Abstract
We consider a simple model of decision under uncertainty, in which a representative risk-averse agent maximizes the expected utility of a random wealth. The wealth is postulated in a quite general form, especially concerning the effect of the decision variable, so that other decision problems under uncertainty can be considered as particular cases of this model. In this general framework, we propose a new method to easily obtain both properties of the optimal solution and comparative-static effects. We illustrate the usefulness of this formulation by applying it to some models from the theory of the firm under uncertainty. In these models we are able to easily derive their key properties, and also new results.
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Notes
- 1.
- 2.
In order to simplify the notation, if possible we will write simply W instead of W(x).
- 3.
Actually, we would only have that \(g^{{\prime\prime}}f^{{\prime}}- g^{{\prime}}f^{{\prime\prime}}\geqslant 0\), because \(f^{{\prime\prime}}\leqslant 0\) and \(g^{{\prime\prime}}\geqslant 0\). But if f ′ ′ and g ′ ′ were simultaneously null (so that the function h turns out to be constant), the problem (5.5) would admit the solution x c ∗ if and only if \(I = \left [0,x_{\mathrm{c}}^{{\ast}}\right ]\), and in this case the result would also hold.
- 4.
We are implicitly assuming that there is a full loss offset. See [7, p. 70] for a discussion about the plausibility of this hypothesis at least in the case of a firm.
- 5.
As usual in the literature, these acronyms mean increasing, constant or decreasing (respectively) relative risk aversion.
- 6.
We specify here a slightly simplified version of the original model presented in [3]. For our purposes, that will be enough.
References
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Acknowledgements
Álvarez-López would like to thank the financial support provided by the Spanish Interministerial Commission of Science and Technology (CICYT: Comisión Interministerial de Ciencia y Tecnología), under the Projects with the reference numbers ECO2008-06395-C05-03 and ECO2012-39553-C04-01.
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Álvarez-López, A.A., Rodríguez-Puerta, I., Sebastiá-Costa, F., Buendía, M. (2014). Properties and Comparative-Static Effects in Models of Decision Under Uncertainty: Applications to the Theory of the Firm. In: Pinto, A., Zilberman, D. (eds) Modeling, Dynamics, Optimization and Bioeconomics I. Springer Proceedings in Mathematics & Statistics, vol 73. Springer, Cham. https://doi.org/10.1007/978-3-319-04849-9_5
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