<p>It depends on the definition of homotheticity. One definition states that consumer's preferences are homotethic if and only if his or her preferences can be represented by a utility function which is homogenous of degree 1. </p>

<p>For example, if we have two goods $x$ and $y$, then the utility function $u:\mathbb{R}^2\to\mathbb{R}$ represents homothetic preferences if and only if $$u(t\cdot x,t\cdot y)=t\cdot u(x,y)$$ for all $t&gt;0$. Hence, to see if $u(x,y)=12x^{0.2}y^{0.8}$ is homothetic, check if $u(t\cdot x,t\cdot y)=t\cdot u(x,y)$.</p>

<p>Another definition states that a homothetic function is a function that can be written as a monotonic transformation of a homogenous function. According to this definition, $u(x,y)=x^3y^3+xy$ is homoethetic since $u(x,y)=g(v(x,y))$ with $g(z)=z^3+z$ and $v(x,y)=xy$.</p>

CC BY-SA 3.0
2016-11-13T14:39:35.800Z
2016-11-13T14:39:35.800Z
Elias
<p>Does the utility function U(X, Y) = 12x^0.2 y^0.8 represent homothetic preferences?</p>

<p>I was given this question but I have no idea how to work it out. I understand what homothetic preferences are in theory, but I don't know how to calculate if a utility function represents homothetic preferences.</p>

<p>Any help would be hugely appreciated - I've been stuck on this for over an hour.</p>

<microeconomics><utility>
Does the utility function U(X, Y) = 12x^0.2 y^0.8 represent homothetic preferences?
Mike
# Does the utility function U(X, Y) = 12x^0.2 y^0.8 represent homothetic preferences?

(Asked by: username_0 on 2016-11-13)

Does the utility function U(X, Y) = 12x^0.2 y^0.8 represent homothetic preferences?

I was given this question but I have no idea how to work it out. I understand what homothetic preferences are in theory, but I don't know how to calculate if a utility function represents homothetic preferences.

Any help would be hugely appreciated - I've been stuck on this for over an hour.

**Tags:** microeconomics, utility


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## 1 answers

### Answer by username_1 on 2016-11-13. Score: 2

It depends on the definition of homotheticity. One definition states that consumer's preferences are homotethic if and only if his or her preferences can be represented by a utility function which is homogenous of degree 1.

For example, if we have two goods $x$ and $y$, then the utility function $u:\mathbb{R}^2\to\mathbb{R}$ represents homothetic preferences if and only if $$u(t\cdot x,t\cdot y)=t\cdot u(x,y)$$ for all $t>0$. Hence, to see if $u(x,y)=12x^{0.2}y^{0.8}$ is homothetic, check if $u(t\cdot x,t\cdot y)=t\cdot u(x,y)$.

Another definition states that a homothetic function is a function that can be written as a monotonic transformation of a homogenous function. According to this definition, $u(x,y)=x^3y^3+xy$ is homoethetic since $u(x,y)=g(v(x,y))$ with $g(z)=z^3+z$ and $v(x,y)=xy$.


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