I'm a calculus teacher trying to construct a realistic example of a smooth utility function $U(x,y)$ that has a local maximum at some point $(x_0,y_0)$. This requires two goods, X and Y, such that the single-variable functions $U(x_0,y)$ and $U(x,y_0)$ have local maxima at $y_0$ and $x_0$, respectively. I can't think of a meaningful example of a good/service where, after acquiring a certain amount, the utility actually decreases as more of the good/service is acquired. This condition seems quite strange, but perhaps there are some standard examples I've never heard of given my thin background in economics.
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1$\begingroup$ The quantities in the utility function are the amounts of the goods consumed, not the amounts of the goods aquired. I challenge you to name a food where you have no satiation quantity. $\endgroup$– GiskardCommented Apr 21, 2022 at 20:16
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$\begingroup$ Also, I am not quite sure what you mean by "realistic example". $\endgroup$– GiskardCommented Apr 21, 2022 at 20:20
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Such a point where utility is a local maxima is called a satiating point. Think of a good that turns into a bad beyond a certain amount. Most non-tradable consumables goods are of such kind, where beyond a certain point you don't need them, and they may only create inventory or wastage costs.
Usually though, we don't talk much about satiating points, as a utility maximizing individual in making a choice of a consumption bundle will never optimally choose to buy a bundle that had a negative marginal utility for any of the goods in the bundle. Or given an initial endowment, one can reformulate the utility maximizing problem facing the consumer in the way that does not need to consider the satiating point.
