# Infinite marginal utility for vanishing consumption: Is this always true?

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.

• 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. Commented yesterday

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.