Lecture 06

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

---

``` r
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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

Suppose the regulator chooses emissions for `$N$` firms: `$E_1,\dots,E_N$`

Let firm `$i$`'s abatement cost be `$C_i(E_i)$`

Let total damages depend on aggregate emissions:
`$$D = D\!\left(\sum_{i=1}^N E_i\right)$$`

Regulator problem:
`$$\min_{E_1,\dots,E_N}\;\sum_{i=1}^N C_i(E_i) + D\!\left(\sum_{i=1}^N E_i\right)$$`

---

# Optimal policy with multiple firms

Take one FOC for each firm `$i=1,\dots,N$`:
`$$\frac{\partial}{\partial E_i}\left[\sum_{j=1}^N C_j(E_j) + D\!\left(\sum_{j=1}^N E_j\right)\right] = 0$$`
`$$C_i'(E_i) + D'\!\left(\sum_{j=1}^N E_j\right) = 0$$`

`$$\underbrace{-C_i'(E_i)}_{MAC_i} = \underbrace{D'\!\left(\sum_{j=1}^N E_j\right)}_{MD}$$`

---

# Optimal policy with multiple firms

`$$\underbrace{-C_i'(E_i)}_{MAC_i} = \underbrace{D'\!\left(\sum_{j=1}^N E_j\right)}_{MD}$$`

At the optimum:
`$$MAC_1 = MAC_2 = \cdots = MAC_N = MD$$`

This gives the target condition that we try to achieve through a decentralized permit market

---

# 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

]

---

# 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 `$\bar{E}_2 + 1$`

Total emissions didn't change, but how much each firm gets to emit does through voluntary trade

---

# 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

]

---
count: false

# Tradable permits: graphical

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

]

---
count: false

# Tradable permits: graphical

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

]

---
count: false

# Tradable permits: graphical

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-save2-1.png)
]

Step 1: allow firm 2 (high MAC) to emit more, from `$\bar{E}/2$` to `$E_2$`

The blue shaded area is firm 2's cost saving

]

---

# Tradable permits: graphical

![permits](06-slides-permits_files/figure-html/permits-dwl-cost1-1.png)
]

Step 2: firm 1 (low MAC) emits less, from `$\bar{E}/2$` to `$E_1$`

The orange shaded area is firm 1's cost increase

]

---

# Tradable permits: graphical

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

Step 3: net cost change from trading

The grey rectangles are equal (same height and same width `$\Delta E$`), so they cancel out

What remains is the two red triangles: the net cost reduction from reallocation

]

---

# 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 permit, the MAC of the selling firm

or it is the MB of freeing up one more permit (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

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

---

# Tradable permits: numerical example

Suppose two firms trade permits with total cap:
`$$E_1 + E_2 = \bar{E} = 90$$`

Abatement cost functions:
`$$C_1(E_1) = \frac{1}{2}(90-E_1)^2,\qquad C_2(E_2) = \frac{1}{4}(150-2E_2)^2$$`

So marginal abatement costs are:
`$$MAC_1(E_1) = -C_1'(E_1) = 90 - E_1$$`
`$$MAC_2(E_2) = -C_2'(E_2) = 150 - 2E_2$$`

---

# Tradable permits: numerical example

In a competitive permit market:
`$$MAC_1(E_1^*) = p,\qquad MAC_2(E_2^*) = p,\qquad E_1^*+E_2^* = \bar{E} = 90$$`

Plug in the MAC expressions:
`$$90-E_1^* = p,\qquad 150-2E_2^* = p,\qquad E_1^*+E_2^* = 90$$`

Solve:
`$$E_1^* = 40,\qquad E_2^* = 50,\qquad p^* = 50$$`

Firm 2 has higher MAC at an equal split, so it buys permits and emits more in equilibrium.

---

# Tradable permits: numerical example

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

Numerical solution:
`$$E_1^* = 40,\quad E_2^* = 50,\quad p^* = 50$$`

Starting from equal allocation (`$\frac{\bar{E}}{2} = 45$` each):

- firm 1 sells permits and emits less
- firm 2 buys permits and emits more

]

---

# 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 auctions

Recall the firm's problem when a permit costs `$p$`:
`$$\min_E\; C(E) + pE \quad\Rightarrow\quad MAC(E) = p$$`

Flip it around: at price `$p$`, the firm wants permits up to where `$MAC(E) = p$`

Each permit lets the firm emit one more unit, so it is worth the abatement cost it avoids

---

# Permit auctions: MAC is permit demand

![MAC as permit demand](06-slides-permits_files/figure-html/permits-auction-demand-1.png)
]

At `$p = 70$`: wants 20 permits<br>At `$p = 50$`: wants 40 permits

Firm 2: `$MAC_2 = 150 - 2E_2$`

Setting `$MAC_2 = p$` gives `$E_2 = 75 - p/2$`<br>At `$p = 70$`: wants 40 permits<br>At `$p = 50$`: wants 50 permits
]

---

# Permit auctions: supply meets demand

<img src="06-slides-permits_files/figure-html/permits-auction-supply-demand-1.png" alt="permit auction supply and demand" style="width:90%;">
]

---

# Permit auctions: supply meets demand

.pull-left[
<img src="06-slides-permits_files/figure-html/permits-auction-supply-demand-1.png" alt="permit auction supply and demand" style="width:90%;">
]

Set equal: `$165 - 1.5p = 90$`, so `$p = 50$`

`$E_1 = 40$`, `$E_2 = 50$`: .hi[the trading equilibrium]
]

---

# Permit auctions: the rules

Major permit auctions (RGGI, California-Quebec, EU ETS) are .hi[uniform-price sealed-bid] auctions. These are our Round 1 rules:<sup>1</sup>

1. Each firm privately bids the most it would pay for each permit

2. The regulator ranks all bids from highest to lowest

3. The top `$\bar{E}$` bids win

4. .hi[Every] winner pays the same price: the lowest winning bid

.footnote[
<sup>1</sup> The EU ETS and California-Quebec also charge the lowest winning bid; RGGI charges the highest losing bid; EPA's Acid Rain Program auctions charged each winner its own bid.
]

---

# Permit auctions: example

Three firms bid for .hi[5 permits]:

|        | Permit 1 | Permit 2 | Permit 3 | Permit 4 |
|--------|---------:|---------:|---------:|---------:|
| Firm A | 12       | 9        | 6        | 3        |
| Firm B | 10       | 7        | 4        | 1        |
| Firm C | 8        | 5        | 2        |          |

Which bids win? What is the price? What does each firm pay?

---

# Permit auctions: example

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

Price = lowest winning bid = .hi[7]

All winners pay only .hi[7 per permit], even when they bid higher

Your bid mostly decides .hi[whether] you win a permit; the price is the lowest winning bid, usually someone else's
]

---

# Permit auctions: the gain to the firm

.pull-left[
![clearing the example auction](06-slides-permits_files/figure-html/permits-auction-clearing-1.png)
]

.pull-right[
A whole permit avoids one unit of abatement. Its value is the .hi[area under MAC] over that unit:

`$$\text{Gain} = \text{avoided abatement cost} - p$$`

If a permit saves 12 in abatement cost and costs 7, the firm gains .hi[5]

The bars show submitted bids
]

---

# Let's trade: the permit market game

---

# The permit market game: auctions and trading

After .hi[each auction]: your MAC shifts, then the open market begins

.hi[Round 3: transaction costs.] Restart Round 2 trading: same starting permits and post-shock MACs, plus a .hi[$3 buyer fee per permit]

---

# The permit market game: what should happen?

- Different bidding incentives under uniform-price and pay-as-bid auctions
- Trades from low-MAC to high-MAC firms after costs change
- Lower total abatement costs as firms trade toward the cost-effective allocation
- In Rounds 1 and 2, equal MACs when the class reaches that allocation
- In Round 3, the $3 fee blocks trades whose abatement-cost savings are less than $3

After trading closes, compare the MAC histogram with the cost-effective price line

---

# The permit market game: scoring

`$$\text{Score} = \text{avoided abatement cost} + \text{sales} - \text{purchases} - \text{fees}$$`

Avoided cost is relative to abating all baseline emissions, using your .hi[post-shock MAC]. Purchases include auction payments

The benchmark values your cost-effective permits at the no-fee price. Round 3 reuses Round 2's auction payments and deducts $3 per permit bought

---

# The permit market game: class cost reductions

`$$100 \times \frac{C_{\text{start}}-C_{\text{end}}}{C_{\text{start}}-C^*}$$`

- `$C_{\text{start}}$`: total abatement cost at the auction allocation
- `$C_{\text{end}}$`: total abatement cost after trading
- `$C^*$`: cost-effective total abatement cost

All three use the same .hi[post-shock MACs] and exclude permit payments

100% means the no-fee gap closed. A zero initial gap has no percentage

Round 3 also reports .hi[abatement cost + transaction costs]

---

# The permit market game: bids and orders

---

# Join the permit market game

Or open .hi[[aem4510.ivanrudik.com/permits](https://aem4510.ivanrudik.com/permits)]
]

---

# After the permit game

---

# Debrief: monopsony power in the auction

In a uniform-price auction, lowering a bid that sets the price can lower payments on .hi[all] the buyer's winning permits

For example, A cuts its second bid from 9 to 6. The price falls to 6, and A still wins 2 permits, paying 12 instead of 14

This .hi[demand reduction] uses monopsony power. It can leave high-value buyers with too few permits

---

# Debrief: why trade after an auction?

If bids equal the abatement costs each whole permit saves, the auction reaches the cost-effective allocation

In our game, MACs shift after each auction, creating new gains from trade

Firms whose MAC rose can buy from firms whose MAC fell. Trading can also correct an auction's misallocation

---

# Debrief: pay as you bid

EPA's Acid Rain Program auctions (from 1993) were .hi[pay-as-bid]; they sold about 2.8% of allowances, and the rest were given away

In pay-as-bid, firms .hi[shade bids down] below their willingness to pay

A lower winning bid means paying less. Bidding too low risks losing the permit

Bids can therefore differ from avoided abatement costs, leaving gains from later trading

---

# Permit auctions: bidding and cost-effectiveness

In a uniform-price auction, many small buyers have little .hi[monopsony power]: each bid is unlikely to set the price

Bids close to each permit's avoided abatement cost help allocate permits to the firms that value them most

When bids equal those values, the auction produces a .hi[cost-effective allocation]

Pay-as-bid gives even small buyers a reason to shade bids down, because each winning bid sets its own payment

---

# 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

]

---

# When the uniform standard has higher welfare

![High damages at firm 2](06-slides-permits_files/figure-html/permits-welfare-standard-1.png)
]

The welfare ranking depends on the MACs and MDs

Raise `$MD_2$` (solid line); the dashed line shows its previous level

]

---

# When permit trading has higher welfare

![Lower damages at firm 2](06-slides-permits_files/figure-html/permits-welfare-trading-1.png)
]

Lower `$MD_2$` (solid line); the dashed line shows its previous level

Changing the MACs and MDs can make either policy better, or leave them tied

]

---

# 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.

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# Damages caused by ARP

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# 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

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# AB32

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# EU-ETS

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# China ETS

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# Banking and borrowing in permit markets

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# Banking and borrowing in permits

In cap-and-trade, permits may be shifted across time if we allow for banking and borrowing:

Without borrowing, the firm can only use permits it already has

---

# The firm's permit balance

`$$B_t = B_{t-1} + \bar{E}_t - E_t$$`

- `$\bar{E}_t$`: new permits you receive in year `$t$`
- `$B_{t-1}$`: permits saved from last year, or debt if negative
- `$E_t$`: permits used for this year's emissions

`$B_t$` is what remains after adding new permits and subtracting those used

- `$B_t > 0$`: extra permits you are saving for later
- `$B_t < 0$`: borrowed permits you must give back later

---

# Banking and borrowing in permits

Consider how one firm allocates its abatement effort over two periods:

`$$\min_{E_1,E_2} C_1(E_1) + C_2(E_2)$$`
`$$\text{subject to: }E_1 + E_2 = \bar{E}_1 + \bar{E}_2$$`

FOC:
`$$-C_1'(E_1^*) = -C_2'(E_2^*)$$`

In MAC terms:
`$$MAC_1(E_1^*) = MAC_2(E_2^*)$$`

A firm equalizes its MAC over time!

---

# Banking and borrowing in permits

Intuition:

Standard cap-and-trade allows .hi-blue[within-period trade across firms]

Banking and borrowing add .hi-blue[across-period trade within a firm]

So a firm can effectively trade with itself over time

This shifts abatement away from high-MAC periods and toward low-MAC periods

---

# Banking and borrowing in permits

Why might a firm's MAC change over time?

1. Fuel and input prices move (e.g., coal vs natural gas prices)

2. Output demand changes (business cycle, weather-driven load)

3. Abatement equipment has temporary outages or maintenance

4. New technology adoption lowers future abatement cost

5. Learning-by-doing improves operational efficiency

With these shocks, banking/borrowing lowers compliance cost relative to a rigid period-by-period cap

---

# Banking and borrowing in permits

<div class="permit-time-note-1">Compare the firm's MAC at each period's original emissions limit</div>

<div class="permit-time-note-2">If MAC<sub>2</sub>(Ē<sub>2</sub>) &gt; MAC<sub>1</sub>(Ē<sub>1</sub>), period 2 abatement is more expensive</div>

<div class="permit-time-note-3">Banking is optimal: E<sub>1</sub><sup>*</sup> &lt; Ē<sub>1</sub> and E<sub>2</sub><sup>*</sup> &gt; Ē<sub>2</sub></div>

---

# Banking and borrowing in permits

<div class="permit-time-note-1">Compare the firm's MAC at each period's original emissions limit</div>

<div class="permit-time-note-2">If MAC<sub>1</sub>(Ē<sub>1</sub>) &gt; MAC<sub>2</sub>(Ē<sub>2</sub>), period 1 abatement is more expensive</div>

<div class="permit-time-note-3">Borrowing is optimal: E<sub>1</sub><sup>*</sup> &gt; Ē<sub>1</sub> and E<sub>2</sub><sup>*</sup> &lt; Ē<sub>2</sub></div>

---

# Banking and borrowing in practice: Acid Rain Program

U.S. `$SO_2$` trading (Acid Rain Program) allowed banking across years

Firms over-complied in the 1990s and built an allowance bank before tighter caps

]

---

# Banking and borrowing in practice: EU-ETS

Phase I (2005-2007): limited inter-phase banking; surplus allowances led to a price collapse near phase end

From Phase II onward, banking became a key margin for compliance planning

Large cumulative surplus carried forward across phases and affected prices

Borrowing is limited in practice; EU design relies much more on banking than borrowing

]

---

# Banking and borrowing: design lessons

1. Banking is widely used because it lowers costs while preserving a cumulative cap
2. Borrowing is often constrained because of default, enforcement, and credibility concerns
3. Rules on banking/borrowing strongly affect allowance price dynamics
4. Without banking across phase boundaries, end-of-phase price volatility can be severe

---

# The permit game: banking and borrowing

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# The permit game: decisions across rounds

We now allow permits to move between the two rounds

.hi[Borrow:] emit more than your permits cover in Round 1 and repay the difference with Round 2 permits

Use the emissions choice in Round 1. Unrepaid borrowing pays a penalty at the end of Round 2

---

# The permit game: predictions across rounds

How should expected Round 2 conditions change your Round 1 decisions?

- Bank when permits are more valuable in Round 2
- Borrow when permits are more valuable in Round 1, accounting for repayment
- Trading across rounds brings marginal abatement costs closer together

Compare these results with the game without banking and borrowing

The trading cost-reduction percentage is omitted here. Separate-round benchmarks do not measure cost-effectiveness across time

---

# Rejoin the permit game: banking and borrowing

Or open .hi[[aem4510.ivanrudik.com/permits](https://aem4510.ivanrudik.com/permits)]
]

---

# Debrief: banking and borrowing

Did teams bank before the tighter Round 2 cap? Did any teams borrow instead?

Compare your MAC across rounds: could shifting one more permit have reduced your total abatement cost?

Banking and borrowing extend the gains from trade across time

---

# Comparison of standards, taxes, permits

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# 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 `$E^*$` with auctioned permits at price `$p$`

]

---

# 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 if all permits are auctioned

]

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# 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

Key identity:
`$$E = \left(\frac{E}{q}\right)\times q$$`

Firms can reduce total emissions by lowering emissions intensity `$E/q$`, output `$q$`, or both

---

# Output effects: the firm's choices

The firm chooses output `$q>0$` and emissions intensity `$e=E/q$`, so `$E=eq$`

Let `$R(q)$` be revenue and `$C_0(q)$` production cost before abatement

Let `$a(e)$` be abatement cost per unit of output, so total cost is
`$$C_0(q)+q a(e)$$`

Unregulated intensity is `$e_0$`, with `$a(e_0)=0$`

For `$e<e_0$`, assume `$a'(e)<0$`: cleaner production costs more

---

# Deriving the MAC

Holding `$q$` fixed, MAC is the cost of avoiding one more unit of emissions
`$$MAC(E,q)=-\frac{\partial[q a(E/q)]}{\partial E}$$`

Applying the chain rule gives
`$$MAC(E,q)=-q a'(E/q)\frac{1}{q}=-a'(e)$$`

---

# Output effects: choosing intensity

Hold output fixed and use the familiar abatement rules

- A tax saves `$\tau$` for each unit of emissions avoided
- A subsidy pays `$s$` for each unit of emissions avoided

Stop abating when `$MAC(e^*)=\tau$` under the tax, or `$MAC(e^*)=s$` under the subsidy

For this comparison, the standard limits emissions *per unit of output*, such as one ton per MWh

This is an .hi[intensity standard], `$e\leq\bar e$`, whereas the earlier cap limited total emissions

Set `$\bar e=e^*$` and `$\tau=s=MAC(e^*)$` to match emissions intensity across policies

---

# Output effects: intensity then output

<div class="output-walkthrough">
<img src="06-slides-permits_files/figure-html/output1-1.png" alt="Emissions intensity and costs per unit of output">
<div class="output-walkthrough-text">
<p>All areas are dollars per unit of output</p>
<p>The firm chooses e<sup>*</sup> where MAC equals the tax rate</p>
</div>
</div>

<div class="output-walkthrough">
<div class="output-walkthrough-text">
<p>Abatement and tax payments raise the cost per unit of output by A+B</p>
</div>
<img src="06-slides-permits_files/figure-html/output2-1.png" alt="Tax raises marginal production cost by A+B and reduces output">
</div>

<div class="output-walkthrough">
<div class="output-walkthrough-text">
<p>Abatement and tax payments raise the cost per unit of output by A+B</p>
<p>At MR = MC, profit-maximizing output falls from q<sub>0</sub> to q<sup>*</sup></p>
</div>
<img src="06-slides-permits_files/figure-html/output2-1.png" alt="Tax raises marginal production cost by A+B and reduces output">
</div>

<div class="output-walkthrough">
<div class="output-walkthrough-text">
<p>Abatement and tax payments raise the cost per unit of output by A+B</p>
<p>At MR = MC, profit-maximizing output falls from q<sub>0</sub> to q<sup>*</sup></p>
<p>Lower intensity and lower output both reduce total emissions</p>
</div>
<img src="06-slides-permits_files/figure-html/output2-1.png" alt="Tax raises marginal production cost by A+B and reduces output">
</div>

---

# The output results

Emission tax:
- The firm chooses intensity `$e^*$` where MAC equals the tax rate
- Abatement cost A plus tax payment B raises output MC by A+B
- Profit-maximizing output satisfies `$MR=MC$`, so higher MC lowers output
- Both lower intensity and lower output reduce total emissions

---

# The output results

Emissions-intensity standard:
- Require `$e \leq e^*$`, where `$e=E/q$`
- The firm pays A in abatement cost per unit of output, with no tax payment
- This raises the marginal cost of output by A
- Output and total emissions fall, but by less than under the tax

]

![standard output effect](06-slides-permits_files/figure-html/output2_standard-1.png)
]

---

# The output results

Abatement subsidy:
- Pay `$s(e_0-e)$` per unit of output relative to baseline intensity `$e_0$`
- Gross subsidy A+C minus abatement cost A leaves net gain C per unit of output
- Output MC falls by C, so output rises
- Total emissions `$E=e^*q^*$` can rise or fall

A subsidy `$s(E_0-E)$` from a fixed total baseline instead has the same marginal output effect as a tax

]

![subsidy output effect](06-slides-permits_files/figure-html/output2_subsidy-1.png)
]

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

.dynamic-motivation[
Firms may innovate or adopt new technologies that lower their marginal abatement costs, shifting MAC down

What incentives do different policy instruments give firms to make these improvements?

To determine this, we need to work out how a lower MAC changes the firm's abatement costs and tax or subsidy payments
]

.pull-left.dynamic-comparison[

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

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

]

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# Dynamic incentives

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

]

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# 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

]

---

# How do carbon markets work?

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