Leibniz

12.1.1 External effects of pollution

When the production or consumption of a good affects anyone other than the buyers and sellers, the market allocation of the good will not be Pareto efficient. We demonstrate this mathematically in the case of rice stubble burning, which pollutes the air in the neighbouring city. The amount of rice produced using this system is greater than the Pareto-efficient level.

Pareto efficient

An allocation with the property that there is no alternative technically feasible allocation in which at least one person would be better off, and nobody worse off.

In Leibniz 8.5.1, we analysed the gains from trade in the bread market by calculating the total surplus, which was equal to the sum of the consumer surplus and the producer surplus. We showed that the allocation of a market equilibrium in perfect competition maximized the total surplus. It follows that, at this allocation, it is not possible to make any of the consumers or firms better off (that is, to increase the surplus of any individual) without making at least another one of them worse off. Assuming that what happens in this market does not affect anyone other than the participating buyers and sellers, we can say that the equilibrium allocation is Pareto efficient.

To analyse the case of rice production using the rice–wheat cropping system (RWCS), we adopt the same approach, working out the total surplus resulting from the production and sale of rice. But there is an important difference. Rice production affects the buyers and sellers of rice, but also it has a negative external effect on the city-dwellers—the cost of air pollution from the smoke caused due to stubble burning. So when we calculate the total surplus, we need to include the city-dwellers’ costs too.

There is another difference between the bread market model and the rice production model. We assume that however much rice is produced, it can be sold in the wholesale rice market at a constant price, \(P^W\). This is a reasonable assumption as long as the farmers produce a small fraction of the world’s rice. It means that decisions about rice production in and near the city do not change the surplus obtained by consumers, whether they live in the city, or elsewhere in the world. So we don’t need to include consumer surplus in our analysis. Note that this assumption is a useful simplification, but it would be straightforward to adapt the model to the case where a lot of rice was produced.

Overall, then, the total surplus (often called the social surplus in this context) from growing rice will be the sum of the producer surplus and the surplus obtained by city-dwellers. But since the effect on the city-dwellers is negative, we write the social surplus as:

\[\begin{align*}\text{social surplus} = \\\text{producer surplus to the farmers} - \\\text{external cost to the city-dwellers}\end{align*}\]

The producer surplus is calculated just as in Leibniz 8.5.1. It is equal to the farmers’ revenue, minus their total cost of production. If \(Q\) tonnes of rice is produced and sold, producer surplus is:

\[P^W Q - C_p (Q)\]

Here, \(C_p(Q)\) is the total private cost of producing rice. In economic terminology, the private costs and benefits of a decision are costs and benefits experienced by the decision-maker. Let \(C_e(Q)\) be the external cost imposed on the city-dwellers when \(Q\) tonnes of rice is produced. We can say that the social cost of rice production is \(C(Q)= C_p(Q)+ C_e(Q)\), the sum of the private and external costs.

The social surplus \(N(Q)\) is then:

\[N(Q) = (P^W Q - C_p (Q)) - C_e(Q) = P^W Q - C(Q)\]

The derivatives of the three cost functions, \(C'(Q)\), \(C'_{p}(Q)\) and \(C'_{e}(Q)\), denote marginal social cost (MSC), marginal private cost (MPC), and marginal external cost (MEC), respectively. We assume that \(C'_{p}(Q) \lt C'(Q)\) for all \(Q\), which means that MEC is positive, and that MSC and MPC are increasing in output. Thus \(C\) and \(C_p\) are convex functions.

Hence \(N'(Q)\) is a concave function, and the social surplus is maximized at the level of output \(Q^*\) which satisfies the first-order condition \(N'(Q^*) =0\). Differentiating the expression above for \(N(Q)\), we find that \(Q^*\) is the output level at which the marginal social cost of rice is equal to the price:

\[C'(Q^*)=P^W\]

Since the social surplus is maximized at \(Q^*,\) we know that if a different level of output were chosen, for example to make the city-dwellers better off (by lowering their costs), then it must make the farmers worse off. So \(Q^*\) is the Pareto-efficient level of output.

However, \(Q^*\) is not the level of output that will be produced in equilibrium. The farmers want to maximize profits and they are price-takers, because whatever the quantity of rice they produce, the price at which each tonne of rice can be sold is \(P^W\). So they each choose their output so that their marginal private cost is equal to the market price, and the market supply curve is therefore the marginal private cost curve. Thus the total output \(Q_p\) of the farmers satisfies the equation:

\[C_p'(Q_p)=P^W\]

To compare the private equilibrium level of output \(Q_p\) with the Pareto efficient level \(Q^*\), consider the derivative of the social surplus:

\[N'(Q_{p }) = P^W - C'(Q_{p}) = C'_{p}(Q_{p}) - C'(Q_{p}) = - C'_{ e}(Q^{p})\]

Since MEC is positive, it follows that:

\[N'(Q_{p }) \lt 0 = N'(Q^*)\]

And since \(N\) is a concave function (\(N'(Q)\) falls as \(Q\) increases), we deduce that:

\[Q_{p } \gt Q^*\]

This inequality tells us that the farmers produce too much output, according to the criterion of Pareto efficiency. This result can be seen in Figure 12.2 of the text, reproduced as Figure 1 below. In the figure, you can see the features of the problem we have analysed mathematically above:

• The marginal social cost \(C'(Q)\), and the marginal private cost \(C'_{p}(Q)\), increase with output.
• The marginal social cost is greater than the marginal private cost, and the difference is the marginal external cost.
• The Pareto-efficient level of output \(Q^*\) is where the marginal social cost is equal to the price. In the diagram, \(P^W=30,000\), and \(Q^*=30,000\).
• The farmers will produce \(Q_p\), where the marginal private cost is equal to the price. In the diagram \(Q_p^*=60,000\); they produce more than the Pareto-efficient level.