The preceding chapter showed that the conventional wisdom concerning the direction in which to modify output in the presence of externalities is likely, at least sometimes, to be misleading. The problem can arise whenever the relevant convexity conditions break down.

In this chapter, however, we will show that detrimental externalities of sufficient strength will produce a breakdown in the concavity-convexity conditions (the so-called second-order conditions) usually postulated for a social maximum, so that instead of a unique optimum, society may have the difficult task of choosing among a set, and, sometimes, a substantial set of discrete local maxima. Indeed, in a system otherwise characterized by constant returns everywhere (that is, a linear model), any detrimental externalities, however minor, can produce a nonconvexity. This problem is no mere theoretical curiosity. We will see that it produces some very real and difficult issues in the choice of policy.

Moreover, even in theory, prices and taxes cannot help with this matter. Prices and taxes (which, in general, influence the first-order maximum conditions) can affect the decisions of individuals and firms and thereby determine the location of the economy in relation to its production-possibility set. However, prices or taxes cannot change the shape of the possibility set itself to transform it from a nonconvex into a convex region, for that is essentially a technological matter. Moreover, as we will see later in this chapter, in the presence of nonconvexities, these prices may also give the wrong signals – directing the economy away from the social optimum.