Value trade-offs are an integral part of human decision-making and language use, however, current tools for interpreting such dynamic and multi-faceted notions of values in language models are limited. In cognitive science, so-called "cognitive models" provide formal accounts of such trade-offs in humans, by modeling the weighting of a speaker's competing utility functions in choosing an action or utterance. Here, we show that a leading cognitive model of polite speech can be used to systematically evaluate alignment-relevant trade-offs in language models via two encompassing settings: degrees of reasoning "effort" and system prompt manipulations in closed-source frontier models, and RL post-training dynamics of open-source models. Our results show that LLMs' behavioral profiles under the cognitive model a) shift predictably when they are prompted to prioritize certain goals, b) are amplified by a small reasoning budget, and c) can be used to diagnose other social behaviors such as sycophancy. Our findings from LLMs' post-training dynamics reveal large shifts in values early on in training and persistent effects of the choice of base model and pretraining data, compared to feedback dataset or alignment method. Our framework offers a flexible tool for probing behavioral profiles across diverse model types and gaining insights for shaping training regimes that better control trade-offs between values during model development.