The position of a chess problem must be original. If there is a prior similar position, then the problem is considered useless and is called anticipated.
There is an optimal placement of the chessmen, which presents the beauty of the idea of the composer, (who patiently might work even thirty years on it!).
The position must possess beautiful aesthetics. The aesthetics is a personal matter, thus each problemist prefers certain kinds of problems.
The position must be characterized by economy. Generally, the economy is an asset in a chess composition. The exact description of the element which must be economical is a matter of discussion. All aspects of economy are considered important:
(1) the economy of material or the economy of forces (Only the absolutely essential pieces are used. Every piece in a position is there for a reason. If something can be accompliced with a bishop, we do not use a queen in its place),
(2) the economy of space (All the chessboard is used. The pieces are not tightened in a corner),
(3) the economy of motivation (The variations are limited to those related with the theme of the problem),
(4) the economy of moves (The theme is developed with the minimum necessary moves. If something can be shown with a two-mover, we do not make it a more-mover).
[This Post in Greek language]