Definition
s.t. 
x: commodity bundle, 
p: price vector, 
w: wealth, x > 0
Existence of solution
If price are strictly positive and u is continuous, then the UMP has a solution (not necessarily unique)
Proposition
Suppose that u is continuous and locally non-satiated. Then x(p, w) is:
(1) Homogeneity of degree 0:
for all 
Explain
if the price and wealth change to the same degree (times the same number 
), the budget line on utility graph does not change because unit of goods purchases does not change.
Note
the units of utility graph's x, y axis are unit of good 1 and 2 purchased!
(2) Walras's Law
for all 
Note: implies , but not implies !
We need 
and 
to have 
. See property 3-1.
Interpretation
It means the maximum is always found on the budget line.
(3-1) If 
is convex, so that u is quasi-concave, then x(p,w) is a convex set.
Proof
suppose x, 
both solve UMP,
thus 
because
Moreover, 
by convexity, so 
(3-2) If is strictly convex, so that u is strictly quasi-concave, then x(p,w) is a singleton.
Proof
suppose x, 
both solve UMP, which implies 
strictly convexity implies 
and 
Then x, 
, so it is a contradiction.
Note
- argmax means that the parameter (x) that makes the function (u(x)) to achieve maximum.
is competitive budget set