Demonstration 1 of 4
Fewer children, more schooling
Why does a higher schooling standard make an extra child more expensive?
Because every child receives the same investment, the term d n q makes quantity and quality raise each other's price. A higher standard raises the price of a child even when total resources could still pay for a larger family at the old standard.
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n is the number of children and q the common investment per child. Child-related cost is C = a n + b q + d n q with a = 18, b = 3 and d = 2. P_n = a + d q is the shadow price of another child and P_q = b + d n the shadow price of one more unit of investment per child.
Predict first. When the standard rises from q = 20 to q = 50, does another child become cheaper or dearer at the margin?
Choose an example
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Constructed example: the chapter's invented household (a = 18, b = 3, d = 2, resources 420) and its bundles (3, 20), (2, 50), (3, 50) and (5, 20); the bundles (2, 20) and (5, 50) complete the grid, and P_q at n = 2 and n = 5 is derived from b + d n, not printed in the book.
Calculated values
- Cost C
- 386
- Left over from 420
- 34
- Shadow price of a child P_n
- 118
- Shadow price of quality P_q
- 7
- Affordable
- yes
C = 18(2) + 3(50) + 2(2)(50) = 36 + 150 + 200 = 386, which fits in 420 with 34 left over. Another child costs P_n = 18 + 2(50) = 118 at this standard, and one more unit of investment per child costs P_q = 3 + 2(2) = 7. The interaction term d n q is why a higher standard makes each extra child dearer.
Worked steps
- C = 18(2) + 3(50) + 2(2)(50) = 36 + 150 + 200 = 386
- Left over = 420 - 386 = 34
- P_n = 18 + 2 x 50 = 118
- P_q = 3 + 2 x 2 = 7
Use the idea
Compare the marginal cost of another child at the old and new investment standards, not only whether total resources cover the bundle.
Where the conclusion applies
Linear cost with a fixed interaction coefficient and a common investment for every child. Without a utility function the costs do not show which bundle is optimal.
Check your understanding: Is three children at q = 50 affordable with 420?
Chapter 63 source: section "Quantity-quality tradeoff in children".
Demonstration 2 of 4
A larger working-age share
How much richer per resident is a population with more people of working age?
Output per resident is a product of three ratios, so a larger working-age share raises it in proportion even with no change in productivity or employment. The gold block is the part due to the age structure, measured at this state's productivity and employment.
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Y/N is output per resident, Y/E output per employed worker, E/W the employment rate of the working-age population and W/N the working-age share, so Y/N = (Y/E)(E/W)(W/N).
Predict first. With output per worker 50 and employment 0.70 fixed, how much does Y/N rise when the working-age share goes from 0.45 to 0.60?
Choose an example
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Constructed example: the chapter's hypothetical country (1,200 residents, shares 0.45 and 0.60, employment 0.70 and 0.75, output per worker 50 and 54); mixed combinations of the book's values complete the grid.
Calculated values
- Y/N
- 21.000
- Baseline Y/N
- 15.75
- Change from baseline
- 33.3%
- Y/N at the old 0.45 share
- 15.750
- Age-share contribution
- 5.250
Y/N = 50 x 0.70 x 0.60 = 21.000, against the baseline 50 x 0.70 x 0.45 = 15.75, a change of 33.3%. With the same output per worker and employment rate but the old 0.45 share, Y/N would be 15.750, so the age structure contributes 21.000 - 15.750 = 5.250.
Worked steps
- Y/N = 50 x 0.70 x 0.60 = 21.000
- Baseline = 50 x 0.70 x 0.45 = 15.75
- Change = (21.000 - 15.75) / 15.75 = 33.3%
- At share 0.45: 50 x 0.70 x 0.45 = 15.750
- Age-share contribution = 21.000 - 15.750 = 5.250
Use the idea
Split a change in income per head into productivity, employment and age-share pieces before crediting policy with the whole gain.
Where the conclusion applies
Total population is held at 1,200 to isolate composition. The gain reverses when the large cohort retires, and the example is not a forecast of any real population.
Check your understanding: With employment 0.75 and output per worker 54, what is the age-share contribution?
Chapter 63 source: section "Demographic dividend".
Demonstration 3 of 4
Train if enough others train
Why can one missing commitment turn a profitable program into a loss?
The private return jumps at the threshold, so both zero enrollment and wide enrollment are equilibria. A coordinator needs only enough binding commitments to cross the threshold.
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c is the private training cost. A trainee earns gross return R_0 = 7 when fewer than 48 adults train and R_1 = 15 when at least 48 train, because only then does a specialized employer enter. n is the number of committed participants among 80 adults.
Predict first. An adult expects 46 others to enroll. Should she enroll?
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Constructed example: the chapter's hypothetical community (80 adults, cost 10, returns 7 and 15, threshold 48, participation 0, 47, 48 and 80); training costs 8 and 12 are added for comparison.
Calculated values
- Net return per trainee
- 5
- Group payoff
- 240
- Employer enters
- yes
48 trainees is at or above the threshold of 48, so the gross return is 15 and each trainee nets 15 - 10 = 5. The group payoff is 48(15 - 10) = 240.
Worked steps
- 48 is at or above 48, so R = 15
- Net per trainee = 15 - 10 = 5
- Group payoff = 48 x 5 = 240
Use the idea
Before subsidizing participation, check whether returns depend on how many others participate and how far below the threshold current participation sits.
Where the conclusion applies
A sharp threshold and fixed returns are invented for instruction; a real program must show that employer entry truly depends on trained-worker density.
Check your understanding: What is the group payoff with 47 commitments at cost 10?
Chapter 63 source: section "Threshold-externality poverty trap".
Demonstration 4 of 4
The big push
When does a plant profit only if the others open too?
Each plant's demand depends on how many others operate, so the same payoff schedule supports both no entry and full entry. Only a commitment by all four reaches the profitable outcome.
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pi(m) is one plant's profit when m other plants operate. Each plant has cost 150 and base revenue 105, and every other plant that opens adds the demand spillover (20 in the book).
Predict first. With spillover 20, will a coalition of three plants hold together?
Choose an example
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Constructed example: the chapter's invented region (four plants, cost 150, base revenue 105, spillover 20); spillovers of 15 and 25 are added for comparison.
Calculated values
- pi(m)
- 15
- pi(0), lone plant
- -45
- pi(2), three-plant coalition
- -5
- Aggregate profit, all four
- 60
- Three-plant coalition
- unravels
pi(3) = 105 + 20(3) - 150 = 15. A lone plant earns pi(0) = -45, so zero entry is an equilibrium; with all four in, each earns pi(3) = 15, so full entry is also an equilibrium, and aggregate profit is 4(15) = 60. A three-plant coalition gives each member pi(2) = -5, so it unravels.
Worked steps
- pi(3) = 105 + 20 x 3 - 150 = 15
- pi(0) = 105 - 150 = -45
- pi(2) = 105 + 40 - 150 = -5
- pi(3) = 105 + 60 - 150 = 15
- Aggregate = 4 x 15 = 60
Use the idea
Check whether a project's revenue depends on complementary projects starting together before judging it unprofitable on its own.
Where the conclusion applies
The spillover is assumed to be genuine new demand from modern employment; if it only displaces purchases from traditional producers, the welfare conclusion fails.
Check your understanding: With spillover 25, does a three-plant coalition survive?
Chapter 63 source: section "Big-push coordination failure".