Externalities and Public Goods

Externalities

Externality (arising from a given good/service)

A benefit (positive externality) or cost (negative externality) imposed on third-party agent(s) (not the seller or buyer of the good) by the production/consumption, who did not choose to incur the benefit or cost. Thus, externalities arise when the actions of one agent directly affect the welfare or production possibilities of another agent, and this effect is not transmitted through market prices.

So far, we assumed that there were no externalities:

In consumer theory, the consumption of one consumer does not affect the utility level of another consumer. Of course, this rules out envy or guilt.
In producer theory (perfect competition), the production of one firm does not affect the profits of another firm. Thus, in perfect competition (without externalities), firms can ignore other firms' behavior and focus only on the market price.
Without externalities, the market equilibrium maximizes total surplus (we say that the market equilibrium delivers an efficient outcome).

Externalities

Example

Consider two firms such that firm A imposes a negative externality on firm B: when firm A increases production, the marginal cost of firm B also increases. In particular, assume their marginal cost functions are:

MC_A=2q_{A} \quad \text{and} \quad MC_B=q_A+2q_B

Remember that to maximize profits, a firm will increase production if its marginal cost is below the market price, and decrease production if the marginal cost is above the market price. Thus, the individual supply coincides with the firm's marginal cost curve (provided the MC is increasing).

Thus, the supply curves are:

q_{A}^{S}=P_{A}/2 \quad \text{and} \quad q_{B}^{S}=(P_{B}-q_{A})/2

Externalities Example (cont.)

The Market Equilibrium

The supply curves are:

q_{A}^{S}=P_{A}/2 \quad \text{and} \quad q_{B}^{S}=(P_{B}-q_{A})/2

Now, assume the following demand functions:

q_{A}^{D}=120-P_{A} \quad \text{and} \quad q_{B}^{D}=200-2P_{B}

Notice that in a market equilibrium (supply = demand) for good A:

q_{A}=40 \quad \text{and} \quad P_{A}=80

Using this information, $q_{B}^{S}=(P_{B}-40)/2$ and so equilibrium in the market for good B yields:

q_{B}=24 \quad \text{and} \quad P_{B}=88

Externalities

Example cont.

Notice that an increase in the production of firm A shifts the supply curve of firm B: In order for firm B to keep producing the same quantity (since the marginal cost is now higher), the market price for firm B must also increase. Otherwise, if the price $P_{B}$ remains fixed, firm B will lower its production.

Using the market equilibrium, compute the total surplus: consumer and producer surplus. In this case, however, due to the externalities the total surplus is not maximized at the market equilibrium because firm A ignores the cost imposed on firm B by A's production (the externality).

	Market A	Market B	Total
CS	800	144	944
PS	1600	576	2176
Total	2400	720	3120

Check the following: If firm A were to decrease its production by a "little", the total surplus would increase.

	Market A	Market B	Total
CS	612.5	169	781.5
PS	1750	676	2426
Total	2362.5	845	3207.5

The Efficient Outcome Under Externalities

Maximizing The Social Surplus

Before finding the efficient quantities, let's quantify the externality imposed on firm B by firm A. The externality raises the unit cost of production of good B by $q_A$ and thus, the total cost of firm B is increased by $q_A\times q_B$. That amount is the externality imposed on firm B by firm A.

Internalizing the externality.
Say we impose a tax, $t=q_B$ on good A. The tax revenue will be $q_B\times q_A$, which matches the externality imposed on firm B by firm A. However, now firm A has to pay for it. We internalized the externality.

The social marginal cost of producing A is:

SMC_A=\underbrace{2q_{A}}_{\text{A's private MC}}+\underbrace{q_{B}}_{\text{marginal external cost}}

The social marginal cost of producing B is:

$SMC_B= q_A+2q_B$

Notice that the $SMC_B$ coincides with the private marginal cost of producing B, $MC_B$, since the production of B does not generate any externalities.

To find the efficient quantities we need to solve the following system of equations:

P_A=SMC_A \quad \text{and}\quad P_B=SMC_B.

	Market A	Market B	Total
CS	473.4	191.7	665.1
PS	1798.8	766.9	2565.7
Total	2272.2	958.6	3230.8

Coase's Theorem

statement

The socially efficient quantities can be reached if:

Property rights are clearly assigned and
Bargaining costs are low

The Coase theorem implicitly assumes that firms are price takers, so in the examples below, you should assume $P_A=80$ and $P_B=88$ regardless of the quantity produced by firm A and B.

Coase's Theorem in action

Assume that legal rights are such that firm B has the right to avoid the negative externalities. In this case, firm B will require that firm A cease production so $q_A=0$.

Find:

Firm B's quantity and profit when $q_A=0$.
What is the minimum amount of compensation that firm B would require from firm A to allow firm A to produce at the socially efficient level $q_A=30$?

Coase's Theorem in action

Consider now the alternative case in which firm A has the right to produce as much as it wants. You already saw the equilibrium in this case: $q_A=40$ and $q_B=24$.

Find:

What is the maximum payment that firm B is willing to pay as compensation to firm A so that A would reduce its production to the efficient level $q_A=30$?
What is the minimum amount firm A demands in this case?

Coase's Theorem in action

State $q_A=0$ — Firm B: $\pi_B=1936$

State $q_A=30$ — Firm B: $\pi_B=841$

Assume that legal rights are such that firm B has the right to avoid the negative externalities. In this case, firm B will require that firm A cease production so $q_A=0$.

Assuming firm B has the right to avoid externalities, find:

Firm B's profit at $q_A=0$

At $P_B=88$ (price taker) and without externality, $MC_B=2q_B=88 \Rightarrow q_B=44$. Total cost $C_B$ is the triangle under $MC_B$:

$$\begin{align*} & C_B = \tfrac{1}{2}\cdot 44\cdot 88 = 1936 \\ & \pi_B = 88(44) - 1936 = 1936 \end{align*}$$

Min. comp. B requires to allow $q_A=30$

At $q_A=30$, $MC_B=30+2q_B=88 \Rightarrow q_B=29$. Total cost $C_B$ = rectangle + triangle:

$$\begin{align*} & C_B = \underbrace{30\cdot 29}_{\text{rect}} + \underbrace{\tfrac{1}{2}\cdot 29\cdot 58}_{\text{tri}} = 1711 \\ & \pi_B = 88(29) - 1711 = 841 \\ & \text{Min. comp.} = 1936 - 841 = 1095 \end{align*}$$

Firm A's profit at $q_A=0$

A produces nothing:

$$\begin{align*} & \pi_A = 0 \end{align*}$$

Max. A offers to produce $q_A=30$

At $P_A=80$ (price taker), total cost $C_A$ is the triangle under $MC_A=2q_A$:

$$\begin{align*} & C_A = \tfrac{1}{2}\cdot 30\cdot 60 = 900 \\ & \pi_A = 80(30) - 900 = 1500 \\ & \text{Max. offer} = 1500 - 0 = 1500 \end{align*}$$

← / → to slide

State $q_A=0$ — Firm A: $\pi_A=0$

State $q_A=30$ — Firm A: $\pi_A=1500$

Coase's Theorem in action

State $q_A=40$ — Firm A: $\pi_A=1600$

State $q_A=30$ — Firm A: $\pi_A=1500$

Consider now the alternative case in which firm A has the right to produce as much as it wants. You already saw the equilibrium in this case: $q_A=40$ and $q_B=24$.

Assuming firm A has the right of imposing externalities, find:

Firm A's profit at $q_A=40$

At $P_A=80$ (price taker), $MC_A=2q_A=80 \Rightarrow q_A=40$. Total cost $C_A$ is the triangle under $MC_A$:

$$\begin{align*} & C_A = \tfrac{1}{2}\cdot 40\cdot 80 = 1600 \\ & \pi_A = 80(40) - 1600 = 1600 \end{align*}$$

Min. amount A demands to $\searrow q_A=30$

At $q_A=30$, still $P_A=80$:

$$\begin{align*} & C_A = \tfrac{1}{2}\cdot 30\cdot 60 = 900 \\ & \pi_A = 80(30) - 900 = 1500 \\ & \text{Min. demand} = 1600 - 1500 = 100 \end{align*}$$

← / → to slide

Firm B's profit at $q_A=40$

At $P_B=88$, $MC_B=40+2q_B=88 \Rightarrow q_B=24$. Total cost $C_B$ = rectangle + triangle:

$$\begin{align*} & C_B = \underbrace{40\cdot 24}_{\text{rect}} + \underbrace{\tfrac{1}{2}\cdot 24\cdot 48}_{\text{tri}} = 1536 \\ & \pi_B = 88(24) - 1536 = 576 \end{align*}$$

Max. payment B offers for $q_A=30$

At $q_A=30$, $MC_B=30+2q_B=88 \Rightarrow q_B=29$:

$$\begin{align*} & C_B = \underbrace{30\cdot 29}_{\text{rect}} + \underbrace{\tfrac{1}{2}\cdot 29\cdot 58}_{\text{tri}} = 1711 \\ & \pi_B = 88(29) - 1711 = 841 \\ & \text{Max. payment} = 841 - 576 = 265 \end{align*}$$

State $q_A=40$ — Firm B: $\pi_B=576$

State $q_A=30$ — Firm B: $\pi_B=841$

Coase's Theorem in action

Summary of the two cases. In both assignments of rights, moving to $q_A=30$ generates gains from trade that both firms can share:

B has the right ($q_A=0 \to q_A=30$)

A is willing to pay up to $1500$; B requires at least $1095$. Any transfer $1095 \le T \le 1500$ from A to B is individually rational and makes both strictly better off.

A has the right ($q_A=40 \to q_A=30$)

B is willing to pay up to $265$; A requires at least $100$. Any transfer $100 \le T \le 265$ from B to A is individually rational and leaves both strictly better off.

Either way, the parties can agree on a compensation scheme that moves $q_A$ to $30$ — the precise split depends on bargaining power, but a deal always exists.

Caveat. Because prices are constant in this example (both firms are price takers), $q_A=30$ is not the socially efficient quantity — the welfare-maximizing level is $q_A=24$. What the example illustrates is the logic of the Coase Theorem: when property rights are well-defined and bargaining is costless, private negotiation exhausts all welfare-improving trades, independently of who holds the right.

Public Goods

Public goods are a particular case of positive (consumption) externality. A public good is:

Non-rival

If one person consumes the good, the amount that others can consume of the good remains the same.

Non-excludable

It is too costly to exclude one person from consuming the good.

	Rival	Non-Rival
Excludable	? Private Good	? Club Good
Non-Excludable	? Common Resource	? Public Good

As in the case of positive externalities (production of the public good is a particular case of positive externalities), the market equilibrium level of the public good is below the socially efficient level.