Cost-Effective Policy Design

[Econ 2277](
[Prof. Richard L. Sweeney](

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Policy design criteria

Categorizing pollutants: Temporal dimension

Are damages at a point in time $t$ driven primarily by current emissions or earlier emissions?

Categorizing pollutants: Spatial dimension

What is the degree of mixing of the pollutant?


How should we achieve our environmental objectives?

Cost Effectiveness

Policy criterion: Cost-effectiveness

Finding a cost-effective allocation graphically

How many reductions (q) should come from firm 1 ($q_1$) and how many should come from firm 2 ($q_2$)?

Uniform allocation will generally not be cost-effective


Takeaway from graph

Can also show this analytically

Algebraic solution

What is the cost-effective way to reduce pollution by $\bar{Q}$ units?

Constrained minimization Lagrangian:

First order conditions:

$\partial L /\partial q_1:$ $C’_1(q_1) - \lambda = 0 $

$\partial L /\partial q_2:$ $C’_2(q_2) - \lambda = 0 $

$\partial L /\partial \lambda :$ $\bar{Q} -q_{1}-q_{2}=0 $

Cost-effective solution must satisfy 2 equations

Eq 1: Cost-effectiveness (efficiency) Constraint

$q_{1}^{\star}$ and $q_{1}^{\star}$ equate marginal costs:

Eq 2: Policy Constraint

Total reductions hit the policy target

Numerical Example

Numeric example: solution

Find $q_{1}$ and $q_{2}$ that minimize total cost of reducing emissions by 30.

We can check to see if our solution is cost effective

Properties of Cost-Effective Allocations

Properties of Cost-Effective Allocations

Graphical Intuition: Marginal Costs


Graphical Intuition: Total Costs


Graphical Intuition: Total Costs


Getting to cost-effectiveness

Problem: The government needs to know the exact details of every source’s marginal cost curve


An alternative approach is to use incentives

This is a “market based” approach

For example, rather than prescribing how much every firm must abate, instead make firms pay a tax for every unit of pollution they emit.

Intuition for why taxes are cost effective

How Pigouvian taxes work

Question: How much should the firm abate? [draw graph]

Pigouvian taxes: producer’s problem

Costs of the policy to the firm are:

Firm’s problem: minimize TOTAL policy compliance costs:

First order condition:

Implication: the firm want’s to reduce pollution up until the point where it is cheaper to pay the tax

Graph of firm’s total policy costs


Cost-effectiveness of Pigouvian taxes

Key Result: Any policy that charges firms the same tax will be cost effective

Numeric example: Impose a $60 tax on the firms above

Firm 1’s problem:

Firm 2’s problem:

Same solution as above!

How does each firm’s total costs compare across policies?

What are the firm’s total costs under these approaches?

Command and control:

$60 emission tax:

Why are these so different?

Incentives for technological change

Market based instruments provide greater incentives for innovation


Choosing the tax level

Answer requires the social aggregate cost curve


Need to aggregate the private costs curves horizontally


  1. Solve for $q_{i}$ in terms of $MC_{i}$:
    • $MC_{1}=6q_{1}$ $\rightarrow$ $q_{1}=\frac{1}{6}MC_{1}$
    • $MC_{2}=3q_{2}$ $\rightarrow$ $q_{2}=\frac{1}{3}MC_{2}$
  2. Calculate aggregate emission reductions $q$ as a function of aggregate marginal costs $MC$.
    • $Q=q_{1}+q_{2}=\frac{1}{6}MC_{1}+\frac{1}{3}MC_{2}$

  1. Under a cost effective allocation, we know that marginal costs are equal across sources. So we’ll drop the subscripts.

    • $Q=\frac{1}{6}MC+\frac{1}{3}MC=\frac{1}{2}MC$
  2. Now solve for aggregate marginal costs as a function of $Q$:

    • $Q=\frac{1}{2}MC$ $\rightarrow$ $MC(Q)=2Q$

This is the aggregate cost curve for pollution reductions

Cap-and-trade (tradable permits)

Firms respond by trading permits (and reducing their emissions)

Tradeable permits: producer’s problem

Tradeable permits: solution

Cap-and-trade vs. taxes

Important implication: Little economic distinction between tax and equivalently strict cap-and-trade (in this simple model)

Tradeable permits: Independence property

Note: $P$ is really a function of $\bar{E}$

Possible trades under cap-and-trade


Steps for solving a cap and trade problem

Revisiting our numerical example using permits

How should the government allocate permits?

Example allocation 1: $\bar{e}_{1}=35$ ; $\bar{e}_2=35$

To solve C\&T problem, two conditions:

  1. Cost-effectiveness

    • $MC_{1}(q_{1}^{\star})=MC_{2}(q_{2}^{\star})$
  2. policy constraint (cap) condition

    • $e_{1}^{\star}+e_{2}^{\star}= 70 = (u_1 - q_1^{\star}) + (u_2 - q_2^{\star}) $
    • $(u_1 + u_2) - 70 = 30 = (q_1^{\star} + q_{2}^{\star})=30$

Since these are the same as the tax case, solution is the same!

How much does this cost each firm?

What if we gave firm 1 all the permits?

Example allocation 2: $\bar{e}_{1}=70$ ; $\bar{e}_2=0$

What if we auctioned all the permits?

Example allocation 3: $\bar{e}_{1}=0$ ; $\bar{e}_2=0$

This is a key result

Question: How should the government allocate permits ($\bar{e}{1}$ and $\bar{e}{2}$) to the two firms?

Answer: It doesn’t matter for cost effectiveness - as long as we assume zero transaction costs….

Taxes vs Cap-and-Trade

A cap-and-trade program with full auctioning of permits is fully equivalent to a tax … As long as:


For any target emission reductions $\bar{Q}:$

Required Readings

KO Ch 5, 9, 10