Climate is a persistent asset, bar none: changes in climate-related stocks have consequences spanning over centuries or possibly millennia to the future. To reconcile the discounting of such far-distant impacts and realism of the shorter-term decisions, we consider hyperbolic time-preferences in a climate-economy model. Time-changing utility discount rates have unexplored general-equilibrium effects: carbon prices exceed the pure carbon externality costs - the Pigouvian tax level - by multiple factors in our quantitative assessment. The climate-economy model is rich in details but can be solved in closed-form yielding Markov carbon prices dependent on climate system parameters, damage estimates, technology parameters, and both short- and long-term time preferences. The equilibrium time discount rate is endogenous, and it can justify high carbon taxes as advocated by Stern while maintaining the realism of the macroeconomic outcome, thus providing a solution for the dilemma centering the carbon tax-discount rate debate. The welfare ranking of the policy alternatives is unambiguous: enforcing the Pigouvian tax decreases a consistently-defined welfare measure vis-a-vis the Markov equilibrium.