Abstract
Monotonicity assumptions of preferences are natural and useful. A strictly monotonic preference is such that an increase in even only one commodity consumption is always strictly preferred. However, when we consider a continuum of commodities, it is not easy to find examples of strictly monotonic preferences. We survey some previous results in order to show that purely strictly monotonic preferences always exist but, if the commodity space is rich enough, they cannot be continuous in any linear topology defined on the consumption set and they cannot be represented by a utility function.
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Notes
- 1.
Following Debreu [9], a commodity is a good or a service completely specified physically, temporally, and spatially. The same good or service in different dates or locations is a different commodity. The date, the location or the quality of commodities could be treated as continuous variables. A consumption plan, or a bundle, is a specification for each commodity of the quantity that she will make available or that will be made available to her, i.e, a complete listing of the quantities of her inputs and of her outputs. The commodity space is the vector space that contains all possible consumption plans. The consumption set, for a given consumer, is the set of consumption plans that are available for that consumer.
- 2.
- 3.
See Kelley [20], pp. 48–49.
- 4.
See Steen and Seebach [32], pp. 71–72.
- 5.
The preference \(\succsim \) is locally insatiable if, for any point g in the consumption set X, and for any neighborhood V of f, there is another consumption \(f\in V\) such that \(f\succ g\).
- 6.
This is Theorem 6 on page 153 of Monteiro [25].
- 7.
Note that \(\geqslant _{K}\) is a total order on K, whereas \(\geqslant \) represent the natural partial order in the set of functions X.
- 8.
Linearity guarantee that the path \(t\rightarrow (1-t)a+tb\) joining a to b is continuous.
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Acknowledgements
We thank two anonymous referees for their carefully reading and helpful comments. Hervés-Beloso acknowledges the financial support of Research Grants ECO2016-75712-P (AEI/FEDER, UE) and RGEAF-ECOBAS (Xunta de Galicia). Monteiro acknowledges the financial support of CNPq–Brazil.
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Hervés-Beloso, C., Monteiro, P.K. (2020). Strictly Monotonic Preferences. In: Bosi, G., Campión, M., Candeal, J., Indurain, E. (eds) Mathematical Topics on Representations of Ordered Structures and Utility Theory. Studies in Systems, Decision and Control, vol 263. Springer, Cham. https://doi.org/10.1007/978-3-030-34226-5_10
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