Universities in the United States are remarkably diverse in their efficiency, both in terms of research output and educational achievement. Recent work has highlighted how important this heterogeneity is by showing that different broad categories of institutions demonstrate different scaling relationships for various features as a function of the number of students. These differences in scaling relationships reflect differences in organizational goals, constraints, and strategies. In the existing literature, this heterogeneity is under-explored and often ignored due to the lack of appropriate data and methodological limitations. In this paper, we address this problem by exploiting a newly consolidated dataset and adopting a neural-network based method to infer cost functions for universities. Our analyses reveal (1) the specific economy of scale in two distinct output types (education and research), (2) the nature of the trade-off between research and education efficiency, and (3) significant efficiency differences across universities. Particularly, we show that while both research and education outputs generally exhibit an economy of scale, their scalability largely depends on their size and other institutional characteristics. Similarly, research and education activities are complementary to each other (economy of scope) only in some situations, particularly when the scale of production is small to medium. Consequently, the cost isoclines of universities are highly non-convex, implying the possibility of multiple optima that may explain the diverse strategies universities adopt, and potential efficiency gains from specialization. It also suggests that some basic assumptions of microeconomic models may not be empirically supported.