pareto improvements under matching mechanisms in a public good economy
TRANSCRIPT
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
Overview
• Motivation
• Framework
• Results
• Conclusions
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.
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.
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 Pareto-improving equilibria.
• 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?
• Focus on Pareto-improving moves from the initial Nash equilibrium.
• The condition of the Pareto optimal equilibrium: μ1μ2=1
The model • One private good, one pure public good, and two heterogeneous players.
• 𝑢 𝑥𝑖, 𝐺 = 𝑥𝑖
𝑎𝑖𝐺, α𝑖 ≥ 1, 𝑖 = 1,2
– 𝑥𝑖 is private good consumption.
– 𝐺 is the total public good provision.
– α𝑖 is the weight of value player 𝑖 attaches to the private good relative to the public good.
• Player 𝑖 has an initial income of 𝑤𝑖 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 𝑦𝑖≥0
– Indirect matching contribution
• Total public good provision is 𝐺 = (1 + μ2)𝑦1+(1 + μ1)𝑦2
• 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 +μ1𝑦2 = 𝑤1
• 𝑥2 +𝑦2 +μ2𝑦1 = 𝑤2
• Two states of the economy:
• The initial equilibrium
• The matching equilibrium
• Definition 1 A pairwise (𝑦1∗, 𝑦2
∗) is a matching equilibrium in flat
contributions if, for any player 𝑖, the flat contribution 𝑦𝑖∗ maximizes
))1()1( ,(),( jiijjiiiiii yyyywuGxu
• Definition 2
(i) An interior equilibrium is an equilibrium where each player chooses a
strictly positive flat contribution, 𝑦𝑖>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 Pareto-
improving if, for any player 𝑖, the utility under the matching scheme is
higher than without matching, 𝑢𝑖 𝑥𝑖, 𝐺 > 𝑢𝑖 𝑥𝑖𝑁, 𝐺𝑁 .
Nash Equilibria
• The initial equilibrium
𝐺𝑁 =𝑊
α1+α2+1, 𝑥𝑖
𝑁 =α𝑖𝑊
α1+α2+1, 𝑦𝑖
𝑁 = 𝑤𝑖 −α𝑖𝑊
α1+α2+1
• The matching equilibrium
𝐺 =𝑊
α11 + μ2
+α2
1 + μ1+ 1
, 𝑥𝑖 =α𝑖
1 + μ𝑗
𝑊α1
1 + μ2+
α21 + μ1
+ 1
𝑦𝑖 =
𝑤𝑖 − μ𝑖𝑤𝑗 + μ𝑖
α𝑗
1 + μ𝑖−
α𝑖1 + μ𝑗
𝑊α1
1 + μ2+
α21 + μ1
+ 1
1 − μ1μ2
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, ∞)
w1 w2
k=0 k=∞
Neutrality zone
• Conditions of interior equilibria:𝑦𝑖 > 0
• Proposition 1 Given 𝑢 𝑥𝑖, 𝐺 = 𝑥𝑖
𝑎𝑖𝐺, α𝑖 ≥ 1, 𝑖 = 1,2 and any
income distribution within the neutrality zone α1
α1+α2< 𝑘 <
α1+α2
α2 would achieve the same equilibrium at which the public
good provision is 𝐺 =𝑊
α1+α2+1.
• Proposition 2 Given 𝑢 𝑥𝑖, 𝐺 = 𝑥𝑖
𝑎𝑖𝐺, α𝑖 ≥ 1, 𝑖 = 1,2 and any
income distribution within the neutrality zone α1
α1+α2< 𝑘 <
α1+α2
α2, there always exist small matching rates (μ1μ2<1) to
generate Pareto-improving interior equilibria.
• Interior equilibria • Pareto-improving equilibria
1 1 0.50-2.00
2 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 1.0-2.0
5 1 3.0-6.0
10 1 5.5-11.0
20 1 10.5-21.0
1 2
2
1
2
1 1
1
k
21
21
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.
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.