# 23.4 Production Externalities

In the example of Ken smoking, we had two agents, one of whom was engaged in an activity that harmed the other. In that example, it was a *consumption externality*, because the externality (the smoke Chris had to smell) was a byproduct of a consumption activity (Ken enjoying his cigarettes). We can also think of *production externalities*, which are byproducts of production processes, like pollution.

The canonical example of this is a steel mill which lies upstream along a river from a fishery, and as part of its production process it dumps toxic waste in the river, harming the fishery. The more steel the mill produces, the more polluted the water becomes. Let’s say that each ton of steel produces waste which causes $€2$ of damage to the fishery.

The steel mill has a cost function given by
\(c_S(S) = \frac{S^2}{200}\)
and the fishery has a cost function given by
\(c_F(F,S) = \frac{F^2}{10}+2S\)
The $S^2/200$ is the *total private cost* to the steel mill from producing $S$ tons of steel. The $F^2/10$ is the *total private cost* to the fishery of producing fish. The $2S$ is the *total external cost* imposted by the steel mill on the fishery.

Both the steel mill and the fishery sell their products in competitive markets; steel sells at a price of $p_S = 10$, and fish sell at a price of $p_F = 50$. We can think of this meaning that the *marginal benefit* of steel to consumers is $€10$ per ton, and the *marginal benefit* of fish to consumers is $€50$ per fish.

### Efficient Outcome

What is the optimal quantity of steel for this economy, if we define “optimal” as meaning the quantity that maximizes total combined profits? These profits are \(\begin{aligned} \Pi(S,F) &= \pi_S(S) & &+ \pi_F(F,S)\\\\ &= p_SS - c_S(S) & &+ p_FF - c_F(F,S)\\\\ &= 10S - \frac{S^2}{200} & &+ 50F - \left[\frac{F^2}{10} + 2S\right] \end{aligned}\) The first-order conditions of this are \(\begin{aligned} \frac{\partial \Pi(S,F)}{\partial S} &= 10 - \frac{S}{100} - 2 = 0 & \Rightarrow S^\star &= 800\\\\ \frac{\partial \Pi(S,F)}{\partial F} &= 50 - \frac{F}{5} = 0 & \Rightarrow F^\star &= 250 \end{aligned}\) With this level of production, the profits of the two fims are \(\begin{aligned} \pi_S &= 10 \times 800 - \frac{800^2}{200} = 4,800\\\\ \pi_F &= 50 \times 250 - \frac{250^2}{10} - 2 \times 800 = 4,650\\\\ \end{aligned}\) so total profits are 9,450.

### Equilibrium Outcome

Because the negative externality just affects the fishery’s fixed cost, its production decision remains the same; it will produce $F = 250$. The steel mill will maximize its own profits of \(\pi_S(S) = 10S - \frac{S^2}{200}\) by setting marginal profit equal to zero \(\pi^\prime_S(S) = 10 - \frac{S}{100} = 0 \Rightarrow S = 1000\) Therefore profits for the two firms will be \(\begin{aligned} \pi_S &= 10 \times 1000 - \frac{1000^2}{200} = 5,000\\\\ \pi_F &= 50 \times 250 - \frac{250^2}{10} - 2 \times 1000 = 4,250\\\\ \end{aligned}\) so total profits are 9,250.

## Analysis of the Problem

Compared to the social optimum, the equilibrium outcome has 200 less in combined profits: specifically, the steel mill makes 200 more in profits, and the fishery makes 400 less. The steel mill produces up until the point where its marginal benefit (the price, 10) equals its marginal private cost. However, it imposes a cost of 2 for each unit of steel it produces; so once its marginal cost exceeds 8, it’s making less marginal profit than the cost it imposes on the fishery.

## Solutions to the Problem

We could, as before, impose a “Pigouvian tax” on the steel mill; but it’s not clear that *every* steel mill imposes exactly 2 dollars of marginal external cost on a neighbor, so that may not be efficient.

However, there’s an alternative solution: assigning property rights to pollution. Let’s say we give the fishery the *right* to clean water, and that it can charge the steel mill a price $c$ per ton of steel. Then the two firms’ profit functions become
\(\begin{aligned}
\pi_S(S) &= 10S - \frac{S^2}{200} - cS\\\\
\pi_F(F,S) &= 50F - \left[\frac{F^2}{10} + 2S\right] + cS = 50F - \frac{F^2}{10} + (c-2)S
\end{aligned}\)
The fishery has the *right* to deny the steel mill any production at all; but if it charged any price $c \ge 2$, it would be happy to let the steel mill pollute as much as it liked, because each unit of pollution would earn the fishery an additional $c - 2$ in profits. Therefore the lowest price it would accept per ton of steel would be $c = 2$. This would result in the steel mill perfectly internalizing its externality, so it would produce $S = 800$. It would earn a profit of 3,200 and the fishery would earn a profit of 6,250.

Now let’s suppose we said the steel mill had the right to produce as much as it wanted to, but the steel mill could charge the fishery $c$ for every ton of steel it cut back production below its equilibrium level of 1000. Then the two firms’ profit functions become
\(\begin{aligned}
\pi_S(S) &= 10S - \frac{S^2}{200} + (1000-S)c\\\\
\pi_F(F,S) &= 50F - \left[\frac{F^2}{10} + 2S\right] - (1000-S)c = 50F - \frac{F^2}{10} - 1000c + (c-2)S
\end{aligned}\)
We can see that the most the fishery would be willing to pay would be $c = 2$; and if the steel mill could earn 2 for each ton of steel it *didn’t* produce, it would again choose the optimal quantity of $S = 800$. In this case, the steel mill would earn a profit of 5,200, and the fishery would earn a profit of 4,250.

## Coase Theorem

In this case, we can note that the assignment of property rights results in the efficient quantity, *regardless of who has the property rights*. This is an illustration of the Coase Theorem, which states that, under certain circumstances (including low transaction costs), bargaining between parties can lead to the Pareto efficient outcome. However, it’s worth noting that the solutions weren’t symmetric: assigning the right to clean water to the fishery, while improving overall welfare, hurt the steel mill relative to the equilibrium outcome. Assiging property rights to the steel mill, on the other hand, made the steel mill better off and the fishery no worse off than in the equilibrium outcome. So even though assigning property rights can represent Pareto improvement, it can also make one party much better off and the other slightly less worse off.