Across languages of the world, some grammatical patterns have been argued to be more common than expected by chance. These are sometimes referred to as (statistical) language universals. One such universal is the correlation between constituent order freedom and the presence of a case system in a language. Here, we explore whether this correlation can be explained by a bias to balance production effort and informativity of cues to grammatical function. Two groups of learners were presented with miniature artificial languages containing optional case marking and either flexible or fixed constituent order. Learners of the flexible order language used case marking significantly more often. This result parallels the typological correlation between constituent order flexibility and the presence of case marking in a language and provides a possible explanation for the historical development of Old English to Modern English, from flexible constituent order with case marking to relatively fixed order without case marking. In addition, learners of the flexible order language conditioned case marking on constituent order, using more case marking with the cross‐linguistically less frequent order, again mirroring typological data. These results suggest that some cross‐linguistic generalizations originate in functionally motivated biases operating during language learning.