|
|
| Line 1: |
Line 1: |
| {{JEL code|[[JEL classification codes#Mathematical_and_quantitative_methods_JEL:_C_Subcategories|C65]]}}
| | His title is Taylor Lipton. My spouse doesn't like it the way I do but what I truly like performing is playing badminton and I will never stop doing it. The job I've been occupying for many years is a human sources assistant but the [http://Www.Reddit.com/r/howto/search?q=marketing marketing] by no means comes. For many years he's been living in California. My spouse and I preserve a website. You may want to verify it out right here: http://Www.[http://Www.Wrbl.com/story/26820425/read-my-honest-diabetes-miracle-cure-review-and-find-out-everything-you-need-to-know-about-diabetes-miracle-cure-program Wrbl.com]/story/26820425/read-my-honest-diabetes-miracle-cure-review-and-find-out-everything-you-need-to-know-about-diabetes-miracle-cure-program |
| {{Economics sidebar}}
| |
| '''Convexity''' is an important topic in '''economics'''.<ref name="Newman1987c" >{{harvtxt|Newman|1987c|}}</ref> In the [[Arrow–Debreu model]] of [[general equilibrium|general economic equilibrium]], agents have convex [[budget set]]s and [[convex preferences]]: At equilibrium prices, the budget [[supporting hyperplane|hyperplane supports]] the best attainable [[indifference curve]].<ref name="Newman1987d" >{{harvtxt|Newman|1987d|}}</ref> The [[profit (economics)|profit function]] is the [[convex conjugate]] of the [[Cost curve|cost function]].<ref name="Newman1987c"/><ref name="Newman1987d" /> [[Convex analysis]] is the standard tool for analyzing textbook economics.<ref name="Newman1987c"/> Non‑convex phenomena in economics have been studied with [[subgradient|nonsmooth analysis]], which generalizes [[convex analysis]].<ref name="Khan"/>
| |
| | |
| ==Preliminaries==
| |
| {{offtopic|convex analysis|date=August 2013}}
| |
| The economics depends upon the following definitions and results from [[convex geometry]].
| |
| | |
| ===Real vector spaces===
| |
| {{multiple image
| |
| | width = 107
| |
| | footer = [[Line segment]]s test [[convex set|convexity]].
| |
| | image1 = Convex polygon illustration1.png
| |
| | alt1 = Illustration of a convex set, which looks somewhat like a disk: A (green) convex set contains the (black) line segment joining the points ''x'' and ''y''. The entire line segment lies in the interior of the convex set.
| |
| | caption1 = A [[convex set]] [[cover (topology)|covers]] the [[line segment]] connecting any two of its points.
| |
| | image2 = Convex polygon illustration2.png
| |
| | alt2 = Illustration of a non‑convex set, which looks somewhat like a boomerang or cashew nut. A (green) non‑convex set contains the (black) line segment joining the points ''x'' and ''y''. Part of the line segment lies outside of the (green) non‑convex set.
| |
| | caption2 = A non‑convex set fails to [[cover (topology)|cover]] a point in some [[line segment]] joining two of its points.
| |
| }}
| |
| | |
| A [[real number|real]] [[vector space]] of two [[dimension (vector space)|dimension]]s 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 ''[[Euclidean_vector#Addition_and_subtraction|added]]'' coordinate-wise
| |
| : (''x''<sub>1</sub>, ''y''<sub>1</sub>) + (''x''<sub>2</sub>, ''y''<sub>2</sub>) = (''x''<sub>1</sub>+''x''<sub>2</sub>, ''y''<sub>1</sub>+''y''<sub>2</sub>);
| |
| further, a point can be ''[[scalar multiplication|multiplied]]'' by each real number ''λ'' coordinate-wise
| |
| : ''λ'' (''x'', ''y'') = (''λx'', ''λy'').
| |
| | |
| More generally, any real vector space of (finite) dimension ''D'' can be viewed as the [[set (mathematics)|set]] of all possible lists of ''D'' real numbers {{nowrap|{ (''v''<sub>1</sub>, ''v''<sub>2</sub>, . . . , ''v''<sub>D</sub>)}} } together with two [[operation (mathematics)|operation]]s: [[vector addition]] and [[scalar multiplication|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.
| |
| | |
| ===Convex sets===
| |
| [[File:Extreme points illustration.png|thumb|right|alt=A picture of a smoothed triangle, like a triangular tortilla-chip or a triangular road-sign. Each of the three rounded corners is drawn with a red curve. The remaining interior points of the triangular shape are shaded with blue.|In the [[convex hull]] of the red set, each blue point is a [[convex combination]] of some red points.]]
| |
| In a real vector space, a set is defined to be ''[[convex set|convex]]'' if, for each pair of its points, every point on the [[line segment]] that joins them is [[cover (mathematics)|covered]] by the set. For example, a solid [[cube (geometry)|cube]] is convex; however, anything that is hollow or dented, for example, a [[crescent]] shape, is non‑convex. [[Vacuous truth|Trivially]], the [[empty set]] is convex.
| |
| | |
| More formally, a set ''Q'' is convex if, for all points ''v''<sub>0</sub> and ''v''<sub>1</sub> in ''Q'' and for every real number ''λ'' in the [[unit interval]] {{closed-closed|0,1}}, the point
| |
| : (1 − ''λ'') ''v''<sub>0</sub> + ''λv''<sub>1</sub>
| |
| is a [[element (mathematics)|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 {''v''<sub>0</sub>, ''v''<sub>1</sub>, . . . , ''v''<sub>D</sub>} of a vector space is any [[weighted mean|weighted average]] {{nowrap|''λ''<sub>0</sub>''v''<sub>0</sub> + ''λ''<sub>1</sub>''v''<sub>1</sub> + . . . + ''λ''<sub>D</sub>''v''<sub>D</sub>,}} for some indexed set of non‑negative real numbers {''λ''<sub>d</sub>} satisfying the [[affine combination|equation]] {{nowrap|''λ''<sub>0</sub> + ''λ''<sub>1</sub> + . . . + ''λ''<sub>D</sub>}} = 1.
| |
| | |
| The definition of a convex set implies that the ''[[intersection (set theory)|intersection]]'' of two convex sets is a convex set. More generally, the intersection of a family of convex sets is a convex set. <!-- In this proposition, the family can be empty, finite, countably infinite, or uncountably infinite. -->
| |
| | |
| ===Convex hull===
| |
| For every subset ''Q'' of a real vector space, its {{nowrap|''[[convex hull]]'' Conv(''Q'')}} is the [[minimal element|minimal]] convex set that contains ''Q''. Thus Conv(''Q'') is the intersection of all the convex sets that [[cover (mathematics)|cover]] ''Q''. The convex hull of a set can be equivalently defined to be the set of all convex combinations of points in ''Q''.
| |
| | |
| ==Duality: Intersecting half-spaces==
| |
| | |
| [[File:Supporting hyperplane1.svg|right|thumb|A [[convex set]] <math>S</math> (in pink), a supporting hyperplane of <math>S</math> (the dashed line), and the half-space delimited by the hyperplane which contains <math>S</math> (in light blue). ]]
| |
| ''Supporting hyperplane'' is a concept in [[geometry]]. A [[hyperplane]] divides a space into two [[Half-space (geometry)|half-space]]s. A hyperplane is said to '''support''' a [[Set (mathematics)|set]] <math>S</math> in the [[real coordinate space|real ''n''-space]] <math>\mathbb R^n</math> if it meets both of the following:
| |
| * <math>S</math> is entirely contained in one of the two [[closed set|closed]] half-spaces determined by the hyperplane
| |
| * <math>S</math> has at least one point on the hyperplane.
| |
| Here, a closed half-space is the half-space that includes the hyperplane.
| |
| | |
| ===Supporting hyperplane theorem===
| |
| [[File:Supporting hyperplane2.svg|right|thumb|A convex set can have more than one supporting hyperplane at a given point on its boundary.]]
| |
| This [[theorem]] states that if <math>S</math> is a closed [[convex set]] in <math>\mathbb R^n,</math> and <math>x</math> is a point on the [[boundary (topology)|boundary]] of <math>S,</math> then there exists a supporting hyperplane containing <math>x.</math>
| |
| | |
| The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set <math>S</math> is not convex, the statement of the theorem is not true at all points on the boundary of <math>S,</math> as illustrated in the third picture on the right. | |
| | |
| [[File:Supporting hyperplane3.svg|right|thumb|A supporting hyperplane containing a given point on the boundary of <math>S</math> may not exist if <math>S</math> is not convex.]]
| |
| | |
| ===Economics===
| |
| [[File:Indifference curves showing budget line.svg|thumb|right|The consumer prefers the vector of goods (''Q''<sub>''x''</sub>, ''Q''<sub>''y''</sub>) over other affordable vectors. At this optimal vector, the budget line supports the [[indifference curve]] ''I''<sub>2</sub>.]]
| |
| An optimal basket of goods occurs where the consumer's convex [[convex preferences|preference set]] is [[supporting hyperplane|supported]]<!-- "tangent" is wrong, unless differentiability is needlessly supposed, and ambiguous even then, if foliation happens --> by the budget constraint, as shown in the diagram. If the preference set is convex, then the consumer's set of optimal decisions is a convex set, for example, a unique optimal basket (or even a line segment of optimal baskets).
| |
| | |
| For simplicity, we shall assume that the preferences of a consumer can be described by a [[utility function]] that is a [[continuous function]], which implies that the [[convex preferences|preference set]]s are [[closed set|closed]]. (The meanings of "closed set" is explained below, in the subsection on optimization applications.)
| |
| | |
| ==Non‑convexity==
| |
| {{Main|Non-convexity (economics)}}
| |
| {{See also|Shapley–Folkman lemma}}
| |
| [[File:NonConvex.gif|thumb|right|When consumer preferences have concavities, then the linear budgets need not support equilibria: Consumers can jump between allocations.]]
| |
| If a preference set is non‑convex, then some prices produce a budget supporting two different optimal consumption decisions. For example, we can imagine that, for zoos, a lion costs as much as an eagle, and further that a zoo's budget suffices for one eagle or one lion. We can suppose also that a zoo-keeper views either animal as equally valuable. In this case, the zoo would purchase either one lion or one eagle. Of course, a contemporary zoo-keeper does not want to purchase a half an eagle and a {{nowrap|half a lion}} (or a [[griffin]])! Thus, the contemporary zoo-keeper's preferences are non‑convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both.
| |
| | |
| Non‑convex sets have been incorporated in the theories of general economic equilibria,<ref>Pages 392–399 and page 188: {{cite book |last1=Arrow |first1=Kenneth J. |authorlink1=Kenneth Arrow |last2=Hahn |first2=Frank H. |authorlink2=Frank Hahn |year=1971 |chapter=Appendix B: Convex and related sets |title=General competitive analysis |publisher=Holden-Day, Inc. [North-Holland] |pages=375–401 |mr=439057 |series=Mathematical economics texts [Advanced textbooks in economics] |number=6 [12] |location=San Francisco, CA |isbn=0-444-85497-5}}</p> <p> Pages 52–55 with applications on pages 145–146, 152–153, and 274–275: {{cite book |last=Mas-Colell |first=Andreu |authorlink=Andreu Mas-Colell |year=1985 |chapter=1.L Averages of sets |title=The Theory of General Economic Equilibrium: A ''Differentiable'' Approach |series=Econometric Society Monographs |number=9 |publisher=Cambridge UP |isbn=0-521-26514-2 |mr=1113262}}</p> <p> Theorem C(6) on page 37 and applications on pages 115-116, 122, and 168: {{cite book |last=Hildenbrand |first=Werner |authorlink=Werner Hildenbrand |title=Core and equilibria of a large economy |series=Princeton studies in mathematical economics |number=5 |publisher=Princeton University Press |location=Princeton, N.J. |year=1974 |pages=viii+251 |isbn=978-0-691-04189-6 |mr=389160}} </p></ref> of [[market failure]]s,<ref>Pages 112–113 in Section 7.2 "Convexification by numbers" (and more generally pp. 107–115): {{cite book |last=Salanié |first=Bernard |chapter=7 Nonconvexities <!-- Not "Non–convexities" --> |title=Microeconomics of market failures |edition=English translation of the (1998) French ''Microéconomie: Les défaillances du marché'' (Economica, Paris) |year=2000 |publisher=MIT Press |location=Cambridge, MA |pages=107–125 |isbn=0-262-19443-0|id=ISBN 978-0-262-19443-3}}</ref> and of [[public economics]].<ref>Pages 63–65: {{cite book |last=Laffont |first=Jean-Jacques |authorlink=Jean-Jacques Laffont |year=1988 |chapter=3 Nonconvexities <!-- Not "Non–convexities" --> |title=Fundamentals of public economics |url=http://books.google.com/books?q=editions:ISBN 0-262-12127-1&id=O5MnAQAAIAAJ |publisher=[http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=7534 MIT]
| |
| |isbn=0-262-12127-1|id=ISBN 978-0-262-12127-9}}</ref> These results are described in graduate-level textbooks in [[microeconomics]],<ref>{{cite book |authorlink=Hal Varian |last=Varian |first=Hal R. |chapter=21.2 Convexity and size |pages=393–394 |title=Microeconomic Analysis |publisher=W. W. Norton & Company |edition=3rd |year=1992 |isbn=978-0-393-95735-8 |mr=1036734}} <p> Page 628: {{cite book |last1=Mas–Colell |first1=Andreu |authorlink=Andreu Mas-Colell |last2=Whinston |first2=Michael D. |first3=Jerry R. |last3=Green |chapter=17.1 Large economies and nonconvexities |title=Microeconomic theory |publisher=Oxford University Press |year=1995 |pages=627–630 |isbn=978-0-19-507340-9}}</ref> general equilibrium theory,<ref>Page 169 in the first edition: {{cite book |last=Starr |first=Ross M. |chapter=8 Convex sets, separation theorems, and non‑convex sets in '''R'''<sup>N</sup> |title=General equilibrium theory: An introduction |edition=Second |publisher=Cambridge University Press |location=Cambridge |year=2011 |pages= |isbn=978-0-521-53386-7 |mr=1462618}} <p> 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): {{cite book |title=Competitive equilibrium: Theory and applications |first=Bryan |last=Ellickson |publisher=Cambridge University Press |isbn=978-0-521-31988-1 |doi=10.2277/0521319889 |year=1994 |pages=420}} </p>
| |
| </ref> [[game theory]],<ref>Theorem 1.6.5 on pages 24–25: {{cite book |last=Ichiishi |first=Tatsuro |title=Game theory for economic analysis |series=Economic theory, econometrics, and mathematical economics |publisher=Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers] |location=New York |year=1983 |pages=x+164 |isbn=0-12-370180-5 |mr=700688}}</ref> [[mathematical economics]],<ref>Pages 127 and 33–34: {{cite book |last=Cassels |first=J. W. S. |authorlink=J. W. S. Cassels |chapter=Appendix A Convex sets |title=Economics for mathematicians |series=London Mathematical Society lecture note series |volume=62 |publisher=Cambridge University Press |location=Cambridge, New York |year=1981 |pages=xi+145 |isbn=0-521-28614-X |mr=657578}}</ref>
| |
| and applied mathematics (for economists).<ref>Pages 93–94 (especially example 1.92), 143, 318–319, 375–377, and 416: {{cite book |last=Carter |first=Michael |title=Foundations of mathematical economics |publisher=MIT Press |location=Cambridge, MA |year=2001 |pages=xx+649 |isbn=0-262-53192-5 |mr=1865841}} <p> Page 309: {{cite book |last=Moore |first=James C. |title=Mathematical methods for economic theory: Volume '''I'''
| |
| |series=Studies in economic theory |volume=9 |publisher=Springer-Verlag |location=Berlin |year=1999 |pages=xii+414 |isbn=3-540-66235-9 |mr=1727000}} <p> Pages 47–48: {{cite book |mr=1878374 |last1=Florenzano |first1=Monique |last2=Le Van |first2=Cuong |title=Finite dimensional convexity and optimization |others=in cooperation with Pascal Gourdel |series=Studies in economic theory |volume=13 |publisher=Springer-Verlag |location=Berlin |year=2001 |pages=xii+154 |isbn=3-540-41516-5}}</ref> The [[Shapley–Folkman lemma]] results establish that non‑convexities are compatible with approximate equilibria in markets with many consumers; these results also apply to [[production (economics)|production economies]] with many small [[business|firm]]s.<ref>Economists have studied non‑convex sets using advanced mathematics, particularly [[differential geometry]] and [[differential topology|topology]], [[Baire category]], [[measure (mathematics)|measure]] and [[integral|integration theory]], and [[ergodic theory]]: {{cite book |last=Trockel |first=Walter |title=Market demand: An analysis of large economies with nonconvex preferences |series=Lecture Notes in Economics and Mathematical Systems |volume=223 |publisher=Springer-Verlag |location=Berlin |year=1984 |pages=viii+205 |isbn=3-540-12881-6 |mr=737006}}</ref>
| |
| | |
| In "[[oligopoly|oligopolies]]" (markets dominated by a few producers<!-- with increasing [[returns to scale]] -->), especially in "[[monopoly|monopolies]]" (markets dominated by one producer), non‑convexities remain important.<ref name="GuesnerieNonConvex" ><!-- Guenserie discusses public economies a bit later, but he introduces "indivisibilities" on page 1, and economists would recognize that this synopsis is faithful and accessible to the public, imho -->Page 1: {{cite journal |last=Guesnerie |first=Roger |authorlink=Roger Guesnerie |title=Pareto optimality in non‑convex economies |journal=Econometrica |volume=43 |year=1975 |pages=1–29 |jstor=1913410 |doi=10.2307/1913410 |mr=443877.{{jstor|1913410}}|ref=harv}} ({{cite news |<!-- last=Guesnerie |first=Roger |authorlink=Roger Guesnerie --> |title=Errata<!-- : "Pareto optimality in non‑convex economies" (''Econometrica'' 43 (1975), 1–29) --> |journal=Econometrica |volume=43 |year=1975 |number=5–6 |page=1010 |jstor=1911353 |doi=10.2307/1911353 |mr=443878.{{jstor|1911353}}}})</ref> Concerns with large producers exploiting market power in fact initiated the literature on non‑convex sets, when [[Piero Sraffa]] wrote about on firms with increasing [[returns to scale]] in 1926,<ref>{{cite news |last=Sraffa |first=Piero |authorlink=Piero Sraffa |year=1926 |title=The Laws of returns under competitive conditions |journal=Economic Journal |volume=36 |number=144 |pages=535–550 |jstor=2959866 |ref=harv}}</ref> after which [[Harold Hotelling]] wrote about [[marginal cost pricing]] in 1938.<ref>{{cite news |
| |
| first=Harold |last=Hotelling |authorlink=Harold Hotelling |title=The General welfare in relation to problems of taxation and of railway and utility rates |title=Econometrica |volume=6 |number=3 |date=July 1938 |pages=242–269 |jstor=1907054}}</ref> Both Sraffa and Hotelling illuminated the [[market power]] of producers without competitors, clearly stimulating a literature on the supply-side of the economy.<ref>Pages 5–7: {{cite book |last=Quinzii |first=Martine |title=Increasing returns and efficiency |location=New York |publisher=Oxford University Press |year=1992 |edition=Revised translation of (1988) ''Rendements croissants et efficacité economique''. Paris: Editions du Centre National de la Recherche Scientifique |pages=viii+165 |isbn=0-19-506553-0}}</ref>
| |
| Non‑convex sets arise also with [[environmental economics|environmental goods]] (and other [[externality|externalities]]),<ref>Pages 106, 110–137, 172, and 248: {{cite book |title=The Theory of environmental policy |edition=Second |first1=William J. |last1=Baumol |authorlink1=William Baumol |last2=Oates |first2=Wallace E. |isbn=978-0-521-31112-0 |doi=10.2277/0521311128 |year=1988 |pages=x+299 |publisher=Cambridge University Press |location=Cambridge |author3=with contributions by V. S. Bawa and David F. Bradford |chapter=8 Detrimental externalities and nonconvexities in the production set |ref=harv}}
| |
| </ref><ref>{{cite news |mr=449575 |last=Starrett |first=David A. |title=Fundamental nonconvexities in the theory of externalities |journal=Journal of Economic Theory |volume=4 |year=1972 |number=2 |pages=180–199 |url=http://www.sciencedirect.com/science/article/B6WJ3-4CYGBWD-NX/2/0f7447ebad01895b6e454dfee4ac481b |doi=10.1016/0022-0531(72)90148-2 |ref=harv}}<p>Starrett discusses non‑convexities in his textbook on [[public economics]] (pages 33, 43, 48, 56, 70–72, 82, 147, and 234–236): {{cite book |last=Starrett |first=David A. |title=Foundations of public economics |series=Cambridge economic handbooks |volume= |year=1988 |number= |pages= |publisher=Cambridge University Press |location=Cambridge |url=http://books.google.com/books?id=R35yljdyyIkC&pg=PR11&dq=David+A.+Starrett,+public+economics#v=onepage&q=nonconvex%20OR%20nonconvexities&f=false |ref=harv}}<p/></ref> with [[information economics]],<ref>{{cite news |first=Roy |last=Radner |authorlink=Roy Radner |title=Competitive equilibrium under uncertainty |journal=Econometrica |volume=36 |year=1968 |pages=31–53 |ref=harv}} <!-- Apparently not [[Radner Equilibrium]]: {{cite journal |authorlink=Roy Radner |last=Radner |first=R. |year=1967 |title=Equilibre des marchés à terme et au comptant en cas d’incertitude [Equilibrium of temporal sequences of markets under uncertainty] |language=French |journal=Cahiers du Séminaire d’Econométrie |volume=17 |pages=35-52 |}} and {{cite article |authorlink=Roy Radner |first=Roy |last=Radner |mr=381655 |title=Existence of equilibrium of plans, prices, and price expectations in a sequence of markets |journal=Econometrica |volume=40 |year=1972 |pages=289–304 |}} -->
| |
| </ref> and with [[stock market]]s<ref name="GuesnerieNonConvex"/> (and other [[incomplete markets]]).<ref>Page 270: {{cite book |mr=926685 |last=Drèze |first=Jacques H. |authorlink=Jacques H. Drèze |title=Essays on economic decisions under uncertainty |publisher=Cambridge University Press |editor-last=Drèze |editor-first=J. H. |<!-- editor-link=Jacques H. Drèze -->location=Cambridge |year=1987 | pages=261–297<!-- xxviii+424 --> |isbn=0-521-26484-7 |chapter=14 Investment under private ownership: Optimality, equilibrium and stability |ref=harv}} (Originally published as {{cite book |last=Drèze |first=Jacques H. |authorlink=Jacques H. Drèze |year=1974 |chapter=Investment under private ownership: Optimality, equilibrium and stability |editor-last=Drèze |editor-first=J. H.|title=Allocation under Uncertainty: Equilibrium and Optimality |publisher=Wiley |location=New York |pages=129–165 |ref=harv}})</ref><ref>Page 371: {{cite book |last1=Magill |first1=Michael |last2=Quinzii |first2=Martine |year=1996 |chapter=6 Production in a finance economy, Section 31 Partnerships|pages=329–425 |title=The Theory of incomplete markets |publisher=MIT Press |location=Cambridge, Massachusetts |ref=harv}}</ref> Such applications continued to motivate economists to study non‑convex sets.<ref name="MCNC">{{cite book |last=Mas-Colell |first=A. |authorlink=Andreu Mas-Colell |chapter=Non‑convexity |title=The New Palgrave: A Dictionary of Economics |editor1-first=John |editor1-last=Eatwell |editor2-first=Murray |editor2-last=Milgate |editor3-first=Peter |editor3-last=Newman |publisher=Palgrave Macmillan |year=1987 |edition=first|doi=10.1057/9780230226203.3173 |pages=653–661 |url=http://www.econ.upf.edu/~mcolell/research/art_083b.pdf |ref=harv}}
| |
| </ref>
| |
| | |
| ===Nonsmooth analysis===
| |
| {{cleanup|section|date=August 2013|reason=Relationship between subderivatives and non‑convexity remains cryptic}}
| |
| Economists have increasingly studied non‑convex sets with [[subderivative|nonsmooth analysis]], which generalizes [[convex analysis]]. "Non‑convexities in [both] production and consumption ... required mathematical tools that went beyond convexity, and further development had to await the invention of non‑smooth calculus" (for example, Francis Clarke's [[Rademacher's theorem|locally Lipschitz]] calculus), as described by {{harvtxt|Rockafellar|Wets|1998}}<ref>{{cite book |last1=Rockafellar |first1=R. Tyrrell |authorlink1=R. Tyrrell Rockafellar |last2=Wets |first2=Roger <!-- NO PERIODS -->J-B |authorlink2=Roger J-B Wets |title=Variational analysis |series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] |volume=317 |publisher=Springer-Verlag |location=Berlin |year=1998 |pages=xiv+733 |isbn=3-540-62772-3 |mr=1491362 |ref=harv}}</ref> and {{harvtxt|Mordukhovich|2006}},<ref name=Mordukhovich2>Chapter 8 "Applications to economics", especially Section 8.5.3 "Enter nonconvexity" (and the remainder of the chapter), particularly page 495: <p>{{cite book |authorlink=Boris Mordukhovich |first=Boris S. |last=Mordukhovich |title=Variational analysis and generalized differentiation '''II''': Applications |series=Grundlehren Series (Fundamental Principles of Mathematical Sciences) |volume=331 |publisher=Springer |year=2006 |pages=i–xxii and 1–610 |mr=2191745 |ref=harv}}<p/></ref> according to {{harvtxt|Khan|2008}}.<ref name="Khan" >{{cite book |last=Khan |first=M. Ali |chapter=Perfect competition |title=The New Palgrave Dictionary of Economics |editor-first=Steven N. |editor-last=Durlauf |editor2-first=Lawrence E., ed. |editor2-last=Blume |publisher=Palgrave Macmillan |year=2008 |edition=Second |pages= |url=http://www.dictionaryofeconomics.com/article?id=pde2008_P000056 |doi=10.1057/9780230226203.1267 |ref=harv}}</ref> {{harvtxt|Brown|1995|pp=1967–1968}} wrote that the "major methodological innovation in the general equilibrium analysis of firms with pricing rules" was "the introduction of the methods of non‑smooth analysis, as a [synthesis] of global analysis (differential topology) and [of] convex analysis." According to {{harvtxt|Brown|1995|p=1966}}, "Non‑smooth analysis extends the local approximation of manifolds by tangent planes<!-- , --> [and extends] the analogous approximation of convex sets by tangent cones to sets" that can be non‑smooth or non‑convex.<!-- "which [deprecated] are neither smooth nor convex [false: Ioffe notes all subdifferentials specialize to the convex-analysis subdifferential for convex functions] -->.<ref>{{cite book |last=Brown |first=Donald J. |chapter=36 Equilibrium analysis with non‑convex technologies |doi=10.1016/S1573-4382(05)80011-6 |url=http://www.sciencedirect.com/science/article/B7P5Y-4FKY4C6-C/2/0bd1a73374b0b690f702691e1f7fe671 |title=Handbook of mathematical economics, Volume '''IV''' |pages=1963–1995 [1966] |mr=1207195 |editor1-first=Werner |editor1-last=Hildenbrand |editor1-link=Werner Hildenbrand |editor2-first=Hugo |editor2-last=Sonnenschein |editor2-link=Hugo Sonnenschein |series=Handbooks in Economics |volume=1 |publisher=North-Holland Publishing Co |location=Amsterdam |year=1991 |isbn=0-444-87461-5<!-- The terminal "5" is correct, while Elsevier's on-line "0" is for volume 2 --> |ref=harv}}</ref> Economists have also used [[<!-- reduced -->singular homology|algebraic topology]].<ref>{{cite news |mr=1218037 |last=Chichilnisky |first=G. |authorlink=Graciela Chichilnisky |title=Intersecting families of sets and the topology of cones in economics |journal=Bulletin of the American Mathematical Society (New Series) |volume=29 |year=1993 |number=2 |pages=189–207 |doi=10.1090/S0273-0979-1993-00439-7 |url=http://www.ams.org/journals/bull/1993-29-02/S0273-0979-1993-00439-7/S0273-0979-1993-00439-7.pdf |ref=harv}}
| |
| </ref>
| |
| | |
| ==Notes==
| |
| <references/>
| |
| | |
| ==References==
| |
| * {{cite book |last=Blume |first=Lawrence E. |authorlink=Lawrence E. Blume |editor2-link=Lawrence E. Blume |chapter=Convexity |year=2008c |title=The New Palgrave Dictionary of Economics |editor-first=Steven N. |editor-last=Durlauf |editor2-first=Lawrence E<!-- . --> |editor2-last=Blume |publisher=Palgrave Macmillan |edition=Second |pages= |url=http://www.dictionaryofeconomics.com/article?id=pde2008_C000508 |doi=10.1057/9780230226203.0315 |ref=harv}}
| |
| * {{cite book |last=Blume |first=Lawrence E. |chapter=Convex programming |year=2008cp |authorlink=Lawrence E. Blume |editor2-link=Lawrence E. Blume |title=The New Palgrave Dictionary of Economics |editor-first=Steven N. |editor-last=Durlauf |editor2-first=Lawrence E<!-- . --> |editor2-last=Blume |publisher=Palgrave Macmillan |edition=Second |pages= |url=http://www.dictionaryofeconomics.com/article?id=pde2008_C000348 |doi=10.1057/9780230226203.0314 |ref=harv}}
| |
| * {{cite book |last=Blume |first=Lawrence E. |authorlink=Lawrence E. Blume |editor2-link=Lawrence E. Blume |chapter=Duality |year=2008d |title=The New Palgrave Dictionary of Economics |editor-first=Steven N. |editor-last=Durlauf |editor2-first=Lawrence E<!-- . --> |editor2-last=Blume |publisher=Palgrave Macmillan |edition=Second |pages= |url=http://www.dictionaryofeconomics.com/article?id=pde1987_X000626 |doi=10.1057/9780230226203.0411 |ref=harv}}
| |
| * {{cite book |last=Crouzeix |first=J.-P. |chapter=Quasi-concavity |title=The New Palgrave Dictionary of Economics |editor-first=Steven N. |editor-last=Durlauf |editor2-first=Lawrence E<!-- . --> |editor2-last=Blume |editor2-link=Lawrence E. Blume |publisher=Palgrave Macmillan |year=2008 |edition=Second |pages= |url=http://www.dictionaryofeconomics.com/article?id=pde2008_Q000008 |doi=10.1057/9780230226203.1375 |ref=harv}}
| |
| * {{cite book |first=W. E. |last=Diewert |chapter=12 Duality approaches to microeconomic theory
| |
| |pages=535–599
| |
| |url=http://www.sciencedirect.com/science/article/B7P5Y-4FDF0FN-R/2/dcc0f8c9352eb054c96b3ff481976ce7
| |
| |doi=10.1016/S1573-4382(82)02007-4
| |
| |title=Handbook of mathematical economics, Volume '''II''' |editor1-link=Kenneth Arrow |editor1-first=Kenneth Joseph |editor1-last=Arrow |editor2-first=Michael D<!-- . --> |editor2-last=Intriligator |series=Handbooks in economics |volume=1 |publisher=North-Holland Publishing Co. |location=Amsterdam |year=1982 |isbn=978-0-444-86127-6 |mr=648778 |ref=harv}}
| |
| * {{cite book |first1=Jerry |last1=Green |first2=Walter P. |last2=Heller |chapter=1 Mathematical analysis and convexity with applications to economics |pages=15–52 |url=http://www.sciencedirect.com/science/article/B7P5Y-4FDF0FN-5/2/613440787037f7f62d65a05172503737 |doi=10.1016/S1573-4382(81)01005-9 |title=Handbook of mathematical economics, Volume '''I''' |editor1-link=Kenneth Arrow |editor1-first=Kenneth Joseph |editor1-last=Arrow |editor2-first=Michael D<!-- . --> |editor2-last=Intriligator |series=Handbooks in economics |volume=1 |publisher=North-Holland Publishing Co. |location=Amsterdam |year=1981 |isbn=0-444-86126-2 |mr=634800 |ref=harv}}
| |
| * Luenberger, David G. ''Microeconomic Theory'', McGraw-Hill, Inc., New York, 1995.
| |
| * {{cite book |last=Mas-Colell |first=A. |authorlink=Andreu Mas-Colell |chapter=Non‑convexity |title=The New Palgrave: A Dictionary of Economics |editor1-first=John |editor1-last=Eatwell |editor2-first=Murray |editor2-last=Milgate |editor3-first=Peter |editor3-last=Newman |editor3-link=Peter Kenneth Newman |publisher=Palgrave Macmillan |year=1987 |edition=first|doi=10.1057/9780230226203.3173 |pages=653–661 |url=http://www.econ.upf.edu/~mcolell/research/art_083b.pdf |ref=harv}}
| |
| * {{cite book |last=Newman |first=Peter |authorlink=Peter Kenneth Newman |chapter=Convexity |title=The New Palgrave: A Dictionary of Economics |editor1-first=John |editor1-last=Eatwell |editor2-first=Murray |editor2-last=Milgate |editor3-first=Peter |editor3-last=Newman |editor3-link=Peter Kenneth Newman |publisher=Palgrave Macmillan |year=1987c |edition=first|doi=10.1057/9780230226203.2282<!-- SNAFU at NP? 30 Jan 2011--> |pages= |url=http://www.dictionaryofeconomics.com/article?id=pde1987_X000453 |ref=harv}}
| |
| * {{cite book |last=Newman |first=Peter |authorlink=Peter Kenneth Newman |chapter=Duality |title=The New Palgrave: A Dictionary of Economics |editor1-first=John |editor1-last=Eatwell |editor2-first=Murray |editor2-last=Milgate |editor3-first=Peter |editor3-last=Newman |editor3-link=Peter Kenneth Newman |publisher=Palgrave Macmillan |year=1987d |edition=first|doi=10.1057/9780230226203.2412<!-- SNAFU at NP? 30 Jan 2011--> |pages= |url=http://www.dictionaryofeconomics.com/article?id=pde1987_X000626 |ref=harv}}
| |
| * {{cite book |last=Rockafellar |first=R. Tyrrell |authorlink=R. Tyrrell Rockafellar|title=Convex analysis |edition=Reprint of the 1979 Princeton mathematical series '''28''' |series=Princeton landmarks in mathematics |publisher=Princeton University Press |location=Princeton, NJ |year=1997|isbn=0-691-01586-4|id={{MR|1451876}}, {{MR|274683}}|ref=harv}}
| |
| * {{cite book |last=Schneider |first=Rolf |title=Convex bodies: The Brunn–Minkowski theory |series=Encyclopedia of mathematics and its applications |volume=44 |publisher=Cambridge University Press |location=Cambridge |year=1993 |pages=xiv+490 |ref=harv |isbn=0-521-35220-7 |mr=1216521}}
| |
| | |
| {{Geometry-footer|state=collapsed}}
| |
| {{Microeconomics|state=collapsed}}
| |
| | |
| {{DEFAULTSORT:Convexity In Economics}}
| |
| [[Category:Convex hulls]]
| |
| [[Category:Convex geometry]]
| |
| [[Category:Mathematical and quantitative methods (economics)]]
| |
| [[Category:Mathematical economics]]
| |
| [[Category:General equilibrium and disequilibrium]]
| |
| [[Category:Convexity in economics| ]]
| |