I'm working on some economics homework and there is a problem I'm finding particularly difficult. The first question asks us to find the function that a firm will use to determine how much CO2 to emit under a cap-and-trade system, given a certain cost function for reducing its emissions. I've arrived at the expression e = (2αi-p)/2α²), where α is a constant, i is a numerical value assigned to each firm and p is the price of emitting 1 unit of CO2.
The next question is where it gets difficult. The question asks us to determine ∑(i=1, n) e, and it also tells us that ∑(i=1, n) i = N(N + 1)/2. Naturally, I substituted N(N + 1)/2 in for i, only to hit a brick wall. The TA for the course informed me that I can't directly substitute in N(N + 1)/2 for i if there is a fraction in the e-term, and advised me to "try separating the two terms so that one part has the sum i expression and the other part doesn't have i". I've been digging into the rules about summation and come up short; I also tried rearranging the equation first and then substituting for i, rather than the other way around, but I just got the same solution as when I substituted, then rearrange. Is there some summation rule I'm missing here?
Would appreciate any insight. Thanks for the attention!