Competitive Market Achieves the Greatest
Happiness of the Minimum Number
Yosuke YASUDA ¦ Osaka University
[email protected]
- June 2016 -
1
Motivation
• Inequality at the forefront of public debate!
2
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
3
Natural Questions
•
What if redistribution is NOT available?
•
The set of feasible allocations change
(Pareto frontier & PE allocations also change)
•
Can competitive market still be optimal?
•
Is egalitarian or equitable market design needed?
•
Changing a market less competitive may help.
=> These are what I study in this project!
4
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 # or 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))
5
Summary (2)
•
Converse result also holds.
•
Unless a demand or supply curve is completely flat, there
always exists a PE and IR allocation that entails strictly
larger number of trades than the equilibrium quantity.
•
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 PE and IR.
•
# of left-behind agents from trade is maximized!
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 PE and IR.
•
The assortative matching assumption is often imposed in
applied works, e.g., labor markets and marriage markets.
7
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
8
Supply-Demand
10
8 7 6 5 3
1 2 3 4
Supply
Curve
Demand
Curve
Equilibrium
Price
0 Q
P
9
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
($) 3.5 1.5 0 0
10
Supply-Demand
10
8 7 6 5 3
1 2 3 4
Supply
Curve
Demand
Curve
0 Q
P
Left-behind
Agents
11
Alternative: X
• Trade pairs: B1-S3, B2-S2, B3-S1: p = V+C/2
Buyer B1 B2 B3 B4
Surplus
($) 1.5 1.5 1.5 0
Seller S1 S2 S3 S4
Surplus
($) 1.5 1.5 1.5 0
12
Alternative: Y
• Trade pairs: B1-S4, B2-S3, B3-S2, B4-S1
Buyer B1 B2 B3 B4
Surplus
($) 0.5 0.5 0.5 0.5
Seller S1 S2 S3 S4
Surplus
($) 0.5 0.5 0.5 0.5
13
Comparison
• Trade-off: efficiency vs. equity
Allocation CE X Y
Total Surplus 10 9 4
# of Trading
Agents 4 (50%) 6 (75%) 8 (100%)
PE and IR Yes Yes Yes
Unique Price Yes No No
14
Why are X and Y PE?
• Not Pareto dominated by CE.
Buyer B1 B2 B3 B4
Surplus
($) 3.5 1.5 0 0
Seller S1 S2 S3 S4
Surplus
($) 3.5 1.5 0 0
Redistribution
• Transfer from B1/S1 to B3&B4/S3&S4.
Buyer B1 B2 B3 B4
Surplus
($) 3.5 1.5 0 0
Seller S1 S2 S3 S4
Surplus
($) 3.5 1.5 0 0
$0.5
$0.5
$1.5
$1.5
Not PE with Side Payment
• CE + side payment Pareto dominates X & Y.
Buyer B1 B2 B3 B4
Surplus
($) 1.5 1.5 1.5 0.5
Seller S1 S2 S3 S4
Surplus
($) 1.5 1.5 1.5 0.5
Discussion
• How to implement alternative allocations?
• Unclear if they can be achieved in centralized mechanisms.
• Any implications to positive analysis?
• Matching services, middle-men, social custom, etc.
• Experiments: Chamberlin (1948) vs. Smith (1962)
• Our results may explain excess quantity phenomenon.
• What could be the policy target?
• E.g., Maximize total surplus times # of trades
18
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
19
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 bilateral transactions. (no side
payment beyond each buyer-seller pair is allowed)
•
Preferences: larger surplus is better (unit demand).
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 , which
• Pareto dominates z, and
• is achieved through mutually profitable bilateral transactions (requires no transfer).
• PE allocation is always PENT, but not vice versa.
• e.g., X and Y are PENT but not PE (if transfers are feasible).
Main Theorem
•
Theorem 1
The number of agents who engage in trades under
the market equilibrium is minimum among all
feasible allocations that are Pareto efficient and
individually rational.
•
Alternative Statement
The number of agents who engage in trades under
the market equilibrium is minimum among all
allocations that are PENT and individually rational.
22
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. V
B*is (weakly) larger than p* which is also larger than C
S*.
4. B*-S* pair generates positive surplus. <= V
B*> C
S*5. Contradicts to our presumption that z is PENT.
23
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 PENT and individually rational 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.
24
Equilibrium (k = 2)
10
8 7 6 5 3
1 2 3 4
Supply
Curve
Demand
Curve
Equilibrium
Price
0 Q
P
25
Illustration
10
8 7 6 5 3
1 2 3 4
Supply
Curve
Demand
Curve
Equilibrium
Quantity
0 Q
P
26
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.
27
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 individually rational.
28
Continuous Case
• Theorem 2
There exists a PENT and individually rational
allocation that entails strictly larger mass of
trades than that of CE if and only if
• neither demand nor supply curve is
completely virtual at the CE (trivial), and
• neither demand nor supply curve is
completely flat to the left of the CE.
29
Graphical Intuition
OK NG
30
Graphical Intuition
OK NG
Possible to find
their partners Impossible to find
their partners
31
Remarks
•
Why focusing on # of trades or trading agents?
•
# looks relevant in some market, e.g., labor market.
•
Provides some benchmark: Clarifies that competitive market
results in unequal/biased allocation.
•
What is the largest # of (IR) trades?
•
Easily derived by demand and supply curves.
•
The corresponding allocation must be PENT.
•
Our result may change the interpretation of bisection method.
32
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.
33
Matching Market
•
Stable matching 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.
34
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)
35
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
36
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
H1H1
D1D1
2nd H2 - D2 D2
37
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
38
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 D12nd
H2- D1 D2
39
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
40
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
41
Example 3a
•
2 workers, 2 firms
•
Unique Core: W1-F1 (W2, F2 single)
•
Alternative: W1-F2, W2-F1 <= PE and IR
F1 F2
W1
104
W2 4 -5
(5 - 5)
(2 - 2) (2 - 2)
42
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
43
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
8W2
4-5
(5 - 5)
(7 - 1) (1 - 3)
44
Example 4
•
Revisit (reformulate) Example 1 <= A
ij:= V
i- C
j•
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
45
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
53
1B2
53
1-1
B3 3
1-1 -3
B4
1-1 -3 -5
46
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 individually rational.
• What is assortative stable matching (ASM)?
47
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, A
ij, is (weakly) decreasing in i and j.
• (2) Stable matching induces pairs of
• W
1-F
1, W
2-F
2, …, W
k-F
kfor some k.
48
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.
49
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 individually rational matching outcomes.
•
Proof idea: The set of agents who have partners
under (different) stable matchings is identical.
•
Known as Rural Hospital Theorem.
50
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
51
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
H1H1
D1D1
2nd -
H2D2
D252
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
allocation while CE always does.
•
No negative result on stable matching in NTU case
=> Not imply there is NO cost for pursuing stability.
53
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
P1P1
D1D1
2nd P2 - D2 D2
54
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
•
We economists may take this trade-off more seriously…
=> Redistribution is crucial when market is competitive.
=> May better consider equitable market design.
55
