Want to max net benefits (Kaldor-Hicks)
But other criteria often matter in the real world
Important design considerations
Are damages at a point in time $t$ driven primarily by current emissions or earlier emissions?
What is the degree of mixing of the pollutant?
Traditional approach is ``command and control''
Traditional economics approach is Pigovian
Coasian approach is create property rights
How should we choose between these? What are the implications?
Practical challenges:
A more modest policy criterion: cost effectiveness
Useful even if policy sets inefficient pollution target
Imagine there are two firms, 1 and 2
Firm 1 has a lower cost of pollution reduction than firm 2
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 ($q_2$)?
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.
Let $U$ denote aggregate baseline ("unconstrained") emissions
Let $Q$ denote aggregate emission reductions under the policy
Let $E$ denote resulting policy emissions
What is the cost-effective way to reduce pollution by $\bar{Q}$ units?
Constrained minimization Lagrangian:
$\partial L /\partial q_1:$ $C'_1(q_1) - \lambda = 0$
$\partial L /\partial q_2:$ $C'_2(q_2) - \lambda = 0$
$\partial L /\partial \lambda :$ $\bar{Q} -q_{1}-q_{2}=0$
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
Two firms that emit 100 units of baseline pollution
Total costs of abatement for each firm:
Government would like to reduce overall emissions by 30 units.
Find $q_{1}$ and $q_{2}$ that minimize total cost of reducing emissions by 30.
Know the cost effective solution must satisfy two conditions:
We have two equations and two variables. Simplify and solve:
$q_{1}^{\star}=10$ and $q_{2}^{\star}=20$
Marginal costs should be equal at the two firms:
And the pollution constraint should be satisfied:
Plug back in to cost curves to get total policy cost (900)
Marginal private costs are equal at every firm:
The marginal social cost of pollution reduction is equal to the marginal private cost of pollution reduction at any firm:
The marginal social cost curve is equal to the horizontal sum of the marginal private cost curves.
The total social cost curve is always below (or equal to)
the lowest total private cost curve:
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:
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
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 $q_{2}=20$
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.
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!
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 $C(q$)
Question: How much should the firm abate?
[draw graph]
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
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
Key Result:
Any policy that charges firms the same tax will be cost effective
Firm 1's problem:
Firm 2's problem:
Same solution as above!
We just showed that we can achieve cost-effectiveness by either:
requiring $q_{1}^{\star}=10$ and $q_{2}^{\star}=20$
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?
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.
First best: set $\tau$ equal to marginal benefits (Pigouvian approach)
But we are rarely in a first best world
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?
Steps:
Solve for $q_{i}$ in terms of $MC_{i}$:
Calculate aggregate emission reductions $q$ as a function of aggregate marginal costs $MC$.
Under a cost effective allocation, we know that marginal costs are
equal across sources. So we'll drop the subscripts.
Now solve for aggregate marginal costs as a function of $Q$:
This is the aggregate cost curve for pollution reductions
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.
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!
Firms must provide a permit for every unit of emissions ($e$)
Firms are given $\bar{e}$ permits to start
Costs of the policy to the firm are:
Abatement costs: $C(q)$
The net cost of permits: $P(e-\bar{e})=P(u-q-\bar{e})$
As with taxes, firms minimize TOTAL policy compliance costs.
Total cost = Abatement cost + Permit Cost
Permit costs are emissions less allocated permits
Solution (FOC): $MC(q)-P=0$
This is identical to the tax solution, with $P$ instead of $\tau$
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)
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
Take the graphical example from last class.
Wanted 100 units of abatement from firms A and B.
What happens if we give firm B 100 permits? Firm $A$ gets 0?
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
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.
two firms that emit 100 of baseline pollution:
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.
Example allocation 1: $\bar{e}_{1}=35$ ; $\bar{e}_2=35$
To solve C&T problem, two conditions:
Cost-effectiveness
policy constraint (cap) condition
Since these are the same as the tax case, solution is the same!
Solution gives us implied emissions:
We know permit price is equal to marginal costs:
Total costs = Abatements costs + Permit Costs
Firm 1: $TC_{1}=3(10)^{2}+60(50-35)=1200$
Firm 2: $TC_{2}=\frac{3}{2}(20)^{2}+60(20-35)=-300$
Example allocation 2: $\bar{e}_{1}=70$ ; $\bar{e}_2=0$
Apply cost effectiveness and cap conditions to get $q^{\star}$
Allocated permits ($\bar{e}$) do not appear in these equations.
This implies that the solution is the same!
However, the distributional effects are very different:
Example allocation 3: $\bar{e}_{1}=0$ ; $\bar{e}_2=0$
Government agrees to sell 70 permits to highest bidder
Again, this doesn't impact $MC$ or $\bar{Q}$, so solution is the same
Now firm costs are higher:
However the government also collected $70 \times 60=4200$ in auction revenue
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!
Note that formally we also need the emitters to all be small
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 $\bar{Q}:$
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