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.