Lecture 4

class: center, middle, inverse, title-slide

# Lecture 4
## Imperfect information
### Ivan Rudik
### AEM 6510

---

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```r
if (!require("pacman")) install.packages("pacman")
pacman::p_load(
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)
options(htmltools.dir.version = FALSE)
knitr::opts_hooks$set(fig.callout = function(options) {
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knitr::opts_chunk$set(echo = TRUE, fig.align="center")
  options
})
```

```
## Warning: 'xaringanExtra::style_panelset' is deprecated.
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## Warning in style_panelset_tabs(...): The arguments to `syle_panelset()` changed in xaringanExtra 0.1.0. Please refer to the documentation to update your slides.
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---

# Roadmap

What happens when the regulator has imperfect information about:
  - Marginal abatement costs?
  - Marginal damages?

So we far have assumed perfect information which is unlikely to be true

We will continue assuming that:

- Firms know their own marginal abatement cost
- Regulators observe firm-level emissions

---

# Imperfect information

How do we think about this?

Firms are unlikely to tell regulators exactly what their marginal abatement costs are

Therefore regulators will have to try to estimate it when designing policy

Estimation will naturally result in some error

Does this error matter for policy design?

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# Price vs quantities (Weitzman 1974)

Let's start by comparing emissions taxes and tradable permits

We'll start by looking at damage function uncertainty with known abatement costs

Then we look at abatement cost uncertainty with known damage functions

We will mainly be focused on efficiency outcomes and want to understand which policy delivers the highest welfare and why

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# Damage function uncertainty

- `$D(E)$` is the social damage function
--

- `$C(E)$` is the aggregate abatement cost function
--

- `$E^*$` is the optimal level of pollution
  - This is defined by `$-C'(E) = D'(E)$`
--
  
- Let `$\tilde{D}(E)$` and `$\tilde{C}(E)$` denote the estimated damage and abatement cost function

First suppose the regulator estimates `$-C'(E)$` correctly, but underestimates marginal damages: `$\tilde{D}'(E) < D'(E) \,\, \forall E$`

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# Damage function uncertainty

The regulator chooses policy based on her estimate `$\tilde{D}'(E)$`

This means she sets:
`$$-C'(E) = \tilde{D}'(E)$$`

which is solved by some `$\tilde{E} > E^*$` because she underestimated marginal damages

What is the welfare loss from targeting `$\tilde{E}$` instead of `$E^*$`?

Does the size of the loss depend on the policy instrument?

---

# Damage function uncertainty

Define welfare loss as the difference in total social costs at `$\tilde{E}$` versus the efficient level `$E^*$`:

`\begin{align}
WL &= [D(\tilde{E}) + C(\tilde{E})] - [D(E^*) + C(E^*)] \notag\\
  &= [D(\tilde{E}) -D(E^*)] + [ C(\tilde{E}) - C(E^*)]
\end{align}`

This is equivalent to the area under the marginal damage and abatement cost curves between the two emission levels:
`$$WL = \int_{E^*}^{\tilde{E}}D'(E) dE - \int_{E^*}^{\tilde{E}} - C'(E)dE$$`

---

# Damage function uncertainty with permits

.pull-left[

![Uncertain MD with permits](4-imperfect-information_files/figure-html/uncertain_damages_permit_nodwl-1.png)
]

.pull-right[

Here is the set up

Solve for `$\tilde{E}, E^*$` and the welfare loss from setting the number of permits to be `$L = \tilde{E}$`

]

---

# Damage function uncertainty with permits

.pull-left[

![Uncertain MD with permits](4-imperfect-information_files/figure-html/uncertain_damages_permit-1.png)
]

.pull-right[

With a permit scheme the regulator fixes the total amount of emissions at `$\tilde{E}$`

Since she underestimates `$D'$`, she lets firms emit too much

She incurs welfare loss `$A$` from emissions where marginal damage `$>$` marginal abatement cost

]

---

# Damage function uncertainty with taxes

.pull-left[

![Uncertain MD with permits](4-imperfect-information_files/figure-html/uncertain_damages_permit_nodwl_tax-1.png)
]

.pull-right[

Here is the set up

Solve for `$\tilde{E}, E^*$` and `$E(\tau)$` which is the firm's choice of emissions given `$\tau$`, and the welfare loss from setting the tax to be `$\tau(\tilde{E})$` which achieves `$E=\tilde{E}$` given `$-\tilde{C}'(E)$`

]

---

# Damage function uncertainty with taxes

.pull-left[

![Uncertain MD with taxes](4-imperfect-information_files/figure-html/uncertain_damages_tax-1.png)
]

.pull-right[

The regulator sets the tax as a function of her target emissions `$\tau(\tilde{E})$`

Since she underestimates `$D'$`, she sets `$\tau(\tilde{E})$` too low

The firm then selects `$E(\tau) = \tilde{E}$`

She incurs welfare loss `$A$` from emissions where marginal damage `$>$` marginal abatement cost

]

---

# Damage function uncertainty

What did we learn?

With damage function uncertainty,  
the regulator will set the incorrect level of policy

If she underestimates MD:
--

- She sets the total number of permits too high
  - She sets too low of a tax

Both lead to the exact same welfare loss  
so both policies have the same efficiency

---

# Abatement cost function uncertainty

Suppose the regulator estimates `$D'(E)$` correctly and underestimates marginal abatement cost: `$-\tilde{C}'(E) < -C'(E) \,\, \forall E$`

The regulator chooses policy based on her estimate `$-\tilde{C}'(E)$`

This means she sets:
`$$-\tilde{C}'(E) = D'(E)$$`
which is solved by some `$\tilde{E} < E^*$` because she underestimated marginal costs

What is the welfare loss from targeting `$\tilde{E}$` instead of `$E^*$`?

---

# Abatement cost function uncertainty

.pull-left[

![Uncertain MAC breaks the equivalence between taxes and permits](4-imperfect-information_files/figure-html/uncertain_abatement_nodwl-1.png)
]

.pull-right[

Here's the uncertain MAC problem

Solve for `$\tilde{E}, E^*, \sigma(\tilde{E}),$` and the welfare loss from setting the number of permits to be `$L = \tilde{E}$`

Solve for `$\tilde{E}, E^*,$` and `$E(\tau)$` which is the firm's choice of emissions given `$\tau$`, and the welfare loss from setting the tax to be `$\tau(\tilde{E})$`

]

---

# Abatement cost function uncertainty

.pull-left[

![Uncertain MAC breaks the equivalence between taxes and permits](4-imperfect-information_files/figure-html/uncertain_abatement-1.png)
]

.pull-right[

With permits, the regulator allows `$\tilde{E}$` permits which results in a permit price of `$\sigma(\tilde{E})$` where `$\tilde{E}$` intersects the true MAC

This yields a welfare loss of `$C$`

Firm behavior sets the price even though quantity is fixed by the regulator

]

---

# Abatement cost function uncertainty

.pull-left[
![Uncertain MAC breaks the equivalence between taxes and permits](4-imperfect-information_files/figure-html/uncertain_abatement-1.png)
]

.pull-right[

With a tax, the regulator sets a price `$\tau(\tilde{E})$` per unit of emissions, and the firms choose the quantity of emissions where `$\tau(\tilde{E}) = -C'(E)$` which causes total emissions to be `$E(\tau)$`

This yields a welfare loss of `$B$`

Firm behavior sets the quantity even though price is fixed by the regulator

]

---

# Abatement cost function uncertainty

.pull-left[
![Uncertain MAC breaks the equivalence between taxes and permits](4-imperfect-information_files/figure-html/uncertain_abatement-1.png)
]

.pull-right[

Since `$E(\tau) \neq \tilde{E}$`, abatement cost uncertainty matters: tradable permits and taxes give us different emission outcomes

Is there any systematic difference in the efficiency properties of permits and taxes?

]

---

# Abatement cost function uncertainty

In general it seems like sometimes the welfare loss of taxes will be higher, and other times the welfare losses of permits will be higher

What we will do next is try to understand the characteristics of the MAC and MD curves that tend to drive one policy to be better than the other

---

# Steep versus flat MD

.center[
![steep vs flat md curves](4-imperfect-information_files/figure-html/prices_vs_quantities_nodwl-1.png)
]
Solve for the permit and tax DWLs in both of these scenarios where the only difference is the steepness of the marginal damage curve

---

# Steep versus flat MD

.center[
![Initial distribution matters for how costs and benefits are spread out](4-imperfect-information_files/figure-html/prices_vs_quantities-1.png)
]

The only difference between the two plots is how steep the MD is relative to the MAC, and subsequently the policy that the regulator sets

Welfare loss for taxes is given by .red[A], welfare loss for permits is given by .blue[B]

---

# Steep versus flat MD

.center[
![Initial distribution matters for how costs and benefits are spread out](4-imperfect-information_files/figure-html/prices_vs_quantities-1.png)
]

Permits do better with steep MD, taxes do better with flat MD!

Why?

---

# Steep versus flat MD

With a steep (and known) MD:
--

- The difference between `$E^*$` and `$\tilde{E}$` is restricted: steeper curves are more inelastic, errors in estimating MAC lead to small errors in `$\tilde{E}-E^*$`

- The difference between `$E^*$` and `$E(\tau)$` can be very large: errors in estimating the MAC lead to bigger errors in `$\tau(\tilde{E}) - \tau(E^*)$` because MD is very steep

Steep MD: optimal .hi[quantity] of emissions is inelastic with respect to the true MAC

Think about the corner case of a vertical MD

---

# Steep versus flat MD

With a flat (and known) MD:

- The difference between `$E^*$` and `$\tilde{E}$` is can be very large: flatter curves are more elastic, errors in estimating MAC lead to large errors in `$\tilde{E}-E^*$`

- The difference between `$E^*$` and `$E(\tau)$` is restricted: errors in estimating the MAC lead to smaller errors in `$\tau(\tilde{E}) - \tau(E^*)$` because MD is very flat

Flat MD: optimal .hi[price] of emissions is inelastic with respect to the true MAC

Think about the corner case of constant MD