<h3>Are homothetic preferences strictly monotonically increasing?</h3>

<p>Homotheticity requires that </p>

<p>$$ \alpha^\gamma U(x,y) = U(\alpha x, \alpha y) $$</p>

<p>This is not defined over the "increasing" part of strictly monotonously increasing. Indeed, you can have decreasing preferences $U(x,y) = -x -y$ for whom homotheticity holds. </p>

<h3>Are nomothetic preferences weakly monotonic?</h3>

<p>However, the "strict" part (in both jointly) is required, i.e. $U(\alpha x, \alpha y) \neq U(x,y)$ for $\alpha \neq 0$.. You cannot have homothetic and weakly increasing or weakly decreasing preferences. </p>

<p><strong>Proof outline</strong> To prove that, we assume otherwise. There is some homothetic $U$ with a strictly increasing part and a flat part. You will see that the $\gamma$ corresponding to the increasing part can potentially be any number (it depends on the functional form). However, the $\gamma$ corresponding to the flat part <em>needs</em> to be 0.[1] Hence, there cannot be a $\gamma$ that holds for the utility function as a whole, for any general increasing part.</p>

<p>[1]: Unless of course, $U(x,y) = 0$ at the flat part. However, as utility functions are immune to monotone transformations, we can ignore such cases.</p>

CC BY-SA 3.0
2015-05-24T12:34:34.723Z
By weakly monotonic do you mean monotonic but not strictly monotonic?
If yes, what about the counterexample in the answer below?
Giskard
CC BY-SA 3.0
2015-05-24T12:41:49.287Z
@denesp: Weakly increasing in *both jointly*. With Leontief preferences you have it that your utility *always* increases when you increase both $x$ and $y$.
FooBar
CC BY-SA 3.0
2015-05-24T12:22:24.840Z
2015-06-16T09:16:33.877Z
2015-06-16T09:16:33.877Z
FooBar
<p>I'm trying to understand intuitively what a homothetic preference is, and I am still not quite there. I understand the definition, that a homothetic preference implies that the slope of the indifference curve remains constant along any ray from origin.</p>

<p>But intuitively, like in real life, how would we know that a preference is not homothetic? If its not strictly monotonically increasing (i.e. you don't prefer more of a product to less of it always), does it mean that the preference is not homothetic? Can someone give an example of what exactly is not a homothetic preference so I understand what is?</p>

<microeconomics><mathematical-economics><preferences>
Are homothetic preferences monotonic?
dexter
# Are homothetic preferences monotonic?

(Asked by: username_0 on 2015-05-24)

I'm trying to understand intuitively what a homothetic preference is, and I am still not quite there. I understand the definition, that a homothetic preference implies that the slope of the indifference curve remains constant along any ray from origin.

But intuitively, like in real life, how would we know that a preference is not homothetic? If its not strictly monotonically increasing (i.e. you don't prefer more of a product to less of it always), does it mean that the preference is not homothetic? Can someone give an example of what exactly is not a homothetic preference so I understand what is?

**Tags:** microeconomics, mathematical-economics, preferences

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## 1 answers

### Answer by username_1 on 2015-05-24. Score: 9

### Are homothetic preferences strictly monotonically increasing?

Homotheticity requires that

$$ \alpha^\gamma U(x,y) = U(\alpha x, \alpha y) $$

This is not defined over the "increasing" part of strictly monotonously increasing. Indeed, you can have decreasing preferences $U(x,y) = -x -y$ for whom homotheticity holds.

### Are nomothetic preferences weakly monotonic?

However, the "strict" part (in both jointly) is required, i.e. $U(\alpha x, \alpha y) \neq U(x,y)$ for $\alpha \neq 0$.. You cannot have homothetic and weakly increasing or weakly decreasing preferences.

**Proof outline** To prove that, we assume otherwise. There is some homothetic $U$ with a strictly increasing part and a flat part. You will see that the $\gamma$ corresponding to the increasing part can potentially be any number (it depends on the functional form). However, the $\gamma$ corresponding to the flat part *needs* to be 0.[1] Hence, there cannot be a $\gamma$ that holds for the utility function as a whole, for any general increasing part.

[1]: Unless of course, $U(x,y) = 0$ at the flat part. However, as utility functions are immune to monotone transformations, we can ignore such cases.

> **Comments:**
> **username_2**: By weakly monotonic do you mean monotonic but not strictly monotonic? If yes, what about the counterexample in the answer below?  
> **username_1**: @denesp: Weakly increasing in *both jointly*. With Leontief preferences you have it that your utility *always* increases when you increase both $x$ and $y$.  

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