Lecture 08

.title[
# Lecture 08
]
.subtitle[
## Multiple Market Failures and Second-Best Policies
]
.author[
### Ivan Rudik
]
.date[
### AEM 4510
]

---

``` r
if (!require("pacman")) install.packages("pacman")
```

```
## Loading required package: pacman
```

``` r
pacman::p_load(
  tidyverse, xaringanExtra, rlang, patchwork, nycflights13, vembedr
)
source("R/08-second-best-intensity-figures.R")
source("R/08-second-best-diagram.R")
source("R/08-perfect-competition-figures.R")
options(htmltools.dir.version = FALSE)
knitr::opts_hooks$set(fig.callout = function(options) {
  if (options$fig.callout) {
    options$echo <- FALSE
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knitr::opts_chunk$set(
  cache = TRUE,
  cache.extra = list(tools::md5sum(c(
    "R/08-second-best-intensity-figures.R", "R/08-second-best-diagram.R",
    "R/08-perfect-competition-figures.R"
  ))),
  echo = TRUE,
  fig.align = "center"
)
  options
})
```

```
## Warning in xaringanExtra::style_panelset(panel_tab_color_active = "red"): 'xaringanExtra::style_panelset' is deprecated.
## Use 'style_panelset_tabs' instead.
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## Warning in style_panelset_tabs(...): The argument names of `style_panelset()`
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## NULL
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---

# Roadmap

1. What happens when we have another market failure like market power?
3. How do second-best policies like output taxes or intensity standards work?

---

# Market power and pollution

---

# Market power

Lets consider two extreme cases to understand whether and how market power matters

1. Perfect competition
2. Monopoly

In both cases we will assume that:

1. Marginal costs of production are constant `$MC$`
2. Emissions are perfectly proportional to output, so each additional unit of output creates the same marginal damage and gives us constant social marginal cost `$SMC = MC + MD$`

---

# Perfect competition

A tax of `$SMC - MC$` moves output from `$q_0$` to `$q^*$`

Track the welfare effects using areas A, B, and C

]

---

# Perfect competition

A tax of `$SMC - MC$` moves output from `$q_0$` to `$q^*$`

]

---

# Perfect competition

A tax of `$SMC - MC$` moves output from `$q_0$` to `$q^*$`

Loss in CS: `$-(A+B)$`

]

---

# Perfect competition

A tax of `$SMC - MC$` moves output from `$q_0$` to `$q^*$`

Loss in CS: `$-(A+B)$`

Avoided damages: `$B+C$`

]

---

# Perfect competition

A tax of `$SMC - MC$` moves output from `$q_0$` to `$q^*$`

Loss in CS: `$-(A+B)$`

Avoided damages: `$B+C$`

Tax revenue: `$A$`

]

---

# Monopoly

Now consider a monopolist with the same marginal cost and marginal damage structure

What is the difference with a monopolist?

The monopolist can set the price

This means that the MR curve lies beneath the demand curve

Why?

The monopolist accounts for how additional output lowers the market price on inframarginal units

---

# Monopoly

![perfect competition](08-slides-second-best_files/figure-html/monopoly-1.png)
]

The socially efficient allocation is where social marginal cost is equal to the social marginal benefit

This is where SMC crosses the demand curve: `$(q^*,p^*)$`

What is the welfare outcome under the unregulated monopolist outcome?

]

---

# Monopoly

![perfect competition](08-slides-second-best_files/figure-html/monopoly2-1.png)
]

In the absence of regulation, the monopolist maximizes profit where MR = MC: `$(q^m, p^m)$`

This results in deadweight loss equal to the .hi-red[red] area

Now what happens if we set a Pigouvian tax equal to marginal damage?

]

---

# Monopoly

![monopoly](08-slides-second-best_files/figure-html/monopoly3-1.png)
]

The Pigouvian tax restricts output even more, adding deadweight loss equal to the .hi-blue[blue] area on top of the deadweight loss in the .hi-red[red] area

The tax actually made us worse off by the blue area!

Why?

]

---

# Monopoly

We now have two market failures:

1. Market power
2. Pollution externality

]

---

# Monopoly

We now have two market failures:

1. Market power
2. Pollution externality

With market power, the unregulated equilibrium quantity is .hi[too low]

]

---

# Monopoly

We now have two market failures:

1. Market power
2. Pollution externality

With market power, the unregulated equilibrium quantity is .hi[too low]

With a pollution externality, the unregulated equilibrium quantity is .hi[too high]

]

---

# Monopoly

They have opposing forces on quantities, so the market failures offset each other (partially)

]

---

# Monopoly

They have opposing forces on quantities, so the market failures offset each other (partially)

This means that if we fully correct the pollution externality, we no longer get the off-setting benefit and have the full welfare cost of market power

]

---

# Monopoly

![monopoly](08-slides-second-best_files/figure-html/monopoly4-1.png)
]

What is the actual optimal thing to do here?

]

---

# Monopoly

![monopoly](08-slides-second-best_files/figure-html/monopoly5-1.png)
]

What is the actual optimal thing to do here?

]

---

# Monopoly

What is the actual optimal thing to do here?

In this example, the market power externality dominates the pollution externality: we need to increase output

]

---

# Monopoly

What did we learn?

With multiple market failures we don't necessarily want to .hi[fully] correct for a single market failure

In this example we actually wanted to do the .hi-red[opposite] of what you likely thought

You can always draw this example in a different way where you should tax output

You just need marginal damages to be sufficiently large relative to the market power effect on quantity

---

# Monopoly

![monopoly](08-slides-second-best_files/figure-html/monopoly6_base-1.png)
]

What if marginal damages are larger?

Suppose SMC is high enough that demand and SMC intersect to the .hi[left] of `$q^m$`

]

---

# Monopoly

What if marginal damages are larger?

Suppose SMC is high enough that demand and SMC intersect to the .hi[left] of `$q^m$`

Then the efficient quantity is lower than monopoly output: `$q^* < q^m$`

]

---

# Monopoly

What if marginal damages are larger?

Suppose SMC is high enough that demand and SMC intersect to the .hi[left] of `$q^m$`

Then the efficient quantity is lower than monopoly output: `$q^* < q^m$`

]

---

# Monopoly

![monopoly](08-slides-second-best_files/figure-html/monopoly6_tax-1.png)
]

Policy implication flips: .hi[tax output] at rate `$t$` so `$MC+t$` crosses `$MR$` at `$q^*$`

]

---

# Monopoly

![monopoly](08-slides-second-best_files/figure-html/monopoly6_tax_gain-1.png)
]

Policy implication flips: .hi[tax output] at rate `$t$` so `$MC+t$` crosses `$MR$` at `$q^*$`

Welfare gain from this correction is the reduction in overproduction between `$q^m$` and `$q^*$`

]

---

# Monopoly: more generally

This was a .hi[special case]: marginal damage from output was constant

When emissions and output are separate choices, both market failures remain

We can no longer write `$E = f(q)$` or marginal damage as a function of output alone

A pollution tax/subsidy alone cannot correct both market failures

---

# Monopoly: more generally

Typically we must follow the .hi[Tinbergen rule:] you need as many policy instruments as you have market failures to achieve the efficient outcome

What does that mean here?

We need:

1. Pollution tax
--

2. Output subsidy

The tax incentivizes abatement, the subsidy incentivizes production

---

# Output taxes

Sometimes emission taxes and abatement subsidies are difficult to administer because monitoring is hard

What can we do instead?

We can tax the output of the production process

e.g. tax on electricity, cement

Will this be efficient?

If so, what assumptions do we need about the production process?

---

# Output taxes

![monopoly](08-slides-second-best_files/figure-html/output-1.png)
]

Assume emissions are proportional to output

And MD is constant

The firm chooses to produce/emit at `$q^u$` in the unregulated equilibrium

If we tax output equal to MD we can achieve the socially optimal allocation `$q^*$`

]

---

# Output taxes

An output tax can be efficient, .hi[if we assume that emissions are proportional to output]

Now let's break the link between output and emissions by writing down a slightly more complicated model where the firm chooses emissions and output separately

]

---

# Output taxes, part two

Here's our model:

- Production uses labor `$L$` and energy `$E$`: `$Q = f(L,E)$`
- Labor costs `$w$` per unit and energy costs `$r$` per unit
- The firm receives a price `$p$` per unit of output
- 1 unit of energy causes 1 unit of emissions, with marginal damage `$d$` per unit of emissions

The key point is that the firm can adjust `$L$` and `$E$` separately

What does an output tax `$\tau_o$` do versus a regular emission tax `$\tau_e$`?

---

# Output taxes, part two

The regulator wants the firm to internalize its social costs:
`$$\max_{L,E} p f(L,E) - wL - rE - dE$$`

The first-order conditions for a socially efficient allocation of `$L$` and `$E$` are:
`$$p\frac{\partial f(L,E)}{\partial L} = w \qquad p\frac{\partial f(L,E)}{\partial E} = r + d$$`

For a social optimum, labor is priced at its private cost, but emissions must face the extra damage term `$d$`

---

# Output taxes, part two

The firm's problem for the output tax is:
`$$\max_{L,E} (p-\tau_o)f(L,E) - wL - rE$$`

The firm's profit-maximizing choices are given by the first-order conditions:
`$$(p-\tau_o)\frac{\partial f(L,E)}{\partial L} = w \qquad (p-\tau_o)\frac{\partial f(L,E)}{\partial E} = r$$`

What do these conditions mean?

---

# Output taxes, part two

The output tax reduces the value of the marginal product of .hi[both] inputs

Rearranging the labor condition gives:
`$$p\frac{\partial f(L,E)}{\partial L} = w + \tau_o\frac{\partial f(L,E)}{\partial L}$$`

So the output tax implicitly taxes clean labor too, which is .hi-red[not efficient]

---

# Output taxes, part two

How does this compare to a pure emission tax?

The firm's problem when facing an emission tax is:
`$$\max_{L,E} p f(L,E) - wL - (r+\tau_e)E$$`
--

The firm's profit-maximizing choices are given by the first-order conditions:
`$$p\frac{\partial f(L,E)}{\partial L} = w \qquad p\frac{\partial f(L,E)}{\partial E} = r + \tau_e$$`

A tax of `$\tau_e = d$` can achieve the efficient allocation!

---

# Output taxes takeaways

If emissions are perfectly determined by output, we can use output taxes to achieve the efficient outcome

If emissions can be chosen separately from outcome by the firm, this is no longer true

In this case an output tax incorrectly taxes our clean inputs

---

# Intensity standards

Another common second-best policy is an .hi[intensity standard]

Instead of pricing each unit of emissions directly, the regulator sets an emissions rate per unit of output

These policies are common when direct emissions monitoring is difficult but output composition is observable

---

# Intensity standards: RPS example

---

# Intensity standards: the constraint

Let `$H$` be the dirtier product and `$L$` the cleaner product

An intensity standard requires:
`$$\frac{\beta_H Q_H + \beta_L Q_L}{Q_H + Q_L} \leq \sigma$$`

- `$\beta_i$`: emissions per unit of product `$i$`
- `$\sigma$`: maximum average emissions per unit of output
- Assume `$\beta_H > \sigma > \beta_L$`: `$H$` is above the standard, `$L$` below

The standard limits average emissions; total emissions can still rise

---

# Intensity standards: firm's problem

Suppose the firm takes output prices `$p_H$` and `$p_L$` as given

Its private problem is:
`$$\max_{Q_H,Q_L} \pi(Q_H,Q_L) = p_H Q_H + p_L Q_L - C_H(Q_H) - C_L(Q_L)$$`

subject to the intensity standard:
`$$\frac{\beta_H Q_H + \beta_L Q_L}{Q_H + Q_L} \leq \sigma$$`

---

# Intensity standards: Lagrangian setup

Rewrite the constraint as:
`$$\sigma(Q_H + Q_L) - \beta_H Q_H - \beta_L Q_L \geq 0$$`

The Lagrangian is:
`$$\mathcal{L} = p_H Q_H + p_L Q_L - C_H(Q_H) - C_L(Q_L) + \lambda\left[\sigma(Q_H + Q_L) - \beta_H Q_H - \beta_L Q_L\right]$$`

with `$\lambda \geq 0$`; if the standard binds, `$\lambda$` is the shadow value of relaxing it

---

# Intensity standards: firm incentives

Taking first-order conditions from the Lagrangian gives:
`$$p_i = MC_i(Q_i) + \lambda(\beta_i-\sigma), \quad i \in \{H,L\}$$`

If `$\beta_i > \sigma$`, the policy creates an implicit .hi-red[tax] on that output

If `$\beta_i < \sigma$`, the policy creates an implicit .hi-blue[subsidy] on that output

---

# Intensity standards: equivalent policy mix

Rewrite the wedge term as:
`$$\lambda(\beta_i-\sigma) = \underbrace{\lambda\beta_i}_{\text{emissions tax piece}} - \underbrace{\lambda\sigma}_{\text{output subsidy piece}}$$`

An .hi-red[emissions tax] of `$\lambda$` per unit of emissions costs `$\lambda\beta_i$` per unit of output, so dirtier goods pay more

An .hi-blue[output subsidy] of `$\lambda\sigma$` per unit is the same for every good, regardless of `$\beta_i$`

---

# Intensity standards: cleaner is not clean

Suppose `$\beta_H = 1.0$`, `$\beta_L = 0.4$`, and `$\sigma = 0.6$`

Then the policy wedges are:
- for `$H$`: `$\lambda(1.0-0.6)=+0.4\lambda$` (implicit tax)
- for `$L$`: `$\lambda(0.4-0.6)=-0.2\lambda$` (implicit subsidy)

The lower-intensity good still pollutes, but it can get subsidized anyway

That is the core problem: we generally do .hi-red[not] want to subsidize positive emissions

---

# Intensity standards: baseline DWL

---
count: false

# Intensity standards: baseline DWL

---
count: false

# Intensity standards: baseline DWL

---
count: false

# Intensity standards: baseline DWL

---
count: false

# Intensity standards: baseline DWL

---

# Intensity standards: the implicit tax

---
count: false

# Intensity standards: the implicit tax

---
count: false

# Intensity standards: the implicit subsidy

---
count: false

# Intensity standards: the implicit subsidy

---

# Intensity standards: baseline DWL

---

# Intensity standards: DWL after the standard

.market-caption[
In `$H$`, the smaller blue triangle shows the remaining DWL after the implicit tax
]

---
count: false

# Intensity standards: DWL after the standard

---

# Intensity standards: why DWL changes

---

# Why this is second-best

An emissions tax prices all emissions at the same marginal damage

An intensity standard instead mixes:
- an implicit tax on dirtier output
- an implicit subsidy on cleaner output

So it can improve on no policy, but usually cannot hit the socially optimal allocation

---

# Intensity standards: scale vs composition

The policy helps on .hi[composition]: firms shift away from dirtier output

But it can hurt on .hi[scale]: the implicit output subsidy encourages more total production

Total emissions are:
`$$E = \beta_H Q_H + \beta_L Q_L$$`

So total emissions can rise if the scale effect dominates the composition effect

---

# Intensity standards: perverse incentives

Horizontal axis: dirty output `$Q_H$`

Vertical axis: cleaner output `$Q_L$`

Both outputs pollute: `$0 < \beta_L < \beta_H$`

]

---
count: false

# Intensity standards: perverse incentives

At given prices, rising MC eventually exceeds price, so producing more is not always better

This makes the curves ovals, with higher profit toward the center

]

---
count: false

# Intensity standards: perverse incentives

A is the .hi[unregulated profit-maximizing choice]

It maximizes profit, ignoring emissions damages

Moving away from A lowers profit

]

---

# Intensity standards: perverse incentives

The red line holds .hi[total emissions] fixed at A's level
`$$\beta_H Q_H + \beta_L Q_L = K$$`

Its slope is `$-\beta_H/\beta_L$`

]

---
count: false

# Intensity standards: perverse incentives

Above the red line: .hi-red[more emissions than A]

Below the line: fewer emissions

The shaded region has `$E > K$`

]

---

# Intensity standards: perverse incentives

The blue line is the .hi[binding standard]
`$$Q_L = \frac{\beta_H-\sigma}{\sigma-\beta_L}Q_H$$`

On the line, emissions intensity equals `$\sigma$`

]

---
count: false

# Intensity standards: perverse incentives

The blue line is the .hi[binding standard]
`$$Q_L = \frac{\beta_H-\sigma}{\sigma-\beta_L}Q_H$$`

A violates the standard; feasible choices lie on or above the blue line

]

---
count: false

# Intensity standards: perverse incentives

C is the .hi[regulated profit-maximizing choice]

The highest attainable iso-profit curve is tangent to the standard at C

C lies above the red line: emissions increase

]

---

# Intensity standards: intuition

Dirty output falls: `$Q_H^{IS} < Q_H^0$`

Cleaner output rises: `$Q_L^{IS} > Q_L^0$`

]

---
count: false

# Intensity standards: intuition

Tall ovals mean expanding cleaner output reduces profit less than an equal cut in dirty output

Here, .hi[total output rises] while emissions intensity falls

]

---

# Intensity standards: intuition

At C, the purple emissions line lies above A's red line

Extra emissions from `$L$` exceed the savings from `$H$`

]

---