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The large majority of climate change mitigation scenarios that hold warming below 2 °C show high deployment of carbon dioxide removal (CDR), resulting in a peak-and-decline behavior in global temperature. This is driven by the assumption of an exponentially increasing carbon price trajectory which is perceived to be economically optimal for meeting a carbon budget. However, this optimality relies on the assumption that a finite carbon budget associated with a temperature target is filled up steadily over time. The availability of net carbon removals invalidates this assumption and therefore a different carbon price trajectory should be chosen. We show how the optimal carbon price path for remaining well below 2 °C limits CDR demand and analyze requirements for constructing alternatives, which may be easier to implement in reality. We show that warming can be held at well below 2 °C at much lower long-term economic effort and lower CDR deployment and therefore lower risks if carbon prices are high enough in the beginning to ensure target compliance, but increase at a lower rate after carbon neutrality has been reached.