<p>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 <span class="math-container">$u'(x)=c$</span> have a unique solution for all <span class="math-container">$c&gt;0$</span>. So it's an assumption made mostly for convenience and simplicity.</p>

CC BY-SA 4.0
2024-07-18T08:15:29.210Z
Thanks, but this doesn't really answer my question. I would like to understand if unbounded marginal utility at $0$ consumption is a reasonable boundary condition in general, since I would like to use this property in another context.
Anthony
CC BY-SA 4.0
2024-07-12T12:35:37.783Z
2024-07-12T12:35:37.783Z
VARulle
<p>I am considering a utility function over a single good for an intertemporal allocation problem. The class of power utility functions
<span class="math-container">$$
u(x) = \frac{x^\gamma}{\gamma}
$$</span>
where <span class="math-container">$\gamma &lt; 1$</span>, <span class="math-container">$\gamma \neq 0$</span> has derivative
<span class="math-container">$$
 u'(x) = x^{\gamma-1}
$$</span>
and thus satisfies
<span class="math-container">$$
 \lim_{x \to 0} u'(x) = \infty\ .
$$</span>
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.</p>
<p>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.</p>

CC BY-SA 4.0
2024-07-12T15:48:10.167Z
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.
Jesper Hybel
<reference-request><utility><consumer-theory>
Infinite marginal utility for vanishing consumption: Is this always true?
Anthony
# Infinite marginal utility for vanishing consumption: Is this always true?

(Asked by: username_0 on 2024-07-09)

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.

**Tags:** reference-request, utility, consumer-theory

> **Comments:**
> **username_1**: 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.  

---

## 1 answers

### Answer by username_2 on 2024-07-12. Score: 3

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.

> **Comments:**
> **username_0**: Thanks, but this doesn't really answer my question. I would like to understand if unbounded marginal utility at $0$ consumption is a reasonable boundary condition in general, since I would like to use this property in another context.  

---

