Chapter 72: Redistribution, Social Insurance, Giving, and Patronage. 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

Median voter and the mean-median gap

How does a richer top earner change what a linear tax and equal transfer give the median voter?

The transfer is paid out of mean taxable earnings, but the median voter pays tax only on her own earnings. The further the mean sits above the median, the more a given rate returns to her than it takes.

$Equation, written in LaTeX: \bar z=\frac{18+28+38+58+108}{5}=50,$

Equation, written in LaTeX: y_i(t)=(1-0.1t)z_i.

$Equation, written in LaTeX: b(0.20)=0.20\times49=9.80.$

$Equation, written in LaTeX: c_m(0.20)=0.80\times37.24+9.80=39.592.$

Scroll sideways for the whole equation

z_i is voter i's baseline taxable earnings and t the linear tax rate. Taxable earnings respond as y_i(t) = (1 - 0.1t) z_i. The revenue is paid back as an equal transfer b, so the median voter's disposable resources are c_m = (1 - t) y_m + b.

Predict first. If the top income rises from 108 to 158 and the median is unchanged, does the median voter gain more or less from t = 0.20?

Your prediction More Less The same

Top earner's income

Tax rate

Figure: Median voter and the mean-median gap. Left: baseline earnings 18, 28, 38, 58 and 108, with the mean 50 above the median 38. Right: the median voter's baseline 38, kept earnings 29.792, transfer 9.80 and disposable resources 39.592 at t = 0.20. — Top earner's income: 108, Tax rate: 0.20
Constructed example: the chapter's hypothetical five voters (18, 28, 38, 58, 108), response 0.1, tax rate 0.20 and top income 158 are the book's; top incomes 133 and 208 and rates 0.10 and 0.30 are added for comparison.

Mean earnings: 50
Response factor 1 - 0.1t: 0.98
Mean taxable earnings: 49.00
Transfer b: 9.80
Median taxable earnings y_m: 37.24
Median disposable resources c_m: 39.592
Change from baseline 38: 1.592

The top earner's 108 pulls the mean to 50, while the median stays at 38. At t = 0.20 taxable earnings shrink by the factor 0.98, so the equal transfer is 0.20 x 49.00 = 9.80. The median voter keeps 0.80 x 37.24 = 29.792 and ends with c_m = 39.592, above the baseline 38. This is a mechanical comparison: it leaves the value of extra leisure uncounted and does not show that this rate is the voter's best one.

Worked steps

Mean = (18 + 28 + 38 + 58 + 108) / 5 = 50
Response factor = 1 - 0.1 x 0.20 = 0.98
Mean taxable earnings = 0.98 x 50 = 49.00
b = 0.20 x 49.00 = 9.80
y_m = 0.98 x 38 = 37.24
c_m = 0.80 x 37.24 + 9.80 = 29.792 + 9.80 = 39.592

Use the idea

When comparing places or periods, look at the gap between mean and median income, not only at the mean, before predicting the political demand for redistribution.

Where the conclusion applies

Five voters, one linear tax with an equal transfer, a fixed earnings response of 0.1 per unit of t and no value placed on leisure. Choosing the voter's best rate needs utility across rates.

Check your understanding: With top income 158, what is the median voter's disposable resources at t = 0.20?

Mean = 60, taxable mean = 0.98 x 60 = 58.80, b = 0.20 x 58.80 = 11.76, and c_m = 0.80 x 37.24 + 11.76 = 29.792 + 11.76 = 41.552.

Chapter 72 source: section "Meltzer-Richard redistribution mechanism".

Demonstration 2 of 4

Samaritan's dilemma and relief coverage

When does generous relief stop a household from preparing for a loss?

Relief shrinks the loss the household bears, and it shrinks the larger loss by more. Preparation then buys a smaller private saving than its full resource saving, so a household can rationally skip a step that would save resources overall.

$Equation, written in LaTeX: J_N=(1-0.75)\times120=30.$

$Equation, written in LaTeX: J_P=24+(1-0.75)\times50=36.50.$

Scroll sideways for the whole equation

J_N is the household's own cost without preparation and J_P its cost with preparation. Relief covers a share of whichever loss occurs: 120 without preparation, 50 with it. Preparation itself costs the amount shown.

Predict first. Will cutting coverage from 75 to 50 percent flip the household to preparing?

Your prediction Yes, it prepares No, it still does not prepare

Relief coverage share

Preparation cost

Figure: Samaritan's dilemma and relief coverage. Paired bars. Without preparation the household pays 30.00 of a 120 resource loss; with preparation it pays 36.50 of 74. Choice: does not prepare. — Relief coverage share: 75%, Preparation cost: 24
Constructed example: the chapter's hypothetical household (preparation 24, losses 120 and 50, coverage 75 and 50 percent) are the book's; coverage of 25 and 100 percent and preparation costs 16 and 32 are added for comparison.

Own cost without preparation J_N: 30.00
Own cost with preparation J_P: 36.50
Household choice: Does not prepare
Resource use without preparation: 120
Resource use with preparation: 74
Resource saving from preparing: 46

With relief covering 75 percent, the household bears 0.25 of each loss: J_N = 0.25 x 120 = 30.00 and J_P = 24 + 0.25 x 50 = 36.50. Because 30.00 < 36.50, the household does not prepare, even though preparing would save 46 of total resources. The comparison shows a response margin; it does not pick a real coverage rule, which also needs liquidity, inequality and catastrophic risk.

Worked steps

J_N = (1 - 0.75) x 120 = 0.25 x 120 = 30.00
J_P = 24 + (1 - 0.75) x 50 = 24 + 12.50 = 36.50
Resources: 120 without preparation, 24 + 50 = 74 with it
Resource saving = 120 - 74 = 46

Use the idea

When designing relief, check whether the coverage share leaves the beneficiary's own cost of preparing below the own cost of not preparing.

Where the conclusion applies

One household, a known loss in each case, relief as a fixed share of either loss and no liquidity limit. Differences in control over the loss are not modelled.

Check your understanding: At 50 percent coverage, what are J_N and J_P, and which is chosen?

J_N = 0.50 x 120 = 60 and J_P = 24 + 0.50 x 50 = 49. Since 49 < 60, the household prepares.

Chapter 72 source: section "Samaritan's dilemma".

Demonstration 3 of 4

Benefit phaseout and the reward to earning more

How much of a raise does a household keep when its benefit is withdrawn as it earns?

Inside the phaseout range each extra unit earned costs p units of benefit, so the household keeps 1 - p of a raise. Once the benefit reaches zero the household keeps everything, so the kept share over a wide raise can exceed 1 - p.

Equation, written in LaTeX: T(12{,}000)=9{,}000-0.40(12{,}000)=4{,}200,

Equation, written in LaTeX: T(18{,}000)=9{,}000-0.40(18{,}000)=1{,}800,

Scroll sideways for the whole equation

T(e) is the benefit at earnings e: the guarantee B minus the phaseout rate p times earnings, and never below zero. Disposable income is earnings plus T(e).

Predict first. Halving the phaseout to 0.20, what share of the extra 6,000 does the household keep?

Your prediction 60 percent 80 percent 100 percent

Phaseout rate

Guarantee

Benefit at 12,000: 4,200
Benefit at 18,000: 1,800
Disposable income at 12,000: 16,200
Disposable income at 18,000: 19,800
Gain from 6,000 more earnings: 3,600
Share of the 6,000 kept: 60%

With guarantee 9,000 and phaseout 0.40, the benefit is 4,200 at 12,000 and 1,800 at 18,000, so disposable income rises from 16,200 to 19,800: 19,800 - 16,200 = 3,600. Each extra unit earned withdraws 0.40 of benefit, so the household keeps 60 percent of the extra 6,000. A gentler phaseout raises payments, so choosing a schedule needs a financing plan and a stated objective.

Worked steps

T(12,000) = 9,000 - 0.40 x 12,000 = 4,200
T(18,000) = 9,000 - 0.40 x 18,000 = 1,800
Disposable incomes: 12,000 + 4,200 = 16,200 and 18,000 + 1,800 = 19,800
Gain = 19,800 - 16,200 = 3,600
Share kept = 3,600 / 6,000 = 60%

Use the idea

Before advising on a raise or extra hours for someone on a means-tested benefit, compute the benefit at both earnings levels rather than assuming the full raise is kept.

Where the conclusion applies

One benefit with a linear phaseout, no other taxes or credits and no change in hours from the schedule itself. Financing and response estimates are left out.

Check your understanding: At p = 0.20, how much does disposable income rise from earnings 12,000 to 18,000?

T = 9,000 - 2,400 = 6,600 and 9,000 - 3,600 = 5,400; disposable income is 18,600 and 23,400, a gain of 4,800 (80 percent). Benefit payments rise by 2,400 and 3,600.

Chapter 72 source: section "Welfare Economics, Redistribution, and Subsidies".

Demonstration 4 of 4

Tax wedge, deadweight loss, and welfare weights

When does a distorting tax and transfer raise measured welfare?

The levy drives a wedge between buyer and seller prices and shrinks quantity, which costs the triangle D. Moving revenue from payers to recipients adds welfare only through the difference in weights, so the ranking turns on a normative choice, not on the distortion.

Equation, written in LaTeX: Q_D=120-P_b

Equation, written in LaTeX: Q_S=P_s-20.

$Equation, written in LaTeX: R=18\times41=738.$

$Equation, written in LaTeX: D=\frac{1}{2}\times18\times9=81.$

$Equation, written in LaTeX: \Delta W=1.4(738)-738-81-30=184.2.$

Scroll sideways for the whole equation

P_b is the buyer price and P_s the seller price, with P_b = P_s + t for a per-unit levy t. R is revenue, D the deadweight triangle and 30 the administration cost. Each recipient dollar counts with the chosen weight and each payer dollar with weight 1.

Predict first. With equal weights, can this transfer ever raise measured welfare?

Your prediction Yes, at a small enough levy No, never

Per-unit levy

Recipient welfare weight

Figure: Tax wedge, deadweight loss, and welfare weights. Left: demand 120 - Q and supply 20 + Q with a levy of 18, buyer price 79, seller price 61 and quantity 41; the revenue rectangle is 738 and the triangle 81. Right: weighted gains and losses summing to 184.2. — Per-unit levy: 18, Recipient welfare weight: 1.4
Constructed example: the chapter's hypothetical market (Q_D = 120 - P_b, Q_S = P_s - 20), levy 18, administration 30 and weights 1 and 1.4 are the book's; levies 6, 12 and 24 and weight 1.2 are added for comparison.

Seller price P_s: 61
Buyer price P_b: 79
Quantity Q: 41
Revenue R: 738
Deadweight loss D: 81
Administration: 30
Welfare change Delta W: 184.2

A levy of 18 splits evenly: buyers pay 79 and sellers get 61, each 9 away from 70. It raises 738 at a distortion of 81, and administration uses 30. With recipient weight 1.4, Delta W = 1.4 x 738 - 738 - 81 - 30 = 184.2, so the transfer raises measured welfare. The 81 and 30 are resource losses under any weights; only the normative weight moves the ranking.

Worked steps

120 - (P_s + 18) = P_s - 20, so P_s = 61, P_b = 79, Q = 41
R = 18 x 41 = 738
D = 0.5 x 18 x 9 = 81
Delta W = 1.4 x 738 - 738 - 81 - 30 = 1,033.2 - 849 = 184.2

Use the idea

State the welfare weights explicitly when arguing for a tax-financed transfer, and report the deadweight and administration losses separately.

Where the conclusion applies

Linear demand and supply, a per-unit levy, full transfer of revenue and fixed weights. The triangle is a local measure.

Check your understanding: At weight 1.0 and levy 18, what is Delta W?

Delta W = 738 - 738 - 81 - 30 = -111.

Chapter 72 source: section "Welfare Economics and Redistribution".

Median voter and the mean-median gap

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Samaritan's dilemma and relief coverage

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Benefit phaseout and the reward to earning more

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Tax wedge, deadweight loss, and welfare weights

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies