Abstract
Nonlinear prices are commonly observed in market economies. This paper investigates nonlinear pricing under general conditions. It explores how nonlinear pricing can arise under nonconvexity. The arguments are presented in the context of an optimization problem, where a separating hypersurface provides information on pricing under general nonconvexity. The analysis applies to efficiency assessments, noting that Pareto efficiency can be expressed as the maximization of aggregate benefit. When nonconvexity requires a nonlinear separating hypersurface, this implies that nonlinear pricing becomes an integral part of efficiency analysis. This evaluation applies to nonmarket goods (e.g., the pricing of carbon emission) as well as market goods. We show how nonlinear pricing depends on the nature of nonconvexity. We discuss how associated price discrimination schemes can be implemented to support efficient allocations.
Highlights
Studying prices is a cornerstone of economic analysis
When nonconvexity requires a nonlinear separating hypersurface, this implies that nonlinear pricing becomes an integral part of efficiency analysis
This includes the evaluation of nonmarket goods, their prices being assessed as the marginal value of the goods (e.g., the evaluation of the price of carbon emission; see Nordhaus (2019))
Summary
Studying prices is a cornerstone of economic analysis. The economics of prices is well understood under convexity. The separating hyperplane theorem does not hold under nonconvexity, raising questions about the validity of the standard Lagrangian approach and of the associated price evaluation Such arguments have stimulated interest in revisiting and generalizing previous approaches under nonconvexity (e.g., Gould, 1969; Giannessi, 1984, 2005). It is well known that Lagrange multipliers can be interpreted as “marginal values of the constraints” under convexity (e.g., Takayama, 1985) This interpretation continues to apply to a Generalized Lagrangian approach under nonconvexity. This paper studies the nature of nonlinear pricing in the context of a general constrained optimization problem. This means that nonlinear pricing becomes an explicit part of efficiency evaluation This argument is relevant to nonmarket allocation in which case our analysis applies to the shadow prices of nonmarket goods (Rosen, 1974) and contracts (Salanié, 1999).
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