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!
It seems that you have to calculate $\sum\limits_{i=1}^{n} \frac{2ai-p}{2a^2}$. The first step is factoring out the factor which does not depend on index $i$. This is the denominator. Thus we get $\frac{1}{2a^2}\cdot \sum\limits_{i=1}^{n} (2ai-p)$. Then we have two summands in the brackets. I ommit the factor $\frac{1}{2a^2}$ for the next steps. We can write for each summand a sigma sign.
$$\sum\limits_{i=1}^{n} 2ai-\sum\limits_{i=1}^{n}p$$
Here we can again factor out constant factors.
$$2 a\cdot \sum\limits_{i=1}^{n} i-p\cdot \sum\limits_{i=1}^{n}1$$
You know what the result of the first sigma sign is. And $\sum\limits_{i=1}^{n}1 $ is just $n$. After some simplifications you put a bracket around the term and divide it by $2a^2$.