We consider a fitness model assumed to generate simple graphs with power-law heavy-tailed degree sequence: P(k) \propto k^{-1-α} with 0 < α< 1, in which the corresponding distributions do not posses a mean. We discuss the situations in which the model is used to produce a multigraph and examine what happens if the multiple edges are merged to a single one and thus a simple graph is built. We give the relation between the (normalized) fitness parameter r and the expected degree νof a node and show analytically that it possesses non-trivial intermediate and final asymptotic behaviors. We show that the model produces P(k) \propto k^{-2} for large values of k independent of α. Our analytical findings are confirmed by numerical simulations.