Non-Convexity
Non-convexity refers to the absence of convexity in preferences, production sets, or constraints. In economics, convexity assumptions are widely used because they ensure that optimisation problems have well-behaved solutions, that equilibria exist and are unique, and that the welfare theorems hold. When convexity fails, these results may not hold, and the analysis becomes more complex.
Convexity in Production and Consumption
A production set is convex if any weighted average of two feasible input-output combinations is also feasible, which corresponds to the absence of increasing returns to scale and of fixed costs. Non-convexity arises from increasing returns, from indivisible inputs, and from technological features that make larger scale more efficient.
Preferences are convex if mixtures of preferred bundles are also preferred, which corresponds to diminishing marginal rates of substitution and to a preference for variety. Non-convex preferences, such as those in which consumers prefer extremes, violate this property.
Consequences
Non-convexity has significant consequences. In production, increasing returns imply that average cost declines with output, which can lead to natural monopoly and to the failure of competitive markets to allocate resources efficiently. In consumption, non-convex preferences can generate discontinuous demand and multiple optima.
The existence and uniqueness of equilibrium may fail under non-convexity. Multiple equilibria can arise, and the welfare theorems, which rely on convexity, may not apply. The analysis of such economies requires different tools, including the study of externalities and of the role of institutions.
Non-Convexity and Market Failure
Non-convexity is a source of market failure. Increasing returns to scale mean that marginal cost pricing may not cover total cost, so that competitive equilibrium may not exist or may not be efficient. Public goods and externalities also involve non-convexities, since the benefits or costs of additional provision may not be divisible or appropriately priced.
The analysis of these failures motivates a range of policies, including regulation, public provision, and the design of pricing mechanisms that address the non-convexity. The appropriate response depends on the source of the non-convexity and on the available instruments.
Extensions and Empirical Relevance
Non-convexity is important in many modern industries, particularly those with high fixed costs, network effects, and knowledge production. In these industries, increasing returns and indivisibilities are pervasive, and the competitive model is a poor guide to outcomes.
The analysis of non-convex economies has developed tools to handle the resulting complexities, including the theory of contests, the analysis of platforms and network effects, and the study of the dynamics of technology adoption. The insights inform the design of policy in industries where scale and innovation are central.
Fixed Costs and Indivisibilities
Fixed costs and indivisibilities are sources of non-convexity in production. A fixed cost does not vary with output, so average cost declines as output rises, generating increasing returns over the relevant range. An indivisible input must be used in a minimum quantity, so production below a certain level is infeasible or inefficient.
These features are pervasive in modern economies. Industries with large investments in plant, equipment, or research and development exhibit fixed costs and increasing returns. The resulting non-convexity implies that competitive markets may not allocate resources efficiently and that scale and market structure matter.
Natural Monopoly and Regulation
Natural monopoly arises when the cost structure is subadditive, so that a single firm can produce the total output at lower cost than multiple firms. This typically occurs where fixed costs are large relative to the market and marginal cost is low, so that average cost declines over the range of demand.
Because a natural monopolist would charge prices above marginal cost and restrict output, regulation is used to align prices with costs while allowing the firm to recover its fixed costs. The design of regulation must contend with the information problems involved in determining costs and demand, and with the incentives created by different regulatory mechanisms.
Network Effects and Platforms
Non-convexity also arises from network effects, where the value of a product increases with the number of users. Network effects generate increasing returns on the demand side, leading to market tipping, in which a single platform or standard dominates. The resulting concentration raises questions about competition policy and about the regulation of dominant platforms.
The analysis of platform markets combines the economics of network effects with the analysis of two-sided markets, in which platforms serve distinct groups whose participation depends on each other. The design of policy in these markets is an active area of research and debate.
Public Goods and Externalities
Public goods and externalities involve non-convexities because the benefits or costs of provision are not appropriately divisible or priced. A public good, once provided, is available to all, so that the private incentive to provide it is insufficient. An externality imposes costs or benefits on third parties, so that private decisions diverge from the social optimum.
The analysis of these problems informs the design of policies, including public provision, Pigouvian taxes and subsidies, and the assignment of property rights. The non-convexity means that the standard conditions for efficiency do not hold, and that the design of policy must address the underlying market failure.