Lecture 02

.title[
# Lecture 02
]
.subtitle[
## Market Failures
]
.author[
### Ivan Rudik
]
.date[
### AEM 4510
]

---

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

# Roadmap

- What are market failures?
- When do they happen?
- What are the consequences?

---

# Market failures and the environment

---

# The ideal world: graphical

In the best case scenario, a market equilibrium leads to the efficient allocation

Example: bread

We have a private bread supply curve (private MC)

We have a private bread demand curve (private MB)

In equilibrium: supply = demand so PMC = PMB = price

For bread, the private costs and benefits are very likely the social costs and benefits

---

# Market equilibrium

![CS is area below demand, above price](02-slides-market-failures_files/figure-html/pmb-pmb-1.png)
]

]

---

# Market equilibrium

.pull-left[
![CS is area below demand, above price](02-slides-market-failures_files/figure-html/pmb-pmb-1.png)
]

What does this mean about the market allocation?
]

---

# Market equilibrium

.pull-left[
![CS is area below demand, above price](02-slides-market-failures_files/figure-html/pmb-pmb-1.png)
]

Why?

]

---

# Market equilibrium

.pull-left[
![CS is area below demand, above price](02-slides-market-failures_files/figure-html/pmb-pmb-1.png)
]

Why?

Consider deviating from `$(P^*, Q^*)$`
]

---

# Market equilibrium

.pull-left[
![CS is area below demand, above price](02-slides-market-failures_files/figure-html/pmb-pmb-1.png)
]

Benefit of last unit `$\geq$` cost of last unit before `$Q^*$`

Competitive market allocations are efficient for private goods

]

---

# Externalities

---

# Externalities

If the world consisted only of competitive markets, private goods, and no third parties, there would be no reason to do anything after Econ 101

That's not the case in the real world

In the real world we have .hi-blue[externalities]

> An externality exists whenever an individual or firm undertakes an action that impacts another individual or firm in an unintended way for which the latter is not compensated (a negative externality), or for which the latter does not pay (a positive externality)

---

# Externalities

What is the problem with externalities for market outcomes and efficiency?

There is not a market for the externality

E.g. at Wegmans or Agway you will not find some important goods on sale:
--

- Cleaner air outside
--

- Biodiversity in the Amazon

The central problem is that there are goods that are .hi-red[not priced], why is this a problem?

Markets rely on prices to signal the social value of goods

---

# Arizona's water problems share a market-failure structure

- Arizona shares a limited river supply with other states, Tribal Nations, and Mexico
- Each user's withdrawal increases scarcity for other users
]

- Pumping from a shared aquifer lowers the water table and can cause land subsidence
- Pumpers do not bear all the costs imposed on nearby wells and future users
]

---

# Colorado River rules are being rewritten for 2027

---

# The post-2026 Colorado River regime is not yet settled

As of August 26, 2026:

- Current operating guidelines and drought plans expire at the end of 2026
- The July 31 final environmental impact statement identifies an adaptive framework through 2036
- The Department of the Interior expects final guidelines before October 1

---

# Saudi farming exposed Arizona groundwater rules

---

# Arizona has begun regulating Ranegras groundwater

- Annual withdrawals have exceeded natural recharge by roughly .hi-red[900%]
- One monitored well has fallen more than .hi-red[240 feet] since the 1980s, and groundwater loss has caused land subsidence
- Arizona designated Ranegras Plain as its eighth Active Management Area in January 2026
- The new rules restrict irrigation on new acreage and require large users to measure and report pumping

---

# Groundwater scarcity is also moving through the courts

- Arizona sued Fondomonte in December 2024, alleging that excessive pumping threatened nearby communities and infrastructure
- In February 2026, the attorney general argued that the new Active Management Area complements rather than replaces the lawsuit
- The state is seeking remedies targeted at the alleged harm, while the Active Management Area regulates pumping throughout the basin

---

# Groundwater enforcement now uses settlements as well as lawsuits

---

# Externalities: several classifications

We can classify externalities in a few ways:

.hi[Production externalities:] 
--
generated by a firm in the process of producing some output (e.g. pollution, innovation)

.hi[Consumption externalities:] 
--
generated by an individual in the process of consuming an output (e.g. congestion, vaccination)

---

# Negative externalities: what is this?

---

# Negative externalities: DDT, shockingly bad for you

Widely used to eradicate Typhus and Malaria

Used to treat lice
]

---

# Negative externalities: DDT, gives you cancer

> A relationship between DDT exposure and reproductive effects in humans is suspected, based on studies in animals. In addition, some animals exposed to DDT in studies developed liver tumors. As a result, today, DDT is classified as a probable human carcinogen.

]

---

# The birth of the environmental movement

---

# Negative externalities: graphical

![externalities shift SMC up](02-slides-market-failures_files/figure-html/neg-ext0-1.png)
]

.hi-red[Social marginal cost (SMC)] is the sum of private marginal cost (PMC) and the external marginal cost (EMC)

Where is the SMC?

]

---

# Negative externalities: graphical

![externalities shift SMC up](02-slides-market-failures_files/figure-html/neg-ext-1.png)
]

.hi-red[Social marginal cost (SMC)] is the sum of private marginal cost (PMC) and the external marginal cost (EMC)

The PMC curve only reflects the **private costs** of making the DDT

It does not account for the external health and wildlife costs

]

---

# Negative externalities: graphical

.pull-left[
![externalities shift SMC up](02-slides-market-failures_files/figure-html/neg-ext-1.png)
]

Adding the private and external marginal costs together gives us the SMC, what we care about from the social planner or regulator's perspective

]

---

# Negative externalities: graphical

.pull-left[
![externalities shift SMC up](02-slides-market-failures_files/figure-html/neg-ext-1.png)
]

Adding the private and external marginal costs together gives us the SMC, what we care about from the social planner or regulator's perspective

The unregulated market gives us `$(P^u,Q^u)$` as an outcome when we want `$(P^*,Q^*)$`

]

---

# Negative externalities: graphical

.pull-left[
![externalities shift SMC up](02-slides-market-failures_files/figure-html/neg-ext-1.png)
]

Adding the private and external marginal costs together gives us the SMC, what we care about from the social planner or regulator's perspective

The unregulated market gives us `$(P^u,Q^u)$` as an outcome when we want `$(P^*,Q^*)$`

What's the social cost of this market failure?

]

---

# Negative externalities: graphical

.pull-left[
![externalities shift SMC up](02-slides-market-failures_files/figure-html/neg-ext-1.png)
]

Negative externalities generate deadweight loss equal to...

]

---

# Negative externalities: graphical

![externalities generate DWL](02-slides-market-failures_files/figure-html/neg-ext-dwl-1.png)
]

Negative externalities generate deadweight loss equal to the .hi-red[red] area

]

---

# Negative externalities: graphical

.pull-left[
![externalities generate DWL](02-slides-market-failures_files/figure-html/neg-ext-dwl-1.png)
]

Negative externalities generate deadweight loss equal to the .hi-red[red] area

This is the difference in SMC and SMB for units bought/sold where SMC > SMB:

Total SMC - SMB from `$Q^*$` to `$Q^u$`

]

---

# Negative externalities: graphical

.pull-left[
![externalities generate DWL](02-slides-market-failures_files/figure-html/neg-ext-dwl-1.png)
]

Negative externalities generate deadweight loss equal to the .hi-red[red] area

This is the difference in SMC and SMB for units bought/sold where SMC > SMB

This is the loss to society caused by the externality in the unregulated private market

]

---

# Negative externalities: graphical

.pull-left[
![externalities generate DWL](02-slides-market-failures_files/figure-html/neg-ext-dwl-1.png)
]

Key takeaway:

]

---

# Negative externalities: graphical

.pull-left[
![externalities generate DWL](02-slides-market-failures_files/figure-html/neg-ext-dwl-1.png)
]

Key takeaway:

The private market produces too much DDT

]

---

# Negative externalities: graphical

.pull-left[
![externalities generate DWL](02-slides-market-failures_files/figure-html/neg-ext-dwl-1.png)
]

Key takeaway:

The private market produces too much DDT

The private actors are not accounting for the .hi[external costs] they are imposing on people who are not in the DDT transaction (e.g. third parties whose health is affected)

]

---

# Estimating marginal damages with EZ-Pass

---

# Positive externalities

---

# Positive externalities

Getting a vaccine or wearing a mask can benefit people who are not involved in that decision
]

---

# Positive externalities: graphical

![externalities shift SMB up](02-slides-market-failures_files/figure-html/pos-ext0-1.png)
]

.hi-red[Social marginal benefit (SMB)] is the sum of private marginal benefit (PMB) and the external marginal benefit (EMB)

Where does the SMB curve lie?

]

---

# Positive externalities: graphical

![externalities shift SMB up](02-slides-market-failures_files/figure-html/pos-ext-1.png)
]

.hi-red[Social marginal benefit (SMB)] is the sum of private marginal benefit (PMB) and the external marginal benefit (EMB)

]

---

# Positive externalities: graphical

.pull-left[
![externalities shift SMB up](02-slides-market-failures_files/figure-html/pos-ext-1.png)
]

.hi-red[Social marginal benefit (SMB)] is the sum of private marginal benefit (PMB) and the external marginal benefit (EMB)

The PMB curve only reflects the **private benefits** of getting a vaccine

]

---

# Positive externalities: graphical

.pull-left[
![externalities shift SMB up](02-slides-market-failures_files/figure-html/pos-ext-1.png)
]

.hi-red[Social marginal benefit (SMB)] is the sum of private marginal benefit (PMB) and the external marginal benefit (EMB)

The PMB curve only reflects the **private benefits** of getting a vaccine

It does not account for the external herd immunity benefits

]

---

# Positive externalities: graphical

.pull-left[
![externalities shift SMB up](02-slides-market-failures_files/figure-html/pos-ext-1.png)
]

Adding the private and external marginal benefits together gives us the SMB, what we care about from the social planner or regulator's perspective

]

---

# Positive externalities: graphical

.pull-left[
![externalities shift SMB up](02-slides-market-failures_files/figure-html/pos-ext-1.png)
]

Adding the private and external marginal benefits together gives us the SMB, what we care about from the social planner or regulator's perspective

The unregulated market gives us `$(P^u,Q^u)$` as an outcome when we want `$(P^*,Q^*)$`

]

---

# Positive externalities: graphical

.pull-left[
![externalities shift SMB up](02-slides-market-failures_files/figure-html/pos-ext-1.png)
]

Adding the private and external marginal benefits together gives us the SMB, what we care about from the social planner or regulator's perspective

The unregulated market gives us `$(P^u,Q^u)$` as an outcome when we want `$(P^*,Q^*)$`

What's the social cost of this market failure?

]

---

# Positive externalities: graphical

.pull-left[
![externalities shift SMB up](02-slides-market-failures_files/figure-html/pos-ext-1.png)
]

Positive externalities generate deadweight loss equal to...

]

---

# Positive externalities: graphical

![pos externalities generate DWL](02-slides-market-failures_files/figure-html/pos-ext-dwl-1.png)
]

Positive externalities generate deadweight loss equal to the .hi-blue[blue] area

]

---

# Positive externalities: graphical

.pull-left[
![pos externalities generate DWL](02-slides-market-failures_files/figure-html/pos-ext-dwl-1.png)
]

Positive externalities generate deadweight loss equal to the .hi-blue[blue] area

This is the difference in SMB and SMC for units where SMC < SMB:

Total SMB - SMC from `$Q^u$` to `$Q^*$`

]

---

# Positive externalities: graphical

.pull-left[
![pos externalities generate DWL](02-slides-market-failures_files/figure-html/pos-ext-dwl-1.png)
]

Positive externalities generate deadweight loss equal to the .hi-blue[blue] area

This is the difference in SMB and SMC for units where SMC < SMB

This is the loss to society caused by the externality in the unregulated private market
]

---

# Positive externalities: graphical

.pull-left[
![pos externalities generate DWL](02-slides-market-failures_files/figure-html/pos-ext-dwl-1.png)
]

The private market produces too few vaccines

The private actors are not accounting for the external benefits they confer on people who are not in the vaccine transaction (e.g. third parties whose health is affected)

]

---

# News report: the first U.S. measles deaths of 2026

---

# MMR coverage was near 95% before COVID

![U.S. kindergarten two-dose MMR coverage, 2011–12 through 2025–26](02-slides-market-failures_files/figure-html/mmr-coverage-trend-1.png)

---

# Measles returned as vaccination rates fell

On August 25, 2026, Pennsylvania confirmed:

- .hi-red[Two measles-associated deaths], both among unvaccinated people
- The state's first measles-associated deaths in 35 years
- 393 confirmed cases across 28 counties during 2026

Two doses of the MMR vaccine provide approximately .hi-blue[97% protection]

---

# Vaccination and infection create externalities

- It lowers an individual's probability of infection
- When it also lowers transmission, it protects other people

- Exposure shifts uncompensated risk to other people
- Private choices omit some expected health costs

---

# Why do externalities arise?

Typically one of two reasons:

1. Poorly defined property rights
  - Who owns the right to the air?
--

2. High transactions costs
  - Hard to bargain over desired air quality with millions of people

Let's conceptualize a model of efficient bargaining using an Edgeworth Box

---

# Why do externalities arise? Edgeworth Box

- Two individuals: A and B
- Two private goods: X and Y

Each individual begins with an initial endowment of each good:
- `$A: w_X^A, w_Y^A$`
- `$B: w_X^B, w_Y^B$`

This gives us a total endowment:
- `$W_X = w_X^A + w_X^B$`
- `$W_Y = w_Y^A + w_Y^B$`

---

# Why do externalities arise? Edgeworth Box

Each person starts with an ordinary consumption diagram

Rotate B's diagram 180 degrees and place it over A's so the endowment points coincide

The result is a box with width `$W_X$` and height `$W_Y$`

---

# Why do externalities arise? Edgeworth Box

---

# Why do externalities arise? Edgeworth Box

.pull-left[
<center>
<img src="02-slides-market-failures_files/figure-html/edgeworth1-box-1.png" alt="" width="100%" />
</center>
]

Total horizontal distance is `$W_X$`

Initial endowment is given by the empty circle

Initial indifference curves for A and B are `$U_A(0)$` and `$U_B(0)$`
]

---

# Why do externalities arise? Edgeworth Box

.pull-left[
<center>
<img src="02-slides-market-failures_files/figure-html/edgeworth1-box-1.png" alt="" width="100%" />
</center>
]

e.g. can we make both A and B better off?
]

---

# Why do externalities arise? Edgeworth Box

<img src="02-slides-market-failures_files/figure-html/edgeworth1-lens-1.png" alt="" width="100%" />
</center>
]

The shaded lens is the set of Pareto improvements

If we move anywhere in the lens of their initial indifference curves we have a Pareto improvement
]

---

# Why do externalities arise? Edgeworth Box

<img src="02-slides-market-failures_files/figure-html/edgeworth1-optimum-1.png" alt="" width="100%" />
</center>
]

At these allocations no further Pareto improvement is possible

This is where the indifference curves are .hi[tangent] to one another, like the filled-in point
]

---

# Why do externalities arise? Edgeworth Box

<img src="02-slides-market-failures_files/figure-html/edgeworth1-tangency2-1.png" alt="" width="100%" />
</center>
]

---

# Why do externalities arise? Edgeworth Box

<img src="02-slides-market-failures_files/figure-html/edgeworth1-tangency3-1.png" alt="" width="100%" />
</center>
]

---

# Why do externalities arise? Edgeworth Box

<img src="02-slides-market-failures_files/figure-html/edgeworth1-paretoset-1.png" alt="" width="100%" />
</center>
]

.pull-right[
This segment is the set of Pareto-efficient allocations reachable by voluntary trade from the endowment
]

---

# Why do externalities arise? Edgeworth Box

In a properly functioning market:

- The endowment point is well-established
--

- A and B can trade X and Y to some Pareto improving point
--

- They continue trading until they achieve a Pareto optimal allocation
--

- This allocation lies on the .hi[contract curve]: the line consisting of all Pareto efficient allocations

---

# Why do externalities arise? Edgeworth Box

The contract curve collects every allocation where A's and B's indifference curves are tangent

---

# Why do externalities arise? Edgeworth Box

Starting from `$c$`, voluntary trade reaches only the thick segment

This is the achievable Pareto set for that endowment

---

# Why do externalities arise? Edgeworth Box

A different initial allocation, `$c'$`, gives a different achievable Pareto set on the same contract curve

---

# Why do externalities arise? Edgeworth Box

Now suppose Y is not a private good, but a public good/bad, e.g. smoke

This means that A and B consume the .hi[exact same level of Y]

Unlike our regular Edgeworth Box, now Y increases for .hi[both] A and B as we move to the top of the slide (before Y increased for B as we moved to the bottom)

Suppose that A likes Y, but B does not

Suppose both start off with the same quantity of X

---

# Why do externalities arise? Edgeworth Box

<img src="02-slides-market-failures_files/figure-html/edgeworth3-endowments-1.png" alt="" width="100%" />
</center>
]

- W1 (B has property rights)
- W2 (A has property rights)

Think about why these are where we must start

]

---

# Why do externalities arise? Edgeworth Box

.pull-left[
<center>
<img src="02-slides-market-failures_files/figure-html/edgeworth3-endowments-1.png" alt="" width="100%" />
</center>
]

]

---

# Why do externalities arise? Edgeworth Box

<img src="02-slides-market-failures_files/figure-html/edgeworth3-w1-conflict-1.png" alt="" width="100%" />
</center>
]

A wants to have more Y, but this imposes a cost on B

]

---

# Why do externalities arise? Edgeworth Box

<img src="02-slides-market-failures_files/figure-html/edgeworth3-w1-pay-1.png" alt="" width="100%" />
</center>
]

A wants to have more Y, but this imposes a cost on B

Therefore, A has to .hi[pay] B to get more Y

]

---

# Why do externalities arise? Edgeworth Box

<img src="02-slides-market-failures_files/figure-html/edgeworth3-w1-opt-1.png" alt="" width="100%" />
</center>
]

A wants to have more Y, but this imposes a cost on B

Therefore, A has to .hi[pay] B to get more Y

A pays B in units of X, move to Z1, Pareto optimum

]

---

# Why do externalities arise? Edgeworth Box

<img src="02-slides-market-failures_files/figure-html/edgeworth3-w2-start-1.png" alt="" width="100%" />
</center>
]

]

---

# Why do externalities arise? Edgeworth Box

<img src="02-slides-market-failures_files/figure-html/edgeworth3-w2-conflict-1.png" alt="" width="100%" />
</center>
]

B wants to have less Y, but this imposes a cost on A
]

---

# Why do externalities arise? Edgeworth Box

<img src="02-slides-market-failures_files/figure-html/edgeworth3-w2-pay-1.png" alt="" width="100%" />
</center>
]

B wants to have less Y, but this imposes a cost on A

Therefore, B has to .hi[pay] A to get less Y

]

---

# Why do externalities arise? Edgeworth Box

<img src="02-slides-market-failures_files/figure-html/edgeworth3-w2-opt-1.png" alt="" width="100%" />
</center>
]

B wants to have less Y, but this imposes a cost on A

Therefore, B has to .hi[pay] A to get less Y

B pays A in units of X, move to Z2, Pareto optimum

]

---

# Why do externalities arise? Edgeworth Box

In the previous example we were able to achieve the Pareto optimum even with a public good / externality

Why?

1. Property rights were assigned to either A or B

2. Transactions costs were low (didn't have to pay a fee to trade X)

---

# Property rights and externalities

A solution to many externalities is to just assign property rights and let the market do its thing

We'll talk about a few ways that we can assign property rights

---

# Transactions costs and externalities

Now suppose there were many non-smokers

Even if they were assigned the property rights, it might be hard for them to bargain
  - Takes a lot of time to find something that works for everyone
  - Negotiating over how much X each person gets

The costs of bargaining may exceed the benefits and we end up stuck at W2

---

# Transactions costs and externalities

Road noise: drivers implicitly have property rights to make noise around roads

Even if you prefer quiet, you cannot negotiate a payment with every loud car that might pass

---

# The free-rider problem

Externalities and public goods/bads often exhibit many of the same features

Both are subject to the .hi[Free-Rider Problem]

> A type of market failure that occurs when those who benefit from resources, public goods (such as public roads or hospitals), or services of a communal nature do not pay for them or under-pay

e.g.

- people don't pay their taxes for publicly-provided services
- non-smokers will wait for others to pay in order to reduce smoke

---

# The provision of public goods

---

# Public goods

How do we efficiently provide public goods?

We know:

- Private goods: PMB = PMC `$\leftrightarrow$` SMB = SMC
- Goods with negative externalities: PMB = SMC `$\leftrightarrow$` SMB = SMC
- goods with positive externalities: SMB = PMC `$\leftrightarrow$` SMB = SMC

Suppose we have a public good, e.g. air quality in a city

How do we decide the socially efficient level of air quality?

---

# Public goods

Optimal provision is .hi-blue[always] given by: SMB = SMC

What are the SMB and SMC for a public good?

Public goods have two defining characteristics:

---

# Optimal provision of public goods

What does this mean for how we determine the optimal provision?

When we count up the SMB, we need to add up .hi-blue[everyone's] PMB:

If we ignore the fact that public goods are non-rival, we get underprovision of the good

e.g. the free market underprovides clean air, national defense, etc

---

# Modeling the provision of public goods

How do we model public goods?

First: how do we aggregate marginal benefit or demand curves?

Here's how to think about it:

Private goods are rival, only one person can consume each unit

At each price, what is the total quantity that is demanded?

At each price, we need to add up quantities

Private goods: we add demand curves horizontally

---

# Modeling the provision of public goods

Public goods are non-rival, multiple people can consume each unit

At each quantity, what is the total marginal benefit?

At each quantity, we need to add up PMBs/prices

Public goods: we add demand curves vertically

---

# Public goods: graphical

![public good demand curves](02-slides-market-failures_files/figure-html/pub-good-1.png)
]

3 groups: asthmatic families (F), outdoor workers (W), healthy adults (H)

Each has a different marginal benefit for air quality:
- Families: MB = 12-Q
- Workers: MB = 10-Q
- Adults: MB = 6-Q
- MC of abatement: MC = Q

]

---

# Public goods: graphical

![aggregate public good demand](02-slides-market-failures_files/figure-html/pub-good-agg-1.png)
]

Now we need to aggregate them to get the .hi[social marginal benefit]

We do so by adding up the demand curves vertically:

At each Q, sum the MBs

]

---

# Public goods: graphical

.pull-left[
![aggregate public good demand](02-slides-market-failures_files/figure-html/pub-good-agg-1.png)
]

Why is the aggregate demand curve kinked?

Because at each level of air quality, only certain groups are willing to pay

Kinks are at the dotted lines, where PMBs hit zero

]

---

# Public goods: graphical

.pull-left[
![aggregate public good demand](02-slides-market-failures_files/figure-html/pub-good-agg-1.png)
]

At quantities > 10, only families are willing to pay

At quantities > 6 and <= 10, only families and workers are willing to pay

At quantities `$Q \leq 6$`, all groups are willing to pay

]

---

# Public goods: graphical

.pull-left[
![aggregate public good demand](02-slides-market-failures_files/figure-html/pub-good-agg-1.png)
]

The SMB curve is:

28 - 3Q for Q <= 6

22 - 2Q for 6 < Q <= 10

12 - Q for 10 < Q <= 12

Summing the PMBs over the relevant range of Q

]

---

# Public goods: graphical

![aggregate public good demand](02-slides-market-failures_files/figure-html/pub-good-optim-1.png)
]

The optimal provision of the public good is where the MC curve crosses the SMB curve

This is across the middle segment

Q = 22 - 2Q `$\Rightarrow$` Q = 22/3

The optimal quantity of Q = 22/3 is greater than the quantity any individual group would be willing to purchase

]

---

# Public goods financing

The socially optimal quantity is greater than the individual privately optimal quantities

This means that the MC of provision, MC = 22/3

Is this the price the groups pay?

If the government is able to provide the good, how does it finance the cost of raising air quality above the unregulated level?

---

# Public goods financing

It can charge each group a constant share of this price

What share is everyone charged?

Families pay: 14/3
Workers pay: 8/3
Adults: free

Notice that the prices sum to the marginal cost!

Since the good is non-rival, this is enough to finance the cost

---

# Public goods financing

What is a key problem with Lindahl pricing?

People can lie about which group they're in

Workers might say they're healthy adults

It requires perfect information on behalf of the regulator

---

# A second public good: flood protection

.pull-left[
<center>
<img src="files/02-levee-new-orleans.jpg" alt="" width="100%" />
</center>
.smallest[Levee and floodwall, 17th Street Canal, New Orleans]

]

- Non-rival: my protection does not reduce yours
- Non-excludable: the wall cannot keep the flood out for payers only

How high should the levee be?
]

---

# Flood protection: graphical

![levee marginal benefit curves](02-slides-market-failures_files/figure-html/levee-1.png)
]

3 groups: homeowners (H), businesses (B), and hill residents (R)

Marginal benefits of levee height:
- Homeowners: MB = 16-2Q
- Businesses: MB = 9-Q
- Hill residents: MB = 4-Q
- MC of provision: MC = 2Q

]

---

# Flood protection: graphical

.pull-left[
![levee marginal benefit curves](02-slides-market-failures_files/figure-html/levee-1.png)
]

3 groups: homeowners (H), businesses (B), and hill residents (R)

Marginal benefits of levee height:
- Homeowners: MB = 16-2Q
- Businesses: MB = 9-Q
- Hill residents: MB = 4-Q
- MC of provision: MC = 2Q

]

---

# Flood protection: graphical

![aggregate levee demand](02-slides-market-failures_files/figure-html/levee-agg-1.png)
]

The SMB curve is:

29 - 4Q for Q <= 4

25 - 3Q for 4 < Q <= 8

9 - Q for 8 < Q <= 9

Kinks where R's and H's MBs hit zero

]

---

# Flood protection: graphical

The SMB curve is:

29 - 4Q for Q <= 4

25 - 3Q for 4 < Q <= 8

9 - Q for 8 < Q <= 9

]

---

# Flood protection: graphical

![optimal levee height](02-slides-market-failures_files/figure-html/levee-optim-1.png)
]

Optimal height: MC crosses SMB on the middle segment

2Q = 25 - 3Q `$\Rightarrow$` Q = 5

Lindahl prices at Q = 5:
- Homeowners pay: 6
- Businesses pay: 4
- Hill residents: free

The prices again sum to MC = 10

]