<p>We know that the CES utility function approximates a Leontief utility function as in the following:</p>
<p><span class="math-container">$$u(x_1,x_2) = (x_1^{-r}+x_2^{-r})^{-1/r}\overset{r\rightarrow\infty}{\longrightarrow}\min\{x_1, x_2\}$$</span></p>
<p>However, this limit only holds for <span class="math-container">$x_i\ge0$</span>. Is there a CES-like utility function, which also holds for negative <span class="math-container">$x_i$</span> and large <span class="math-container">$r$</span>?</p>
<p>I am particularly interested in functions of the form <span class="math-container">$u(f_1(x_1),f_2(x_2))$</span> with <span class="math-container">$f_i(x_i) = \frac{x_i^{1-\eta_i}-1}{1-\eta_i}, \eta_i&gt;0$</span>. Adding a large constant to <span class="math-container">$f_i$</span> is not feasible since <span class="math-container">$f_i$</span> can get arbitrarily small (at least for <span class="math-container">$\eta_i\ge1$</span>). So for <span class="math-container">$f_i(x_i)&lt;0$</span>, the function <span class="math-container">$u(f_1(x_1),f_2(x_2))$</span> does not behave like <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> are complements. Is there a different formulation of CES utility which does not have this problem?</p>

CC BY-SA 4.0
2021-08-13T13:55:56.690Z
You want to be able to hold negative amounts of $x_i$, but do you also want to have positive amounts? If everyone wants negative amounts, if $x_i$ is a "bad", maybe you can just define $y_i=-x_i$ and proceed as normal.
BKay
CC BY-SA 4.0
2021-08-16T11:13:15.323Z
Yes, unfortunately positive and negative values have to be accounted for. In my case, I am constructing a nested utility function, so the interpretation of $x_i$ as good or "bad" is not so straightforward anymore. If this family of utility functions would be defined only on one half plane, some trickery might be possible. But here, $f_i(x_i)$ are defined on all $\mathbb{R}$...
jabberwocky
CC BY-SA 4.0
2021-08-16T14:08:23.037Z
Elasticities are hard to interpret in situations where quantities are sometimes negative and sometimes positive. See for example this question: 
https://economics.stackexchange.com/questions/29696/how-do-you-use-a-log-linear-model-when-you-have-negative-xs
BKay
CC BY-SA 4.0
2021-08-19T11:12:17.470Z
That's good to know, thanks! But does it prevent us from defining sensible elasticities of substitution for these cases? Or is it inherently impossible because we have
$$u(x_1,x_2) \overset{\eta\rightarrow0^+}\longrightarrow\min\{x_1,x_2\},\\
u(x_1,x_2) \overset{\eta\rightarrow0^-}\longrightarrow\max\{x_1,x_2\}$$
(for $-r = \frac{\eta-1}{\eta}$)?
jabberwocky
<utility><leontief><ces-function>
CES utility with negative values
jabberwocky
# CES utility with negative values

(Asked by: username_0 on 2021-08-12)

We know that the CES utility function approximates a Leontief utility function as in the following:

$$u(x\_1,x\_2) = (x\_1^{-r}+x\_2^{-r})^{-1/r}\overset{r\rightarrow\infty}{\longrightarrow}\min\{x\_1, x\_2\}$$

However, this limit only holds for $x\_i\ge0$. Is there a CES-like utility function, which also holds for negative $x\_i$ and large $r$?

I am particularly interested in functions of the form $u(f\_1(x\_1),f\_2(x\_2))$ with $f\_i(x\_i) = \frac{x\_i^{1-\eta\_i}-1}{1-\eta\_i}, \eta\_i>0$. Adding a large constant to $f\_i$ is not feasible since $f\_i$ can get arbitrarily small (at least for $\eta\_i\ge1$). So for $f\_i(x\_i)<0$, the function $u(f\_1(x\_1),f\_2(x\_2))$ does not behave like $x\_1$ and $x\_2$ are complements. Is there a different formulation of CES utility which does not have this problem?

**Tags:** utility, leontief, ces-function

> **Comments:**
> **username_1**: You want to be able to hold negative amounts of $x_i$, but do you also want to have positive amounts? If everyone wants negative amounts, if $x_i$ is a "bad", maybe you can just define $y_i=-x_i$ and proceed as normal.  
> **username_0**: Yes, unfortunately positive and negative values have to be accounted for. In my case, I am constructing a nested utility function, so the interpretation of $x_i$ as good or "bad" is not so straightforward anymore. If this family of utility functions would be defined only on one half plane, some trickery might be possible. But here, $f_i(x_i)$ are defined on all $\mathbb{R}$...  
> **username_1**: Elasticities are hard to interpret in situations where quantities are sometimes negative and sometimes positive. See for example this question:  https://economics.stackexchange.com/questions/29696/how-do-you-use-a-log-linear-model-when-you-have-negative-xs  
> **username_0**: That's good to know, thanks! But does it prevent us from defining sensible elasticities of substitution for these cases? Or is it inherently impossible because we have $$u(x_1,x_2) \overset{\eta\rightarrow0^+}\longrightarrow\min\{x_1,x_2\},\\ u(x_1,x_2) \overset{\eta\rightarrow0^-}\longrightarrow\max\{x_1,x_2\}$$ (for $-r = \frac{\eta-1}{\eta}$)?  

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## 0 answers

