Demand functions homogeneous of degree zero in prices and income - how this relates to budget exhaustion (solving a consumer's problem) I have the following demand system:

 





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







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



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.
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.