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Classification of Graph Surfaces Induced by Weighted-Homogeneous Functions Exhibiting Vanishing Gaussian Curvature

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Abstract

Developable surfaces are surfaces in three-dimensional Euclidean space with zero Gaussian curvature. If these surfaces are explicitly defined in the functional form \(z=f(x,y)\), then f is nothing but a solution of the homogeneous Monge–Ampère equation. The main aim of this paper is to classify developable surfaces defined as graphs of weighted-homogeneous functions and to apply the result in economic analysis. We establish a complete classification of weighted-homogeneous production models through associated production surfaces, proving that there exist five classes of weighted-homogeneous production functions exhibiting vanishing Gaussian curvature, generalizing the result established in Chen and Vîlcu (Appl Math Comput 225, 345–351, 2013), where it was stated that only two classes of homogeneous production functions define developable surfaces, namely those having constant return to scale and those defined by binomial functions. We also propose some challenging problems for further research.

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References

  1. Abe, K., Okamoto, H., Tawada, M.: A note on the production possibility frontier with pure public intermediate goods. Canad. J. Econ. 19(2), 351–356 (1986)

    Article  Google Scholar 

  2. Alodan, H., Chen, B.-Y., Deshmukh, S., Vîlcu, G.-E.: On some geometric properties of quasi-product production models. J. Math. Anal. Appl. 474(1), 693–711 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alodan, H., Chen, B.-Y., Deshmukh, S., Vîlcu, G.-E.: Solution of the system of nonlinear PDEs characterizing CES property under quasi-homogeneity conditions. Adv. Differ. Equ. 2021, 257 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anosov, D.V., Aranson, S.K., Arnold, V.I., Bronshtein, I.U., Grines, V.Z., Il’yashenko, Y.S.: Ordinary Differential Equations and Smooth Dynamical Systems. Springer-Verlag, Berlin (1997)

    MATH  Google Scholar 

  5. Aydin, M.E., Ergüt, M.: Composite functions with Allen determinants and their applications to production models in economics. Tamkang J. Math. 45(4), 427–435 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Aydin, M.E., Mihai, A.: Classification of quasi-sum production functions with Allen determinants. Filomat 29(6), 1351–1359 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bayanjargal, D., Yerkyebulan, B., Battsukh, T.: A new class of production function. Theor. Econ. Lett. 10, 356–365 (2020)

    Article  Google Scholar 

  8. Chen, B.-Y.: On some geometric properties of quasi-sum production models. J. Math. Anal. Appl. 392(2), 192–199 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, B.-Y.: Solutions to homogeneous Monge-Ampère equations of homothetic functions and their applications to production models in economics. J. Math. Anal. Appl. 411, 223–229 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, B.-Y., Vîlcu, G.-E.: Geometric classifications of homogeneous production functions. Appl. Math. Comput. 225, 345–351 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Cheng, M., Han, Y.: Application of a modified CES production function model based on improved PSO algorithm. Appl. Math. Comput. 387, 125178 (2020)

    MathSciNet  MATH  Google Scholar 

  12. Cobb, C.W., Douglas, P.H.: A theory of production. Am. Econ. Rev. 18, 139–165 (1928)

    Google Scholar 

  13. Courant, R., Hilbert, D.: Methods of Mathematical Physics. Partial Differential Equations, Interscience Publishers, New York, London (1962)

    MATH  Google Scholar 

  14. Debreu, G.: Mathematical Economics: Twenty Papers, vol. 4. Cambridge University Press, Cambridge (1983)

    Book  MATH  Google Scholar 

  15. Decu, S., Verstraelen, L.: A note on the isotropical geometry of production surfaces. Kragujevac J. Math. 37, 217–220 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Donato, J.: Minimal surfaces in economic theory. In: Geometry in Partial Differential Equations (pp. 68–90). World Scientific, Singapore (1994)

  17. Eichhorn, W.: Theorie Der Homogenen Produktions funktion. Springer-Verlag, Berlin, Heidelberg, New York (1970)

    Book  MATH  Google Scholar 

  18. Eichhorn, W., Oettli, W.: Mehrproduktunternehmungen mit linearen expansionswegen. Oper. Res. Verfahren. 6, 101–117 (1969)

    MATH  Google Scholar 

  19. Färe, R.: Ray-homothetic production functions. Econometrica 45, 133–146 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fu, Y., Wang, W.G.: Geometric characterizations of quasi-product production models in economics. Filomat 31(6), 1601–1609 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Goriely, A.: Integrability and Nonintegrability of Dynamical Systems. Advanced Series in Nonlinear Dynamics:Advanced Series in Nonlinear Dynamics, vol. 19. World Scientific, Singapore (2001)

    MATH  Google Scholar 

  22. Greenwood, J.P., Magleby, S.P., Howell, L.L.: Developable mechanisms on regular cylindrical surfaces. Mech. Mach. Theory 142, 103584 (2019)

    Article  Google Scholar 

  23. Hankey, A., Stanley, H.E.: Systematic application of generalized homogeneous functions to static scaling, dynamic scaling, and universality. Phys. Rev. B 6(9), 3515 (1972)

    Article  Google Scholar 

  24. Haraux, A., Pham, T.S.: On the Lojasiewicz exponents of quasi-homogeneous functions. J. Singul. 11, 52–66 (2015)

    MathSciNet  MATH  Google Scholar 

  25. Hasanis, T., López, R.: Classification of separable surfaces with constant Gaussian curvature. Manuscr. Math. 166, 403–417 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Howell, L.L., Lang, R.J., Magleby, S.P., Zimmerman, T.K., Nelson, T.G.: Developable mechanisms on developable surfaces. Sci. Robot. 4(27), eaau5171 (2019)

    Article  Google Scholar 

  27. Hyatt, L.P., Magleby, S.P., Howell, L.L.: Developable mechanisms on right conical surfaces. Mech. Mach. Theory 149, 103813 (2020)

    Article  Google Scholar 

  28. Ioan, C.A., Ioan, G.: A generalization of a class of production functions. Appl. Econ. Lett. 18, 1777–1784 (2011)

    Article  Google Scholar 

  29. Inoue, T.: On the shape of the production possibility frontier with more commodities than primary factors. Intern. Econ. Review 25(2), 409–424 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  30. Inoue, T., Wegge, L.L.: On the geometry of the production possibility frontier. Intern. Econ. Review 27(3), 727–737 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  31. Jensen, B.: The Dynamic Systems of Basic Economic Growth Models, Mathematics and Its Applications. Springer, Dordrecht, Netherlands (1994)

    Book  Google Scholar 

  32. Kamke, E.: Losungmethoden und Losungen. B.G. Teubner, Stuttgart (1983)

    Google Scholar 

  33. Kemp, M.C., Khang, C., Uekawa, Y.: On the flatness of the transformation surface. J. Intern. Econ. 8(4), 537–542 (1978)

    Article  Google Scholar 

  34. Khatskevich, G.A., Pranevich, A.F.: On quasi-homogeneous production functions with constant elasticity of factors substitution. J. Belarus. State Univ. Econ. 1, 46–50 (2017)

    Google Scholar 

  35. Lawrence, S.: Developable surfaces: their history and application. Nexus Netw. J. 13, 701–714 (2011)

    Article  MATH  Google Scholar 

  36. Liu, H.: Translation surfaces with constant mean curvature in 3-dimensional spaces. J. Geom. 64, 141–149 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. López, R., Moruz, M.: Translation and homothetical surfaces in Euclidean space with constant curvature. J. Korean Math. Soc. 52(3), 523–535 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Losonczi, L.: Production functions having the CES property. Acta Math. Acad. Paedagog. Nyházi. N.S. 26(1), 113–125 (2010)

    MathSciNet  MATH  Google Scholar 

  39. Mak, K.-T.: General homothetic production correspondences. In: Dogramaci, A., Färe, R. (eds.) Applications of modern Production Theory: Efficiency and Productivity. Springer, Dordrecht, Netherlands (1988)

    Google Scholar 

  40. Mishra, S.K.: A brief history of production functions. IUP J. Manage. Econ. 8, 6–34 (2010)

    Google Scholar 

  41. Moruz, M., Munteanu, M.I.: Minimal translation hypersurfaces in \({E}^4\). J. Math. Anal. Appl. 439(2), 798–812 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Quevedo, H., Quevedo, M.N., Sánchez, A.: Quasi-homogeneous black hole thermodynamics. Eur. Phys. J. C 79, 229 (2019)

    Article  MATH  Google Scholar 

  43. Reynes, F.: The Cobb-Douglas function as a flexible function. A new perspective on homogeneous functions through the lens of output elasticities. Math. Soc. Sci. 97, 11–17 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  44. Shephard, R.: Some remarks on the theory of homogeneous production functions. Z. Nationalökon. 31, 251–256 (1971)

    Article  Google Scholar 

  45. Solow, R.M.: A contribution to the theory of economic growth. Q. J. Econ. 70(1), 65–94 (1956)

    Article  Google Scholar 

  46. Struik, D.J.: Lectures on Classical Differential Geometry. Dover, New York (1988)

    MATH  Google Scholar 

  47. Toponogov, V.A.: Differential geometry of curves and surfaces. A concise guide with the editorial assistance of Vladimir Y. Rovenski. Birkhäuser. Boston Inc, Boston, MA (2006)

    Google Scholar 

  48. Vîlcu, A.D., Vîlcu, G.-E.: On some geometric properties of the generalized CES production functions. Appl. Math. Comput. 218(1), 124–129 (2011)

    MathSciNet  MATH  Google Scholar 

  49. Vîlcu, A.D., Vîlcu, G.-E.: Some characterizations of the quasi-sum production models with proportional marginal rate of substitution. C. R. Math. Acad. Sci. Paris 353, 1129–1133 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. Vîlcu, A.D., Vîlcu, G.-E.: On quasi-homogeneous production functions. Symmetry 11(8), 1–11 (2019)

    Article  Google Scholar 

  51. Vîlcu, G.-E.: A geometric perspective on the generalized Cobb-Douglas production functions. Appl. Math. Lett. 24(5), 777–783 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  52. Vîlcu, G.-E.: On a generalization of a class of production functions. Appl. Econ. Lett. 25(2), 106–110 (2018)

    Article  Google Scholar 

  53. Wang, X.: A geometric characterization of homogeneous production models in economics. Filomat 30(13), 3465–3471 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  54. Wang, X., Fu, Y.: Some characterizations of the Cobb-Douglas and CES production functions in microeconomics, Abstr. Appl. Anal. 2013, Article ID 761832 (2013)

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Acknowledgements

The third author was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDI—UEFISCDI, project number PN-III-P4-ID-PCE-2020-0025, within PNCDI III.

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Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, Michigan, 48824-1027, USA

    Bang-Yen Chen

  2. Department of Information Technology, Mathematics and Physics, Petroleum-Gas University of Ploieşti, Bd. Bucureşti 39, Ploieşti, 100680, Romania

    Alina-Daniela Vîlcu

  3. “Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Bucharest, 050711, Romania

    Gabriel-Eduard Vîlcu

  4. Research Center in Geometry, Topology and Algebra, University of Bucharest, Str. Academiei 14, Bucharest, 70109, Romania

    Gabriel-Eduard Vîlcu

  5. Department of Mathematics and Informatics, University Politehnica of Bucharest, Splaiul Independenţei 313, Bucharest, 060042, Romania

    Gabriel-Eduard Vîlcu

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  1. Bang-Yen Chen
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  2. Alina-Daniela Vîlcu
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  3. Gabriel-Eduard Vîlcu
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Correspondence to Gabriel-Eduard Vîlcu.

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Chen, BY., Vîlcu, AD. & Vîlcu, GE. Classification of Graph Surfaces Induced by Weighted-Homogeneous Functions Exhibiting Vanishing Gaussian Curvature. Mediterr. J. Math. 19, 162 (2022). https://doi.org/10.1007/s00009-022-02106-2

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  • DOI: https://doi.org/10.1007/s00009-022-02106-2

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