a simple economics of inequality: market design approach
TRANSCRIPT
A Simple Economics of Inequality-Market Design Approach-
Yosuke YASUDA | Osaka University [email protected]
- October 2016 -
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Motivation• Inequality at the forefront of public debate!
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Global Wealth: Top 1% > Bottom 99 %
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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
Motivation• OK, inequality should be fixed, but how?
• Does redistribution (from rich to poor) work?
• Efficiency loss: distortion on incentives
• Not so effective: capital gains, tax haven
• Difficult to enforce: lobbying by rich
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Natural Questions• What if redistribution is NOT available?
• Then, how should we (re)evaluate market economy?
• Can competitive market still be optimal?
• Does market or globalization bring on inequality?
• Is egalitarian or equitable market design needed?
=> These are what I study in this project!
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Summary (1) • I consider the relationship between efficiency and trade volume
for homogenous good markets, assuming that
• (i) each buyer/seller has a unit demand/supply, and
• (ii) redistribution (by the third party) is infeasible.
• Show that competitive market minimizes the # of trades.
• The quantity of goods traded under the competitive market equilibrium is minimum among all feasible allocations that are Pareto efficient and individually rational. (given assumptions (i) and (ii))
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Summary (2)• Converse result also holds.
• Unless a demand or supply curve is completely flat, there always exists a PENT and IR allocation that entails strictly larger number of trades than the equilibrium quantity.
• PENT = Pareto Efficient with No Transfer; weaker than PE
• Given unit demand, equilibrium may be seen most unequal:
• The number of agents who engage in trades under the market equilibrium is minimum among all feasible allocations that are PENT and IR.
• # of left-behind agents from trade is maximized!
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Summary (3)• Our result NOT always holds for one-to-one two-sided
matching markets (with or without monetary transfers).
• But holds for assortative stable matching:
• The number of agents who are matched with their partners under the assortative stable matching is minimum among all feasible matching outcomes that are PENT and IR.
• The assortative matching assumption is often imposed in applied works, e.g., labor markets and marriage markets.
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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.
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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
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Chamberlin (1948)
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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.
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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
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Supply-Demand10
876
5
3
1 2 3 4
Supply Curve
Demand Curve
Equilibrium Price (p*)
0Q
P
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Comp. 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 ($) 3.5 1.5 0 0
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Supply-Demand10
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3
1 2 3 4
Supply Curve
Demand Curve
0Q
P
Left-behind Agents
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Alternative: X • Trade pairs: B1-S3, B2-S2, B3-S1: p = V+C/2
Buyer B1 B2 B3 B4
Surplus ($) 1.5 1.5 1.5 0
Seller S1 S2 S3 S4
Surplus ($) 1.5 1.5 1.5 0
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Alternative: Y• Trade pairs: B1-S4, B2-S3, B3-S2, B4-S1
Buyer B1 B2 B3 B4
Surplus ($) 0.5 0.5 0.5 0.5
Seller S1 S2 S3 S4
Surplus ($) 0.5 0.5 0.5 0.5
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Comparison• Trade-off: efficiency vs. equity
Allocation CE X Y
Total Surplus 10 9 4
# of Trading Agents 4 (50%) 6 (75%) 8 (100%)
PENT & IR Yes Yes Yes
Unique Price Yes No No
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Why are X and Y PENT?• Not Pareto dominated by CE.
Buyer B1 B2 B3 B4
Surplus ($) 3.5 1.5 0 0
Seller S1 S2 S3 S4
Surplus ($) 3.5 1.5 0 0
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Redistribution• Transfer from B1/S1 to B3&B4/S3&S4.
Buyer B1 B2 B3 B4
Surplus ($) 3.5 1.5 0 0
Seller S1 S2 S3 S4
Surplus ($) 3.5 1.5 0 0
$0.5
$0.5
$1.5
$1.5
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Not PE with Redistribution• CE + side payment Pareto dominates X & Y.
Buyer B1 B2 B3 B4
Surplus ($) 1.5 1.5 1.5 0.5
Seller S1 S2 S3 S4
Surplus ($) 1.5 1.5 1.5 0.5
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Discussion• Why focusing on # of trades or trading agents?
• # looks relevant in some market, e.g., labor market.
• Benchmark: how market economy works without redistribution => importance of redistribution.
• How to implement alternative allocations?
• Through centralized or decentralized mechanisms?
• Any implications to positive analysis (of imperfect market)?
• Matching services, middle-men, social custom, etc.
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Bisection Method• 1st: Increase the pie
• Partial Equilibrium — Total surplus maximization
• General Equilibrium — Pareto efficiency
• 2nd: Redistribute (if necessary)
• PE — Compensation principle
• GE — Second Welfare Theorem
=> Make sense only if effectual redistribution is “feasible.”
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Market Economy• Homogenous good market
• Finitely many buyers and sellers
• Each has unit demand/supply
• Other simplifying assumptions:
A. 0 utility for non-trading agents
B. No buyer-seller pair generates 0 surplus
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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 mutually profitable transactions. (no one just receives/gives good alone or money alone or both from the initial endowment.)
• Preferences: larger surplus is better (unit demand).
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Another Interpretation• PE in our environment can be seen as a weaker version of PE concept
in the standard market environment.
• Allocation z is called Pareto efficient with No transfer (PENT) if and only if there exists no other allocation z’ such that
• z’ Pareto dominates z, and
• no one receives larger/smaller consumption bundle than his/her initial endowment, i.e.,
• PE allocation is always PENT, but NOT vice versa.
• e.g., X and Y are PENT but not PE (if transfers are feasible).
@i, z0i � ei or z0i ei.
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Main Theorem• Theorem 1 (Greatest happiness of the minimum number)
For any market economy, the number of trading agents under the market equilibrium is minimum among all allocations that are PENT and IR.
• Def. of trading agents
• Those who consumes an indivisible good strictly less/more than their initial endowments.(less => seller / more => buyer)
• Other agents end up receiving initial endowments.
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Proof (Theorem 1)1. Suppose not. Then, there must exist a PENT and IR 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 PENT.
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Converse• Order agents in each side as follows:
• Buyer/seller with smaller number has higher value/lower cost.
• Suppose that quantity traded under CE is k.
• Uniquely determined with our (simplifying) assumption B.
• Theorem 2 There exists a feasible, PENT 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.
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Equilibrium (k = 2)10
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3
1 2 3 4
Supply Curve
Demand Curve
Equilibrium Price
0Q
P
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Illustration10
8765
3
1 2 3 4
Supply Curve
Demand Curve
Equilibrium Quantity
0Q
P
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Proof (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 PENT 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.
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Continuous Case• Continuous demand and supply curves
• No vertical jump at any points
• continuum agents with unit demand/supply
• mass of agents with V = C is 0 (simplification)
• Theorem 1’ The mass of agents who engage in trades under the market equilibrium is minimum among all allocations that are PENT and IR.
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Continuous Case• Theorem 2’
There exists a feasible, PENT and IR allocation that entails strictly larger mass of trades than that of CE if and only if
• neither demand nor supply curve is completely vertical at the CE (trivial), and
• neither demand nor supply curve is completely flat to the left of the CE.
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Graphical Intuition
OK NG
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Graphical Intuition
OK NG
Possible to find their partners Impossible to find
their partners
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Matching Market• Stable matching (Core) may induce minimum pairs.
=> Examples 2a, 3a, 4, 6
• However, Theorem 1 does NOT hold.
• # of Stable matching pairs not always minimum. => Examples 2b, 3b, 5.
• NTU — Anything can happen. (PE = PENT)
• TU — Assortative stable matching is minimum.
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One-to-one Matching• Finitely many doctors and hospitals
• Each matched with at most one agent
• Being single is strictly different from matched with any mate (simplifying assumption)
• having mate is strictly better/worse than alone
• No monetary transfer at all (NTU)
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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
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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
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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
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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
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Assignment Game• Finitely many workers and firms
• Each matched with at most one agent
• Receive 0 utility if unmatched
• Monetary transfers allowed (TU)
• Paris arbitrarily divide production surplus
• No side payment beyond each worker-firm pair
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Example 3a• 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
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Example 3a• 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)46
Example 3b• 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
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Example 3b• 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)
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Example 4• 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
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Example 4• 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
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Result in TU Case
• Theorem 3 The number of worker-firm pairs under the assortative stable matching is minimum among all matching outcomes that are PENT and IR.
• What is assortative stable matching (ASM)?
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Definition of ASM• (1) Agents in each side can be ordered:
• Worker/firm with smaller number is better.
• Production surplus between worker i and firm j, Aij, is (weakly) decreasing in i and j.
• (2) Stable matching induces pairs of
• W1-F1, W2-F2, …, Wk-Fk for some k.
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Proof (Theorem 3)1. Suppose not. Then, there must exist a PENT 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 PENT.
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Slight Extension• Claim
Suppose that the set of stable matchings contains a assortative stable matching (but possibly other stable matchings). Then, the number of worker-firm pairs under any stable matching is minimum among all PENT and IR matching outcomes.
• Proof idea: The set of agents who have partners under (different) stable matchings is identical.
• Known as “Rural Hospital Theorem.”
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Example 5• NTU: 2 doctors, 2 hospitals
• Unique Stable Matching = ASM: D1-H1, D2-H2
• An Alternative: D2-H1 (D1, H2 single) <= PE and IR
=> All agents find their mates under ASM.
Agent D1 D2 H1 H2
1st H1 H1 D1 D1
2nd - H2 D2 D2
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Example 5• NTU: 2 doctors, 2 hospitals
• Unique Stable Matching = ASM: D1-H1, D2-H2
• An Alternative: D2-H1 (D1, H2 single) <= PE and IR
=> All agents find their mates under ASM.
Agent D1 D2 H1 H2
1st H1 H1 D1 D1
2nd - H2 D2 D2
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Interpretation• Theorems 1-3 clarify limitation/weakness of market
economy even if no market failure is presupposed. <= Feasibility of redistribution is crucial!
• Real-life market, e.g., via middle-men or social custom, may achieve more trading pairs than competitive market.
• However, these market systems may fail to achieve PE (even PENT) allocation while CE always does.
• No “negative” result on stable matching in NTU case => Not imply there is NO cost for pursuing stability.
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Example 6• 2 doctors, 2 patients (P2: poor patient)
• Unique Stable Matching: D1-P1 (D2, P2 single)
• An Alternative: D1-P2, D2-P1 <= PE and IR
=> What if patient 2 would die if he/she cannot find any doctor…
Agent D1 D2 P1 P2
1st P1 P1 D1 D1
2nd P2 - D2 D2
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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
• Redistribution is crucial when market is competitive.
• Imperfect markets may achieve equitable allocations.
=> May better consider equitable market design.
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New!