Abstract
Let g and u be C 1-class real-valued functions that satisfy the Lagrange multiplier condition Du = λ g and Du ≠ 0. In this paper, we show that u is quasi-concave if and only if g satisfies an inequality which is related to the Bordered Hessian condition even if both of u and g are C 1 rather than C 2.
JEL Classification: D11
Mathematics Subject Classification (2010): 91B08, 91B16
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Notes
- 1.
A real-valued function f defined on the convex set is said to be quasi-concave if every upper contour set {x | f(x) ≥ a} is convex.
- 2.
This condition is known as the Bordered Hessian condition.
- 3.
Debreu [1] introduces an example of C 1-class function g such that it satisfies the integrability condition and there is no C 2-function u such that Du = λ g for some C 1-function λ. Hence, our extension is meaningful.
References
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Hosoya, Y.: Elementary form and proof of the Frobenius theorem for economists. Adv. Math. Econ. 16, 39–52 (2012)
Otani, K.: A characterization of quasi-concave functions. J. Econ. Theory 31, 194–196 (1983)
Acknowledgements
We are grateful to Shinichi Suda and Toru Maruyama for their helpful comments and suggestions.
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Hosoya, Y. (2014). A Characterization of Quasi-concave Function in View of the Integrability Theory. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics Volume 18. Advances in Mathematical Economics, vol 18. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54834-8_4
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DOI: https://doi.org/10.1007/978-4-431-54834-8_4
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