Equilibrium and hurdle rates
I have a question regarding the use of hurdle rates. I'm wondering whether the use of hurdle rates (technology specific discount rates) actually reflects the decision making by private investors. 
Below, I’ve included the reasoning I used. Based on this reasoning, I seem to come to the conclusion that, when technology specific discount rates are used, the results of the optimization model (TIMES) do not necessarily correspond to the the equilibrium that would be formed if private investors would value their investments with their respective technology-specific discount rates. And even, there seems to be no way to compute the equilibrium using an optimization model whenever technology-specific discount rates are used. As this seems to touch the essence of the idea of technology-specific discount rates and I’m not aware of any discussion about this, I’m wondering if my reasoning is correct..
If no technology specific discount rates are used, the costs and value (~revenues) related to an investment will equal. Assuming for simplicity a technology with no costs related to the activity, i.e., only an investment cost (e.g., wind turbines). This technology generates electricity and generates revenues in year a equal to revenue(a) which can be calculated based on the dual variables of the COMBAL constraint for electricity and the generation in each time slice.
- the contribution of a unit investment in this technology to the objective function equals NCAP_COST (assuming the objective function discounts to the year of the investment and no lead time is considered)
- the value perceived by the model equals: sum_a=1^TLIFE (revenue(a) * r^(a-1))
If the hurdle rate equals the general discount rate r (meaning no hurdle rate is used in the model), both the cost and the value related to an investment are perceived the same by the model and by a private investor.
Now, assume that a hurdle rate r_s is introduced for this technology which differs from the general discount rate r.
- The cost perceived by the model now equals: NCAP_COST*(CRF_s/CRF)   (1) . Here, CRF_s/CRF can be considered as a cost mark-up for the investment cost.
- The value perceived by the model again equals: sum_a=1^TLIFE (revenue(a) * r^(a-1))    (2)
The costs and values perceived by a private investor are the following:
- The costs perceived by a private investor equals: NCAP_COST     (3)
- The value perceived by a private investor equals: sum_a=1^TLIFE (revenue(a) * r_s^(a-1))     (4)
So, both the costs and the value perceived by the model differ from those perceived by a private investor. The value perceived by the model is directly related to the general discount rate and thus cannot be easily adapted. The costs perceived by the model can be adapted such that, whenever costs = value from an investor perspective, this is also true for the model (such that the model would have similar decision making to a private agent).
The perceived value of the model can be related to the perceived value of the private investor:
Value model = (sum_a=1^TLIFE (revenue(a) * r^(a-1)) / sum_a=1^TLIFE (revenue(a) * r_s^(a-1))) * value perceived by investor.
When the revenue term is assumed constant over all years the investment is operational, this gives:
Value model = (CRF_s/CRF)*value perceived by investor.
For the model to have the same decision making as the investor, the costs in the investment costs in the model should also be increased by this factor. This is exactly what is done in TIMES (see (1)).
The problem I face is that this no longer holds whenever the revenue streams are not constaint over the years. For example, assuming d_s > d and decreasing revenue streams over the lifetime of the investment, value_model/value_investor < CRF_s/CRF. Thus, the revenues are not sufficiently valued by the model (or seen differently, the investment cost mark-up used is too high). 

Sorry for the long post, but I did not manage to find a more concise way to formulate the question.
Dear K_Poncelet,

I think your question was good, and I was keen to see how the expert modelers respond.
Checking now again after a long time, to my dismay I see that nobody has answered.

Possibly the expert modelers consider the issue less important, or perhaps they are just mostly ignoring this Forum, which is a bit quiet and also mostly technical anyway.

As I think the question is not particular to TIMES, but a more general one, I guess there could be more active Forums on economic modelling where the question could be raised. Have you tried any other Forums?  Or do you know of any good solutions that avoid the issue you raised, which are used in some other models than TIMES?
Dear Antti-L,

Thanks for your reply.

After having some discussions and readings, I found that this issue is indeed true and described in a paper titled 'Risk adjusted discounted cash flows in capacity expansion models' of Ehrenmann and Smeers in Math Programming journal.

Unless you would revert to using mixed complementarity problems or solving the model iteratively, there is no exact solution to this problem.

What I conclude is that:
- the hurdle rate (technology specific discount rate) in TIMES cannot be directly be interpreted as such, but it remains a way of increasing the capital cost.
- the deviations between the IRR you would calculate ex-post and the hurdle rate you specified ex-ante are likely to be small unless:
- your revenues are strongly decreasing over the lifetime of the technology (e.g., an investment in solar PV or batteries where investment costs will drop significantly in the later
years). In this case, the hurdle rate you specify in the model should be lower than the one you effectively want to implement
- your revenues are strongly increasing over the lifetime of the technology (e.g., an investment in a wind turbine in a scenario with a stringent target for renewables, but the
potential of wind turbines quickly becomes depleted.
Thanks for sharing these insights, very useful!
Hi, Antti, and K_Poncelet
I read the first question but wasn't in a position to answer with certainty unless I studied it further.
I have been thinking for a long time that a general modelling forum would be very useful to me. Do you know of any?
I have tried a couple, with good results in terms of participation but extremely technical for what I needed.
K_Poncelet, if you'd ever think discussing a paper like the one you read may be helpful to you. I'd be happy to do it if you send it to me so that i can read it. It may be helpful, who knows? And I'll learn something new Smile
Thanks to both of you for insights

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