Marginal cost of CO2

I specified a CO2 tax over the model horizon. After checking the results I noticed that in some years marginal cost of CO2 is not equal to the specified tax (see the table below).


Can anybody explain how exactly this works (or give a reference to documentation)? I can see that whether or not marginal cost is equal to the specified tax in a particular year depends on the values specified for the preceeding and the following data year.


Well, I think this is quite easily explained: The marginal cost represents an average over the period, while the tax you have specified is not constant within periods, but is changing within many of the periods. In TIMES, all costs are densely interpolated over all individual years. Hence, if the tax is 40.00 in 2040 and 80.00 in 2050, the tax is increasing quite rapidly from 40.00 already in the last part of the period containing 2040. Therefore, the resulting marginal cost in the period 2040 is not 40.00, but is higher. In the same way, the marginal cost in 2050 is not 80.00, but is somewhat lower, because the tax is increasing more steeply between 2040 and 2050 compared to 2050-2060, and so the average (weighted by the discount factors) will be lower.

I hope this explanation makes the differences understandable. You should be able to get a "full match" by specifying a tax that is constant within each period (by specifying the tax at the first and last year of each period).
Note that the marginal costs in the LP model results are period-specific discounted values. The TIMES reporting routines convert these into the undiscounted ANNUAL values, which are reported. The conversion factor is simply the sum of the present value factors of the years in each period. I am afraid that this conversion formula is only briefly described in the documentation (see parameter VDA_DISC on page 126 of the documentation, Part II). However, it is easy to verify (both theoretically and experimentally) that the resulting undiscounted marginal costs are such that if you would replace your original taxes with these marginal costs (and make sure they are constant within each period), the value of the objective function will not change.
Thanks a lot, Antti! I suspected the first part, but because I did not take into account weighting by the discount factors I could never get from the values I had specified to the reported marginal costs: the values were either slighly higher or lower. This clears it up. :-)
The model generator could.

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