Final answer:
The ventromedial prefrontal cortex is primarily responsible for integrating value for inter-temporal choices and responding negatively to delay costs, making it a key region in assessing risk, reward, and making judgments regarding delayed gratification.
Step-by-step explanation:
Among the cortical brain regions that integrate value for inter-temporal choices and respond negatively to delay costs, the ventromedial prefrontal cortex (VMPC) plays a crucial role. This area of the prefrontal cortex is associated with the assessment of risk and reward, making judgments, and controlling impulsive behavior. It helps to process the values of different actions or stimuli when deciding on a course of action, taking into account the future consequences or rewards of those actions. The VMPC integrates different types of information to make choices that will maximize benefits over time, and in the case of delayed gratification, it evaluates the 'cost' of waiting.
While the amygdala, hippocampus, and basal ganglia also play roles related to emotions, memory, and movement respectively, they are not primarily responsible for integrating value for inter-temporal choices and responding to delay costs in the same way that the ventromedial prefrontal cortex is.