Child-directed speech (CDS) is a particular type of speech that adults use when addressing young children. Its properties also change as a function of extralinguistic factors, such as age of the child being addressed. Access to large amounts of representative and varied CDS would be useful for child language research, as this would enable controlled computational modeling experiments of infant language acquisition with realistic input in terms of quality and quantity. In this study, we describe an approach to model age-dependent linguistic properties of CDS using a language model (LM) trained on CDS transcripts and ages of the recipient children, as obtained from North American English corpora of the CHILDES database. The created LM can then be used to stochastically generate synthetic CDS transcripts in an age-appropriate manner, thereby scaling beyond the original datasets in size. We compare characteristics of the generated CDS against the real speech addressed at children of different ages, showing that the LM manages to capture age-dependent changes in CDS, except for a slight difference in the effective vocabulary size. As a side product, we also provide a systematic characterization of age-dependent linguistic properties of CDS in CHILDES, illustrating how all measured aspects of the CDS change with children's age.