The following example shows the "price" of fPO. The integral allocation maximizing the product of utilities (also called the Nash welfare) is PE but not fPO. Moreover, the product of utilities in any fPO allocation is at most 1/3 of the maximum product. There are five goods {h1,h2,g1,g2,g3} and 3 agents with the following values (where C is a large constant and d is a small positive constant):
