Workers in some countries are a lot more productive than workers in other countries.  One of the main differences is that people in more productive countries have more education.  When we hear that education in a country is going up, we normally take this as a sign that productivity is rising.  Question: If the signaling model of education is right, does this mean that rising education is not a sign of progress?  Does it mean that rising education is actually a sign of decay?!

No and no.  Holding policy constant, the signaling model of education specifically implies that rising education is a symptom – though not a cause – of rising productivity.  I’ve long taken this for granted, but yesterday I created a simple model to prove it.  Alex Tabarrok and Garett Jones have checked my math, but any errors are my own.

The model:

1. IQ ~ U[A, 100+A], with A

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≥0. 

In English, intelligence is uniformly distributed between A and 100+A, like so:

iqsig.jpg

The higher A, the higher the average IQ of the workforce.  We’ll define IQ* momentarily.

2. Worker productivity is proportional to IQ.  For convenience, in fact, productivity equals IQ.

3. The only signal of IQ is a diploma, and a worker can only get one diploma.

4. Due to competition and imperfect information, employers pay a wage equal to the average productivity for people with their education level.  WN=wage of workers without a diploma.  WD=wage of workers with a diploma.

5. Each worker maximizes U=W – K/IQ, where W is the worker’s wage, K is a constant, and IQ is the worker’s IQ.  K/IQ is a signaling cost; the smarter you are, the less painful you find it to get your diploma.  (A≥0 because this signaling cost function wouldn’t make sense with negative A).

Given the set-up, there’s clearly going to be an IQ level such that everyone above it gets a diploma and everyone below it doesn’t.  Let’s call that IQ level IQ* and solve for it.  The formula for a midpoint of a line implies that for the person with IQ*:

WN=(A+IQ*)/2
WD=(IQ*+100+A)/2

The person with IQ* is indifferent about getting a diploma when:

(A+IQ*)/2=(IQ*+100+A)/2 – K/IQ*

Subtracting (A+IQ*)/2 from both sides and simplifying leaves:

IQ*=K/50

In equilibrium, everyone with IQ greater than K/50 gets a diploma.  Notice that A, the ability parameter, drops out; IQ*=K/50 for all A.  If A equals 0, and K=2500, then IQ is uniformly distributed between 0 and 100 and IQ*=50, so 50% of the population gets a degree.  If A=40, then IQ is uniformly distributed between 40 and 140 and IQ*=50 (still!), so 90% of the population gets a degree.  If you could increase IQ by sprinkling pixie dust on the population, education levels would rise despite the fact that education has zero effect on productivity.

Is it possible that the social costs of signaling are so great that more social ability leads to lower social utility?  At least in this model, no.  The deadweight loss of signaling is just the area under the signaling cost function K/IQ for everyone who chooses to signal:

iqsig2.jpg

This integral equals K*ln IQ.  Plugging in the lower and upper limits yields a deadweight cost equal to:

K[ln(100+A)-ln(K/50)]

Total Social Welfare is just the population (=100) times the average wage (=average IQ = 50+A) minus the deadweight cost of signaling:

SW=100*(50+A) – K[ln(100+A)-ln(K/50)]

To verify that higher IQ increases SW, differentiate SW with respect to A:

dSW/dA=100 – K/(100+A)

Is this always greater than 0?  Yes.  In any interior solution, the diploma cutoff IQ* is less than or equal to the maximum IQ: K/50

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≤100+A.  So K/(100+A)≤50, implying that 100 – K/(100+A)≥50 for all A≥0.  Higher IQ unambiguously raises Social Welfare for interior solutions. 

The same holds for corner solutions.  If the cost of signaling is so high that no one gets a diploma (K/50>100+A), there’s zero deadweight cost of signaling, and raising A automatically raises Social Welfare by 100

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ΔA (the population*the change in A).  If the cost of signaling is so low that everyone gets a diploma (K/50<A), it’s impossible for the deadweight cost of signaling to increase any further.  Raising A once again raises Social Welfare by 100ΔA.

Bottom line: When IQ rises, total signaling costs go up, but the net effect of more productivity plus more signaling is still positive.  More diplomas, like bigger wedding rings, are indeed a symptom of progress.  But only a symptom.  Insofar as the signaling model of education (or the signaling model of wedding rings!) is true, policy analysts have to be careful not to conflate a mere symptom of progress with progress itself.

P.S. If I were submitting this model to a journal, I’d spend a lot more time checking my math.  If you do find a mistake, please share it with no more than moderate scorn. 🙂

P.P.S. It wouldn’t be too hard to show that this model also implies that increasing educational equality is a genuine symptom – but not a cause – of increasing labor market equality.