Qeyri-qabarıqlılıq (iqtisadiyyat) (Azerbaijani Wikipedia)

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Green, Jerry; Heller, Walter P. 1 Mathematical analysis and convexity with applications to economics // Arrow, Kenneth Joseph; Intriligator, Michael D (redaktorlar ). Handbook of mathematical economics, Volume I. Handbooks in economics. 1. Amsterdam: North-Holland Publishing Co. 1981. 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800.
Page 1: Guesnerie, Roger. "Pareto optimality in non-convex economies". Econometrica. 43. 1975. 1–29. doi:10.2307/1913410. JSTOR 1913410. MR 0443877. ("Errata". Econometrica. 43 (5–6). 1975. səh. 1010. doi:10.2307/1911353. JSTOR 1911353. MR 0443878.)
Chapter 8 "Applications to economics", especially Section 8.5.3 "Enter nonconvexity" (and the remainder of the chapter), particularly page 495:
Mordukhovich, Boris S. Variational analysis and generalized differentiation II: Applications. Grundlehren Series (Fundamental Principles of Mathematical Sciences). 331. Springer. 2006. i–xxii and&nbsp, 1–610. MR 2191745.
Brown, Donald J. 36 Equilibrium analysis with non-convex technologies // Hildenbrand, Werner; Sonnenschein, Hugo (redaktorlar ). Handbook of mathematical economics, Volume IV. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. 1991. 1963–1995 [1966]. doi:10.1016/S1573-4382(05)80011-6. ISBN 0-444-87461-5. MR 1207195.
Diewert, (1982. səh. 552–553): Diewert, W. E. 12 Duality approaches to microeconomic theory // Arrow, Kenneth Joseph; Intriligator, Michael D (redaktorlar ). Handbook of mathematical economics, Volume II. Handbooks in economics. 1. Amsterdam: North-Holland Publishing Co. 1982. 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778.
Samuelson, (1950. səh. 359–360):
It will be noted that any point where the indifference curves are convex rather than concave cannot be observed in a competitive market. Such points are shrouded in eternal darkness—unless we make our consumer a monopsonist and let him choose between goods lying on a very convex "budget curve" (along which he is affecting the price of what he buys). In this monopsony case, we could still deduce the slope of the man's indifference curve from the slope of the observed constraint at the equilibrium point.
Samuelson, Paul A. "The problem of integrability in utility theory". Economica. New Series. 17. 1950. 355–385. doi:10.2307/2549499. JSTOR 2549499. MR 0043436.For the epigraph to their seventh chapter, "Markets with non-convex preferences and production" presenting Starr, (1969), Arrow, Hahn, (1971. səh. 169) quote John Milton's description of the (non-convex) Serbonian Bog in Paradise Lost (Book II, lines 592–594):
A gulf profound as that Serbonian Bog

Betwixt Damiata and Mount Casius old,

Where Armies whole have sunk.
Pages 231 and 239 (Figure 10 a–b: Illustration of lemma 5 [page 240]): Wold, Herman. "A synthesis of pure demand analysis II". Skandinavisk Aktuarietidskrift [Scandinavian Actuarial Journal]. 26. 1943b. 220–263. MR 0011939.
Exercise 45, page 146: Wold, Herman; Juréen, Lars (in association with Wold). 8 Some further applications of preference fields (pp. 129–148) // Demand analysis: A study in econometrics. Wiley publications in statistics. New York: John Wiley and Sons, Inc. Stockholm: Almqvist and Wiksell. 1953. xvi+358. MR 0064385.
Aumann, (1966. səh. 1–2): Aumann, Robert J. "Existence of competitive equilibrium in markets with a continuum of traders". Econometrica. 34 (1). January 1966: 1–17. doi:10.2307/1909854. JSTOR 1909854. MR 0191623. Aumann, (1966) builds on two papers: Şablon:Harvs
Aumann, Robert J. "Markets with a continuum of traders". Econometrica. 32 (1–2). January–April 1964: 39–50. doi:10.2307/1913732. JSTOR 1913732. MR 0172689.
Aumann, Robert J. "Integrals of set-valued functions". Journal of Mathematical Analysis and Applications. 12 (1). August 1965: 1–12. doi:10.1016/0022-247X(65)90049-1. MR 0185073.
Pages 392–399 and page 188: Arrow, Kenneth J.; Hahn, Frank H. Appendix B: Convex and related sets // General competitive analysis. Mathematical economics texts [Advanced textbooks in economics]. San Francisco, CA: Holden-Day, Inc. [North-Holland]. 1971. 375–401. ISBN 0-444-85497-5. MR 0439057.
Pages 52–55 with applications on pages 145–146, 152–153, and 274–275: Mas-Colell, Andreu. 1.L Averages of sets // The Theory of General Economic Equilibrium: A Differentiable Approach. Econometric Society Monographs. Cambridge UP. 1985. ISBN 0-521-26514-2. MR 1113262.

Theorem C(6) on page 37 and applications on pages 115-116, 122, and 168: Hildenbrand, Werner. Core and equilibria of a large economy. Princeton studies in mathematical economics. Princeton, N.J.: Princeton University Press. 1974. viii+251. ISBN 978-0-691-04189-6. MR 0389160.
Varian, Hal R. 21.2 Convexity and size // Microeconomic Analysis (3rd). W. W. Norton & Company. 1992. 393–394. ISBN 978-0-393-95735-8. MR 1036734.
Page 628: Mas–Colell, Andreu; Whinston, Michael D.; Green, Jerry R. 17.1 Large economies and nonconvexities // Microeconomic theory. Oxford University Press. 1995. 627–630. ISBN 978-0-19-507340-9.
Page 169 in the first edition: Starr, Ross M. 8 Convex sets, separation theorems, and non-convex sets in R^N // General equilibrium theory: An introduction (Second). Cambridge: Cambridge University Press. 2011. doi:10.1017/CBO9781139174749. ISBN 978-0-521-53386-7. MR 1462618.
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. Competitive equilibrium: Theory and applications. Cambridge University Press. 1994. 420. doi:10.2277/0521319889. ISBN 978-0-521-31988-1.
Pages 93–94 (especially example 1.92), 143, 318–319, 375–377, and 416: Carter, Michael. Foundations of mathematical economics. Cambridge, MA: MIT Press. 2001. xx+649. ISBN 0-262-53192-5. MR 1865841.
Page 309: Moore, James C. Mathematical methods for economic theory: Volume I. Studies in economic theory. 9. Berlin: Springer-Verlag. 1999. xii+414. doi:10.1007/978-3-662-08544-8. ISBN 3-540-66235-9. MR 1727000.
Pages 47–48: Florenzano, Monique; Le Van, Cuong. Finite dimensional convexity and optimization. Studies in economic theory. 13. in cooperation with Pascal Gourdel. Berlin: Springer-Verlag. 2001. xii+154. doi:10.1007/978-3-642-56522-9. ISBN 3-540-41516-5. MR 1878374.
Economists have studied non-convex sets using advanced mathematics, particularly differential geometry and differential topology, Baire category, measure (mathematics) and integration theory, and ergodic theory: Trockel, Walter. Market demand: An analysis of large economies with nonconvex preferences. Lecture Notes in Economics and Mathematical Systems. 223. Berlin: Springer-Verlag. 1984. viii+205. doi:10.1007/978-3-642-46488-1. ISBN 3-540-12881-6. MR 0737006.
Starrett, David A. "Fundamental nonconvexities in the theory of externalities". Journal of Economic Theory. 4 (2). 1972. 180–199. doi:10.1016/0022-0531(72)90148-2. MR 0449575.
Page 270: Drèze, Jacques H. 14 Investment under private ownership: Optimality, equilibrium and stability // Drèze, J. H. (redaktor). Essays on economic decisions under uncertainty. Cambridge: Cambridge University Press. 1987. 261–297. doi:10.1017/CBO9780511559464. ISBN 0-521-26484-7. MR 0926685. (Originally published as Drèze, Jacques H. Investment under private ownership: Optimality, equilibrium and stability // Drèze, J. H. (redaktor). Allocation under Uncertainty: Equilibrium and Optimality. New York: Wiley. 1974. 129–165.)
Troutman, John L. With the assistance of William Hrusa (redaktor). Variational calculus and optimal control: Optimization with elementary convexity. Undergraduate Texts in Mathematics (Second). New York: Springer-Verlag. 1996. xvi+461. doi:10.1007/978-1-4612-0737-5. ISBN 0-387-94511-3. MR 1363262.
Craven, B. D. Control and optimization. Chapman and Hall Mathematics Series. London: Chapman and Hall, Ltd. 1995. x+193. doi:10.1007/978-1-4899-7226-2. ISBN 0-412-55890-4. MR 1349574.
Vinter, Richard. Optimal control. Systems & Control: Foundations & Applications. Boston, MA: Birkhäuser Boston, Inc. 2000. xviii+507. ISBN 0-8176-4075-4. MR 1756410.
Rockafellar, R. Tyrrell; Wets, Roger J-B. Variational analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 317. Berlin: Springer-Verlag. 1998. xiv+733. doi:10.1007/978-3-642-02431-3. ISBN 3-540-62772-3. MR 1491362.
Algebraic topology has also been used to study convex and non-convex sets in economics:Chichilnisky, G. "Intersecting families of sets and the topology of cones in economics" (PDF). Bulletin of the American Mathematical Society. New Series. 29 (2). 1993. 189–207. doi:10.1090/S0273-0979-1993-00439-7. MR 1218037. 2022-01-23 tarixində arxivləşdirilib (PDF). İstifadə tarixi: 2020-10-12.

archive.org

Salanié, Bernard. 7 Nonconvexities // Microeconomics of market failures (English translation of the (1998) French Microéconomie: Les défaillances du marché (Economica, Paris)). Cambridge, MA: MIT Press. 2000. 107–125. ISBN 0-262-19443-0.
Page 1: Guesnerie, Roger. "Pareto optimality in non-convex economies". Econometrica. 43. 1975. 1–29. doi:10.2307/1913410. JSTOR 1913410. MR 0443877. ("Errata". Econometrica. 43 (5–6). 1975. səh. 1010. doi:10.2307/1911353. JSTOR 1911353. MR 0443878.)
Farrell, M. J. "The Convexity assumption in the theory of competitive markets". The Journal of Political Economy. 67 (4). August 1959: 371–391. doi:10.1086/258197. JSTOR 1825163. Farrell, M. J. "On Convexity, efficiency, and markets: A Reply". Journal of Political Economy. 69 (5). October 1961a: 484–489. doi:10.1086/258541. JSTOR 1828538. Farrell, M. J. "The Convexity assumption in the theory of competitive markets: Rejoinder". Journal of Political Economy. 69 (5). October 1961b: 493. doi:10.1086/258544. JSTOR 1828541.
Koopmans, Tjalling C. "Convexity assumptions, allocative efficiency, and competitive equilibrium". The Journal of Political Economy. 69 (5). October 1961: 478–479. doi:10.1086/258539. JSTOR 1828536.
Koopmans, (1961. səh. 478) and others—for example, Farrell, (1959. səh. 390–391) and Farrell, (1961a. səh. 484), Bator, (1961. səh. 482–483), Rothenberg, (1960. səh. 438), and Starr, (1969. səh. 26)—commented on Koopmans, (1957. səh. 1–126, especially 9–16 [1.3 Summation of opportunity sets], 23–35 [1.6 Convex sets and the price implications of optimality], and 35–37 [1.7 The role of convexity assumptions in the analysis]):
Tjalling C., Koopmans. Allocation of resources and the price system // Koopmans, Tjalling C (redaktor). Three essays on the state of economic science. New York: McGraw–Hill Book Company. 1957. 1–126. ISBN 0-07-035337-9.
Rothenberg, (1960. səh. 447): Rothenberg, Jerome. "Non-convexity, aggregation, and Pareto optimality". The Journal of Political Economy. 68 (5). October 1960: 435–468. doi:10.1086/258363. JSTOR 1830308. (Rothenberg, Jerome. "Comments on non-convexity". Journal of Political Economy. 69 (5). October 1961: 490–492. doi:10.1086/258543. JSTOR 1828540.)
Aumann, (1966. səh. 1–2): Aumann, Robert J. "Existence of competitive equilibrium in markets with a continuum of traders". Econometrica. 34 (1). January 1966: 1–17. doi:10.2307/1909854. JSTOR 1909854. MR 0191623. Aumann, (1966) builds on two papers: Şablon:Harvs
Aumann, Robert J. "Markets with a continuum of traders". Econometrica. 32 (1–2). January–April 1964: 39–50. doi:10.2307/1913732. JSTOR 1913732. MR 0172689.
Aumann, Robert J. "Integrals of set-valued functions". Journal of Mathematical Analysis and Applications. 12 (1). August 1965: 1–12. doi:10.1016/0022-247X(65)90049-1. MR 0185073.
Pages 392–399 and page 188: Arrow, Kenneth J.; Hahn, Frank H. Appendix B: Convex and related sets // General competitive analysis. Mathematical economics texts [Advanced textbooks in economics]. San Francisco, CA: Holden-Day, Inc. [North-Holland]. 1971. 375–401. ISBN 0-444-85497-5. MR 0439057.
Pages 52–55 with applications on pages 145–146, 152–153, and 274–275: Mas-Colell, Andreu. 1.L Averages of sets // The Theory of General Economic Equilibrium: A Differentiable Approach. Econometric Society Monographs. Cambridge UP. 1985. ISBN 0-521-26514-2. MR 1113262.

Theorem C(6) on page 37 and applications on pages 115-116, 122, and 168: Hildenbrand, Werner. Core and equilibria of a large economy. Princeton studies in mathematical economics. Princeton, N.J.: Princeton University Press. 1974. viii+251. ISBN 978-0-691-04189-6. MR 0389160.
Varian, Hal R. 21.2 Convexity and size // Microeconomic Analysis (3rd). W. W. Norton & Company. 1992. 393–394. ISBN 978-0-393-95735-8. MR 1036734.
Page 628: Mas–Colell, Andreu; Whinston, Michael D.; Green, Jerry R. 17.1 Large economies and nonconvexities // Microeconomic theory. Oxford University Press. 1995. 627–630. ISBN 978-0-19-507340-9.
Pages 93–94 (especially example 1.92), 143, 318–319, 375–377, and 416: Carter, Michael. Foundations of mathematical economics. Cambridge, MA: MIT Press. 2001. xx+649. ISBN 0-262-53192-5. MR 1865841.
Page 309: Moore, James C. Mathematical methods for economic theory: Volume I. Studies in economic theory. 9. Berlin: Springer-Verlag. 1999. xii+414. doi:10.1007/978-3-662-08544-8. ISBN 3-540-66235-9. MR 1727000.
Pages 47–48: Florenzano, Monique; Le Van, Cuong. Finite dimensional convexity and optimization. Studies in economic theory. 13. in cooperation with Pascal Gourdel. Berlin: Springer-Verlag. 2001. xii+154. doi:10.1007/978-3-642-56522-9. ISBN 3-540-41516-5. MR 1878374.
Sraffa, Piero. "The Laws of returns under competitive conditions". Economic Journal. 36 (144). 1926. 535–550. JSTOR 2959866.
Hotelling, Harold. "The General welfare in relation to problems of taxation and of railway and utility rates". Econometrica. 6 (3). July 1938: 242–269. doi:10.2307/1907054. JSTOR 1907054.
Pages 5–7: Quinzii, Martine. Increasing returns and efficiency (Revised translation of (1988) Rendements croissants et efficacité economique. Paris: Editions du Centre National de la Recherche Scientifique). New York: Oxford University Press. 1992. viii+165. ISBN 0-19-506553-0.
Pages 106, 110–137, 172, and 248: Baumol, William J.; Oates, Wallace E.; with contributions by V. S. Bawa and David F. Bradford. 8 Detrimental externalities and nonconvexities in the production set // The Theory of environmental policy (Second). Cambridge: Cambridge University Press. 1988. x+299. doi:10.2277/0521311128. ISBN 978-0-521-31112-0.
Radner, Roy. "Competitive equilibrium under uncertainty". Econometrica. 36. 1968. 31–53. doi:10.2307/1909602. JSTOR 1909602.
Page 270: Drèze, Jacques H. 14 Investment under private ownership: Optimality, equilibrium and stability // Drèze, J. H. (redaktor). Essays on economic decisions under uncertainty. Cambridge: Cambridge University Press. 1987. 261–297. doi:10.1017/CBO9780511559464. ISBN 0-521-26484-7. MR 0926685. (Originally published as Drèze, Jacques H. Investment under private ownership: Optimality, equilibrium and stability // Drèze, J. H. (redaktor). Allocation under Uncertainty: Equilibrium and Optimality. New York: Wiley. 1974. 129–165.)
Craven, B. D. Control and optimization. Chapman and Hall Mathematics Series. London: Chapman and Hall, Ltd. 1995. x+193. doi:10.1007/978-1-4899-7226-2. ISBN 0-412-55890-4. MR 1349574.
Vinter, Richard. Optimal control. Systems & Control: Foundations & Applications. Boston, MA: Birkhäuser Boston, Inc. 2000. xviii+507. ISBN 0-8176-4075-4. MR 1756410.
Dixit, Avinash; Pindyck, Robert. Investment Under Uncertainty. Princeton Univ. Press. 1994. ISBN 0-691-03410-9.

books.google.com

Pages 63–65: Laffont, Jean-Jacques. 3 Nonconvexities // Fondements de L'economie Publique [Fundamentals of public economics]. MIT. 1988. ISBN 0-262-12127-1. 2020-08-06 tarixində arxivləşdirilib. İstifadə tarixi: 2020-10-12.
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. Foundations of public economics. Cambridge economic handbooks. Cambridge: Cambridge University Press. 1988. ISBN 9780521348010.

columbia.edu

www2.gsb.columbia.edu

Heal, (1999. səh. 4 in preprint): Heal, G. M. Introduction (PDF) // The economics of increasing returns. The International Library of Critical Writings in Economics. Edward Elgar. 1999. 640. ISBN 978-1-85898-160-4. 15 September 2015 tarixində orijinalından (PDF) arxivləşdirilib. İstifadə tarixi: 5 March 2011.

dictionaryofeconomics.com

Khan, M. Ali. Perfect competition // Durlauf, Steven N.; Blume, Lawrence E., ed. (redaktorlar ). The New Palgrave Dictionary of Economics (Second). Palgrave Macmillan. 2008. 354–365. doi:10.1057/9780230226203.1267. ISBN 978-0-333-78676-5. 2017-09-14 tarixində arxivləşdirilib. İstifadə tarixi: 2020-10-12.

doi.org

Mas-Colell, A. Non-convexity (PDF) // Eatwell, John; Milgate, Murray; Newman, Peter (redaktorlar ). The New Palgrave: A Dictionary of Economics (first). Palgrave Macmillan. 1987. 653–661. doi:10.1057/9780230226203.3173. ISBN 9780333786765. 2018-08-09 tarixində arxivləşdirilib (PDF). İstifadə tarixi: 2020-10-12.
Green, Jerry; Heller, Walter P. 1 Mathematical analysis and convexity with applications to economics // Arrow, Kenneth Joseph; Intriligator, Michael D (redaktorlar ). Handbook of mathematical economics, Volume I. Handbooks in economics. 1. Amsterdam: North-Holland Publishing Co. 1981. 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800.
Page 1: Guesnerie, Roger. "Pareto optimality in non-convex economies". Econometrica. 43. 1975. 1–29. doi:10.2307/1913410. JSTOR 1913410. MR 0443877. ("Errata". Econometrica. 43 (5–6). 1975. səh. 1010. doi:10.2307/1911353. JSTOR 1911353. MR 0443878.)
Khan, M. Ali. Perfect competition // Durlauf, Steven N.; Blume, Lawrence E., ed. (redaktorlar ). The New Palgrave Dictionary of Economics (Second). Palgrave Macmillan. 2008. 354–365. doi:10.1057/9780230226203.1267. ISBN 978-0-333-78676-5. 2017-09-14 tarixində arxivləşdirilib. İstifadə tarixi: 2020-10-12.
Brown, Donald J. 36 Equilibrium analysis with non-convex technologies // Hildenbrand, Werner; Sonnenschein, Hugo (redaktorlar ). Handbook of mathematical economics, Volume IV. Handbooks in Economics. 1. Amsterdam: North-Holland Publishing Co. 1991. 1963–1995 [1966]. doi:10.1016/S1573-4382(05)80011-6. ISBN 0-444-87461-5. MR 1207195.
Hotelling, (1935. səh. 74): Hotelling, Harold. "Demand functions with limited budgets". Econometrica. 3 (1). January 1935: 66–78. doi:10.2307/1907346. JSTOR 1907346.
Diewert, (1982. səh. 552–553): Diewert, W. E. 12 Duality approaches to microeconomic theory // Arrow, Kenneth Joseph; Intriligator, Michael D (redaktorlar ). Handbook of mathematical economics, Volume II. Handbooks in economics. 1. Amsterdam: North-Holland Publishing Co. 1982. 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778.
Samuelson, (1950. səh. 359–360):
It will be noted that any point where the indifference curves are convex rather than concave cannot be observed in a competitive market. Such points are shrouded in eternal darkness—unless we make our consumer a monopsonist and let him choose between goods lying on a very convex "budget curve" (along which he is affecting the price of what he buys). In this monopsony case, we could still deduce the slope of the man's indifference curve from the slope of the observed constraint at the equilibrium point.
Samuelson, Paul A. "The problem of integrability in utility theory". Economica. New Series. 17. 1950. 355–385. doi:10.2307/2549499. JSTOR 2549499. MR 0043436.For the epigraph to their seventh chapter, "Markets with non-convex preferences and production" presenting Starr, (1969), Arrow, Hahn, (1971. səh. 169) quote John Milton's description of the (non-convex) Serbonian Bog in Paradise Lost (Book II, lines 592–594):
A gulf profound as that Serbonian Bog

Betwixt Damiata and Mount Casius old,

Where Armies whole have sunk.
Bator, Francis M. "On convexity, efficiency, and markets". The Journal of Political Economy. 69 (5). October 1961a: 480–483. doi:10.1086/258540. JSTOR 1828537. Bator, Francis M. "On convexity, efficiency, and markets: Rejoinder". Journal of Political Economy. 69 (5). October 1961b: 489. doi:10.1086/258542. JSTOR 1828539.
Farrell, M. J. "The Convexity assumption in the theory of competitive markets". The Journal of Political Economy. 67 (4). August 1959: 371–391. doi:10.1086/258197. JSTOR 1825163. Farrell, M. J. "On Convexity, efficiency, and markets: A Reply". Journal of Political Economy. 69 (5). October 1961a: 484–489. doi:10.1086/258541. JSTOR 1828538. Farrell, M. J. "The Convexity assumption in the theory of competitive markets: Rejoinder". Journal of Political Economy. 69 (5). October 1961b: 493. doi:10.1086/258544. JSTOR 1828541.
Koopmans, Tjalling C. "Convexity assumptions, allocative efficiency, and competitive equilibrium". The Journal of Political Economy. 69 (5). October 1961: 478–479. doi:10.1086/258539. JSTOR 1828536.
Koopmans, (1961. səh. 478) and others—for example, Farrell, (1959. səh. 390–391) and Farrell, (1961a. səh. 484), Bator, (1961. səh. 482–483), Rothenberg, (1960. səh. 438), and Starr, (1969. səh. 26)—commented on Koopmans, (1957. səh. 1–126, especially 9–16 [1.3 Summation of opportunity sets], 23–35 [1.6 Convex sets and the price implications of optimality], and 35–37 [1.7 The role of convexity assumptions in the analysis]):
Tjalling C., Koopmans. Allocation of resources and the price system // Koopmans, Tjalling C (redaktor). Three essays on the state of economic science. New York: McGraw–Hill Book Company. 1957. 1–126. ISBN 0-07-035337-9.
Rothenberg, (1960. səh. 447): Rothenberg, Jerome. "Non-convexity, aggregation, and Pareto optimality". The Journal of Political Economy. 68 (5). October 1960: 435–468. doi:10.1086/258363. JSTOR 1830308. (Rothenberg, Jerome. "Comments on non-convexity". Journal of Political Economy. 69 (5). October 1961: 490–492. doi:10.1086/258543. JSTOR 1828540.)
Shapley, Shubik, (1966. səh. 806): Shapley, L. S.; Shubik, M. "Quasi-cores in a monetary economy with nonconvex preferences". Econometrica. 34 (4). October 1966: 805–827. doi:10.2307/1910101. JSTOR 1910101. Zbl 0154.45303.
Aumann, (1966. səh. 1–2): Aumann, Robert J. "Existence of competitive equilibrium in markets with a continuum of traders". Econometrica. 34 (1). January 1966: 1–17. doi:10.2307/1909854. JSTOR 1909854. MR 0191623. Aumann, (1966) builds on two papers: Şablon:Harvs
Aumann, Robert J. "Markets with a continuum of traders". Econometrica. 32 (1–2). January–April 1964: 39–50. doi:10.2307/1913732. JSTOR 1913732. MR 0172689.
Aumann, Robert J. "Integrals of set-valued functions". Journal of Mathematical Analysis and Applications. 12 (1). August 1965: 1–12. doi:10.1016/0022-247X(65)90049-1. MR 0185073.
Page 169 in the first edition: Starr, Ross M. 8 Convex sets, separation theorems, and non-convex sets in R^N // General equilibrium theory: An introduction (Second). Cambridge: Cambridge University Press. 2011. doi:10.1017/CBO9781139174749. ISBN 978-0-521-53386-7. MR 1462618.
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. Competitive equilibrium: Theory and applications. Cambridge University Press. 1994. 420. doi:10.2277/0521319889. ISBN 978-0-521-31988-1.
Pages 93–94 (especially example 1.92), 143, 318–319, 375–377, and 416: Carter, Michael. Foundations of mathematical economics. Cambridge, MA: MIT Press. 2001. xx+649. ISBN 0-262-53192-5. MR 1865841.
Page 309: Moore, James C. Mathematical methods for economic theory: Volume I. Studies in economic theory. 9. Berlin: Springer-Verlag. 1999. xii+414. doi:10.1007/978-3-662-08544-8. ISBN 3-540-66235-9. MR 1727000.
Pages 47–48: Florenzano, Monique; Le Van, Cuong. Finite dimensional convexity and optimization. Studies in economic theory. 13. in cooperation with Pascal Gourdel. Berlin: Springer-Verlag. 2001. xii+154. doi:10.1007/978-3-642-56522-9. ISBN 3-540-41516-5. MR 1878374.
Economists have studied non-convex sets using advanced mathematics, particularly differential geometry and differential topology, Baire category, measure (mathematics) and integration theory, and ergodic theory: Trockel, Walter. Market demand: An analysis of large economies with nonconvex preferences. Lecture Notes in Economics and Mathematical Systems. 223. Berlin: Springer-Verlag. 1984. viii+205. doi:10.1007/978-3-642-46488-1. ISBN 3-540-12881-6. MR 0737006.
Hotelling, Harold. "The General welfare in relation to problems of taxation and of railway and utility rates". Econometrica. 6 (3). July 1938: 242–269. doi:10.2307/1907054. JSTOR 1907054.
Starrett, David A. "Fundamental nonconvexities in the theory of externalities". Journal of Economic Theory. 4 (2). 1972. 180–199. doi:10.1016/0022-0531(72)90148-2. MR 0449575.
Pages 106, 110–137, 172, and 248: Baumol, William J.; Oates, Wallace E.; with contributions by V. S. Bawa and David F. Bradford. 8 Detrimental externalities and nonconvexities in the production set // The Theory of environmental policy (Second). Cambridge: Cambridge University Press. 1988. x+299. doi:10.2277/0521311128. ISBN 978-0-521-31112-0.
Radner, Roy. "Competitive equilibrium under uncertainty". Econometrica. 36. 1968. 31–53. doi:10.2307/1909602. JSTOR 1909602.
Page 270: Drèze, Jacques H. 14 Investment under private ownership: Optimality, equilibrium and stability // Drèze, J. H. (redaktor). Essays on economic decisions under uncertainty. Cambridge: Cambridge University Press. 1987. 261–297. doi:10.1017/CBO9780511559464. ISBN 0-521-26484-7. MR 0926685. (Originally published as Drèze, Jacques H. Investment under private ownership: Optimality, equilibrium and stability // Drèze, J. H. (redaktor). Allocation under Uncertainty: Equilibrium and Optimality. New York: Wiley. 1974. 129–165.)
Ramsey, F. P. "A Mathematical Theory of Saving". Economic Journal. 38 (152). 1928: 543–559. doi:10.2307/2224098. JSTOR 2224098.
Hotelling, Harold. "The Economics of Exhaustible Resources". JPE. 39 (2). 1931: 137–175. doi:10.1086/254195. JSTOR 1822328. 2023-07-15 tarixində arxivləşdirilib. İstifadə tarixi: 2020-10-12.
Troutman, John L. With the assistance of William Hrusa (redaktor). Variational calculus and optimal control: Optimization with elementary convexity. Undergraduate Texts in Mathematics (Second). New York: Springer-Verlag. 1996. xvi+461. doi:10.1007/978-1-4612-0737-5. ISBN 0-387-94511-3. MR 1363262.
Craven, B. D. Control and optimization. Chapman and Hall Mathematics Series. London: Chapman and Hall, Ltd. 1995. x+193. doi:10.1007/978-1-4899-7226-2. ISBN 0-412-55890-4. MR 1349574.
Merton, Robert C. "An Intertemporal Capital Asset Pricing Model". Econometrica. 41 (5). 1973: 867–887. doi:10.2307/1913811. JSTOR 1913811.
Rockafellar, R. Tyrrell; Wets, Roger J-B. Variational analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 317. Berlin: Springer-Verlag. 1998. xiv+733. doi:10.1007/978-3-642-02431-3. ISBN 3-540-62772-3. MR 1491362.
Algebraic topology has also been used to study convex and non-convex sets in economics:Chichilnisky, G. "Intersecting families of sets and the topology of cones in economics" (PDF). Bulletin of the American Mathematical Society. New Series. 29 (2). 1993. 189–207. doi:10.1090/S0273-0979-1993-00439-7. MR 1218037. 2022-01-23 tarixində arxivləşdirilib (PDF). İstifadə tarixi: 2020-10-12.

e-elgar.com

Hotelling, Harold. "The Economics of Exhaustible Resources". JPE. 39 (2). 1931: 137–175. doi:10.1086/254195. JSTOR 1822328. 2023-07-15 tarixində arxivləşdirilib. İstifadə tarixi: 2020-10-12.

jstor.org

Page 1: Guesnerie, Roger. "Pareto optimality in non-convex economies". Econometrica. 43. 1975. 1–29. doi:10.2307/1913410. JSTOR 1913410. MR 0443877. ("Errata". Econometrica. 43 (5–6). 1975. səh. 1010. doi:10.2307/1911353. JSTOR 1911353. MR 0443878.)
Hotelling, (1935. səh. 74): Hotelling, Harold. "Demand functions with limited budgets". Econometrica. 3 (1). January 1935: 66–78. doi:10.2307/1907346. JSTOR 1907346.
Samuelson, (1950. səh. 359–360):
It will be noted that any point where the indifference curves are convex rather than concave cannot be observed in a competitive market. Such points are shrouded in eternal darkness—unless we make our consumer a monopsonist and let him choose between goods lying on a very convex "budget curve" (along which he is affecting the price of what he buys). In this monopsony case, we could still deduce the slope of the man's indifference curve from the slope of the observed constraint at the equilibrium point.
Samuelson, Paul A. "The problem of integrability in utility theory". Economica. New Series. 17. 1950. 355–385. doi:10.2307/2549499. JSTOR 2549499. MR 0043436.For the epigraph to their seventh chapter, "Markets with non-convex preferences and production" presenting Starr, (1969), Arrow, Hahn, (1971. səh. 169) quote John Milton's description of the (non-convex) Serbonian Bog in Paradise Lost (Book II, lines 592–594):
A gulf profound as that Serbonian Bog

Betwixt Damiata and Mount Casius old,

Where Armies whole have sunk.
Bator, Francis M. "On convexity, efficiency, and markets". The Journal of Political Economy. 69 (5). October 1961a: 480–483. doi:10.1086/258540. JSTOR 1828537. Bator, Francis M. "On convexity, efficiency, and markets: Rejoinder". Journal of Political Economy. 69 (5). October 1961b: 489. doi:10.1086/258542. JSTOR 1828539.
Farrell, M. J. "The Convexity assumption in the theory of competitive markets". The Journal of Political Economy. 67 (4). August 1959: 371–391. doi:10.1086/258197. JSTOR 1825163. Farrell, M. J. "On Convexity, efficiency, and markets: A Reply". Journal of Political Economy. 69 (5). October 1961a: 484–489. doi:10.1086/258541. JSTOR 1828538. Farrell, M. J. "The Convexity assumption in the theory of competitive markets: Rejoinder". Journal of Political Economy. 69 (5). October 1961b: 493. doi:10.1086/258544. JSTOR 1828541.
Koopmans, Tjalling C. "Convexity assumptions, allocative efficiency, and competitive equilibrium". The Journal of Political Economy. 69 (5). October 1961: 478–479. doi:10.1086/258539. JSTOR 1828536.
Koopmans, (1961. səh. 478) and others—for example, Farrell, (1959. səh. 390–391) and Farrell, (1961a. səh. 484), Bator, (1961. səh. 482–483), Rothenberg, (1960. səh. 438), and Starr, (1969. səh. 26)—commented on Koopmans, (1957. səh. 1–126, especially 9–16 [1.3 Summation of opportunity sets], 23–35 [1.6 Convex sets and the price implications of optimality], and 35–37 [1.7 The role of convexity assumptions in the analysis]):
Tjalling C., Koopmans. Allocation of resources and the price system // Koopmans, Tjalling C (redaktor). Three essays on the state of economic science. New York: McGraw–Hill Book Company. 1957. 1–126. ISBN 0-07-035337-9.
Rothenberg, (1960. səh. 447): Rothenberg, Jerome. "Non-convexity, aggregation, and Pareto optimality". The Journal of Political Economy. 68 (5). October 1960: 435–468. doi:10.1086/258363. JSTOR 1830308. (Rothenberg, Jerome. "Comments on non-convexity". Journal of Political Economy. 69 (5). October 1961: 490–492. doi:10.1086/258543. JSTOR 1828540.)
Shapley, Shubik, (1966. səh. 806): Shapley, L. S.; Shubik, M. "Quasi-cores in a monetary economy with nonconvex preferences". Econometrica. 34 (4). October 1966: 805–827. doi:10.2307/1910101. JSTOR 1910101. Zbl 0154.45303.
Aumann, (1966. səh. 1–2): Aumann, Robert J. "Existence of competitive equilibrium in markets with a continuum of traders". Econometrica. 34 (1). January 1966: 1–17. doi:10.2307/1909854. JSTOR 1909854. MR 0191623. Aumann, (1966) builds on two papers: Şablon:Harvs
Aumann, Robert J. "Markets with a continuum of traders". Econometrica. 32 (1–2). January–April 1964: 39–50. doi:10.2307/1913732. JSTOR 1913732. MR 0172689.
Aumann, Robert J. "Integrals of set-valued functions". Journal of Mathematical Analysis and Applications. 12 (1). August 1965: 1–12. doi:10.1016/0022-247X(65)90049-1. MR 0185073.
Sraffa, Piero. "The Laws of returns under competitive conditions". Economic Journal. 36 (144). 1926. 535–550. JSTOR 2959866.
Hotelling, Harold. "The General welfare in relation to problems of taxation and of railway and utility rates". Econometrica. 6 (3). July 1938: 242–269. doi:10.2307/1907054. JSTOR 1907054.
Radner, Roy. "Competitive equilibrium under uncertainty". Econometrica. 36. 1968. 31–53. doi:10.2307/1909602. JSTOR 1909602.
Ramsey, F. P. "A Mathematical Theory of Saving". Economic Journal. 38 (152). 1928: 543–559. doi:10.2307/2224098. JSTOR 2224098.
Hotelling, Harold. "The Economics of Exhaustible Resources". JPE. 39 (2). 1931: 137–175. doi:10.1086/254195. JSTOR 1822328. 2023-07-15 tarixində arxivləşdirilib. İstifadə tarixi: 2020-10-12.
Merton, Robert C. "An Intertemporal Capital Asset Pricing Model". Econometrica. 41 (5). 1973: 867–887. doi:10.2307/1913811. JSTOR 1913811.

semanticscholar.org

Shapley, Shubik, (1966. səh. 806): Shapley, L. S.; Shubik, M. "Quasi-cores in a monetary economy with nonconvex preferences". Econometrica. 34 (4). October 1966: 805–827. doi:10.2307/1910101. JSTOR 1910101. Zbl 0154.45303.
Pages 93–94 (especially example 1.92), 143, 318–319, 375–377, and 416: Carter, Michael. Foundations of mathematical economics. Cambridge, MA: MIT Press. 2001. xx+649. ISBN 0-262-53192-5. MR 1865841.
Page 309: Moore, James C. Mathematical methods for economic theory: Volume I. Studies in economic theory. 9. Berlin: Springer-Verlag. 1999. xii+414. doi:10.1007/978-3-662-08544-8. ISBN 3-540-66235-9. MR 1727000.
Pages 47–48: Florenzano, Monique; Le Van, Cuong. Finite dimensional convexity and optimization. Studies in economic theory. 13. in cooperation with Pascal Gourdel. Berlin: Springer-Verlag. 2001. xii+154. doi:10.1007/978-3-642-56522-9. ISBN 3-540-41516-5. MR 1878374.
Ramsey, F. P. "A Mathematical Theory of Saving". Economic Journal. 38 (152). 1928: 543–559. doi:10.2307/2224098. JSTOR 2224098.
Merton, Robert C. "An Intertemporal Capital Asset Pricing Model". Econometrica. 41 (5). 1973: 867–887. doi:10.2307/1913811. JSTOR 1913811.
Rockafellar, R. Tyrrell; Wets, Roger J-B. Variational analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 317. Berlin: Springer-Verlag. 1998. xiv+733. doi:10.1007/978-3-642-02431-3. ISBN 3-540-62772-3. MR 1491362.

upf.edu

econ.upf.edu

Mas-Colell, A. Non-convexity (PDF) // Eatwell, John; Milgate, Murray; Newman, Peter (redaktorlar ). The New Palgrave: A Dictionary of Economics (first). Palgrave Macmillan. 1987. 653–661. doi:10.1057/9780230226203.3173. ISBN 9780333786765. 2018-08-09 tarixində arxivləşdirilib (PDF). İstifadə tarixi: 2020-10-12.

utdallas.edu

Sethi, S. P.; Thompson, G. L. Optimal Control Theory: Applications to Management Science and Economics (2nd). Berlin: Springer. 2000. ISBN 0-387-28092-8. Slides are available at http://www.utdallas.edu/~sethi/OPRE7320presentation.html Arxivləşdirilib 2017-08-12 at the Wayback Machine

utexas.edu

eco.utexas.edu

Adda, Jerome; Cooper, Russell, Dynamic Economics, MIT Press, 2003, 2008-12-05 tarixində orijinalından arxivləşdirilib, İstifadə tarixi: 2020-10-12

web.archive.org

Mas-Colell, A. Non-convexity (PDF) // Eatwell, John; Milgate, Murray; Newman, Peter (redaktorlar ). The New Palgrave: A Dictionary of Economics (first). Palgrave Macmillan. 1987. 653–661. doi:10.1057/9780230226203.3173. ISBN 9780333786765. 2018-08-09 tarixində arxivləşdirilib (PDF). İstifadə tarixi: 2020-10-12.
Khan, M. Ali. Perfect competition // Durlauf, Steven N.; Blume, Lawrence E., ed. (redaktorlar ). The New Palgrave Dictionary of Economics (Second). Palgrave Macmillan. 2008. 354–365. doi:10.1057/9780230226203.1267. ISBN 978-0-333-78676-5. 2017-09-14 tarixində arxivləşdirilib. İstifadə tarixi: 2020-10-12.
Pages 63–65: Laffont, Jean-Jacques. 3 Nonconvexities // Fondements de L'economie Publique [Fundamentals of public economics]. MIT. 1988. ISBN 0-262-12127-1. 2020-08-06 tarixində arxivləşdirilib. İstifadə tarixi: 2020-10-12.
Hotelling, Harold. "The Economics of Exhaustible Resources". JPE. 39 (2). 1931: 137–175. doi:10.1086/254195. JSTOR 1822328. 2023-07-15 tarixində arxivləşdirilib. İstifadə tarixi: 2020-10-12.
Adda, Jerome; Cooper, Russell, Dynamic Economics, MIT Press, 2003, 2008-12-05 tarixində orijinalından arxivləşdirilib, İstifadə tarixi: 2020-10-12
Sethi, S. P.; Thompson, G. L. Optimal Control Theory: Applications to Management Science and Economics (2nd). Berlin: Springer. 2000. ISBN 0-387-28092-8. Slides are available at http://www.utdallas.edu/~sethi/OPRE7320presentation.html Arxivləşdirilib 2017-08-12 at the Wayback Machine
Heal, (1999. səh. 4 in preprint): Heal, G. M. Introduction (PDF) // The economics of increasing returns. The International Library of Critical Writings in Economics. Edward Elgar. 1999. 640. ISBN 978-1-85898-160-4. 15 September 2015 tarixində orijinalından (PDF) arxivləşdirilib. İstifadə tarixi: 5 March 2011.
Algebraic topology has also been used to study convex and non-convex sets in economics:Chichilnisky, G. "Intersecting families of sets and the topology of cones in economics" (PDF). Bulletin of the American Mathematical Society. New Series. 29 (2). 1993. 189–207. doi:10.1090/S0273-0979-1993-00439-7. MR 1218037. 2022-01-23 tarixində arxivləşdirilib (PDF). İstifadə tarixi: 2020-10-12.

wikisource.org

az.wikisource.org

Samuelson, (1950. səh. 359–360):
It will be noted that any point where the indifference curves are convex rather than concave cannot be observed in a competitive market. Such points are shrouded in eternal darkness—unless we make our consumer a monopsonist and let him choose between goods lying on a very convex "budget curve" (along which he is affecting the price of what he buys). In this monopsony case, we could still deduce the slope of the man's indifference curve from the slope of the observed constraint at the equilibrium point.
Samuelson, Paul A. "The problem of integrability in utility theory". Economica. New Series. 17. 1950. 355–385. doi:10.2307/2549499. JSTOR 2549499. MR 0043436.For the epigraph to their seventh chapter, "Markets with non-convex preferences and production" presenting Starr, (1969), Arrow, Hahn, (1971. səh. 169) quote John Milton's description of the (non-convex) Serbonian Bog in Paradise Lost (Book II, lines 592–594):
A gulf profound as that Serbonian Bog

Betwixt Damiata and Mount Casius old,

Where Armies whole have sunk.

zbmath.org

Shapley, Shubik, (1966. səh. 806): Shapley, L. S.; Shubik, M. "Quasi-cores in a monetary economy with nonconvex preferences". Econometrica. 34 (4). October 1966: 805–827. doi:10.2307/1910101. JSTOR 1910101. Zbl 0154.45303.