Non-convexity (economics) (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Non-convexity (economics)" in English language version.

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Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of mathematical economics, Volume I. Handbooks in economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800.
Starrett, David A. (1972). "Fundamental nonconvexities in the theory of externalities". Journal of Economic Theory. 4 (2): 180–199. doi:10.1016/0022-0531(72)90148-2. MR 0449575.
Page 1: Guesnerie, Roger (1975). "Pareto optimality in non-convex economies". Econometrica. 43 (1): 1–29. doi:10.2307/1913410. JSTOR 1913410. MR 0443877. ("Errata". Econometrica. Vol. 43, no. 5–6. 1975. p. 1010. doi:10.2307/1911353. JSTOR 1911353. MR 0443878.)
Mordukhovich, Boris S. (2006). "Chapter 8 Applications to economics". Variational analysis and generalized differentiation II: Applications. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 331. Springer. especially Section 8.5.3 "Enter nonconvexity" (and the remainder of the chapter), particularly page 495. ISBN 978-3-540-25438-6. MR 2191745.
Brown, Donald J. (1991). "36 Equilibrium analysis with non-convex technologies". In Hildenbrand, Werner; Sonnenschein, Hugo (eds.). Handbook of mathematical economics, Volume IV. Handbooks in Economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 1963–1995 [1966]. doi:10.1016/S1573-4382(05)80011-6. ISBN 0-444-87461-5. MR 1207195.
Pages 231 and 239 (Figure 10 a–b: Illustration of lemma 5 [page 240]): Wold, Herman (1943b). "A synthesis of pure demand analysis II". Skandinavisk Aktuarietidskrift [Scandinavian Actuarial Journal]. Vol. 26. pp. 220–263. MR 0011939.
Exercise 45, page 146: Wold, Herman; Juréen, Lars (in association with Wold) (1953). "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. MR 0064385.
Samuelson (1950, pp. 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.
For the epigraph to their seventh chapter, "Markets with non-convex preferences and production" presenting Starr (1969), Arrow & Hahn (1971, p. 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.
Samuelson, Paul A. (1950). "The problem of integrability in utility theory". Economica. New Series. 17 (68): 355–385. doi:10.2307/2549499. JSTOR 2549499. MR 0043436.
Diewert (1982, pp. 552–553). Diewert, W. E. (1982). "12 Duality approaches to microeconomic theory". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of mathematical economics, Volume II. Handbooks in economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778.
Aumann (1966, pp. 1–2): Aumann, Robert J. (January 1966). "Existence of competitive equilibrium in markets with a continuum of traders". Econometrica. 34 (1): 1–17. doi:10.2307/1909854. JSTOR 1909854. MR 0191623. S2CID 155044347. Aumann (1966) builds on two papers: Aumann (1964, 1965)
Aumann, Robert J. (January–April 1964). "Markets with a continuum of traders". Econometrica. 32 (1–2): 39–50. doi:10.2307/1913732. JSTOR 1913732. MR 0172689.

Aumann, Robert J. (August 1965). "Integrals of set-valued functions". Journal of Mathematical Analysis and Applications. 12 (1): 1–12. doi:10.1016/0022-247X(65)90049-1. MR 0185073.
Taking the convex hull of non-convex preferences had been discussed earlier by Wold (1943b, p. 243) and by Wold & Juréen (1953, p. 146), according to Diewert (1982, p. 552).
Diewert, W. E. (1982). "12 Duality approaches to microeconomic theory". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of mathematical economics, Volume II. Handbooks in economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778.
Pages 392–399 and page 188: Arrow, Kenneth J.; Hahn, Frank H. (1971). "Appendix B: Convex and related sets". General competitive analysis. Mathematical economics texts [Advanced textbooks in economics]. San Francisco, CA: Holden-Day, Inc. [North-Holland]. pp. 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 (1985). "1.L Averages of sets". The Theory of General Economic Equilibrium: A Differentiable Approach. Econometric Society Monographs. Cambridge University Press. ISBN 0-521-26514-2. MR 1113262.
Theorem C(6) on page 37 and applications on pages 115-116, 122, and 168: Hildenbrand, Werner (1974). Core and equilibria of a large economy. Princeton studies in mathematical economics. Princeton, NJ: Princeton University Press. ISBN 978-0-691-04189-6. MR 0389160.
Varian, Hal R. (1992). "21.2 Convexity and size". Microeconomic Analysis (3rd ed.). W. W. Norton & Company. pp. 393–394. ISBN 978-0-393-95735-8. MR 1036734.
Page 628: Mas–Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995). "17.1 Large economies and nonconvexities". Microeconomic theory. Oxford University Press. pp. 627–630. ISBN 978-0-19-507340-9.
Page 169 in the first edition: Starr, Ross M. (2011). "8 Convex sets, separation theorems, and non-convex sets in R^N". General equilibrium theory: An introduction (Second ed.). Cambridge: Cambridge University Press. doi:10.1017/CBO9781139174749. ISBN 978-0-521-53386-7. MR 1462618.^{[clarification needed]}
Ellickson (1994, p. xviii), and especially Chapter 7 "Walras meets Nash" (especially section 7.4 "Nonconvexity" pages 306–310 and 312, and also 328–329) and Chapter 8 "What is Competition?" (pages 347 and 352): Ellickson, Bryan (1994). Competitive equilibrium: Theory and applications. Cambridge University Press. ISBN 978-0-521-31988-1.
Theorem 1.6.5 on pages 24–25: Ichiishi, Tatsuro (1983). Game theory for economic analysis. Economic theory, econometrics, and mathematical economics. New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. ISBN 0-12-370180-5. MR 0700688.
Cassels, J. W. S. (1981). "Appendix A Convex sets". Economics for mathematicians. London Mathematical Society lecture note series. Vol. 62. Cambridge, New York: Cambridge University Press. pp. 33–34 and 127. ISBN 0-521-28614-X. MR 0657578.
Pages 93–94 (especially example 1.92), 143, 318–319, 375–377, and 416: Carter, Michael (2001). Foundations of mathematical economics. Cambridge, MA: MIT Press. ISBN 0-262-53192-5. MR 1865841.
Page 309: Moore, James C. (1999). Mathematical methods for economic theory: Volume I. Studies in economic theory. Vol. 9. Berlin: Springer-Verlag. doi:10.1007/978-3-662-08544-8. ISBN 3-540-66235-9. MR 1727000.
Pages 47–48: Florenzano, Monique; Le Van, Cuong (2001). Finite dimensional convexity and optimization. Studies in economic theory. Vol. 13. in cooperation with Pascal Gourdel. Berlin: Springer-Verlag. doi:10.1007/978-3-642-56522-9. ISBN 3-540-41516-5. MR 1878374. S2CID 117240618.
Economists have studied non-convex sets using advanced mathematics, particularly differential geometry and topology, Baire category, measure and integration theory, and ergodic theory: Trockel, Walter (1984). Market demand: An analysis of large economies with nonconvex preferences. Lecture Notes in Economics and Mathematical Systems. Vol. 223. Berlin: Springer-Verlag. doi:10.1007/978-3-642-46488-1. ISBN 3-540-12881-6. MR 0737006.
Page 270: Drèze, Jacques H. (1987). "14 Investment under private ownership: Optimality, equilibrium and stability". In Drèze, J. H.location=Cambridge (ed.). Essays on economic decisions under uncertainty. Cambridge University Press. pp. 261–297. doi:10.1017/CBO9780511559464. ISBN 0-521-26484-7. MR 0926685. (Originally published as Drèze, Jacques H. (1974). "Investment under private ownership: Optimality, equilibrium and stability". In Drèze, J. H. (ed.). Allocation under Uncertainty: Equilibrium and Optimality. New York: Wiley. pp. 129–165.)
Troutman, John L. (1996). With the assistance of William Hrusa (ed.). Variational calculus and optimal control: Optimization with elementary convexity. Undergraduate Texts in Mathematics (Second ed.). New York: Springer-Verlag. doi:10.1007/978-1-4612-0737-5. ISBN 0-387-94511-3. MR 1363262.
Craven, B. D. (1995). Control and optimization. Chapman and Hall Mathematics Series. London: Chapman and Hall, Ltd. doi:10.1007/978-1-4899-7226-2. ISBN 0-412-55890-4. MR 1349574.
Vinter, Richard (2000). Optimal control. Systems & Control: Foundations & Applications. Boston, MA: Birkhäuser Boston, Inc. ISBN 0-8176-4075-4. MR 1756410.
Rockafellar, R. Tyrrell; Wets, Roger J-B (1998). Variational analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 317. Berlin: Springer-Verlag. doi:10.1007/978-3-642-02431-3. ISBN 3-540-62772-3. MR 1491362. S2CID 198120391.
Algebraic topology has also been used to study convex and non-convex sets in economics:Chichilnisky, G. (1993). "Intersecting families of sets and the topology of cones in economics" (PDF). Bulletin of the American Mathematical Society. New Series. 29 (2): 189–207. doi:10.1090/S0273-0979-1993-00439-7. MR 1218037.

ams.org

Algebraic topology has also been used to study convex and non-convex sets in economics:Chichilnisky, G. (1993). "Intersecting families of sets and the topology of cones in economics" (PDF). Bulletin of the American Mathematical Society. New Series. 29 (2): 189–207. doi:10.1090/S0273-0979-1993-00439-7. MR 1218037.

archive.org

Pages 392–399 and page 188: Arrow, Kenneth J.; Hahn, Frank H. (1971). "Appendix B: Convex and related sets". General competitive analysis. Mathematical economics texts [Advanced textbooks in economics]. San Francisco, CA: Holden-Day, Inc. [North-Holland]. pp. 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 (1985). "1.L Averages of sets". The Theory of General Economic Equilibrium: A Differentiable Approach. Econometric Society Monographs. Cambridge University Press. ISBN 0-521-26514-2. MR 1113262.
Theorem C(6) on page 37 and applications on pages 115-116, 122, and 168: Hildenbrand, Werner (1974). Core and equilibria of a large economy. Princeton studies in mathematical economics. Princeton, NJ: Princeton University Press. ISBN 978-0-691-04189-6. MR 0389160.
Varian, Hal R. (1992). "21.2 Convexity and size". Microeconomic Analysis (3rd ed.). W. W. Norton & Company. pp. 393–394. ISBN 978-0-393-95735-8. MR 1036734.
Page 628: Mas–Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995). "17.1 Large economies and nonconvexities". Microeconomic theory. Oxford University Press. pp. 627–630. ISBN 978-0-19-507340-9.
Craven, B. D. (1995). Control and optimization. Chapman and Hall Mathematics Series. London: Chapman and Hall, Ltd. doi:10.1007/978-1-4899-7226-2. ISBN 0-412-55890-4. MR 1349574.
Dixit, Avinash; Pindyck, Robert (1994). Investment Under Uncertainty. Princeton Univ. Press. ISBN 0-691-03410-9.
Dasgupta & Heal (1979, pp. 96–97, 285, 404, 420, 422, and 429) Dasgupta, P. S.; Heal, G. M. (1979). Economic Theory and Exhaustible Resources. Cambridge University Press.
Dasgupta & Heal (1979, pp. 51, 64–65, 87, and 91–92) Dasgupta, P. S.; Heal, G. M. (1979). Economic Theory and Exhaustible Resources. Cambridge University Press.

books.google.com

Laffont, Jean-Jacques (1988). "3 Nonconvexities". Fondements de L'economie Publique [Fundamentals of public economics]. MIT. pp. 63–65. ISBN 0-262-12127-1.
Starrett discusses non-convexities in his textbook on public economics (pages 33, 43, 48, 56, 70–72, 82, 147, and 234–236): Starrett, David A. (1988). Foundations of public economics. Cambridge economic handbooks. Cambridge: Cambridge University Press. ISBN 9780521348010.

columbia.edu

www2.gsb.columbia.edu

Heal (1999, p. 4 in preprint): Heal, G. M. (1999). "Introduction" (PDF). The economics of increasing returns. The International Library of Critical Writings in Economics. Edward Elgar. ISBN 978-1-85898-160-4. Retrieved 5 March 2011.

dictionaryofeconomics.com

Khan, M. Ali (2008). "Perfect competition". In Durlauf, Steven N.; Blume, Lawrence E. (eds.). The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. pp. 354–365. doi:10.1057/9780230226203.1267. ISBN 978-0-333-78676-5.

doi.org

Mas-Colell, A. (1987). "Non-convexity" (PDF). In Eatwell, John; Milgate, Murray; Newman, Peter (eds.). The New Palgrave: A Dictionary of Economics (first ed.). Palgrave Macmillan. pp. 653–661. doi:10.1057/9780230226203.3173. ISBN 9780333786765.
Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of mathematical economics, Volume I. Handbooks in economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 15–52. doi:10.1016/S1573-4382(81)01005-9. ISBN 0-444-86126-2. MR 0634800.
Starrett, David A. (1972). "Fundamental nonconvexities in the theory of externalities". Journal of Economic Theory. 4 (2): 180–199. doi:10.1016/0022-0531(72)90148-2. MR 0449575.
Page 1: Guesnerie, Roger (1975). "Pareto optimality in non-convex economies". Econometrica. 43 (1): 1–29. doi:10.2307/1913410. JSTOR 1913410. MR 0443877. ("Errata". Econometrica. Vol. 43, no. 5–6. 1975. p. 1010. doi:10.2307/1911353. JSTOR 1911353. MR 0443878.)
Khan, M. Ali (2008). "Perfect competition". In Durlauf, Steven N.; Blume, Lawrence E. (eds.). The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. pp. 354–365. doi:10.1057/9780230226203.1267. ISBN 978-0-333-78676-5.
Brown, Donald J. (1991). "36 Equilibrium analysis with non-convex technologies". In Hildenbrand, Werner; Sonnenschein, Hugo (eds.). Handbook of mathematical economics, Volume IV. Handbooks in Economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 1963–1995 [1966]. doi:10.1016/S1573-4382(05)80011-6. ISBN 0-444-87461-5. MR 1207195.
Hotelling (1935, p. 74): Hotelling, Harold (January 1935). "Demand functions with limited budgets". Econometrica. 3 (1): 66–78. doi:10.2307/1907346. JSTOR 1907346.
Samuelson (1950, pp. 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.
For the epigraph to their seventh chapter, "Markets with non-convex preferences and production" presenting Starr (1969), Arrow & Hahn (1971, p. 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.
Samuelson, Paul A. (1950). "The problem of integrability in utility theory". Economica. New Series. 17 (68): 355–385. doi:10.2307/2549499. JSTOR 2549499. MR 0043436.
Diewert (1982, pp. 552–553). Diewert, W. E. (1982). "12 Duality approaches to microeconomic theory". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of mathematical economics, Volume II. Handbooks in economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778.
Farrell, M. J. (August 1959). "The Convexity assumption in the theory of competitive markets". The Journal of Political Economy. 67 (4): 371–391. doi:10.1086/258197. JSTOR 1825163. S2CID 153653926. Farrell, M. J. (October 1961a). "On Convexity, efficiency, and markets: A Reply". Journal of Political Economy. 69 (5): 484–489. doi:10.1086/258541. JSTOR 1828538. S2CID 154398283. Farrell, M. J. (October 1961b). "The Convexity assumption in the theory of competitive markets: Rejoinder". Journal of Political Economy. 69 (5): 493. doi:10.1086/258544. JSTOR 1828541. S2CID 154200859.
Bator, Francis M. (October 1961a). "On convexity, efficiency, and markets". The Journal of Political Economy. 69 (5): 480–483. doi:10.1086/258540. JSTOR 1828537. S2CID 153979194. Bator, Francis M. (October 1961b). "On convexity, efficiency, and markets: Rejoinder". Journal of Political Economy. 69 (5): 489. doi:10.1086/258542. JSTOR 1828539. S2CID 154255876.
Koopmans, Tjalling C. (October 1961). "Convexity assumptions, allocative efficiency, and competitive equilibrium". The Journal of Political Economy. 69 (5): 478–479. doi:10.1086/258539. JSTOR 1828536. S2CID 154831335.
Koopmans (1961, p. 478) and others—for example, Farrell (1959, pp. 390–391) and Farrell (1961a, p. 484), Bator (1961a, pp. 482–483), Rothenberg (1960, p. 438), and Starr (1969, p. 26)—commented on Koopmans (1957, pp. 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]):
Koopmans, Tjalling C. (1957). "Allocation of resources and the price system". In Koopmans, Tjalling C (ed.). Three essays on the state of economic science. New York: McGraw–Hill Book Company. pp. 1–126. ISBN 0-07-035337-9.
Rothenberg (1960, p. 447): Rothenberg, Jerome (October 1960). "Non-convexity, aggregation, and Pareto optimality". The Journal of Political Economy. 68 (5): 435–468. doi:10.1086/258363. JSTOR 1830308. S2CID 154192326. (Rothenberg, Jerome (October 1961). "Comments on non-convexity". Journal of Political Economy. 69 (5): 490–492. doi:10.1086/258543. JSTOR 1828540. S2CID 154070123.)
Shapley & Shubik (1966, p. 806): Shapley, L. S.; Shubik, M. (October 1966). "Quasi-cores in a monetary economy with nonconvex preferences". Econometrica. 34 (4): 805–827. doi:10.2307/1910101. JSTOR 1910101. S2CID 46271184. Zbl 0154.45303.
Aumann (1966, pp. 1–2): Aumann, Robert J. (January 1966). "Existence of competitive equilibrium in markets with a continuum of traders". Econometrica. 34 (1): 1–17. doi:10.2307/1909854. JSTOR 1909854. MR 0191623. S2CID 155044347. Aumann (1966) builds on two papers: Aumann (1964, 1965)
Aumann, Robert J. (January–April 1964). "Markets with a continuum of traders". Econometrica. 32 (1–2): 39–50. doi:10.2307/1913732. JSTOR 1913732. MR 0172689.

Aumann, Robert J. (August 1965). "Integrals of set-valued functions". Journal of Mathematical Analysis and Applications. 12 (1): 1–12. doi:10.1016/0022-247X(65)90049-1. MR 0185073.
Taking the convex hull of non-convex preferences had been discussed earlier by Wold (1943b, p. 243) and by Wold & Juréen (1953, p. 146), according to Diewert (1982, p. 552).
Diewert, W. E. (1982). "12 Duality approaches to microeconomic theory". In Arrow, Kenneth Joseph; Intriligator, Michael D. (eds.). Handbook of mathematical economics, Volume II. Handbooks in economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 535–599. doi:10.1016/S1573-4382(82)02007-4. ISBN 978-0-444-86127-6. MR 0648778.
Page 169 in the first edition: Starr, Ross M. (2011). "8 Convex sets, separation theorems, and non-convex sets in R^N". General equilibrium theory: An introduction (Second ed.). Cambridge: Cambridge University Press. doi:10.1017/CBO9781139174749. ISBN 978-0-521-53386-7. MR 1462618.^{[clarification needed]}
Ellickson (1994, p. xviii), and especially Chapter 7 "Walras meets Nash" (especially section 7.4 "Nonconvexity" pages 306–310 and 312, and also 328–329) and Chapter 8 "What is Competition?" (pages 347 and 352): Ellickson, Bryan (1994). Competitive equilibrium: Theory and applications. Cambridge University Press. ISBN 978-0-521-31988-1.
Pages 93–94 (especially example 1.92), 143, 318–319, 375–377, and 416: Carter, Michael (2001). Foundations of mathematical economics. Cambridge, MA: MIT Press. ISBN 0-262-53192-5. MR 1865841.
Page 309: Moore, James C. (1999). Mathematical methods for economic theory: Volume I. Studies in economic theory. Vol. 9. Berlin: Springer-Verlag. doi:10.1007/978-3-662-08544-8. ISBN 3-540-66235-9. MR 1727000.
Pages 47–48: Florenzano, Monique; Le Van, Cuong (2001). Finite dimensional convexity and optimization. Studies in economic theory. Vol. 13. in cooperation with Pascal Gourdel. Berlin: Springer-Verlag. doi:10.1007/978-3-642-56522-9. ISBN 3-540-41516-5. MR 1878374. S2CID 117240618.
Economists have studied non-convex sets using advanced mathematics, particularly differential geometry and topology, Baire category, measure and integration theory, and ergodic theory: Trockel, Walter (1984). Market demand: An analysis of large economies with nonconvex preferences. Lecture Notes in Economics and Mathematical Systems. Vol. 223. Berlin: Springer-Verlag. doi:10.1007/978-3-642-46488-1. ISBN 3-540-12881-6. MR 0737006.
Hotelling, Harold (July 1938). "The General welfare in relation to problems of taxation and of railway and utility rates". Econometrica. 6 (3): 242–269. doi:10.2307/1907054. JSTOR 1907054.
Radner, Roy (1968). "Competitive equilibrium under uncertainty". Econometrica. 36 (1): 31–53. doi:10.2307/1909602. JSTOR 1909602.
Page 270: Drèze, Jacques H. (1987). "14 Investment under private ownership: Optimality, equilibrium and stability". In Drèze, J. H.location=Cambridge (ed.). Essays on economic decisions under uncertainty. Cambridge University Press. pp. 261–297. doi:10.1017/CBO9780511559464. ISBN 0-521-26484-7. MR 0926685. (Originally published as Drèze, Jacques H. (1974). "Investment under private ownership: Optimality, equilibrium and stability". In Drèze, J. H. (ed.). Allocation under Uncertainty: Equilibrium and Optimality. New York: Wiley. pp. 129–165.)
Ramsey, F. P. (1928). "A Mathematical Theory of Saving". Economic Journal. 38 (152): 543–559. doi:10.2307/2224098. JSTOR 2224098. S2CID 154223797.
Hotelling, Harold (1931). "The Economics of Exhaustible Resources". Journal of Political Economy. 39 (2): 137–175. doi:10.1086/254195. JSTOR 1822328. S2CID 222432341.
Troutman, John L. (1996). With the assistance of William Hrusa (ed.). Variational calculus and optimal control: Optimization with elementary convexity. Undergraduate Texts in Mathematics (Second ed.). New York: Springer-Verlag. doi:10.1007/978-1-4612-0737-5. ISBN 0-387-94511-3. MR 1363262.
Craven, B. D. (1995). Control and optimization. Chapman and Hall Mathematics Series. London: Chapman and Hall, Ltd. doi:10.1007/978-1-4899-7226-2. ISBN 0-412-55890-4. MR 1349574.
Merton, Robert C. (1973). "An Intertemporal Capital Asset Pricing Model". Econometrica. 41 (5): 867–887. doi:10.2307/1913811. JSTOR 1913811. S2CID 1504746.
Rockafellar, R. Tyrrell; Wets, Roger J-B (1998). Variational analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 317. Berlin: Springer-Verlag. doi:10.1007/978-3-642-02431-3. ISBN 3-540-62772-3. MR 1491362. S2CID 198120391.
Algebraic topology has also been used to study convex and non-convex sets in economics:Chichilnisky, G. (1993). "Intersecting families of sets and the topology of cones in economics" (PDF). Bulletin of the American Mathematical Society. New Series. 29 (2): 189–207. doi:10.1090/S0273-0979-1993-00439-7. MR 1218037.

e-elgar.com

Hotelling, Harold (1931). "The Economics of Exhaustible Resources". Journal of Political Economy. 39 (2): 137–175. doi:10.1086/254195. JSTOR 1822328. S2CID 222432341.

jstor.org

Page 1: Guesnerie, Roger (1975). "Pareto optimality in non-convex economies". Econometrica. 43 (1): 1–29. doi:10.2307/1913410. JSTOR 1913410. MR 0443877. ("Errata". Econometrica. Vol. 43, no. 5–6. 1975. p. 1010. doi:10.2307/1911353. JSTOR 1911353. MR 0443878.)
Hotelling (1935, p. 74): Hotelling, Harold (January 1935). "Demand functions with limited budgets". Econometrica. 3 (1): 66–78. doi:10.2307/1907346. JSTOR 1907346.
Samuelson (1950, pp. 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.
For the epigraph to their seventh chapter, "Markets with non-convex preferences and production" presenting Starr (1969), Arrow & Hahn (1971, p. 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.
Samuelson, Paul A. (1950). "The problem of integrability in utility theory". Economica. New Series. 17 (68): 355–385. doi:10.2307/2549499. JSTOR 2549499. MR 0043436.
Farrell, M. J. (August 1959). "The Convexity assumption in the theory of competitive markets". The Journal of Political Economy. 67 (4): 371–391. doi:10.1086/258197. JSTOR 1825163. S2CID 153653926. Farrell, M. J. (October 1961a). "On Convexity, efficiency, and markets: A Reply". Journal of Political Economy. 69 (5): 484–489. doi:10.1086/258541. JSTOR 1828538. S2CID 154398283. Farrell, M. J. (October 1961b). "The Convexity assumption in the theory of competitive markets: Rejoinder". Journal of Political Economy. 69 (5): 493. doi:10.1086/258544. JSTOR 1828541. S2CID 154200859.
Bator, Francis M. (October 1961a). "On convexity, efficiency, and markets". The Journal of Political Economy. 69 (5): 480–483. doi:10.1086/258540. JSTOR 1828537. S2CID 153979194. Bator, Francis M. (October 1961b). "On convexity, efficiency, and markets: Rejoinder". Journal of Political Economy. 69 (5): 489. doi:10.1086/258542. JSTOR 1828539. S2CID 154255876.
Koopmans, Tjalling C. (October 1961). "Convexity assumptions, allocative efficiency, and competitive equilibrium". The Journal of Political Economy. 69 (5): 478–479. doi:10.1086/258539. JSTOR 1828536. S2CID 154831335.
Koopmans (1961, p. 478) and others—for example, Farrell (1959, pp. 390–391) and Farrell (1961a, p. 484), Bator (1961a, pp. 482–483), Rothenberg (1960, p. 438), and Starr (1969, p. 26)—commented on Koopmans (1957, pp. 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]):
Koopmans, Tjalling C. (1957). "Allocation of resources and the price system". In Koopmans, Tjalling C (ed.). Three essays on the state of economic science. New York: McGraw–Hill Book Company. pp. 1–126. ISBN 0-07-035337-9.
Rothenberg (1960, p. 447): Rothenberg, Jerome (October 1960). "Non-convexity, aggregation, and Pareto optimality". The Journal of Political Economy. 68 (5): 435–468. doi:10.1086/258363. JSTOR 1830308. S2CID 154192326. (Rothenberg, Jerome (October 1961). "Comments on non-convexity". Journal of Political Economy. 69 (5): 490–492. doi:10.1086/258543. JSTOR 1828540. S2CID 154070123.)
Shapley & Shubik (1966, p. 806): Shapley, L. S.; Shubik, M. (October 1966). "Quasi-cores in a monetary economy with nonconvex preferences". Econometrica. 34 (4): 805–827. doi:10.2307/1910101. JSTOR 1910101. S2CID 46271184. Zbl 0154.45303.
Aumann (1966, pp. 1–2): Aumann, Robert J. (January 1966). "Existence of competitive equilibrium in markets with a continuum of traders". Econometrica. 34 (1): 1–17. doi:10.2307/1909854. JSTOR 1909854. MR 0191623. S2CID 155044347. Aumann (1966) builds on two papers: Aumann (1964, 1965)
Aumann, Robert J. (January–April 1964). "Markets with a continuum of traders". Econometrica. 32 (1–2): 39–50. doi:10.2307/1913732. JSTOR 1913732. MR 0172689.

Aumann, Robert J. (August 1965). "Integrals of set-valued functions". Journal of Mathematical Analysis and Applications. 12 (1): 1–12. doi:10.1016/0022-247X(65)90049-1. MR 0185073.
Sraffa, Piero (1926). "The Laws of returns under competitive conditions". Economic Journal. Vol. 36, no. 144. pp. 535–550. JSTOR 2959866.
Hotelling, Harold (July 1938). "The General welfare in relation to problems of taxation and of railway and utility rates". Econometrica. 6 (3): 242–269. doi:10.2307/1907054. JSTOR 1907054.
Radner, Roy (1968). "Competitive equilibrium under uncertainty". Econometrica. 36 (1): 31–53. doi:10.2307/1909602. JSTOR 1909602.
Ramsey, F. P. (1928). "A Mathematical Theory of Saving". Economic Journal. 38 (152): 543–559. doi:10.2307/2224098. JSTOR 2224098. S2CID 154223797.
Hotelling, Harold (1931). "The Economics of Exhaustible Resources". Journal of Political Economy. 39 (2): 137–175. doi:10.1086/254195. JSTOR 1822328. S2CID 222432341.
Merton, Robert C. (1973). "An Intertemporal Capital Asset Pricing Model". Econometrica. 41 (5): 867–887. doi:10.2307/1913811. JSTOR 1913811. S2CID 1504746.

semanticscholar.org

api.semanticscholar.org

Farrell, M. J. (August 1959). "The Convexity assumption in the theory of competitive markets". The Journal of Political Economy. 67 (4): 371–391. doi:10.1086/258197. JSTOR 1825163. S2CID 153653926. Farrell, M. J. (October 1961a). "On Convexity, efficiency, and markets: A Reply". Journal of Political Economy. 69 (5): 484–489. doi:10.1086/258541. JSTOR 1828538. S2CID 154398283. Farrell, M. J. (October 1961b). "The Convexity assumption in the theory of competitive markets: Rejoinder". Journal of Political Economy. 69 (5): 493. doi:10.1086/258544. JSTOR 1828541. S2CID 154200859.
Bator, Francis M. (October 1961a). "On convexity, efficiency, and markets". The Journal of Political Economy. 69 (5): 480–483. doi:10.1086/258540. JSTOR 1828537. S2CID 153979194. Bator, Francis M. (October 1961b). "On convexity, efficiency, and markets: Rejoinder". Journal of Political Economy. 69 (5): 489. doi:10.1086/258542. JSTOR 1828539. S2CID 154255876.
Koopmans, Tjalling C. (October 1961). "Convexity assumptions, allocative efficiency, and competitive equilibrium". The Journal of Political Economy. 69 (5): 478–479. doi:10.1086/258539. JSTOR 1828536. S2CID 154831335.
Koopmans (1961, p. 478) and others—for example, Farrell (1959, pp. 390–391) and Farrell (1961a, p. 484), Bator (1961a, pp. 482–483), Rothenberg (1960, p. 438), and Starr (1969, p. 26)—commented on Koopmans (1957, pp. 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]):
Koopmans, Tjalling C. (1957). "Allocation of resources and the price system". In Koopmans, Tjalling C (ed.). Three essays on the state of economic science. New York: McGraw–Hill Book Company. pp. 1–126. ISBN 0-07-035337-9.
Rothenberg (1960, p. 447): Rothenberg, Jerome (October 1960). "Non-convexity, aggregation, and Pareto optimality". The Journal of Political Economy. 68 (5): 435–468. doi:10.1086/258363. JSTOR 1830308. S2CID 154192326. (Rothenberg, Jerome (October 1961). "Comments on non-convexity". Journal of Political Economy. 69 (5): 490–492. doi:10.1086/258543. JSTOR 1828540. S2CID 154070123.)
Shapley & Shubik (1966, p. 806): Shapley, L. S.; Shubik, M. (October 1966). "Quasi-cores in a monetary economy with nonconvex preferences". Econometrica. 34 (4): 805–827. doi:10.2307/1910101. JSTOR 1910101. S2CID 46271184. Zbl 0154.45303.
Aumann (1966, pp. 1–2): Aumann, Robert J. (January 1966). "Existence of competitive equilibrium in markets with a continuum of traders". Econometrica. 34 (1): 1–17. doi:10.2307/1909854. JSTOR 1909854. MR 0191623. S2CID 155044347. Aumann (1966) builds on two papers: Aumann (1964, 1965)
Aumann, Robert J. (January–April 1964). "Markets with a continuum of traders". Econometrica. 32 (1–2): 39–50. doi:10.2307/1913732. JSTOR 1913732. MR 0172689.

Aumann, Robert J. (August 1965). "Integrals of set-valued functions". Journal of Mathematical Analysis and Applications. 12 (1): 1–12. doi:10.1016/0022-247X(65)90049-1. MR 0185073.
Pages 93–94 (especially example 1.92), 143, 318–319, 375–377, and 416: Carter, Michael (2001). Foundations of mathematical economics. Cambridge, MA: MIT Press. ISBN 0-262-53192-5. MR 1865841.
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Pages 47–48: Florenzano, Monique; Le Van, Cuong (2001). Finite dimensional convexity and optimization. Studies in economic theory. Vol. 13. in cooperation with Pascal Gourdel. Berlin: Springer-Verlag. doi:10.1007/978-3-642-56522-9. ISBN 3-540-41516-5. MR 1878374. S2CID 117240618.
Ramsey, F. P. (1928). "A Mathematical Theory of Saving". Economic Journal. 38 (152): 543–559. doi:10.2307/2224098. JSTOR 2224098. S2CID 154223797.
Hotelling, Harold (1931). "The Economics of Exhaustible Resources". Journal of Political Economy. 39 (2): 137–175. doi:10.1086/254195. JSTOR 1822328. S2CID 222432341.
Merton, Robert C. (1973). "An Intertemporal Capital Asset Pricing Model". Econometrica. 41 (5): 867–887. doi:10.2307/1913811. JSTOR 1913811. S2CID 1504746.
Rockafellar, R. Tyrrell; Wets, Roger J-B (1998). Variational analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 317. Berlin: Springer-Verlag. doi:10.1007/978-3-642-02431-3. ISBN 3-540-62772-3. MR 1491362. S2CID 198120391.

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Adda, Jerome; Cooper, Russell (2003), Dynamic Economics, MIT Press, archived from the original on 2008-12-05, retrieved 2011-03-01

wikisource.org

en.wikisource.org

Samuelson (1950, pp. 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.
For the epigraph to their seventh chapter, "Markets with non-convex preferences and production" presenting Starr (1969), Arrow & Hahn (1971, p. 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.
Samuelson, Paul A. (1950). "The problem of integrability in utility theory". Economica. New Series. 17 (68): 355–385. doi:10.2307/2549499. JSTOR 2549499. MR 0043436.

zbmath.org

Shapley & Shubik (1966, p. 806): Shapley, L. S.; Shubik, M. (October 1966). "Quasi-cores in a monetary economy with nonconvex preferences". Econometrica. 34 (4): 805–827. doi:10.2307/1910101. JSTOR 1910101. S2CID 46271184. Zbl 0154.45303.