Lecture 06

.title[
# Lecture 06
]
.subtitle[
## Tradable Permits
]
.author[
### Ivan Rudik
]
.date[
### AEM 4510
]

---

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---

# Roadmap

1. How do tradable permit systems work in theory and in the real world?
2. What happens under a tradable permit system?

---

---

# Tradable permits

How do tradable permit systems work?<sup>1</sup>

First, recall a regular emission standard: we set `$\bar{E}$` at the point where MAC = MD

This is easy with one firm, but what if we have several, or hundreds?

---

# Optimal policy with multiple firms

![taxes](06-slides-permits_files/figure-html/multifirm-1.png)
]

Firm #2 is 'dirty': has higher MAC

Firm #1 is 'clean': has lower MAC

If we use a regular emission standard: it has to be firm-specific!

Mandate `$E^*_1$` for 1 and `$E^*_2$` for 2

This requires .hi[a lot] of info and political capital on behalf of the regulator

]

---

# Optimal policy with multiple firms

![taxes](06-slides-permits_files/figure-html/multi-tax-1.png)
]

Regulating multiple heterogeneous firms with a tax can be easy:

If MD is constant, then since firms select MAC = `$\tau$`, as long as we set `$\tau = MD$`, we can achieve the efficient outcome (MAC = MD) without knowing anything about the firms!

]

---

# Optimal policy with multiple firms

Taxes also achieve the .hi[cost-effective] outcome: achieving a given emission level at least-cost

Let's see why

Suppose we want to minimize the total cost of achieving emission level `$\bar{E}$` by abating across two different sources, plant 1 and plant 2

The plants have abatement cost functions: `$C_1(E_1)$` and `$C_2(E_2)$`

Write down the regulator's problem

---

# Optimal policy with multiple firms

`$$\min_{E_1,E_2} C_1(E_1) + C_2(E_2) \,\,\,\, \text{subject to:} \, E_1 + E_2 = \bar{E}$$`

Solve the constraint for `$E_2 = \bar{E} - E_1$` so we have a simpler problem:
`$$\min_{E_1} C_1(E_1) + C_2(\bar{E} - E_1)$$`

Take the first-order condition to find what is necessary for a cost minimum:
`$$C'_1(E_1) + C'_2(\bar{E} - E_1) \times (-1) = 0$$`

---

# Optimal policy with multiple firms

This gives us:
`$$\underbrace{-C'_1(E_1)}_{\text{MAC}_1} = \underbrace{-C'_2(\overbrace{\bar{E} - E_1}^{E_2})}_{\text{MAC}_2}$$`

The marginal abatement costs across the sources must be equal at the cost-effective pollution level

This is called the .hi-blue[equimarginal principle]

---

# Optimal policy with multiple firms

Taxes .hi[always] achieve the equimarginal principle and get us the given amount of emission reductions at least-cost

Why?

We know firms optimally select MAC equal to the emission tax

This means all firms' MACs are equal!

Even if we don't set the tax equal to MD, whatever emission reduction we get will be as cheap as possible

---

# Optimal policy with multiple firms

The big problem is political feasibility

Firms resist taxation because they have to pay a fine for each unit of emissions

Tradable permit systems are a way to make emission standards flexible enough to handle heterogeneous firms

]

---

# Tradable permit systems

So how do these systems make standards more flexible?

They allow firms to trade their emission allowances

E.g. if firms are restricted to `$\bar{E}_1$` and `$\bar{E}_2$`, we can allow the firms to trade

If firm 1 sells an allowance/permit to firm 2, their new restrictions are: `$\bar{E}_1 - 1$` and `$\bar{E}_2 + 1$`

---

# Tradable permit systems: example

The US Acid Rain Program is the classic example

Permit = license to create 1 ton of SO2

Phase I (1995-2000):  
- 6.3 million permits issued per year
- affected 263 generating units at 110 dirtiest power plants

Phase II (2000+):  
- 9 million permits issued per year
- affects all power plants over some minimum size

---

# Tradable permit systems: example

---

# Tradable permit systems: example

---

# Tradable permit systems: example

---

# Tradable permit systems: example

---

# Tradable permit systems: example

---

# Tradable permits: graphical

![permits](06-slides-permits_files/figure-html/permits-1.png)
]

Suppose we want to limit to `$\bar{E}$` total emissions so each firm gets `$\bar{E}/2$` permits, but cant trade them

.hi-red[It also can't be cost-effective:] it doesn't minimize the cost of achieving `$\bar{E}$` total emissions

]

---

# Tradable permits: cost-effectiveness

For cost-effectiveness, we need total costs to be minimized for achieving a given level of emissions:
`$$\min_{E_1, E_2} C_1(E_1) + C_2(E_2) \,\,\, \text{subject to: } E_1 + E_2 = \bar{E}$$`

This is the same problem as:
`$$\min_{E_1, E_2} C_1(E_1) + C_2(\bar{E} - E_1)$$`

which has a solution where:
`$$-C_1'(E_1^*) = -C_2'(\bar{E} - E^*_1)$$`

---

# Tradable permits: cost-effectiveness

Cost-effectiveness requires:
`$$-C_1'(E_1^*) = -C_2'(\bar{E} - E^*_1) \leftrightarrow MAC_1 = MAC_2$$`

That marginal abatement costs are equal across all emitters

If firms have different MAC curves, giving them the same amount of emissions/permits cannot be optimal

How do we fix this?

Let them trade the permits

---

# Tradable permits: graphical

![permits](06-slides-permits_files/figure-html/permits-dwl0-1.png)
]

We can reduce costs by increasing abatement at which firm, and decreasing abatement at which firm?

]

---

# Tradable permits: graphical

![permits](06-slides-permits_files/figure-html/permits-dwl1-1.png)
]

We can reduce costs by increasing emissions at high MAC firm 2 and decreasing emissions at low MAC firm 1 until they are equal

]

---

# Tradable permits: graphical

![permits](06-slides-permits_files/figure-html/permits-dwl-1.png)
]

We can reduce costs by increasing emissions at high MAC firm 2 and decreasing emissions at low MAC firm 1 until they are equal

This allows us to recover DWL equal to the red area

The red area is the difference in areas under MAC2 and MAC1 over the range of emissions changes

]

---

# Tradable permits: graphical

![permits](06-slides-permits_files/figure-html/permits-reallo-1.png)
]

We want to equalize MACs for cost-effectiveness, but does the permit market cause this to happen?

]

---

# Tradable permits: graphical

We want to equalize MACs for cost-effectiveness, but does the permit market cause this to happen?

Firm 2 is willing to pay a price up to the blue point (1.25) to be able to emit 1 more unit

Firm 1 can abate 1 more unit at cost equal to the red point (0.83)

]

---

# Tradable permits: graphical

Firm 2 can buy the right to emit 1 unit of pollution from firm 1 for anywhere between 1.25 and 0.83 and .hi[both will be better off] [very Coasean!]

These trades can be done until the MACs are equal at a value of 1

This would be the prevailing permit price in a tradable permit system

]

---

# Tradable permits: graphical

An alternative way to think about it:

the prevailing permit price is the MC of freeing up one more unit, the MAC of the selling firm

or it is the MB of freeing up one more unit (avoided MAC), the MAC of the buying firm

]

---

# Tradable permits: firm

We can also see this result mathematically

Suppose there is a permit price `$p$` in the competitive tradable permit market

Firms are price-takers in the permit market

Let's set up the firm problem: they want to minimize the cost of satisfying the policy

---

# Tradable permits: firm

The firm's problem is then:
$$ \min_E C(E) + p E$$

The firm's first-order condition to minimize costs is:
`$$-C'(E^*) = p$$`

The firm minimizes costs by choosing emissions `$E^*$` so that its MAC equals the permit price

---

# Tradable permits: cost-effectiveness

This makes sense!

The permit price is the MC of emitting, the MAC is the MB of emitting (reduced abatement cost)

Alternatively: the permit price is the MB of abating, the MAC is the MC of abating

Costs are minimized when these two things are equal

---

# Tradable permits: cost-effectiveness

What else does firm behavior tell us about permits?

If firms all set their MACs equal to `$p$` then all their MACs are equal to one another, .hi[we have cost-effectiveness:]
`$$-C_1'(E^*_1) = -C_2'(E^*_2) = \dots = -C_N'(E^*_N) = p$$`

Taxes and permits both achieve cost-effectiveness

Why?

Because firms treat permit prices and a tax identically in decisionmaking

---

# Tradable permits: cost-effectiveness

Tradable permit systems are .hi[always] cost-effective: whatever emissions limit you set, it will be achieved at least-cost<sup>1</sup>
.footnote[
<sup>1</sup> Try to see if you can use the same mathematical derivation to show that taxes are also always cost-effective.
]

This does not mean that it is necessarily efficient!

---

# Tradable permits: efficiency

For efficiency, we also need MAC = MD

To keep things simple suppose MD is constant and the same across all firms: `$MD = d$`

If we can set `$\bar{E}$` such that the equilibrium permit price `$p = d$`, then we also have efficiency

---

# Tradable permits in practice

Knowing MD is often difficult in practice

Tradable permit systems are nice because we can just let politicians choose a `$\bar{E}$` that is politically feasible, and then the permit market ensures that we get the associated emissions reductions at least-cost

What often happens in practice is `$\bar{E}$` starts high, giving us a low `$p$`, and then `$\bar{E}$` gets ratcheted down over time

---

# Political economy of permits

Firms are often more willing to accept a higher `$p$` than `$\tau$` because they are often endowed with (some) permits for free

This means that *in practice*, we might expect to get greater emissions reductions under a permit system than a tax because of these political economy reasons

This is one of the key reasons the 1990 CAA amendments were able to be passed

---

# Permit market challenges

.hi-blue[Auction] charge firms for each permit they hold, let price be set by marketplace, revenues can be used in other ways by the government, auction price will be the same as a Pigouvian tax

---

# Permit market challenges

We can decentralize trading market to cut down on transaction costs

Do trades need to be validated by central authority to ensure permit validity?

---

# Permit market challenges

Transactions costs create a wedge that prevents otherwise surplus-increasing trades from being made

Examples?

Search, information, bargaining, monitoring/enforcement

Lots of these costs are fixed, prohibit small trades

---

# Permit systems and heterogeneous MDs

So far we assumed all firms faced the same MD

This is true for things like climate change, less true for things like `$SO_2$` or `$NO_x$`

How well do permit systems perform with heterogeneous MD?

---

# Permit systems and heterogeneous MDs

Let's think about a setting with two firms: 1 and 2

The firms have different MACs: `$MAC_1 < MAC_2$` for all `$E$`

And the firms have different marginal damages as well: `$MD_1 < MD_2$` for all `$E$`

How well does a permit system work?

---

# Permit systems and heterogeneous MDs

Well we know the following:

Firms set `$MAC = p$` so we will have `$MAC_1 = MAC_2 = p$`

But for efficiency we also want `$MAC = MD$`: `$MAC_1 = MD_1$` and `$MAC_2 = MD_2$`

If `$MD_1 \neq MD_2$` then the permit system does .hi-red[not] deliver efficiency!

---

# Permit systems and heterogeneous MDs: graphical

![permits heterogeneous damages](06-slides-permits_files/figure-html/permits-hetero-1.png)
]

Suppose we have the two firms with different MACs and MDs:
- `$MAC_1 = 200-E_1$`
- `$MAC_2 = 300-E_2$`
- `$MD_1 = 100$`
- `$MD_2 = 150$`

]

---

# Permit systems and heterogeneous MDs: graphical

.pull-left[
![permits heterogeneous damages](06-slides-permits_files/figure-html/permits-hetero-1.png)
]

Suppose we have the two firms with different MACs and MDs:
- `$MAC_1 = 200-E_1$`
- `$MAC_2 = 300-E_2$`
- `$MD_1 = 100$`
- `$MD_2 = 150$`

The efficient emissions allocation is: `$E^* = 250: E_1^* = 100, E_2^* = 150$`

The regulator sets `$\bar{E} = 250$`

]

---

# Permit systems and heterogeneous MDs: graphical

.pull-left[
![permits heterogeneous damages](06-slides-permits_files/figure-html/permits-hetero-1.png)
]

We can solve for the permit market allocation and price using:
`$MAC_1 = MAC_2$` and `$E_1 + E_2 = 250$`

These two conditions tell us:

]

---

# Permit systems and heterogeneous MDs: graphical

.pull-left[
![permits heterogeneous damages](06-slides-permits_files/figure-html/permits-hetero-1.png)
]

We can solve for the permit market allocation and price using:
`$MAC_1 = MAC_2$` and `$E_1 + E_2 = 250$`

These two conditions tell us:

`$E^p_1 = 75, E^p_2 = 175, p = 125$`

but efficiency is at:

]

---

# Permit systems and heterogeneous MDs: graphical

.pull-left[
![permits heterogeneous damages](06-slides-permits_files/figure-html/permits-hetero-1.png)
]

We can solve for the permit market allocation and price using:
`$MAC_1 = MAC_2$` and `$E_1 + E_2 = 250$`

These two conditions tell us:

`$E^p_1 = 75, E^p_2 = 175, p = 125$`

but efficiency is at:

`$E^*_1 = 100, E^*_2 = 150$`

]

---

# Permit systems and heterogeneous MDs: graphical

![permits heterogeneous damages](06-slides-permits_files/figure-html/permits-hetero2-1.png)
]

We can solve for the permit market allocation and price using:
`$MAC_1 = MAC_2$` and `$E_1 + E_2 = 250$`

These two conditions tell us:

`$E^p_1 = 75, E^p_2 = 175, p = 125$`

but efficiency is at:

`$E^*_1 = 100, E^*_2 = 150$`

]

---

# Permit systems and heterogeneous MDs: graphical

.pull-left[
![permits heterogeneous damages](06-slides-permits_files/figure-html/permits-hetero2-1.png)
]

Relative to the optimal allocation, the permit system has DWL equal to the .hi-red[red area]

The permit allocation is not an .hi[efficient] allocation, but is it a .hi[Pareto improvement] over:

1. No policy?
2. A uniform standard of `$\bar{E}/2$`?

]

---

# Permit systems and heterogeneous MDs: graphical

![permits heterogeneous damages](06-slides-permits_files/figure-html/permits-hetero3-1.png)
]

The blue area is the DWL under the uniform standard

.hi[In this specific case,] a uniform standard and the permit system have the same efficiency since the red and blue areas are equal

The only difference is what kind of welfare loss is occurring where

]

---

# Permit systems and heterogeneous MDs: graphical

![permits heterogeneous damages](06-slides-permits_files/figure-html/permits-hetero4-1.png)
]

The DWL without any policy is the two large green triangles

These are clearly larger than the DWL under the permit system

The permit system can deliver a welfare improvement

]

---

# Permit systems and heterogeneous MDs: graphical

.pull-left[
![permits heterogeneous damages](06-slides-permits_files/figure-html/permits-hetero2-1.png)
]

What if the high MAC firm was the low MD firm?

i.e: what if the correlation between MAC and MD was .hi-red[negative] instead of .hi-blue[positive]?

What might we expect the correlation to be?

]

---

# Permit systems and heterogeneous MDs

What is the problem with permit systems and heterogeneous MD?

We want to have firms pay a price equal to their MD

Firms have different MDs 
--
but there's only one common permit price

This means we can't make all firms correctly account for their externalities

One way around this is to use .hi[trading ratios]: firms in high damage areas need to procure more permits for the same amount of emissions

Another way is .hi[zonal trading]: firms can only trade in similar MD areas

---

# Trading ratios: Acid Rain Program

Below are estimates of efficient trading ratios for the Acid Rain Program

Muller and Mendelsohn (2009)

---

# PM2.5 damages

Trading ratios are required because damages are heterogeneous across space

Muller and Mendelsohn (2009)

---

# Damages caused by ARP

The Acid Rain Program .hi-red[increased] damages in the eastern US

Chan et al. (2018)

---

# Damages caused by ARP

Chan et al. (2018) JEEM:

> We also compare health damages associated with observed SO2 emissions from all ARP units in 2002 with damages from a no-trade counterfactual. Damages under the ARP are `$2.1 billion (1995$`) higher than under the no-trade scenario, reflecting allowance transfers from units in the western US to units in the eastern US with larger exposed populations.

---

# Damages caused by ARP

---

# Zonal trading: RECLAIM

Regional Clean Air Management (RECLAIM) Program

California South Coast Air Quality Management District (SCAQMD)

Started in 1994, goal of cutting NOx and SOx  emissions by 80% by 2003

RECLAIM is a facility-level tradable permit system

---

# Zonal trading: RECLAIM

14% of permits allocated to power generators

Permit prices driven by electricity sector

During 2000 electricity price spikes, lots of very dirty plants brought on-line to meet demand

Permit prices rose dramatically for everyone else

$4,284 per ton of NOx in 1999

$39,000 per ton of NOx in 2000

---

# Zonal trading: RECLAIM

LA Basin has two distinct zones with very different MD’s

1. Old heavy industry (high MAC) and mountains trap NOx emissions and heat them up `$\rightarrow$` smog (high MD)

2. Newer firms (low MAC) close to the ocean, breezes dissipate pollution before it can turn into smog (low MD)

]

---

# Other permit market examples

Tradble permit systems are increasingly common:

1. Acid Rain Program
2. NOx Budget Program
3. Regional Greenhouse Gas initative
4. California AB32
5. EU Emission Trading System
6. China's National Carbon Cap and Trade

---

# RGGI

---

# RGGI

---

# AB32

---

# EU-ETS

---

# AB32

---

# Comparison of standards, taxes, permits

---

# What do we know so far

So far we have seen that:

1. Standards, taxes, and tradable permits can all achieve the efficient allocation
2. Taxes and tradable permits are cost-effective .hi[no matter what]
  - (all firms set MAC = `$\tau$` and MAC = p)
  
--

This still leaves a few questions to answer:

1. What are the equity effects?
2. What are the output effects?
3. What are the administrative burdens?
4. What are the .hi[dynamic] incentives under these policies?

---

# The equity set up

![base set up](06-slides-permits_files/figure-html/base-1.png)
]

Lets consider this our base set up for 1 firm

The regulator can achieve `$E^*$` through:

- an emission standard of `$E^*$`
- a tax of `$\tau$`
- an abatement subsidy of `$s$`
- "tradable permit" cap of `$E^*$`

]

---

# The equity set up

First let's look at .hi[equity]

How do the costs and benefits of the policies fall on different groups?

From here on we will roll the tax and permit system into 1: they are actually identical in terms of their impacts

]

---

# The distributional outcomes

|            	| Tax/Permits      	| Standard 	| Subsidy 	| Ranking         	|
|------------	|----------	|----------	|---------	|-----------------	|
| Firm       	| -(A+B+C) 	| -A       	| E       	| Sub > Std > Tax 	|
| Households 	| A+D+E    	| A+D+E    	| A+D+E   	| Indifferent     	|
| Government 	| B+C      	| 0        	| -(E+A)  	| Tax > Std > Sub 	|
| Total      	| D+E      	| D+E      	| D+E     	|                 	|

The total welfare gain is the same for all policies

The difference is in the .hi[distribution]

The standard strikes a middle ground out of the three

]

---

# Output effects

So far we have assumed that actual firm output is not affected by abatement/emission decisions

This won't always be true in the real world

Different policies have different implications for total cost and can thus affect production

To keep things simple lets suppose the firm has constant returns to scale technology and chooses the emissions rate / emissions per unit of output: `$E/q$`, this means that if they cut back on emissions it raises the MC of output

---

# The output set up

![base set up](06-slides-permits_files/figure-html/output1-1.png)
]

![base set up](06-slides-permits_files/figure-html/output2-1.png)
]

---

# The output results

Emission tax:
- Firm chooses `$E^*/q^*$`
- Firm pays A+B in tax and abatement cost .hi[per unit of output]
- This raises the MC of production by A+B to MC<sup>*</sup>
- Output `$q^*$` falls
- Pollution `$(E^*/q^*)*q^*$` falls even more since the tax lowers the optimal `$E^*/q^*$`, and increased MC lowers `$q^*$`

---

# The output results

Emission standard:
- Firm pays A in abatement cost per unit of output
- This raises the MC of production by A
- Output and `$(E^*/q^*)*q^*$` fall, but not by as much as under the tax

---

# The output results

Abatement subsidy:
- Reduces firm costs per unit of output by C
- This reduces the MC of production by C
- This .hi[raises] output
- Even though `$E/q$` goes down because the subsidy induces a lower emission intensity, total emissions may go up because `$q$` will rise

Cliffnotes:

- Output falls under taxes and standards
- This raises output prices
- Can have regressive effects through necessities like electricity or gas

---

# Administration

These policies are not equivalent in their cost of implementation

Enforcing a technology standard is very easy: you can inspect to see if they have installed the tech or not

Emission standards, taxes, subsidies require monitoring of emissions

Monitoring systems cost money

Firms have incentives to try to cheat!

---

# Administration

Pigouvian policies will not work as well for .hi[non-point sources] like cars or farms

With non-point sources we observe total emissions but not who emitted it

With non-point sources it often makes sense to use technology standards

Point sources like power plants are much easier to handle with Pigouvian policies like taxes

---

# Administration

Technology and emission standards typically guarantee some amount of emissions reductions

Taxes and subsidies guarantee firms pay a certain price but doesn't deliver us a guaranteed quantity

This might make things more politically difficult to pass

---

# Administration

When does C&C / technology standards make sense?

1. If there's a dominant technology where there's benefits to coordination or scale economies from production of the technology
2. High costs of monitoring/enforcement
3. High admin costs and little heterogeneity across firms

---

# Dynamic incentives

![taxes](06-slides-permits_files/figure-html/dynamic-1.png)
]

What are the gains to the firm from moving from `$MAC_1$` to `$MAC_2$`?

]

---

# Dynamic incentives

What are the gains to the firm from moving from `$MAC_1$`  to `$MAC_2$`?

]

---

# Dynamic incentives

What are the gains to the firm from moving to `$MAC_2$`?

Taxes and subsidies give greater incentives to innovate!

Once a firm meets a standard, there's no additional incentive beyond reducing abatement costs

Taxes and subsidies give the firm extra benefits for further reductions

]