Chapter 7: Social Welfare, Inequality, and Compensation. 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

A transfer from rich to poor, with and without leakage

When does moving income from a richer to a poorer household raise summed welfare?

With concave utility a unit is worth more to the poorer household, so a costless transfer raises the sum. Leakage destroys resources and can turn the gain into a loss.

$Equation, written in LaTeX: W_0=\sqrt{25}+\sqrt{100}=5+10=15.$

$Equation, written in LaTeX: W_1=\sqrt{34}+\sqrt{91}\approx 5.831+9.539=15.370.$

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Income is in thousands. L starts with 25 and H with 100; welfare is the sum of sqrt(y). The transfer is what H gives up; leakage is lost to collection and reduced production before L receives the rest.

Predict first. Is a leaky transfer of 16 (5 lost) better or worse than the starting welfare of 15?

Your prediction Better Worse

Amount taken from H

Amount lost in transit

Figure: A transfer from rich to poor, with and without leakage. Square-root utility curve with L moving from 25 to 34 and H from 100 to 91; welfare 15.370 against 15. — Amount taken from H: 9, Amount lost in transit: 0
Constructed example: the chapter's hypothetical households (25 and 100, transfer 9, leakage 5); transfers of 16 and 25 are added. The book rounds the leaky drop to about 0.076; the exact drop is 0.0754.

Incomes after (L, H): (34, 91)
Welfare W: 15.370
Change from W0 = 15: +0.370
Total income: 125
Marginal utility at the start (L, H): 0.10, 0.05

H gives up 9. Incomes become (34, 91), so W = sqrt(34) + sqrt(91) = 5.831 + 9.539 = 15.370. Evaluated welfare rises by 0.370 from 15. At the start a unit is worth 1 / (2 x 5) = 0.10 to L and 1 / (2 x 10) = 0.05 to H.

Worked steps

L: 25 + 9 - 0 = 34; H: 100 - 9 = 91
W = 5.831 + 9.539 = 15.370
Change = 15.370 - 15 = +0.370

Use the idea

Weigh the gap in marginal value against the share of each transfer lost along the way.

Where the conclusion applies

A chosen welfare representation u = sqrt(y), equal weights, and no effect of the transfer on incomes other than the stated leakage.

Check your understanding: With a transfer of 16 and leakage of 5, what is evaluated welfare?

Incomes (36, 84): sqrt(36) + sqrt(84) = 6 + 9.165 = 15.165, above 15, so it passes.

Chapter 7 source: section "Diminishing marginal utility of income".

Demonstration 2 of 4

How much mean income to trade for equality?

Does a reform that lowers mean income but narrows the gap raise welfare?

The more the evaluator dislikes inequality, the further the EDE falls below the mean for an unequal distribution. Somewhere between 0.5 and 1 the ranking flips from P to Q.

$Equation, written in LaTeX: A(\varepsilon)=1-\frac{y_{\mathrm{EDE}}}{\mu}.$

$Equation, written in LaTeX: y_{\mathrm{EDE},P}=\sqrt{20\cdot80}=\sqrt{1600}=40.$

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P = (20, 80) and Q = (30, 60) in thousands, for two equally weighted households. The EDE is the equal income giving the same welfare; epsilon is the inequality aversion and mu the mean.

Predict first. At epsilon = 0.5, which distribution ranks higher?

Your prediction P Q

Inequality aversion epsilon

Figure: How much mean income to trade for equality? Bars of mean income and EDE for P and Q at epsilon 1: P 40.00 against 50, Q 42.43 against 45. — Inequality aversion epsilon: 1
Constructed example: the chapter's hypothetical distributions P and Q (epsilon 1 and 0); epsilon 0.5 and 2 are added for comparison.

EDE of P: 40.00
Atkinson index of P: 0.200
EDE of Q: 42.43
Atkinson index of Q: 0.057
Ranking: Q above P

With epsilon = 1 the EDE is the geometric mean: sqrt(20 x 80) = 40.00 and sqrt(30 x 60) = 42.43. Q ranks higher. The index is 1 - EDE / mean: 1 - 40.00 / 50 = 0.200 for P and 1 - 42.43 / 45 = 0.057 for Q.

Worked steps

EDE(P) = 40.00, EDE(Q) = 42.43
A(P) = 1 - 40.00 / 50 = 0.200
A(Q) = 1 - 42.43 / 45 = 0.057
Ranking: Q higher

Use the idea

Report the inequality aversion with any welfare ranking; the ranking can depend on it.

Where the conclusion applies

Two equally weighted households and the Atkinson family of welfare functions.

Check your understanding: What are the two EDEs at epsilon = 0.5?

P: ((4.472 + 8.944) / 2)^2 = 45.00. Q: ((5.477 + 7.746) / 2)^2 = 43.71. P ranks higher.

Chapter 7 source: section "Atkinson inequality aversion".

Demonstration 3 of 4

A side payment that ends envy

Which payment from the window office holder makes both workers content with their packages?

Each worker compares whole packages of office plus money. Too small a payment leaves B envious; too large a payment makes A envious. Between them both prefer what they have.

Equation, written in LaTeX: 120-15=105

Equation, written in LaTeX: 80+15=95

Equation, written in LaTeX: 90+15=105

Equation, written in LaTeX: 100-15=85

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A gets the window office W and pays t to B, who gets the quiet office Q. A values W at 120 and Q at 80; B values W at 100 and Q at 90.

Predict first. Which worker becomes envious if the transfer is too large?

Your prediction A, the payer B, the receiver

Transfer from A to B

Figure: A side payment that ends envy. Package values against the transfer, with the envy-free band from 5 to 20 and the current transfer 15. — Transfer from A to B: 15
Constructed example: the chapter's hypothetical offices (transfers 0 and 15); transfers of 5 and 25 are added for comparison.

A: own vs other: 105 vs 95
B: own vs other: 105 vs 85
Envy-free: yes
Envy-free transfers: 5 to 20

A: 120 - 15 = 105 against 80 + 15 = 95; A prefers its own (105 > 95). B: 90 + 15 = 105 against 100 - 15 = 85; B prefers its own (105 > 85). The assignment is envy-free.

Worked steps

A own = 120 - 15 = 105; A's view of B = 80 + 15 = 95
B own = 90 + 15 = 105; B's view of A = 100 - 15 = 85
No envy for A needs t <= (120 - 80) / 2 = 20; for B t >= (100 - 90) / 2 = 5

Use the idea

When dividing indivisible items, look for a money adjustment inside the band where no one would swap.

Where the conclusion applies

Quasilinear values in a common money unit, one office each and a single transfer.

Check your understanding: At t = 25, is the assignment envy-free?

No. A gets 120 - 25 = 95 against 80 + 25 = 105, so A envies B. The envy-free range is 5 to 20.

Chapter 7 source: section "Envy-freeness".

Demonstration 4 of 4

Could the winners pay the losers?

Does the bypass pass the compensation test, and is it an actual Pareto improvement?

The Kaldor-Hicks test asks whether winners could compensate losers and still gain. Paying the compensation turns a potential improvement into an actual one, but the administration uses real resources.

Equation, written in LaTeX: S=G-L-R=125-80-30=15.

Equation, written in LaTeX: S'=125-80-50=-5.

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G is winners' gross willingness to pay (80 + 45), L losers' required compensation (55 + 25) and R the real resource cost, all in thousands a year. Admin cost is what paying compensation uses up.

Predict first. With resource cost 40 and compensation actually paid, does the project still pass?

Your prediction Yes, narrowly No

Annual construction cost

Compensation actually paid

Figure: Could the winners pay the losers? Waterfall from gains 125 through losses 80, resource cost 30 and admin cost 0 to a residual of 15. — Annual construction cost: 30, Compensation actually paid: No (admin cost 0)
Constructed example: the chapter's hypothetical bypass (R = 30 and 50, admin 4); R = 40 is added for comparison.

Gains G: 125
Losses L: 80
Resource cost R: 30
Admin cost: 0
Residual S: 15
Passes Kaldor-Hicks: yes

S = 125 - 80 - 30 - 0 = 15. It passes the potential test, but without payments the losers are worse off, so it is not a Pareto improvement.

Worked steps

G = 80 + 45 = 125
L = 55 + 25 = 80
S = 125 - 80 - 30 - 0 = 15

Use the idea

Separate transfers between people from real resource costs when you add up a project.

Where the conclusion applies

Money valuations of gains and losses that can be added across people, and payments that work as stated.

Check your understanding: What is the residual at a resource cost of 40 with admin cost 4?

125 - 80 - 40 - 4 = 1, so it passes narrowly.

Chapter 7 source: section "Kaldor-Hicks compensation principle".

A transfer from rich to poor, with and without leakage

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

How much mean income to trade for equality?

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

A side payment that ends envy

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Could the winners pay the losers?

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies