Lecture 9
Under the conditions of the First Welfare Theorem, competitive markets deliver Pareto-efficient outcomes. But those conditions are demanding: complete markets, well-defined property rights, price-taking behavior, and no spillover effects between agents.
We studied one source of market failure: asymmetric information. In this lecture we study two more classic sources of market failure:
In both cases, private incentives diverge from social incentives and there is a potential role for policy.
An externality arises when an economic actor does not face the correct price for taking a specific action. The correct price is the marginal social cost of that action.
When markets work properly, they align private costs and benefits with social costs and benefits. When private benefits differ from social benefits (either higher or lower), externalities result.
Put differently: externalities occur when one agent’s action directly affects another’s welfare, and this effect is not reflected in market prices.
Traffic congestion: When I take the highway, I increase congestion for other drivers. But my decision to drive is based on my private cost (time, fuel) and does not account for the additional delay I impose on others.
Disease transmission: When deciding whether to vaccinate, individuals are most likely to weigh only private costs and benefits, ignoring the positive externality of reduced disease spread to others.
Pollution: Because clean air is not priced, we pay essentially no cost to pollute. The marginal cost of our polluting actions do not incorporate the social cost of additional pollution, so we pollute more than is socially optimal.
Until the publication of Ronald Coase’s 1960 paper, most economists would have answered no.
Coase made them reconsider that view. Coase gave the example of a doctor and a baker who share an office building. The baker’s loud machinery disturbed the doctor’s medical practice.
The standard reasoning (at the time): the baker should compensate the doctor for the harm, since the baker was “causing” the externality.
But Coase pointed out a re-framing: suppose the doctor sets up his office after the baker is already there, and then demands the baker shut down. Who is responsible for the externality now?
One can argue that the doctor is creating an externality by requiring the baker to bake in silence. The baker’s noise is an “input” into his production of baked goods.
From a legal point of view, the answer may be clear. From an economic point of view, the answer is indeterminate based only on this information. We need to know the costs of the different solutions.
Suppose:
Economic efficiency demands the lowest-cost solution. The baker should buy quieter machinery.
But does this mean the baker should pay to abate? Not necessarily.
Scenario 1: The town council assigns the doctor the right to control noise. The baker spends $50 for quieter machinery.
Scenario 2: The town council assigns the baker the right to make noise. Will the doctor spend $100 to soundproof? If the parties can negotiate, they should arrange for the doctor to pay the baker $50 for quieter machinery instead.
In both cases, the efficient outcome occurs: quieter machinery is purchased. The assignment of rights determines only who pays, not what happens.
If (1) property rights are complete (one party clearly owns the relevant right) and (2) parties can negotiate costlessly, then the parties will always negotiate an efficient solution to the externality.
The law determines who pays the cost, but the outcome is the same.
Note the parallel with the Welfare Theorems: efficiency and distribution are separable problems.
The Coase theorem implies the market will solve externalities unless:
The theorem is often misinterpreted to suggest markets will solve all externalities. This is not true. The market can potentially solve externalities if property rights are clearly assigned and negotiation is feasible.
And in many cases, Coasian bargaining is infeasible due to large numbers of affected parties, high transaction costs, or asymmetric information. In those cases, government intervention may be necessary to correct the externality.
Consider two oil refineries that both produce fuel.
The amount of smog per gallon differs at the two plants:
\[s_1 = y_1^2, \qquad s_2 = \frac{1}{2}y_2^2\]
where \(y_1, y_2\) are gallons of fuel produced at each plant.
Plant 2 pollutes only \(\frac{1}{2}\) as much as Plant 1 for a given level of production.
Without any legal framework for resolving the externality, each firm solves:
\[\max_{y_1} \pi_1 = y_1 \cdot (3-2) \;\; s.t. \;\; y_1 \leq 200\] \[\max_{y_2} \pi_2 = y_2 \cdot (3-2) \;\; s.t. \;\; y_2 \leq 200\]
Result: \(y_1^* = y_2^* = 200\). Each firm produces at capacity.
Pollution: \(s_1 = 40{,}000\), \(s_2 = 20{,}000\) cubic feet. The externality costs are $400 and $200 respectively.
To find the efficient level, equate marginal social benefit to marginal social cost:
\[MB_s = MC_s\]
The social benefit of a gallon of fuel is $3 (from the demand curve). The social cost is $2 in inputs plus the pollution damage.
At the margin, no plant should produce more than $1 of environmental damage per gallon. So no plant should produce more than 100 cubic feet of smog per gallon.
If each plant faced private plus social costs:
\[\max_{y_1} \pi_1 = y_1(3-2) - 0.01 \cdot y_1^2 \;\;\; s.t. \;\; y_1 \leq 200\]
\[\max_{y_2} \pi_2 = y_2(3-2) - 0.01 \cdot \tfrac{1}{2}y_2^2 \;\;\; s.t. \;\; y_2 \leq 200\]
Efficient output: \(y_1^{**} = 50, \;\; y_2^{**} = 100\)
Plant 1 produces less because it is the dirtier plant. Optimal pollution is not zero. It reflects the tradeoff between production benefits and environmental harm.
What regulation can achieve this outcome? We will discuss three approaches: command and control, Pigouvian taxes, and cap-and-trade.
The traditional approach: set numerical quantity limits on polluting activities.
If the regulator knows each plant’s cost structure, they could mandate: “Plant 1 may produce 50 gallons, Plant 2 may produce 100 gallons.” This achieves the efficient outcome.
This kind of regulation is clumsy:
If the law cannot differentiate across plants and must impose a uniform cap, further inefficiencies result. (The optimal uniform cap is not simply 75 gallons; work this out as an exercise.)
Despite these weaknesses, command and control is the most common approach to regulating externalities.
An alternative: use the price system to internalize the externality.
If we set a tax of \(t = \$0.01\) per cubic foot of smog, each plant’s problem becomes:
\[\max_{y_1} \pi = y_1(3-2) - t \cdot y_1^2 \implies y_1^p = 50\] \[\max_{y_2} \pi = y_2(3-2) - t \cdot \tfrac{1}{2}y_2^2 \implies y_2^p = 100\]
The tax achieves the efficient result with less information: we don’t need to know firms’ production functions, only the marginal social damage of pollution.
Advantage: Each firm optimally chooses its own level of pollution given the tax. No need to write a separate law for each plant.
Risk: If marginal damage varies with the quantity of pollution (e.g., pollution above a threshold causes mass extinction), setting the tax slightly wrong could be catastrophic.
When marginal damage is constant, the Pigouvian tax is straightforward. When it varies, setting the right tax schedule becomes much harder.
The Pigouvian tax doesn’t fully use the Coase theorem. The state sets the price, not the market. Can we do better by assigning property rights to pollution?
The optimal total pollution is \(50^2 + \frac{1}{2}(100^2) = 7{,}500\) cubic feet. The government could issue 7,500 permits, each allowing 1 cubic foot of smog. Permits can be traded.
Plant 2 (the more efficient refinery) receives all 7,500 permits. It could:
Plant 2 cannot do better than option 2. The market leads it to the efficient allocation.
Now Plant 1 (the dirtier refinery) receives all 7,500 permits. It could:
Again, the efficient allocation emerges. Regardless of who receives the permits, the key economic outcome (fuel produced, pollution produced, and the allocation across plants) is identical.
Only which plant makes the profits (a transfer among plant owners).
This is the power of the Coase theorem in action. By assigning property rights to pollution, the government allows the market to correct the externality.
The initial allocation of permits is a major political question, but it has no effect on economic efficiency.
| Feature | Command & Control | Pigouvian Tax | Cap & Trade |
|---|---|---|---|
| Sets | Quantity for each firm | Price of pollution | Total quantity |
| Information needed | Each firm’s costs | Marginal damage | Total efficient pollution |
| Self-correcting? | No | Yes | Yes |
| Firms optimize? | No | Yes | Yes |
The SO\(_2\) cap-and-trade program in the U.S. dramatically exceeded expectations: it reduced emissions at far lower cost than anticipated, because the market revealed that firms’ true abatement costs were lower than what they had told regulators under command-and-control.
Cap and trade may not work well if the regulator cares about which plant does the polluting.
If all the low-cost polluters happen to be concentrated in one geographic area, cap and trade could lead to localized pollution hotspots, even if total pollution is efficient.
Goods can be classified along two dimensions:
| Excludable | Non-excludable | |
|---|---|---|
| Rival | Private good | Common good |
| Non-rival | Club good | Public good |
The social planner chooses \(X_i\) for each household and \(G\) to maximize a weighted sum of utilities subject to the resource constraint:
\[\max_{X_1, \ldots, X_N, G} \sum_{i=1}^N \beta_i U_i(X_i, G) \quad s.t. \quad \sum_{i=1}^N X_i + G = \sum_{i=1}^N Y_i\]
First-order conditions: \[\begin{align*} \beta_i \frac{\partial U_i}{\partial X_i} &= \lambda \quad \text{for each } i \\ \sum_{i=1}^N \beta_i \frac{\partial U_i}{\partial G} &= \lambda \end{align*}\]
Plugging \(\beta_i = \lambda / \frac{\partial U_i}{\partial X_i}\) from the first condition into the second condition gives:
\[ \sum_{i=1}^N \underbrace{\ \frac{\partial U_i / \partial G}{\partial U_i / \partial X_i}}_{\text{MRS}_i} = 1 \]
This is the Samuelson condition for optimal provision of a public good.
Remember: \(MRS_i\) is the amount of private good \(X\) that household \(i\) is willing to give up for one more unit of public good \(G\).
The Samuelson condition says that the total willingness to pay for an additional unit of \(G\) across all households must equal the relative cost of providing that unit (which is 1 in this case, since \(G\) uses up one unit of resources that could have been used for \(X\)).
With no intervention, individuals choose their own contributions to the public good, taking others’ contributions as given (Nash equilibrium). Each household solves:
\[\max_{X_i, g_i} U_i(X_i, g_1+g_2+...+g_N) \quad s.t. \quad X_i + g_i = Y_i\]
Nash equilibrium outcome will satisfy: \[MRS_i = \frac{\partial U_i / \partial G}{\partial U_i / \partial X_i} = 1\]
Since \(MRS_i=1\), total willingness to pay is \(\sum_{i=1}^N MRS_i = N>1\). This means the Nash equilibrium will under-provide the public good compared to the social optimum. This is the free-rider problem.
Each roommate takes the other’s contribution as given and chooses their own contribution to maximize utility. For roommate 1: \[ \max_{X_1, g_1} \; X_1 \cdot (g_1 + g_2) \quad s.t. \quad X_1 + g_1 = 100 \]
First-order conditions: (1) \(g_1 + g_2 = \lambda\), (2) \(X_1 = \lambda\), (3) \(X_1 + g_1 = 100\).
Solving these gives us the best response function for roommate 1: \[ g_1^* = 50 - \frac{g_2}{2} \]
By symmetry, \(g_2^* = 50 - \frac{g_1}{2}\). So Nash equilibrium contributions are given by: \[ g^*_1 = g^*_2 = 33.33 \]
The planner chooses \(X_1, X_2, G\) to maximize total welfare:
\[ \max_{X_1, X_2, G} \; X_1 G + X_2 G \quad s.t. \quad X_1 + X_2 + G = 200 \]
Note, this is equivalent to maximizing \(XG\) subject to \(X + G = 200\). The solution is \(X = 100\), \(G = 100\).
Intuition: The planner internalizes the fact that one unit of \(G\) benefits both roommates, so the efficient level of \(G\) is higher than in the Nash equilibrium.
Free-riding occurs because each roommate benefits from the public good regardless of their own contribution, which leads to under-provision of the public good compared to the social optimum.
Are individuals really free-riders?
Early lab experiments testing free-rider behavior (Marwell & Ames, 1981)
Key question: How can we solve the underprovision problem?
Government provision: Set \(G\) directly, funded by mandatory taxes. Solves free-riding but requires knowing preferences to choose \(G\) efficiently.
Subsidizing private contributions: Reduce the private cost of contributing (e.g., tax deductions for charitable giving). Narrows the gap but may not eliminate underprovision.
Lindahl pricing: Charge each person a personalized price equal to their \(MRS_i\). Achieves efficiency in theory, but people have incentives to misrepresent their willingness to pay.
All three face the same fundamental challenge: the government needs information about individual preferences that people have incentives to hide.