Demonstration 1 of 4
Fallback power picks the allocation
How does raising one partner's separation utility change which allocation the couple chooses?
The Nash product rewards allocations that give each partner a large gain over the fallback. Raising Ana's fallback shrinks her gain under both allocations, but proportionally more under Y, so agreement moves toward X without any new resources.
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X and Y are two efficient allocations giving utility pairs (U_A, U_B) = (78, 52) and (60, 70). d_A and d_B are the disagreement (separation) utilities. The Nash product N is the product of each partner's gain over the fallback; the larger product is chosen.
Predict first. A reform raises Ana's fallback from 35 to 45 with the same resources. Which allocation wins?
Choose an example
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Constructed example: the book's hypothetical couple has X = (78, 52), Y = (60, 70) and fallbacks (35, 32) and 45 for Ana; fallbacks of 40 and 61 for Ana and 40 for Ben are added.
Calculated values
- N_X
- 860
- N_Y
- 950
- Chosen allocation
- Y
With fallbacks d_A = 35 and d_B = 32, N_X = (78 - 35)(52 - 32) = 43(20) = 860 and N_Y = (60 - 35)(70 - 32) = 25(38) = 950. So Y has the larger Nash product, 950 against 860. Resources and the two utility pairs are fixed; only the fallbacks move.
Worked steps
- N_X = (78 - 35)(52 - 32) = 43(20) = 860
- N_Y = (60 - 35)(70 - 32) = 25(38) = 950
- Result: Y has the larger Nash product, 950 against 860
Use the idea
When a policy changes who controls an asset or income after separation, ask how it moves each partner's fallback before predicting spending changes.
Where the conclusion applies
Two feasible allocations, cardinal utilities and the Nash bargaining rule. A fallback of 61 makes Y violate Ana's individual rationality (60 < 61).
Check your understanding: With d_A = 40 and d_B = 32, which allocation is chosen?
Chapter 88 source: section "Cooperative household bargaining".
Demonstration 2 of 4
Comparative advantage at home
Who should supply the household's care, and how much income does the right choice add?
The partner with the lower opportunity cost per care unit should supply care, whoever has the higher wage. A change in relative productivity reverses the efficient direction.
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Each partner has 12 flexible hours. Ana earns $32 an hour or makes 2 care units an hour; Ben earns $45 or makes h_B units. The household needs 18 care units. Cost per care unit is wage over care productivity.
Predict first. If a care technology lets Ben make 4 units an hour, who should do the care?
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Constructed example: the book's hypothetical week has wages 32 and 45, productivities 2, 1.5 and 4, 12 hours and 18 care units; a productivity of 3 for Ben is added.
Calculated values
- Ana's cost per care unit
- $16.00
- Ben's cost per care unit
- $30.00
- Income under this plan
- $636.00
- Gain from specializing
- $126.00
With Ben producing 1.50 units an hour, care costs Ana 32 / 2 = 16.00 and Ben 45 / 1.50 = 30.00 dollars of forgone earnings per unit, so Ana has the comparative advantage in care. Ana, the lower-cost carer, makes all 18 units: income is $96.00 + $540.00 = $636.00. Specializing gains $126.00 over the equal split, with the same 18 care units.
Worked steps
- Opportunity costs: Ana 32 / 2 = $16.00, Ben 45 / 1.50 = $30.00
- Ana: 18.0 units take 9.00 hours; 3.00 market hours x 32 = $96.00
- Ben: 0.0 units take 0.00 hours; 12.00 market hours x 45 = $540.00
- Total = $96.00 + $540.00 = $636.00
- Specializing beats the equal split by $636.00 - $510.00 = $126.00
Use the idea
Compare forgone earnings per unit of home output, not wages alone, when dividing household tasks; then add dynamic costs and risk, which the static gain ignores.
Where the conclusion applies
Linear technologies, fixed wages, a fixed care need and no value placed on who performs a task. The book notes career costs and insurance that the static gain leaves out.
Check your understanding: With h_B = 3 and specialization, who does the care and what is income?
Chapter 88 source: section "Gains from marital specialization".
Demonstration 3 of 4
How many children for old-age security?
How many births does a parent need for a target chance of at least one supporting child?
Each extra child cuts the chance of having no supporter by the factor 1 - q. A lower q or a higher target pushes the required family size up; credible pensions lower the target.
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q is the independent chance that each child can and will provide care; the target is the required chance that at least one of n children does. P(n) = 1 - (1 - q)^n.
Predict first. If migration lowers q to 0.35, how many births does the 92 percent target need?
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Constructed example: the book's hypothetical parents have q = 0.55 and 0.35 and targets of 92 and 85 percent; q = 0.75 and a 95 percent target are added.
Calculated values
- P(3 children)
- 90.89%
- P(4 children)
- 95.90%
- Children needed
- 4
Each child can support with chance 0.55, so the chance at least one of n can is 1 - 0.45^n; for example 1 - 0.45^3 = 0.9089. With 3 children it is 90.89%, short of 92%; with 4 it is 95.90%, so the security rule asks for 4 children.
Worked steps
- P(n) = 1 - (1 - 0.55)^n = 1 - 0.45^n
- n = 3: 1 - 0.45^3 = 90.89%, below 92%
- n = 4: 1 - 0.45^4 = 95.90%, at or above 92%
- Smallest family meeting the target: 4 children
Use the idea
When judging whether pensions will change fertility, compare the security target and the chance each child provides care, not fertility alone.
Where the conclusion applies
Independent outcomes across children and a single type of support; correlated sibling outcomes make extra births diversify far less, as the chapter warns.
Check your understanding: With q = 0.75 and target 95 percent, how many children are needed?
Chapter 88 source: section "Old-age-security fertility motive".
Demonstration 4 of 4
When public transfers are crowded out
How much of a public parent-to-child transfer does an altruistic parent undo?
While the gift is positive, the parent chooses the final allocation, so a public transfer between the same budgets is offset dollar for dollar. Once the gift hits zero, further transfers change who consumes what.
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Y_D = $84,000 and Y_R = $26,000 are parent and child incomes, b the public transfer from parent to child, g the private gift and c_D, c_R final consumption. With no program the parent gives $12,000.
Predict first. At a $15,000 program, is crowd-out still one for one?
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Constructed example: the book's hypothetical family has incomes 84,000 and 26,000, a desired gift of 12,000 and programs of 4,500 and 15,000; a 12,000 program is added.
Calculated values
- Private gift
- $7,500
- Parent consumes c_D
- $72,000
- Child consumes c_R
- $38,000
- Crowded out
- $4,500
The parent wants the child to consume 38,000, so with a public transfer of $4,500 the gift becomes $7,500. Then c_D = 84,000 - 4,500 - 7,500 = 72,000 and c_R = 26,000 + 4,500 + 7,500 = 38,000: the gift falls one for one, so consumption stays at $72,000 and $38,000: full crowd-out.
Worked steps
- Gift g = max(0, 12,000 - 4,500) = 7,500
- c_D = 84,000 - 4,500 - 7,500 = 72,000
- c_R = 26,000 + 4,500 + 7,500 = 38,000
- Crowd-out = 12,000 - 7,500 = 4,500
Use the idea
Before predicting that a program shifts resources to recipients, ask whether a linked private donor is already giving and could simply give less.
Where the conclusion applies
One altruistic donor, a cash gift with no service attached, and a program taxed from that donor. General financing or exchange-motivated gifts break neutrality, as the chapter notes.
Check your understanding: At b = 12,000, what are the gift and consumption levels?
Chapter 88 source: section "Altruistic-transfer neutrality".