Convexity in economics

Convexity is an important topic in economics. In the Arrow–Debreu model of general economic equilibrium, agents have convex budget sets and convex preferences: At equilibrium prices, the budget hyperplane supports the best attainable indifference curve. The profit function is the convex conjugate of the cost function. Convex analysis is the standard tool for analyzing textbook economics. Non‑convex phenomena in economics have been studied with nonsmooth analysis, which generalizes convex analysis. Page 628: Mas–Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995). '17.1 Large economies and nonconvexities'. Microeconomic theory. Oxford University Press. pp. 627–630. ISBN 978-0-19-507340-9. In Ellickson, page xviii, and especially Chapter 7 'Walras meets Nash' (especially section 7.4 'Nonconvexity' pages 306–310 and 312, and also 328–329) and Chapter 8 'What is Competition?' (pages 347 and 352): Ellickson, Bryan (1994). Competitive equilibrium: Theory and applications. Cambridge University Press. p. 420. doi:10.2277/0521319889. ISBN 978-0-521-31988-1. Page 309: Moore, James C. (1999). Mathematical methods for economic theory: Volume I. Studies in economic theory. 9. Berlin: Springer-Verlag. pp. xii+414. ISBN 978-3-540-66235-8. MR 1727000. Starrett discusses non‑convexities in his textbook on public economics (pages 33, 43, 48, 56, 70–72, 82, 147, and 234–236): Starrett, David A. (1988). Foundations of public economics. Cambridge economic handbooks. Cambridge: Cambridge University Press. ISBN 9780521348010.Mordukhovich, Boris S. (2006). Variational analysis and generalized differentiation II: Applications. Grundlehren Series (Fundamental Principles of Mathematical Sciences). 331. Springer. pp. i–xxii and&nbsp, 1–610. MR 2191745. Convexity is an important topic in economics. In the Arrow–Debreu model of general economic equilibrium, agents have convex budget sets and convex preferences: At equilibrium prices, the budget hyperplane supports the best attainable indifference curve. The profit function is the convex conjugate of the cost function. Convex analysis is the standard tool for analyzing textbook economics. Non‑convex phenomena in economics have been studied with nonsmooth analysis, which generalizes convex analysis. The economics depends upon the following definitions and results from convex geometry. A real vector space of two dimensions may be given a Cartesian coordinate system in which every point is identified by a list of two real numbers, called 'coordinates', which are conventionally denoted by x and y. Two points in the Cartesian plane can be added coordinate-wise further, a point can be multiplied by each real number λ coordinate-wise More generally, any real vector space of (finite) dimension D can be viewed as the set of all possible lists of D real numbers { (v1, v2, . . . , vD) } together with two operations: vector addition and multiplication by a real number. For finite-dimensional vector spaces, the operations of vector addition and real-number multiplication can each be defined coordinate-wise, following the example of the Cartesian plane. In a real vector space, a set is defined to be convex if, for each pair of its points, every point on the line segment that joins them is covered by the set. For example, a solid cube is convex; however, anything that is hollow or dented, for example, a crescent shape, is non‑convex. Trivially, the empty set is convex. More formally, a set Q is convex if, for all points v0 and v1 in Q and for every real number λ in the unit interval , the point is a member of Q. By mathematical induction, a set Q is convex if and only if every convex combination of members of Q also belongs to Q. By definition, a convex combination of an indexed subset {v0, v1, . . . , vD} of a vector space is any weighted average λ0v0 + λ1v1 + . . . + λDvD, for some indexed set of non‑negative real numbers {λd} satisfying the equation λ0 + λ1 + . . . + λD = 1.