Tax It, Trade It, Govern It: Pigou, Coase, Ostrom as Three Switches on MST

Pigou, Coase, and Ostrom make the same missing coordinate visible in different ways

The paper uses the Missing System Theorem to restate externality economics as a problem of representation: system welfare W is absent from player-indexed payoff vectors under axiom A2.

• Pigou adds the missing system price λ_W from outside the game through a planner.
• Coase removes the MST problem at the zero-transaction-cost boundary by making W tradable.
• Ostrom changes the game so rules, monitoring, and sanctions make W-degradation costly to players.

Axiom A2 excludes system welfare from every player payoff vector

MST starts from the claim that W is not in the span of any player-indexed payoff vector. Externality problems follow because private maximization can succeed while the system coordinate deteriorates.

• The structural defect is not a bad preference or a missing moral norm; W is absent by construction.
• The Hollow Win is the outcome where players maximize private payoffs while system welfare falls.
• The paper compares Pigou, Coase, and Ostrom by asking how each makes W visible inside or around the game.

Pigou works when the planner knows λ_W and can enforce τ = λ_W

Proposition E.1 defines a Pigovian regime as MST with the system price supplied exogenously. The planner observes marginal damage λ_W* and imposes a tax equal to it.

• The tax τ folds W into each utility function from outside the game.
• The efficient Pigovian outcome obtains iff the planner knows λ_W* and can enforce τ against the players.
• Pigou is the case where the system price exists but is set by fiat.

Capture can push the imposed tax below marginal damage

Because W is not in span(u), players have an incentive to influence the rate-setting process. The paper treats capture as the native failure mode of the imposed-price switch.

• When capture intensity φ exceeds threshold φ*, the imposed tax τ collapses below λ_W*.
• The Hollow Win returns because private incentives again outrun the system coordinate.
• Stigler, Peltzman, and Laffont-Tirole enter as the regulation-capture teaching arc.

Coase is the boundary where A2 fails

Proposition E.2 says the Coase theorem is the case where MST stops binding. If a property right over W can be assigned and bargained at zero transaction cost, W enters the payoff vector.

• With A2 off, there is no missing coordinate for MST to diagnose.
• The parties bargain to a system-preserving efficient outcome regardless of initial allocation.
• The paper states this as Coase = MST in the transaction-cost-zero and costlessly partitionable limit.

Positive transaction costs put the case back inside MST

The paper does not treat Coase as a general escape from externality problems. It treats Coase as the limiting case where W can be traded without friction.

• With positive transaction costs, A2 holds and the assignment of rights matters.
• With a non-partitionable system, W cannot be costlessly carved into tradable claims.
• This is the positive-cost world Coase identified as the interesting one.

Ostrom repairs the game through institutions rather than fiat or frictionless trade

Proposition E.3 defines self-governed commons as instances of the game-change repair R: G → G'. The institution changes the payoffs so system degradation is privately costly.

• Monitoring makes W-degradation observable to the community.
• Graduated sanctions make degradation costly without relying on an outside tax authority.
• Boundary rules define who participates in the commons and who is subject to the institution.

The repaired game preserves the system at equilibrium

In the paper's notation, Ostromian repair folds c back into each u_i endogenously. The equilibrium of the repaired game G' preserves the system.

• The formal preservation condition is c(σ'*) = 1.
• A2 is restored by the players' own institution, not by an external planner.
• Ostrom's eight design principles are treated as the empirical structure of successful repairs R.

The three switches are impose the price, drop A2, or repair the game

The paper's comparison is structural. It does not rank tax, bargain, and govern as universal policy tools.

• Pigou: impose λ_W through a planner, with failure risk from capture of τ.
• Coase: make W tradable, but only at zero transaction cost or under costless partition.
• Ostrom: build rules that make W-degradation individually costly inside G'.

One externality can be taught as three restoration designs

The classroom move is to hold one domain fixed and ask how each tradition would restore the system coordinate.

• Pigovian design: identify λ_W, set τ = λ_W, and ask whether enforcement can withstand capture.
• Coasean design: assign rights over W and test whether transaction costs and partitionability are near zero.
• Ostromian design: specify boundary rules, monitoring, and graduated sanctions for the community.

The paper is a theory map, not a domain calibration

The manuscript positions this result above applied β_W domain ranking. Its claim is a representation of the externality canon, not an empirical ranking of fields or instruments.

• No domain β_W tile is supplied.
• The status is draft manuscript, with peer review pending.
• The target journal named in the record is JLEO.