Cost-Effective Policy Design

[Econ 2277](http://www.richard-sweeney.com/intro_env_econ)
[Prof. Richard L. Sweeney](http://www.richard-sweeney.com/)

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Outline


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?

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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

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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

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Graphical Intuition: Total Costs

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Graphical Intuition: Total Costs

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Getting to cost-effectiveness


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


Taxes


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


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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


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Choosing the tax level


Answer requires the social aggregate cost curve

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Need to aggregate the private costs curves horizontally

Steps:

  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

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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:


Implications:

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

Required Readings

KO Ch 5, 9, 10