Definition
The demand change when price and wealth change are based on Comparative Statics
Change in demand when wealth change
Denote the change in demand when wealth change as 
Note: my understanding: D: change; w: when wealth change, x(p, w): change of demand.
is the vector of each demand 
with respect to w, its ith element is 
The set of points 
that are demanded, with fixed price but changing wealth, forms an income expansion path.
Change in demand when price change
Denote the change in demand when price change as the matrix
, its ij entry is 
, and it gives the response of demand for every good to every price change.
The set of points that are demanded, with fixed wealth and one price but changing the other price, forms an offer curve.
See also
Price and wealth effects must cancel out for each good
Interpretation: when all price and wealth increases by the same scale, the demand x does not change, which means that the effect from price change and effect from wealth change cancels out.
Implication of homogeneity of degree 0 of x(p, w).
, or
, or
Where 
: cross price elasticity of demand of l with respect to 
.
Note it is different from the elasticity of demand.
: elasticity of demand with respect to w.
Explain:
From 
, we have:
By the definition of elasticity of demand, we have:
So
See also