The blade element efficiency is higest in the inboard half of the propeller. This alone would indicate that the propeller chord should be greatest there. The drop to zero at the tip is the same effect as the lift at the tip of a wing going to zero. The blade diameter is needed to keep blade loading down but, because efficiency is low, the tips should be rounded with thickness a low as practical and reduced pitch.
On large aircraft if this happens the propeler is geared to reduce rotational speed and the most efficient place on the blade moves out near the tip.