# Leibniz

## 12.3.1 Pigouvian taxes

A Pigouvian tax may be used by the government to address the problem of market failure due to an externality such as pollution. In this Leibniz, we show mathematically how to find the Pigouvian tax that achieves Pareto efficiency in our model of rice production causing air pollution.

In our analysis of the external effects of air pollution due to RWCS (Leibniz 12.1.1), we show that the profit-maximizing farmers choose their output $Q_p$ so that their marginal private cost is equal to the market price:

But the social surplus is maximized at the level of output $Q^*(\lt Q_p)$ at which the marginal social cost of rice is equal to the price:

$Q^*$ is the Pareto-efficient level of output. Remember also that the social cost $C(Q)$ can be written as the sum of the private cost $C_p(Q)$ and the external cost $C_e(Q)$ imposed on the city-dwellers by stubble burning. So we can write the equation for the Pareto efficient output $Q^*$ as:

Now suppose that the government imposes a tax of $x$ units of money for each tonne of rice produced. The farmers’ cost of producing $Q$ tonnes of rice is now $C_p(Q) + xQ$. Differentiating with respect to $Q$, we see that the marginal cost incurred by the farmers is $C'_{p}(Q) +x$: taxes raise the marginal cost of production. As before, farmers choose their output so that the marginal cost is equal to the price, but since the marginal cost has changed, so does their choice of output. They will produce $Q^{\dagger}$, where:

Since the private marginal cost $C'_{p}(Q)$ is an increasing function of $Q$, $Q^{\dagger}$ is smaller than $Q_{p}$ if $x$ is positive—and the higher the tax, the lower the output produced.

Comparing this equation with the previous one, we can see how the government can achieve Pareto efficiency. If the tax $x$ is equal to $C_e '(Q^*)$, then the equation that determines $Q^{\dagger}$ is satisfied when $Q^{\dagger}=Q^*$. So by choosing a tax rate

the government can induce the farmers to choose the Pareto-efficient level of output $Q^*$. $x^*$ is called the Pigouvian tax rate.

The Pigouvian tax rate is the marginal external cost (MEC) at the Pareto-efficient output level. It addresses the externality problem and achieves Pareto efficiency by changing the marginal costs faced by the farmers so that they take into account the full social costs of their decisions, including the costs they impose on others.

An alternative way of thinking about the Pigouvian tax is to say that it works by changing the price that the farmers obtain for their rice, rather than their costs. Then they will choose their output so that their marginal private cost is equal to the after-tax price $P^W-x^*$. So again they choose $Q^*$, because:

This is illustrated in Figure 12.5 of the text, reproduced as Figure 1 below. The Pareto-efficient output of rice is 30,000 tonnes, where the marginal social cost is equal to the price (Rs. 30,000). The tax is equal to the difference between the marginal social cost and the marginal private cost at this level of output, which is Rs. 7,500. The after-tax price, $P_1$, is Rs. 30,000, and they choose an output of 30,000 tonnes because that is where the marginal private cost is equal to Rs. 30,000.

Figure 1 Using a tax to achieve Pareto efficiency.

Finally, remember that we found the Pareto-efficient quantity of rice by looking for the quantity that maximized the social surplus. We calculated the social surplus as the producer surplus minus the costs incurred by the city-dwellers. You may have noticed that the tax reduces the surplus of the city-dwellers, and wondered whether this changes the Pareto efficient quantity. The answer is that it does not do so, because with a tax $x$ on each tonne of rice, the social surplus at quantity $Q$ is:

The first term, in square brackets, is the producer surplus, taking into account the tax producers have to pay, and the second is the costs borne by the city-dwellers. The third term is the tax revenue obtained by the government which, provided that the tax revenue is used to benefit society, also contributes to the social surplus. But the two $xQ$ terms cancel each other out. Thus the social surplus is unaffected by the tax, and the Pareto-efficient output level $Q^*$ remains the same, whether or not a tax is imposed.