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I have the following not locally non-satiated utility function:

$$U(x,y)=-(x-1)^2-(y-2)^2$$

where $U(x,y): \, \!R^n_+ \rightarrow \!R $

The 3D plot of this function is an infinite paraboloid; therefore, $U(x,y)$ can be considered as a not locally non-satiated utility function.

After I've found the Marshallian demand and the indirect utility functions, I'd like to determine the Hicksian demand curve and the expenditure functions using the duality theorem. Can be the duality theorem applied to this type of not locally non-satiated utility functions?

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This system doesn´t have solution, because the consumer want to maximize their utility, so there is not a possible solution because is better consume x=0 that any other bundle

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    $\begingroup$ (-1) This answer does not really address the question, and it is also incorrect. $\endgroup$
    – Giskard
    Sep 27, 2019 at 19:31

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