The w-value: A Market-Based Rule for Sharing Pollution-Control Gains

Pollution agreements must address two connected questions: how much pollution should participants produce, and how should they divide the benefits of cooperation? Juanjuan Fan and Ying Wang propose a solution called the w-value, which combines a welfare-sharing rule with prices and transfers intended to implement it.

Their central claim is conditional: within their cooperative pollution-control model, the mechanism selects an efficient welfare allocation that no coalition can improve upon by leaving, under a specified assumption about how outsiders behave.

A model of benefits and shared damage

Each participant earns revenue from an activity that emits pollution. Its welfare equals that revenue minus damage from the total pollution stock, before policy-related payments. Each unit of the activity adds one unit of pollution, and the stock also includes pollution already present.

Acting alone, a participant weighs its own revenue against its own pollution damage. Coordinating participants instead account for damage across their membership. The fully cooperative outcome maximizes total welfare by balancing each participant’s marginal emission-related revenue against the marginal damage suffered by everyone.

The assumptions matter. Revenue functions are strictly increasing and strictly concave: more emissions bring additional revenue, but progressively smaller increments. Damage functions are strictly increasing and strictly convex: pollution causes increasing damage at an increasing marginal rate. The model also assumes marginal revenue becomes arbitrarily large as an individual’s emissions approach zero.

Welfare is measured in money and can be transferred between participants. This allows an agreement to separate the choice of efficient emissions from the distribution of the resulting gains. Participants maximize their own welfare; a coalition maximizes its members’ combined welfare.

Separate incentives from distribution

The authors divide the problem into two sub-games. The first establishes prices that align individual emission choices with the collective optimum. The second determines initial permit endowments, which affect who receives the benefits.

Their selected pricing arrangement, called Price Constraint I, sets a common emission price equal to total marginal pollution damage at the optimum. It also uses an individual transfer rate equal to each participant’s own marginal damage. Lump-sum taxes or subsidies are zero in this arrangement.

Participants’ payments depend on the difference between their initial permits and actual emissions, together with an adjustment tied to aggregate emissions relative to aggregate initial permits. The authors show that the resulting transfers balance across participants: the mechanism redistributes money rather than requiring a net outside contribution.

In their model, changing initial permit allocations changes individual welfare without changing the efficient emissions outcome or total welfare under this pricing arrangement. That makes permit allocation the central distributional question, rather than an additional source of efficiency gains.

What defines the w-value?

The proposed starting allocation gives every participant permits equal to its emissions in the fully non-cooperative Nash equilibrium. This is the situation in which everyone acts independently, choosing emissions to maximize individual welfare given others’ choices.

The authors call this baseline the w-initial pollution-emission permit allocation, or w-IPA. Applying their pricing mechanism to that allocation produces each participant’s w-value.

The calculation has four stages: find non-cooperative emissions, calculate efficient emissions, determine the optimal prices, and calculate welfare including transfers. The authors argue that their equilibrium reasoning selects this allocation uniquely. This is not a claim that the broader set of stable allocations contains only one possible distribution.

Stability depends on outsider behavior

The paper evaluates stability using the γ-core. An allocation belongs to this set if it distributes the total efficient welfare and gives every coalition at least what that coalition could obtain by departing under the specified deviation scenario.

Crucially, outsiders to a departing coalition remain separate participants rather than forming another coalition. They choose emissions to maximize their own welfare. Fan and Wang claim that the w-value satisfies the resulting coalition-by-coalition comparisons.

The stability claim therefore concerns this particular model of departure and outsider behavior, not every possible response to an agreement breaking down.

A four-participant illustration

The paper illustrates the calculation with four hypothetical participants whose revenue and damage functions differ. It reports total emissions falling from approximately 1.380 in the non-cooperative equilibrium to 0.452 at the efficient outcome.

Reported total welfare rises from 24.284 to 33.545. After transfers, the four w-values are 17.583, 4.279, 2.940 and 8.743; each exceeds that participant’s non-cooperative welfare. The authors also tabulate coalition comparisons supporting γ-core membership in the example.

These are model calculations, not measured policy outcomes; the paper states that no data were used. Its contribution is a theoretical proposal connecting efficient pollution pricing with a specific distribution of cooperation gains. The claimed existence, uniqueness and stability of that distribution rest on the model’s assumptions and equilibrium arguments.