I have been trying to recreate the TIMES OBJINV calculation for processes with lead times.
The process in question has TLIFE=40, ELIFE=30 and ILED=-5. The model base year is 2010 and the periods are 2010, 2011, 2012, 2013-2017, 2018-2022, further 5-year periods.
I've calculated OBJINV coeffs for 2011 and 2020 vintages. See the attached workbook for the discount rates and calculation.
For 2020, I can almost recreate the TIMES calculation (I'd be interested to know the cause of the 0.05% error, but am not worrying about it).
For 2011, when the lead time would be prior to the base year, I can't recreate the TIMES calculation. Do you know what I'm doing incorrectly?

Thanks for the interesting question.
Let's focus on the 2011 case which worried you. I can see in your Excel file a row with the following entries:
Cost in UK TIMES 9.07597 <-- calculated here
I assume that this value is what you would also expect TIMES to generate? At least I can see it is the result of your Excel calculation.
I tested with a simple test model and with the parameters you have disclosed, and I got an objective coefficient 9.07597369 for the 2011 investment costs. It looks exactly the same value as in your Excel calculation, and so I can only conclude that I can see no difference to your calculation.
Could you please confirm that this is also the value you expected to see? If so, could you clarify how you expect me to be able to reproduce the problem you are seeing?

Thanks, Antti.
Looking again, I think I must have made a mistake extracting the 2011 vintage coefficient from the objective equation - I also have the correct number from the model equations and I can't recreate the 8.32 value I extracted previously. It's very odd as I'm sure I checked that I had the right value. I'm glad my calculation was correct, but sorry that I wasted your time with the question.
I hope other people will find the workbook useful for understanding how TIMES calculates investment costs for processes with lead times.

pauldodds Wrote:For 2020, I can almost recreate the TIMES calculation (I'd be interested to know the cause of the 0.05% error, but am not worrying about it).

The cause of the 0.05% error is the use of time-dependent general discount rates (G_DRATE). I think one of the main ideas behind the annualization of the investment costs is that, assuming zero risk premium (no NCAP_DRATE specified) and zero IDC (no NCAP_ILED specified), the lump-sum present value of the investment payments should be equal to the original investment cost (NCAP_COST), when the annual payments are discounted back to the beginning of the commissioning year.
However, your calculation is inconsistent with this equivalence principle.
For some extreme examples, consider buying an electric car in 2012, which has a price

£30,000 and ELIFE=10. Assume first that G_DRATE is constant, with the values 1%, 10% and 100%. With all these discount rates, the lump-sum investment cost obtained by discounting the annual investment payments (£3700, £4439, £15015) back to the beginning of the commissioning year is still in all cases exactly

£30,000. But now, assume that G_DRATE is 10% in 2012, but then either A) decreases to 1% or B) increases to 100% starting from 2013. Using

your calculation method, in the first case A), the lump-sum investment cost would have a value of £42,459, and in case B) it would have a value of only £8,868! So, in case B) the electric car, which is bought in 2012 with the price £30,000, would effectively cost only £8,868, just because of the annualization of the investment cost. I think such a price reduction would be inconsistent, but perhaps you disagree?
To avoid such artificial alteration in prices when G_DRATE is changing, TIMES currently does not follow your calculation, but discounts the payments back to the commissioning year by the G_DRATE of that year, and therefore in TIMES the lump-sum payment is consistent with NCAP_COST. Would you prefer that your approach is adopted instead?

Thanks for the explanation, Antti. I can now repeat the TIMES investment cost calculation.
This is an interesting philosophical question that I haven't considered before.
Perhaps it depends on the point of the discount rate. The technology-specific rate can be considered an up-front loan cost with staged payments, so that is unlikely to change. In the UK model, we treat the global discount rate as a “social time preference rate”, based on the HM Treasury Green Book. This comprises:

- A ‘time preference’, which is the rate at which consumption and public spending are discounted over time, assuming no change in per capita consumption. This captures the preference for value now rather than later.
- A ‘wealth effect’, which reflects expected growth in per capita consumption over time, where future consumption will be higher relative to current consumption and is expected to have a lower utility.

Both cases reflect that the value of a particular investment changes over time. One might argue that if the discount rate changes, that reflects the value of the investment changing, and hence the lump investment would not need to be the same. So the cost of the car might be different to the value to the owner. Following this argument, time-varying discount factors would be most appropriate for calculating investment costs.
However, I’m by no means an expert in this area and I’d be interested in the views of others. I imagine the differences between the two methods are very small for most models.
I have attached an updated version of the file for the benefit of others who are looking at this question and interested in how TIMES calculates the investment cost.

Thanks for the follow-up.
Indeed, I can see that one may argue for changing discount rates affecting the investment cost of technology. But I am somewhat confused about that possibility. If we assume so, and let's say the car would have an NPV of £42,459 when paying annually, and only £30,000 if paying the full amount at once, I think it would be clearly optimal to pay at once. So, shouldn't the model then include the option to pay the full price at once?
On the other hand, if the NPV would be only of £8,868 when paying annually a car bought in 2012, but the full costs of the manufacturers and retailers correspond to the NPV price £30,000, wouldn't they be selling at a big loss if they would accept annual payments that are worth of only £8,868? As a car seller with perfect foresight, don't think I would accept such.
The TIMES documentation on the objective function (written by prof. Richard Loulou AFAIK), states that "if the technology discount rate is equal to the general discount rate, then the stream of ELIFE yearly payments is equivalent to a single payment of the whole investment cost located at year k, inasmuch as both have the same discounted present value." And by definition, if the technology discount rate is not specified, it is equal to the general discount rate. So, to me that would seem to confirm the equivalence principle, although the documentation does not seem to explicitly give much consideration about time-variant discount rates, other than mentioning that they are supported.
The current handling of time-dependent discount rates in the TIMES code was settled by about 15 years ago, in agreement by me and Uwe, who was at that time the primary maintainer. It can of course be changed, if there are good arguments and agreement about that.