Cumulative emissions drive peak global warming and determine the carbon budget needed to keep temperature below 2 or 1.5 °C. This safe carbon budget is low if uncertainty about the transient climate response is high and risk tolerance (willingness to accept risk of overshooting the temperature target) is low. Together with energy costs, this budget determines the optimal carbon price and how quickly fossil fuel is abated and replaced by renewable energy. This price is the sum of the present discounted value of all future losses in aggregate production due to emitting one ton of carbon today plus the cost of peak warming that rises over time to reflect the increasing scarcity of carbon as temperature approaches its upper limit. If policy makers ignore production losses, the carbon price rises more rapidly. If they ignore the peak temperature constraint, the carbon price rises less rapidly. The alternative of adjusting damages upwards to factor in the peak warming constraint leads initially to a higher carbon price which rises less rapidly.