<p>If a consumer has the utility function <span class="math-container">$U(x,y) = x + y$</span>, and <span class="math-container">$p_x &gt; p_y$</span>, then the optimal bundle is <span class="math-container">$(x,y) = (0,m/p_y)$</span>. This seems to result in a vertical income expansion path.</p>
<p>Note that the same is true for
<span class="math-container">$$\hat{U}(x,y) = y - x,$$</span>
and
<span class="math-container">$$\tilde{U}(x,y) = \ln (x+1) + y.$$</span></p>
<p>Using the definitions, one can easily ascertain which of the above utility functions are homothetic or strongly monotonic.</p>
<p>I am not sure what is meant by &quot;representative consumer&quot; in the question.</p>

CC BY-SA 4.0
2024-01-06T09:48:11.357Z
2024-01-06T11:27:48.343Z
2024-01-06T11:27:48.343Z
Giskard
<p>On the horizontal axis we have good x, on the vertical axis good y. The income expansion path is vertical. Which of the following assumptions is wrong?</p>
<ul>
<li>no homotheticity</li>
<li>no strong monoticity</li>
<li>represantative consumer</li>
</ul>

CC BY-SA 4.0
2024-01-05T21:29:44.943Z
The assumption is "no homotheticity", as in we assume the utility function is *not* homothetic? Can you flesh out the question a bit? Also, seems like a homework problem. Where do you get stuck?
Giskard
CC BY-SA 4.0
2024-01-05T21:35:23.857Z
I'm not sure of my answer. With no homotheticity, it is assumed that the preferences of the consumer are not homothetic. Since in homotheticity there is an income expansion path that goes through the origin, I think that in this case ther is no homotheticity. But im not sure about monoticity. I think there is monoticity since the IC are negativly sloped, but don't know wheter is is a correct assumption, since the concept op strong monoticity remains vague for me ("monoticity = IC's are decreasing since more is better?)
EcoSTUD233
<microeconomics><consumer-theory>
Vertical income expansion path: explanation?
EcoSTUD233
# Vertical income expansion path: explanation?

(Asked by: username_0 on 2024-01-05)

On the horizontal axis we have good x, on the vertical axis good y. The income expansion path is vertical. Which of the following assumptions is wrong?

* no homotheticity
* no strong monoticity
* represantative consumer

**Tags:** microeconomics, consumer-theory

> **Comments:**
> **username_1**: The assumption is "no homotheticity", as in we assume the utility function is *not* homothetic? Can you flesh out the question a bit? Also, seems like a homework problem. Where do you get stuck?  
> **username_0**: I'm not sure of my answer. With no homotheticity, it is assumed that the preferences of the consumer are not homothetic. Since in homotheticity there is an income expansion path that goes through the origin, I think that in this case ther is no homotheticity. But im not sure about monoticity. I think there is monoticity since the IC are negativly sloped, but don't know wheter is is a correct assumption, since the concept op strong monoticity remains vague for me ("monoticity = IC's are decreasing since more is better?)  

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## 1 answers

### Answer by username_1 on 2024-01-06. Score: 1

If a consumer has the utility function $U(x,y) = x + y$, and $p\_x > p\_y$, then the optimal bundle is $(x,y) = (0,m/p\_y)$. This seems to result in a vertical income expansion path.

Note that the same is true for
$$\hat{U}(x,y) = y - x,$$
and
$$\tilde{U}(x,y) = \ln (x+1) + y.$$

Using the definitions, one can easily ascertain which of the above utility functions are homothetic or strongly monotonic.

I am not sure what is meant by "representative consumer" in the question.


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