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This article presents the reply of the author and co-authors to an article about the analysis of differences in the fertility patterns between national populations and over time. In their comments to our paper Osona and Kohler make a number of interesting and potentially helpful points. While their comments do not, in general, appear to be in conflict with our opinions, we are concerned that they have misunderstood one specific point. In their first section, on the analytical relationship between Hadwiger parameters and demographic measures, the authors re-present some known results. Those results, of course, were not the essential point of our paper, which was concerned primarily with the analysis of differences in fertility patterns. The functions were fitted in order to gain understanding of the differences in the patterns between countries and periods. The correspondences between variables to which they refer were used in order to investigate the effects of departures from the models in cases of distorted distributions. In their section dealing with reparameterization the authors propose an alternative to our presentation of the Hadwiger mixture model. Their explicit interpretation of the Hadwiger parameters as the total fertility, the mean age at childbearing, and the variance of the fertility schedule is possibly a step forward for the use of the model in demography.