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Let $f(x)$ and $g(x)$ be two increasing continuous functions.

Given that $x_1 + x_2 = k$, show that the minimum of $f(x_1) + g(x_2)$ occurs where $f(x_1) = g(x_2)$

Trent Gm
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  • Let $f(x) = g(x) = x-e^{-x}$. Then $f(x_1)+g(k-x_1) = k - e^{-x_1} - e^{x_1-k}$, which has no minimum. – Joey Zou Nov 23 '16 at 03:18
  • This question came from trying to show how the document at http://energy.gov/sites/prod/files/oeprod/DocumentsandMedia/review_of_congestion_costs_october_03.pdf can claim on page 6 that the minimum cost is achieved where the two functions intersect (it's not obvious to me). What extra constraints are required in order to prove the claim on page 6 (or is the claim just incorrect?) – Trent Gm Nov 23 '16 at 23:47
  • The curves in the above document represent the "marginal cost of generation vs. supply", i.e. if $f(x)$ and $g(x)$ are the costs of generation given $x$ supply in areas $A$ and $B$, respectively, then the curves drawn are $f'$ and $g'$. So the correct question to ask would be: given $x_1+x_2=k$ (and that $f'$ and $g'$ are increasing and continuous), show that the minimum of $f(x_1)+g(x_2)$ occurs where $f'(x_1)=g'(x_2)$. – Joey Zou Nov 24 '16 at 21:16

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I'm not sure so check it please, but I think that the cost is evaluated by the integral of the two functions (in that case the area under the graph of the functions), hence if we must supply $1000$MW as constraint, the minimum of the area is when the two functions intersect each other; any other choice has a greater cost, for instance: if Area A supplies $800$MW and Area B $200$MW, the constraint is right but the cost (the integral) is equal to the minimum one plus the area between the two functions in the range $[700$MW$,800$MW$]$, so we would have a higher cost compared to the following choice: Area A supplies $700$MW and Area B $300$MW. I hope it helps.

MattG88
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  • Hi Matt. Thanks for your answer. However another missing piece of the puzzle is that in the US power markets the price all generators are paid is the marginal cost of supply, so the cost would be a constant 1000MW x $25/MWh – Trent Gm Nov 28 '16 at 00:59
  • Yes I read something and I think I understood that in the "new market" (market clearing price) the cost is a constant (25$/MWh, that is the intersection between the two previous functions), but this is different from the first case...maybe I didn't understand the new "problem" :-) – MattG88 Nov 28 '16 at 01:55