Non - Convexities:

The intuitive meaning of convexity is that you can combine goods (or production inputs) in any proportion as you wish within the feasible set. This assumption is often taken for granted. In its absence, however, optimality may become unachievable.

Suppose you want to buy bread and cheese. If the smallest package of eachis so large that you would have to spend your entire budget on only one item, you cannot consume both goods. Although this example sounds trivial, indivisibilities are sufficient to prevent the attainment of optimality. Such a nonconvexity may be created by increasing returns to scale. It may mean a project of size Ais feasible, but a project one-half the size ofA is too expensive. For example, usually the minimum-efficient scale of refineries or steel mills is too high. It causes problems in small countries; small plants are not cost efficient, big plants may have insufficient demand, and no plant means that the country would be entirely dependent on imports. The same applies to infrastructure investments such as bridges, health programs, and railways. A country may desperately need infkastructure but find the available systems excessively large and prohibitively expensive. The absence of intermediate scale solutions implies that a decentralized market will not lead to an optimum.

Typically, external effects also create this kind of non-convexity. Consider two goods, A and B, that have strong negative externalities between them. You can have A or B, but obtaining both is difficult. Similarly by definition, public goods imply that everyone has to consume the same quantity: there can be one state of the economyin which everyone gets none of the public good and there can be a state in which everyone gets a quantity z of the same good, but there cannot be a state in which some people get none and some get z. This restriction, too, means that the decentralized market will not lead to an optimum. Other causes of non-convexities include common pool resources, congestion and joint production.