I understand that monotonic preferences imply non-satiation. But I am not sure 100% if non-monotonic functions always have satiation. An intuitive and mathematical explanation would be very helpful.
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1$\begingroup$ Defining the exact type of monotonicity and satiation you mean may be helpful. $\endgroup$ – Giskard Oct 18 at 6:01
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1$\begingroup$ Are you asking about local satiation or global satiation? Are you asking about strongly monotonic or weakly monotonic functions and their complements? $\endgroup$ – Henry Oct 18 at 7:43
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$\begingroup$ I am asking about local-satiation and strongly monotonic functions. $\endgroup$ – Frodo Baggins Oct 18 at 14:43
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$\begingroup$ Then my answer does not answer your question. $\endgroup$ – Giskard Oct 19 at 9:19
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It depends on the function. A non-monotonic function with satiation: $$ U(x_1,x_2) = 0. $$ A non-monotonic function without (global) satiation: $$ U(x_1,x_2) = \left \lfloor{x_1}\right \rfloor + \left \lfloor{x_2}\right \rfloor . $$
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$\begingroup$ Does non-satiation imply monotonicity always? $\endgroup$ – Frodo Baggins Oct 22 at 3:48
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$\begingroup$ I wouldn't have asked these questions had I not first tried myself. You may be experienced in this field, but I am a novice. So, if you don't want to answer that fine, but I would politely remind you not to impede others from asking pertinent questions. $\endgroup$ – Frodo Baggins Oct 22 at 6:05
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$\begingroup$ @FrodoBaggins What exactly have you tried? Could you edit some of your efforts into your questions? Into this one and also any other questions you may want to ask. It would help us understand better where you get stuck. If it is the mathematical notation that you find difficult I recommend reading a math textbook on analysis. $\endgroup$ – Giskard Oct 22 at 8:08
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$\begingroup$ I haven't asked homework questions, but merely conceptual questions. Plus, there is no limit on the number of questions anyone can ask in the forum. $\endgroup$ – Frodo Baggins Oct 22 at 15:04
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A non-monotonic function with local non-satiation: $$u(x, y) = x - y$$
