Abstract
This note shows that a utility function of a homothetic preference relation satisfying u(0) = 0 is a least concave utility function if and only if it is homogeneous of degree one.
Received: July 24, 2010
Received: October 22, 2010
JEL classification: D11
Mathematics Subject Classification (2010): 91B16, 91B08
We are grateful to Toru Maruyama and an anonymous referee for helpful comments and suggestions.
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Notes
- 1.
The definition of homothetic preference is in the next section.
- 2.
The uniqueness up to a positive affine transformation means the following fact: if both u 1 and u 2 are the least concave utility function, then there exist some a > 0 and b ∈ ℝ such that \({u}_{1} = a{u}_{2} + b\).
- 3.
Since Ω is convex and w is continuous, w(Ω) must be connected. In general, any connected subset of ℝ is convex. Therefore, we have w(Ω) is convex.
- 4.
The condition P ≿ ≠∅ is so mild that we could not find any example in which P ≿ = ∅, except a trivial example: P ≿ = ∅ if x ∼ y for any x, y ∈ Ω.
- 5.
This result is partially shown in Kihlstrom and Mirman [3].
- 6.
It can be shown by the same argument as the proof of Proposition 3.C.1 of Mas-Colell, Whinston and Green [4].
- 7.
It can be verify by the same argument as Exercise 2-1 of Stokey and Lucas [5].
References
Debreu, G.: Least concave utility functions. J. Math. Econ. 3, 121–129 (1976)
Kannai, Y.: The ALEP definition of complementarity and least concave utility functions. J. Econ. Theory 22, 115–117 (1980)
Kihlstrom, R.E., Mirman, L.J.: Constant, increasing, and decreasing risk aversion with many commodities. Rev. Econ. Stud. 48, 271–280 (1981)
Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, Oxford (1995)
Stokey, N., Lucas, R.: Recursive Methods in Economic Dynamics. Harvard University Press, Cambridge, MA (1989)
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Hosoya, Y. (2011). An existence result and a characterization of the least concave utility of homothetic preferences. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 15. Springer, Tokyo. https://doi.org/10.1007/978-4-431-53930-8_6
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