<p>I think what you need is that if $U(x,y)$ is homothetic then
$$
\forall \alpha \in \mathbb{R}_{++}, \forall (x,y) : \hskip 6pt
\frac{\frac{\partial U(x,y)}{\partial x}}{\frac{\partial U(x,y)}{\partial y}} = 
\frac{\frac{\partial U(\alpha \cdot x,\alpha \cdot y)}{\partial x}}{\frac{\partial U(\alpha \cdot x,\alpha \cdot y)}{\partial y}}
$$
and love.</p>

CC BY-SA 3.0
2015-10-05T12:30:16.940Z
Then it should be "_all_ you need", I guess.
Alecos Papadopoulos
CC BY-SA 3.0
2015-10-05T13:21:13.873Z
@AlecosPapadopoulos that was my first draft but since the proof is not complete I decided to do away with the "all".
Giskard
CC BY-SA 3.0
2015-10-04T14:32:06.980Z
2015-10-04T14:32:06.980Z
Giskard
<p><span class="math-container">$$D(p,I) = \arg\max {U(x):px≤I}.$$</span>
The problem
<span class="math-container">$$\arg\max_x {U(x/I):px/I≤1}$$</span>
has solution
<span class="math-container">$$x/I = D(p,1)$$</span> hence <span class="math-container">$$D(p,I)/I = D(p,1)$$</span> or <span class="math-container">$$D(p,I) = ID(p,1)$$</span></p>

CC BY-SA 4.0
2018-12-05T14:19:48.520Z
2019-10-22T08:15:39.343Z
2019-10-22T08:15:39.343Z
Thijs ten Raa
<p>1.Since there exists an <strong><em>assumption</em></strong> that "the preference is homothetic", according to the exact definition, <strong>U(t<em>x)=t</em>U(x)</strong>, where <em>U(x)</em> is the direct utility of goods x and <strong><em>U(x) is homogeneous of degree one.</em></strong> under the previous <strong><em>assumption</em></strong>.
(<a href="https://en.wikipedia.org/wiki/Homothetic_preferences" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Homothetic_preferences</a>)</p>

<p>2.I need to prove <strong><em>x(p,m)</em></strong> is also <strong><em>homogeneous of degree one</em></strong> in <strong><em>m</em></strong>. We can prove it by contradiction.</p>

<p><strong><em>Suppose x(p,t</em>m)≠ t<em>x(p,m)</em></strong> and <strong><em>x(p,m)</em></strong> is the <strong><em>optimal choice</em></strong> which maximizes utility based on the budget constraint <strong><em>m</em></strong>. </p>

<p>It's noticed that both consumptions <strong><em>x(p,t</em>m)*</strong> and <strong><em>t</em>x(p,m)*</strong> are <strong><em>feasible</em></strong> based on the budget <strong><em>t</em>m*</strong>. But <strong><em>x(p,t</em>m)*</strong> is the <strong><em>optimal</em></strong> choice, we can get <strong><em>U(x(p,t</em>m)) ＞ U(t<em>x(p,m)).</em></strong></p>

<p>We can <strong><em>multiply by 1/t</em></strong> both sides, which turns out to be <strong><em>U(x(p,t</em>m))/ t＞U(t<em>x(p,m))/ t</em></strong>. 
Since U(x) is <strong><em>homogeneous of degree one</em></strong> in x,  we can <strong><em>put 1/t inside U(x),</em></strong> which is <strong><em>U(X(p,t</em>m)/t)＞U(t<em>x(p,m)/t)=U(x(p,m)).</em></strong></p>

<p>Notice that <strong><em>x(p,m)</em></strong> is already <strong><em>the optimal choice</em></strong> based on the budget constraint <strong><em>m</em></strong>, but here we find <strong><em>another optimal consumption x(p,t</em>m)/t</strong> . Contradicts.</p>

<p>So <strong><em>x(p,t</em>m) ＝ t<em>x(p,m).</em></strong> And x(p,m) is  homogeneous of degree one in m.</p>

<p>3.Because the preference is homothetic, <strong><em>x(p,t</em>m)=t<em>x(p,m),</em></strong> which means that <strong><em>m</em>x(p)=m<em>x(p,1)=x(p,m)</em>.</strong></p>

<p>So <strong><em>x(p,m)=m</em>x(p)</strong> where the demand is the linear function of the income.</p>

<p>I apologize for my rudeness.</p>

CC BY-SA 3.0
2016-11-10T07:29:57.657Z
This answer does not use the fact that the utility function is homothetic. What it does say is that if demand is linear in income, then demand is linear in income.
Giskard
CC BY-SA 3.0
2016-11-12T01:24:55.087Z
If you guys are not able to understand my proof, I won't blame you for it. But still you guys need to prove this question in your ways and convince everyone with your proof, not just a well-known but useless formula and nothing else, which can't prove anything.
Li.shen
CC BY-SA 3.0
2016-11-12T01:32:33.677Z
BTW, recommend an advanced microeconomics textbook called Microeconomics Analysis by Hal.Varian. That's what you guys and your professors really need.
Li.shen
CC BY-SA 3.0
2016-11-12T08:19:20.543Z
Thank you for your kind words wise Li.shen. Your new proof, which is as I can still see in the edit history very different from your old proof, is mostly correct. Now if you only had the wisdom not to blame us for not writing this  one in the first place. Btw. not all utility functions representing homothetic preferences are homogeneous of degree one. All homothetic preferences have a utility representation which is homogeneous of degree one, but other representations exist as well.
Giskard
CC BY-SA 3.0
2016-11-10T04:56:27.663Z
2016-11-12T16:42:05.677Z
2016-11-12T16:42:05.677Z
Li.shen
<p>A homothetic utility function is one which is a monotonic transformation of a homogeneous utility function. </p>

<p>I am asked to show that if a utility function is homothetic then the associated demand functions are linear in income.</p>

<p>In general if $H$ is monotonic and we compose it with a function $g$ which is homogeneous we get $H(g(x,y))$ which is homothetic and so the ratio of derivatives of $H$ with respect to $x$ and $y$ is the same as the ratio of the derivatives of $g$ with respect to $x$ and $y$ since the outer function cancels out. This means all income expansion paths are rays from the origin and the slope of indifference curves (level sets) have the same slope along the income expansion path.</p>

<p>My question is how does one show generally that the demand functions are linear in income? If you don't have the functional form of the utility function then what can you substitute into the budget constraint to solve for $x$ or $y$ as functions of prices and income only?</p>

<p>All you know is that the negative ratio of derivatives is equal to $p_1/p_2$ but you can't substitute anything into the budget constraint for $x$ or $y$ since you only have general partial derivatives.</p>

<p>The only way I can think to do that is take the inverse of the derivative of the inner (homogenous) function g in order to solve for $x$ (or $y$).</p>

CC BY-SA 3.0
2015-10-04T02:05:10.533Z
Hi, welcome to Economics SE. Usually people here don't like straight up homework questions without any indication of work done on it, though it seems you are quite lost. Is this supposed to be a more general proof question? If I may ask, what context are you trying to answer this question in? Are you in a graduate masters or doctorate program?
Kitsune Cavalry
CC BY-SA 3.0
2015-10-13T22:42:08.993Z
I voted to leave this homework question open, because the OP does present his thoughts on the matter.
Alecos Papadopoulos
<microeconomics><utility><self-study>
How to show that a homothetic utility function has demand functions which are linear in income
MHall
# How to show that a homothetic utility function has demand functions which are linear in income

(Asked by: username_0 on 2015-10-04)

A homothetic utility function is one which is a monotonic transformation of a homogeneous utility function.

I am asked to show that if a utility function is homothetic then the associated demand functions are linear in income.

In general if $H$ is monotonic and we compose it with a function $g$ which is homogeneous we get $H(g(x,y))$ which is homothetic and so the ratio of derivatives of $H$ with respect to $x$ and $y$ is the same as the ratio of the derivatives of $g$ with respect to $x$ and $y$ since the outer function cancels out. This means all income expansion paths are rays from the origin and the slope of indifference curves (level sets) have the same slope along the income expansion path.

My question is how does one show generally that the demand functions are linear in income? If you don't have the functional form of the utility function then what can you substitute into the budget constraint to solve for $x$ or $y$ as functions of prices and income only?

All you know is that the negative ratio of derivatives is equal to $p\_1/p\_2$ but you can't substitute anything into the budget constraint for $x$ or $y$ since you only have general partial derivatives.

The only way I can think to do that is take the inverse of the derivative of the inner (homogenous) function g in order to solve for $x$ (or $y$).

**Tags:** microeconomics, utility, self-study

> **Comments:**
> **username_1**: Hi, welcome to Economics SE. Usually people here don't like straight up homework questions without any indication of work done on it, though it seems you are quite lost. Is this supposed to be a more general proof question? If I may ask, what context are you trying to answer this question in? Are you in a graduate masters or doctorate program?  
> **username_2**: I voted to leave this homework question open, because the OP does present his thoughts on the matter.  

---

## 3 answers

### Answer by username_3 on 2015-10-04. Score: 11

I think what you need is that if $U(x,y)$ is homothetic then
$$
\forall \alpha \in \mathbb{R}\_{++}, \forall (x,y) : \hskip 6pt
\frac{\frac{\partial U(x,y)}{\partial x}}{\frac{\partial U(x,y)}{\partial y}} =
\frac{\frac{\partial U(\alpha \cdot x,\alpha \cdot y)}{\partial x}}{\frac{\partial U(\alpha \cdot x,\alpha \cdot y)}{\partial y}}
$$
and love.

> **Comments:**
> **username_2**: Then it should be "_all_ you need", I guess.  
> **username_3**: @AlecosPapadopoulos that was my first draft but since the proof is not complete I decided to do away with the "all".  

---

### Answer by username_4 on 2018-12-05. Score: 2

$$D(p,I) = \arg\max {U(x):px≤I}.$$
The problem
$$\arg\max\_x {U(x/I):px/I≤1}$$
has solution
$$x/I = D(p,1)$$ hence $$D(p,I)/I = D(p,1)$$ or $$D(p,I) = ID(p,1)$$


---

### Answer by username_5 on 2016-11-10. Score: 0

1.Since there exists an ***assumption*** that "the preference is homothetic", according to the exact definition, **U(t*x)=t*U(x)**, where *U(x)* is the direct utility of goods x and ***U(x) is homogeneous of degree one.*** under the previous ***assumption***.
(<https://en.wikipedia.org/wiki/Homothetic_preferences>)

2.I need to prove ***x(p,m)*** is also ***homogeneous of degree one*** in ***m***. We can prove it by contradiction.

***Suppose x(p,t*m)≠ t*x(p,m)*** and ***x(p,m)*** is the ***optimal choice*** which maximizes utility based on the budget constraint ***m***.

It's noticed that both consumptions ***x(p,t*m)\*** and ***t*x(p,m)\*** are ***feasible*** based on the budget ***t*m\***. But ***x(p,t*m)\*** is the ***optimal*** choice, we can get ***U(x(p,t*m)) ＞ U(t*x(p,m)).***

We can ***multiply by 1/t*** both sides, which turns out to be ***U(x(p,t*m))/ t＞U(t*x(p,m))/ t***.
Since U(x) is ***homogeneous of degree one*** in x, we can ***put 1/t inside U(x),*** which is ***U(X(p,t*m)/t)＞U(t*x(p,m)/t)=U(x(p,m)).***

Notice that ***x(p,m)*** is already ***the optimal choice*** based on the budget constraint ***m***, but here we find ***another optimal consumption x(p,t*m)/t** . Contradicts.

So ***x(p,t*m) ＝ t*x(p,m).*** And x(p,m) is homogeneous of degree one in m.

3.Because the preference is homothetic, ***x(p,t*m)=t*x(p,m),*** which means that ***m*x(p)=m*x(p,1)=x(p,m)*.**

So ***x(p,m)=m*x(p)** where the demand is the linear function of the income.

I apologize for my rudeness.

> **Comments:**
> **username_3**: This answer does not use the fact that the utility function is homothetic. What it does say is that if demand is linear in income, then demand is linear in income.  
> **username_5**: If you guys are not able to understand my proof, I won't blame you for it. But still you guys need to prove this question in your ways and convince everyone with your proof, not just a well-known but useless formula and nothing else, which can't prove anything.  
> **username_5**: BTW, recommend an advanced microeconomics textbook called Microeconomics Analysis by Hal.Varian. That's what you guys and your professors really need.  
> **username_3**: Thank you for your kind words wise username_5. Your new proof, which is as I can still see in the edit history very different from your old proof, is mostly correct. Now if you only had the wisdom not to blame us for not writing this  one in the first place. Btw. not all utility functions representing homothetic preferences are homogeneous of degree one. All homothetic preferences have a utility representation which is homogeneous of degree one, but other representations exist as well.  

---

