1) The document discusses how matching mechanisms can be used to provide public goods in a voluntary contribution system. It focuses on whether small matching rates can lead to Pareto-improving equilibria. 2) It presents a model with two players who value private and public goods differently and have initial incomes. It defines Nash, matching, and Pareto-improving equilibria in this setting. 3) The results show that as long as the income difference between players is not too large, lying within a "neutrality zone", there always exist small matching rates that can generate Pareto-improving outcomes compared to the initial Nash equilibrium.

1. Pareto Improvements under
Matching Mechanisms in a
Public Good Economy
Larry Liu
Centre for Applied Macroeconomic Analysis (CAMA)
Australian National University
November 4, 2013
Crawford School PhD Conference

2. Overview
•
•
•
•
Motivation
Framework
Results
Conclusions

3. Motivation
• Public goods are underprovided in voluntary contribution.
• Matching mechanisms (Guttman 1978, 1987)
– Two-stage game
• Stage 1: Announce a matching rate
• Stage 2: Decide how much to provide
– Non-cooperative mode of public good provision.
– Subsidize individual public good contributions.
– The sub-game perfect equilibrium is Pareto optimal.
• Literature focuses on the Pareto optimal equilibrium.

4. Reaching Pareto optimal equilibria is very ambitious in practice:
1. Information
• Knowledge about preferences is difficult to obtain.
• Pareto optimal policy requires global knowledge of relevant
parameters while Pareto-improving reform only needs local
information (Deaton, 1987; Myles, 1995).
• Agents are more uncertain about larger changes.
• Gradually Pareto-improving reform is more desirable.
• More severe at the international level because it is more difficult
to know the aggregate preference of one country.

5. 2. Commitments
• Credibility of commitments in the second stage should not be
taken for granted (Boadway, Song and Tremblay 2007, 2011).
• No supranational authority can force sovereign countries to
implement their commitments at the international level.
• The Pareto optimal equilibrium requires larger matching rates
and hence more ambitious commitments than some Paretoimproving equilibria.

6. • Focus on Pareto-improving moves from the initial Nash
equilibrium.
• The condition of the Pareto optimal equilibrium: μ1μ2=1
• Without knowledge of
preferences, what players can do
is no more than trying arbitrarily
any small matching rates whose
product is smaller than one.
• What is the chance of reaching
Pareto-improving outcomes?

7. The model
•
One private good, one pure public good, and two heterogeneous players.
•
𝑢 𝑥 𝑖 , 𝐺 = 𝑥 𝑖 𝑖 𝐺, α 𝑖 ≥ 1, 𝑖 = 1,2
𝑎
–
𝑥 𝑖 is private good consumption.
–
𝐺 is the total public good provision.
– α 𝑖 is the weight of value player 𝑖 attaches to the private good relative to the public good.
•
Player 𝑖 has an initial income of 𝑤 𝑖 units of the private good
– Total income is 𝑊 = 𝑤1 +𝑤2 (𝑊 is constant)
– Income ratio is 𝑘 = 𝑤1 /𝑤2
•
Player 1’s matching rate μ1 ≥0, Player 2’s matching rate μ2 ≥0
•
A matching scheme: 𝑚(μ1 , μ2 )
•
Each player’s contribution to the public good:
– Direct flat contribution 𝑦 𝑖 ≥0
– Indirect matching contribution
•
Total public good provision is 𝐺 = (1 + μ2 )𝑦1 +(1 + μ1 )𝑦2

8. • Assume that two players have the same linear production
technology of the public good.
• Prices of the private good and the public good are both
normalized to one.
• Budget constraints
• 𝑥1 +𝑦1 +μ1 𝑦2 = 𝑤1
• 𝑥2 +𝑦2 +μ2 𝑦1 = 𝑤2
• Two states of the economy:
• The initial equilibrium
• The matching equilibrium

9. •
Definition 1 A pairwise (𝑦1∗ , 𝑦2∗ ) is a matching equilibrium in flat
contributions if, for any player 𝑖, the flat contribution 𝑦 𝑖 ∗ maximizes
ui ( xi , G )  ui ( wi  yi  i y  , (1   j ) yi  (1  i ) y  )
j
j
•
Definition 2
(i) An interior equilibrium is an equilibrium where each player chooses a
strictly positive flat contribution, 𝑦 𝑖 >0;
(ii) A corner equilibrium is an equilibrium where at least one player
chooses a zero flat contribution, 𝑦1 =0 or 𝑦2 =0.
•
Focus on interior equilibria: Both players provide positive flat contributions.
•
Definition 3 An equilibrium under a matching scheme 𝑚(μ1 , μ2 ) is Paretoimproving if, for any player 𝑖, the utility under the matching scheme is
higher than without matching, 𝑢 𝑖 𝑥 𝑖 , 𝐺 > 𝑢 𝑖 𝑥 𝑖 𝑁 , 𝐺 𝑁 .

10. Nash Equilibria
• The initial equilibrium
𝐺𝑁=
𝑊
,
α1 +α2 +1
𝑥𝑖 𝑁 =
α𝑖 𝑊
,
α1 +α2 +1
𝑦𝑖 𝑁 = 𝑤𝑖 −
α𝑖 𝑊
α1 +α2 +1
• The matching equilibrium
𝑊
α𝑖
𝑊
𝐺= α
, 𝑥𝑖 =
α2
1
1 + μ 𝑗 α1 + α2 + 1
+
+1
1 + μ2 1 + μ1
1 + μ2 1 + μ1
α𝑗
α𝑖
𝑊
𝑤 𝑖 − μ 𝑖 𝑤𝑗 + μ 𝑖
−
α1
α2
1 + μ𝑖 1 + μ 𝑗
+
+1
1 + μ2 1 + μ1
𝑦𝑖 =
1 − μ1 μ2

11. Neutrality zone
• Neutrality (Warr, 1982, 1983; Bergstrom, Blume and Varian,
1986; Cornes and Sandler, 1996).
• The total public good provision and the private consumption
is unaffected by small income redistribution among
contributors.
Income ratio kϵ(0, ∞)
k=0
w1
k=∞
Neutrality zone
w2

12. • Conditions of interior equilibria:𝑦 𝑖 > 0
𝑎
• Proposition 1 Given 𝑢 𝑥 𝑖 , 𝐺 = 𝑥 𝑖 𝑖 𝐺, α 𝑖 ≥ 1, 𝑖 = 1,2 and any
α
income distribution within the neutrality zone 1 < 𝑘 <
α1 +α2
α2
α1 +α2
would achieve the same equilibrium at which the public
good provision is 𝐺 =
𝑊
.
α1 +α2 +1
𝑎𝑖
• Proposition 2 Given 𝑢 𝑥 𝑖 , 𝐺 = 𝑥 𝑖 𝐺, α 𝑖 ≥ 1, 𝑖 = 1,2 and any
α
income distribution within the neutrality zone 1 < 𝑘 <
α1 +α2
,
α2
α1 +α2
there always exist small matching rates (μ1μ2<1) to
generate Pareto-improving interior equilibria.

13. • Interior equilibria
• Pareto-improving equilibria

14. 1
1  1
2
1
1
0.50-2.00
2
0.67-1.50
5
5
0.83-1.20
10
10
0.91-1.10
20
20
0.95-1.05
2
1   2
1
2
1   2
2
1
1.0-2.0
5
1
3.0-6.0
10
1
5.5-11.0
20
1
10.5-21.0
2 1
k

15. Policy implications
• If the income heterogeneity is not too large, two players
can always implement some small matching schemes to
make them both better off.
• The less ambitious the matching scheme, the more likely
they reach Pareto-improving outcomes.

16. Conclusions
• Reaching Pareto optimal equilibria is very ambitious in
practice.
• Given the Cobb-Douglas utility function, there is always a
neutrality zone.
• Within the neutrality zone there always exist small matching
schemes to generate Pareto-improving outcomes.
• The higher the weights of value to the private good, the
smaller the neutrality zone.
• The less ambitious the matching scheme, the more likely
they reach Pareto-improving outcomes.