A common assumption in psychology, economics, and other fields holds that higher performance will result if extrinsic rewards (such as money) are offered as an incentive. While this principle seems to work well for tasks that require the execution of the same sequence of steps over and over, with little uncertainty about the process, in other cases, especially where creative problem solving is required due to the difficulty in finding the optimal sequence of actions, external rewards can actually be detrimental to task performance. Furthermore, they have the potential to undermine intrinsic motivation to do an otherwise interesting activity. In this work, we extend a computational model of the dorsomedial and dorsolateral striatal reinforcement learning systems to account for the effects of extrinsic and intrinsic rewards. The model assumes that the brain employs both a goal-directed and a habitual learning system, and competition between both is based on the trade-off between the cost of the reasoning process and value of information. The goal-directed system elicits internal rewards when its models of the environment improve, while the habitual system, being model-free, does not. Our results account for the phenomena that initial extrinsic reward leads to reduced activity after extinction compared to the case without any initial extrinsic rewards, and that performance in complex task settings drops when higher external rewards are promised. We also test the hypothesis that external rewards bias the competition in favor of the computationally efficient, but cruder and less flexible habitual system, which can negatively influence intrinsic motivation and task performance in the class of tasks we consider.