and “B”: i.e.,

A = Number of people drawing or newly applying for unemployment benefits.

B = Number of people currently working.

From these we obtain…

C = A + B (total current workforce)

and R = A/C (current unemployment rate)

Simple, eh? It recalls my old teacher’s motto: “For every problem there’s a solution that is simple, elegant and completely wrong.” (Grandpa’s “lies, damned lies, and statistics” also comes to mind.)

Quantities A and B come readily to hand for government officials. Unfortunately, A is not a correct count of the total unemployed population, which means that C doesn’t really count all people who are working, could work and want to work. Economists say that at the national level A understates the unemployed population – perhaps by as much as 50% during bad economic times, and hardly at all during good times. During bad times, after long-time unemployed workers use up their unemployment benefits they are no longer included in A. Ditto for people who couldn’t apply for those benefits in the first place, because they were self-employed, etc. To show a correct unemployment rate, we would need the true number of unemployed people: i.e.,

U = A + X

where X = Number of unemployed people outside the unemployment benefits system

and A = Number of people drawing or applying for unemployment benefits (as above).

Then we could calculate the true unemployment rate:

R_{T} = U/(C+ X)

Eyes glazed over now? Relax. The eyes of the government’s green-eyeshade corps are a little glazed-over too. It’s not their fault, as the aptly-named quantity “X” isn’t really available. It can only be estimated. That’s why government statisticians have stuck with the simpler (but incomplete) formulation, R = A/C.

All this would be OK if everyone agreed that the published statistics had limited accuracy and shouldn’t be micro-examined for trends. But when politics gets involved, caveats about statistical limitations go out the window. This happened consistently during the Obama administration with respect to statistic “R” (the official unemployment rate). Because the economic news was mostly bad, early in Mr. Obama’s tenure, the current administration was desperate to see some encouraging trend in unemployment, which had reached a high of 10.2% in November 2009. Then it dropped to 10% in December, and fell to 9.7% in January 2010. This was hailed as an indicator that the economy was finally “turning around.”

In
both of those monthly reports, however, the number of people working (i.e., “B,”
above) actually declined – in January, by 20,000 people. This should have meant
a higher unemployment rate. But the number of people on (or seeking)
unemployment benefits *also declined*. This
pushed *down* the unemployment rate, R,
causing White House poobahs to shout hosannas over the “shrinking” jobless rate.
It’s an arithmetic anomaly, however. (Does 20,000 fewer people at work really
sound like better times?)

The result is confusion for both the People and Mainstream Media reporters. To some extent, what we might call “mathematical illiteracy” is also to blame. It’s a mathematical fact that reducing a fraction’s numerator and denominator by the same amount produces a fraction of lower value. That’s really why we got lower unemployment rates for those two months.

For instance, take the fraction 50/100 (i.e., 0.500). Now subtract 10 from both numerator and denominator. The new fraction is 40/90 = 0.444… which is obviously less than 0.500.

This works for any fraction, but bigger numbers can confuse some people. The rule is still true, but it’s not as obvious. Say 10,000,000 people are on (or applying for) unemployment benefits, and 90,000,000 are working. Then A=10,000,000, B=90,000,000; and the declared unemployment rate is:

R = (10,000,000)/(100,000,000) = 10%

So far, so good. But if next month 200,000 of those unemployed workers quit looking for work, or their benefits run out, the government says the number of unemployed workers is now 9,800,000, and the work-force totals 99,800,000. Thus, the unemployment rate is:

R = (9,800,000)/(99,800,000) = 9.82%

This looks (slightly) encouraging, but nothing has been gained. Not one additional person is working. Some bureaucrats might posture about the “recovery taking hold,” or some such nonsense, but it’s just an illusion. Government statistical conventions have contrived to produce deceptive numbers.

(By
the way, in case some readers suspect that the fraction-reduction principle
isn’t always true, and that I might have contrived some special numbers here, please
consult the algebraic proof in the footnote.** ^{1}** This isn’t rocket science. If you can understand the
proof, thank your old algebra teacher. If you can’t, you’ve helped demonstrate another
serious national problem.)

There is no substitute for good data. Incomplete data furnish only a blurred picture. I have seen estimates of the true unemployment rate, R_{T}, ranging from 15% to as high as 22%, when the stated rate (“R”) was just 10%. I have no idea which figure was accurate because “X” was unknown. The likelihood is strong, however, that the official value of R vastly understated reality. Around 2010 I knew many unemployed people, personally. In the past I didn’t know anyone unemployed, so it was empirically evident that things were not good.

Were things getting better during those few months which I cited? Doubtful, but no one really knew. I do know, however, that the country would be much better served with accurate data. This issue isn’t a big deal now, with the economy at (or near) full employment. But it will be baaaack when there’s a dip. All those high-priced federal economists ought to be smart enough to find X (and U). Americans need to know the real score, not a politically motivated “misunderestimate” (as George W. used to say).

*******

[1]
**Theorem:** Reducing a fraction’s
numerator and denominator by the same amount always produces a fraction of
lesser value.

**Conditions:** b
> a > x > 0.

**Proof:**

- We postulate that: a/b > (a-x)/(b-x)
- Then multiplying each fraction by b(b-x): b(b-x)a/b > b(b-x)(a-x)/(b-x)
- Cancelling terms gives: a(b-x) > b(a-x)
- Expanding: ab-ax > ab-bx
- Simplifying: -ax > -bx, or bx-ax > 0
- Or x(b-a) > 0.
- This inequality must be true because both x and the quantity (b-a) are positive.

QED.