Cost-Effective Policy Design

Econ 2277

Prof. Richard L. Sweeney

(print this presentation)


  • 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

Uniform allocation will generally not be cost-effective


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:


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.

Graphical Intuition: Marginal Costs


Graphical Intuition: Total Costs


Graphical Intuition: Total Costs


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?


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

Graph of firm's total policy costs


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:



  • FOC: $6q_{1}=60$$\rightarrow$$q_{1}^{\star}=10$

Firm 2's problem:


  • 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:

  • $TC_{1}=3(q_{1})^{2}=3(10)^{2}=300$

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

$60 emission tax:

  • $TC_{1}=3(q_{1})^{2}+\tau(u_{1}-q_{1})=3(10)^{2}+60(50)=3300$

  • $TC_{2}=\frac{3}{2}(q_{2})^{2}+\tau(u_{2}-q_{2})=\frac{3}{2}(20)^{2}+60(20)=1800$

Why are these so different?

  • The tax penalizes emissions, C&C just mandates abatement
  • This has important dynamic implications

Incentives for technological change

  • Imagine some technology exists which can reduce firm abatement costs ($C(q)$) for a fee

  • Which regime provides a higher willingness to pay for that innovation?

  • Under C&C regulation, firms only care about meeting the regulation (costs just a function of reductions)

  • Under the tax, a firm also cares about reducing its tax burden.

  • Thus, even if the government has perfect information on cost curves, it will be more cost effective in the long run to use incentive based regulation.

Market based instruments provide greater incentives for innovation


Choosing the tax level

  • First best: set $\tau$ equal to marginal benefits (Pigouvian approach)

    • Remember net benefits where $MB=MC$
  • But we are rarely in a first best world

    • either don't know what the benefit function is or can't achieve the social optimum politically
  • So we often have some target $\bar{Q}$ in mind instead

  • How can we choose $\tau$ to achieve a total of $\bar{Q}$ reductions across all sources?

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)

  • To solve the previous problem, government still needed to know the aggregate MC curve

  • An alternative would be for the government to just set the pollution level directly by issuing pollution permits, and requiring firms to relinquish one permit for every unit of pollution they emit.

Firms respond by trading permits (and reducing their emissions)

  • If a firm's marginal cost is less than the market permit price, then it could make money by reducing its emissions by one unit and selling one of its emissions permits.

  • If its marginal cost is greater than the permit price, then it could reduce its costs by emitting an extra unit and purchasing a permit to cover those emissions.

  • So firm's buy and sell permits until their marginal cost of pollution control is equal to the market price of permits.

  • Since everyone faces the same permit price, the result is cost effectiveness!

Tradeable permits: producer's problem

  • Firms must provide a permit for every unit of emissions ($e$)

    • the price of permit is $P$
  • Firms are given $\bar{e}$ permits to start

  • Costs of the policy to the firm are:

    1. Abatement costs: $C(q)$

    2. The net cost of permits: $P(e-\bar{e})=P(u-q-\bar{e})$

Tradeable permits: solution

  • As with taxes, firms minimize TOTAL policy compliance costs.

  • Total cost = Abatement cost + Permit Cost

  • Permit costs are emissions less allocated permits


  • Solution (FOC): $MC(q)-P=0$

    • reduce pollution until it is cheaper to buy permits

Cap-and-trade vs. taxes

  • This is identical to the tax solution, with $P$ instead of $\tau$

    • So $P(\bar{Q})=\tau(\bar{Q})$
  • Imagine we pick some tax $\tau'$, and firms respond by reducting emissions by $Q' = Q(\tau')$

  • If we had instead issued a fixed number of pollution permits $E'$ such that $U - Q' = E'$, the resulting permit price would have been $P=\tau'$

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

Tradeable permits: Independence property

  • Solution (FOC): $MC(q)-P=0$

  • Note that the firm's first order condition does not depend on the number of permits it is allocated ($\bar{e}$)

  • Why not?

  • This is the independence property

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

  • we have assumed firms take it as given (ie firms are small)

  • if one firm has a large share of permits, we need to account for this ``market power'' in the permit market

Possible trades under cap-and-trade

  • Take the graphical example from last class.

  • Wanted 100 units of abatement from firms A and B.

    • Both firms initially produce 100 units of pollution
  • What happens if we give firm B 100 permits? Firm $A$ gets 0?


Steps for solving a cap and trade problem

  • Previous example suggests profitable trades will exist whenever $MC_{1}\ne MC_{2}$

  • Writ large this says that trading will continue until $MC$s are equal across all firms

    • This no profitable trades condition is identical to the cost effectiveness condition under taxation
  • We also know that the total amount of emissions is capped at $\bar{E}$

    • Permits capped: $e_{1}+e_{2}\leq\bar{E}$

    • implies $(u_{1}-q_{1})-(u_{2}-q_{2})=U-Q$

  • This gives us two equations and two unknowns, just like last time.

Revisiting our numerical example using permits

  • two firms that emit 100 of baseline pollution:

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

    • $C_{1}(q_{1})=3(q_{1})^{2}$

    • $C_{2}(q_{2})=\frac{3}{2}(q_{2})^{2}$

  • Government wants to reduce overall emissions by 30 units.

    • Set the cap equal the baseline emissions minus the desired emissions reductions.
    • $\bar{E}=U-\bar{Q}=100-30=70$

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!

  • $q_{1}^{\star}=10$ and $q_{2}^{\star}=20$

How much does this cost each firm?

  • Solution gives us implied emissions:

    • $e_{1}=60-10=50$ and $e_{2}=40-20=20$
  • We know permit price is equal to marginal costs:

    • $P=MC_{1}(q_{1}^{\star})=MC_{2}(q_{2}^{\star})=60$
  • Total costs = Abatements costs + Permit Costs

    • $TC(q^{\star}) = C(q^{\star})+P(e^{\star}-\bar{e})$
  • Firm 1: $TC_{1}=3(10)^{2}+60(50-35)=1200$

  • Firm 2: $TC_{2}=\frac{3}{2}(20)^{2}+60(20-35)=-300$

What if we gave firm 1 all the permits?

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

  • Apply cost effectiveness and cap conditions to get $q^{\star}$

    • $MC_{1}(q_{1}^{\star})=MC_{2}(q_{2}^{\star})$ and $q_{1}^{\star}+q_{2}^{\star}=30$
  • Allocated permits ($\bar{e}$) do not appear in these equations.

  • This implies that the solution is the same!

    • $q_{1}^{\star}=10$ and $q_{2}^{\star}=20$
  • However, the distributional effects are very different:

    • Firm 1: $TC_{1}=3(10)^{2}+60(50-70)=-900$
    • Firm 2: $TC_{2}=\frac{3}{2}(20)^{2}+60(20-0)=1800$

What if we auctioned all the permits?

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

  • Government agrees to sell 70 permits to highest bidder

    • $e^{\star}_1 + e^{\star}_2 = \bar{e}_G = 70$
  • Again, this doesn't impact $MC$ or $\bar{Q}$, so solution is the same

    • $q_{1}^{\star}=10$ and $q_{2}^{\star}=20$
  • Now firm costs are higher:

    • Firm 1: $TC_{1}=3(10)^{2}+60(50-0)=3300$
    • Firm 2: $TC_{2}=\frac{3}{2}(20)^{2}+60(20-0)=1800$
  • However the government also collected $70 \times 60=4200$ in auction revenue

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....

  • This is just the Coase Theorem!

    • As long as property rights are clearly defined and there are no transaction costs, the efficient allocation is obtained.
  • Note that formally we also need the emitters to all be small

    • ie price takers

Taxes vs Cap-and-Trade

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

  • there are no transaction costs
  • polluters are price takers (small)
  • and there is no uncertainty (covering this next week) then


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

  • we know that this implies an equilibrium permit price $P$ were $MC$
    is equal for all firms

  • if we simply set the tax $\tau$ equal to that permit price, firms will abate until $MC=P$

  • this will give us $\bar{Q}$ emission reductions across firms