  The fossil-fuel induced contribution to further warming over the 21st century
will be determined largely by integrated CO2 emissions over time rather than
the precise timing of the emissions, with a relation of near-proportionality
between global warming and cumulative CO2 emissions. This paper examines
optimal abatement pathways under an exogenous constraint on cumulative
emissions. Least cost abatement pathways have carbon tax rising at the
risk-free interest rate, but if endogenous learning or climate damage costs are
included in the analysis, the carbon tax grows more slowly. The inclusion of
damage costs in the optimization leads to a higher initial carbon tax, whereas
the effect of learning depends on whether it appears as an additive or
multiplicative contribution to the marginal cost curve. Multiplicative models
are common in the literature and lead to delayed abatement and a smaller
initial tax. The required initial carbon tax increases with the cumulative
abatement goal and is higher for lower interest rates. Delaying the start of
abatement is costly owing to the increasing marginal abatement cost. Lower
interest rates lead to higher relative costs of delaying abatement because
these induce higher abatement rates early on. The fraction of business-as-usual
emissions (BAU) avoided in optimal pathways increases for low interest rates
and rapid growth of the abatement cost curve, which allows a lower threshold
global warming goal to become attainable without overshoot in temperature. Each
year of delay in starting abatement raises this threshold by an increasing
amount, because the abatement rate increases exponentially with time.
