<p>The only utility function that comes to mind is the Stone-Geary utility function. For 2 goods, <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, this takes the form:
<span class="math-container">$$
u(x,y) = (x - a)^\alpha (y- b)^{1- \alpha}.
$$</span>
This is a Cobb-Douglas type of utility function where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are subsistence levels, i.e. you need to consume at least <span class="math-container">$a$</span> from <span class="math-container">$x$</span> and <span class="math-container">$b$</span> from <span class="math-container">$y$</span> to survive. It is the utility function that leads to the Linear expenditure system.</p>
<p>To see that it is linear homothetic notice that:
<span class="math-container">$$
\begin{align*}
u(\beta \tilde x + a, \beta \tilde y + b) &amp;= (\beta \tilde x + a - a)^\alpha (\beta \tilde y + b - b)^{1-\alpha},\\
&amp;=\beta (\tilde x)^\alpha (\tilde y)^{1-\alpha},\\
&amp;=\beta ((\tilde x + a) - a)^\alpha ((\tilde y + b) - b)^{1-\alpha},\\
&amp;= \beta u(\tilde x + a, \tilde y + b).
\end{align*}
$$</span>
It could be that there is more work on utility functions with subsistence levels that lead to other preferences that are also linear homothetic.</p>
<p>For example you could define a CES utility function with subsistence levels:
<span class="math-container">$$
u(x,y) = (\alpha_x(x -a)^\sigma + \alpha_y(y-b)^\sigma)^{1/\sigma}
$$</span>
This will also satisfy linear homogeneity. <a href="https://link.springer.com/article/10.1007/s10640-015-9976-z" rel="noreferrer">This paper of Baumgärtner, Drupp &amp; Quaas</a> does something like this.</p>
<p>In general, if you take any homothetic utility function <span class="math-container">$u(x,y)$</span> then the modified 'subsistence-augmented' function:
<span class="math-container">$$
\tilde u(x, y) \equiv u(x - a, y - b),
$$</span>
will be linear homothetic.</p>

CC BY-SA 4.0
2021-05-08T18:06:16.460Z
2021-05-08T19:15:00.913Z
2021-05-08T19:15:00.913Z
tdm
<p>A Homothetic Utility is where
<span class="math-container">$$
\forall x,y, \forall a \in \mathbb{R}_+: \ u(ax,ay)=au(x,y) 
$$</span> (or its monotonic transformation).</p>
<p>A linear Homothetic utility is defined as <span class="math-container">$$
\forall x,y, \forall a \in \mathbb{R}_+: \ u(ax+b,ay+c)=au(x+b,y+c) 
$$</span>
where <span class="math-container">$b,c$</span> are constants.</p>
<p>This preference has very similar property as the homothetic preference. In fact, if we simply translate the coordinate system in the direction of (b,c), then the preference becomes homothetic.</p>
<p>Are there any works covering this property? I've checked a lot of theory papers in homothetic preference but found no luck.</p>
<blockquote>
<p>Homothetic Preferences by James DOW· and Sergio Ribeiro da Costa
WERLANG</p>
<p>Homothetic and weakly homothetic preferences by J.C. Candeal, E.
Indurain</p>
<p>Linear-homothetic  preferences, by B Datta, H Dixon</p>
</blockquote>

CC BY-SA 4.0
2019-05-27T23:53:53.667Z
in the first line, should $au(x +y)$ be $au(x,y)$, as $u$ seems to take 2 arguments?
201p
CC BY-SA 4.0
2019-05-28T00:03:09.287Z
@201p You are right, that was a typo
High GPA
CC BY-SA 4.0
2019-05-30T14:26:45.620Z
@Giskard You are right about a,x,y. b,c are constants
High GPA
CC BY-SA 4.0
2019-05-30T17:13:39.337Z
Can you give an example of a function $u \neq 0$ satisfying this identity?
Bertrand
CC BY-SA 4.0
2019-06-02T10:44:23.380Z
If you want to allow for $(b,c)$-translations, your then the preference should instead satisfy
$$
\forall x,y, \forall a \in \mathbb{R}_+: \ u(a(x+b),a(y+c))=au(x+b,y+c), 
$$
but this is equivalent to homotheticity.
Bertrand
<microeconomics><mathematical-economics><utility><supply-and-demand><preferences>
Linear Homothetic Utility
High GPA
# Linear Homothetic Utility

(Asked by: username_0 on 2019-05-27)

A Homothetic Utility is where
$$
\forall x,y, \forall a \in \mathbb{R}\_+: \ u(ax,ay)=au(x,y)
$$ (or its monotonic transformation).

A linear Homothetic utility is defined as $$
\forall x,y, \forall a \in \mathbb{R}\_+: \ u(ax+b,ay+c)=au(x+b,y+c)
$$
where $b,c$ are constants.

This preference has very similar property as the homothetic preference. In fact, if we simply translate the coordinate system in the direction of (b,c), then the preference becomes homothetic.

Are there any works covering this property? I've checked a lot of theory papers in homothetic preference but found no luck.

> Homothetic Preferences by James DOW· and Sergio Ribeiro da Costa
> WERLANG
>
> Homothetic and weakly homothetic preferences by J.C. Candeal, E.
> Indurain
>
> Linear-homothetic preferences, by B Datta, H Dixon

**Tags:** microeconomics, mathematical-economics, utility, supply-and-demand, preferences
> **Comments:**
> **username_1**: in the first line, should $au(x +y)$ be $au(x,y)$, as $u$ seems to take 2 arguments?  
> **username_0**: @201p You are right, that was a typo  
> **username_0**: @Giskard You are right about a,x,y. b,c are constants  
> **username_2**: Can you give an example of a function $u \neq 0$ satisfying this identity?  
> **username_2**: If you want to allow for $(b,c)$-translations, your then the preference should instead satisfy $$ \forall x,y, \forall a \in \mathbb{R}_+: \ u(a(x+b),a(y+c))=au(x+b,y+c),  $$ but this is equivalent to homotheticity.  

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## 1 answers

### Answer by username_3 on 2021-05-08. Score: 7 | Accepted answer

The only utility function that comes to mind is the Stone-Geary utility function. For 2 goods, $x$ and $y$, this takes the form:
$$
u(x,y) = (x - a)^\alpha (y- b)^{1- \alpha}.
$$
This is a Cobb-Douglas type of utility function where $a$ and $b$ are subsistence levels, i.e. you need to consume at least $a$ from $x$ and $b$ from $y$ to survive. It is the utility function that leads to the Linear expenditure system.

To see that it is linear homothetic notice that:
$$
\begin{align\*}
u(\beta \tilde x + a, \beta \tilde y + b) &= (\beta \tilde x + a - a)^\alpha (\beta \tilde y + b - b)^{1-\alpha},\\
&=\beta (\tilde x)^\alpha (\tilde y)^{1-\alpha},\\
&=\beta ((\tilde x + a) - a)^\alpha ((\tilde y + b) - b)^{1-\alpha},\\
&= \beta u(\tilde x + a, \tilde y + b).
\end{align\*}
$$
It could be that there is more work on utility functions with subsistence levels that lead to other preferences that are also linear homothetic.

For example you could define a CES utility function with subsistence levels:
$$
u(x,y) = (\alpha\_x(x -a)^\sigma + \alpha\_y(y-b)^\sigma)^{1/\sigma}
$$
This will also satisfy linear homogeneity. [This paper of Baumgärtner, Drupp & Quaas](https://link.springer.com/article/10.1007/s10640-015-9976-z) does something like this.

In general, if you take any homothetic utility function $u(x,y)$ then the modified 'subsistence-augmented' function:
$$
\tilde u(x, y) \equiv u(x - a, y - b),
$$
will be linear homothetic.


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