The "developmental hourglass'' describes a pattern of increasing morphological divergence towards earlier and later embryonic development, separated by a period of significant conservation across distant species (the "phylotypic stage''). Recent studies have found evidence in support of the hourglass effect at the genomic level. For instance, the phylotypic stage expresses the oldest and most conserved transcriptomes. However, the regulatory mechanism that causes the hourglass pattern remains an open question. Here, we use an evolutionary model of regulatory gene interactions during development to identify the conditions under which the hourglass effect can emerge in a general setting. The model focuses on the hierarchical gene regulatory network that controls the developmental process, and on the evolution of a population under random perturbations in the structure of that network. The model predicts, under fairly general assumptions, the emergence of an hourglass pattern in the structure of a temporal representation of the underlying gene regulatory network. The evolutionary age of the corresponding genes also follows an hourglass pattern, with the oldest genes concentrated at the hourglass waist. The key behind the hourglass effect is that developmental regulators should have an increasingly specific function as development progresses. Analysis of developmental gene expression profiles from Drosophila melanogaster and Arabidopsis thaliana provide consistent results with our theoretical predictions.