<p>From &quot;<a href="https://www.online-vwl.de/english/micro/dynvwlengse29.html#x54-6300010.3" rel="nofollow noreferrer">Dynamic economics An online textbook with dynamic graphics for the introduction to economics</a>&quot; by Prof. Dr. Christian Bauer:</p>
<blockquote>
<p>A function <span class="math-container">$f:\mathbb{R} \to \mathbb{R},(x,y)\to f(x,y)$</span> is called homogeneous of the degree <span class="math-container">$n\in \mathbb{R}$</span>, if for all <span class="math-container">$(x,y)\in\mathbb{R}^2$</span> the following is valid:
<span class="math-container">$$
f(kx,ky) = k^n f(x,y) \ \forall k\in\mathbb{R}_0^+
$$</span></p>
</blockquote>
<p>A <a href="https://economics.stackexchange.com/questions/40682/homogeneous-utilities-anything-other-than-ces">lot of utility functions</a> (all CES, including C-D) have this property. It is also well-known that for <a href="https://economics.stackexchange.com/questions/8519/how-to-show-that-a-homothetic-utility-function-has-demand-functions-which-are-li">functions with this property the demand for the goods is linear in income</a>, thus none will be 'more necessary' than another.</p>
<p>You could use something like quasilinear utility, which does not have this property:
<span class="math-container">$$
U(x,y) = v(x) + y
$$</span>
where <span class="math-container">$v()$</span> is usually a concave function with an initially steep slope, e.g.; <span class="math-container">$\ln()$</span>. In this case below a certain income level all income is spent on <span class="math-container">$x$</span>, above the income level no income is spent on <span class="math-container">$x$</span>. The amount consumed also depends on the price ratio. Whether such a simple function is useful for your model depends on what you are trying to achieve. This type of function is often used in IO literature, though personally I think its use is not always justified.</p>

CC BY-SA 4.0
2023-12-15T11:32:58.250Z
Thank you! What is "IO" literature? Also, would you be able to supplement your answer with a brief discussion of [Stone-Geary utility](https://en.wikipedia.org/wiki/Stone%E2%80%93Geary_utility_function)? I would just like some intuition on how the quasilinear approach differs from CD with a subsistence level.
Anthony
CC BY-SA 4.0
2023-12-15T11:36:32.323Z
IO stands for Industrial Organization. I will defer to others on questions of Stone-Geary utility, I don't know enough about it :)
Giskard
CC BY-SA 4.0
2023-12-15T11:16:10.893Z
2023-12-15T11:37:36.803Z
2023-12-15T11:37:36.803Z
Giskard
<p>I am trying to formulate a decision problem for an agent involving their heating-energy consumption <span class="math-container">$c$</span>. Let <span class="math-container">$x$</span> denote all other consumption. What would be a reasonable utility function to employ here? I want to capture the intuition that heating energy is a necessary good, whereas everything else might be modelled as a normal good.</p>
<p>Something like Cobb-Douglas doesn't work since <span class="math-container">$x$</span> and <span class="math-container">$c$</span> would then both be normal goods. Is CES utility an option? Would be grateful for any references that consider similar setups.</p>

CC BY-SA 4.0
2023-12-15T10:17:56.457Z
Hi! Do you know what a homothetic function is? Also, have you studied microeconomics? I am guessing yes, but I want to make sure the answer is digestible.
Giskard
CC BY-SA 4.0
2023-12-15T10:29:01.187Z
Hi @Giskard I'm coming more from an optimisation background, so the more straightforward the explanation the better! Thanks.
Anthony
CC BY-SA 4.0
2023-12-16T02:49:04.243Z
Also see this: https://x.com/amit_k_goyal/status/1717156615315427769?s=20
Amit
<utility><consumer-theory>
Utility function for a combination of a normal good and necessary good
Anthony
# Utility function for a combination of a normal good and necessary good

(Asked by: username_0 on 2023-12-15)

I am trying to formulate a decision problem for an agent involving their heating-energy consumption $c$. Let $x$ denote all other consumption. What would be a reasonable utility function to employ here? I want to capture the intuition that heating energy is a necessary good, whereas everything else might be modelled as a normal good.

Something like Cobb-Douglas doesn't work since $x$ and $c$ would then both be normal goods. Is CES utility an option? Would be grateful for any references that consider similar setups.

**Tags:** utility, consumer-theory

> **Comments:**
> **username_1**: Hi! Do you know what a homothetic function is? Also, have you studied microeconomics? I am guessing yes, but I want to make sure the answer is digestible.  
> **username_0**: Hi @username_1 I'm coming more from an optimisation background, so the more straightforward the explanation the better! Thanks.  
> **username_2**: Also see this: https://x.com/amit_k_goyal/status/1717156615315427769?s=20  

---

## 1 answers

### Answer by username_1 on 2023-12-15. Score: 2 | Accepted answer

From "[Dynamic economics An online textbook with dynamic graphics for the introduction to economics](https://www.online-vwl.de/english/micro/dynvwlengse29.html#x54-6300010.3)" by Prof. Dr. Christian Bauer:

> A function $f:\mathbb{R} \to \mathbb{R},(x,y)\to f(x,y)$ is called homogeneous of the degree $n\in \mathbb{R}$, if for all $(x,y)\in\mathbb{R}^2$ the following is valid:
> $$
> f(kx,ky) = k^n f(x,y) \ \forall k\in\mathbb{R}\_0^+
> $$

A [lot of utility functions](https://economics.stackexchange.com/questions/40682/homogeneous-utilities-anything-other-than-ces) (all CES, including C-D) have this property. It is also well-known that for [functions with this property the demand for the goods is linear in income](https://economics.stackexchange.com/questions/8519/how-to-show-that-a-homothetic-utility-function-has-demand-functions-which-are-li), thus none will be 'more necessary' than another.

You could use something like quasilinear utility, which does not have this property:
$$
U(x,y) = v(x) + y
$$
where $v()$ is usually a concave function with an initially steep slope, e.g.; $\ln()$. In this case below a certain income level all income is spent on $x$, above the income level no income is spent on $x$. The amount consumed also depends on the price ratio. Whether such a simple function is useful for your model depends on what you are trying to achieve. This type of function is often used in IO literature, though personally I think its use is not always justified.

> **Comments:**
> **username_0**: Thank you! What is "IO" literature? Also, would you be able to supplement your answer with a brief discussion of [Stone-Geary utility](https://en.wikipedia.org/wiki/Stone%E2%80%93Geary_utility_function)? I would just like some intuition on how the quasilinear approach differs from CD with a subsistence level.  
> **username_1**: IO stands for Industrial Organization. I will defer to others on questions of Stone-Geary utility, I don't know enough about it :)  

---

