In this paper, we propose an agent-based model for investigating possible scenarios of genetic and cultural language evolution based on an integrated gene-culture coevolutionary framework. We focused on the following problems: (1) how communicative ability can evolve directionally under positive frequency-dependent selection and (2) how much of the directional effect there is between language and biological evolution. In our evolutionary experiments and analysis, we discovered a coevolutionary scenario involving the biological evolution of phenotypic plasticity and a cyclic coevolutionary dynamic between genetic and cultural evolution that is mediated by phenotypic plasticity. Furthermore, we discovered that the rates of cultural change are usually faster than the biological rates and fluctuate on a short time scale; on a long time scale, however, cultural rates tend to be slow. This implies that biological evolution can maintain the pace with language evolution. Finally, we analyzed the transfer entropy for a quantitative discussion of the directional effects between both evolutions. The results showed that biological evolution appears to be unable to maintain the pace with language evolution on short time scales, while their mutual directional effects are in the same range on long time scales. This implies that language and the relevant biology could coevolve.