<p>If a utility function is homothethic,
$-\frac {\frac {\partial u}{\partial x} (tx)} {\frac {\partial u} {\partial y} (ty)} = -\frac {\frac {\partial u}{\partial x} (x)} {\frac {\partial u} {\partial y} (y)} $</p>

<p>In other words, the MRS must be homogenous of degree zero.</p>

<p>Hence, for $MRS(tx,ty)=2(tx+ty)=2t(x,y)=t MRS(x,y)$ is homogenous of degree 1</p>

<p>While $MRS(tx,ty)=\frac{(tx)^{2}+(ty)^{2}}{(tx)(ty)}=\frac{t^{2}(x+y)}{t^{2}(xy)}=\frac{x^{2}+y^{2}}{xy}=MRS(x,y)$ is homogenous of degree 0.</p>

CC BY-SA 3.0
2016-10-25T06:17:08.957Z
2016-10-25T07:34:15.207Z
2016-10-25T07:34:15.207Z
Walrasian Auctioneer
<blockquote>
  <p>Suppose we have the following equations for the MRS of a utility function. $$U(x, y)$$
  Which of the following corresponds to a homothetic utility function?</p>
  
  <ol>
  <li><p>$$MRS (x, y) = \frac{x^2+y^2}{xy}$$</p></li>
  <li><p>$$ MRS(x, y) = 2(x + y)$$</p></li>
  </ol>
</blockquote>

<p>The answer is 1. But why the second equation is not homothetic?</p>

<microeconomics><mathematical-economics>
What is a homothetic function?
EconDude
# What is a homothetic function?

(Asked by: username_0 on 2016-10-25)

> Suppose we have the following equations for the MRS of a utility function. $$U(x, y)$$
> Which of the following corresponds to a homothetic utility function?
>
> 1. $$MRS (x, y) = \frac{x^2+y^2}{xy}$$
> 2. $$ MRS(x, y) = 2(x + y)$$

The answer is 1. But why the second equation is not homothetic?

**Tags:** microeconomics, mathematical-economics

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## 1 answers

### Answer by username_1 on 2016-10-25. Score: 8 | Accepted answer

If a utility function is homothethic,
$-\frac {\frac {\partial u}{\partial x} (tx)} {\frac {\partial u} {\partial y} (ty)} = -\frac {\frac {\partial u}{\partial x} (x)} {\frac {\partial u} {\partial y} (y)} $

In other words, the MRS must be homogenous of degree zero.

Hence, for $MRS(tx,ty)=2(tx+ty)=2t(x,y)=t MRS(x,y)$ is homogenous of degree 1

While $MRS(tx,ty)=\frac{(tx)^{2}+(ty)^{2}}{(tx)(ty)}=\frac{t^{2}(x+y)}{t^{2}(xy)}=\frac{x^{2}+y^{2}}{xy}=MRS(x,y)$ is homogenous of degree 0.


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