### Cost-Effective Policy Design

Econ 2277

Prof. Richard L. Sweeney

(print this presentation)

# Outline

• Policy design criteria
• Cost effectiveness
• Getting to cost effectiveness

## Policy design criteria

• Want to max net benefits (Kaldor-Hicks)

• But other criteria often matter in the real world

• Distributional equity
• Politically feasible
• Practically implementable and enforceable
• Important design considerations

• Not all pollutants are the same
• Cost-effectiveness
• Dynamic efficiency/ technological change
• Performance under uncertainty

## Categorizing pollutants: Temporal dimension

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

• $E_{t}$= emissions of pollutant
• $D_{t}$= decay of pollutant
• $S_{t}$= stock of pollutant

$$S_{t}=S_{0}+\sum_{i=1}^{t}E_{i}-\sum_{i=1}^{t}D_{i}$$

• Pure stock pollutant: $D_{t}=0$
• climate change
• Pure flow pollutant: $D_{t}=E_{t}$
• noise

## Categorizing pollutants: Spatial dimension

What is the degree of mixing of the pollutant?

## How should we achieve our environmental objectives?

• Traditional approach is command and control''

• Traditional economics approach is Pigovian

• set taxes equal to marginal cost, then let firm max profits
• Coasian approach is create property rights

• fix amount of pollution but allow firms to trade this right
• How should we choose between these? What are the implications?

# Cost Effectiveness

## Policy criterion: Cost-effectiveness

• Economic efficiency $\rightarrow$ Kaldor-Hicks $\rightarrow$ benefit-cost analysis

$$max_{\{q_{i}\}}\sum[B_{i}(q_{i})-C_{i}(q_{i})]$$

• Practical challenges:

• May not know the benefit function
• Efficient level may not be politically implementable
• A more modest policy criterion: cost effectiveness

• Does the policy achieve it's goal in the least costly way?
$$min_{\{q_{i}\}}\sum[C_{i}(q_{i})]\hspace{1em}s.t.\sum q_{i}\ge\bar{Q}$$
• Useful even if policy sets inefficient pollution target

## Finding a cost-effective allocation graphically

• Imagine there are two firms, 1 and 2

• Both produce 100 units of pollution
• Firm 1 has a lower cost of pollution reduction than firm 2

• Say $MC_2 = 1.5 * MC_1$
• So at $MC_1(50) = 100$ ; $MC_2(50)= 150$
• Government wants to reduce pollution by 100 units total

• Assume the pollutant is a uniformly mixing, flow pollutant

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

• First graph MCs
• Then add second axis to impose 100 unit target

## Takeaway from graph

• Should be able to show the total costs under any allocation of effort

• Unless marginal costs are equal, arbitrage lowers total costs.

• Logic is the arbitrage argument from before: if the the marginal costs of abatement at the two firms weren't equal, then we could shift one unit of abatement from the higher marginal cost firm to the lower marginal cost firm. This would lower the overall cost without changing the quantity of pollution reduced.

## Can also show this analytically

• Let $U$ denote aggregate baseline ("unconstrained") emissions

• $u_{i}$ are the baseline emissions at firm $i$
• Let $Q$ denote aggregate emission reductions under the policy

• $q_{i}$ are the reductions are firm $i$
• Let $E$ denote resulting policy emissions

• $E=U-Q$
• $e_{i}=u_{i}-q_{i}$

## Algebraic solution

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

• ie what choice of $q_{1}$ and $q_{2}$ will solve the problem:

\begin{aligned}\min_{q_{1},q_{2}} & [C_{1}(q_{1})+C_{2}(q_{2})]\\ & \qquad s.t.\qquad q_{1}+q_{2}\ge\bar{Q} \end{aligned}

Constrained minimization Lagrangian:

$$\min_{q_{1},q_{2,}\lambda} L= [ C_{1}(q_{1})+C_{2}(q_{2})+\lambda(\bar{Q} -q_{1}-q_{2}) ]$$

### 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:

$$MC_{1}(q_{2}^{\star})=\lambda=MC_{2}(q_{2}^{\star})$$

Eq 2: Policy Constraint

Total reductions hit the policy target

$$\bar{Q} = q_{1}+q_{2}$$

## Numerical Example

• Two firms that emit 100 units of baseline pollution

• $u_{1}=60$
• $u_{2}=40$
• Total costs of abatement for each firm:

• $C_{1}(q_{1})=3(q_{1})^{2}$
• $C_{2}(q_{2})=\frac{3}{2}(q_{2})^{2}$
• where $q$ are units of emission reduction
• Government would like to reduce overall emissions by 30 units.

• What is the cost-effective allocation of emissions reductions to achieve this goal?

## Numeric example: solution

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

• Know the cost effective solution must satisfy two conditions:

1. Cost effectiveness: $MC_{1}(q_{1}^{\star})=MC_{2}(q_{2}^{\star})$
2. Policy target: $q_{1}^{\star}+q_{2}^{\star}=30$
• We have two equations and two variables. Simplify and solve: