<p>Converting my comment to an answer...</p>

<p>I suppose the expression you wrote is <em>almost</em> correct, except perhaps for a notational issue. I would write $u(x,y)=[w(x,y)]^{-1}$, because $w^{-1}(x,y)$ is usually used to denote the <strong>pre-image</strong> of $w(x,y)$.</p>

<p>More generally, inverse proportionality between $u$ and $w$ is defined up to a multiplicative constant $K$; that is, $u(x,y)\cdot w(x,y)=K$ for some $K\in\mathbb R$. In the above case, you implicitly assumed $K=1$. So generally, you'd have 
$$
u(x,y)=K[w(x,y)]^{-1}=\frac{K}{w(x,y)}.
$$</p>

CC BY-SA 3.0
2016-01-13T23:05:55.090Z
2016-01-13T23:05:55.090Z
Herr K.
<p>If I have two products $x$ and $y$ and two users $A$ and $B$ whose utility functions for those products $u$ and $w$ are inversely proportional, would it be ok to express this relationship as: $u(x,y) = w^{-1}(x,y)$? Thanks.</p>

CC BY-SA 3.0
2016-01-13T04:51:13.877Z
Two variables $p,q$ are **inversely proportional** if $pq=K$ where $K$ is a constant. So if $u$ and $w$ are inversely proportional if for any pair of products $(x,y)$, $$u(x,y)=\frac{K}{w(x,y)}$$ for some fixed $K$.
Herr K.
CC BY-SA 3.0
2016-01-13T11:19:08.893Z
This is somewhat a strange relationship. It means that $A$ dislikes consuming the goods that $B$ likes. Is it what you have in mind? Otherwise, could you provide a little more context?
Oliv
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2016-01-13T22:38:05.533Z
@HerrK could you please explain me why you need $pq$ to define $K$? I don't know much about utility functions, but I am trying to focus on only two products $xy$. Or in your example $K$ is a constant of $xy$?
vabm
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2016-01-13T22:41:45.900Z
@Oliv yes, that's exactly the kind of relationship I am looking to establish. For example if I have two products that may have different colours $A$ dislikes exactly what $B$ likes and vice versa. Does $w^{-1}$ implies that?
vabm
CC BY-SA 3.0
2016-01-13T22:46:18.307Z
@vabm: You can think of $p,q$ in my case as two utility numbers; for instance $u(x,y)=p$ and $w(x,y)=q$. If $p$ and $q$ are inversely proportional, then $p=K/q$, for some $K$. In your case, $K=1$, but it could take other values too.
Herr K.
CC BY-SA 3.0
2016-01-13T22:49:14.920Z
@vabm: I suppose the expression you wrote is almost correct, except perhaps for a notational issue. I would write $u(x,y)=[w(x,y)]^{-1}$, because $w^{-1}(x,y)$ is usually used to denote the **pre-image** of $w(x,y)$.
Herr K.
CC BY-SA 3.0
2016-01-14T07:07:33.667Z
@vabm: in that case, you can simply posit $u(x,y)=-w(x,y)$ to guarantee that, $u(x,y)>u(x',y')$,implies $w(x,y)<w(x',y')$. Writing $u(x,y)=\frac{1}{w(x,y)}$ is ok as well if the utility functions take positive values.
Oliv
<microeconomics><utility>
inversely proportional utility functions
vabm
# inversely proportional utility functions

(Asked by: username_0 on 2016-01-13)

If I have two products $x$ and $y$ and two users $A$ and $B$ whose utility functions for those products $u$ and $w$ are inversely proportional, would it be ok to express this relationship as: $u(x,y) = w^{-1}(x,y)$? Thanks.

**Tags:** microeconomics, utility

> **Comments:**
> **username_1**: Two variables $p,q$ are **inversely proportional** if $pq=K$ where $K$ is a constant. So if $u$ and $w$ are inversely proportional if for any pair of products $(x,y)$, $$u(x,y)=\frac{K}{w(x,y)}$$ for some fixed $K$.  
> **username_2**: This is somewhat a strange relationship. It means that $A$ dislikes consuming the goods that $B$ likes. Is it what you have in mind? Otherwise, could you provide a little more context?  
> **username_0**: @HerrK could you please explain me why you need $pq$ to define $K$? I don't know much about utility functions, but I am trying to focus on only two products $xy$. Or in your example $K$ is a constant of $xy$?  
> **username_0**: @username_2 yes, that's exactly the kind of relationship I am looking to establish. For example if I have two products that may have different colours $A$ dislikes exactly what $B$ likes and vice versa. Does $w^{-1}$ implies that?  
> **username_1**: @username_0: You can think of $p,q$ in my case as two utility numbers; for instance $u(x,y)=p$ and $w(x,y)=q$. If $p$ and $q$ are inversely proportional, then $p=K/q$, for some $K$. In your case, $K=1$, but it could take other values too.  
> **username_1**: @username_0: I suppose the expression you wrote is almost correct, except perhaps for a notational issue. I would write $u(x,y)=[w(x,y)]^{-1}$, because $w^{-1}(x,y)$ is usually used to denote the **pre-image** of $w(x,y)$.  
> **username_2**: @username_0: in that case, you can simply posit $u(x,y)=-w(x,y)$ to guarantee that, $u(x,y)>u(x',y')$,implies $w(x,y)<w(x',y')$. Writing $u(x,y)=\frac{1}{w(x,y)}$ is ok as well if the utility functions take positive values.  

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## 1 answers

### Answer by username_1 on 2016-01-13. Score: 3 | Accepted answer

Converting my comment to an answer...

I suppose the expression you wrote is *almost* correct, except perhaps for a notational issue. I would write $u(x,y)=[w(x,y)]^{-1}$, because $w^{-1}(x,y)$ is usually used to denote the **pre-image** of $w(x,y)$.

More generally, inverse proportionality between $u$ and $w$ is defined up to a multiplicative constant $K$; that is, $u(x,y)\cdot w(x,y)=K$ for some $K\in\mathbb R$. In the above case, you implicitly assumed $K=1$. So generally, you'd have
$$
u(x,y)=K[w(x,y)]^{-1}=\frac{K}{w(x,y)}.
$$


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