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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
2383:, 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 of global analysis (differential topology) and convex analysis." According to
2721:
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):
2387:, p. 1966), "Non‑smooth analysis extends the local approximation of manifolds by tangent planes the analogous approximation of convex sets by tangent cones to sets" that can be non‑smooth or non‑convex. Economists have also used
2143:
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).
2363:. "Non‑convexities in 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
2192:
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
1334:. 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.
2246:" (markets dominated by one producer), non‑convexities remain important. Concerns with large producers exploiting market power in fact initiated the literature on non‑convex sets, when
5069:
960:
Convexity is a key simplifying assumption in many economic models, as it leads to market behavior that is easy to understand and which has desirable properties. For example, the
1925:
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2200:)! 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.
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1995:
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2017:
1945:
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Chapter 8 "Applications to economics", especially
Section 8.5.3 "Enter nonconvexity" (and the remainder of the chapter), particularly page 495:
35:
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3213:. with contributions by V. S. Bawa and David F. Bradford (Second ed.). Cambridge: Cambridge University Press. pp. x+299.
5055:
1034:
941:. Informally, an economic phenomenon is convex when "intermediates (or combinations) are better than extremes". For example, an economic agent with
2227:
results establish that non‑convexities are compatible with approximate equilibria in markets with many consumers; these results also apply to
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When consumer preferences have concavities, then the linear budgets need not support equilibria: Consumers can jump between allocations.
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of producers without competitors, clearly stimulating a literature on the supply-side of the economy. Non‑convex sets arise also with
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4402:
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3184:. Paris: Editions du Centre National de la Recherche Scientifique ed.). New York: Oxford University Press. pp. viii+165.
2791:. London Mathematical Society lecture note series. Vol. 62. Cambridge, New York: Cambridge University Press. pp. xi+145.
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3526:. Grundlehren Series (Fundamental Principles of Mathematical Sciences). Vol. 331. Springer. pp. i–xxii and 1–610.
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in which every point is identified by a list of two real numbers, called "coordinates", which are conventionally denoted by
4351:
4087:. Encyclopedia of mathematics and its applications. Vol. 44. Cambridge: Cambridge University Press. pp. xiv+490.
2890:. Studies in economic theory. Vol. 13. in cooperation with Pascal Gourdel. Berlin: Springer-Verlag. pp. xii+154.
5508:
3880:
Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In
5264:
4153:
954:
4036:
3998:
3960:
3610:
3441:
3280:
2581:
2342:
1069:
3341:(1987). "14 Investment under private ownership: Optimality, equilibrium and stability". In Drèze, J. H. (ed.).
2964:. Lecture Notes in Economics and Mathematical Systems. Vol. 223. Berlin: Springer-Verlag. pp. viii+205.
1999:
The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set
1637:. The convex hull of a set can be equivalently defined to be the set of all convex combinations of points in
1593:
of two convex sets is a convex set. More generally, the intersection of a family of convex sets is a convex set.
5284:
5201:
3378:(1974). "Investment under private ownership: Optimality, equilibrium and stability". In Drèze, J. H. (ed.).
914:
2754:. Economic theory, econometrics, and mathematical economics. New York: Academic Press, Inc. . pp. x+164.
252:
3468:. Grundlehren der Mathematischen Wissenschaften . Vol. 317. Berlin: Springer-Verlag. pp. xiv+733.
1180:
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3597:. Handbooks in Economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 1963–1995 .
3209:; Oates, Wallace E. (1988). "8 Detrimental externalities and nonconvexities in the production set".
3137:(July 1938). "The General welfare in relation to problems of taxation and of railway and utility rates".
1167:
1051:
863:
83:
73:
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The branch of mathematics which supplies the tools for convex functions and their properties is called
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3849:. Handbooks in economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 535–599.
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Pages 112–113 in
Section 7.2 "Convexification by numbers" (and more generally pp. 107–115):
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Pages 93–94 (especially example 1.92), 143, 318–319, 375–377, and 416:
2658:; Whinston, Michael D.; Green, Jerry R. (1995). "17.1 Large economies and nonconvexities".
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2163:. (The meanings of "closed set" is explained below, in the subsection on optimization applications.)
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3891:. Handbooks in economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 15–52.
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2517:. Princeton studies in mathematical economics. Princeton University Press. pp. viii+251.
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4053:. Princeton landmarks in mathematics (Reprint of the 1979 Princeton mathematical series
3634:
3265:(pages 33, 43, 48, 56, 70–72, 82, 147, and 234–236):
3234:
Starrett, David A. (1972). "Fundamental nonconvexities in the theory of externalities".
2947:
2939:
2935:
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Starr, Ross M. (2011). "8 Convex sets, separation theorems, and non‑convex sets in
2508:
Theorem C(6) on page 37 and applications on pages 115–116, 122, and 168:
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A convex set can have more than one supporting hyperplane at a given point on its boundary.
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Pages 52–55 with applications on pages 145–146, 152–153, and 274–275:
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Non‑convex sets have been incorporated in the theories of general economic equilibria, of
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Brown, Donald J. (1991). "36 Equilibrium analysis with non‑convex technologies". In
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For simplicity, we shall assume that the preferences of a consumer can be described by a
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is not convex, the statement of the theorem is not true at all points on the boundary of
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2855:. Studies in economic theory. Vol. 9. Berlin: Springer-Verlag. pp. xii+414.
2120:) over other affordable vectors. At this optimal vector, the budget line supports the
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Economists have studied non‑convex sets using advanced mathematics, particularly
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Diewert, W. E. (1982). "12 Duality approaches to microeconomic theory". In
2286:). Such applications continued to motivate economists to study non‑convex sets.
1696:(the dashed line), and the half-space delimited by the hyperplane that contains
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posits that if preferences are convex and there is perfect competition, then
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2449:. Mathematical economics texts . San Francisco: Holden-Day, Inc. . pp.
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Here, a closed half-space is the half-space that includes the hyperplane.
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Market demand: An analysis of large economies with nonconvex preferences
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3275:. Cambridge economic handbooks. Cambridge: Cambridge University Press.
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494:
3639:"Intersecting families of sets and the topology of cones in economics"
3399:(1996). "6 Production in a finance economy, Section 31 Partnerships".
2320:
Relationship between subderivatives and non‑convexity remains cryptic.
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is convex; however, anything that is hollow or dented, for example, a
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The economics depends upon the following definitions and results from
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A supporting hyperplane containing a given point on the boundary of
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2487:. Econometric Society Monographs. Cambridge University Press.
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An optimal basket of goods occurs where the consumer's convex
4695:
4577:
4557:
4334:
3811:. In Durlauf, Steven N.; Blume, Lawrence E (eds.).
3780:. In Durlauf, Steven N.; Blume, Lawrence E (eds.).
3749:. In Durlauf, Steven N.; Blume, Lawrence E (eds.).
3718:. In Durlauf, Steven N.; Blume, Lawrence E (eds.).
2211:. These results are described in graduate-level textbooks in
5077:
3091:(1926). "The Laws of returns under competitive conditions".
2183:
1285:
More generally, any real vector space of (finite) dimension
4852:
4131:
3424:. In Eatwell, John; Milgate, Murray; Newman, Peter (eds.).
3345:. Cambridge: Cambridge University Press. pp. 261–297.
2555:(Economica, Paris) ed.). MIT Press. pp. 107–125.
3520:
Variational analysis and generalized differentiation
2692:(Second ed.). Cambridge: Cambridge University Press.
2628:(3rd ed.). W. W. Norton & Company. pp.
2355:
Economists have increasingly studied non‑convex sets with
937:
is a geometric property with a variety of applications in
3815:(Second ed.). Palgrave Macmillan. pp. 815–816.
3784:(Second ed.). Palgrave Macmillan. pp. 551–555.
3753:(Second ed.). Palgrave Macmillan. pp. 220–225.
3722:(Second ed.). Palgrave Macmillan. pp. 225–226.
3553:(Second ed.). Palgrave Macmillan. pp. 354–365.
3549:. In Durlauf, Steven N.; Blume, Lawrence E. (eds.).
2242:" (markets dominated by a few producers), especially in "
3947:(first ed.). Palgrave Macmillan. pp. 653–661.
3428:(first ed.). Palgrave Macmillan. pp. 653–661.
3403:. Cambridge, Massachusetts: MIT Press. pp. 329–425.
3380:
Allocation under
Uncertainty: Equilibrium and Optimality
4057: ed.). Princeton, NJ: Princeton University Press.
2654:
3261:
Starrett discusses non‑convexities in his textbook on
2850:
Mathematical methods for economic theory: Volume
3302:(1968). "Competitive equilibrium under uncertainty".
3008:(1975). "Pareto optimality in non‑convex economies".
2999:
2997:
2075:
2055:
2025:
2005:
1980:
1974:
then there exists a supporting hyperplane containing
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if, for each pair of its points, every point on the
2262:in 1938. Both Sraffa and Hotelling illuminated the
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for some indexed set of non‑negative real numbers {
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2042:as illustrated in the third picture on the right.
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1630:) is the intersection of all the convex sets that
4123:: CS1 maint: DOI inactive as of September 2024 (
4023:(first ed.). Palgrave Macmillan. p. 1.
3985:(first ed.). Palgrave Macmillan. p. 1.
5490:
3886:Handbook of mathematical economics, Volume
3844:Handbook of mathematical economics, Volume
3592:Handbook of mathematical economics, Volume
2886:Florenzano, Monique; Le Van, Cuong (2001).
2885:
2726:Competitive equilibrium: Theory and applications
2223:, and applied mathematics (for economists). The
1854:
1587:The definition of a convex set implies that the
3456:
2368:
2102:The consumer prefers the vector of goods (
1358:In a real vector space, a set is defined to be
4084:Convex bodies: The Brunn–Minkowski theory
3343:Essays on economic decisions under uncertainty
3182:Rendements croissants et efficacité economique
2553:Microéconomie: Les défaillances du marché
2481:The Theory of General Economic Equilibrium: A
5063:
4396:
4147:
3720:The New Palgrave Dictionary of Economics
3646:Bulletin of the American Mathematical Society
3506:
3394:
2888:Finite dimensional convexity and optimization
2662:. Oxford University Press. pp. 627–630.
2547:Salanié, Bernard (2000). "7 Nonconvexities".
915:
3633:
3514:
2787:(1981). "Appendix A Convex sets".
2372:
4045:
4021:The New Palgrave: A Dictionary of Economics
3983:The New Palgrave: A Dictionary of Economics
3945:The New Palgrave: A Dictionary of Economics
3879:
3426:The New Palgrave: A Dictionary of Economics
3205:
2728:. Cambridge University Press. p. 420.
2690:General equilibrium theory: An introduction
2509:
1178:. Two points in the Cartesian plane can be
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5056:
4403:
4389:
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3413:
2551:(English translation of the (1998) French
2475:
2442:"Appendix B: Convex and related sets"
2413:
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2343:Learn how and when to remove this message
1904:
1847:has at least one point on the hyperplane.
1778:
1070:Learn how and when to remove this message
1006:; non-convex phenomena are studied under
3813:The New Palgrave Dictionary of Economics
3806:
3782:The New Palgrave Dictionary of Economics
3751:The New Palgrave Dictionary of Economics
3551:The New Palgrave Dictionary of Economics
3266:
3233:
2749:
2748:Theorem 1.6.5 on pages 24–25:
2182:
2097:
2044:
1858:
1825:half-spaces determined by the hyperplane
1821:is entirely contained in one of the two
1648:
1341:
953:sort of good; this represents a kind of
5160:Locally convex topological vector space
4007:
3969:
3884:; Intriligator, Michael D (eds.).
3842:; Intriligator, Michael D (eds.).
3837:
3172:
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2419:
1035:not related to the topic of the article
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2515:Core and equilibria of a large economy
2430:Pages 392–399 and page 188:
1676:(in pink), a supporting hyperplane of
5051:
4384:
4135:
4015:. In Eatwell, John; Milgate, Murray;
3977:. In Eatwell, John; Milgate, Murray;
3939:. In Eatwell, John; Milgate, Murray;
3581:
3374:
3337:
2847:
2823:Foundations of mathematical economics
2683:
2384:
2380:
2289:
2250:wrote about on firms with increasing
1350:of the red set, each blue point is a
1089:
4352:List of differential geometry topics
3925:, McGraw-Hill, Inc., New York, 1995.
3544:
3538:
3407:
3382:. New York: Wiley. pp. 129–165.
2682:Page 169 in the first edition:
2376:
2293:
1017:
976:for every commodity in the economy.
3044:Guesnerie, Roger (1975). "Errata".
1798:if it meets both of the following:
1498:, . . . ,
13:
4959:Microfoundations of macroeconomics
4410:
3211:The Theory of environmental policy
3204:Pages 106, 110–137, 172, and 248:
957:of having more of the same good.
949:of goods over having a lot of any
14:
5525:
3776:Blume, Lawrence E. (2008c).
3745:Blume, Lawrence E. (2008b).
3178:Increasing returns and efficiency
2752:Game theory for economic analysis
2549:Microeconomics of market failures
1920:{\displaystyle \mathbb {R} ^{n},}
1645:Duality: Intersecting half-spaces
1372:by the set. For example, a solid
1116:connecting any two of its points.
5032:
5031:
5020:
4365:
4364:
3401:The Theory of incomplete markets
3267:Starrett, David A. (1988).
2588:Fundamentals of public economics
2479:(1985). "1.L Averages of sets".
2298:
2166:
1791:{\displaystyle \mathbb {R} ^{n}}
1121:
1098:
1022:
1013:
889:
877:
62:
5265:Ekeland's variational principle
3686:10.1090/S0273-0979-1993-00439-7
3627:
3575:
3450:
3387:
3330:
3292:
3271:Foundations of public economics
3227:
3198:
3180:(Revised translation of (1988)
3165:
3127:
3081:
2928:
1468:is convex if and only if every
1332:multiplication by a real number
164:Concepts, theory and techniques
2825:. MIT Press. pp. xx+649.
2813:
2776:
2742:
2676:
2619:"21.2 Convexity and size"
2606:
2569:
2539:
2424:
2215:, general equilibrium theory,
1596:
1337:
1038: and should be moved to
16:Significant topic in economics
1:
3897:10.1016/S1573-4382(81)01005-9
3855:10.1016/S1573-4382(82)02007-4
3704:
3603:10.1016/S1573-4382(05)80011-6
2848:Moore, James C. (1999).
2369:Rockafellar & Wets (1998)
1855:Supporting hyperplane theorem
1395:is convex if, for all points
1050:or discuss this issue on the
4161:
4047:Rockafellar, R. Tyrrell
3458:Rockafellar, R. Tyrrell
3287:nonconvex OR nonconvexities.
3248:10.1016/0022-0531(72)90148-2
2789:Economics for mathematicians
2447:General competitive analysis
2093:
1605:of a real vector space, its
966:general economic equilibrium
955:diminishing marginal utility
7:
5285:Hermite–Hadamard inequality
4904:Civil engineering economics
4889:Statistical decision theory
4529:Income elasticity of demand
2785:Cassels, J. W. S.
2394:
2367:calculus), as described by
2318:. The specific problem is:
1505:} of a vector space is any
1484:of an indexed subset {
1168:Cartesian coordinate system
10:
5530:
5509:General equilibrium theory
4539:Price elasticity of supply
4534:Price elasticity of demand
4524:Cross elasticity of demand
4029:10.1057/9780230226203.2412
3991:10.1057/9780230226203.2282
3953:10.1057/9780230226203.3173
3882:Arrow, Kenneth Joseph
3840:Arrow, Kenneth Joseph
3821:10.1057/9780230226203.1375
3790:10.1057/9780230226203.0411
3759:10.1057/9780230226203.0314
3728:10.1057/9780230226203.0315
3559:10.1057/9780230226203.1267
3545:Khan, M. Ali (2008).
3434:10.1057/9780230226203.3173
3236:Journal of Economic Theory
2750:Ichiishi, Tatsuro (1983).
2176:
2170:
1735:. A hyperplane is said to
1413:and for every real number
1322: } together with two
1136:joining two of its points.
1128:A non‑convex set fails to
5470:
5437:
5392:
5323:
5249:
5173:
5115:
5089:
5015:
4982:
4861:
4418:
4360:
4327:
4296:
4246:
4194:
4169:
3474:10.1007/978-3-642-02431-3
3373:(Originally published as
2970:10.1007/978-3-642-46488-1
2896:10.1007/978-3-642-56522-9
2861:10.1007/978-3-662-08544-8
2724:Ellickson, Bryan (1994).
2173:Non-convexity (economics)
2155:, which implies that the
1731:divides a space into two
1622:convex set that contains
1293:of all possible lists of
5471:Applications and related
5275:Fenchel-Young inequality
4595:Income–consumption curve
4093:10.1017/CBO9780511526282
4081:Schneider, Rolf (1993).
3807:Crouzeix, J.-P. (2008).
3351:10.1017/CBO9780511559464
2960:Trockel, Walter (1984).
2821:Carter, Michael (2001).
2698:10.1017/CBO9781139174749
2406:
1248:further, a point can be
152:JEL classification codes
29:JEL classification codes
5231:Legendre transformation
5155:Legendre transformation
4929:Industrial organization
4347:List of geometry topics
4095:(inactive 2024-09-13).
3712:Blume, Lawrence E.
3207:Baumol, William J.
338:Industrial organization
195:Computational economics
5478:Convexity in economics
5412:(lower) ideally convex
5270:Fenchel–Moreau theorem
5260:Carathéodory's theorem
3516:Mordukhovich, Boris S.
3376:Drèze, Jacques H.
3339:Drèze, Jacques H.
2625:Microeconomic Analysis
2433:Arrow, Kenneth J.
2221:mathematical economics
2188:
2132:
2090:
2083:
2063:
2036:
2013:
1991:
1968:
1941:
1921:
1885:
1864:
1841:
1815:
1792:
1756:
1717:
1710:
1690:
1670:
1462:mathematical induction
1380:shape, is non‑convex.
1355:
190:Experimental economics
23:Convexity in economics
5400:Convex series related
5300:Shapley–Folkman lemma
4899:Engineering economics
4494:Cost–benefit analysis
4319:Differential geometry
3921:Luenberger, David G.
3547:"Perfect competition"
2936:differential geometry
2782:Pages 127 and 33–34:
2578:Laffont, Jean-Jacques
2276:information economics
2260:marginal cost pricing
2254:in 1926, after which
2225:Shapley–Folkman lemma
2186:
2179:Shapley–Folkman lemma
2101:
2084:
2064:
2048:
2037:
2014:
1992:
1969:
1942:
1922:
1886:
1862:
1842:
1816:
1793:
1757:
1721:Supporting hyperplane
1711:
1691:
1671:
1652:
1428:(1 −
1391:More formally, a set
1345:
1289:can be viewed as the
1033:may contain material
5290:Krein–Milman theorem
5083:variational analysis
4716:Price discrimination
4610:Intertemporal choice
3923:Microeconomic Theory
3747:"Convex programming"
3466:Variational analysis
3462:Wets, Roger J-B
2660:Microeconomic theory
2359:, which generalizes
2325:improve this section
2314:to meet Knowledge's
2229:production economies
2073:
2053:
2023:
2003:
1978:
1955:
1931:
1899:
1875:
1831:
1805:
1773:
1746:
1700:
1680:
1660:
1536:+ . . . +
1254:by each real number
1048:improve this section
417:Social choice theory
5514:Convex optimization
5280:Jensen's inequality
5150:Lagrange multiplier
5140:Convex optimization
5135:Convex metric space
5027:Business portal
4964:Operations research
4791:Substitution effect
3668:1993math.....10228C
3584:Hildenbrand, Werner
2614:Varian, Hal R.
2511:Hildenbrand, Werner
2437:Hahn, Frank H.
2373:Mordukhovich (2006)
2268:environmental goods
2153:continuous function
1480:. By definition, a
1368:that joins them is
1354:of some red points.
983:is associated with
884:Business portal
205:Operations research
185:National accounting
26:is included in the
5408:(cs, bcs)-complete
5379:Algebraic interior
5097:Convex combination
4605:Indifference curve
4573:Goods and services
4514:Economies of scope
4509:Economies of scale
4314:Algebraic geometry
3588:Sonnenschein, Hugo
2952:integration theory
2656:Mas–Colell, Andreu
2582:"3 Nonconvexities"
2575:Pages 63–65:
2477:Mas-Colell, Andreu
2389:algebraic topology
2357:nonsmooth analysis
2290:Nonsmooth analysis
2284:incomplete markets
2189:
2133:
2122:indifference curve
2091:
2079:
2059:
2035:{\displaystyle S,}
2032:
2009:
1990:{\displaystyle x.}
1987:
1967:{\displaystyle S,}
1964:
1947:is a point on the
1937:
1917:
1881:
1865:
1837:
1811:
1788:
1752:
1718:
1706:
1686:
1666:
1482:convex combination
1470:convex combination
1356:
1352:convex combination
1090:Real vector spaces
1008:nonsmooth analysis
970:aggregate supplies
962:Arrow–Debreu model
943:convex preferences
215:Industrial complex
210:Middle income trap
5486:
5485:
5045:
5044:
5007:Political economy
4806:Supply and demand
4686:Pareto efficiency
4378:
4377:
4102:978-0-521-35220-8
4064:978-0-691-01586-6
3906:978-0-444-86126-9
3864:978-0-444-86127-6
3830:978-0-333-78676-5
3809:"Quasi-concavity"
3799:978-0-333-78676-5
3768:978-0-333-78676-5
3737:978-0-333-78676-5
3568:978-0-333-78676-5
3483:978-3-540-62772-2
3395:Magill, Michael;
3360:978-0-521-26484-6
3220:978-0-521-31112-0
3191:978-0-19-506553-4
3135:Hotelling, Harold
2979:978-3-540-12881-6
2905:978-3-540-41516-9
2870:978-3-540-66235-8
2832:978-0-262-53192-4
2798:978-0-521-28614-5
2761:978-0-12-370180-0
2735:978-0-521-31988-1
2707:978-0-521-53386-7
2669:978-0-19-507340-9
2639:978-0-393-95735-8
2599:978-0-262-12127-9
2562:978-0-262-19443-3
2524:978-0-691-04189-6
2494:978-0-521-26514-0
2460:978-0-444-85497-1
2365:locally Lipschitz
2353:
2352:
2345:
2316:quality standards
2307:This section may
2082:{\displaystyle S}
2069:may not exist if
2062:{\displaystyle S}
2012:{\displaystyle S}
1940:{\displaystyle x}
1884:{\displaystyle S}
1840:{\displaystyle S}
1814:{\displaystyle S}
1755:{\displaystyle S}
1716:(in light blue).
1709:{\displaystyle S}
1689:{\displaystyle S}
1669:{\displaystyle S}
1601:For every subset
1558:} satisfying the
1258:coordinate-wise
1080:
1079:
1072:
993:market equilibria
989:supply and demand
974:aggregate demands
932:
931:
5521:
5404:(cs, lcs)-closed
5350:Effective domain
5305:Robinson–Ursescu
5181:Convex conjugate
5072:
5065:
5058:
5049:
5048:
5035:
5034:
5025:
5024:
4767:Returns to scale
4625:Market structure
4405:
4398:
4391:
4382:
4381:
4368:
4367:
4156:
4149:
4142:
4133:
4132:
4128:
4122:
4114:
4076:
4042:
4004:
3966:
3938:
3918:
3876:
3834:
3803:
3772:
3741:
3698:
3697:
3679:
3661:
3643:
3635:Chichilnisky, G.
3631:
3625:
3624:
3579:
3573:
3572:
3542:
3536:
3535:
3510:
3504:
3503:
3454:
3448:
3447:
3423:
3411:
3405:
3404:
3397:Quinzii, Martine
3391:
3385:
3383:
3372:
3334:
3328:
3327:
3296:
3290:
3289:
3274:
3263:public economics
3259:
3231:
3225:
3224:
3202:
3196:
3195:
3174:Quinzii, Martine
3169:
3163:
3162:
3131:
3125:
3124:
3099:(144): 535–550.
3093:Economic Journal
3085:
3079:
3077:
3041:
3006:Guesnerie, Roger
3001:
2992:
2991:
2932:
2926:
2925:
2882:
2844:
2817:
2811:
2810:
2780:
2774:
2773:
2746:
2740:
2739:
2719:
2680:
2674:
2673:
2651:
2621:
2610:
2604:
2603:
2591:
2573:
2567:
2566:
2543:
2537:
2536:
2506:
2472:
2444:
2428:
2422:
2417:
2348:
2341:
2337:
2334:
2328:
2302:
2301:
2294:
2256:Harold Hotelling
2252:returns to scale
2231:with many small
2209:public economics
2195:
2149:utility function
2088:
2086:
2085:
2080:
2068:
2066:
2065:
2060:
2041:
2039:
2038:
2033:
2018:
2016:
2015:
2010:
1996:
1994:
1993:
1988:
1973:
1971:
1970:
1965:
1946:
1944:
1943:
1938:
1926:
1924:
1923:
1918:
1913:
1912:
1907:
1890:
1888:
1887:
1882:
1846:
1844:
1843:
1838:
1820:
1818:
1817:
1812:
1797:
1795:
1794:
1789:
1787:
1786:
1781:
1761:
1759:
1758:
1753:
1723:is a concept in
1715:
1713:
1712:
1707:
1695:
1693:
1692:
1687:
1675:
1673:
1672:
1667:
1617:
1584: = 1.
1583:
1550:
1507:weighted average
1476:also belongs to
1423:
1321:
1184:coordinate-wise
1132:a point in some
1125:
1102:
1075:
1068:
1064:
1061:
1055:
1026:
1025:
1018:
991:differ or where
924:
917:
910:
896:Money portal
894:
893:
892:
882:
881:
378:Natural resource
170:Economic systems
66:
43:
42:
30:
25:
5529:
5528:
5524:
5523:
5522:
5520:
5519:
5518:
5504:Convex geometry
5489:
5488:
5487:
5482:
5466:
5433:
5388:
5319:
5245:
5236:Semi-continuity
5221:Convex function
5202:Logarithmically
5169:
5130:Convex geometry
5111:
5102:Convex function
5085:
5079:Convex analysis
5076:
5046:
5041:
5019:
5011:
4978:
4857:
4499:Deadweight loss
4436:Consumer choice
4414:
4409:
4379:
4374:
4356:
4323:
4292:
4249:
4242:
4197:
4190:
4165:
4160:
4116:
4115:
4103:
4065:
4051:Convex analysis
4039:
4001:
3963:
3936:
3934:"Non‑convexity"
3907:
3865:
3831:
3800:
3769:
3738:
3707:
3702:
3701:
3677:10.1.1.234.3909
3641:
3632:
3628:
3613:
3580:
3576:
3569:
3543:
3539:
3513:
3511:
3507:
3484:
3455:
3451:
3444:
3421:
3419:"Non‑convexity"
3412:
3408:
3392:
3388:
3361:
3335:
3331:
3316:10.2307/1909602
3297:
3293:
3283:
3260:
3232:
3228:
3221:
3203:
3199:
3192:
3170:
3166:
3151:10.2307/1907054
3132:
3128:
3105:10.2307/2959866
3086:
3082:
3058:10.2307/1911353
3022:10.2307/1913410
3002:
2995:
2980:
2933:
2929:
2906:
2883:
2871:
2845:
2833:
2818:
2814:
2799:
2781:
2777:
2762:
2747:
2743:
2736:
2720:
2708:
2681:
2677:
2670:
2652:
2640:
2611:
2607:
2600:
2574:
2570:
2563:
2544:
2540:
2525:
2507:
2495:
2473:
2461:
2429:
2425:
2418:
2414:
2409:
2397:
2375:, according to
2361:convex analysis
2349:
2338:
2332:
2329:
2322:
2303:
2299:
2292:
2205:market failures
2193:
2181:
2175:
2169:
2157:preference sets
2130:
2119:
2110:
2096:
2074:
2071:
2070:
2054:
2051:
2050:
2024:
2021:
2020:
2004:
2001:
2000:
1979:
1976:
1975:
1956:
1953:
1952:
1932:
1929:
1928:
1908:
1903:
1902:
1900:
1897:
1896:
1876:
1873:
1872:
1871:states that if
1857:
1832:
1829:
1828:
1806:
1803:
1802:
1782:
1777:
1776:
1774:
1771:
1770:
1747:
1744:
1743:
1701:
1698:
1697:
1681:
1678:
1677:
1661:
1658:
1657:
1647:
1606:
1599:
1582:
1575:
1568:
1562:
1557:
1548:
1542:
1535:
1529:
1522:
1516:
1510:
1504:
1497:
1490:
1445:
1438:
1421:
1408:
1401:
1340:
1328:vector addition
1319:
1312:
1305:
1298:
1243:
1236:
1229:
1222:
1215:
1208:
1201:
1194:
1166:may be given a
1153:
1152:
1151:
1150:
1139:
1138:
1137:
1126:
1118:
1117:
1103:
1092:
1084:convex geometry
1076:
1065:
1059:
1056:
1045:
1041:convex analysis
1027:
1023:
1016:
1004:convex analysis
985:market failures
928:
890:
888:
876:
869:
868:
839:
829:
828:
827:
826:
590:von Böhm-Bawerk
478:
467:
466:
228:
220:
219:
175:Economic growth
165:
157:
156:
98:
96:classifications
41:
40:
39:
28:
21:
17:
12:
11:
5:
5527:
5517:
5516:
5511:
5506:
5501:
5484:
5483:
5481:
5480:
5474:
5472:
5468:
5467:
5465:
5464:
5459:
5457:Strong duality
5454:
5449:
5443:
5441:
5435:
5434:
5432:
5431:
5396:
5394:
5390:
5389:
5387:
5386:
5381:
5372:
5367:
5365:John ellipsoid
5362:
5357:
5352:
5347:
5333:
5327:
5325:
5321:
5320:
5318:
5317:
5312:
5307:
5302:
5297:
5292:
5287:
5282:
5277:
5272:
5267:
5262:
5256:
5254:
5252:results (list)
5247:
5246:
5244:
5243:
5238:
5233:
5228:
5226:Invex function
5223:
5214:
5209:
5204:
5199:
5194:
5188:
5183:
5177:
5175:
5171:
5170:
5168:
5167:
5162:
5157:
5152:
5147:
5142:
5137:
5132:
5127:
5125:Choquet theory
5121:
5119:
5113:
5112:
5110:
5109:
5104:
5099:
5093:
5091:
5090:Basic concepts
5087:
5086:
5075:
5074:
5067:
5060:
5052:
5043:
5042:
5040:
5039:
5029:
5016:
5013:
5012:
5010:
5009:
5004:
5002:Macroeconomics
4999:
4998:
4997:
4986:
4984:
4980:
4979:
4977:
4976:
4971:
4966:
4961:
4956:
4951:
4946:
4941:
4936:
4931:
4926:
4921:
4916:
4911:
4906:
4901:
4896:
4891:
4886:
4881:
4876:
4871:
4865:
4863:
4859:
4858:
4856:
4855:
4850:
4849:
4848:
4843:
4833:
4828:
4827:
4826:
4817:
4803:
4798:
4793:
4788:
4779:
4774:
4769:
4764:
4759:
4754:
4749:
4744:
4739:
4738:
4737:
4732:
4723:
4718:
4713:
4708:
4703:
4701:Price controls
4693:
4688:
4683:
4682:
4681:
4676:
4671:
4666:
4665:
4664:
4659:
4649:
4644:
4643:
4642:
4637:
4622:
4620:Market failure
4617:
4612:
4607:
4602:
4597:
4592:
4587:
4586:
4585:
4580:
4570:
4565:
4560:
4555:
4554:
4553:
4543:
4542:
4541:
4536:
4531:
4526:
4516:
4511:
4506:
4501:
4496:
4491:
4490:
4489:
4484:
4479:
4474:
4473:
4472:
4462:
4457:
4447:
4438:
4433:
4428:
4422:
4420:
4416:
4415:
4412:Microeconomics
4408:
4407:
4400:
4393:
4385:
4376:
4375:
4373:
4372:
4361:
4358:
4357:
4355:
4354:
4349:
4344:
4343:
4342:
4331:
4329:
4325:
4324:
4322:
4321:
4316:
4311:
4306:
4300:
4298:
4294:
4293:
4291:
4290:
4285:
4280:
4275:
4270:
4265:
4260:
4254:
4252:
4248:Non-Euclidean
4244:
4243:
4241:
4240:
4238:Solid geometry
4235:
4234:
4233:
4228:
4221:Plane geometry
4218:
4213:
4208:
4202:
4200:
4192:
4191:
4189:
4188:
4183:
4182:
4181:
4170:
4167:
4166:
4159:
4158:
4151:
4144:
4136:
4130:
4129:
4101:
4078:
4063:
4043:
4037:
4005:
3999:
3967:
3961:
3930:Mas-Colell, A.
3926:
3919:
3905:
3877:
3863:
3835:
3829:
3804:
3798:
3773:
3767:
3742:
3736:
3706:
3703:
3700:
3699:
3652:(2): 189–207.
3648:. New Series.
3626:
3611:
3574:
3567:
3537:
3524:: Applications
3505:
3482:
3449:
3442:
3415:Mas-Colell, A.
3406:
3386:
3359:
3329:
3291:
3281:
3242:(2): 180–199.
3226:
3219:
3197:
3190:
3164:
3145:(3): 242–269.
3126:
3080:
2993:
2978:
2956:ergodic theory
2944:Baire category
2927:
2904:
2869:
2831:
2812:
2797:
2775:
2760:
2741:
2734:
2706:
2675:
2668:
2638:
2605:
2598:
2568:
2561:
2538:
2523:
2493:
2483:Differentiable
2459:
2423:
2420:Newman (1987c)
2411:
2410:
2408:
2405:
2404:
2403:
2401:Convex duality
2396:
2393:
2351:
2350:
2306:
2304:
2297:
2291:
2288:
2213:microeconomics
2171:Main article:
2168:
2165:
2137:preference set
2128:
2115:
2106:
2095:
2092:
2089:is not convex.
2078:
2058:
2031:
2028:
2008:
1986:
1983:
1963:
1960:
1936:
1916:
1911:
1906:
1880:
1856:
1853:
1849:
1848:
1836:
1826:
1810:
1785:
1780:
1751:
1705:
1685:
1665:
1646:
1643:
1598:
1595:
1580:
1573:
1566:
1555:
1546:
1540:
1533:
1527:
1520:
1514:
1502:
1495:
1488:
1472:of members of
1447:
1446:
1443:
1436:
1406:
1399:
1339:
1336:
1317:
1310:
1303:
1283:
1282:
1246:
1245:
1241:
1234:
1227:
1220:
1213:
1206:
1199:
1192:
1141:
1140:
1127:
1120:
1119:
1104:
1097:
1096:
1095:
1094:
1093:
1091:
1088:
1078:
1077:
1030:
1028:
1021:
1015:
1012:
930:
929:
927:
926:
919:
912:
904:
901:
900:
899:
898:
886:
871:
870:
867:
866:
861:
851:
846:
840:
835:
834:
831:
830:
825:
824:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
642:
637:
632:
627:
622:
617:
612:
607:
602:
597:
592:
587:
582:
577:
572:
567:
562:
557:
552:
547:
542:
537:
532:
527:
522:
517:
512:
507:
502:
497:
492:
487:
481:
480:
479:
473:
472:
469:
468:
465:
464:
459:
454:
449:
444:
439:
434:
429:
424:
419:
410:
405:
400:
395:
390:
385:
383:Organizational
380:
375:
370:
365:
360:
355:
350:
345:
340:
335:
330:
325:
320:
315:
310:
305:
300:
295:
290:
285:
280:
275:
270:
265:
260:
255:
250:
245:
240:
235:
229:
227:By application
226:
225:
222:
221:
218:
217:
212:
207:
202:
197:
192:
187:
182:
177:
172:
166:
163:
162:
159:
158:
155:
154:
149:
144:
139:
134:
129:
120:
115:
110:
105:
99:
93:
92:
89:
88:
87:
86:
81:
76:
68:
67:
59:
58:
52:
51:
20:
19:
18:
15:
9:
6:
4:
3:
2:
5526:
5515:
5512:
5510:
5507:
5505:
5502:
5500:
5497:
5496:
5494:
5479:
5476:
5475:
5473:
5469:
5463:
5460:
5458:
5455:
5453:
5450:
5448:
5445:
5444:
5442:
5440:
5436:
5429:
5427:
5421:
5419:
5413:
5409:
5405:
5401:
5398:
5397:
5395:
5391:
5385:
5382:
5380:
5376:
5373:
5371:
5368:
5366:
5363:
5361:
5358:
5356:
5353:
5351:
5348:
5346:
5342:
5338:
5334:
5332:
5329:
5328:
5326:
5322:
5316:
5313:
5311:
5308:
5306:
5303:
5301:
5298:
5296:
5295:Mazur's lemma
5293:
5291:
5288:
5286:
5283:
5281:
5278:
5276:
5273:
5271:
5268:
5266:
5263:
5261:
5258:
5257:
5255:
5253:
5248:
5242:
5241:Subderivative
5239:
5237:
5234:
5232:
5229:
5227:
5224:
5222:
5218:
5215:
5213:
5210:
5208:
5205:
5203:
5200:
5198:
5195:
5193:
5189:
5187:
5184:
5182:
5179:
5178:
5176:
5172:
5166:
5163:
5161:
5158:
5156:
5153:
5151:
5148:
5146:
5143:
5141:
5138:
5136:
5133:
5131:
5128:
5126:
5123:
5122:
5120:
5118:
5117:Topics (list)
5114:
5108:
5105:
5103:
5100:
5098:
5095:
5094:
5092:
5088:
5084:
5080:
5073:
5068:
5066:
5061:
5059:
5054:
5053:
5050:
5038:
5030:
5028:
5023:
5018:
5017:
5014:
5008:
5005:
5003:
5000:
4996:
4993:
4992:
4991:
4988:
4987:
4985:
4981:
4975:
4972:
4970:
4967:
4965:
4962:
4960:
4957:
4955:
4952:
4950:
4947:
4945:
4942:
4940:
4937:
4935:
4934:Institutional
4932:
4930:
4927:
4925:
4922:
4920:
4917:
4915:
4912:
4910:
4907:
4905:
4902:
4900:
4897:
4895:
4892:
4890:
4887:
4885:
4882:
4880:
4879:Computational
4877:
4875:
4872:
4870:
4867:
4866:
4864:
4860:
4854:
4851:
4847:
4844:
4842:
4839:
4838:
4837:
4834:
4832:
4829:
4825:
4824:Law of supply
4821:
4818:
4816:
4815:Law of demand
4812:
4809:
4808:
4807:
4804:
4802:
4801:Social choice
4799:
4797:
4794:
4792:
4789:
4787:
4786:Excess supply
4783:
4780:
4778:
4775:
4773:
4772:Risk aversion
4770:
4768:
4765:
4763:
4760:
4758:
4755:
4753:
4750:
4748:
4745:
4743:
4740:
4736:
4733:
4731:
4727:
4724:
4722:
4719:
4717:
4714:
4712:
4709:
4707:
4706:Price ceiling
4704:
4702:
4699:
4698:
4697:
4694:
4692:
4689:
4687:
4684:
4680:
4677:
4675:
4672:
4670:
4667:
4663:
4662:Complementary
4660:
4658:
4655:
4654:
4653:
4650:
4648:
4645:
4641:
4638:
4636:
4633:
4632:
4631:
4628:
4627:
4626:
4623:
4621:
4618:
4616:
4613:
4611:
4608:
4606:
4603:
4601:
4598:
4596:
4593:
4591:
4588:
4584:
4581:
4579:
4576:
4575:
4574:
4571:
4569:
4566:
4564:
4561:
4559:
4556:
4552:
4549:
4548:
4547:
4544:
4540:
4537:
4535:
4532:
4530:
4527:
4525:
4522:
4521:
4520:
4517:
4515:
4512:
4510:
4507:
4505:
4502:
4500:
4497:
4495:
4492:
4488:
4485:
4483:
4480:
4478:
4475:
4471:
4468:
4467:
4466:
4463:
4461:
4458:
4456:
4453:
4452:
4451:
4448:
4446:
4445:non-convexity
4442:
4439:
4437:
4434:
4432:
4429:
4427:
4424:
4423:
4421:
4417:
4413:
4406:
4401:
4399:
4394:
4392:
4387:
4386:
4383:
4371:
4363:
4362:
4359:
4353:
4350:
4348:
4345:
4341:
4338:
4337:
4336:
4333:
4332:
4330:
4326:
4320:
4317:
4315:
4312:
4310:
4307:
4305:
4302:
4301:
4299:
4295:
4289:
4286:
4284:
4281:
4279:
4276:
4274:
4271:
4269:
4266:
4264:
4261:
4259:
4256:
4255:
4253:
4251:
4245:
4239:
4236:
4232:
4229:
4227:
4224:
4223:
4222:
4219:
4217:
4214:
4212:
4209:
4207:
4206:Combinatorial
4204:
4203:
4201:
4199:
4193:
4187:
4184:
4180:
4177:
4176:
4175:
4172:
4171:
4168:
4164:
4157:
4152:
4150:
4145:
4143:
4138:
4137:
4134:
4126:
4120:
4112:
4108:
4104:
4098:
4094:
4090:
4086:
4085:
4079:
4074:
4070:
4066:
4060:
4056:
4052:
4048:
4044:
4040:
4038:9780333786765
4034:
4030:
4026:
4022:
4018:
4017:Newman, Peter
4014:
4010:
4009:Newman, Peter
4006:
4002:
4000:9780333786765
3996:
3992:
3988:
3984:
3980:
3979:Newman, Peter
3976:
3972:
3971:Newman, Peter
3968:
3964:
3962:9780333786765
3958:
3954:
3950:
3946:
3942:
3941:Newman, Peter
3935:
3931:
3927:
3924:
3920:
3916:
3912:
3908:
3902:
3898:
3894:
3890:
3889:
3883:
3878:
3874:
3870:
3866:
3860:
3856:
3852:
3848:
3847:
3841:
3836:
3832:
3826:
3822:
3818:
3814:
3810:
3805:
3801:
3795:
3791:
3787:
3783:
3779:
3774:
3770:
3764:
3760:
3756:
3752:
3748:
3743:
3739:
3733:
3729:
3725:
3721:
3717:
3713:
3709:
3708:
3695:
3691:
3687:
3683:
3678:
3673:
3669:
3665:
3660:
3655:
3651:
3647:
3640:
3636:
3630:
3622:
3618:
3614:
3612:0-444-87461-5
3608:
3604:
3600:
3596:
3595:
3589:
3585:
3578:
3570:
3564:
3560:
3556:
3552:
3548:
3541:
3533:
3529:
3525:
3523:
3517:
3509:
3501:
3497:
3493:
3489:
3485:
3479:
3475:
3471:
3467:
3463:
3459:
3453:
3445:
3443:9780333786765
3439:
3435:
3431:
3427:
3420:
3416:
3410:
3402:
3398:
3390:
3381:
3377:
3370:
3366:
3362:
3356:
3352:
3348:
3344:
3340:
3333:
3325:
3321:
3317:
3313:
3309:
3305:
3301:
3295:
3288:
3284:
3282:9780521348010
3278:
3273:
3272:
3264:
3257:
3253:
3249:
3245:
3241:
3237:
3230:
3222:
3216:
3212:
3208:
3201:
3193:
3187:
3183:
3179:
3175:
3168:
3160:
3156:
3152:
3148:
3144:
3140:
3136:
3130:
3122:
3118:
3114:
3110:
3106:
3102:
3098:
3094:
3090:
3089:Sraffa, Piero
3084:
3075:
3071:
3067:
3063:
3059:
3055:
3052:(5–6): 1010.
3051:
3047:
3039:
3035:
3031:
3027:
3023:
3019:
3015:
3011:
3007:
3000:
2998:
2989:
2985:
2981:
2975:
2971:
2967:
2963:
2957:
2953:
2949:
2945:
2941:
2937:
2931:
2923:
2919:
2915:
2911:
2907:
2901:
2897:
2893:
2889:
2884:Pages 47–48:
2880:
2876:
2872:
2866:
2862:
2858:
2854:
2853:
2842:
2838:
2834:
2828:
2824:
2816:
2808:
2804:
2800:
2794:
2790:
2786:
2779:
2771:
2767:
2763:
2757:
2753:
2745:
2737:
2731:
2727:
2717:
2713:
2709:
2703:
2699:
2695:
2691:
2687:
2679:
2671:
2665:
2661:
2657:
2649:
2645:
2641:
2635:
2631:
2627:
2626:
2620:
2615:
2609:
2601:
2595:
2590:
2589:
2583:
2579:
2572:
2564:
2558:
2554:
2550:
2542:
2534:
2530:
2526:
2520:
2516:
2512:
2504:
2500:
2496:
2490:
2486:
2482:
2478:
2470:
2466:
2462:
2456:
2452:
2448:
2443:
2438:
2434:
2427:
2421:
2416:
2412:
2402:
2399:
2398:
2392:
2390:
2386:
2382:
2378:
2374:
2370:
2366:
2362:
2358:
2347:
2344:
2336:
2326:
2321:
2317:
2313:
2312:
2305:
2296:
2295:
2287:
2285:
2281:
2280:stock markets
2277:
2273:
2272:externalities
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2241:
2236:
2234:
2230:
2226:
2222:
2218:
2214:
2210:
2206:
2201:
2199:
2185:
2180:
2174:
2167:Non-convexity
2164:
2162:
2158:
2154:
2150:
2145:
2142:
2138:
2127:
2123:
2118:
2114:
2109:
2105:
2100:
2076:
2056:
2047:
2043:
2029:
2026:
2006:
1997:
1984:
1981:
1961:
1958:
1950:
1934:
1914:
1909:
1894:
1878:
1870:
1861:
1852:
1834:
1827:
1824:
1808:
1801:
1800:
1799:
1783:
1769:
1767:
1749:
1742:
1738:
1734:
1730:
1726:
1722:
1703:
1683:
1663:
1656:
1651:
1642:
1640:
1636:
1633:
1629:
1625:
1621:
1615:
1611:
1610:
1604:
1594:
1592:
1591:
1585:
1579:
1572:
1565:
1561:
1554:
1545:
1539:
1532:
1526:
1519:
1513:
1508:
1501:
1494:
1487:
1483:
1479:
1475:
1471:
1467:
1463:
1458:
1456:
1452:
1442:
1435:
1431:
1427:
1426:
1425:
1424:, the point
1422:[0,1]
1420:
1419:unit interval
1416:
1412:
1405:
1398:
1394:
1389:
1387:
1383:
1379:
1375:
1371:
1367:
1363:
1362:
1353:
1349:
1344:
1335:
1333:
1329:
1325:
1316:
1309:
1302:
1297:real numbers
1296:
1292:
1288:
1280:
1276:
1272:
1268:
1264:
1261:
1260:
1259:
1257:
1253:
1252:
1240:
1233:
1226:
1219:
1212:
1205:
1198:
1191:
1187:
1186:
1185:
1183:
1182:
1177:
1173:
1169:
1165:
1161:
1158:
1148:
1144:
1143:Line segments
1135:
1131:
1124:
1115:
1111:
1108:
1101:
1087:
1085:
1074:
1071:
1063:
1053:
1049:
1043:
1042:
1037:
1036:
1031:This section
1029:
1020:
1019:
1014:Preliminaries
1011:
1009:
1005:
1000:
998:
994:
990:
986:
982:
981:non-convexity
979:In contrast,
977:
975:
971:
967:
963:
958:
956:
952:
948:
944:
940:
936:
925:
920:
918:
913:
911:
906:
905:
903:
902:
897:
887:
885:
880:
875:
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873:
872:
865:
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859:
855:
852:
850:
847:
845:
842:
841:
838:
833:
832:
823:
822:
818:
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813:
811:
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806:
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801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
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773:
771:
768:
766:
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753:
751:
748:
746:
743:
741:
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736:
733:
731:
728:
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723:
721:
718:
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708:
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698:
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693:
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688:
686:
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678:
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631:
628:
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623:
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583:
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531:
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491:
488:
486:
485:de Mandeville
483:
482:
477:
471:
470:
463:
460:
458:
455:
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450:
448:
445:
443:
440:
438:
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430:
428:
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423:
420:
418:
414:
413:Public choice
411:
409:
406:
404:
401:
399:
396:
394:
391:
389:
388:Participation
386:
384:
381:
379:
376:
374:
371:
369:
366:
364:
361:
359:
356:
354:
351:
349:
348:Institutional
346:
344:
341:
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336:
334:
331:
329:
326:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
299:
298:Expeditionary
296:
294:
291:
289:
288:Environmental
286:
284:
281:
279:
276:
274:
271:
269:
266:
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261:
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150:
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133:
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128:
124:
121:
119:
118:International
116:
114:
111:
109:
106:
104:
101:
100:
97:
94:Branches and
91:
90:
85:
82:
80:
77:
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72:
71:
70:
69:
65:
61:
60:
57:
54:
53:
49:
45:
44:
38:
37:
31:
24:
5499:Convex hulls
5477:
5462:Weak duality
5425:
5417:
5337:Orthogonally
4969:Optimization
4954:Mathematical
4914:Experimental
4909:Evolutionary
4894:Econometrics
4752:Public goods
4726:Price system
4721:Price signal
4635:Monopolistic
4504:Distribution
4440:
4419:Major topics
4304:Trigonometry
4083:
4054:
4050:
4020:
3982:
3944:
3922:
3887:
3885:
3845:
3843:
3812:
3781:
3750:
3719:
3659:math/9310228
3649:
3645:
3629:
3593:
3591:
3577:
3550:
3540:
3521:
3519:
3508:
3465:
3452:
3425:
3409:
3400:
3389:
3379:
3342:
3332:
3310:(1): 31–53.
3307:
3304:Econometrica
3303:
3294:
3286:
3270:
3239:
3235:
3229:
3210:
3200:
3181:
3177:
3167:
3142:
3139:Econometrica
3138:
3129:
3096:
3092:
3083:
3049:
3046:Econometrica
3045:
3013:
3010:Econometrica
3009:
2961:
2930:
2887:
2851:
2849:
2822:
2815:
2788:
2778:
2751:
2744:
2725:
2689:
2685:
2678:
2659:
2624:
2608:
2587:
2571:
2552:
2548:
2541:
2514:
2484:
2480:
2446:
2426:
2415:
2354:
2339:
2330:
2323:Please help
2319:
2308:
2264:market power
2258:wrote about
2248:Piero Sraffa
2237:
2202:
2190:
2146:
2134:
2125:
2116:
2112:
2107:
2103:
1998:
1891:is a closed
1866:
1850:
1765:
1736:
1720:
1719:
1638:
1634:
1627:
1626:. Thus Conv(
1623:
1613:
1607:
1602:
1600:
1590:intersection
1588:
1586:
1577:
1570:
1563:
1552:
1543:
1537:
1530:
1524:
1517:
1511:
1499:
1492:
1485:
1481:
1477:
1473:
1465:
1459:
1454:
1448:
1440:
1433:
1429:
1414:
1410:
1403:
1396:
1392:
1390:
1366:line segment
1359:
1357:
1314:
1307:
1300:
1294:
1286:
1284:
1278:
1274:
1270:
1266:
1262:
1255:
1249:
1247:
1238:
1231:
1224:
1217:
1210:
1203:
1196:
1189:
1179:
1175:
1171:
1160:vector space
1154:
1134:line segment
1114:line segment
1081:
1066:
1057:
1046:Please help
1039:
1032:
1001:
978:
959:
950:
947:combinations
946:
938:
934:
933:
854:Publications
819:
442:Sociological
415: /
313:Geographical
293:Evolutionary
268:Digitization
233:Agricultural
137:Mathematical
108:Econometrics
33:
22:
5452:Duality gap
5447:Dual system
5331:Convex hull
4919:Game theory
4884:Development
4831:Uncertainty
4711:Price floor
4691:Preferences
4630:Competition
4600:Information
4563:Externality
4546:Equilibrium
4487:Transaction
4465:Opportunity
4426:Aggregation
3975:"Convexity"
3716:"Convexity"
3300:Radner, Roy
3171:Pages 5–7:
3016:(1): 1–29.
2385:Brown (1991
2381:Brown (1991
2377:Khan (2008)
2333:August 2013
2327:if you can.
2282:(and other
2278:, and with
2270:(and other
2240:oligopolies
2217:game theory
2194:half a lion
1733:half-spaces
1609:convex hull
1597:Convex hull
1388:is convex.
1348:convex hull
1338:Convex sets
1060:August 2013
997:inefficient
972:will equal
690:von Neumann
343:Information
283:Engineering
263:Development
258:Demographic
200:Game theory
142:Methodology
5493:Categories
5375:Radial set
5345:Convex set
5107:Convex set
4949:Managerial
4869:Behavioral
4742:Production
4679:Oligopsony
4519:Elasticity
4431:Budget set
4288:Riemannian
4283:Projective
4268:Symplectic
4263:Hyperbolic
4196:Euclidean
3705:References
3393:Page 371:
3336:Page 270:
2846:Page 309:
2653:Page 628:
2244:monopolies
2177:See also:
2151:that is a
1893:convex set
1729:hyperplane
1655:convex set
1576:+ . . . +
1324:operations
1313:, . . . ,
1251:multiplied
1164:dimensions
1107:convex set
849:Economists
720:Schumacher
625:Schumpeter
595:von Wieser
515:von ThĂĽnen
476:economists
452:Statistics
447:Solidarity
368:Managerial
333:Humanistic
328:Historical
273:Ecological
238:Behavioral
132:Mainstream
34:JEL:
5360:Hypograph
4990:Economics
4862:Subfields
4757:Rationing
4674:Oligopoly
4669:Monopsony
4657:Bilateral
4590:Household
4441:Convexity
4309:Lie group
4273:Spherical
4119:cite book
4013:"Duality"
4011:(1987d).
3973:(1987c).
3778:"Duality"
3714:(2008a).
3672:CiteSeerX
3500:198120391
2922:117240618
2207:, and of
2141:supported
2094:Economics
1386:empty set
1382:Trivially
1147:convexity
1052:talk page
939:economics
935:Convexity
765:Greenspan
730:Samuelson
710:Galbraith
680:Tinbergen
620:von Mises
615:Heckscher
575:Edgeworth
393:Personnel
353:Knowledge
318:Happiness
308:Financial
278:Education
253:Democracy
147:Political
113:Heterodox
56:Economics
5384:Zonotope
5355:Epigraph
5037:Category
4983:See also
4874:Business
4846:Marginal
4841:Expected
4782:Shortage
4777:Scarcity
4652:Monopoly
4558:Exchange
4470:Implicit
4460:Marginal
4370:Category
4258:Elliptic
4250:geometry
4231:Polyform
4216:Discrete
4198:geometry
4179:Timeline
4163:Geometry
4049:(1997).
4019:(eds.).
3981:(eds.).
3943:(eds.).
3932:(1987).
3637:(1993).
3590:(eds.).
3518:(2006).
3464:(1998).
3417:(1987).
3176:(1992).
3003:Page 1:
2940:topology
2616:(1992).
2580:(1988).
2513:(1974).
2485:Approach
2439:(1971).
2395:See also
2309:require
2274:), with
1949:boundary
1725:geometry
1560:equation
1464:, a set
1453:of
1378:crescent
1044:instead.
987:, where
945:prefers
858:journals
844:Glossary
795:Stiglitz
760:Rothbard
740:Buchanan
725:Friedman
715:Koopmans
705:Leontief
685:Robinson
570:Marshall
474:Notable
422:Regional
398:Planning
373:Monetary
303:Feminist
248:Cultural
243:Business
48:a series
46:Part of
5439:Duality
5341:Pseudo-
5315:Ursescu
5212:Pseudo-
5186:Concave
5165:Simplex
5145:Duality
4995:Applied
4974:Welfare
4836:Utility
4796:Surplus
4735:Pricing
4647:Duopoly
4640:Perfect
4583:Service
4551:General
4455:Average
4226:Polygon
4174:History
4111:1216521
4073:0274683
3915:0634800
3873:0648778
3694:1218037
3664:Bibcode
3621:1207195
3532:2191745
3492:1491362
3369:0926685
3324:1909602
3256:0449575
3159:1907054
3121:6458099
3113:2959866
3074:0443878
3066:1911353
3038:0443877
3030:1913410
2988:0737006
2948:measure
2914:1878374
2879:1727000
2841:1865841
2807:0657578
2770:0700688
2716:1462618
2648:1036734
2630:393–394
2592:. MIT.
2533:0389160
2503:1113262
2469:0439057
2451:375–401
2311:cleanup
2198:griffin
2111:,
1869:theorem
1762:in the
1737:support
1620:minimal
1618:is the
1491:,
1432:)
1417:in the
1370:covered
1346:In the
1209:,
1195:,
1162:of two
995:can be
864:Schools
856: (
815:Piketty
810:Krugman
675:Kuznets
665:Kalecki
640:Polanyi
530:Cournot
525:Bastiat
510:Ricardo
500:Malthus
490:Quesnay
462:Welfare
432:Service
103:Applied
79:Outline
74:History
5422:, and
5393:Series
5310:Simons
5217:Quasi-
5207:Proper
5192:Closed
4820:Supply
4811:Demand
4747:Profit
4615:Market
4477:Social
4278:Affine
4211:Convex
4109:
4099:
4071:
4061:
4035:
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2954:, and
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2467:
2457:
2196:(or a
2161:closed
2124:
1823:closed
1768:-space
1509:
1451:member
1384:, the
1361:convex
1110:covers
800:Thaler
780:Ostrom
775:Becker
770:Sowell
750:Baumol
655:Myrdal
650:Sraffa
645:Frisch
635:Knight
630:Keynes
605:Fisher
600:Veblen
585:Pareto
565:Menger
560:George
555:Jevons
550:Walras
540:Gossen
408:Public
403:Policy
358:Labour
323:Health
180:Market
5250:Main
4939:Labor
4924:Green
4696:Price
4578:Goods
4568:Firms
4340:Lists
4335:Shape
4328:Lists
4297:Other
4186:Lists
3937:(PDF)
3654:arXiv
3642:(PDF)
3496:S2CID
3422:(PDF)
3320:JSTOR
3155:JSTOR
3117:S2CID
3109:JSTOR
3062:JSTOR
3026:JSTOR
2918:S2CID
2407:Notes
2233:firms
1867:This
1764:real
1632:cover
1612:Conv(
1449:is a
1273:) = (
1216:) = (
1202:) + (
1181:added
1145:test
1130:cover
837:Lists
805:Hoppe
790:Lucas
755:Solow
745:Arrow
735:Simon
700:Lange
695:Hicks
670:Röpke
660:Hayek
610:Pigou
580:Clark
495:Smith
457:Urban
437:Socio
427:Rural
127:Macro
123:Micro
84:Index
5370:Lens
5324:Sets
5174:Maps
5081:and
4853:Wage
4762:Rent
4730:Free
4482:Sunk
4450:Cost
4443:and
4125:link
4097:ISBN
4059:ISBN
4033:ISBN
3995:ISBN
3957:ISBN
3901:ISBN
3859:ISBN
3825:ISBN
3794:ISBN
3763:ISBN
3732:ISBN
3607:ISBN
3563:ISBN
3478:ISBN
3438:ISBN
3355:ISBN
3277:ISBN
3215:ISBN
3186:ISBN
2974:ISBN
2950:and
2938:and
2900:ISBN
2865:ISBN
2827:ISBN
2793:ISBN
2756:ISBN
2730:ISBN
2702:ISBN
2664:ISBN
2634:ISBN
2594:ISBN
2557:ISBN
2519:ISBN
2489:ISBN
2455:ISBN
2371:and
2238:In "
2159:are
1927:and
1727:. A
1402:and
1374:cube
1330:and
1174:and
1157:real
1112:the
821:more
545:Marx
535:Mill
520:List
5424:(Hw
4944:Law
4089:doi
4025:doi
3987:doi
3949:doi
3893:doi
3851:doi
3817:doi
3786:doi
3755:doi
3724:doi
3682:doi
3599:doi
3555:doi
3470:doi
3430:doi
3347:doi
3312:doi
3244:doi
3147:doi
3101:doi
3054:doi
3018:doi
2966:doi
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2857:doi
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2688:".
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1951:of
1895:in
1741:set
1460:By
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785:Sen
505:Say
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1547:D
1544:v
1541:D
1538:λ
1534:1
1531:v
1528:1
1525:λ
1521:0
1518:v
1515:0
1512:λ
1503:D
1500:v
1496:1
1493:v
1489:0
1486:v
1478:Q
1474:Q
1466:Q
1455:Q
1444:1
1437:0
1434:v
1430:λ
1415:λ
1411:Q
1407:1
1404:v
1400:0
1397:v
1393:Q
1320:)
1318:D
1315:v
1311:2
1308:v
1304:1
1301:v
1295:D
1287:D
1271:y
1267:x
1265:(
1263:λ
1256:λ
1242:2
1239:y
1237:+
1235:1
1232:y
1228:2
1225:x
1223:+
1221:1
1218:x
1214:2
1211:y
1207:2
1204:x
1200:1
1197:y
1193:1
1190:x
1188:(
1176:y
1172:x
1149:.
1073:)
1067:(
1062:)
1058:(
1054:.
923:e
916:t
909:v
860:)
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