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This website discusses the finite difference method we apply to the continuous-time analog of DICE2007. The DICE2007 equations are different from the corresponding ones in earlier versions of DICE. We find that
- these changes substantially affect the result
- the changes imply an nonphysical feedback of future CO2 emissions to current warming
- changing the time period from ten years to one is straightforward.
Prof. Nordhaus states on page 44 that:
“Equations (A.13) through (A.15) are the equations of the carbon cycle. These equations have been modified since the last round to remove a problem with the lag structure.”
On page 45, Prof. Nordhaus continues:
“The lags in the system are primarily caused by the diffusive inertia of the different layers. We have changed the timing slightly to improve the match of the impulse-response function with climate models. Additionally, we have adjusted the climate sensitivity to the center of the IPCC range of 3°C for an equilibrium CO2 doubling. The timing is calibrated to match model experiments for the IPCC Third and Fourth Assessment Reports.”
Climate models are run at time resolutions far shorter than ten years; some are run at resolutions measured in minutes. Climate models aim to solve continuous-time models. It is not clear that it is possible to produce a specification of the climate system with a time resolution of ten years that can match important features of a continuous-time model. Therefore, IAM models should move to continuous-time models for the climate system.
Prof. Nordhaus, following standard practice in IAM work, chooses the parameters of his climate system so that a particular finite difference method with a ten-year time period matches some target results. The calibrated parameters will depend on the time step. The underlying problem, a combination of climate and economic systems, is a system of differential equations. Scientists use experiments to pin down the fundamental parameters. If calibration is needed, the calibration should not depend on the finite difference method to be used. The parameters should represent fundamental features of the physical system. If a finite difference scheme is not producing accurate results then it is natural to change it, but only to another theoretically valid finite difference method for the same physical system.
Furthermore, DICE2007 implies nonphysical phenomena. For example, if you let T=2015 in his equations, you find that the warming between 2015 and 2025 is affected by the stock of atmospheric CO2 in 2035, implying that emissions between 2025 and 2035 will increase warming during the ten years between 2015 and 2025.
A more standard response to problems with lags due to large time steps would be to solve the model at a finer time resolution. Doing this for DICE is straightforward. It just requires one to proportionately change those parameters with time unit dimensions; otherwise, no change in parameters or functional form of the equations is necessary.
That is what we did. We took the diffusivity parameters of DICE2007, and formulated the continuous-time climate model. We then discretized it with an Euler explicit finite difference method for both the climate and economic system. When results are different for time steps of ten years and one year, we always reject the results from the ten-year time step. Further analysis is needed to accept the results for one year. To address the quality of the one-year time step results, we also computed the half-year time step model and the quarter-year model. Those results were trivially different from the one-year results. Therefore, the one-year results are validated.
Hence, the ten-year time step in DICE2007 is an unacceptably large discretization, and lead to carbon taxes 50% greater than the results with shorter and more acceptable time steps.
We do not claim that our continuous-time analog of DICE2007 is the best continuous-time model because we annualize the diffusivity coefficients in DICE2007. The proper way to proceed is to find diffusivity parameters for a continuous-time system that gives an acceptable approximation to more complex, continuous-time climate system models.
The main point of this work is that it is easy and straightforward to solve continuous-time specifications with the same software and hardware commonly used in the IAM literature.