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I am considering a utility function over a single good for an intertemporal allocation problem. The class of power utility functions $$ u(x) = \frac{x^\gamma}{\gamma} $$ where $\gamma < 1$, $\gamma \neq 0$ has derivative $$ u'(x) = x^{\gamma-1} $$ and thus satisfies $$ \lim_{x \to 0} u'(x) = \infty\ . $$ This makes sense to me intuitively: if I am not consuming anything at all, the marginal utility I gain from increasing my consumption even a little bit will be unbounded.

I would like to know if this holds in general. That is, can an argument be made that marginal utility must be unbounded at zero for every (single-good) utility function? References are appreciated.

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  • $\begingroup$ The intuition of never having 0 consumption is more likely to hold when considering a composite good than some particular good. Which is ofte the case in intertemporal problems. $\endgroup$ Commented yesterday

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The reason for using a power utility function is typically not that it fits some intuition (and I also don't think this intuition is particularly convincing). It is rather that these utility functions satisfy the Inada conditions and thus equations of the form $u'(x)=c$ have a unique solution for all $c>0$. So it's an assumption made mostly for convenience and simplicity.

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