A Simple Economics of Inequality-Market Design Approach-

Yosuke YASUDA | Osaka University yosuke.yasuda@gmail.com

- November 2018 -

1

Motivation• Inequality at the forefront of public debate!

2

Global Wealth: Top 1% > Bottom 99 %

3

Each of the remaining 383 million adults (8% of

the world) has net worth above USD 100,000.

This group includes 34 million US dollar million-

aires, who comprise less than 1% of the world’s

adult population, yet own 45% of all household

wealth. We estimate that 123,800 individuals

within this group are worth more than USD 50

million, and 44,900 have over USD 100 million.

The base of the pyramid

Each layer of the wealth pyramid has distinctive

characteristics. The base tier has the most even

distribution across regions and countries (see

Figure 2), but it is also very diverse, spanning

a wide range of personal circumstances. In devel-

oped countries, only about 20% of adults fall

within this category, and for the majority of these

individuals, membership is either transient – due

to business losses or unemployment, for example

– or a lifecycle phenomenon associated with

youth or old age. In contrast, more than 90%

of the adult population in India and Africa falls

within this range. The percentage of the popula-

tion in this wealth group is close to 100% in

some low-income countries in Africa. For many

residents of low-income countries, life member-

ship of the base tier is the norm rather than the

exception.

Mid-range wealth

In the context of global wealth, USD 10,000–

100,000 is the mid-range wealth band. It covers

one billion adults. The average wealth holding is

close to the global average for all wealth levels and

the combined net worth of USD 31 trillion provides

the segment with considerable economic clout.

India and Africa are under-represented in this tier,

whereas China’s share is relatively high. The com-

parison between China and India is very interesting.

India accounts for just 3.4% of those with mid-

range wealth, and that share has changed very little

during the past decade. In contrast, China accounts

for 36% of those with wealth between USD 10,000

and USD 100,000, ten times the number of Indi-

ans, and double the number of Chinese in 2000.

Higher wealth segment of the pyramid

The higher wealth segment of the pyramid – those

with net worth above USD 100,000 – had 215

million adult members at the start of the century.

By 2014, worldwide membership had risen above

395 million, but it declined this year to 383 million,

another consequence of the strengthening US

dollar. The regional composition of the group differs

markedly from the strata below. Europe, North

America and the Asia-Pacific region (omitting China

Source: James Davies, Rodrigo Lluberas and Anthony Shorrocks, Credit Suisse Global Wealth Databook 2015

Figure 1

The global wealth pyramid

USD 112.9 trn (45.2%)

34 m(0.7%)

349 m(7.4%)

1,003 m(21.0%)

3,386 m(71.0%)

Number of adults (percent of world population)

Wealthrange

Total wealth(percent of world)

USD 98.5 trn (39.4%)

USD 31.3 trn (12.5%)

USD 7.4 trn (3.0%)

> USD 1 million

USD 100,000 to 1 million

USD 10,000 to 100,000

< USD 10,000

24 Global Wealth Report 2015

Redistributionseems needed…

Redistribution• Transfer from (super) rich to poor seems not work.

• Why is redistribution difficult?

• Efficiency loss: distortion on incentives

• Not so effective: capital gains, tax haven

• Difficult to enforce: lobbying by rich

4

Our Approach• Observation: Redistribution is difficult.

• Our Model: Redistribution is impossible.

• Feasible allocation / welfare evaluation change.

• Better understand limitation of market economy.

• Q: Does market economy accelerate concentration?

• A: Yes (!?): Market tends to reduce trading volume.

5

Summary• We consider the relationship between total surplus

(efficiency) and trade volume (quantity) for homogenous good markets, assuming that

• (i) each buyer/seller has a unit demand/supply

• (ii) redistribution (by the third party) is infeasible.

• Pareto Efficiency with No Side-payment: PENS

• Show that competitive market minimizes # of trades.

6

Example 1• 4 buyers, 4 sellers, unit demand/supply

Buyer B1 B2 B3 B4

Value ($) 10 8 6 4

Seller S1 S2 S3 S4

Cost ($) 3 5 7 9

7

Supply-Demand10

876

5

3

1 2 3 4

Supply Curve

Demand Curve

Equilibrium Price (p*)

0Q

P

8

Competitive Eqm. (CE)• Maximizes total surplus, $10: assume p* = 6.5

Buyer B1 B2 B3 B4

Surplus ($) 3.5 1.5 0 0

Seller S1 S2 S3 S4

Surplus ($) 3.5 1.5 0 0

9

CE Maximizes Surplus, but…

10

8765

3

1 2 3 4

Supply Curve

Demand Curve

0Q

P

Left-behind Agents

10

Alternative: X • Trade pairs: B1-S3, B2-S2, B3-S1: p = (V+C)/2

Buyer B1 B2 B3 B4

Surplus ($) 1.5 1.5 1.5 0

Seller S1 S2 S3 S4

Surplus ($) 1.5 1.5 1.5 0

11

Alternative: Y• Trade pairs: B1-S4, B2-S3, B3-S2, B4-S1

Buyer B1 B2 B3 B4

Surplus ($) 0.5 0.5 0.5 0.5

Seller S1 S2 S3 S4

Surplus ($) 0.5 0.5 0.5 0.5

12

Comparison• Trade-off: efficiency vs. quantity

Allocation CE X Y

Total Surplus 10 9 4

# of Trading Agents 4 (50%) 6 (75%) 8 (100%)

PENS & IR Yes Yes Yes

Unique Price Yes No No

13

Efficiency vs. Quantity

TotalSurplus

TradingVolume

Competitive market maximizes surplus at the expense of trading volume…

Trade-off!

Market Economy• Homogenous good market

• Finitely many buyers and sellers (n total agents)

• Each has unit demand/supply

• Other simplifying assumptions:

A. 0 utility for non-trading agents

B. No buyer-seller pair generates 0 surplus

15

Pareto Efficiency• Allocation z is Pareto efficient if and only if there

exists NO other feasible allocation z’, which makes

• every one weakly better off, and

• someone strictly better off.

• Feasibility: allocation must be achieved through bilateral transactions (buyer-seller pairs).

• Preferences: larger surplus is better (unit demand).

16

Definition of PENS• Consider (bilaterally achievable (BA) allocations):

• (resource constraint), and

• for each agent i, (no trade), or

• there exist agent j such that (bilateral trade).

• Allocation z is called PENS if there exists no allocation z’ in Z such that z’ Pareto dominates z.

• PE allocation (in Z) is always PENS, but NOT vice versa.

• PENS is weaker than standard PE.

17

Z = {x1, x2, . . . , xn}<latexit sha1_base64="T41Kapk8Plrtw2uem4LrWUGAkPU=">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</latexit><latexit sha1_base64="T41Kapk8Plrtw2uem4LrWUGAkPU=">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</latexit><latexit sha1_base64="T41Kapk8Plrtw2uem4LrWUGAkPU=">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</latexit><latexit sha1_base64="T41Kapk8Plrtw2uem4LrWUGAkPU=">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</latexit>

x1 + x2 + · · ·+ xn = e1 + e2 + · · ·+ en<latexit sha1_base64="9p3QAQxwP4BO915WAcRwpgB9Sqs=">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</latexit><latexit sha1_base64="9p3QAQxwP4BO915WAcRwpgB9Sqs=">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</latexit><latexit sha1_base64="9p3QAQxwP4BO915WAcRwpgB9Sqs=">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</latexit><latexit sha1_base64="9p3QAQxwP4BO915WAcRwpgB9Sqs=">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</latexit>

xi = ei<latexit sha1_base64="z7miwy9++m6yl2iLd4782DujuQU=">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</latexit><latexit sha1_base64="z7miwy9++m6yl2iLd4782DujuQU=">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</latexit><latexit sha1_base64="z7miwy9++m6yl2iLd4782DujuQU=">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</latexit><latexit sha1_base64="z7miwy9++m6yl2iLd4782DujuQU=">AAACa3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKRVbEZSrYKqTGZcYLKBvoWVqYGZmYKRjoGRiYGxoZghhG5ibGJgqGQBEQUGaAgoB8geUMMQwpDPkMyQylDLkMqQx5DCVAdg5DIkMxEEYzGDIYMBQAxWIZqoFiRUBWJlg+laGWgQuotxSoKhWoIhEomg0k04G8aKhoHpAPMrMYrDsZaEsOEBcBdSowqBpcNVhp8NnghMFqg5cGf3CaVQ02A+SWSiCdBNGbWhDP3yUR/J2grlwgXcKQgdCF180lDGkMFmC3ZgLdXgAWAfkiGaK/rGr652CrINVqNYNFBq+B7l9ocNPgMNAHeWVfkpcGpgbNZuACRgAslBVwM0KN9Cz1DANNlB0soDHBwSDNoMSgAQxucwYHBg+GAIZQoLW5DJMZ5jDMZXzHJMIkxSQDUcrECNUjzIACmFQB9+eMxA==</latexit>

xi + xj = ei + ej<latexit sha1_base64="CbA1mcKohJhW4WwscVSavjJ+VY4=">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</latexit><latexit sha1_base64="CbA1mcKohJhW4WwscVSavjJ+VY4=">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</latexit><latexit sha1_base64="CbA1mcKohJhW4WwscVSavjJ+VY4=">AAACeXichVHLSsNAFD2N7/qqiiC4CRZFFMpNLbYKguDGpa+qoLYkcdTRNAlJWtTiD/gDLlwpiKif4cYfcOEniEsFNy68SSviQr3DzJw5c8+dMzOGa0k/IHqKKQ2NTc0trW3x9o7Oru5ET++q75Q9U+RNx3K8dUP3hSVtkQ9kYIl11xN6ybDEmnEwF+6vVYTnS8deCY5csVXSd225I009YKqY6D8sSHVcPSzsqzOqiLAo7BcTSUpN5SbTmUmVUkRZLa2FIJ3NTGRUjZkwkqjHgpO4xia24cBEGSUI2AgYW9Dhc9uABoLL3BaqzHmMZLQvcII4a8ucJThDZ/aAx11ebdRZm9dhTT9Sm3yKxd1jpYpheqQbeqUHuqNn+vi1VjWqEXo54tmoaYVb7D4dWH7/V1XiOcDet+pPzwF2kIu8SvbuRkx4C7OmrxyfvS5PLw1XR+iSXtj/BT3RPd/ArryZV4ti6Rxx/oCvV1Z/B/l0aiqlLWaSs7n6T7RiEEMY5efOYhbzWECej63iAre4i30oQ8qoMlZLVWJ1TR9+hDLxCfbzkLA=</latexit><latexit sha1_base64="CbA1mcKohJhW4WwscVSavjJ+VY4=">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</latexit>

Why are X and Y PENS?• CE allocation Pareto dominates neither X nor Y.

Buyer B1 B2 B3 B4

Surplus ($) 3.5 1.5 0 0

Seller S1 S2 S3 S4

Surplus ($) 3.5 1.5 0 0

18

If Side-Payment Possible• Transfer from B1 to B3, B4 and S1 to S3, S4.

Buyer B1 B2 B3 B4

Surplus ($) 3.5 1.5 0 0

Seller S1 S2 S3 S4

Surplus ($) 3.5 1.5 0 0

$0.5

$0.5

$1.5

$1.5

19

X and Y are Not PE• CE + side-payment Pareto dominates X & Y.

Buyer B1 B2 B3 B4

Surplus ($) 1.5 1.5 1.5 0.5

Seller S1 S2 S3 S4

Surplus ($) 1.5 1.5 1.5 0.5

20

Main Theorem

• Lemma 1Any CE allocation is BA and PENS.

• Theorem 1 The number of trading agents (trading volume) under a CE allocation is minimum among all BA allocations that are PENS.

21

Proof of Theorem 11. Suppose not. Then, there must exist a PENS allocation, say z,

which has strictly fewer (trading) buyer-seller pairs than the competitive equilibrium.

2. There are at least a buyer, say B*, and a seller, S*, who would receive non-negative surplus in CE but cannot engage in any trade, i.e., receive zero surplus, in the alternative allocation z.

3. VB* is (weakly) larger than p* which is also larger than CS*.

4. B*-S* pair generates positive surplus. <= VB* > CS*

5. Contradicts to our presumption that z is PENS.

22

Converse• Theorem 2

Let k be the trading volume under a CE. Then, there exists a BA, PENS and IR allocation that entails strictly larger number of trades than k if and only if

• (i) value of B1 exceeds the cost of Sk+1, and

• (ii) value of Bk+1 exceeds the cost of S1,

where buyer/seller with smaller number has higher value/lower cost.

23

Equilibrium (k = 2)10

8765

3

1 2 3 4

Supply Curve

Demand Curve

Equilibrium Price

0Q

P

24

Illustration10

8765

3

1 2 3 4

Supply Curve

Demand Curve

Equilibrium Quantity

0Q

P

25

Proof of Theorem 2• If part (<=)

• B1-Sk+1 and Bk+1-S1 pairs generate positive surplus.

• Let B2, …, Bk trade with S2, …, Sk.

• This is a PENS and IR allocation with k+1 trades.

• Only if part (=>)

• If (i) is not satisfied, Sk+1 cannot engage in any profitable trade.

• If (ii) is not satisfied, Bk+1 cannot engage in any profitable trade.

• Impossible to make k+1 (or more) profitable trading pairs.

26

Graphical Intuition

OK NG

27

Graphical Intuition

OK NG

Possible to find their partners Impossible to find

their partners

28

　Pioneering Experiments• Connection to the experimental studies:

• Chamberlin (1948) vs. Smith (1962)

• Chamberlin, E. H. (1948). “An experimental imperfect market.” - The Journal of Political Economy, 95-108.

• Smith, V. L. (1962). “An experimental study of competitive market behavior.” - The Journal of Political Economy, 111-137.

29

　Pioneering Experiments• Chamberlin (1948) vs. Smith (1962)

• In Chamberlin, buyers and sellers engage in bilateral bargaining, transaction price is recorded on the blackboard as contracts made; single period.=> Imperfect market: Excess quantities

• In Smith’s double auctions, each trader’s quotation is addressed to the entire trading group one quotation at a time; multiple periods (learning).=> Converge to perfectly competitive market

30

Chamberlin (1948)

31

Excess Quantity• Chamberlin’s excess quantity puzzle:

• Sales volume > equilibrium quantity => 42/46

• Sales volume = equilibrium quantity => 4/46

• Sales volume < equilibrium quantity => 0/46

• “price fluctuation render the volume of sales normally greater than the equilibrium amount which is indicated by supply and demand curves”

• Our results may account for Chamberlin’s puzzle.

32

Extension: Matching• Stable matching (Core) induce minimum pairs.

=> Examples 2a, 3, 4a

• # of Stable matching pairs not always minimum. => Examples 2b, 4b

• NTU — Anything can happen. (PE = PENS)

• TU — Assortative stable matching is minimum.

33

NTU: Example 2a• 2 doctors, 2 hospitals

• Unique Stable Matching: D1-H1 (D2, H2 single)

• An Alternative: D1-H2, D2-H1 <= PE and IR

=> All agents find their mates under non-stable outcome.

Agent D1 D2 H1 H2

1st H1 H1 D1 D1

2nd H2 - D2 D2

34

NTU: Example 2a• 2 doctors, 2 hospitals (H2: rural hospital)

• Unique Stable Matching: D1-H1 (D2, H2 single)

• An Alternative: D1-H2, D2-H1 <= PE and IR

=> All agents find their mates under non-stable outcome.

Agent D1 D2 H1 H2

1st H1 H1 D1 D1

2nd H2 - D2 D2

35

NTU: Example 2b• 2 doctors, 2 hospitals

• Unique Stable Matching: D1-H2, D2-H1

• An Alternative: D1-H1 (D2, H2 single) <= PE and IR

=> All agents find their mates under stable outcome.

Agent D1 D2 H1 H2

1st H1 H1 D2 D1

2nd H2 - D1 D2

36

NTU: Example 2b• 2 doctors, 2 hospitals

• Unique Stable Matching: D1-H2, D2-H1

• An Alternative: D1-H1 (D2, H2 single) <= PE and IR

=> All agents find their mates under stable outcome.

Agent D1 D2 H1 H2

1st H1 H1 D2 D1

2nd H2 - D1 D2

37

TU: Assignment Game• Finitely many workers and firms

• Each matched with at most one agent

• Receive 0 utility if unmatched.

• Each pair yields surplus by production.

• Monetary transfers allowed (UT: Transferable Utility)

• Paris arbitrarily divide production surplus.

• No side-payment beyond each worker-firm pair

38

Result in TU Case• Theorem 3

The number of worker-firm pairs under the assortative stable matching is minimum among all BA outcomes that are PENS and IR.

• Def. of assortative stable matching (ASM)

• Agents in both sides are linearly ordered. (Surplus Aij is weakly decreasing in i and j.)

• Matching results in 1st-1st, 2nd-2nd, and so on.

39

Proof (Theorem 3)1. Suppose not. Then, there must exist a PENS and individually rational

outcome, say T, which has strictly fewer worker-firm pairs than ASM.

2. There are at least a worker, say W*, and a firm, F*, that would receive non-negative surplus in ASM but cannot engage in any trade, i.e., receive zero surplus, in the alternative outcome T.

3. Production surplus between W* and F* must be positive.

1. Both W* and F* are (weakly) smaller than k. <= (2)

2. AW*F* must be (weakly) larger than Akk, a positive surplus. <= (1)

4. Contradicts to the presumption that T is PENS.

40

Application: Example 3• Revisit (reformulate) Example 1 <= Aij := Vi - Cj

• Core: B1-S1, B2-S2 or B1-S2, B2-S1

• X: B1-S3, B2-S2, B3-S1 Y: B1-S4, B2-S3, B3-S2. B4-S1

S1 S2 S3 S4

B1 7 5 3 1

B2 5 3 1 -1

B3 3 1 -1 -3

B4 1 -1 -3 -5

41

Application: Example 3• Revisit (reformulate) Example 1

• Core: B1-S1, B2-S2 or B1-S2, B2-S1

• X: B1-S3, B2-S2, B3-S1 Y: B1-S4, B2-S3, B3-S2. B4-S1

S1 S2 S3 S4

B1 7 5 3 1

B2 5 3 1 -1

B3 3 1 -1 -3

B4 1 -1 -3 -5

42

TU: Example 4a• 2 workers, 2 firms

• Unique Core: W1-F1 (W2, F2 single)

• Alternative: W1-F2, W2-F1 <= PE and IR

F1 F2

W1 10 4

W2 4 -5

43

TU: Example 4a• 2 workers, 2 firms

• Unique Core: W1-F1 (W2, F2 single)

• Alternative: W1-F2, W2-F1 <= PE and IR

F1 F2

W1 10 4

W2 4 -5

(5 - 5)

(2 - 2) (2 - 2) 44

TU: Example 4b• 2 workers, 2 firms

• Unique Core: W1-F2, W2-F1

• Alternative: W1-F1 (W2, F2 single) <= PE and IR

F1 F2

W1 10 8

W2 4 -5

45

TU: Example 4b• 2 workers, 2 firms

• Unique Core: W1-F2, W2-F1

• Alternative: W1-F1 (W2, F2 single) <= PE and IR

F1 F2

W1 10 8

W2 4 -5

(5 - 5)

(7 - 1) (1 - 3)

46

Summary: Main Results• Equilibrium allocation may be seen most unequal:

• The quantity of good traded under the competitive market equilibrium is minimum among all feasible allocations that are PENS.

• The converse result also holds:

• Unless a demand or supply curve is completely flat, there always exists a feasible allocation that is PENS , IR and entailing strictly larger number of trades than that of the equilibrium quantity.

47

Heterogenous Goods• Generalization to assignment games

(TU game in one-to-one matching markets).

• Theorem 3 The number of buyer-seller pairs under the assortative stable matching is minimum among all BA outcomes that are PENS.

• The assortative matching assumption is often imposed in labor markets or marriage markets.

48

Last Remarks• Should we aim to design/achieve “competitive” market?

• YES: Efficiency — the greatest happiness

• NO: Equality — of the minimum number

• Trade-off: efficiency vs. equality

• Better understand why market accelerates concentration.

• Redistribution is crucial when market is competitive.

=> May better consider equitable market design.

49

New!

Many Thanks :)Yosuke YASUDA | Osaka University

yosuke.yasuda@gmail.com

Any comments and questions are appreciated.

50