# Pricing a negative externality under high uncertainty and severe non-linearity

Take a negative externality such as excess greenhouse gas emissions.

We know some awkward things about the economic cost of this externality:

There is fairly large uncertainty about the timing of damage associated with a unit of emissions. And about the scale of economic consequences of that damage.

Furthermore, there are significant non-linearities: if the world were to emit just one more tonne of CO2, the unit cost of damage is tiny. If we hit a trillion tonnes total emitted, then we're looking at costs that start to hit percents of global GDP. If we burn most of our remaining fossil fuel resources, then we're looking at damage potentially of the order of tens of percent of global GDP, and potentially threatening the existence of human civilisation as we know it.

Normally, a pigouvian tax or similar would have a unit price that represented the cost of damage: here, the uncertainty means that there would need to be some sort of option pricing within that. But also, it is typically a fixed unit price, which cannot be efficient if the damage costs are highly non-linear. What sort of correction to market pricing could take account both of the uncertainties, and the non-linearities?

I've tagged with , but that's just one possible instrument, so I'll add too (though my own hunch is that Pigou, rather than Coase, holds the answer here - YMMV)

• To clarify, are you asking about how the we should structure the pigouvian tax on businesses? – Mathematician Nov 20 '14 at 20:53
• @Mathematician good question - I've edited to clarify that a Pigouvian tax is just one option, and e.g. Coasian bargaining could be another. As to whether the tax would be applied to primary sources, intermediaries, or consumers, I think that's not directly relevant to the question of how to account for the uncertainty and non-linearity, but I could be wrong about that. – 410 gone Nov 21 '14 at 11:45

This should probably be a comment, but it's too long so I am posting it as an answer.

I'm not sure I necessarily agree with "[A Pigouvian tax] is a fixed unit price, which cannot be efficient if the damage costs are highly non-linear." But I do definitely agree with the fact that non-linearity causes practical problems for a Pigovian implementation, as I note below.

Suppose that the private benefit from emissions is $\pi(e)$, and that emissions cause external damages of $d(e)$. Assume that more emissions lead to greater damage ($d'(e)>0$), that the marginal damages are increasing ($d''(e)>0$—so damages are non-linear), and that profits are concave ($\pi''(e)\leq0$).

The self-interested individual will maximise his payoff by solving $\pi'(e)=0$. In contrast, social welfare is maximised when $\pi'(e)-d'(e)=0$. As usual, there is an inefficiently high level of emissions.

Write $e^*$ for the level of emissions that is socially optimal, and suppose that we set a Pigouvian tax at $t=d'(e^*)$ per unit. This means that the individual's optimisation problem is now $$\max_e \pi(e)-te=\max_e \pi(e)-d'(e^*)e.$$ The corresponding first-order condition is $$\pi'(e)-d'(e^*)=0.$$ This is solved by $e=e^*$, so the private individual is therefore induced to choose the efficient level of emissions even though the external damages are non-linear. Intuitively, if decisions are made at the margin then all the Pigouvian tax needs to do is to ensure that the private marginal benefit is equal to zero exactly when the social marginal benefit it. That the tax does not equal the external cost for all of the infra-marginal units has no bearing on the marginal decision problem.

Although the Pigouvian tax implements the efficient level of emissions, its implementation has important normative consequences. In particular, the polluting individual is paying a tax of $d'(e^*)$ for every unit of emission, even though the marginal external cost of the first $e^*$ units is less than this (because $d$ is convex). This implies that the polluter is required to pay more than the cost that he imposes on society, which some might not consider equitable.

There is also an awkward inter-temporal dimension to this problem that is not considered in the above static "textbook" treatment. Because damages are non-linear, if I allow $e$ units of pollution "today" then the marginal cost (and hence the optimal level) of emissions "tomorrow" will be different to if I had allowed $e+1$ units today. This seems to suggest that what is really needed (if the solution is, indeed, to be Pigouvian) is a time-varying tax and the problem becomes one of optimal control. That is, rather than calculating a single tax rate, we need to find the time-path $\{e_t^*\}_{t=0}^{\infty}$ that characterises the optimal evolution of emissions and change the tax over time to implement this. This increases the burden on the central authority, which is now required to calculate not only the emissions it wants to allow today, but also the emissions that it anticipates allowing into the indefinite future. This makes the informational problems noted in the question significantly more acute.