<p>Iso-elastic utility is defined as a function $U(w)$ where for all $k \gt 0$ you have $$U(kw)=f(k)U(w)+g(k)$$ for some functions $f(k)$ and $g(k)$ independent of $w$, i.e. an affine transformation of the original utility function</p>

<p>This is <em>iso-</em> because this kind of utility function leads to identical (or at least proportionate) decisions: it means that the optimal risk/reward behaviour at given level of wealth $w$ is also optimal for another level of wealth $kw$ if you multiply all the original investments by $k$.  For example, if at some wealth level you would put a third of it into a particular risky investment and two-thirds into a particular safe investment, then with an iso-elastic utility function you would do the same with at any wealth level   </p>

<p>Both your formulations are iso-elastic in this sense, since they only differ by a constant $\frac{1}{1-\mu}$ and so lead to identical distribution decisions.  The difference between them is that the first has $u(0)=0$ and $u(1)=\frac{1}{1-\mu}$ while the second has $u(1)=0$, and since that difference has no real-life implications it might be unreasonable to claim the two utility functions had different properties  </p>

<p>Those who use the second formulation often associate it with a logarithmic utility function as the limit function as $\mu \to 1$.  This limit cannot be done without subtracting the $\frac{1}{1-\mu}$ (no longer a constant, since $\mu$ is changing), so they take the second formulation for arithmetical convenience</p>

CC BY-SA 3.0
2016-12-21T07:13:19.360Z
"This is iso- because..." yes but why is it elastic? By the same reasoning I could also call it *isocost* utility.
Giskard
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2016-12-21T09:34:25.817Z
You regard constant elasticity as something which, for example, leads to a constant proportionate change in demand for a given proportionate income change no matter what income you start from, or a constant proportionate change in supply for a given proportionate price change no matter what price you start from.  Both are observable behaviours if true.  These isoelastic utility functions lead to a constant proportionate distribution of wealth as wealth changes no matter what level of wealth you start from, again an observable behaviour if true. But I did not invent the name
Henry
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2016-12-21T20:54:11.287Z
@denesp look at Henry's comment. He forgot to tag you, I think.
An old man in the sea.
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2016-12-21T23:58:23.500Z
@Anoldmaninthesea. Thank you. I saw it, I think the system sends a notification to the previous commenter.
Giskard
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2016-12-21T01:04:43.407Z
2016-12-21T01:11:12.810Z
2016-12-21T01:11:12.810Z
Henry
<p>If $u(x)=\frac{x^{1-\mu}}{1-\mu}$, then the elasticity defined as $\frac{\partial u}{\partial x}\frac{x}{u}=1-\mu$.</p>

<p>However, when we define $u(x)=\frac{x^{1-\mu}-1}{1-\mu}$ (usual def. for isoelastic utility), we have $\frac{\partial u}{\partial x}\frac{x}{u}=(1-\mu)\frac{x^{1-\mu}}{x^{1-\mu}-1}$. And this doesn't seem to give a constant elasticity. So, why do we call this utility isoelastic?</p>

<p>Any help would be appreciated.</p>

<microeconomics>
Why the name for isoelastic utility?
An old man in the sea.
# Why the name for isoelastic utility?

(Asked by: username_0 on 2016-12-20)

If $u(x)=\frac{x^{1-\mu}}{1-\mu}$, then the elasticity defined as $\frac{\partial u}{\partial x}\frac{x}{u}=1-\mu$.

However, when we define $u(x)=\frac{x^{1-\mu}-1}{1-\mu}$ (usual def. for isoelastic utility), we have $\frac{\partial u}{\partial x}\frac{x}{u}=(1-\mu)\frac{x^{1-\mu}}{x^{1-\mu}-1}$. And this doesn't seem to give a constant elasticity. So, why do we call this utility isoelastic?

Any help would be appreciated.

**Tags:** microeconomics

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## 1 answers

### Answer by username_1 on 2016-12-21. Score: 3 | Accepted answer

Iso-elastic utility is defined as a function $U(w)$ where for all $k \gt 0$ you have $$U(kw)=f(k)U(w)+g(k)$$ for some functions $f(k)$ and $g(k)$ independent of $w$, i.e. an affine transformation of the original utility function

This is *iso-* because this kind of utility function leads to identical (or at least proportionate) decisions: it means that the optimal risk/reward behaviour at given level of wealth $w$ is also optimal for another level of wealth $kw$ if you multiply all the original investments by $k$. For example, if at some wealth level you would put a third of it into a particular risky investment and two-thirds into a particular safe investment, then with an iso-elastic utility function you would do the same with at any wealth level

Both your formulations are iso-elastic in this sense, since they only differ by a constant $\frac{1}{1-\mu}$ and so lead to identical distribution decisions. The difference between them is that the first has $u(0)=0$ and $u(1)=\frac{1}{1-\mu}$ while the second has $u(1)=0$, and since that difference has no real-life implications it might be unreasonable to claim the two utility functions had different properties

Those who use the second formulation often associate it with a logarithmic utility function as the limit function as $\mu \to 1$. This limit cannot be done without subtracting the $\frac{1}{1-\mu}$ (no longer a constant, since $\mu$ is changing), so they take the second formulation for arithmetical convenience

> **Comments:**
> **username_2**: "This is iso- because..." yes but why is it elastic? By the same reasoning I could also call it *isocost* utility.  
> **username_1**: You regard constant elasticity as something which, for example, leads to a constant proportionate change in demand for a given proportionate income change no matter what income you start from, or a constant proportionate change in supply for a given proportionate price change no matter what price you start from.  Both are observable behaviours if true.  These isoelastic utility functions lead to a constant proportionate distribution of wealth as wealth changes no matter what level of wealth you start from, again an observable behaviour if true. But I did not invent the name  
> **username_0**: @denesp look at Henry's comment. He forgot to tag you, I think.  
> **username_2**: @Anoldmaninthesea. Thank you. I saw it, I think the system sends a notification to the previous commenter.  

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