The gene duplication problem seeks a species tree that reconciles given gene trees with the minimum number of gene duplication events, called gene duplication cost. To better assess species trees inferred by the gene duplication problem we study diameters of the gene duplication cost, which describe elementary mathematical properties of this cost. The gene duplication cost is defined for a gene tree, a species tree, and a leaf labeling function that maps the leaf-genes of the gene tree to the leaf-species. The diameters of this cost are its maximal values when one topology or both topologies of the trees involved are fixed under all possible leaf labelings and are fundamental in understanding how gene trees and species trees relate. We describe the properties and formulas for these diameters for bijective and general leaf labelings, and present efficient algorithms to compute the diameters and their corresponding leaf labelings. Finally, we study these diameters under the unrooted reconciliation model in which gene trees are unrooted.11A preliminary version of this article was presented in ISBRA 2014 [11].