Chapter 87: Human Capital, Technology, Signaling, and Discrimination. The Encyclopedia of Economic Principals, interactive companion

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Skill premium after a biased technology shock

How do skill-biased technology and skill supply set the wage premium?

A shift in relative demand toward skill raises the premium most when supply cannot respond. As training expands supply, the premium falls back but need not return to its start.

$Equation, written in LaTeX: \ln(H/L)=\ln(A_H/A_L)-2\ln(w_H/w_L).$

$Equation, written in LaTeX: \ln(0.25)=\ln(2)-2\ln(w_H/w_L),$

Scroll sideways for the whole equation

H/L is the relative supply of skilled to unskilled labor, A_H/A_L the technology ratio and w_H/w_L the skill premium. The elasticity of substitution is 2.

Predict first. If supply rises to 0.40 with technology at 2, is the premium above or below the original 2?

Your prediction Above 2 Below 2

Technology ratio A_H/A_L

Skill supply H/L

Figure: Skill premium after a biased technology shock. Downward relative-demand curve for A_H/A_L = 2 and a dashed neutral curve, cut by supply at H/L = 0.25; the premium is 2.83. — Technology ratio A_H/A_L: 2, Skill supply H/L: 0.25
Constructed example: the chapter's hypothetical values are technology ratios 1 and 2 and supplies 0.25 and 0.40; a ratio of 3 and supply 0.55 are added.

Relative wage w_H/w_L: 2.83
Change vs starting premium 2: +41.4%

Hypothetical market-clearing schedule from the chapter. With technology ratio 2 and skill supply 0.25, the premium is (2 / 0.25)^(1/2) = 2.83, +41.4% against the starting premium of 2. Biased technology raises the premium; added skill supply lowers it.

Worked steps

ln(0.25) = ln(2) - 2 ln(w_H/w_L)
w_H/w_L = (2 / 0.25)^(1/2) = 8.0000^(1/2) = 2.8284
Change: 2.8284 / 2 - 1 = 0.4142, or +41.4%

Use the idea

Decompose a premium change into the part predicted by realized supply with fixed technology and the part predicted by technology with fixed supply.

Where the conclusion applies

A stylized schedule with elasticity 2 and market clearing; the numbers are illustrative, not estimates for any occupation.

Check your understanding: With A_H/A_L = 3 and H/L = 0.55, what is the premium?

(3 / 0.55)^(1/2) = 5.4545^(1/2) = 2.34.

Chapter 87 source: section "Skill-biased technical change".

Demonstration 2 of 4

Is training worth it? The horizon decides

How does the number of years left to use a skill change whether training pays?

The same training is a good investment for a long horizon and a bad one for a short horizon, which is why investment concentrates early in working life.

$Equation, written in LaTeX: H_{t+1}=(1-\delta)H_t+F(s_tH_t,x_t),$

$Equation, written in LaTeX: 3{,}000\frac{1-(1.05)^{-20}}{0.05}\approx37{,}386.$

Scroll sideways for the whole equation

Training costs 20,000 (8,000 tuition plus 12,000 forgone pay) or 15,000 with a tuition cut, and raises earnings by 3,000 a year for the years set here, discounted at 5 percent.

Predict first. Does a 5,000 tuition cut rescue the five-year worker?

Your prediction Yes No

Post-training earning years

Total cost (tuition plus forgone pay)

Figure: Is training worth it? The horizon decides. Bars of cumulative present value over 20 years, reaching 37,387, against a cost line at 20,000. — Post-training earning years: 20, Total cost (tuition plus forgone pay): 20000
Constructed example: the chapter's hypothetical values are the 3,000 gain, 5 percent, costs 20,000 and 15,000 and horizons 20, 5 and 8 years.

PV of benefit: 37,387
Total cost: 20,000
NPV: 17,387
Verdict: Worthwhile

Hypothetical program from the chapter. A 3,000 annual gain over 20 years at 5 percent is worth 3,000 x 12.4622 = 37,387 at the end of training, against a cost of 20,000: NPV 17,387, so the program is worthwhile. The horizon, not the training itself, decides.

Worked steps

Annuity factor = (1 - 1.05^-20) / 0.05 = 12.4622
PV = 3,000 x 12.4622 = 37,387
NPV = 37,387 - 20,000 = 17,387, so the program is worthwhile

Use the idea

Value training as an annuity over the years the skill will actually be used, and compare it with the full cost at the same date.

Where the conclusion applies

A constant 3,000 gain with no depreciation of the skill and a 5 percent discount rate; the chapter's 10 percent obsolescence case is not shown. The book prints 37,386; the exact value 37,386.63 rounds to 37,387 (NPV 17,387).

Check your understanding: With eight years and the full 20,000 cost, is the program worthwhile?

PV = 3,000 x 6.4632 = 19,390 < 20,000; NPV = -610, so no.

Chapter 87 source: section "Ben-Porath life-cycle human-capital model".

Demonstration 3 of 4

Automation, scale, and new tasks

How do displacement, output scale and new tasks combine to set employment after automation?

Automation displaces workers from old tasks; cheaper output can expand scale; new human tasks reinstate labor. The same net change can come from very different mixes.

$Equation, written in LaTeX: \frac{50}{7}\approx7.14.$

Equation, written in LaTeX: (50+15)(1.2)=78

Scroll sideways for the whole equation

The plant has 100 activities per cycle; labor covers 70 before and 50 after automation. Each worker supplies 7 activity equivalents. Scale multiplies cycles; new tasks add human activities per cycle.

Predict first. Without new tasks, does 20 percent scale growth restore baseline employment?

Your prediction Yes No

Output scale

New human tasks

Figure: Automation, scale, and new tasks. Waterfall from 10 workers: displacement -2.86, scale +1.43, reinstatement +2.57, ending at 11.14. — Output scale: 1.2, New human tasks: 15
Constructed example: the chapter's hypothetical values are the 100 activities, 70 and 50 labor tasks, 7 per worker, scales 1.0, 1.2 and 2.0 and 0, 15 and 30 new tasks.

Employment after automation: 7.14
Scale effect: +1.43
Reinstatement effect: +2.57
Final employment: 11.14
Change vs baseline 10: +1.14

Hypothetical plant from the chapter. Automation cuts labor tasks from 70 to 50 (50 / 7 = 7.14 workers). Output scale 1.2 and 15 new tasks give (50 + 15) x 1.2 / 7 = 11.14 workers, +1.14 against the baseline of 10. Labor's share of the original tasks is 50% after automation in every case.

Worked steps

Displacement: 50 / 7 = 7.14, a change of -2.86
Scale: 50 x 1.2 / 7 = 8.57, adding +1.43
Reinstatement: (50 + 15) x 1.2 = 78.0 activities; 78.0 / 7 = 11.14
Net change: 11.14 - 10 = +1.14

Use the idea

Look for separate evidence on each channel: workflow logs for displacement, output for scale, and new duties for reinstatement.

Where the conclusion applies

Equally weighted activities, fixed activities per worker and immediate adjustment; the chapter notes the interim path can differ sharply.

Check your understanding: With scale 2.0 and 15 new tasks, what employment is required?

(50 + 15) x 2.0 / 7 = 130 / 7 = 18.57 workers.

Chapter 87 source: section "Task displacement and reinstatement".

Demonstration 4 of 4

Shrinkage and unequal score reliability

Why can two applicants with the same score get different forecasts when score reliability differs?

A noisier score is shrunk harder toward the group mean. That hurts high scorers in the noisy group and helps low scorers; equal reliability removes the difference.

$Equation, written in LaTeX: E(\theta_i\mid s_i,g)=\lambda_gs_i+(1-\lambda_g)\mu_g,$

$Equation, written in LaTeX: E(\theta\mid80,A)=0.75(80)+0.25(60)=75$

Scroll sideways for the whole equation

s is the applicant's score, mu = 60 the prior mean for both groups and lambda_g the weight on the score (0.75 for A; set here for B). The job requires a forecast of at least 70.

Predict first. At a score of 50, which group gets the higher forecast (lambda_B = 0.25)?

Your prediction Group A Group B

Reliability weight for group B

Applicant score

Figure: Shrinkage and unequal score reliability. Number line from 40 to 100 with the prior at 60, the cutoff at 70, the score 80, and forecasts 75.0 for A and 65.0 for B. — Reliability weight for group B: 0.25, Applicant score: 80
Constructed example: the chapter's hypothetical values are prior 60, cutoff 70, weights 0.75, 0.25 and 0.90 and scores 80 and 50; a weight of 0.50 and a score of 90 are added.

Posterior A: 75.0
Posterior B: 65.0
Hire A: Yes
Hire B: No

Hypothetical hiring exercise from the chapter. With score 80, A's forecast is 0.75 x 80 + 0.25 x 60 = 75.0 and B's is 0.25 x 80 + 0.75 x 60 = 65.0; A's forecast is higher, because a score above the mean of 60 is pulled down more for B, whose score gets less weight. Against the cutoff of 70, A is hired and B is not hired.

Worked steps

A: 0.75 x 80 + 0.25 x 60 = 60.00 + 15 = 75.0
B: 0.25 x 80 + 0.75 x 60 = 20.00 + 45.00 = 65.0
Cutoff 70: A is hired, B is not hired

Use the idea

Audit a decision rule at common scores and check whether improving score reliability closes the gap, before attributing it to taste.

Where the conclusion applies

A linear-normal forecast with equal prior means; the weights are beliefs about reliability, not measured facts about any group.

Check your understanding: With lambda_B = 0.50 and a score of 90, is B hired?

0.5 x 90 + 0.5 x 60 = 75 >= 70, so yes.

Chapter 87 source: section "Statistical discrimination".

Skill premium after a biased technology shock

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Is training worth it? The horizon decides

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Automation, scale, and new tasks

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Shrinkage and unequal score reliability

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies