The Coase Theorem and Bargaining

Why Property Rights Matter for Externalities

Pollution creates a conflict between a polluter’s private gains and the damages imposed on others. A policy can address this conflict, but under particular conditions the affected parties may also be able to negotiate an efficient outcome themselves. The Coase theorem explains when that possibility holds, and what it does—and does not—say about who ultimately benefits.

The key idea is to distinguish efficiency from entitlement. Efficiency concerns whether the available gains from production and pollution have been maximized. Entitlement concerns who has the legal right to pollute or to be free from pollution. Bargaining may change the allocation of resources without changing the initial legal entitlement; a payment can compensate the party that gives up some of its rights.

Definition: The Coase theorem states that, when property rights are clearly defined and transferable, transaction costs are negligible, and the parties can bargain and enforce an agreement, private bargaining can lead to an efficient allocation of resources. In the standard benchmark, the efficient allocation does not depend on which party initially holds the relevant right. The initial assignment can, however, affect the bargaining payments and the distribution of gains.

This is a conditional result, not a claim that pollution always resolves itself through negotiation. Its assumptions matter. For example, the parties must know which actions cause the relevant gains and damages, and the agreement must be enforceable. If the conditions do not hold, the theorem does not establish that bargaining will produce an efficient result. The focus here is the benchmark in which they do.

The theorem is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. To apply it, first identify the efficient level of the activity. Then compare each party’s benefit or cost from moving away from the initial allocation. Those comparisons determine whether there are gains both parties can accept.

Efficiency in a Pollution Model

Let $e$ denote emissions, with $0 \leq e \leq 40$. Suppose the polluting firm’s total benefit from emissions is $B(e)$, while the total external damage to affected residents is $D(e)$. For this example, assume these functions are known and measured in dollars:

B(e)=100e-e^2,\qquad D(e)=20e+e^2.

The net social benefit is the firm’s benefit minus the external damage. The efficient emissions level maximizes this difference over the feasible range. At an interior maximum, the marginal benefit of emissions equals the marginal damage. This is the same marginal comparison used in the earlier material on deriving efficient output with external costs.

Key result: For an interior efficient level $e^*$, the marginal benefit of emissions, $B'(e^*)$, equals the marginal damage, $D'(e^*)$. This condition requires that the functions be differentiable and that the optimum be interior. If the equality has no feasible solution, or if the best outcome is at a boundary, compare total net benefits at the relevant boundary and interior candidates.

The reason for the marginal condition is direct. If marginal benefit exceeds marginal damage, a small increase in emissions adds more benefit than damage, so net social benefit rises. If marginal damage exceeds marginal benefit, a small reduction in emissions raises net social benefit. At an interior maximum, neither small change improves the outcome, so the two marginal effects are equal.

Worked Example: Finding the Efficient Emissions Level

A hypothetical plant’s benefit from emissions $e$, measured in pollution units, is $B(e)=100e-e^2$ dollars. Residents’ total damage is $D(e)=20e+e^2$ dollars. Emissions are restricted to $0 \leq e \leq 40$. Find the efficient level and verify its total net benefit.

Method. Set marginal benefit equal to marginal damage, then check the candidate against the feasible range and calculate total net benefit.

Differentiate the total benefit and damage functions:

B'(e)=100-2e,\qquad D'(e)=20+2e.

Set the two marginal values equal:

100-2e=20+2e \quad\Longrightarrow\quad 80=4e \quad\Longrightarrow\quad e^*=20.

The solution lies in the feasible interval. To verify the outcome, calculate both totals at $e=20$:

B(20)=100(20)-20^2=2{,}000-400=1{,}600, $$ $$ D(20)=20(20)+20^2=400+400=800.

Net social benefit is $1{,}600-800=\$800$. In fact, $B(e)-D(e)=80e-2e^2$, a concave quadratic whose maximum occurs at $e=20$; this confirms that the marginal condition gives the maximum, not a minimum. Common mistake avoided: equating total benefit and total damage would answer the wrong question. Efficiency requires comparing marginal benefit and marginal damage, then checking the total net benefit.

Bargaining When Residents Hold the Right to Clean Air

Suppose residents initially have a right to clean air, so the plant cannot emit without their agreement. At zero emissions, the plant receives no emissions-related benefit and residents suffer no pollution damage. If the parties consider an agreement allowing $e=20$, the plant can offer compensation in exchange for permission to emit.

At that level, the plant’s benefit is $\$1{,}600$, and residents’ damage is $\$800$. For residents to be at least as well off as under zero emissions, compensation must cover their $\$800$ damage. For the plant to be at least as well off as under zero emissions, the payment cannot exceed its $\$1{,}600$ benefit. Thus any transfer from $\$800$ to $\$1{,}600$ can make both parties willing to agree to $e=20$. The joint gain, whatever the transfer within that range, is $\$800$.

Worked Example: Bargaining for Permission to Emit

Use the benefit and damage functions $B(e)=100e-e^2$ and $D(e)=20e+e^2$. Residents hold the right to clean air, and the parties consider permission for the plant to emit 20 units. If they agree on a transfer of $\$1{,}200$ from the plant to residents, calculate each party’s net gain relative to no emissions. Check that the transfer is acceptable to both.

Method. Calculate each party’s position under the agreement, then compare it with the zero-emissions starting point.

The plant’s benefit from 20 units is:

B(20)=100(20)-20^2=2{,}000-400=\$1{,}600.

Residents’ damage is:

D(20)=20(20)+20^2=400+400=\$800.

After paying $\$1{,}200$, the plant retains $1{,}600-1{,}200=\$400$. Residents receive $\$1{,}200$ and bear $\$800$ in damage, leaving them $1{,}200-800=\$400$ better off. Together, the parties gain $\$400+\$400=\$800$, equal to the increase in net social benefit. The transfer is acceptable because it is between the minimum compensation of $\$800$ and the plant’s maximum payment of $\$1{,}600$. Common mistake avoided: treating the transfer as a social cost. It moves money between the parties; it does not erase the real $\$800$ net gain from the agreement.

Changing the Right Changes the Transfer

Now suppose the plant initially holds the right to emit 40 units. The residents can offer payment for a reduction. To compare the new agreement with the initial entitlement, calculate the firm’s lost benefit and the residents’ avoided damage. The efficient target remains 20 units: the property right changes the starting point, but not the marginal comparison that identifies the efficient level.

Worked Example: Residents Bargain for an Emissions Reduction

The plant holds the right to emit 40 units under the same benefit and damage functions. Residents propose that emissions fall to the efficient level of 20. Find the range of payments from residents to the plant that can make both parties willing to accept the change. Then calculate each party’s gain if the payment is $\$1{,}200$.

Method. Compare the plant’s benefit before and after the reduction, and compare residents’ damage before and after it.

At 40 units, the plant’s benefit is $B(40)=100(40)-40^2=4{,}000-1{,}600=\$2{,}400$. At 20 units, its benefit is $\$1{,}600$. The plant gives up $2{,}400-1{,}600=\$800$, so it must receive at least $\$800$.

Residents’ damage at 40 units is $D(40)=20(40)+40^2=800+1{,}600=\$2{,}400$. At 20 units it is $\$800$. They avoid $2{,}400-800=\$1{,}600$ in damage, so they can pay at most $\$1{,}600$.

The mutually acceptable payment range is therefore $\$800$ to $\$1{,}600$. At a transfer of $\$1{,}200$, the plant’s net gain relative to its initial position is $1{,}200-800=\$400$. Residents’ net gain is $1{,}600-1{,}200=\$400$. Their combined gain is $\$800$, the same as in the clean-air-rights case. Common mistake avoided: inferring that the party with the initial right must remain at the initial emissions level. The right affects the starting position and payment direction, not the efficient target under the theorem’s assumptions.

A Marginal Bargaining Test

The same logic applies when the negotiated action is abatement rather than permission to emit. Let $a$ be units of abatement. Residents benefit from reduced damage, while the plant incurs abatement costs. A mutually beneficial increase in abatement is possible when the residents’ marginal benefit from avoided damage exceeds the plant’s marginal abatement cost. Bargaining reaches the efficient abatement level when those marginal values are equal, provided the assumptions of the benchmark apply.

Worked Example: Bargaining Over Abatement

A hypothetical plant’s cost of abating $a$ units is $C(a)=6a+a^2$ dollars. Residents’ total benefit from avoided pollution damage is $G(a)=30a-\tfrac{1}{2}a^2$ dollars, for $0 \leq a \leq 12$. Find the efficient abatement level and the range of resident payments to the plant that can support it when the plant initially has the right to emit. If the payment is $\$160$, calculate each party’s gain relative to no abatement.

Method. Equate the marginal benefit of abatement to its marginal cost, then compare total benefits and costs to find the bargaining range.

The marginal abatement cost is $C'(a)=6+2a$, and the marginal benefit of avoided damage is $G'(a)=30-a$. Equating them gives:

6+2a=30-a \quad\Longrightarrow\quad 3a=24 \quad\Longrightarrow\quad a^*=8.

The result is feasible. At $a=8$, the plant’s cost is $C(8)=6(8)+8^2=48+64=\$112$. Residents’ benefit is $G(8)=30(8)-\tfrac{1}{2}(8^2)=240-32=\$208$. Hence a payment from residents to the plant must be at least $\$112$ to cover the plant’s cost, and no more than $\$208$ to leave residents with a nonnegative gain.

At a payment of $\$160$, the plant gains $160-112=\$48$, and residents gain $208-160=\$48$. The joint gain is $\$96$, which equals $208-112$, the benefit of abatement net of its cost. Common mistake avoided: comparing total benefit with marginal cost, or comparing the two parties’ marginal values at the wrong abatement level. The efficient condition compares marginal benefit and marginal cost at the same quantity.

What the Theorem Does Not Determine

The Coase theorem identifies an efficient allocation under its assumptions; it does not specify a unique payment or bargaining split. In the clean-air example, transfers between $\$800$ and $\$1{,}600$ allow both parties to accept emissions of 20 units, but different transfers leave different shares of the $\$800$ joint gain with each party. Likewise, starting with the right to pollute reverses the direction of payment in the example, without changing the efficient emissions level.

Keep three questions separate when solving a problem:

What is efficient? Compare marginal benefit with marginal damage, or marginal benefit with marginal cost, to identify the activity level.
Who has the initial right? This determines the starting allocation and can determine who pays whom.
Is there a mutually acceptable bargain? Compare each party’s benefit or cost from the proposed change to find the range of transfers that can leave both parties at least as well off as before.

The conclusion depends on the conditions stated at the outset. Clear, transferable rights and negligible transaction costs are central assumptions, alongside the ability to identify the parties, negotiate, and enforce an agreement. The theorem is therefore a benchmark for analyzing bargaining, not a substitute for checking whether its assumptions fit a particular situation.

Key takeaway: Under the Coase theorem’s assumptions, bargaining can move an externality-producing activity to an efficient level regardless of the initial assignment of property rights. The initial rights still matter for the starting allocation and the distribution of gains through transfers.

Check Your Understanding

For each problem, identify the efficient quantity or bargaining range and keep the underlying benefits and costs distinct from transfers.

A plant’s benefit is $B(e)=72e-e^2$, and residents’ damage is $D(e)=12e+2e^2$, for $0 \leq e \leq 18$. Find efficient emissions and net social benefit at that level. (Answer: $e^*=10$; net social benefit is $\$300$.)
A facility has $B(e)=80e-e^2$, and nearby residents have $D(e)=8e+e^2$. Residents hold the right to clean air and consider allowing the efficient emissions level. Find that level and the acceptable payment range from the facility to residents. (Answer: $e^*=18$; payment range is $\$468$ to $\$1{,}116$.)
A plant initially emits 30 units and holds the right to do so. Its benefit is $B(e)=100e-e^2$, and residents’ damage is $D(e)=20e+e^2$. Find the efficient emissions level, then calculate the acceptable payment range from residents to the plant for the reduction. (Answer: $e^*=20$; payment range is $\$500$ to $\$700$.)
For abatement $a$, a plant’s cost is $C(a)=4a+a^2$, and residents’ benefit from avoided damage is $G(a)=28a-\tfrac{1}{2}a^2$, with $0 \leq a \leq 12$. Find efficient abatement, the total net gain, and the acceptable payment range from residents to the plant. (Answer: $a^*=8$; total net gain is $\$96$; payment range is $\$96$ to $\$192$.)
Two hypothetical parties agree on an efficient pollution level that creates a total gain of $\$900$ relative to their initial positions. Residents’ damage at that level is $\$700$, and the polluter’s benefit is $\$1{,}600$. If residents hold the right to clean air, give the acceptable transfer range and explain why changing the transfer within that range does not change total gains. (Answer: $\$700$ to $\$1{,}600$; the transfer reallocates gains between the parties, while total gains remain $\$1{,}600-\$700=\$900$.)

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