<blockquote>
  <p>I would have thought the budget constraint would not have been satisfied only if one of the demand functions was not homogeneous of degree zero in prices and income.</p>
</blockquote>

<p>You should look into this claim. It is true that demand functions are homogeneous of degree zero in prices and income, but not all functions that are homogeneous of degree zero in prices and income are demand functions.</p>

<p>Some examples with only one good, so where <span class="math-container">$p_x x = I$</span> should hold:</p>

<p><span class="math-container">$
x= \frac{I}{7p_x}  
$</span>
is homogeneous of degree zero, but with this the consumer does not spend all their income. </p>

<p><span class="math-container">$
x= 0
$</span>
is homogeneous of degree zero, but with this the consumer does not spend all their income. </p>

<p><span class="math-container">$
x= - \frac{p_x}{I}  
$</span>
is homogeneous of degree zero, but there is definitely something weird going on here.</p>

CC BY-SA 4.0
2020-04-17T04:52:26.250Z
2020-04-17T04:52:26.250Z
Giskard
<p>I have the following demand system:</p>

<p><img src="https://latex.codecogs.com/png.latex?x%28p%2CI%29%3D%5Cfrac%7BIp_%7Bz%7D%7D%7B4p_%7Bx%7Dp_%7By%7D%7D" alt="x(p,I)=\frac{Ip_{z}}{4p_{x}p_{y}}"> </p>

<p><img src="https://latex.codecogs.com/png.latex?y%28p%2CI%29%3D%5Cfrac%7BI%7D%7B4p_%7By%7D%7D" alt="y(p,I)=\frac{I}{4p_{y}}"></p>

<p><img src="https://latex.codecogs.com/png.latex?z%28p%2CI%29%3D0" alt="z(p,I)=0"></p>

<p>It appears to me that each demand function is homogeneous of degree zero in prices and income as:</p>

<p><img src="https://latex.codecogs.com/png.latex?x%28tp%2CtI%29%3D%5Cfrac%7BIp_%7Bz%7D%7D%7B4p_%7Bx%7Dp_%7By%7D%7D" alt="x(tp,tI)=\frac{Ip_{z}}{4p_{x}p_{y}}"></p>

<p><img src="https://latex.codecogs.com/png.latex?y%28tp%2CtI%29%3D%5Cfrac%7BI%7D%7B4p_%7By%7D%7D" alt="y(tp,tI)=\frac{I}{4p_{y}}"></p>

<p><img src="https://latex.codecogs.com/png.latex?z%28tp%2CtI%29%3D0" alt="z(tp,tI)=0"></p>

<p>Why would this demand system <strong>not</strong> satisfy the following budget equation?</p>

<p><img src="https://latex.codecogs.com/gif.latex?p_xx&plus;p_yy&plus;p_zz%3DI" alt="p_xx+p_yy+p_zz=I"></p>

<p>where I is exogenous. I would have thought the budget constraint would <strong>not</strong> have been satisfied <strong>only if</strong> one of the demand functions was not homogeneous of degree zero in prices and income.</p>

<utility>
Demand functions homogeneous of degree zero in prices and income - how this relates to budget exhaustion (solving a consumer's problem)
newbie
# Demand functions homogeneous of degree zero in prices and income - how this relates to budget exhaustion (solving a consumer's problem)

(Asked by: username_0 on 2020-04-17)

I have the following demand system:

![x(p,I)=\frac{Ip_{z}}{4p_{x}p_{y}}](https://latex.codecogs.com/png.latex?x%28p%2CI%29%3D%5Cfrac%7BIp_%7Bz%7D%7D%7B4p_%7Bx%7Dp_%7By%7D%7D)

![y(p,I)=\frac{I}{4p_{y}}](https://latex.codecogs.com/png.latex?y%28p%2CI%29%3D%5Cfrac%7BI%7D%7B4p_%7By%7D%7D)

![z(p,I)=0](https://latex.codecogs.com/png.latex?z%28p%2CI%29%3D0)

It appears to me that each demand function is homogeneous of degree zero in prices and income as:

![x(tp,tI)=\frac{Ip_{z}}{4p_{x}p_{y}}](https://latex.codecogs.com/png.latex?x%28tp%2CtI%29%3D%5Cfrac%7BIp_%7Bz%7D%7D%7B4p_%7Bx%7Dp_%7By%7D%7D)

![y(tp,tI)=\frac{I}{4p_{y}}](https://latex.codecogs.com/png.latex?y%28tp%2CtI%29%3D%5Cfrac%7BI%7D%7B4p_%7By%7D%7D)

![z(tp,tI)=0](https://latex.codecogs.com/png.latex?z%28tp%2CtI%29%3D0)

Why would this demand system **not** satisfy the following budget equation?

![p_xx+p_yy+p_zz=I](https://latex.codecogs.com/gif.latex?p_xx+p_yy+p_zz%3DI)

where I is exogenous. I would have thought the budget constraint would **not** have been satisfied **only if** one of the demand functions was not homogeneous of degree zero in prices and income.

**Tags:** utility

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## 1 answers

### Answer by username_1 on 2020-04-17. Score: 1

> I would have thought the budget constraint would not have been satisfied only if one of the demand functions was not homogeneous of degree zero in prices and income.

You should look into this claim. It is true that demand functions are homogeneous of degree zero in prices and income, but not all functions that are homogeneous of degree zero in prices and income are demand functions.

Some examples with only one good, so where $p\_x x = I$ should hold:

$
x= \frac{I}{7p\_x}
$
is homogeneous of degree zero, but with this the consumer does not spend all their income.

$
x= 0
$
is homogeneous of degree zero, but with this the consumer does not spend all their income.

$
x= - \frac{p\_x}{I}
$
is homogeneous of degree zero, but there is definitely something weird going on here.


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