### Cost-Effective Policy Design

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

[(print this presentation)](http://www.richard-sweeney.com/intro_env_econ/slides/CostEffectiveness.html?print-pdf)

# 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
• 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
• 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:

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:
1. $6q_{1}=3q_{2}$ $\rightarrow$ $2q_{1}=q_{2}$$\rightarrow$
2. $q_{1}+2q_{1}=30$ $\rightarrow$
• $q_{1}^{\star}=10$ and $q_{2}^{\star}=20$

### We can check to see if our solution is cost effective

• Marginal costs should be equal at the two firms:
• $MC_{1}(q_{1}^{\star})=6q_{1}^{\star}=6\cdot10=$ $60 •$MC_{2}(q_{2}^{\star})=3q_{2}^{\star}=3\cdot20=60 (yes!)
• And the pollution constraint should be satisfied:
• $q_{1}^{\star}+q_{2}^{\star}=10+20=30$ (yes!)
• Plug back in to cost curves to get total policy cost (900)
• $C_{1}(q_{1})=3(q_{1})^{2}=300$
• $C_{2}(q_{2})=\frac{3}{2}(q_{2})^{2}=600$

## Properties of Cost-Effective Allocations

• Marginal private costs are equal at every firm:
• $MC_{i}(q_{i})=MC_{j}(q_{j})\quad\forall i,j$.
• Intuition: arbitrage condition
• The marginal social cost of pollution reduction is equal to the marginal private cost of pollution reduction at any firm:
• $MC_{s}(Q^{\star})=MC_{i}(q_{i}^{\star})\quad\forall i$
• Intuition: since they’re all equal by construction, we only need to know one firm’s marginal cost
• The marginal social cost curve is equal to the horizontal sum of the marginal private cost curves.

## Properties of Cost-Effective Allocations

• The total social cost curve is always below (or equal to) the lowest total private cost curve:
• $C_{s}(Q)\leq C_{i}(Q)\quad\forall i,Q$
• Intuition: The social planner assigns emissions reductions to the lowest cost firms.
• So by spreading emissions out, the social cost can’t be higher that the lowest cost firm’s costs of achieving the same target

• Caveat to all of these properties:
• Some firms (with fixed costs or high marginal costs) might not have any reductions.
• Remember: firms can’t reduce more than baseline emissions.

## Getting to cost-effectiveness

• Controlling pollution in a cost-effective way sounds like a good idea……but how do we actually do it?
• Could use a traditional “command and control” approach
• Sets separate pollution control requirements for each individual source of emissions.
• If these requirements are chosen such that all firms have equal marginal costs, the policy will be cost effective

• In previous example, can mandate $q_{1}=10$ and $q_{2}=20$

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

• This is near-impossible. Why?

# 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.

• Each firm will respond by reducing its emissions until its marginal cost of pollution control is equal to the tax.

## Intuition for why taxes are cost effective

• When the firm’s marginal cost is less than the tax, then it is cheaper for the firm to reduce its emissions by one unit than to pay a tax on that unit.
• However, when its marginal cost is greater than the tax, then it is cheaper for the firm just to pay the tax and emit the unit of pollution.

• Remember that a firm’s marginal costs increase as it reduces its emissions

• The result: cost effectiveness!

## How Pigouvian taxes work

• Firm has baseline emissions $u$

• Government implements a tax $\tau$ for every unit of pollution emitted

• If firm does nothing to respond, it’s costs are $u\times\tau$

• But firms can respond by reducing emissions at total cost $C(q$)

• where $q$ is the quantity of emissions reduction

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

## Pigouvian taxes: producer’s problem

Costs of the policy to the firm are:

• Abatement costs: $C(q)$
• PLUS taxes on emissions not abated: $\tau(u-q)$

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

## Cost-effectiveness of Pigouvian taxes

• Just showed: Each firm responds by reducing its emissions until its marginal cost of pollution control is equal to the tax.

• If we charge every firm the same tax, this will ensure that the marginal cost of abatement is the same for all firms

• Necessary condition for cost-effectiveness

• Note that the government does not need to know MC1 and MC2!

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: $min_{q}TC_{1}=C_{1}(q_{1})+\tau(u_{1}-q_{1})$ $min_{q}TC_{1}=3(q_{1})^{2}+60(60-q_{1})$ • FOC:$6q_{1}=60$\rightarrow$q_{1}^{\star}=10$Firm 2’s problem: $min_{q}TC_{2}=\frac{3}{2}(q_{2})^{2}+60(40-q_{2})$ • FOC:$3q_{2}=60$\rightarrow$q_{2}^{\star}=20$Same solution as above! ### How does each firm’s total costs compare across policies? • We just showed that we can achieve cost-effectiveness by either: 1. requiring$q_{1}^{\star}=10$and$q_{2}^{\star}=20$2. taxing firms at$60 per unit of emissions $e$

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

Command and control:

• $TC_{1}=3(q_{1})^{2}=3(10)^{2}=300$

• $TC_{2}=\frac{3}{2}(q_{2})^{2}=\frac{3}{2}(20)^{2}=600$